THEORY OF WING SECTIONS Including a Summary of Airfoil Data By IRA H. ABBOTT DIRECTOR OF AERONAUTICAL AND SPACE RESEARCH NATIONAL AERONAUTICS AND SPACE ADMINISTRATION and ALBERT E. VON DOENHOFF RESEARCH ENGINEER. NASA DOVER PUBLICATIONS, INC. NEW YORK Copyright '© ~949, 1959 by Ira H. Abbott and Albert E. von Doenhoff. All rights reserved under Pan American and Inter­ national Copyright Conventions. Published in Canada by General Publishing Com­ pany, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. This Dover edition, first published in 1959. is an unabridged and corrected republication of the first edition fint published in 1949 by the McGraw-Hill Book Company, Inc. This Dover edition includes a new Preface by the authors. Stand4,-d Book. Number: 486-60586-8 Library 01 Congress Catalog Card Number: 60-1601 Manufactured in the United States of America Dover Publications, Inc. ISO Varlek Street New York, N. Y. 10014 PREFACE TO DOVER EDITION The new edition of this book originally published in 1949 results from the continuing demand for a concise compilation of 'the sub­ sonic aerodynamic characteristics of modern NACA wing sections together with a description of their geometry and associated theory. These wing sections, or their derivatives, continue to be the most commonly used ones for airplanes designed for both subsonic and supersonic speeds, and for application to helicopter rotor blades, propeller blades, and high performance fans. A number of errors in the original version have been corrected in the present publication. The authors are pleased to acknowledge their debt to the many readers who called attention to these errors. Since original publication many new contributions have been made to the understanding of the boundary layer, the methods of boundary-layer control, and the effects of compressibility at super­ critical speeds. Proper treatment of each of these subjects would require a book in itself. Inasmuch as these subjects are only peripherally involved with the main material of this book, and could not, in any ease, be treated adequately in this volume, it was considered best to expedite republication by foregoing extensive revision. CHEVY CHASE, MD. June, 1958 v IRA H. ABBOTT ALBERT E. VON DOENHOFF PREFACE In preparing this book an attempt has been made to present concisely the most important and useful results of research on the aerodynamics of wing sections at suberitical speeds. The theoretical and experimental results included are those found by the authors to be the most useful. Alternative theoretical approaches to the problem and many experimental data have been rigorously excluded to keep the book at a reasonable length. This exclusion of many interesting approaches to the problem prevents any claim to complete coverage of the subject but should permit easier use of the remaining material. The book is intended to serve as a reference for engineers, but it should also be useful to students as a supplementary text. To a large extent, these two uses are not compatible in that they require different arrange­ ments and developments of the material. Consideration has' been given to the needs of students and engineers with a limited background in theoretical aerodynamics and mathematics. A knowledge of differential and integral calculus and of elementary mechanics is presupposed. Care has been taken in the theoretical developments to state the assumptions and to review briefly the elementary principles involved. An attempt has been made to keep the mathematics as simple as is consistent with the difficulties of the problems treated. The material, presented is largely the result of research conducted by the National Advisory Committee for Aeronautics over the last several years. Although the authors have been privileged to participate in this research, their contributions have been no greater than those of other members of the research team. The authors wish to acknowledge es­ pecially the contributions of Eastman N. Jacobs, who inspired and directed much of the research. The authors are pleased to acknowledge the im­ portant contributions of Theodore Theodorsen, I. E. Garrick, H. Julian Allen, Robert M. Pinkerton, John Stack, Robert 1'. Jones, and the many others whose names appear in the list of references. The authors also wish to acknowledge the contributions to the attainment of low-turbulence air streams made by Dr. Hugh L. Dryden and his former coworkers at the National Bureau of Standards, and to express their appreciation for the in­ spiration and support of the late Dr. George W. Lewis. CHEVY CHASE, ~{D. July, 1949 vii IRA H. ABBOTT ALBERT E. VON DOENHOFF CONTENTS v vii PREFACE TO DOVER EDITION . . . . . . . . . . . . PREFACE. . . . . . . . . . . . . . . . . . . . . . . 1. THE SIGNIFICANCE OF WING-SECTION CHARACTERISTICS . . 1 Symbols. The Forces on Wings. Effect of Aspect Ratio. Application of Section Data to Monoplane Wings: a. Basic Concepts of Lifting-line T~eory. b. Solutions for Linear Lift Curves. c. Generalized Solution. Applicability of Section Data. 2. SIMPLE TWO-DIMENSIONAL FLOWS 31 Symbols. Introduction. Concept of a Perfect Fluid. Equations of Motion. Description of Flow Patterns. Simple Two-dimensional Flows: a. Uniform Stream. b. Sources and Sinks. c. Doublets. d. Circular Cylinder in a Uniform Stream. e. Vortex. f. Circular Cylinder with Circulation. 3. THEORY OF WING SECTIONS OF FINITE THICI{NESS 46 . Introduetiou. Complex Variables. Conformal Transformations. Transformation of a Circle into & Wmg Section. Flow about Arbitrary Wing SeetiODS. Empirical Modification of the Theory. Design of Wing Sections. Symbols. 4. THEORY OF TJIIN WING SECTIONS . . . . . . . . . . . . ~. 64 Symbols. Basic Concepts. Angle of Zero Lift and Pitching Moment. De­ sign of Mean Lines. Engineering ApplicatioD8 of Section Theory. 5. THE EFFECTS OF VISCOSITY . . . . . . . . . . . . . . 80 Symbols. Concept of Reynolds Number and Boundary Layer. Flow around Wmg Sections. Characteristics of the Laminar Layer. Laminar Skin Frietion. Momentum Relation. Laminar Separation. Turbulent Flow in Pipes. Turbulent Skin Friction. Calculation of Thickness of the Turbulent Layer. Turbulent Separation. Transition from Laminar to Turbulent Flow. Calculation of Profile Drag. Effect of Mach Number on Skin Friction. III 6. FAMILIES OF WING SECTIONS Symbols. IDtroduction. Method of Combining Mean Lines and Thickness Distributions. NACA F~igit Wing Sections: a. Thickness Distributions. b. Mean Lines. c. Numbering System. d. Approximate Theoretical Characteristics. NACA Five-digit Wing Sections: a. Thickness Distribu­ tions. b. Mean Lines. c. Numbering System. d. Approximate Theoretical Characteristics. Modified NACA Four- and Five-digit Series Wing Sections. N ACA l-Series Wing Sections: Q. Thickness Distributions. b. Mean Lines. ix CONTENTS x c. Numbering System. d. Approximate Theoretical Characteristics. NACA 6-Series Wing Sections: G. Thickness Distributions. b. Mean Lines. c. Numbering System. d. Approximate Theoretical Characteristics. NACA 7-8eries Wing Sections. Special Combinations of Thickness and Camber. 7. EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS . 124 Symbols. Introduction. Standard Aerodynamic Characteristics. Lift Characteristics: 4. Angle of Zero Lift. b. Lift-curve Slope. c. ~faximum Lift. d. Effect of Surface Condition on Lift Characteristics. Drag Charac­ teristics: G. Minimum Drag of Smooth Wing Sections. b. Variation of Profile Drag with Lift Coefficient. c. Effect of Surface Irregularities on Drag Characteristics. d. Unconservative Wing Sections. Pitching­ moment Charncteristics. 8. HIGH-LIFT DEVICES . 188 Symbols. Introduction. Plain Flaps. Split Flaps. Slotted Flaps: 4. De­ scription of Slotted Flaps. b. Single-slotted Flaps. c. Extemal-airfoil Flaps. d. Double-slotted Flaps. Leading-edge High-lift Devices:a. Slats. b. Slots. c. Leading-edge Flape. Boundary-layer Control. The Chordwise Load Distribution ovcr Flapped 'Ving Sections. 9. EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS . . . . 247 Symbols.. Introduction. Steady Flow through a Stream Tube: 4. Adiabatic Law. b. Velocity of Sound. c. Bernoulli's Equation for Compressible Flow. d. Cross-sectional Areas and Pressures in a Stream Tube. e. Relations for a Normal Shook. First-order Compressibility Effects: G. Glauert-Prandtl Rule. b. Effect of Mach Number on the Pressure Coefficient. Flow' about Wing Sections at IIigh Speed: 4. Flow at Subcritical Mach Numbers. b. Flow at Supercritieal ~{ach Numbers.. Experimental Wing Characteris­ tics at High Speeds: 4. Lift Characteristics. b. Drag Characteristics. c. Moment Characteristics. Wmgs for High-speed Applications. REFERENCES 300 APPE~DIX 309 I. Basic Thickness Forms. II. Mean Lines . . . INDJ4~X . 382 I I I. Airfoil Ordinates 406 IV. Aerodynamic Chnracteristies of \Ying Sections 449 CHAPTER 1 THE SIGNIFICANCE O:F WING-SECTION CHARACTERISTICS 1.1. Symbols. A An CD Co, CL CL ma x CAl CJI oe D E E G H J L L. L, M 8 V X," a a. Go ac b C c c' Cd Cd, c, Cl el CI. Cl max c. c.... C. Cc aspect ratio coefficients of the Fourier series for the span-load distribution drag coefficient induced dra.g'"coefficient lift coefficient maximum lift coefficient pitching-moment coefficient pitching-moment coefficient about the aerodynamic center drag Jonesi' edge-velocity factor, equals ratio of the semi perimeter of the plan form of the wing under consideration to the span of the wing a factor (see Fig. 13) a factor (see Fig. 14) a factor (see Fig. 15) a factor (see Fig. 9) lift " additional" loading coefficient U basic" loading coefficient pitching moment wing area speed longitudinal distance between the aerodynamic center of the root section and the aerodynamic center of the wing, positive to the rear wing lift-curve slope effective section lift-curve slope, 40/ E section lift-curve slope aerodynamic center wing span wing chord mean geometric chord, 8/b mean aerodynamic chord section drag coefficient section induced-drag coefficient section lift coefficient local U additional" section lift coefficient for a wing lift coefficient equal to unity local U basic" section lift coefficient section maximum lift coefficient section-moment coefficient section-moment coefficient about the aerodynamic center root chord tip chord 1 THEORY OF WING SECTIONS 2 d. section drag f a factor (see Fig. 8) k a spanwise station l section lift Z. " additional" section lift It ",basic" section lift m seetion moment r an even number of stations used in the Fourier analysis of the span-load distribution u a factor (see Fig. 10) v a factor (see Fig. 11) tD a factor (see Fig. ~) ~ projected distance in the plane of symmetry from· the wing reference point to the aerodynamic center of the wing section, measured parallel to the chord of the root section, positive to the rear 71 distance alang the span ~ projected distance in the plane of symmetry from the wing reference point to the aerodynamic center of the wing section measured perpendicu1U' to the root chord, positive upward ex angle of attack act section angle of attack a. effective angle. of attack eli angle of downwash section angle of attack for zero lift a ... angle of zero lift of the root section a. wing ang1e of attack measured from the chord of the root section 0.<1-0) angle of atteu:k of the root section for zero lift of the wiDi fl angle of sweepbaek e aerodyutUDic twist from root to tip II. multiplier for obtaining the wing cbaracteristies ,_ multiplier for obtaining the span-load distribution a,. , CX»J-l ( - ~) multiplier for obtaining the inducecL-angle distribution ... ratio of the circumference of a circle to ita diameter p mass density of air )..1: 1.2. The Forces- on W-mgs. The surfaces that support the aircraft by means of dynamic reaction on the air are called wings. An aircraft may have several wings which may either be fixed with respect to the fuselage or have any of several motions as in the case of helicopters or omithopters. Regardless of the type of lifting surface, its aerodynamic characteristics will be stronglyatfected by the shape of the wing section. The wing char­ acteristics may be predicted from the known aerodynamic characteristics of the wing section if the span is large with respect to the chord, if the Mach numbers are subcritical, and if the chordwise component of ,\relooity is large compared with the spanwise component. Thus the wing-section characteristics considered in this volume have a 1arge field of applicability. A complete discussion of the application of section characteristics to the THE 8IGNIFICANCE OF WING-8ECTION CHARACTERISTIC8 3 prediction of wing characteristics is beyond the scope of this work, but some indication of the methods is given for the case of the monoplane wing in steady straight flight without roll or sideslip. The monoplane wing supports the airplane by means of "a.lift force generated by the motion through the air. This lift is defined as the com­ ponent of force acting in the plane of symmetry in & direction perpendicular .12 2.4 J 40 I .KJ 20 I I .08 1.6 I/ II ~ I " r I ~ V" ~ ~ 'CL ~ J'I I ....... o 0 0 ',­ /, ~D '-/0 ~~ i' JV f' ~ -­ o ~, ~ 8 16 Angle of t1fIack,~.".. 24 FIG. 1. Typical wing characteristics. to the line of flight. In addition to the lift, a force directly opposing the motion of the wing through the air is always present and is called the "drag." For a given attitude of geometrically similar wings, the forces ten.d to vary directly with the density of the air, the wing area, and the square -of the speed. It is accordingly convenient to express these forces in terms of nondimensional coefficients that are functions primarily of the attitude of_the wing. The lift and drag are given by the following ex­ pressjons: ~pV28CL (1.1) D == ~pVJ8CD (1.2) L == The lift and drag forces may be considered. to act at a fixed point with respect to the wing. A complete specification of the forces acting on the 4 THEORY OF WING SECTIONS wing requires a knowledge of the moment about this fixed point. For a symmetrical wing moving. with translation only in the plane of symmetry, the side force perpendicular to the lift and drag is equal to zero, and the moment acts in the plane of symmetry. This moment tends to change 1.4 ~ 1.2 ~ r--.. r" jA~ VV / 1.0 .8 ~ V V ~/ / V ~ ~ o V FIG. V ~ ~/ ~ /t V V VI V ~ ~ V 6. -.2 ~ A ~ ~ ~ r ~ II. V V ~ V / V V / v / V 0/0­ Angle ofo/fack, tt, (degrees) 2. Lift coefficients plotted as function of angle of attack for aspect ratios of 7 to 1. the angle of attack of the wing. It is accordingly called the ., pitching moment" and may be expressed as follows: (1.3) ..~ convenient way of describing the aerodynamic characteristics of a wing is to plot the values of the coefficients against the angle of attack, which is the angle bet,veen the plane of the wing and the direction of motion. Such a plot is shown in Fig. 1. The lift coefficient increases .ilinost linearly with angle of attack until a maximum value is reached, whereupon the wing is said to U stall." The drag coefficient has a minimum value at a low lift coefficient, and the shape of the curve is approximately parabolic at angles of attack below the stall. H the point about which THE SIGNIFICANCE OF WING-BECTION CHARACTERISTICS 5 the moment is taken is properly chosen (the aerodynamic center), the moment coefficient is essentially constant up to maximum lift. A measure of the efficiency of the wing as a lifting surface is given by the lift-drag 1.4 j 1.0 A~ / ~V' V V V '/ / ~ I FIG. ~ ·It:/~ / (.)......6 -.2 ~~ V .8 o ------ -- A 1.2 1\~ V ~ V" w~ V V v / '>: ~ V V V Y --- ~ ~ V - V i/ r---..... V ~ I ~ ~ ~ ~ ./0 .20 Drog'coefficiert, Co 3. Polar diagrams for seven wings with aspect ratios of 7 to 1. ratio, which is also plotted in Fig. 1. This ratio increases from zero at zero lift to a maximum value at a moderate lift coefficient, after which it decreases relatively slowly as the angle of attack is further increased. It is desirable for the wing to have the smallest possible drag. Inas­ much as the high-speed lift coefficient is usually substantially less than that corresponding to -the best lift-drag ratio, one of the best ways of reducing the wing drag is to reduce the wing area. This reduction of area is usually limited by considerations of stalling speed or maneuverability. These considerations are directly influenced by the maximum lift coeffi­ cient obtainable. The wing should therefore have a high maximum lift 6 THEORY OF WING 8ECTION8 coefficient.combined with low drag coefficients for high-speed and cruising flight. This combination of desirable qualities can be obtained only to a limited extent by a single wing configuration. It is therefore customary to use some retractable device such as flaps to improve the maximum lift characteristics of the wing. 1.3. Effect of Aspect Ratio. Aspect ratio is defined as the ratio of the span squared to the wing area (IJI/S), which reduces to the ratio of the 1.4 .. 1.2 . . 0 -6 o 1.0 0 eJ~ ."\ c lA+ .8 ~. A i"II' ~~ o Vz~ 0=4 61 J -.2 ~ -I---- • : 5 to tI -I-- I +=2 8tf,.,.a lUd' o I e : 6 til : ,,'" cl T - + G p9 -·~/O- 0 10· 20­ Angle ofoltock, a,(degteeS) FIG. 4. Lift coe8iciente as function of aqle of attack, reduced to aspect ratio of 5. span to the chord in the case of & rectangular wing. Early wind-tunnel investigations of wing characteristics showed that the rates of change of the lift and drag coefficients with angle of attack were strongly affected by the aspect ratio of the model. Wings of high aspect ratio were ob­ served to have higher lift-curve slopes and lower drag coefficients at high lift coefficients than wings of low aspect ratio. The effect 9£ aspect ratio on the lift curve is shown in Fig. 2.88 The wings of various aspect ratios are shown to have about the same angle of attack at zero lift, but the slope of the lift curve increases prograNively with increase of aspect THE SIGNIFICANCB OF WING-8ECTION CHARACTERI8TICS 7 ratio. The effect of aspect ratio on the drag coefficient is shown in Fig.3.88 Although the drag coefficients for all the models of various aspect ratios are substantially equal at zero lift, marked reductions in the drag 1.4 1.2 •• A 0 1.0 • 0 •- 0 • ~ of ... 6 0 D .8 + • :» • A C 411 + -2 0 a~ A ~t - - -3 o -4 c· o· • -5 I-- at e -6 b e -7 til • .2 I ~41~ ~ - J, 11 o -.2 • ~ '·1) ftA 0 • 0 • .I .2 Drag coefficienfl Cd FIG. S. Polar diagrams reduced to aspect ratio of S. coefficient occur at the higher lift coefficients as the 8Bpect ratio is in­ creased. As a result of such observations, the Lanchester-Prandtl wing theory was developed. This theory shows that, for wings having elliptical span­ wise distributions of lift, the following simple expressions relate the drag 8 THEORY OF WING SECTIONS coefficients and angles of attack as functions of aspect ratio at constant lift coefficients 1 1) + C (-!.. _!) A' A 2 , CL ( - - CD=CD+1[" A' A a' =a L 1[" (1.4) (1.5) where CD and a' correspond, respectively, to the drag coefficient and angle of attack (radians) of a wing of aspect ratio A'. Application of Eqs. (1.4) and (1.5) to reduce the data of Figs. 2 and 3 to an aspect ratio of five results in the data of Figs. 4 and 5.88 These figures show that the characteristics of a wing of one aspect ratio may be pre­ dicted with considerable accuracy from data obtained from tests of a wing of widely different aspect ratio. Equations (1.4) and (1.5) may be simplified by the concept of infinite aspect ratio. If Cd and ao indicate the drag coefficient and angle of attack of a wing of infinite aspect ratio, the characteristics of an elliptical wing of aspect ratio A may be expressed as (1.6) (1.7) A wing of infinite aspect ratio would have the same flow pattern in all planes perpendicular to the span. In other words, there would be no components of flow along the span, and the flow over the wing section would be two-dimensional, Infinite aspect ratio characteristics are ac­ cordingly commonly called "section characteristics. II The section char­ acteristics are intrinsically associated with the shape of the wing sections as contrasted with 'ling characteristics, which are strongly affected by the wing plan form. The detailed study of wings is greatly simplified by tile concept of wing-section characteristics because wing theory offers a method for obtaining the properties of wings of arbitrary plan form from a summation of the characteristics of the component sections. 1.4. Application of Section nata to Monoplane Vmgs. a. Basic Con­ cepts of Lifting-line Theory. The simplest three-dimensional wing theory is that based on the concept of the lifting line. 88, ~ In this theory the wing is replaced by a straight line. The circulation about the wing associated with the lift is replaced by ~ vortex filament. This vortex filament lies along the straight line; and, at each spanwise station, the strength of the vortex is proportional to the local intensity of the lift. According to Helm­ holtz's theorem, a vortex filament cannot terminate in the fluid. The variation of vortex strength along the straight line is therefore assumed to THE SIGNIFICANCE OF WING-BECTION CHARACTERISTICS 9 result from superposition of a number of horseshoe-shaped vortices, as shown in Fig. 6. The portions of the vortices lying along the span are called the" bound vortices." The portions of the vortices extending down­ stream indefinitely are called the" trailing vortices." The effect of trailing vortices corresponding to a positive lift is to induce a downward component of velocity at and behind the wing. This down­ ward component is called the U downwash." The magnitude of the down­ wash at any section along the span is equal to the sum of the effects of all the trailing vortices along the entire span. The effect of the downwash FIG. 6. Vortex pattern representing a lifting wing. is to change the relative direction of the air stream over the section. The section is assumed to have the same aerodynamic characteristics with re­ spect to the rotated air stream as it has in normal two-dimensional flow, The rotation of the flow effectively reduces the angle of attack. Inasmuch as the downwash is proportional to the lift coefficient, the effect of the trailing vortices is to reduce tile slope of the lift curve. The rotation of the flow also causes a corresponding rotation of the lift vector to produce a drag component in the direction of motion. This induced-drag coeffi­ cient varies as the square of the lift coefficient because the amount of rotation and the magnitude of the lift vector increase simultaneously. The problem of evaluating the downwash at each point is difficult be­ cause of the interrelation of the downwash, lift distribution, and plan form. A comparatively simple solution was obtained by Prandtl'" for an elliptical lift distribution. In this case the downwash is constant between the wing tips and the induced drag is less than that for any other type of lift dis­ tribution. Equations (1.6) and (1.7) give the relation between the section and wing characteristics for an elliptical lift distribution. b. Solutions for Linear Lift Curves. Glauert" applied a Fourier series analysis to the problem and developed methods for obtaining solutions for wings of any plan form and twist. Anderson 10 applied Glauert's methods THEORY OF WING SECTIONS 10 to the determination of the characteristics of wings with a wide range of aspect ratio and straight taper and with a linear spanwise variation of twist. Anderson considered the spanwise lift distribution for any typical wing to consist of two parts. One part, called the U basic distribution" , is the distribution that depends principally on the twist of the wing and occurs when the total lift of the wing is zero; it does not change with the angle of attack of the wing. The second part of the lift distribution, called the additional distribution," is the lift due to change of the wing angle I( 120 ~ 10-­ r--.. -­ ~ - .........,; ~ r--..-l r--­ ~- la c­ ~~ .... ~ ~-. ~ r-..... r-, r-; "-­ <, ........ r-, ~ I ' "' ~ "" r-,"-. ",\ ~ ~ /, o a - ~ ~ r--.... ~ r---. 2 --­ 4 6 8 /0 /2 /4 /6 Distonce along semispon. feet --1/ /8 20 Flo. 7. Typical wmiepan lift diatn1Jution. C£ - 1.2 of attack j it is independent of the twist of the wing and maintains the same form throughout the reasonably straight part of the lift curve. A typical distribution of lift over the semispan of a twisted, tapered wing at a moderately high lift coefficient"is shown in Fig. 7. Anderson presented his solutions in the form of tables and charts that are easy to apply. In order to apply these solutions the wing geometry must be known in the following terms: 8 wing area II span A aspect ratio c,/e. taper ratio where Cc is the tip chord and c. is the root chord E aerodynamic twist in degrees from root to tip, measured between the sero lift directions of the center and tip sections, positive for wash in fj angle of sweepback., measured between the lateral axis aDd a line through the aerodynamic centers of the wing sectiODS Go section lift-eurve slope THE SIGNIFICANCB OF WING-8ECTION CHARACTERISTICS 11 effective lift..curve slope, 04/B B Jones" edge-velocity factor, equals ratio of the semiperimeter of the wing under consideration to the span of the wing CIe c.. section-moment coefficient about the aerodynamic center e chord at any spanwise station Anderson considered only the case of a linear distribution of twist between the center section and the tip. The types of fairing ordinarily used for wings result in nonlinear distributions of twist if the wing is tapered. The departure from the assumed linear distribution may not be negligible if appreciable twist is combined with a large taper ratio. The data required to find the span'vise lift distribution are given in Tables 1 and 2. The basic and additional loading coefficients L b and La are presented at various spanwise stations for a wide range of aspect ratio and taper ratio. The local basic section lift coefficient Cl, for a wing lift coefficient equal to zero is given by the expression Ea,$ Cit == Ci) Lb The local additional section lift coefficient c,.. for a wing lift coefficient equal to unity is given by the expression S == Cb La C,•• The actual or total local section lift coefficient C, for a wing lift coefficient equal to C L is obtained from c, == c,. + CV; lea The lift-curve slope per degree a of the wing is obtained from a;IIK J 1 a. + (57.3a./ ...A ) where the effective section lift-curve slope a~ is taken as the average for the sections composing the wing and the factor J is given in Fig. 8 on page 16. The Jones edge-velocity correctionj" which had not been derived when Anderson!O obtained these results, has boon applied to Anderson's formulas by the use of an effective section lift-curve slope. This effective slope a. equals ao/E where E is· the ratio of the semiperimeter to the span of the wing under consideration. It will be noted that this effective lift-curve slope is not solely a property of the section but that it varies with the plan form of the" wing. The angle of attack of the wing corresponding to any value of the wing lift coefficient CL is given by a. = C - L a + alo' + J E THEORY OF WING SECTIONS 12 TABLE I.-BASIC SPAN LIFT-DISTRIBUTION DATA Values of L. for tapered wings with rounded tips c'. ~ 00110021003100' I 0.6 ~ Spanwise station 'II/(b/2) -O.I~J -0.121 -0.180 -0.192 -0.221 -0.248 -0.269 -0.288 -0.318 -0.342 -oaso -0.3M -0.387 -0.380 -o.3St -0.399 -0.398 -0.411 -0.122 -0.162 -0.197 -0.224 -0.263 -0.275 -0.293 -0.322 -0.080 -0. lOS -0.130 -0.14:8 -0.182 -0.178 -0.189 -0.207 -0.2"26 -0.234 -0.242 -0.2"7 -0.255 -0.082 -0.111 -0.135 -0.156 -0.173 -0.189 -0.200 -0.220 -0.239 -0.248 -0.256 -0.260 I -0.269 2 3 4 6 8 7 8 10 12 1-1 16 18 20 -0.118 -0.163 -0.183 -0.211 -0.235 -0.256 -0.274 -0.3CM -0.329 2 3 -0.076 -0.098 -0.117 -0.131 -0.145 -6.1&6 -0.168 -0.182 -0.197 -0.206 -0.212 -0.219 -0.222 -0.163 -0.199 -0.228 -0.253 -0.276 -0.293 -0.323 -0.349 -0.370 -0.385 -0.403 -0.415 -0.3~ -0.370 -0.388 -0.405 -0.417 -0.122 -0.165 -0.199 -0.226 -0.252 -0.274 -0.291 -0.321 -0.3~ -0.368 -0.382 -0..00 -0.410 = ~ L. -0.121 -o.lM -0.199 -0.225 -0.2&2 -0.272 -0.290 -0.320 -0.34& -0.365 -0.379 -0.393 -0.404 a: 0 -0.121 -o.IH -0.198 -0.224 -o.2S0 -0.270 -0.288 -0.318 -0.341 -0.360 -0.37& -0.387 -0.399 -0.121 -0.183 -0.1\17 -0.224 -0.247 -0.268 -0.285 -0.315 -0.337 -0.355 -0.370 -0.380 -0.392 -0.120 -0.162 -0.1. -0.221 -0.244 -0.264 -0.282 -0.311 -0.331 -0.350 -0.362 -0.376 -0.386 -0.120 -0.161 -0.194 -0.219 -0.243 -0.261 -0.279 -0.306 -0.323 -0.342 -0.358 -0.388 -0.378 -0.085 -0.112 -0.137 -0.157 -0.175 -0.190 -0.205 -0.225 -0.237 -0.248 -0.256 -0.265 -0.272 -O.OM 1-00083 -0.110 -0.108 -0.135 -0.132 -0.166 -0.152 -0.172 -0.170 -0.190 -0.189 -0.204 -0.204 -0.225 -0.225 -0.237 -0.237 -0.248 -O.~48 -0.256 -0.2M -00264 \-00262 -0.272 -0.2i0 -0.018 -0.017 -0.020 -0.021 -00016\-0.0111 -0.018 -0.018 -0.020 -0.021 -0.022 -O.O'J3 -0.024 . -0.026 -0.027 -0.029 -0.030 -0.030 -0.032 -0.032 -0.036 -0.038 -0.040 -0.00 -000'11-0.00 -0.043 -o.CM6 -o.<K8 -0.049 -0.120 -0.160 -0.192 -0.218 -0.242 -0.2M -0.276 -0.299 -0.317 -0.33i -o.M8 -0.360 -0.869 Spanwise station 'II/(b/2) == 0.2 " 6 8 7 8 10 12 14 16 18 20 -0.085 -0.112 -0.138 -0.159 -0.176 -0.192 -0.204 -0.224: -0.240 -0.249 -0.258 -0.264 -0.271 -0.086 -0.113 -0.137 -0.159 -0.176 -0.192 -0.204 -0.225 -0.239 -0.248 -0.257 -0.265 -0.271 -0.086 -0.113 -0.137 -0.158 -0.176 -0.192 -0.205 -0.225 -0.238 -0.248 -0.2M -0.265 -0.271 -0.086 -0.113 -0.137 -0.168 -0.176 -0.191 -o~205 -0.226 -0.238 -0.248 -0.256 -0.:..>&5 -0.272 Spanwise station y/(b/2) == 2 3 4 5 6 7 8 10 12 14 18 18 20 -0.006 -0.002 0 O.OM 0.009 0.012 0.014 0.021 0.028 0.036 0.043 0.049 0.050 -0.011 -0.010 -0.006 -0.004 -0.002 -0.001 0 0.007 0.009 0.013 0.019 0.022 0.023 -0.013 -0.012 -0.011 -0.010 -0.008 -0.010 -0.008 -0.002 -0.001 0 0.002 0.004 0.006 -0.015 -0.015 -0.012 -0.012 -0.012 -0.013 -0.012 -0.010 -0.010 -0.010 -0.008 -0.008 -0.006 -0.016 -0.016 -0.016 -0.016 -0.018 -0.017 -0.017 -0.017 -0.017 -0.017 -0.016 -0.015 -0.014 -0.016 -0.018 -0.016 -0.018 -0.018 -0.018 -0.019 -0.020 -0.021 -0.021 -0.022 -0.022 -0.022 3 4: 6 6 1 8 10 12 1" 16 18 20 0.052 0.070 0.085 0.099 0.109 0.119 0.128 0.139 0.148 0.155 0.160 0.166 0.170 0.Q521 0.069 0.082 0.096 0.107 0.117 0.122 0.138 0.145 0.162 0.158 0.162 0.169 0.051 0.068 0.081 0.092 O.UM 0.114 0.121 0.135 0.141 O.UiO 0.154 0.160 0.165 0.050 0.068 0.080 0.091 0.102 0.112 0.120 0.132 0.140 0.148 0.151 0.158 0.159 00010 0.068 0.080 0.091 1 0.101 0.111 0.120 0.131 0.140 0.145 0.149 0.152 0.152 0.050 0.068 0.080 0.091 0.101 0.110 0.119 0.130 0.139 0.142 0.146 0.148 0.14& 0.4 -0.018 -0.016 -0.018 -0.020 -0.020 -0.020 -0.021 -0.022 -0.025 -0.028 -0.029 -0.031 -0.031 Bpanwise station 1//(b/2) 2 ::z -0.085 -0.113 -0.137 -0.158 -0.176 -0.191 -0.206 -0.226 -0.238 -0.248 -0.256 -0.265 -0.272 -0.016 -0.016 -0.019 -0.021 -0.021 -0.022 -0.025 -0.027 -0.029 -0.031 -0.034: -0.038 -0.038 -O.o-?l -0.025 -O.O".19 -0.030 -0.032 -0.035 -0.038 -0.041 -0.041 0.6 0.050 0.068 0.080 0.091 0.100 0.110 0.119 0.130 0.137 0.141 0.143 0.145 0.147 0.0li0 0.068 0.080 0.091 0.100 0.110 0:118 0.129 0.135 0.140 0.141 0.142 0.1.3 0.049 0.068 0.080 0.090 0.100 0.110 0.118 0.128 0.134 0.139 0.140 0.140 0.141 00049 0.068 0.080 0.090 1 0.100 0.109 (l117 0.126 0.132 0.138 0.139 0.139 0.140 0.048 0.068 0.080 0.090 0.100 0.108 0.116 0.124 0.130 0.135 0.136 0.138 0.140 THE SIGNIFICANCE OF WING-BECTION CHARACTERISTICS 13 TABLE I.-BASIC SPAN LIFT-DISTRIBUTION DATA.-(Continued.) ~I 0 I I I I 0.1 0.2 0.3 0.. LO.S I I I I I 0.6 Spanwise station y/(b/2) 6 6 7 8 10 12 14 16 18 20 0.072 0.088 0.100 0.109 0.115 0.121 0.126 0.136 0.145 0.152 0.159 0.161 0.166 0.079 0.098 0.113 0.125 0.135 0.142 0.149 0.160 0.170 0.182 0.186 0.197 0.201 0.080 0.101 0.120 0.135 0.148 0.158 0.184 0.178 0.188 0.200 0.205 0.215 0.220 2 3 4 6 6 7 8 10 12 14 16 18 20 0.069 0.068 0.074 0.081 0.087 0.090 0.092 0.098 0.100 0.102 0.103 0.105 0.107 0.068 0.083 0.098 0.107 0.117 0.123 0.131 0.139 0.147 0.156 0.161 0.166 0.172 0.072 0.092 0.111 0.122 0.136 0.146 0.153 0.166 0.178 0.188 0.197 0.202 0.211 2 3 4: 0.082 0.102 0.123 0.138 0.152 0.163 0.174 0.188 0.200 0.210 0.216 0.224 0.232 0.083 0.104: 0.125 0.140 0.156 0.169 0.180 0.195 0.208 0.216 0.222 0.230 0.237 0.085 0.108 0.128 0.143 0.160 0.172 0.182 0.200 0.212 0.221 0.22H O.2:J5 0.241 0.073 0.098 0.118 0.131 0.148 0.160 0.170 O.IM 0.198 0.208 0.219 0.228 0.233 0.075 0.099 0.121 0.138 O.IM 0.167 0.179 0.197 0.210 0.220 0.231 0.243 0.248 0.076 0.100 0.122 0.140 0.159 0.171 0.182 0.201 0.213 0.231 0.241 0.252 0.260 2 3 4 5 6 7 8 10 12 14: 16 18 20 0.038 0.044 0.050 0.052 0.0504 0.056 0.057 0.058 0.059 0.060 0.061 0.061 0.061 0.051 0.063 0.072 0.083 0.088 0.093 0.100 0.107 0.112 0.116 0.121 0.126 0.128 0.068 0.073 0.076 0.100 0.109 0.116 0.125 0.138 0.143 0.161 0.169 0.166 0.173 0.019 0.022 0.026 0.029 0.030 0.030 0.030 0.031 0.031 0.031 0.031 0.032 0.032 0.030 0.039 0.043 0.051 0.066 0.060 0.062 0.067 0.069 0.071 0.035 0.045 O.OM 0.065 0.071 0.078 0.081 0.090 0.095 0.102 0.111 0.121 0.128 0.059 0.078 0.092 0.107 0.119 0.130 0.140 0.152 0.165 0.174 0.184 0.194 0.203 0.060 0.079 0.095 0.110 0.122 0.135 0.146 0.162 0.179 0.190 0.203 0.213 0.225 0.060 0.080 0.097 0.112 0.128 0.140 0.152 0.171 0.189 0.202 0.218 0.229 0.239 0.9 1.0 0.086 0.110 0.130 0.148 0.162 0.173 0.183 0.202 0.216 0.227 O.~ 0.2 0.248 U.086 0.110 0.130 0.143 0.163 O.li4: 0.183 ·0.203 0.216 0.228 0.236 0.243 0.248 0.084 0.108 0.130 0.148 0.164 0.174 0.184 0.201 0.214 0.225 0.232 0.242 0.248 0.081 0.106 0.129 0.149 0.165 0.175 0.184 0.075 0.100 0.123 0.141 0.160 0.172 0.184 0.205 0.225 0.241 0.253 0.263 0.273 0.075 0.100 0.123 0.142 0.160 0.172 0.185 0.207 0.228 0.243 0.258 0.269 0.279 0.075 0.100 0.123 0.142 0.160 0.172 0.186 0.209 0.229 0.245 0.259 0.271 0.282 0.075 0.100 0.123 0.142 0.160 0.172 0.187 0.210 0.230 0.246 0.260 0.275 0.285 0.059 0.080 0.100 0.116 0.132 0.150 0.161 0.186 0.202 0.218 0.233 0.248 0.259 0.059 0.079 0.100 0.117 0.131 0.149 0.160 0.187 0.205 0.221 0.236 0.251 0.265 0.058 0.078 0.099 0.116 0.130 0.145 0.159 0.183 0.204 0.222 0.238 0.255 0.271 0.036 0.053 0.069 0.082 0.097 0.110 0.121 0.141 0.160 0.175 0.188 0.200 0.210 0.035 0.052 0.068 0.083 0.097 0.110 0.121 0.142 0.161 0.177 0.190 0.201 0.212 0.034 0.051 0.067 0.083 0.097 0.110 0.121 0.143 0.162 0.178 0.191 0.202 0.213 o.iss 0.210 0.220 0.229 0.238 0.247 = 0.9 0.075 0.100 0.123 0.141 0.160 0.171 0.183 0.203 0.221 0.238 0.249 0.260 0.268 Spanwise station 1I/(bj2) 0.8 = 0.8 0.085 0.109 0.128 0.147 0.160 0.173 0.182 0.201 0.214 0.223 0.232 0.239 0.245 Spanwise station y/(b/2) 0.7 = 0.95 0.060 0.080 0.099 0.113 0.130 0.144 0.158 0.178 0.198 0.211 0.222 0.236 0.245 0.060 0.080 0.100 0.114 0.132 0.148 0.160 0.182 0.200 0.216 0.229 0.241 0.251 Spanwise station 1I/(b/2) = 0.975 2 3 4: 5 6 7 8 10 12 14 16 18 20 OJ117 0.083 0.086 0.037 0.049 0.060 0.070 0.079 0.087 0.091 0.105 0.115 0.127 0.138 0.160 0.168 0.037 0.050 0.062 0.071 0.082 0.091 0.100 0.115 0.131 0.143 0.156 0.169 0.178 0.037 0.051 0.064: 0.075 0.088 0.098 0.107 0.124 0.141 0.155 0.169 0.182 0.193 0.037 0.052 0.068 0.078 0.091 0.101 0.112 0.132 0.149 0.163 0.178 0.191 0.202 0.036 0.054 0.069 0.081 0.094 0.107 0.120 0.138 0.153 0.171 0.182 0.197 0.208 THEORY OF WING SECTIONS 14 TABLE 2.-ADDITIONAL SPAN LIrr-DI8TRIBUTION DATA V~ues of L. for tapered winp·with rounded tips, Cia. ~ o.t - ~ L. I I I I I 0.6 0.8 ~ 0.7 0.8 0.9 1.292 1.276 1.2C0 1.248 "1.237 1.282 1.229 1.222 1.219 1.214 1.208 1.203 1.199 1.200 1.283 1.20 1.221 1.211 1.203 1.198 1.187 1.180 1.172 1.161 1.180 1.162 1.287 1.263 1.228 1.2M 1.187 1.171 1.lea I.W 1.143 1.188 1.127 1.118 1.109 1.282 1.2M 1.211 1.188 1.183 1.149 1.136 1.120 1.109 1.100 1.0lIO 1.0IJ0 1.070 1.211 1.2M 1.220 1.208 1.200 1.193 1.189 1.182 1.172 1.170 1.IM 1.180 1.166 1.263 1.228 l.209 1.1M 1.1M 1.174 1.188 1.118 1.148 1.144 1.135 1.130 1.123 1.280 1.221 1.198 1.181 1.1. 1.167 1.148 1.137 1.126 1.119 1.110 1.103 1.098 1.2a 1.214 1.188 1.1es 1.111 1.138 1.129 1.114 1.102 1.OM 1.087 1.078 1.089 1.169 1.118 1.148 1.138 1.128 1.121 1.111 Lt08 1.102 1.091 1.091 1.087 1.083 1.1. 1.117 1.147 1.1. 1.127 1.120 1.118 1.1M 1.099 1.090 1.080 1.080 1.078 LI88 1.J.a8 1.146 1.1M 1.128 1.119 1.111 1.102 1.094 1.087 1.081 1.076 1.071 1.188 1.111 1.014 1.018 1.014 1.015 1.013 1.012 1.011 1.008 1.008 1.003 1.011 l.0J3 1.023 1.02. 1.024: 1.024 I.OM 1.023 1.022 1.019 1.017 1.015 1.012 1.018 1.080 1.035 1.038 1.039 1.039 1.039 1.039 1.088 1.036 1.033 1.032 1.028 1.019 1.088 1.010 1.013 1.015 Spanwise station 1//(b/2) .. 0 2 3 4 a e 7 8 10 12 14 18 18 20 1.489 1.489 1.127 1.169 1.585 1.809 1.829 1.881 1.188 1.708 1.726 1.741 1.76& 1.«10 1.430 1.412 1.473 1.492 1.610 1.1M I.US 1.678 1.882 1.110 1.823 1.832 1.387 1.386 1.400 1.41" 1.428 1.440 1.319 1.'­ 1.4M 1.419 1.477 1.491 1.502 1.513 1.520 1.527 1.632 1.639 I.M7 1.329 loMe 1.303 1.317 1.388 1.393 1.401 1.411 1.417 1.423 1.428 1.429 1.431 1.300 1.308 1.318 1.324 1.329 1.332 1.338 1.M7 1.349 1.3M I.US 1.369 1.380 1.411 1.413 1.490 1.&02 1.&13 1.62& 1.631 1.339 1.3aO 1.380 1.369 1.378 1.388 1.392 1.409 1.420 1.429 1.433 1.441 1.'­ 1.318 1.322 1.329 1..333 1.338 1.360 1.1"" 1.361 1.881 1.386 1.388 1.310 1.312 1.301 1.302 1.302 1.301 1.300 l.aoo 1.300 1.301 1.308 1.309 1.309 1.308 1.307 I.M 1.288 1.279 1.272 1.287 1.2M 1.2M 1.2M 1.201 1.280 1.265 1.252 1.210 Spanwi8e station 1/1(b/2) .. 0.2 2 3 4 I 8 7 8 10 12 14 18 18 20 1.279 1.279 1.2M 1.288 1.290 1.291 1.294 1.299 1.302 1.307 1.308 1.309 1.311 1.217 1.2GO 1.2C1O 1.280 1.269 1.269 1.211 1.285 1.265 1.288 1.289 1.270 1.271 1.280 1.248 1.243 1.240 1.236 1.236 1.236 1.238 1.233 1.232 1.232 ;.1.231 1.230 1.268 1.241 1.282 1.223 1.218 1.214 1.212 1.209 1.202 1.201 1.199 1.195 1.190 I Spanwiae station 1I/(b/2) == 0.4 2 3 4 5 8 7 8 10 12 J" 11 18 20 1.217 1.220 1.223 1.228 1.229 1.229 1.229 1.228 1.228 1.228 1.228 1.228 1.228 1.190 1.191 LI92 1.193 1.1. 1.193 1.192 1.192 1.192 1.191 1.189 1.188 1.182 1.178 1.176 1.173 1.172 1.171 1.170 1.111 1.187 1.188 1.181 1.1. 1.1&2 1.140 1.172 1.181 1.182 1.159 1.1SS 1.152 I.lao 1.148 1.14& 1.138 1.131 1.129 1.127 Span~ise 2 a t s 8 7 8 10 12 14 18 18 20 0.970 0.950 0.932 0.920 0.909 0.900 0.891 0.881 0.812 0.888 0.881 0.818 0.851 0.978 0.982 O..MB 0.938 0.930 0.920 0.916 0.907 0.901 0.895 0.888 0.883 0.878 O.IM 0.976 0.982 0~9S3 0.949 0.940 0.938 0.929 0.923 0.918 0.912 0.908 0.898 1.172 1.161 1.1ae 1.149 1.146 1.ltO 1.118 1.132 1.125 1.111 1.112 1.111 1.110 O.9S6 0.947 0.941 0.937 0.931 0.925 0.920 1.170 1.169 1.149 1.140 1.132 1.124 1.120 1.113 1.107 1.100 1.097 1.092 1.089 1.1~ 1.133 1.125 1.118 1.110 1.100 1.090 1.082 1.015 1.070 1.015 station 1//(b/2) .. 0.6 - 1.003 0.992 0.985 0.978 0.971 0.966 0.959 1.171 1.180 1.111 1.142 1.138 1.131 1.1. 1.121 1.111 1.104 1.101 1.100 1.098 0.991 0.992 0.988 0.981 0.976 0.972 0.911 0.958 0.9&3 O.MS 0.944 O.NO 1.010 I.OM 1.002 1.000 0.993 0.989 0.988 0.978 0.972 0.919 0.968 0.913 0.98 1.012 1.011 1.008 1.008 1.002 1.000 0.999 0.992 0.989 0.988 983 0. 0.981 0.978 1 1 0.998 .000 0.995 1.OM 1.063 1.052 1.0&1 1.OD l.eNS 1.0&7 1.CN8 THE SIGNIFICANCE OF WING-8ECTION CHARACTERISTICS 15 TABLE 2.-ADDlTIONAL SPAN LIn-DISTRIBUTION DATA.-(Continued.) I I I I I I 0.2 0,3 u 0.5 0.7 0.8 0.9 1.0 0.746 0.772 0.796 0.808 0.820 0.827 0.834 0.842 0.8·0 0.858 0.862 0.870 0.878 0.747 0.782 0.800 0.822 0.838 0.8M 0.868 0.877 0.887 0.894 0.901 0.909 0.747 0.790 0.816 0.834 0.851 0.861 0.872 0.887 0.899 0.911 0.921 0.930 0.937 0.748 0.799 0.824 0.846 0.S62 0.875 0.886 0.905 0.919 0.933 0.944 0.953 0.962 0.637 0.568 0.598 0.822 0.643 0.610 0.673 0.698 0.718 0.739 0.768 0.773 0.791 0.638 0.671 0.603 0.630 0.852 0.671 0.686 0.712 0.736 0.769 0.780 0.800 0.819 0.639 0.676 0.609 0.636 0.669 0.678 0.696 0.723 0.751 0.776 0.801 0.822 0.846 0.386 0.418 0.440 0.483 0.4.82 0.S02 0.621 0.553 0.583 0.609 0.635 0.858 0.680 0.388 0.418 0.444 0.469 0.490 0.610 0.629 0.666 0.598 0.628 0.8M 0.882 0.707 0.390 0.420 0.446 0.471 0.496 0.615 0.634 0.576 0.608 O.MO 0.671 0.702 0.730 0.279 0.298 0.316 0.333 0.3SO 0.386 0.383 0.416 0.448 0.478 0.610 0.639 0.570 0.281 0.300 0.319 0.338 0.367 0.373 0.391 0.428 0.481 0.495 0.629 0.680 0.693 0.282 0.301 0.322 0.342 0.361 0.381 0.400 0."38 0.473 0.610 0.M8 0.580 0.816 0.6 Spanwise station y/(b/2) == 0.8 2 3 4 6 6 7 8 10 12 14 16 18 20 0.460 0.«4 0.678 0.712 0.659 0.700 0.691 0.644 0.632 0.681 0.619 0.676 0.609 0.670 0.663 0.600 0.686 0.853 0.676 0.848 0.569 .0.641 0.664 0.638 0.669 0.636 O.MS 0.629 2 3 4 6 6 7 8 10 12 14 18 18 20 0.378 0.362 0.331 0.814 0.300 0.290 0.282 0.266 0.H3 0.246 0.239 0.234 0.231 0.486 0."7 0.436 0.424 0.416 0.410 0.403 0.383 0.376 0.370 0.381 0.361 0.388 0.61& 0.689 0.668 O.MS 0.631 0.617 0.1i04 0.486 0.472 0.462 O.~ 0.131 0.726 0.723 0.720 0.717 0.713 0.710 0.704 0.702 0.699 0.698 0.698 0.698 0.740 0.743 0.746 0.748 0.748 0.748 0.748 0.748 0.74.8 0.748 0.748 0.750 0.753 0.746 0.7M 0.764 0.769 0.775 0.778 0.779 0.783 0.788 0.789 0.791 0.796 0.801 0.746 0.764 0.781 0.790 0.800 0.802 0.808 0.815 0.821 0.825 0.830 0.835 0.842 0.E45 Spanwise station 1I/(b/2) == 0.9 0.808 O.SOO 0.496 0.490 0.487 0.4M 0.481 0.472 0.469 0.418 0.468 0.410 0.473 0.626 0.628 0.632 0.631 0.631 0.636 0.636 O.Ml 0.M2 0.M6 0.M7 0.652 0.&6Q 0.631 0.M3 0.554 O.li8O 0.565 0.572 0.579 0.590 0.597 0.602 0.609 0.818 0.826 0.634 0.552 0.569 0.583 0.595 0.603 0.612 0.628 0.639 0.648 0.659 0.889 0.879 0.535 0.559 0.581 ·0.600 0.615 0.628 0.638 0.856 0.869 0.684 0.898 0.710 0.722 8panwise station y/(b/?) 4. 6 6 1 8 10 12 1" 18 18 20 0.231 0.209 0.191 0.176 0.166 0.161 0.148 0.138 0.132 0.129 0.128 0.122 0.121 0.298 0.290 0.286 0.281 0.278 0.272 0.261 0.2M 0.2M 0.262 0.262 0.2M 0.2&8 0.834 0.339 0.362 0.1" 0.348 0.346 0.346 0.346 2 3 4 6 6 7 8 10 12 14 16 18 20 0.132 0.119 0.107 0.098 0.089 0.081 0.f117 0.069 0.088 0.088 0.084 0.083 0.082 0.172 0.166 0.163 0.168 0.168 0.168 0.168 0.168 0.181 0.183 0.186 0.189 0.171 0.207 0.210 0.214 0.217 0.219 0.222 0.228 0.233 0.242 0.2i8 0.265 0.288 0.271 2 3 O.MS 0.M9 0.361 0.367 0.3M O.US 0.369 0.378 0.384 0.392 0.398 0.403 0.410 0.419 0.423 0.432 0.439 0."9 0.370 0.389 0.402 0.415 0.428 0.438 0.446 0.460 0.473 0.482 0.495 0.503 0.516 0.379 0.401 0.420 0.436 0.451 0.464 0.475 0.495 0.511 0.529 0.546 0.558 0.569 c:: 0.536 0.664 0.590 0.613 0.631 0.646 0.858 0.679 0.698 0.716 0.729 0.743 0.759 0.95 0.381 0.407 0.428 0.449 0.486 0.481 0.495 0.620 0.M2 0.562 0.681 0.598 0.613 0.383 0.412 0.434 0.468 0.476 0.4904 0.510 0.638 0.686 0.588 0.610 0.829 0.648 8panwise station y/(b/2) -= 0.975 0.239 0.260 0.258 0.269 0.272 0.278 0.283 0.295 0.308 0.320 0.331 0.346 0.363 0.263 0.278 0.288 0.304 0.314 0.320 0.328 0.343 0.380 0.376 0.394 0.412 0.4.35 0.272 0.289 0.304 0.320 0.332 0.342 0.352 0.373 0.395 0.413 0.435 0.461 0.483 0.274 0.291 0.308 0.322 0.340 0.351 0.363 0.390 0.413 0.438 0.463 0.492 0.616 0.277 0.294 0.311 0.328 0.3" 0.359 0.374 0.403 0.430 0.468 0.488 0.618 0.544: THEORY OF WING SECTIONS 16 where a. aloe = wing angle of attack measured from chord of root section = angle of zero lift of root section J = a factor presented in Fig. 9 The angle of zero lift for the wing a.(L-O) is obtained from a.(L-O) = ale. + J E The induced-drag coefficient CD. is given by CD, C L2 == -A + CLEaJJ + (Eae)'tD or u where the effective section lift-curve slope a. is taken lOO --- Ctlc.-1.0)r r--. --..;;::' ~ .98 r ~ .94 .92 roo.90 0 - ~ ~ the average for -- - - -1'6" ~~8 .--.-- ~ ~ .96 -1 - ....-.2 - 88 ---- r'to ~~ ,..... ~- ~ """"'"" ",,- Stroight- toper wing with rounded tips - I I I ~111i?t~o/lwi,?gI I I I I I I I 1- I Z' 4 /2 10 Aspect rot/o 8 6 /4 /6 /8 ~ o 2. Flo. 8. Chart for determining lift,..eune elope. G a. -= 11 + (57.3o,./ ...A ) the sections composing the wing and u, VJ and to are factors presented in Figs. 10, 11, and 12, respectively. The drag coefficient CD of the wing is obtained from 2 {bIt CD = 8}o CdC dll~ CD, where 11 = distance along span Cd == local section drag coefficient corresponding to local section lift coefficient C, The wing pitching-moment coefficient C_. about an axis through the aerodynamic center is given by C.. = Ec.. - GEaeA tan fJ 17 THE BIGNIFICANCB OF WING-8ECTION CHARACTERISTICS -.5 cJ ci=lO -.4 ~ .. . 8 -----­ --.. _t .6 .4­ 't2 ­ Ito -.3 J -.2 -./ Sfraighf- toper wing with rounded tips ~ --I 7 7 £llipficol. wi79 I o 4 6 8 1-­ I 1I I I I m ~ m M Aspect rot/a m ffl 9. Chart for determining angle of attack. CL a. .. a«. J« a.(L-o) = a, JE FIG. + CI /.0 a ~ -- -::::- --::::: ~ .96" / ", 0 /j 0 '. + ~ ~ :::::-..../ I; .9.e / I~ / Ifl V "" ------ "'""- ............. r<, V- ~ ........ <, ............... "'-- I'---. ~ ~ "- ;;"7 -/;.. /0 -- - h9-- .--.. .............. .............. ........... i'--..--; ~ ~~~ /5 ~ Aspecf rafio ~ ':JO-"" ,~ r/f 1/ rl / .84 l II .8 0 .............. ....... ~ /~ If/I .88 -- - - r.:::::- --- ------ ---- vI VI u­ + z~ ~ ~/ o. ......... ~- - - ./ .2 Sfroighf- toper wing with rounded tips Elliptical wing - - .71'6 o .3 .4 .5 Toper .6 ratio .7 .8 FIG. 10. Chart for determining induced-drag fact.or u. CDi .. CLi -A + CLfIJefJ + (<<J.)2tD 1r 1.1 .9 1.0 THBORY OF WING SBCTIONS 18 006 ~ 004 ~ ~ --.8 ~ c,lc.-1.0 ~I' / 002 / j'/ /' / "" 0 "" .-. ~ ~ - ......... ~ ...... r- 1.6 .4 V ­.002 ~ ~ r--. ~ t--" .2 .AI .... Sfra,r- taper wing with rounded fips Jl.. 10- £//ip Col wIng. 11-0 006 """"- 008 o l--~~ 4 2 8 6 ~ .-.~~ ,...... m ~ Aspecf rafio :FIG. 11. Chart for ~~ u. ~ /8 ~ 20 determiniDa inclueecl-drae factor IJ• •ODS ... ~ A .003 [if' •.002 .001 o , .~1.8 .4 .004 ~ .~ ~.;: :::-: .t-O ~ ~ c:-- -.... -r-- ,...... - ~ ... t,...-- ~- ~ ~ - "--- ~~ 1-0. Ce:/c.-O Straight-lope).. 'w~ng with rounded tips, ~ -;- T7·:~lliptico/,wi,?91 I ·f I I I l I I" Z 4 FIG. 6 8 10 Ie Aspect ratio /4 12. Chart for determining induoed-drag factor /6 /8 ~ - ~ EO to. where the section pitching-moment coefficient c,.., is constant along the span. If the sections composing the wing have different pitching-moment coefficients, a weighted average may be used. E and G are factors pre­ sented in Figs. 13 and 14, respectively. 19 THE SIGNIFICANCE OF WING-8ECTION CHARACTERISTICS The longitudinal position of the aerodynamic center of the wing given by ~ ~b=HA tan{J where X ac == longitudinal distance between aerodynamic center of root section and aerodynamic center of wing, positive to the rear H == a factor presented in Fig. 15 /.4 '-.. ~, ~ l2 i'~ '-- ~ to- ,....- ~ A spec t ratio 3 F===:= ~ - - -'---- l--- lO 9 6 .8 E .6 .4 '- .2 o ~ - - - Sfroighf- toper wing wifh rounded lips I - Elliptical wing - ./ .e ..3 .4 Toper .5 .6 ratio, C.,/cs .7 .8 .9 1.0 FIo. 13. Chart for determining pitching moment due to section moment. The maximum lift coefficient-for the wing may be estimated from the assumption that this coefficient is reached when the local section lift c0­ efficient at any position along the span is equal to the local maximum lift coefficient for the corresponding section. This value may be found con­ veniently by the process indicated by Fig. 16. Spanwise variations of the local maximum lift coefficient Cz..a, and of the additional c~ and basic c" lift distributions are plotted. The spanwise variation of Clmax - Cit is then plotted. The minimum value of the ratio Cr..s - Cit Clea is then found. The minimum value of this ratio is considered to be the maximum lift coefficient of the wing. c. Generalized Solution. Although the Glauert" method used by Ander­ THEORY OF WING SECTIONS 20 son l O requires the assumption of a linear variation of lift with angle of attack, methods of successive approximation have been developed by Sherman109 and Tanills that permit application of the actual wing section data. Boshar" applied Tani's method to the solution of this problem. Sivells and Neelylll have developed procedures based on this method which .Q3c . eo. Vfs___ -- ~I- .OcB ~ ./ .024 / .OEO l/ G ~ ~ ~ .0/6 pL/ .Ole ~ ----- ~ .004!--- o ~ -c... .-- ~ ~ ,.-- ~ ~ I...-- ~ I--- ~ - t!2 =::::- - ~ ~ 5 t::== '"'"""'" - poo--' .-- ~ - - ./ FIG. - - ~ ~ ~ - - I- ~ ~ - -r--- ~ I-- -,..... 1- - -~ - 0- - - - - t--- po-- - I--- 1 - ~~ - .~ -,...-- ASRtetjf ratio 2 - - 1-~ I- ~ r- .3 r--- '-' ,.......... ~P'"'"" - - ~ ~ to-- 4 ,..-- - "'"'"""'" ~ --- - 6 8_ ~ ~ I-- ~ ::::---- ~ ~~ -~ .008 ~ V V ~ - :7 k ~ .-.'" ~ ~ t-- ~ ......,;". - - 1- - 10- - !--- 1-"- ~ -'"- - I-- l-- ~ - '-- - ~ -P"- r--- Straight-toper wing wifh rounded fips_ Ellipficol, wi19 t I I I I I I I I - I . .3 .2 14. Chart fOT .5 .4 .6 TODer ratio, c~/Cs .7 .8 .9 1.0 determining pitching moment due to basic lift forces. em". == - GtaoA tan fJ can be carried. out by an experienced computer and which yield highly satisfactory and complete lift, drag, and pitching-moment characteristics of wings, This method will be presented in sufficient detail to permit such calculations to be made. The basic problem is the determination of the downwash at each point along the span from summation of the effects of the trailing vortices. The strengths of the trailing vortices, however, are unknown, but they are intimately associated with the spanwise lift distribution which, in tum, is dependent upon the downwash. The relation between the downwash and the lift distribution as given by Prsndtl" is 04 !!.-.l ... 2... = 180 b /2 (d/d1l)(c,c/4b)d1l -b/2 '1/1 - Y (1.8) 21 THE BIGNI.FICANCB OF WING-8ECTION CHARACTERISTICS °.24 I c;jc. ~ e:::;;­ ~~~ ....­ l..---'" ~ l--­ ~ ~ ­I.° lt ~ ~ - - I-­ ~ ,....­ ~ ,-­ L-­ I~ ~6 - ... - --­ - _1000.­ - - ~ ""-­ .EO ~ H • /8 r--- r--... t--­ -­ --­ 10­ I~C ~ ..... - ~ 'YJ ./6 ./ 4~ Sfroignf- toper wing with rounded tips_ I i I I I I I I - , - --;- --;- -;- Ellipficol,wi'791 1 FIG. I I I I 8 6 4 10 IZ Aspect ratio :ib = HA tan II -- 1.4 I---~- ctma.:- c1b.. . ~ ~- ~. 1"'-'­ ¥ Cl mox• -...... t - L-­ ~ " .... ~t-L36cIQj 1.0 ~ /8 15. Chart for determining aerodynamic-center position. 1.6 1.2 I I /6 14 1.-0- ~ ..... Cla;­ - ... - , ~\. \ r---. '~ \ I "1\\, \ .8 cz Ctmoll.= 1.36 .6 , .4 - .2 ~c1b - o -.20 ./ .2 .3 .4 .5 ~~ .6 r-­ ....... .7 - ~ .8 2Jb Flo. J6. EstimS.tiOD of CLcax for example winz, .9 ~ /.0 eo THEORY OF WING SECTIONS 22 where a. a= angle of downwash (in degrees) at spanwise position 1/1 '1111:1 spanwise position, variable in the integration and measured from center line The effective angle of attack of each section of the wing is (1.9) a.==a-a, where a. == effective angle of attack of any section (angle between chord line and local wind direction) a == geometric angle of attack of any section (angle between chord line and direction of free stream) The effective angle of attack is a function of the lift coefficient of that section !(c,) 80 that the equation for the effective angle of attack becomes a. = !(c,) =a - 180 . .!.1,,/2 (d/dy)(c,c/4b)dy -"12 ... 2... (1.10) 1/1 - 1/ This general integral equation must be solved to determine the downwash and span-load distribution. Equation (1.10) may be solved by assuming a span-load distribution and solving for the corresponding downw• . The load distribution corresponding to this calculated downwash is then found using Eq. (1.9) and the section data. This load distribution is com­ pared with the assumed distribution, and a second approximation is made. The process is continued until the load and downwash distributions are compatible. The span-load distribution is expressed, following Glauert," 88 the Fourier series (1.11) where cos (J == - 2y /b The development of this method will be limited to load distributions symmetri-cal about the center line. Only odd values of n are therefore used. The induced angle from Eq. (1.8) becomes . ISO a, == --;:- ~nA" sin nO sin 6 (1.12) If it is assumed that values of c,c/4b are known at an even number r of stations equally spaced with respect to (J in the range 0 ~ 8 ~ ..., the coefficients A. of the Fourier series (Cit) ~1.. ~.. b r-l == .-1, ... A · n ­m1l" • SID. r where m == I, 2, 3, . . . ,r - 1 may be found by harmonic analysis as THE 8IGNIFICA1!CE OF WING-8ECTION CHARACTERISTICS 'ItA. = 23 r2.sm n rmr Because of the symmetrical relations (:). =(:)~ and . SID mr . (r - m)r n - = SIn n - - ­ r r for odd values of n, the summation of A" needs to be made only for values of m from 1 to T /2. Therefore (1.13) where 4. m1l" 11m" == -smnr r 11m" z:: for m == 1, 2, 3, . . . , (r/2) - 1 and r2.sin n '2 11" for m == r/2 From these relations Sivells and Neely obtained multipliers "." which are presented in Table 3 for r = 20. The coefficients Aft may be obtained by multiplying the known value of clc/4b by the appropriate multipliers and adding the resulting products. By the substitution of the values of Aft from Eq. (1.13) into Eq. (1.12), Sivells and N eel y ll1 obtained the following expression for the downwash at the same stations at which c,c/b is known: (1.14) where k designates the station at which (Xi is to be evaluated, and m is also a station designation that is variable in the summation. Values of 'Xmk for r = 20 are given in Table 4.111 A similar table of values of (41r/I80)~k was also given by Munk. 72 The induced-angle multipliers are independent of the aspect ratio and taper ratio of the wing and thus may be used for any wing whose load distribution is symmetrical. Equation (1.14) permits the solution of Eq. (1.10) by successive ap­ proximations. For one geometric angle of attack of the wing, a lift dis­ ~ TABLE a.-MULTIPLIERS " ..." FOR A" CoEFFICIENTS • m1l" L A"slnn 20 • 19 An A-l,3 2y b o o.1564 o.3090 o.4540 .5878 .7071 o.8090 o.8910 .9511 o.9877 X 10 9 8 7 6 5 4 3 2 1 or = GENERAL FOURIER SERIES L m-l 10 ( CIC) 4b '1m" 1ft 1 3 5 7 9 11 13 15 17 19 0.10000 0.197M 0.19021 0.17820 0.16180 0.14142 0.11756 0.09080 0.06180 0.03129 -0.10000 -0.17820 -0.11756 -0.03129 0.06180 0.14142 0.19021 0.19754 0.16180 0.09080 0.10000 0.14142 0 -0.14142 -0.10000 -0.09080 0.11756 0.19754 0.06180 -0.14142 -0.10021 -0.03120 0.16180 0.17820 0.10000 0.03129 -0.19021 -0.09080 0.16180 0.14142 -0.11756 -0.17820 0.06180 0.19754 -0.10000 0.03129 0.19021 -0.09080 -0.16180 0.14142 0.11756 -0.17820 -0.06180 0.19754 0.10000 -0.09080 -0.11756 0.19754 -0.06180 -0.14142 0.19021 -0.03129 -0.16180 0.17820 -0.10000 0.14142 0 -0.14142 0.20000 -0.14142 0 0.14142 -0.20000 0.14142 0.10000 -0.17820 0.11756 -0.03129 -0.06180 0.14142 -0.19021 0.19754 -0.16180 0.09080 -0.10000 0.19754 -0.19021 0.17820 -0.16180 0.14142 -0.11756 0.09080 -0.06180 0.03129 -0.20000 -0.14142 0 0.14142 0.20000 0.14142 ~ ~ ~ ~ ~ s ~ ~ ?S-s ~ C ~ 25 THE SIGNIFICANCE OF WING-BECTION CHARACTERISTICS TABLE 3a. .". MULTIPLERS - '1t111 FOR DETERMINATION OF WING CoEFFICIENTS 4 2y b 11" - ".1 4 o 0.07854 0.15515 0.14939 0.13996 0.12708 0.11107 0.09233 0.07131 0.04854 0.02457 0.1564 0.3090 0.4540 0.5878 0.7071 0.8090 0.8910 0.9511 0.9877 tribution is assumed from which the load distribution c,c/b is obtained. The corresponding downwash is calculated from Eq. (1.14) using values of Ami: from Table 4. The downwash is subtracted from the geometric angle of attack at each station to give the effective angle of attack of the section [Eq (1.9)]. The section lift coefficients are obtained from curves of the section data plotted against the effective angle of attack a e where ae = E(ao - ala) + alo and E is the Jones" edge-velocity correction. If the resulting "lift distribution does not agree with that assumed, a second approximation to the lift distribution is assumed and the process is repeated until the assumed and calculated lift distributions agree. The entire process must be repeated for each angle of attack for which the wing characteristics are desired. The amount of labor required to obtain the complete characteristics of a wing may be reduced by using Anderson's method" for the range of lift coefficients where the section lift curves are substantially linear. The wing lift coefficient is obtained by spanwise integration of the lift distribution. This process'!' has been reduced to the following summation: CL = ~(CIC) b ... 4: 71... A~ 11" 1 (1.15) m==1 The multipliers r/4 ""'1 are equal to 11'"/4 times the multipliers for A and are presented in Table 3a for r = 20. The section induced-drag coefficient is equal to the product of the sec­ tion lift coefficient and the downwash in radians. ~ TABLE 4.-MULTIPLIERS ,,"'. FOR INDUCED ANGLE OF A'rrAClt al. - to ( e,f.) ~ b ",-1 ~~ 2y b 0 0.1564 0.3090 0.4MO 0.3878 0.7071 0.8090 0.8910 0.9511 0.9877 ~ k m 10 9 8 7 6 6 4 3 2 1 0 0.1864 0.3090 0.4640 '" 0.5878 ~"'~ 0.7071 0.8090 0.8910 0.9611 0.9877 ~ ~ ~ ~ 10 9 8 7 6 5 4 143.239 -116.624 0 -12.884 0 -4.051 0 -1.638 0 -0.459 -58.633 145.026 -64.802 0 -8.320 0 -67.298 IM.611 -82.917 0 -7.872 0 -2.871 0 -0.620 -6.980 0 -67.157 160.761 -61.803 0 -7.208 0 -2.016 0 0 -10.168 0 -72.472 177.054 -71.743 -2.866 0 -9.916 0 -82.083 202.571 -81.434 0 -7.899 0 0 -4.840 0 -10.926 0 -97.965 243.694 -96.962 0 -7.089 -2.880 0 -1.062 0 .... 0 0 -7.370 0 -1.491 3 2 -1.804 0 -3.394 0 -4.968 0 0 -6.812 -18.134 0 -17.888 0 -126.637 0 316.612 -180.628 -122.880 463.533 -167.045 0 1 -1.468 0 -8.768 0 -7.713 0 -26.68li 0 -329.976 915.601 ~ s ~ ~ ~ o ~ THE SIGNIFICANCE OF WIJ.VG-8ECTIOlV CHARAC7'ERIS7'lCS 27 The wing induced-drag coefficient may be obtained by means of a span­ wise integration of the section induced-drag coefficients multiplied by the local chord. AB in the case of the lift coefficient, this process has been reduced to the following summation: (1.16) The section profile drag coefficient can be obtained from the section data for the appropriate wing section and lift coefficient. The "ring profile drag coefficient may be obtained by means of a spanwise integration of the section profile drag coefficient multiplied by the local chord. This process has again been reduced to the following summation: (1.17) where c == mean geometric chord Sib The section pitching-moment coefficient can be obtained from the section data for the appropriate wing section and lift coefficient. For each spanwise station the pitching-moment coefficient is transferred to the wing reference point by the equation em == c... - ~ [cz cos (a. - ai) C + Cdo sin (a. - ai)] - ~ [et sin (a. - a,) C where Cde cos (a. - ai)] c.. == section pitching-moment coefficient about wing reference point a. == geometric angle of attack of root section x == projected distance in plane of symmetry from wing reference point to aerodynamic center of wing section, measured parallel to chord of root section, positive to the rear z == projected distance in plane of symmetry from wing reference point to aerodynamic center of wing section, measured per­ pendicular to root chord, positive upward, The wing pitching-moment coefficient may be obtained by spanwise in­ tegration by use of the multipliers previously used. (1.18) where c' (b/2 = mean aerodynamic chord, 2/8 Jo c2 dy THEORY OF WING SECTIONS 28 1.6. Applicability of Section Data. The applicability of section data. to the prediction of the aerodynamic characteristics of wings is limited by the simplifying assumptions made in the development of wing theory. It is assumed, for instance, that each section acts independently of its neigh­ boring sections except for the induced downwash. Strict compliance with this assumption would require two-dimensional flow, that is, no variation of section, chord, or lift along the span. Such spanwise variations result in spanwise components of flow. H these components are small, the sections act nearly independently. Com­ paratively small spanwise variations of pressure, however, tend to produce large crossflows in the boundary layer. The air adjacent to the surface has lost most of its momentum and therefore tends to flow directly toward the region of lowest pressure rather than in the stream direction. These crossflows become particularly marked under conditions approaching sep­ aration. The flow of this low-energy air from one section to another tends to delay separation in some places and to promote it in others, with the result that the lying characteristics may depart seriously from the cal­ culated ones. In the development of the lifting-line theory, it was assumed that the effect of the trailing vortices was to change the local angle of attack, neglecting any change of downwash along the chord of the section. This variation of downwash is not negligible for sections close to a strong trailing vortex. Strong concentrations of the trailing vortices are obtained when­ ever a large span'vise variation of lift OCCUI'S. Consequently the sections are operating in a curved as well. as a rotated flow field whenever the span­ wise variation of lift is not small. This curved flow field may be interpreted as an effective change in camber which is not considered in ordinary lifting­ line tbeories. This difficulty is avoided by lifting-surface theory, the treat­ ment of which is outside the scope of this volume. Experience has shown that usable results are obtained from lifting-line theory if no spanwise discontinuities or rapid changes of section, chord, or twist are present, and if the wing has no pronounced. sweep, These conditions are obviously not satisfied near the wing tips, near the ex­ tremities partial span flaps or deflected ailerons, near cutouts and large fillets, or if the lying is partly stalled. Since the assumed conditions are not satisfied near the- wing tips, it is obvious that section data-are not applicable to wings of low aspect ratio. In fact, an entirely different theory" applies to wings of very low aspect ratio. The ordinary three­ dimensional wing theories are obviously not applicable at snpereritical speeds when the velocity of sound is exceeded anywhere in the field of flow. Despite these limitations, wing-section data have a wide field of ap­ plication to wings of the aspect ratios and plan forms customarily used for airplanes flying at subcritical speeds. The lift, drag, and moment of 29 THE SIGNIFICANCE OF WING-SECTION CHARACTERISTICS characteristics of such wings may be estimated from the section data with a fair degree of accuracy by Anderson's methods" and with a high degree of accuracy by the method of Sivells and Neely.!" The order of accuracy /.6 1.4 I/~ 1.2 - ~ 1f~ I "'-- r: --- .8 .6 .2 ,J Calculated ~neralized method) Colculofedflineuized method) I/o.. CoIcUIoIed~~ method) ----- Calculated(linearized method) j - i ,,1/1 1I / t> )" \ -.2 ~ L-::!, CExperimenlo1 ~ o \ j 7 7 .4 "0 ...,/ ~ Experimental 1.0 o. ..e.- to oO.a? .04 .06 .08 ~ 12 .10 d-4 .14 0 4 ~ 8 /2 16 .. 0 ~ .1 FlO. 17. Calculated and measured characteristics of a wing of aspect ratio 10 with NACA 44-series sections. 1.4 1.2 c;,iQiiiled... .8 . // c , I ~ / "'-" 1.0 .6 } .4 ~ "of{K~1<lsIs sr,. o -.2 o , CDo r-- 0 -, ~ --- ~ ExptJl'imtJnlrJl- ~ --- - ColculaltJd - i ~I I ~ ~, .. 7 I 1 JT ) .. 0 \ .i i\ · ~ ~ .004.008 .012 ~ --- ~ I t J I ~aNf*t "• .2 z.. I/' / ~ .02 .04 .06 .08.10 -4 / 0 4 CD 8 (& /2 0 -./ em 18. Experimental and calculated characteristics of a wing having NACA- 64-210 airfoil sections, 2-deg wasbout, aspect ratio of 9. and taper ratio of 2.5. R 4,400,000; M = 0.17. FIG. = to be expected from predictions made by the latter methodl" 110 is indicated by Figs. 17 to 20. The three-dimensional wing characteristics were obtained from tests in the NACA 19-foof pressure tunnel. The predicted charac­ teristics were based on section data obtained in the N.L~CA two-dimen­ sional low-turbulence pressure tunnel.P' The agreement between the ex­ THEORY OF WING SECTIONS 30 - 1.4 1.2 1.0 .s -..... /" CoJculoled-.,'/ /( fests /t HtJ\e I ~Force ~ .l'''\ ~ b( I ~ r--....... ..- -~ ...., Experimenlol--r--- r--~ r- - r1 I CoIculoted--- ~a , t/ o -.2.0 SIJfVf!YS ~ I I ~ .)/ t .D04.oos.fJ12 ~~ t o.a.D4.D6 0.8./0 -4 / 0 4 8 0 "12 -:/ ~ ~ 4 ~ Flo. 19. Experimental and calculated characteristics of a wing having NACA 65-210 airfoil sections, 2-deg washout, aspect ratio of 9, and taper ratio of 2.5. R =.; 4,400,000; M = 0.17. 1.4 '" / 1.0 .8 .6 •4 ..2 o • J).... 1.2 rt r ......... f !l'·oFon:tltesls <;;j t , - ..20 .000.DOS.W .............. ~ j ~ ~ ~~ ~ ;~ -~oEx~ ~ 4 ~ --Ct1It:uItIItId----...: r---. j v ,,,/ I l 8 /2 0 -;/ a ~ FIG. 20. Experimental and caleulated characteristics of a wing having NACA 65-210 airfoil aootions. o-deg washout, aspect ratio of 9, and taper ratio of 2.5. R == 4,400,000; J.\f = 0.17. ~ 0 II ~~ V )" ,I, cQiaialed'*"i p -02 .04 .06 .08./0 -4 0 4 ~ perimental and predicted characteristics is so good that the utmost care is required in model construction and experimental technique to obtain em­ pirical wing characteristics as reliable as those predicted from the section data. CHAPTER 2 SIMPLE TWO-DIMENSIONAL FLOWS 2.1. Symbols. H K 8 V X Z a total pressure nondimensional circulation, r /2ra V pressure coefficient, (H - p)/~2PV2 resultant or free-stream velocity component of force in the x direction component of force in the y direction component of force in the 8 direction constant a ~.;V Y b constant c constant e' constant e" constant e base of Naperian logarithm, 2.71828 Z section lift In logarithm to the base e m source strength per unit length p pressure r radial eoordinnte • position of source on z axis t time u component of velocity in the ~ direction u' radial component of velocity " component of velocity Ia the 11 direction ,,' tangential component of velocity, positive counterclockwise tD component of velocity in the • direction z length in Cartesian coordinates 11 length in Cartesian coordinates • length in Cartesian coordinates r circulation, positive clockwise 8 angular coordinate, positive counterclockwise • velocity potential t/I stream function II doublet strength, 2ms ... ratio of the circumference of a circle to its diameter p mass density of air 6J angular velocity, positive clockwise 2.2. Introduction. A considerable body of aerodynamic theory has been developed with which it is possible to calculate some of the important 31 32 THEORY OF lVING SECTIONS characteristics of wing sections. Conversely it is possible to design wing sections to have certain desirable aerodynamic characteristics. The pur­ pose of this chapter is to review the basic fluid mechanics necessary for understanding the theory of lying sections. 2.3. Concept of a Perfect FlUid. The concept of a perfect fluid is an important simplification in fluid mechanics. The perfect fluid is considered to be a continuous homogeneous medium within which no shearing stresses can exist. For the purpose of this chapter the perfect fluid is also con­ sidered to be incompressible. The assumption of zero shearing stresses, or zero viscosity, eliminates the possibility of obtaining any information about the drag of wing sec­ tions or about the separation of the flow from the surface. This assump­ tion is very useful, nevertheless, because it simplifies the equations of motion that otherwise cannot generally be solved and because the resulting solutions represent reasonable approximations to many actual flows, In all cases under consideration, the viscous forces are small compared with the inertia forces except in the layer of fluid adjacent to the surface. The direct effects of viscosity are negligible except in this layer, and viscosity has little effect on the general flow pattern unless the local effects are such as to make the flow separate from the surface. The assumption of incompressibility also results in simplified solutions that are reasonable approximations to actual flows except at high speeds. Although gases such as air are compressible, the relative change in density occurring in a field of flow is small if the variation of pressure is small compared with the absolute pressure. This assumption leads to increas­ ingly important discrepancies as the local velocity anywhere in the field of flow approaches the velocity of sound. The effects of compressibility will be discussed in Chap. 9. 2.4. Equations of Motion. One of the fundamental conditions that must be satisfied is that no fluid can be created or destroyed within the field of flow considered. This condition means that the amount of fluid entering any small element of volume must equal the amount of fluid leaving the element. For an incompressible fluid the amount of fluid may be measured by its volume. The equation of continuity, expressing this condition, may be derived from the following considerations. Figure 21 shows a small element of volume having dimensions dx, dy, and dz. The components of velocity along each of the three axes x, 'Jj, and z are u, u, and w. The volume of fluid entering across each of the faces of the element of volume, perpendicular to the x, y, and z axes, respectively, is u dydz entering yz face v dz dx entering zx face wdxdy entering xy face SIMPLE TWO-D{/1fENSIONAL FWWS 33 To the first order of small quantities, the amount of fluid leaving the element of volume across the corresponding opposing faces is (u + ~: dx ) dy dz (v + :;; d y) dz dx (w + :: dz) dx dy leaving yz face leaving zx face leaving xy face r FIO.21. Z Element of volume considered in derivation of equation of continuity. The condition of continuity requires that the volume of fluid leaving the element must be equal to the volume entering. Hence the equation of continuity for an incompressible fluid is au + aV + ow = 0 ax ay iJz (2.1) The other .fundamental condition to be satisfied is that the motion of the fluid must be in accordance with N ewton's laws of motion. This con­ dition may be stated as follows: Du Dt dx dy dz X = p y= p~ Z =p dxdydz (2.2) Dw Dt ds: dy dz where X, Y, Z == components of force on the element of fluid d» dy dz in" the z, y, and z directions, respectively t == time and the differentiations are performed following the motion of the element. THEORY OF WING SECTIONS 34 For axes stationary with respect to the observer, the components of veloc­ ity u, v, and ware, in general, functions of t, x, 'Y, and e. The total deriva­ tives given in Eqs, (2.2) may therefore be written au au au au Du at + u ax + v ay + W iii == Dt av at av av au"Dv == D..t + u ax + v ay + 1D az aw at + U aw iJx aw 8w Dui + v iJy + w oZ == Dt H gravitational forces are neglected, the only forces acting on the ele­ ment of volume are normal pressure forces. Let p dy dz be the force acting on the face of the element (Fig. 21) in the yz plane. Then to the first order [p + (apjox) dx] dy dz will be the force acting on the opposite face. The resultant force on the element in the x direction is therefore ap X == .- ax dx dy dz Similarly the resultant forces in the y and z directions are y==-iJPdydzdx Oy Z == - ap dz dx dy az The equations stating that the motion of the fluid is in accordance with Newton's laws may be written in the following form by substituting the foregoing expressions for the forces and accelerations in Eqs. (2.2) (au + u au + v au + to au) at ax ay az iJp == p (~ + u au + v av + w av) _ lJp == p ax _ ay - ap - == iJz at p ax iJy iJz (2.3) (ow awax aw aw) -+'1.L-+v-+w­ at ay Bz If we consider only steady motion, the derivatives with respect to time are equal to zero. In this case the differential equations for the path of an element are dx dy dz (2.4) -.=:-==U f) 10 These equations merely state that the displacement along any axis is proportional to the component of velocity along that axis." Preparatory to BIJIPLB TWO-DI},(EN810NAL FWMTS 35 integrating Eqs. (2.3) along the path of motion for steady flow, we shall multiply the equations for the x, 1/, and z components by dx, dy, and dz, respectively. For example, au) ap (u-dx+v-dx+tD-dx au au --dx=p ax ax ay az According to Eqs. (2.4), we have v dx=udy wdx=udz Hence - ap - dx = ax p (u au - dx + u iJu - dy + u au) ax ay . -az dz = p d (1.,) 2 ~ u- Similarly - ap dy = p d(!2 v2) ay - ap dz = 8z pd(!ur) 2 Adding the three foregoing relations, we have - dp = !!. d(u2 + tP + w) = !!- d(V2 ) 2 2 where V = magnitude of velocity. Integrating this equation gives Bernoulli's equation p+%pV2 = H (2.5) where H, the constant of integration, is the total pressure. The application of Eq. (2.5) is limited by the assumptions made in its derivation, namely, 1. Perfect incompressible fluid. 2. Steady motion. 3. Integration along path of motion (streamline). It follows that this equation may be applied only along streamlines in unaccelerated flow where the effects of viscosity and compressibility are negligible. 2.6. Description of .Flow Patterns. It is evident from the foregoing equations that the flow pattern would be completely determined if the values of u, n, and to were known at every point. The theory of wing sections assumes that the flow is two-dimensional, that is, W = o. Even with this simplification, it is still necessary to specify both the components of velocity u and v to define the flow, By the use of the equation ofcon­ tinuity (2.1) it is possible to simplify the problem further so that it is necessary to specify only a single quantity at each point to determine the flow pattern. 36 THEORY OF lVING SECTIONS Consider the flow across the arbitrary line oba that connects the origin o with a point a (Fig. 22). The amount of flow across this line will be the same as that across any other line connecting the two points, for example, line oea, provided that the equation of continuity holds for all points within the region enclosed by the two lines. 'Vith a fixed origin 0 the amount of fluid crossing any line joining the points 0 and a is therefore a function only 'of the position of the point a for a given flow pattern. The amount of fluid passing between the points 0 and a is given by the following expression 1/1= y J: udy-vdx or elf = 11, dy - v dx (2.6) Because the value of 1/1 is independent of the path of integration, Eq. (2.6) remains valid even though dx and dy are varied inde­ pendently. The general expression for the total differential of 1/1 is d.I6 FlO. 22. Derivation of stream function. 'I' = at/! dx + iJl/I d ax ay Y (2.7) This equation is also valid for independent variations of dx and dy. The coefficients of dx and dy in Eqs. (2.6) and (2.7) must therefore be equal, or :~~a~} (2.8) ax _" A single function 1/1 has therefore been found by means of which it is possible to define both components of velocity at all points in the field of flow. This function'" is called the "stream function." Lines in the flow along which t/t is constant are called Cl streamlines," and these lines are the paths of motion of fluid elements for steady motion, In general, the component of velocity in any direction may be obtained by differentiating the stream function in a direction 900 counterclockwise to the component desired. In polar coordinates, therefore, the expressions for the radial and tangential components of velocity are u , =lay, -­ roO v, = - -~ ­ dr radial t tangential .\ (2.9) SIMPLE TWO-DIMENSIONAL FWWB 37 Most flows of a perfect fluid are of the type known as II irrotational motion." In irrotational motion the fluid elements move with translation only. Their angular velocity is zero. This absence of angular velocity does not, of course, prevent the ele­ ments from moving in curved paths. To obtain expressions for the angular velocity of an element w ---,-.-.. in terms of the velocity deriva­ !?doting tives, consider the motion of a disc solid disk. If ,ve assume clock­ wise motion to be positive, the angular velocity of the disk may be expressed as (Fig. 23) CA) = !(au _av) 2 ay ax The expression (~: -:=) FIG. 23. Velocity derivatives for rotatiDg disk. is called It vorticity." The vorticity is seen to be twice the angular velocity. The integral of the tangential component of velocity around any closed curve is defined as the circulation. A simple relation exists between the cir­ culation around a curve and the vorticity over the area enclosed by the curve. Consider the small element shown in Fig. 24. + (u + ~; dy ) dx - at: = v dy dr = (auay _oxau) dx dy or (v + :~ d.'t) dy - u dx That is, the element of circulation is equal to the vorticity multiplied by the element of area. For any finite area, the u+/tdY circulation is given by the following expression: r dy V u = !!(au ay - av) dx dy ox (2.10) The angular velocity of any element can be changed only by the application of tangential FIG. 24. Calculation of cireu(shearing) forces which, by definition, are ab­ lation about element of area. sent in a perfect fluid. It follows that, if the flow is once irrotational, it will rem~in irrotational and the circulation around any closed path will equal zero. If the flow is not irrotational, the circu­ lation around any path moving with the fluid will remain constant. ds THEORY OF WING 8BC7.'IONS 38 H the flow is irrotational, it is possible to derive a second quantity which, like the stream function, can be used to describe the flow pattern completely. Consider the flow field indicated in Fig. 25. The line integral of the velocity over the path oap must be equal to the line integral of the velocity over the path obp if .the motion is irrotational in the region be­ tween the two paths. The value of this integral, called the U velocity potential 4>," therefore depends only on the p0­ sition of the point p relative to the origin o, The value of the velocity potential is I o rP= J:udx+vdy d;=Vcos a; ds Fro. 25. Definition of ve­ locity potential. or d</>=1tdx+vdy (2.11) By a process of reasoning similar to that previously given for the stream function, we obtain 1t == a4>} ax (2.12) aq, v=­ iJy In general, the component of velocity in any direction may be obtained by differentiating the velocity potential in the direction of the desired component. This property of the velocity potential makes it particularly useful for the study of three-dimensional Bows. In polar coordinates, the expressions for the radial and tangential components of velocity are ,aq, ar &dial} U = r v' == ! oq, tangential roO (2.13) The equation of continuity for two-dimensional How au + 8v == 0 ax ay (2.14) assumes a particularly simple form when expressed in terms of the velocity potential a2q, + atq, = 0 (2.15) ax'- ay2 This equation is Laplace's equation in two dimensions. The equation stating that the flow is irrotational is au_ovc:O ay ax SIMPLE TWO-DIMENSIONAL FWWS 39 When written in terms of the stream function, this equation becomes a~ iJ21/I a1/ + ax' = 0 (2.16) In writing Eq. (2.15), it is implicit in the definition of q, that the motion is irrotational and the equation itself states that no fluid is being created or destroyed. In writing Eq. (2.16), it is implicit in the definition of '" that no fluid ia being created or destroyed, and the equation itself states that the flow is irrotational. Equations (2.15) and (2.16) impose rather general conditions on functions chosen to represent actual flow patterns. Any function of cP or t satisfying Eqs. (2.15) or (2.16) represents a possible flow, It can be seen from Eqs. (2.8) and (2.12) that ocP oift u=-=­ ax ay aq, at V=-=-­ ay ox These equations indicate that lines along which ep is constant intersect lines along which 1/1 is constant at right angles. Because Eqs. (2.15) and (2.16) are linear, functions of ep or 1/1 repre­ senting possible flow patterns may be added to obtain new flow patterns. This method of obtaining new flow patterns by superposition of known flows is fundamental to the theory of wing sections because it leads to simple solutions of complicated problems. 2.6. Simple Two-dimensional Flows. A few simple flows upon which the theory of lying sections is based are described in this section. a. Uniform Stream. Consider the functions and If=bY-ax! (2.17) ep == bz + ay which satisfy Eqs. (2.15) and (2.16). The component of velocity along the x axis is iJ1/I iJep u-=-=-=b iJy ox The component of velocity along the y axis is 01/1 iJep V:=:--=-==4 ax ay Equations (2.17) therefore represent a uniform field of flowhaving a velocity V equal to v'a2 + b2 inclined to the x axis at an angle whose tangent is a/b. b. Sources and Sinks. The -eoncept of sources and sinks is one of the building blocks used to construct desired flow patterns. A source is con­ sidered to be a point at which fluid is being created at a given rate. A sink 40 THEORl" OF WING SECTIONS is a point at which fluid is being destroyed. The flow about a point source or sink is assumed to be uniform in all directions and to obey the equations of continuity and of irrotational motion everywhere except at the POint itself. In two-dimensional motion, the flow is assumed to be the same in all planes perpendicular to the z axis. The point source in two-dimensional motion is therefore a line parallel to the z axis from which fluid emanates at a uniform rate throughout its infinite length. In accordance with these assumptions, the radial component of velocity u' at any point at a distance T from a source creating fluid at a rate m per unit length perpendicular to the plane of flow may be expressed as u' ~ 2...r = .(2.18) Because the sink is simply a negative source, the equation for the radial component of velocity is the same as Eq. (2.18) except that the velocity is directed toward the sink. The tangential component of velocity resulting from a source or sink is zero. . Taking the origin at the source ,ve have u == u' cos 8 v == u' sin (J r=Vr+y2 where cos (J == v'x2 + 11 • (J SID == v'z! + y2 y Substituting in Eq. (2.8), we have ih/I mx u = iJy = 211'(x! + V) Integration of this expression with respect to y results in the expression 1/1 = m tan-Ill x 27 (2.19) The same expression may be obtained by a corresponding process using the expression for v and integrating with respect to x. It is seen that f/I is a multiple-valued function of x and y. This peculiarity results from the fact that the equation of continuity is not valid at the source itself. The expression for the flow from a point source in terms of the velocity potential <p may be obtained as follows: oq, mx t t = - = -2- - ­ ax cP = m In 2r 2r(x + y2) vx + y2 2 (2.20) SlJfPLE TWO-DIMENSIOlfAL FWWS 41 From Eqs. (2.19) and (2.20) it is seen that the streamlines for a source situated at the origin are radial lines and that the equipotential lines are circles about the origin. c. Doublets. A doublet is the limiting case of the flow about a single source and a sink of equal strength that approach each other in such a manner that the product of the source strength and the distance between the source and sink remains constant. The axis of the doublet is the line joining the source andsink. To obtain expressions for the flow about a doubtlet, consider the arrangement of "a source and a sink shown in Fig. 26. The source and sink are situated on y the x axis at - 8 and 8, respectively. The flow resulting from the combi­ nation is 1/1 = !!!.(tan-1 ~ 2r x +8- tan-1 _ Y- ) X - 8 Since a-b tan:" a - tarr? b = tan-1 - ­ I +ab we have l/I = m tarr:' - 2ys 2r x 2 + y1. - 8 2 FlO. 26. Streamlines of flow for a source and sink. For values of s small with respect to x and y, 1f = If the product of m and 8 ;: C-; ~y:~) remains constant as 8 approaches zero, (2.21) where p, = 2ms d. Circular Cylinder in a Uniform Stream. The flow about a circular cylinder in a uniform stream is obtained by superposing a uniform stream [Eq. (2.17)] on the flow about a doublet [Eq. (2.21)] with the uniform stream flowing from the source to the sink. 1f = Y( IT - 21r(x:+ if») This equation may be written as follows in terms of polar coordinates: '" = Y r where a2 r2 (1 - ~) sin 8 = JJ/21r l' = x 2 + y2 (2.22) THEORY OF WING SECTIONS 42 It is obvious from Eq. (2.22) that part of the streamline'" == 0 is the circle r = a. At large distances from the origin the flow is uniform and parallel to the x axis. Equation (2.22) therefore represents the flow about a circular cylinder in a uniform stream. The streamlines for this flow pattern are shown in Fig. 27. The velocity distribution about the cylinder may be obtained by finding the tangential component of velocity on the circle T == a. 11 = - : = - V ( 1 + ~) sin (J = - 2V sin (J (2.23) i FIG. 27. Streamlines for the flow about a circular cylinder in a uniform stream. Because the circle r == a is a streamline, \ve may apply Bernoulli's equation (2.5) to obtain the pressure distribution about the cylinder. p + ~2P(4yt sin! 6) == H If we define a pressure coefficient S as S= H-p Mp VJ (2.24) the distribution of S over the surface of the cylinder is given by S == 4 sin 2 8 (2.25) Integration of the pressure distribution over the surface of the cylinder will show that the resultant force is zero. e. Voriex. A vortex is a flow pattern in which the elements of fluid follow circular paths about a point, and the flow is irrotational at all points except the center. The equations defining the flow pattern may be derived directly from Laplace's equation for the stream function. Equation (2.16) may be written 88 follows in terms of polar coordinatesr" (2.26) 8IMPLE TWO-DIMEN8IONAL FWW8 43 Because the radial component of velocity is zero, !ol/l = 0 ro8 and hence lo~ =0 r afP Equation (2.26) therefore becomes lPt/t+!d1/l=O dr rdr Integrating once with respect to r we obtain d ln 1/l + 1n r = c dr or d1/l dr = I' rc' = 2111 == - , v (2.27) where the constant of integration c' is assigned the value I' /21r. The veloc­ ity therefore varies inversely with the disy tance from the center of the vortex. The circulation I' about the vortex is equal to - 2rrv'. The circulation is considered to be positive in the clockwise direction, whereas v' is positive in the counterclockwise direction. The stream function may be obtained by in­ tegration of Eq. (2.27) r 1/1 == 211" In r + e" (2.28) x The pressure in the field of flow may be FIG. 28. Equilibrium conditions for a vortex. calculated from the condition that the pressure gradients must maintain the fluid elements in equilibrium with the acceler­ ations (d'Alembert's principle). The centrifugal force acting on each ele­ ment must be (Fig. 28) pv'" --rd8dr r The pressure force acting on the element is Equating the pressure and acceleration forces, we have pV'2 r= dp dr 44 THEORY OF WIlvG SECTION8 Substituting the value of v' from Eq. (2.27) gives p~ 47N8 p= - = dp dr p~ srr+ H where H = constant of integration This equation may be written in the form P+~pV'2==H which shows that Bernoulli's equation may be applied throughout the field of flow, FIG. 29. Streamlines for the flow about a circular cylinder in a uniform stream with eireu­ lation corresponding to a wing-eection lift coefficient of 0.6. f. Circular Cylinder with Circulation. The flow pattern represented by a circular cylinder with circulation is the basic flow pattern from which the flow about "ring sections of arbitrary shape at various angles of attack is calculated. Such & flow pattern is obtained by superposing the flow pro­ duced by a point vortex upon the flow about a circular cylinder. Adding the flows corresponding to Eqs. (2.22) and (2.28), we obtain '" -= vr(l - a~ sin 8 + 2r~ In !a ~J (2.29) where the constant e" in Eq. (2.28) has been taken equal to -r 1n a 21r A typical flow pattern for a moderate value of the circulation r is given in Fig. 29. The velocity distribution about the cylinder is found by differentiating the expression for the stream function [Eq. (2.29)] as follows: (ar. 2 &/I r - = V 1+-) sin8+­ ar 2rr SIMPLE TTflO-DIMENSIONAL FWWS 45 The tangential component of velocity v' (positive counterclockwise) at the surface of the cylinder is obtained from Eq. (2.9) and the substitution of r == a. Vi == - 2V sin 8 + ~ (2.30) 21ra It is seen that the addition of the circulation I' moves the points of zero velocity (stagnation points) from the positions 8 = 0 and 11" to the positions 8 = sin-1 - r­ 4raV The pressure distribution about the cylinder may be found by applying Bernoulli's equation (2.5) along the streamline 1/1 = o. p + 21 P (4 V2 SIll . 2 8 - 2Vr sin 8 + ra r 2 ) 4ra2 == H (2.31) Setting r 21Ta V = K (2.32) the pressure coefficient S [Eq. (2.24)] becomes S == 4 sin 2 8 - 4K sin 8 + K2 (2.33) The symmetry of Eq. (2.33) about the line 8 = 1f/2 shows that there can be no drag force. The lift on the cylinder can be obtained by integration, over the surface, of the components of pressure normal to the stream. fo2rSa sin 0 dO = 72PV fo2r (4a sin" 0 - l = 72P V2 2 4aK sin" 9 + aJ(2 sin 8) dO l == 72PV2ak[28 - sin 28]~r l == 2pV 2akr == PlTr It can be shown that the relation l == pvr is valid regardless of the shape of the body. 61 (2.34) CHAPTER 3 THEORY OF WING SECTIONS OF FINITE TIDCKNESS 3.1. Symbols. A., B. H 8 V a coefficients of the transformation from x' to z total pressure pressure coefficient (H ­ p)/J-ipVt velocity of free stream radius of circular cylinder Cr section lift coefficient e base of Naperian logarithms, 2.71828 i V=-i In logarithm to the base e m source strength per unit length (~J - (is p 1- p local static pressure radius vector of z (modulus) component of velocity along x axis component of velocity along y axis local velocity at any point on the surface of the wing section complex variable Cartesian coordinate real part of the complex variable % or Cartesian coordinate magnitude of the imaginary part of the complex variable z complex variable complex variable in the near-circle plane complex variable for the flow about a circle whose center is shifted from the center of coordinates circulation, positive clockwise section angle of attack distance of the center of the circle from the center of coordinates complex variable mngnitude of the imaginary part of the complex variable r angular coordinate of the complex variable z (argument) angular coordinate of z' (see Fig. 32) aeA is the radius vector of z doublet strength per unit length real part of the complex variable r ratio of the circumference of a circle to its diameter mass density of air real part of the complex variable w potential function 46 r u v v w x x y y z z' z* r ao E r " 8 fJ ~ p. ~ 11' p q, q, r THEORY OF WING SECTIONS OF FINITE THICKNESS " 1/1 1/1 1/1 1/10 angular coordinate of z magnitude of the imaginary part of the complex variable w stream function ae~ is the radius vector of z' (see Fig. 32) average value of 1/1 00 infinity 47 3.2. Introduction. It was shown in Chap. 2 that the field of flow about a circular cylinder with circulation in a uniform stream is known. It is possible to relate this field of flow to that about an arbitrary "ring section by means of conformal mapping. In relating these fields of flow, the circulation is selected to satisfy the Kutta condition that the velocity at the trailing edge of the section must be finite. Such characteristics as the lift and pressure distribution may then be determined from the known flow about the circular cylinder. The resulting theory permits the ap­ proximate calculation of the angle of zero lift, the moment coefficient, the pressure distribution, and the field of flow about the section under con­ ditions where the flow adheres closely to the surface. 3.3. Complex Variables. Conformal mapping, which makes the cal­ culation of'wing-section characteristics possible, depends on the use of complex variables. A complex number is a number composed of two parts, a real part and a so-called imaginary part. The real part is just an ordinary number that may have any value. The imaginary part contains the factor v=!, which is given the symbol i. A complex variable z may therefore be written in the form z = x+ iy where the symbols in the expression x + iy obey all the usual rules of algebra. It should be remembered that~"2 is equal to - 1. Such a variable z can be represented conveniently by plotting the real part z as the abscissa" of a point and the magnitude y of the imaginary part as the ordinate. By De Moivre's theorem, z may also be written in the form z = reiIJ where rand 8 are the polar coordinates. Let us consider the complex variable w=<I>+il/l and let w = f(z) , that is, 4> + il/l = f(x + iy) Then :2q,2 + i:~.. = /:"r(x + iy) == f'(z) uX u~t and cP (j2 .a2;p ~ ( +.) f"( ) iJ 2 + ~ a y 2 = JJl1J X ~y = z y THEORY OF WING SECTIONS 48 adding, we have 02</J + iJ2q, + i(iJ'-t/I + iJ~) iJ y 2 ax2 iJ y 2 ax2 = 0 In any equation involving complex variables, the real and imaginary parts must be equal to each other independently. Therefore iJ2</J a*lq, or + iJ y 2 = 0 and iJ~+ay=o ar iJlI Because these equations are the same as those of Eqs. (2.15) and (2.16), any differentiable function w = j(z) where and i = x+ iy may be interpreted as a possible case of irrotational fluid motion by giving the meaning of velocity potential and stream function, respectively. The derivative dw/dz has a simple meaning in terms of the velocities in the field of flow. dw = d</>+ i a1/l dz = dx + i dy Further a</J at/> d</J = - dx + - dy </> and'" ax ay a." dl/l = - dx + - dy ax ay 81/1 Therefore [(iJt/>/ax)dx dw dz + (o(j)loll)dy] + i [(iJ~/iJx)dx + (iJ1/I/iJy)dyJ d.x+ i dy = In order for dw/dz to have a definite meaning, it is necessary that the value of du: 'dz be independent of the manner with which dz approaches zero. If dy is assumed to be zero, the value of the differential quotient dw/dz is o</J + i01/! = u - iv ax a~l~ = dw dz (3.]) according to Eqs. (.2.8) and (2.12)./ Similarly, if dx is assumed to be zero, the value of the differential Quotient duridz is ~ iJq, + 0'" = _ iv + 'It toy ay THEORY OF WING SECTIONS OJl~ FINITE THICKNESS 49 The expressions for simple two-dimensional flows given in Sec. 2.6 may be expressed conveniently in terms of complex variables. Uniform stream parallel to x axis to = Vz (3.2) Source at the origin w=m ln z 2r (3.3) Doublet at origin with axis along x axis w=~ (3.4) 2rz Circular cylinder of radius a in a uniform stream w=V(Z+:2) (3.5) Vortex at origin 10 = ir -In z 2r (3.6) Circular cylinder with circulation w = V (z + a2)ir -z + -21r In -az (3.7) 3.4. Conformal Transformations. .1\ conformal transformation con­ sists in mapping a region of one plane on another plane in such a manner that the detailed shape of infinitesimal elements of area is not changed. This restriction does not mean that the shape of finite areas cannot be considerably altered. It has been ShO"·11 previously that the equipotential lines and streamlines intersect at right angles, thus dividing the field of flow into a large number of small rectangles. It has also been shown that the equation w = j(z) represents a possible flow pattern. The equation w = g(r) represents another flow pattern where ~ r is the complex variable + i11 The coordinates in the z plane are considered. to be x and y, and those in the r plane are ~ and 'I. If the equipotential lines and streamlines are plotted in either of the planes, they will divide the plane into a large num­ ber of small rectangles. These rectangles will be similar at .corresponding points in both planes. The corresponding points are found from the relation f(z) = g(t) THEORY OF' WING SECTIONS 50 This equation accordingly represents the conformal transformation from the z plane to the r plane or the converse. In practical use, the Ho\v function in the z plane is known and the corresponding flow function in the r plane is desired. To plot the flow known on the z plane on the r plane, it is necessary to solve this relation for r and to obtain an equation in the form r == h(z) In the transformations we shall consider, this relation will be given, The velocities in the z plane, by Eq. (3.1), are dw _ dz ==u-~v The corresponding velocities in the dw r plane are dwdz (3.8) dt == liz dr As a simple example of a conformal transformation, consider the re­ lations W=V(Z+~)=Vt These relations transform the flow about a circular cylinder on the z plane [Eq. (3.5)] to uniform flow parallel to the ~ axis on the plane [Eq. (3.2)]. Corresponding points of both planes are obtained from the relation r r == z+­atz " and are indicated in Fig. 30. 3.5. Transformation of a Circle into a Wing Section. A circle can be transformed into a shape resembling that of a wing section by substitution of the variable a2 t==z+­z (3.9) into the expression for the flow about a circular cylinder having a radius slightly larger than a, and so placed that the circumference passes through the point x == a. If, in addition, the center of the larger cylinder is placed on the x axis, the transformed curve will be that of a symmetrical wing section (Fig. 31). In the present example, let the center of the larger cylinder be placed at the point x = - E where E is a real quantity. The radius of this cylinder will then be a + E. The equation of flow about the larger cylinder with circulation is then 10 == V(z* + E + (a + E)2) + Z*+E ir In z* + E 211" a+E THEORY OF WIl{G SECTIONS OF FINITE THICKNES8 - - - - - - - ----4~-- + - - ~ 51 --------­ Z PLANE I {=Z+-j- 2 I .I; ( PLANE FIG. 30. Conformal transformation of the flow about a circular cylinder to uniform flow. FIo. 31. Conformal transformation of a circle into a symmetrical wing section. THEOR1' OF WING SECTIONS 52 The more general expression for the flow about the circular cylinder with the flow at infinity inclined at an angle aD to the oX axis is found by substitut­ ing the expression Z + E = (z* + E)eiaro 1» = V [(Z + E)e-iao + (a ~~):eiao] + ;~ In (z ~ 2eE- iao (3.10) Substitution of Eq. (3.9) into Eq. (3.10) would result in the equation for the flow about the wing section but would lead to a complicated ex­ pression. A simple way of obtaining the shape of the 'ling section is to select values of z corresponding to points on the larger cylinder and find the corresponding points in the r plane by the use of Eq. (3.9). The velocity of any point on the wing section can be found from Eq. (3.8). dw dr = [v (e- dw dz = dz dr Wzo _ (a e + E)2 iae ) (z + E)2 + ir ] (~) z~ - a2 2r(z + E) It can be seen from this equation that the velocity at the point z = a is infinite unless the first factor is zero. The point z = a corresponds to the trailing edge of the wing section. The Kutta-Joukowsky condition states that the value of the circulation is such as to make the first factor equal to zero, which is the condition that ensures smooth flow at the trailing edge. The value of the circulation satisfying this condition is found as follows: V + E)2eiaol + ir - 0 iao _ . (a + E)2 J 2"..(a + E) ­ [e- (a ir = 211'"(a + E) V(e iao - e- iao) Since eiao - e- iao • • •• 2 == Sinh tao == t sin ao or I' == 4"..(a + E)V sin Qo The leading edge of the wing section corresponds to the point a2 r=-a-2E­ a + 2E neglecting powers of E greater than one r= - 2a Because the trailing edge corresponds to the point r == 2a THEORY OF WING 8ECTIO~V8 the chord of the wing section is 4a. The lift on the wing section is and the lift coefficient is Ct = 211" ( 1 53 OF FINITE THICKNESS p vr, . + :) sin «XcI In the limiting case as E/a approaches zero, the slope of the lift curve dc,/da o is 2...per radian for small angles of attack. Detailed computations" z-Pkne 1------....------. z~Plme t - - - - - - f ! I i o - - - - - I ~=x+iy 32. Dlustration of transformations used to derive airfoils and calculate pressure dis­ tributions. FIG. will show that the thickness ratio of the wing section is nearly equal to (3V3/4) (E/a). For wing sections approximately 12 per cent thick, the theoretical lift-curve slope is about nine per cent greater than its limiting value for thin sections. 3.6. Flow about Arbitrary Wing Sections. vious section that the transformation a2 s=z+­z It was seen in the pre­ THEORY OF WING SECTIONS 54 transforms a circle in the z plane into a curve resembling a wing section in the plane. Most wing sections have a general resemblance to each other. If the aforementioned transformation is applied to a wing section, the resulting curve in the z plane will therefore be nearly circular in shape. Theodorsen recognized this fact and showed that the flow about the nearly circular curve, and hence about the wing section, can be derived from the flow about the true circle by a rapidly converging process. The basic method is presented in reference 122, and a detailed discussion of the method is given in reference 125. The derivation of Theodorsen's relation for the velocity distribution about the wing is divided into three parts (Fig. 32). 1. Derivation of relations between the flow in the plane of the wing section (r plane) and in the plane of the near circle (z' plane). 2. Derivation of relations between the flow in the z plane and in the plane of the true circle (z plane). 3. Combination of the foregoing relations to obtain the final expression for the velocity distribution in the r plane in terms of the ordinates of the wing section. The basic relation between the z' plane and the r plane is r r = r + --;~z The coordinates of (3.11) r are defined by the relation f=x+iy and the coordinates of z' are defined by the relation z' == tJt!I'+#} By Eq. (3.11), Since e#} and since = cos () + i sin (J r = a(~ + e~) cos 8 + ia(e1' - e-') sin () == 2a cosh '" cos (J + 2ia sinh '" sin 8 x y t == x+iy = 2a cosh 1J' cos = 2a 8} sinh '" sin () (3.12) Expressions for'" and fJ in terms of x and y are desired and may be obtained as follows: x cosh'" == 2a cos 8 sinh '" = 2a SID ~ 6 , I THEORY OF WING SECTIONS OF FINITE THICKNESS 55 Since l cosh l sinh '" - '" =1 (2a :OS S- (2a ~in 8Y= 1 !; ! or 2 sin 8 = P + l ~r + (~y (3.13) where Because sin 8, and hence cos 8, are known in terms of the ordinates x and y of the wing section, the values of sinh'" and cosh '" can be found from Eq. (3.12). The factor relating velocities in the z' plane to those in the r plane is dz'/dr. From Eq. (3.11), dr dz' = 1_ a2 = ,!(z, _ ~) Z'2 = -= = Z' .!(aeY+ iB - z' ~ a(eI' - z z~ Z' ae-+- iB) rI') cos 8 + ia(e" + e-I') sin 8 (2a sinh '" cos 8 + 2ia cosh '" sin 8) (3.14) The second step is to find the relation between the flows in the z' plane and those in the z plane. The coordinates of z are defined by the relation z = aeA+;'" The transformation relating the z' to the z plane is the general transforma­ tion i; (A.+iB.)(l/p) z'=ze 1 By definition Consequently or co '"- }. + i«(J - q» = "\' 4 (An + iBn)..!. (cos rIO 1 where z has been expressed in polar form z = r(cos q> +i sin q» nq> - i sin nq» THEORY OF WING SECTIONS 56 Equating the real and imaginary parts, we obtain the two Fourier ex­ pansions ~(A" 1/1 - ). = L, -;:;- cos n" + B..) r" sm n" 1 and 00 (J-qJ= 1(Bn (3.15) A".) -cosntp--Slnn'l' rn rn 1 These relations show that l/I - ~ and 8 - tp are conjugate functions. In order for the deviation of the near circle from the true circle to be a minimum, the value of ~ corresponding to the radius of the true circle is taken to be 1 2 1/10 ). = = -1 21r .­1/1 0 df) where the values of 1/1 correspond to points on the near circle. The required transformation from the near circle to the true circle is then (3.16) If 1f/ is known as a function of tp, 8 - I{) can be found by a method de­ veloped by Naiman." The 1/1 function is assumed to be approximated by a finite trigonometric series of the form t/I(tp) = 1/10 + Al cos I{) + .. · + An-I cos (n - 1)1{) + A" cos <p + B 1 sin 'P + . · · + Ba-l sin 1 (n - l)cp n-l 1/I(rp) = 1/10 + .--1 (A", cos m" + B", sin mrp) + A" cos ~ If '" is specified at 2n equally spaced intervals in the range 0 ~ cp ~21r: that is, 0, 1r/n, 2r/n, ...., [(2n - l)1r]/n then 2n-l \11o where 1/Ir = value of.p at f/' = = 2~ 1 \IIr r-=O T7r/n 2n-l .4", = ! \" \IIr cos m!:! nL, n rlCO 211-1 B", = ! " \IIr sin m nr nL, n r-O A" = ~ L(- 2n-l r-=O l)r\llr B.=O THEORY OF WING SECTIONS OF FINITE THICKNESS 57 Now, by Eqs. (3.15), 2: 1I-l( 2: 2: "'r 11-1 f{J - (J = E(Ip) = (A.. sin nkp - B m cos mlp) + A .. sin nip m-l 2n-l 1. = 17: mf{J SIll m:=1 cos m r=O rT n 2n-l - cos mlp~ "'r sin m r; 11-1 2n-l = ~ LL 2n-l ) 2~"'t sin nf{J ~ (- 1)r"'r + 2n-l "'r sin m(f{J - r:) + 2~ sin nf{J 2:(- 1)r"'r m=l r=O r=-O Upon interchanging the order of summation, there is obtained 2n-l E(Ip) = ~ .-1 2: 2: "'r r-O' r:) + 2~ sin nip 2: (- 1)r"'r 2n-l sin m(f{J - m~l r==O If E is evaluated at the same points qJ at which 1f is given, that is, at the points qJ = r'r/n, the variable qJ - (r7T/n) becomes [(r' - r)r]/n = - (Kr/n), and the last term becomes zero. "'n: 1fr = rr =,p ("r n + nK 1r) = 1/1 (~ + Knr) ==.pK 2n-1 E(Ip) = - 11-1 1~ ~. n~ "'K ~ sm m n K-O ](r fI&~l The summation over K may also be taken from 0 to 2n - 1 because of the periodicity of 1/Ix. A simple expression can be obtained for the coefficients of 1/IK !(r [( odd . [(r cot -Slnm-= 2n { 11, 0 Keven and therefore 2: 2n-l E(qJ) = - -1 n .pK cot -K7r 2n tc odd E-l or Kodd (3.17) THEORY OF WING SBCTIONS 58 In most cases a value of n equal to 40 gives sufficiently accurate results. Ordinarily 1/1 is known as a function of 6, and first approximations to the values of and (J - f' are obtained by substitution of 6 for fJ in the relations for IJ - fJ and The factor relating the velocities in the true-circle plane to those in the near-circle plane is dz' /dz. From Eq. (3.16), "'0 "'0. ~' dz = ~ {Iz+ d [(1/1 -1J'0) + i(t) dz fP)]} z' ~ [If + i(B - tp) + In z] = But, on the true circle, z=~ from which !s-== dz !i In z = !i (In a + 1/10 + if{J) == !i (kp) dz dz Therefore dz' = z' dz !! [1/1 + i(O dz + ifP] ,,) dz' d - = z' - (l/I + i8) dz dz This expression may be written dr dz = z' !i (1/1 + is) dB dJJ dz But 1 .df(J z dz -=1­ or dz = i d((J = i d(f(J ­ 8) + i dB z and Hence dz' dz = z'd. 1 zdO (- 1,1/1 + 8) 1 + (dEjdIJ) where or z' 1 - i(tU/dD) dz' dz = z 1 + (dE/dB) (3.18) THEORY OF WING SECTIONS OF FINITE THICKNESS 59 The flow in the plane of the true circle of radius a,eh is described by the following equation: W== a2e2h) if Z +-Inz 2r aeh V ( z+-- (3.19) where the value of a in Eq. (3.7) is here taken to be ae/". The velocities in the true-circle plane are 2e2 dw ( 1 a- "") -=V dz Z2 ir +2rz (3.20) In order for the rear stagnation point of the circular cylinder to cor­ respond to the trailing edge of the wing section, the flow about the cylinder is rotated through an angle ao equal to the angle of attack of the wing sec­ tion. This process corresponds to the application of the Kutta-Joukowski condition. At zero angle of attack, the trailing edge corresponds to the point z = a,eh+~T It is therefore necessary for the circulation r to have a value corresponding to a rotation of the stagnation point by an amount ao + ET. This value may be seen from Eq. (2.30) to be I' = 4r~V sin (ao + ET) Substituting the general value of z on the surface of the cylinder into Eq. (3.20) dw == V[I - e-2i (ae+.» dz + 2i sin (ao + ET)e-i<ao+.)] from which the absolute value of dw/dz may be obtained I~;I = 2V[sin (an + q» + sin (ao + ET)] (3.21) The third step is to obtain the velocities in the plane of the wing section from the expression for the velocities in the plane of the true circle. The expression for the velocity in the plane of the wing section is dw dw dz dz' dr = dz dz' dr Multiplying Eqs. (3.14) and (3.18), we obtain drdz' dz' dz = 1[I1-+ i(dl/l/dfJ)]. (dE/dO) Z . · (2a SInh ,pcos 8+ 2w cosh,p sm 8) The absolute value of Idrl == 2v'1 ++ (t#I/d8)2 v':sinh 1/1 + 8m 2 dz eh[1 (de/dO)] 2 (J THEORY OF WING SECTIONS 60 Dividing Eq. (3.21) by this expression, Iddrwl= v = where v V [sin (ao + tp) + sin (ao + Er)][l + (dejdO)]e" -V(sinh2 '" + sinl 6)[1 + (dl/l/d8)I] (3.22) = local velocity at any point on surface of wing section V = free-stream velocity The necessary calculations to obtain the pressure distribution for a given wing section are as follows: 1. The coordinates of the wing ~ction are found with respect to a line joining a point midway between the nose of the section and its center of curvature and its trailing edge for sections having sharp trailing edges. 122 The coordinates of these points are taken as (- 2a, 0) and (2a, 0), respec­ tively. It is usually convenient to let a have the value of unity. 2. sin 8 is found from Eq. (3.13). 3. sinh 11' is found from Eqs. (3.12). 4. '" is found from tables o( hyperbolic functions. 5. '" is plotted as a function of 6. 6. #0 is determined from the relation "'0 =.1 (2" '" dB 2r Jo 7. A first approximation to E is obtained by conjugating the curve of t/I plotted against 8 using Eq. (3.17). 8. From the curves of E and '" plotted against 8, dE/dO and th/I/dB are found. 9. Determine F by the relation F == [1 + (dE/d6»)eJ't v(sinh2 '" + sin2 8)[1 + (dl/I/d8)2J 10. ; = F[sin (8 + ao + E) + sin (ao + ET)] 11. The pressure coefficient 8 == (V/V)2. For most purposes the first approximation to E is sufficiently accurate. If greater accuracy is desired, a second approximation may be found by plotting 1/1 against 8 + E and repeating steps 6 and 7 before proceeding. 3.7. Empirical Modification of the Theory. Although the foregoing theory for computing the pressure distribution about arbitary wing sections is exact for perfect fluid flow, the presence of viscous effects in actual flows leads to discrepancies. Even when the flow is not separated from the surface, the thickness of the boundary layer effectively distorts the shape of the section. One result of this distortion is that the theoretical slope of the lift curve is not realized. A comparison of the actual lift and moment. THEORY OF WING SECTIONS OF FINITE THICKNE8S 61 characteristics with the theoretical ones is made in Fig. 33 for the NACA 4412 wing section.a This comparison is typical in that the experimental lift coefficient is lower than the theoretical value at a given angle of attack and the theoretical value of the pitching-moment coefficient is not realized. Several methods suggest themselves for bringing the theory into closer agreement with experiment. It is obvious that the lift coefficient can be 2.4 J I 2.o~ lJsuoIlheofy / MbWned/~-------j I--- Experiment 0 + 1/ J / V',l 1/ /1 /. e ~ It ~ .1! .8 § ~.4 J ...... i l' v ~ l' // 0+ + ) I -.4 J 0 -. 8 r IJI Cl",..-o J.6 0 If -. - -+- ....-+ emc. r-- ..... + ~ ~ I I -16 -8 0 8 /6 24 Effecfke mgIe ofoIlock,ao,degrees FIo. 33. Comparison between theoretical and measured lift and pitching-moment coefficients, NACA 4412 wing section. brought into agreement by reducing the theoretical angle of attack. This method seriously alters the flow about the leading edge, leading to dis­ crepancies in the pressure distribution over the forward portion of the wing section. Another method tried by Pinkerton'" is to reduce the circulation to the required value by disregarding the Kutta condition. This method leads to fair agreement over .the greater part of the wing section (Fig. 34) but, as would be expected, leads to infinite velocities at the trailing edge. Pinkerton- found that fairly satisfactory agreement with experiment could be obtained by effectively distorting the shape of the section (Fig. 34)~ This distortion is affected by finding an increment aET required to avoid infinite velocities at the trailing edge when the eirculation is 00.­ THEORY OF WING SECTIONS 62 justed to produce the experimentally observed lift coefficient. The altered function fa is arbitrarily assumed to be given by E fa where E = E + aET 2 ( 1 - cos 8) (3.23) original value computed by methods of preceding section modified value used to compute flow about distorted section = Ea = I I I I -3 «=8" x ---r--2 Experiment lJ5uoIlhtf!OtY RedtK:ed Clrculolion Modified theory ~ r-; .. '=' ~ p -I ~ ~-~ ~ .... ~ -~ ~ ~-.::: ~ ~ ~ I ~~ .~ s; o If' o ~ .- -~ ...:::~- 50 -- I'. J , ~ ,.\ 100 Percenlchord 34. Compariaon between measured and various theoretical pressure distributions NACA 4412 wing section. FlO. The agreement obtained by this method is indicated by Fig. 34 and the moment coefficients of Fig. 33. 3.8. Design of WiDg Sections. Section 3.6 presents a method for obtaining the pressure distribution for arbitrary wing sections. A knowl­ edge of the pressure distribution is desirable for structural design and for the estimation of the critical Mach number and the moment coefficient if tests are not available. The pressure distribution also exerts a strong or predominant influence on the boundary-layer flow and, hence, on the sec­ tioJl. characteristics. It is therefore usually advisable to relate the aero­ dynamic characteristics to the pressure distribution rather than directly to the geometry of the wing section. In the experimental development of wing sections, it is accordingly desirable to have a method of determining changes of shape of the section corresponding to desired changes in the 7'BEORY OF WING SECTIONS OF FINITE THICKNESS 63 pressure distribution. A method that has been used successfully for the development of the newer NACA wing sections is presented in-reference 3. Equation (3.22) gives the relation between the velocity distribution over the wing section and the shape parameters 1/1 and 8. It is shown by Theodorsen and Garrick12l that an alternate expression is v sin (ao + Ip) + sin (ao + E'I')e4'o V = v(sinh2 ~ + sin" 8) t[1 - (dE/d'P)]2 + [(dlY/d'P)]2} (3.24) Basic symmetrical shapes are derived by assuming suitable values of dE/dtp as a function of tp. These values are chosen on the basis of previous experience and are subject to the conditions that (1r dE = () 10 dqJ and dE/dtp at qJ is equal to dE/dtp at - qJ. These conditions are necessary for obtaining closed symmetrical shapes. Values of E{qJ) are obtained by in­ tegrating (dE/d'P) dtp. Values of 1/I(tp) are found by obtaining the conjugate curve of E{tp) by means of Eq. (3.17) and adding an arbitrary value 1/10 sufficient to make the value of 1/1 equal to or slightly greater than zero at 'P = ".. This condition assures a sharp trailing-edge shape. If 1/1 equals zero at qJ = ..., the wing section will have a cusplike trailing edge. Slightly positive values of 1/1 result in more conventional trailing-edge shapes. Theodorsen'P has shown that small changes in the velocity distribution at any point on the surface are approximately proportional to 1 + (dE/dqJ). The initially assumed values of dE/dip are accordingly altered by a process of successive approximations until the desired type of velocity distribution is obtained. After the final values of y, and E are obtained, the ordinates of the symmetrical section are computed by Eq. (3.12). Although a similar procedure may be used to design cambered sections, it has been found convenient to design symmetrical sections which may then be cambered by a method to be described in Sec. 4.5. This method is satisfactory for thin or moderately thick sections. Very thick sections must be designed directly with the desired camber. Goldstein17C1 has also developed useful approximate methods for the design of cambered wing sections. CHA~TER 4 THEORY OF THIN WING SECTIONS 4.1. Symbols. A A B L PR S U V a C c, Clo e'i C.L E c.e/4 e k 1 In 1nLB p v v" l1v J1v" x %1 'II r ~ ai a 1~ -0 fJo ../ 8 'h 1£,) r p ~.FJ coefficients of Fourier series multiplier for obtaining the angle of zero lift multiplier for obtaining the pitching-moment coefficient lower surface ordinate in fraction of the chord local load coefficient pressure coefficient (11 - p) fJAp 'Jl1 upper surface ordinate in fraction of the chord velocity of the free stream mean-line designation; fraction of the chord from leading edge over which loading is uniform at the ideal angle of attack chord section lift coefficient value of c, corresponding to calculated value of ~t·4/V ideal lift coefficient section pitching-moment coefficient about the lending edge section pitching-moment coefficient about the quarter-chord point base of Naperian logarithm, 2.71828 multiplier for obtaining the angle of zero lift section lift logarithm to the base e section pitching moment about the leading edge local static pressure local velocity over the surface of a symmetrical wing section at zero lift component of velocity normal to the chon! increment of local velocityover thesurface of a wing section associated with camber increment of local velocity over the surface of a wing section associated with angle of attack distance along chord fixed point on the %axis ordinate of the mean line circulation section angle of attack ideal angle of attack angle of zero lift quantity defined by Eq. (4.13) circulation per unit length along chord angle whose eosine is 1 - (2x/ c) angle whose cosine is 1 - (2x1 / c) quantity defined by Eq. (4.14) ratio of the circumference of a circle to its diameter mass density of air infinity THEORY OF THIN WING SECTIONS 65 4.2. Basic Concepts. Many of. the properties of wing sections are primarily functions of the shape of the mean line, The mean line is con­ sidered to be the locus of points situated halfway between the upper and lower surfaces of the section, these distances being measured normal to the mean line. Although the mean line of an arbitrary lying section is rather difficult to obtain, the construction of cambered lying sections from given mean lines and symmetrical thickness distributions is relatively easy. Among the properties mainly associated with the shape of the mean line are 1. The chordwise load distribution. 2. The angle of zero lift. 3. The pitching-moment coefficient. Other important results obtained from the theory of thin wing sections y ~~-6---------"""'-X 11' FlO. 35. Configuration for analysis of mean linea. are a value of the slope of the lift curve and the approximate position of the aerodynamic center. The theory of thin lying sections was developed by Munk." Later contributions to the theory were made by Birnbaum, Glauert," Theodor­ sen,t21 and Allen." The present treatment follows that given by Glauert." For the purposes of the theory, the lying section is considered to be re­ placed by its mean line. The chordwise distribution of load is assumed to be connected with a chordwise distribution of vortices. Let 'Y be the difference in velocity between the upper and lower surfaces. The total circulation around the section is then given by the relation (4.1) where c is the chord and the camber is assumed to be sufficieutly small for distances along the chord line to be substantially equal to those along the mean line. The configuration for the analysis is given in Fig. 35. The vertical components of velocity caused by an element of the vortex distribution along the mean line have the magnitudes [see Eq. (2.27)] 'Ydx THEORY OF WING SECTIONS 66 where x is the position of the element and Xl is the point where velocity is being calculated. The total vertical component of velocity at. any point Xl is then (4.2) where v" =- component of velocity normal to chord line The thin wing section must of course be a streamline. There is no flow through it. The equation expressing this condition is (4.3) for small angles. Equations (4.2) and (4.3) are the fundamental relations between· the shape of the mean line and its aerodynamic characteristics. It is possible to obtain the characteristics of arbitary thin wing sections from these relations. The pressure distribution or the chordwise load dis­ tribution may, however, be obtained more accurately and just as con­ veniently by the method presented in Sec. 3.6. 4.3. Angle of Zero Lift and Pitching Moment. Simple relations for the angle of zero lift and the pitching moment of arbitrary thin wing sec­ tions may be obtained by the application of a Fourier method of analysis to Eqs. (4.2) and (4.3). Let x c = - (1 ­ 2 cos 6) (4.4) then Assume that 'Y may be represented by a trignometric series as follows: GO 'Y = 2V(Ao cot ~ + LA. sin nO) (4.5) n-l When "Y is expressed in this manner, it will be shown later that the coeffi­ cient Ao depends only on the angle of attack, and the coefficients Aftdepend only on the shape of the mean line. Since cot 2- ~ _ 1 + cos 8 "Y dz sin s = cV[Ao(l + cos 8) + GO LA. sin n8 sin 8]d8 1 THEORY OF THIN WING SECTIONS 67 The lift is l = J:pv-y dx = 10r~pV2[Ao(l + cos 6) + ~ .. sin nO sin 6]dB 1 rcpV2(A o + MAl) = (4.6) and the lift coefficient C, = ~:v2c = 2r(Ao+ ~ AI) (4.7) The expression for the pitching moment about the leading edge is given by the relation Since sin 8(1 - cos 8) = sin 8 - sin 8 cos 8 = sin 8 - ~~ sin 28 mLB 1"'~ pV2 [ Ao(l - =- L 00 cos2 6) + A .. sin nO(sin 6 - ~ sin 2tJ)] dB 1 Upon integration (4.8) Equations (4.7) and (4.8) provide simple relations between the lift and moment and the first few coefficients of the Fourier series. The next step is to determine the relations between the coefficients of the Fourier series and the shape of the mean line. Substituting Eqs. (4.4) and (4.5) into Eq. (4.2), the expression for the vertical component of velocity becomes V 111"{ Vn () 81 = - 11" 0 A o(1 + cos 8) COS 81 - COS 8 HrA.. cos (n - 1)6 -'cos (n+ 1)61}d8 + - -1 - [--------­ cos 61 - cos 8 since ~[CQS (n - 1)8 - cos (n + 1)8] == sin nO sin 8 It is shown by Glauert" that r cos nO dB = Jo cos 8 - cos q, 'lI" ~ nq, SIn q, (4.9) THEORY OF WIJ.VG SECTIONS 68 eonsequently and, dropping the subscript, v V Ao + = - L Aft cos 118 00 (4.10) 1 The required relation between the Fourier coefficients and the shape of the mean line is obtained by substitution of Eq. (4.10) in Eq. (4.3) dy dx = ~ ao - A o + k A n cos n8 (4."11) 1 The standard expressions for the Fourier series coefficients are then 11'IT-dy dB ao - Ao = r 0 dx (4.12) 21'IT An = - r 0 dY d cos nO dB X Equations (4.12) for A o, AI, and 411 2 can be transformed in such a manner that the angle of zero lift and the pitching moment are expressed in terms of the ordinates of the mean line rather than in terms of its slope. The quantities {jo and Po defined as follows are convenient in effecting the change. {jo - and 11.0 = ~1" ll. dB ... 0 c 1+cos8 (4:.13) 1" o ll. cos 9 dB (4.14:) C The value of fJo in terms of the coefficients of the original Fourier series may be found by the following process. Integrating Eq. (4.13) by parts we obtain {J - 2[Y 0-; c 1cos 8]" 2 [" Idydx II - cos 9 d9 1+cos8 -;}o cdxd8V'l+cos8 0 (reference SO, formulas 296 and 578). The first term of this expression equals zero if y approaches zero at 8 = 1(' faster than v'C=X. This is true for nearly all wing sections. From Eq. (4.4), dx = !:. sin 9 = !:. V(1 d8 2 2 + cos 9)(1 - cos 9) THEORY OF THIN lVING SECTIONS Hence 69 li ftd J(1 - cos 8)d8 dx {jo = - - .". 0 and, from Eq. (4.12), Po = Ao - ao + ~Al (4.15) The value of JJ.o in terms of the Fourier coefficients A" is found as follows: Integrating Eq. (4.14) by parts, JJo . 6J1r = [·c~y mn 0 == - lr 0 1 dy dx . - sm 8 dO cdxd8 - - t'.!. dy (l - cos 28)dO Jo 4dx 1) (4.16) + Po) (4.17) 1r( 4" Qo - Ao - 2 A 2 Comparing Eq. (4.7) with Eq. (4.15) 2".(ao C, = The angle of zero lift QL-O is therefore - {3o, and the slope of the lift. curve according to Eq. (4.17) is 211" per radian. The equation for the pitching-moment coefficient about the leading edge is found by substituting Eqs. (4.15) and (4.16) into Eq. (4.8) c",u = (2#lO - ~ Po) - i Cl (4.18) The pitching-moment coefficient about a point one-quarter of the chord behind the leading edge Cm./ equals Cwa L B + (%)c,. From Eq. (4.18), 4 Cwa 1r ~/t = 2#lO - --{3 2 0 (4.19) It should-be noted that the pitching-moment. coefficient about the quarter­ chord point is independent of the lift coefficient. The quarter-chord point is therefore the aerodynamic center of thin wing sections. The aerodynamic center for most commonly used wing sections is found to be approximately the quarter-chord point at speeds where the velocity of sound is not reached in.the field of flow. By substituting the expression x c = - 2 (1 - cos 8) into Eqs. (4.13) and (4.14) {30= i 1y o -/1 (X)dX ­ C C C (4.20) THEORY OF "K'ING SECTIONS 70 and i IJo == 1y (X)dx - f2 - c c (4.21) o c where fl(~) = r[l- (x/c)] ~(x/e)[l f2 1 - (2x/c) (Cx)= V(x/e)[l - (x/c)] (x/e)i In order to avoid infinite velocities at the leading edge, it is necessary that the coefficient A o of Eq. (4.5) be equal to zero. From Eq. (4.12), Ao equals zero when ao = ! ("dy d8 1f'Jo .d,x Substituting the value of dx from Eq. (4.4) 1 {. Qo = ;}o d dB (e/2)Ysin 8 d8 and integrating by parts ao == ![~y/cl· sm S'Jo 1r +.! rr2(y/c~ cos 8 dB 2 11" Jo sln 8 As previously discussed in connection with the integration of Eq. (4.13) the first term of this expression equals zero for most wing sections. Sub­ stitution of the expression x = -2c (1 - cos 8) gives (4.22) where fa ( X) c 1 - (2x/c) = 2-r{ (x/e)[l - (x/e)Jl ~ This angle of attack is called the "ideal angle of attack" by Theodorsen.P' The lift coefficient corresponding to this angle of attack [obtained from Eq. (4.1 i)] is called the "ideal" or ,., design" lift coefficient. In general the angle of sero lift, the pitching moment about the quarter­ chord point, and the ideal angle of attack may be calculated by graphical integration of Eqs. (4.20), (4.21.), and (4.22). For convenience in such calculations, values of the functions /1, !2, and fa corresponding to several values of x/c are presented in Table 5. Although the functions /1, j!, and fa become infinite at both the leading and trailing edges, the integrands of THEORY OF THIN WING SECTIONS 71 Eqs. (4.20) and (4.21) approach zero at the leading edge for mean lines that approach zero faster than VXjC. Most mean lines satisfy this con­ dition. Similarly, the integrand of Eq. (4.21) approaches zero at the trailing edge in most cases. In order to avoid the difficulties at the trailing edge in Eq. (4.20) and at the leading and trailing edges in Eq. (4.22), parts of these integrals are evaluated analytically." The analytical determinaft, TABLE .5.-VALUES OF FUNCTIONS z/c fl(x/c) 0 0.0125 0.0250 0.0500 0.0750 /2, f2(x/c) CO AND fa f.{x/c) CO 00 2.901 2.091 1.537 1.306 1.179 8.774 6.085 4.131 3.226 113.15 39.73 13.84 7.403 4.716 2.447 1.492 0.980 0.662 0.1000 0.15 0.20 0.25 0.30 0.995 0.980 0.992 2.667 1.960 1.602 1.156 0.873 0.40 0.50 0.60 0.70 0.80 1.083 1.273 1.624 2.315 3.979 0.408 0 -0.408 -0.873 -1.502 0.271 0 - 0.271 - 0.662 - 1.492 -2.667 -4.131 - 4.716 - 13.84 1.049 0.90 0.95 10.61 29.21 1.00 00 00 - 00 tion of the increments at the critical points is accomplished by assuming the mean line near the ends to be of the form y x (X)2 -==A+B-+C c c c Evaluation of the increment of the angle of zero lift gives (xC = 0.95 to 1.0) dy J1fJo == 0.964Yo.t, - 0.0954 dx 1 Where Yo." is the ordinate of the mean line at x/c = 0.95 and dy/dx1 is the slope of the mean line at x/c = 1.0. This increment is added to the value of 80 obtained by graphical integration from x/c = 0 to x/c = 0.95. The increments for the ideal angle of attack are dy 1:100i= + O.467Yo.05 + 0.0472 dx o dy { - 0.467Yo.9S + 0.0472 dx 1 (~= 0 to 0.05) (~ = 0.9,) to 1.0) THEORY OF WING SECTIONS 72 These increments are added to the" values of Qi obtained by graphical integration from x/c == 0.05 to x/c == 0.95. By application of Gauss's roles for numerical integration to the expres­ sion for the angle of zero lift, Munk70 obtained a simple approximate solution. - ao' = klYl + kty2 + kaYa + k4'g4 + k,y, where gl, gt, etc., are the ordinates of the mean line expressed as fractions of the chord at the points Xl, X2, etc., as tabulated together with corre­ sponding values of the constants kl , k i , ete., calculated to give the angle of zero lift in degrees. kl 0.99458 0.87426 0.50000 0.12574 0.00542 Xl X2 X, %4 z, 1,252.24 109.048 32.5959 15.6838 5.97817 let k, ~ k, This method is useful for quick calculation of the angle of zero lift when a high order of accuracy is not required. Approximate solutions for the angle of zero lift and the moment coefficient were also obtained by Pankhurst78 in the following form: (XL-O = 2;A(U + L) c..1. == '1:B(U + L) where U,L = upper and lower ordinates of wing section in fractions of chord A,B == constants given in following table x 0 0.025 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 1.00 B A 1.45 2.11 2.41 2.94 - 2.88 3.13 3.67 4.69 6.72 - 0.009 0.045 0.101 0.170 0.273 11.75 21.72 99.85 - 164.90 0.477 0.786 3.026 - 4.289 r.se 0.119 0.156 0.104 0.124 0.074 THEORY OF THIN WING SECTIONS .78 Pankhurst's solution for the angle of zero lift is often more convenient than Munk's because the constants are given for stations at which the ordinates are usually specified. The solution is based on the assumptions that the mean-line ordinates are represented satisfactorily by (U + L)/2 and that the tangent of the mean line at the trailing edge coincides with the last t\VO points. The accuracy is accordingly expected to be limited for sections with large curvatures of the mean line near the trailing edge, or for thick, highly cambered sections. 4:.4. Design of Mean Lines. It is possible to design mean lines to have certain desired load distributions by means of the theory of thin wing sections. The fundamental relations are Eqs. (4.1), (4.2), and (4.3). A simple example of the manner in which a mean line is designed may be HI demonstrated by the design of a mean line to have uniform load. \ For a uniform chordwise load distribution, the value of -y in Eq. (4.2) is a constant over the chord. Equation (4.2) may therefore be written (4.23) The lift is and Cl, equals C, , 21 2-y pV c V = - 2- = ­ or Ve" 'Y=T By substitution, Eq. (4.23) becomes "ft(XI) Cl'1 (x/c)d(x/c) V C 411" - (xI/c) 1 = (4.24) 0 Special means must be used to integrate Eq. (4.24) because the inte­ grand becomes infinite as x approaches Xl. The integration is performed as follows: V"(Xl) V C = = cl'I· 4r ~ [l wc d(x/c) >-, 0 (x/c) - (Xl/C) + r d(x/c) ] - (Xl/C) }CZl/C)+e (x/c) Cl, lim{[ln(~ - ~)~:Q-' IJo + [In(~C - ~)Jl 411" e-+O C C = ~~[ln(l-~)-ln(:l)~ C } St+e THEORY OF WING SECTIONS 74 The angle of attack at which the uniform load distribution is realized is zero because the assumed load distribution, and accordingly the mean line, is symmetrical about the mid-chord point. Equation (4.3) becomes ~ = :; [ In(1-~) -In(~)J (4.25) Integration of this expression gives y == - ~~[(l- ~)In(l- ~)+~ In ~J (4.26) where the constant of integration has been selected to make the ordinates zero at the leading and trailing edges. By similar but more complicated processes, the mean lines corre­ sponding to other types of load distribution may be calculated. In this con­ nection, it is interesting to note that the distribution of l' and the slope of the mean line are conjugate functions at the ideal angle of attack [see Eqs. (4.5) and (4.11)]. The slope of the mean line can therefore be obtained from the assumed load distribution by the proper application of Naiman's coefficientg8.73 as in Eq. (3.17). The mean-line ordinates can then be ob­ tained by graphical or analytical methods. Mean lines have been caleulatedv corresponding to a load distribution that is uniform from x/c = 0 to x/c = a and decreases linearly to zero at x/c = 1. The ordinates are given bythe expression ~ - ~r(:'+ 1) {I ~ a[~(a - ~rln \a - ~I- ~(1 - ~rln(l -~) +~(l-~r -~(a-~)1-~ln~+g-h~} where (4.27) J) +-4IJ - 1 [ 0)(12 g:::z-- a· - lna-- I-a h= 4 n 1~ a (1 - aYIn(1 - a) - ~ (1 - a)] + 0 The ideal angle of attack for these mean lines is Cl,}" a; = 21r(a + 1) It \\;11 be noted that the ordinates of the mean lines are directly pro­ portional to the design lift coefficient. The fact that the load distributions and the ordinates of the corresponding mean lines are additive may be seen from examination of Eqs.. (4.5) and (4.11). This property is, of course, a direct consequence of the linearizing assumptions made in writing the fundamental expressions (4.1), (4.2), and (4.3).. This linearity is con­ venient because ordinates of a mean line having a load distribution made THEORY OF THIN WING SECTIONS 75 up of the sums of load distributions for which the corresponding ordinates are known can be obtained by simply adding the ordinates of the com­ ponent mean lines. The linearizing assumptions lead to increasing errors as the slope of the mean line increases. For geometrically similar mean lines, the slope in­ creases with the design lift coefficient, and the errors are consequently greater for the larger design lift coefficients. In the case of the uniform­ load mean line [Eq. (4.26)], the slope is infinite at the leading and trailing edges, and it is not to be expected that the assumed load will be maintained 2.0 -"-­ / I( °0 -~ ~ " .2 .6 .4 r--. r---.... .8 1\ LO ~ FIo. 36. Typical basic load distribution at ideal angle of attack. near these points. The region in which the slope is large is extremely small, however, because of the manner in which the slope approaches infinity [Eq. (4.25)]. 4.6. Engineering Applications of Section Theory. The theory of wing sections presented in Sec. 3.6 permits the calculation of the pressure distri­ bution and certain other characteristics of arbitrary sections with con­ siderable accuracy. Although this method is not unduly laborious, the computations required are too long to permit quick and easy calculations for large numbers of wing sections. The need for a simple method of quickly obtaining pressure distributions with engineering accuracy has led to the development of a method! combining features of thin and thick wing-section theory. This simple method makes use of previously cal­ culated characteristics of a limited number of mean lines and thickness forms that may be combined to form large numbers of related wing sections. The theory of thin wing sections as presented in Sec. 4.3 and reference 121 shows that the load distribution of a thin section may be considered to consist of 1. A basic distribution at the ideal angle of attack (Fig. 36). 2. An additional distribution proportional to the angle of attack as measured from the ideal angle of attack (Fig. 37). The first load distribution is a function only of the shape of the thin wing section or (if.the thin wing section is considered to be a mean line) of the mean-line geometry. Integration of this load distribution along the THEORY OF WING SECTIONS 76 chord results in a normal force coefficient which, at small angles of attack, is substantially equal to the design lift coefficient. If, moreover, the camber of the mean line is changed by multiplying the mean-line ordinates by a constant factor, the resulting load distribution, the ideal or design angle of attack as, and the design lift coefficient Ck may be obtained simply by multiplying the original values by the same factor. The second load distribution, which results from changing the angle of attack, is designated the U additional load distribution," and the cor­ responding lift coefficient is designated the Ct additional lift coefficient." 5.0 4.0 ~O \ \ \ ~ 1.0 00 "~ .2 ~ ............. .4 ~ .6 - r--. r--.- r-, .8 LO ~ Flo. 37. Additional load distribution 8880Ciated with anale of attack.. This additional load distribution contributes no moment about the quarter­ chord point and, according to the theory of thin wing sections, is independ­ ent of the geometry of the wing section except for angle of attack. The additional load distribution obtained from the theory of thin wing sections is of limited practical application, however, because this simple theory leads to infinite values of the velocity at the leading edge. This difficulty is obviated by the exact theory of wing sections presented in Sec. 3.6, which also shows that the additional load distribution is neither com­ pletely independent of the airfoil shape nor exactly a linear function of the angle of attack. Suitable additional load distributions can be calculated by the methods of Sec. 3.6 for each thickness form (symmetrical section) and a typical value of the additional lift coefficient.. The variation of the additional load distribution with the additional lift coefficient is arbitrarily assumed to be linear. In addition to the pressure distributions associated with these two load distributions, another pressure distribution exists which is associated with the thickness form, or basic symmetrical section, at zero angle of attack and which is calculated by the methods of Sec. 3.6. J THEORY OF THIN WING SECTIONS 77 The velocity distribution about the wing section is thus considered to be composed of three separate and independent components as follows: 1. The distribution corresponding to the velocity distribution over the basic thickness form at zero angle of attack. 2. The distribution corresponding to the load distribution of the mean line at its ideal angle of attack. 3. The distribution corresponding to the additional load distribution associated with angle of attack. The local load at any chordwise position is caused by a difference of velocity bet,veen the upper and the lower surfaces. It is assumed? that the velocity increment on one surface is equal to the velocity decrement on the other surface. This assumption is in accord with the basic concept of the distribution of circulation used in the theory of thin wing sections. It can be shown that the relation between the local load coefficient P R 'and the velocity increment ratios !:1v/V or !:1vo/V is !:1v V ~Va or 1 PR V =4v1V where v/V is the velocity ratio at the corresponding point on the surface of the basic thickness form. In actual practice the value of !:"v IV used is that corresponding to the thin wing section or mean line where v/V equals unity: The velocity increment ratios !J.v/V and I1vo / V corresponding to com­ ponents 2 and 3 are added to the velocity ratio corresponding to component 1 to obtain the total velocity at one point from which the pressure coefficient S is obtained, thus S = (::. ± .1v ± I1Va V V )2 V, , (4.28) This procedure is illustrated in Fig. 38. "Then this formula is used, values of the ratios corresponding to one value of x/c are added together and the resulting value of the pressure coefficient S is assigned to the surface of the wing section at the same value of x/c. The values of !:,.v/V' and I1vo / V are positive on the upper surface and negative on the lower surface for positive values of the corresponding local loads. When the ratio I1va/ V has the value of zero, integration of the resulting pressure-distribution diagram will give approximately the design lift coefficient Cli of the mean line. In general, however, the value of CI will be greater than ct, by an amount dependent on the thickness ratio of the basic thickness form. This discrepancy is caused by applying the values of !:1v/V obtained for the mean line to the sections of finite thickness where v/V is greater than unity over most of the surface. The pressure distribution will usually be desired at some specified lift coefficient not corresponding to cu: For this purpose, the ratio ava/V must THEORY OF WIJtiG SECTIONS 78 1.2 E __---~ ~ C A LYJN; SECT1CJ'.J ATZERO AtG...E OFATTACK .8 v V .4 .4 [ - MEAN l.f£ AT IlEAL ANiLE OFATTACK E CM'SEREO YiN; SECTOJAT lEAL ANiLE (F ATTACK - - - _ ~ I~' .4 0'------------­ ~ E·_u===:::::'-­ VELOCITY IfCREMENT RESlLTING FRQ.1 ANGLE rF ATTACK FOR SYtAETRCAL ¥lING ~ - 0[ 1.6 V 1.6 1.2 YJ..Ar.+- ~ V~ V - V .4 °0 .2 .4 .6 .8 LO .Jt FIG. 38. Synthesis of pressure distribution. be assigned some value by multiplying the originally calculated values oi this ratio by a factor J(a). For a first approximation, this factor may be assigned the value f{a) = C, - Cl, Clo where CI is the lift coefficient for which the pressure distribution is desired and Clo is the lift coefficient for which the values of L1va / V were originally calculated. If greater accuracy is desired, the value of J(a) may be ad­ THEORY OF THIN WING SECTIONS 79 justed by trial and error to produce the actual desired life coefficient as determined by integration of the pressure-distribution diagram. Although this method of superposition of velocities has inadequate theoretical justification, experience has shown that the results obtained are ~ 1. c/ ~ ~ -~ .: r- Upper .UI"t'aoe ~ ~Lowr.Ul"1"_ ·1'" .o, \ - r-, 1\ ~ ~ -~OrJ' "~ ~, o Kx:perJ.-ent o .2 .,. .& .8 &.0 x/o FIO.39. Comparison of theoretical and experimental pressure distributions for the NACA 66(215)-216 airfoil; Cl == 0.23 adequate for many engineering uses. In fact the results of even the first approximation agree well with experimental data and are adequate for at least preliminary consideration and selection of wing sections. A com­ parison of a first-approximation theoretical pressure distribution with an experimental distribution isshown in Fig. 39. Some discrepancy naturally occurs between the results of experiment and the results of any theoretical method based on potential flow because of the presence of the boundary layer. These effects are small, however, over the range of lift coefficients for which the boundary layer is thin and the drag coefficient is low. € H APTER 5 THE EFFECTS OF VISCOSITY 6.1. Symbols. A F F H H K L £, B Be R,. Rz Ra* R, U U« UtI U. Uz V c Cd ep c/ d " l l In log n p q go r u u v v 11* v. constant velocity gradient skin friction total pressure outside boundary layer ratio of the displacement thickness to the momentum thickness constant characteristic length length of mean free path of molecules Reynolds number pVL/p Reynolds number pVc/p. Reynolds number pur/ p. Reynolds number pVx/p. Reynolds number pV6*/p. or pU6*/p. Reynolds number pU6/p. velocity outside boundary layer maximum velocity velocity outside boundary layer at the point Xo velocity outside boundary layer at the laminar separation point value of U at the point :z; velocity of the free stream chord section drag coefficient skin-friction coefficient local skin-friction coefficient diameter of pipe base of Naperian logarithms, 2.71828 length wave length logarithm to the base e logarithm to the base 10 exponent local static pressure dynamic pressure dynamic pressure of the free stream ~p yt radius of pipe . component of velocity in boundary la.yer along the x axis average velocity of flow in pipe component of velocity in boundary layer along the y axis mean molecular velocity friction velocity v:;jp velocity of slip 80 THE EFFECTS OF VISCOSIT1' z 81 Cartesian coordinate (distance along surface) z. effective length of flat plate %0 11 a position on surface at start of integration Cartesian coordinate (distance normal to surface) 21r/l fJi time rate of amplification of disturbance a boundary-layer thickness a* displacement thickness of the boundary layer " nondimensional variable (y/x)Vifz '1 8 80 ~ ~ p. ~ 11" p .,. q, wit 1/11 GO distance parameter plJ*y/ Il momentum thickness of the boundary layer momentum thickness at the point Zo pressure-loss coefficient laminar boundary layer shape parameter viscosity of the fluid coefficient of slip ratio of the circumference of a circle to its diameter mass density of the fluid shearing stress at the wall (skin friction) velocity parameter u/v· stream function nondimensional stream function infinity 6.2. Concept of Reynolds Number and Boundary Layer. In order to be able to compare directly the forces acting on geometrically similar bodies of various sizes at various air speeds, it is customary to express the forces in terms of nondimensional coefficients, as explained in Chap. 1. For geometrically similar configurations, these coefficients tend to remain constant, and they would remain exactly constant if all factors influencing the flow about the body were properly accounted for. It is well known, however, that some characteristics such as the drag and maximum lift coefficients vary with the size of the wing for a given air speed, and with the air speed for a given size of wing. The two most important factors that are neglected in defining the, coefficients are effects associated with the compressibility and viscosity of air. An examination of the complete equations of motion- shows the significant parameters to be the ratio of the air speed to the speed of sound in air (the Mach number) and the Reynolds number pVL/p, where p is the mass density of the fluid, V is the velocity of the free stream, L is the characteristic length, and, "" is the viscosity of the fluid. It can be shown that similarity of the flows about different bodies is obtained only if the bodies are geometrically similar and if the Reynolds number and the Mach number are the same. At speeds where the pressure variations around the body are small compared with the absolute pressure (low Mach numbers) the effects of compressibility are negligible, and the viscous effects may be considered independently. THEORY OF WING SECTIONS 82 Some concept of the physical significance of the Reynolds number may be obtained by expressing this number in terms of the mean velocity and the mean free path of the molecules, as suggested by von Karman. 137 The kinetic theory of gases indicates that the coefficient of viscosity p, of a homogeneous gas may be expressed by a formula of the type66 p, = KtWL, where K = constant == mean molecular velocity LI == mean free path of the molecules Substituting this value of p. into the formula for the Reynolds number; we obtain 1 V L v R=--K V t; where R = Reynolds number The Reynolds number is thus proportional to the product of the ratio of the speed of the body to the mean speed of the molecules and of the ratio of the size of the body to the mean free path of the molecules. For bodies of ordinary size moving at low Mach numbers in air of ordinary density, the ratio V (V is small and the ratio L/L/ is very large. Under these conditions, the flow around the body corresponds to the conditions under which this formula was derived, namely, that the velocities are not large compared with the mean velocity and that the spatial dimensions of the phenomenon are large compared with the mean free path. The mean velocity v is of the same order of magnitude as the velocity of sound, and consequently the ratio V IV is of the same order as the Mach number. It accordingly be inferred that the Reynolds and Mach numbers are not independent parameters and that the scale effects ex­ perienced at high Mach numbers would be different from those at low Mach numbers even though the Reynolds numbers were the same in each case. Moreover, in a rarefied gas, as at extremely high altitudes, the mean free path may be of the same order as, or even much larger than, the characteristic length of the body. The concept of Reynolds number as presented here presupposes a mean free path that is small compared with the thickness of the boundary layer. Consequently the significance of this concept as applied to flows in very rarefied gases or at high Mach numbers is uncertain. Under such conditions, it is preferable to compare flows on the basis of the independent parameters V IV and L/L,. The effects of viscosity are of primary importance in a thin region near the surface of the wing called the U boundary layer." Boundary layers, in general, are of two types, namely, laminar and turbulent. The llow in the laminar layer is smooth and free from any eddying motion. The flow in the turbulent layer is characterized by the presence of a large number of relatively small eddies. The eddies in the turbulent layer produce a trans­ may THE EFFECTS OF VISCOSITY 83 fer of momentum from the relatively fast moving outer parts of the boundary layer to the portions closer to the surface. Consequently the distribution of average velocity is characterized by relatively higher velocities near the surface and a greater total boundary-layer thickness in a turbulent than in a laminar boundary layer developed under otherwise identical conditions. Skin friction is therefore higher for turbulent boundary-layer flow than for laminar flow. Aerodynamically, the concept of the boundary layer is based on the premise of a continuous homogeneous viscous fluid. According to this concept the velocity within the boundary layer varies from zero at the surface to the full local stream value at the outer edge of the layer. Such a concept is valid only if the density of the gas is sufficiently great to limit the mean free path of the molecules to a length very small compared with the thickness of the boundary layer. The kinetic theory of gases" indi­ cates that the velocity at the surface is not exactly zero but that there is a velocity of slip proportional to the velocity gradient. du v. = t dy The coefficient of slip ~ has the dimension of length and may be con­ sidered as a backward displacement of the wall with the velocity gradient extending effectively right up to the displaced wall where the velocity is zero. It has been shown by Maxwell, Millikan, and others that, for most surfaces, the coefficient of slip is very nearly equal to the mean free path of the molecules. At ordinary altitudes, this distance is so small that it may properly be neglected. At very high altitudes, this slip velocity may have large effects even though the mean free path is still only a fraction of the boundary-layer thickness. At still higher altitudes where the mean free path is long, the entire concept of the boundary layer and viscosity as presented here becomes invalid. 6.3. Flow around Wing Sections. When the pressures along the wing surfaces are increasing in the direction of flow, a general deceleration takes place. At the outer limits of the boundary layer, this deceleration takes place in accordance with Bernoulli's law. Closer to the surface, no such simple law can be given because of the action of the viscous forces within the boundary layer. In general, however, the relative loss of speed is somewhat greater for the particles of fluid within the boundary layer than for those at the outer limits of the layer, because the reduced kinetic .energy of the boundary-layer air limits its ability to flow against the adverse pres­ sure gradient. If the rise in pressure is sufficiently great, portions of the fluid within the boundary layer may actually have their direction of motion reversed and may start moving upstream. When this reversal occurs, the boundary layer is said to be "separated." Because of the increased inter­ 84 THEORY OF WING SECTIONS change of momentum from different parts of the layer, turbulent boundary layers are much more resistant to separation than are laminar layers. Except under very special circumstances laminar boundary layers can exist for only a relatively short distance in a region in which the pressure increases in the direction of flow. After laminar separation occurs, the flow may either leave the surface permanently or reattach itself in the form of a turbulent layer. Not much is known concerning the factors controllmg this phenomenon. Laminar separation on wings is usually not permanent at flight values of the Rey­ nolds number except when it occurs on some wing sections neai the leading edge under conditions corresponding to maximum lift. The size of the locally separated region that is formed when the laminar boundary layer separates and the flow returns to the surface decreases with increasing Reynolds number at a given angle of attack. The flow over aerodynamically smooth wings at low and moderate lift coefficients is characterized by laminar boundary layers from the leading edge back to approximately the location of the first minimum-pressure point on both upper and Iower surfaces. If the region of laminar flow is extensive, separation oecurs immediately downstream from the location of minimum pressure and the flow returns to the surface almost immedi­ ately at flight Reynolds numbers as a turbulent layer. This turbulent boundary layer extends to the trailing edge. If the surfaces are not suf­ ficiently smooth and fair, if the air stream is turbulent, or perhaps if the Reynolds number is sufficiently large, transition from laminar to turbulent flow may occur anywhere upstream of the calculated laminar separation point. ­ For low and moderate lift coefficients where inappreciable separation occurs, the wing profile drag is largely caused by skin friction, and the value of the drag coefficient depends mostly on the relative amounts of laminar and turbulent flow, If the location of transition is known or as­ sumed, the drag coefficient may be calculated with reasonable accuracy from boundary-layer theory. As the lift coefficient of the wing is increased by changing the angle of attack, the resulting application of the additional type of lift distribution moves the minimum-pressure point upstream on the upper surface, and the possible extent of laminar flow is thus reduced. The resulting greater proportion of turbulent flow, together with the larger average velocity of flow over the surfaces, causes the drag to increase with lift coefficient. At high lift coefficients, a large part of the drag is contributed by pres­ sure or form drag resulting from separation of the flow from the surface. The flow over the upper surface is characterized by a negative pressure peak near the leading edge, which causes laminar separation. The onset of turbulence causes the flow to return to the S1ma~e LCJ a turbulent boundary THE EFFECTS OF VISCOSITY 85 layer. High Reynolds numbers are favorable to the development of tur­ bulence and aid in this process. If the lift coefficient is sufficiently high or if the reestablishment of the flow following laminar separation is unduly delayed by low Reynolds numbers, the turbulent layer will separate from the surface near the trailing edge with resulting large drag increases. The eventual loss of lift with increasing angle of attack may result either from relatively sudden failure of the boundary layer to reattach itself to the surface following separation of the laminar boundary layer near the leading edge or from progressive forward movement of turbulent separation. Under the latter condition, the flow over a relatively large portion of the surface may be separated prior to maximum lift. 5.4. Characteristics of the Laminar Layer. The characteristics of the laminar boundary layer may be deduced from detailed consideration of the general equations of motion. For two-dimensional steady motion neglect­ ing the effects of compressibility, these equations are28 u au +. v au dX u ay av + v iJv ay oX = _ ~ ap + ~ (a u2 + 02U)} p iJx = _ ! dp + J!' 2 (iJ V 2 pax p iJy d y2 dX2 p + iJ2v) (5.1) iJ y 2 These equations are known as the "K"avier-Stokes equations." Their general solution is difficult. A simple integration can be performed only in special cases. These equations, however, may be simplified by the use of dimensionless variables and consideration of the order of magnitude of each term. Prandtl" has shown that, if the viscous effects are assumed to be confined to a thin layer over the surface, the following approximate relation applies in this region u au ax + v au = ay _ ! iJp + ~ a2u p ax p iJ y 2 (5.2) This equation is a simplified form of the first relation of Eq. (5.1). Con­ sideration of the order of magnitude of the various terms of the second relation of Eq. (5.1) indicates that op;iJy is small and hence that pressures are transmitted unchanged through the boundary layer. An interesting conclusion is reached by substitution of the following nondimensional variables in Eq. (5.2). X Xl =- c U UI =-y PI = 1­2 pV Yl = 'U.VR, C vl=VVR THEORY OF WING SECTIONS 86 Equation (5.2) becomes V 8(Ul V) a(XIC) 11,1 + VI V a(UIV) = _ VR a(Ylc/VR) ! a(PtPV 2 ) + a2 (Ul V) p a (XIC) a(Ylc/VR)" and, upon simplification, UI au! + VI OUI = _ apl + a2ul aXI aYl ax! aYl2 (5.3) It will be noted that the viscosity does not appear directly in Eq. (5 3). Solution of this equation will give ttl and VI in terms of Xl and 'Ut. Because YI equals (y/c)vo" the effect of variations of the Reynolds number will be merely to change y in a manner inversely proportional to the square root u y FIG. 40. Effect of pressure gradient on velocity distribution in laminar bounda.ry IS1'er. of the Reynolds number. It follows directly that the thickness of the laminar boundary layer varies inversely with the square root of the Rey­ nolds number. It may also be concluded that the shape of the velocity distribution through the boundary layer is independent of the Reynolds number, unless, of course, the pressure distribution is a function of the­ Reynolds number. It also follows that the position of the laminar separa­ tion point, characterized by the condition that au/ay is zero at the surface, is independent of the Reynolds number. A considerable amount of information concerning the velocity distribu­ tion in the laminar boundary layer can be derived from Eq. (5.3) without actually attempting a solution. _~t the surface Ul and VI must both equal zero. At the outer limit of the boundary layer Ul approaches a constant value. Consequently near the outer edge of the layer a2u/all must be negative and approach zero with increasing distance from the surface, as shown in Fig. 40. At the surface, Eq. (5.3) simplifies to the form 82u1 apt a yl2 = aXI When apI/aXI is positive (increasing pressure in the direction of flow), atujayt must be positive at the surface, and the shape of the velocity dis­ tribution will resemble that shown in Fig. 40 for this condition. Similarly the velocity distribution must be linear near the surface for zero pressure THE EFFECTS OF VISCOSITY 87 gradient, and the velocity profile will be continuously convex upward as shown in Fig. 40 for decreasing pressures in the direction of HOlV. 5.6. Laminar Skin Friction. Blasius'vobtained a solution for Eq. (5.2) for the case where op/ox is zero, that is, for a flat plate with uniform velocity outside the boundary layer. Blasius found that Eq. (5.2) could be greatly simplified by assuming a variable oi the form == v.. v'Rz x 11 where B; = pVx/p. and by the introduction of the nondimensional stream function "'1 = fo~ Ul dT/ = j(T/) where Ul = u/V The true stream function is ,p= fo"UdY '" = U = Vx f"Ul ~.Io dT/ at/! == at/! °11 oy o"loy = = Vx j(T/) = v'R z ~J! xVj(,,) P [~J!p XVj/(T/)] (VRz) = x Vj'(l1) The velocity v perpendicular to the flat plate is given by &p 01/1 0 71 v == - - == - - -. ox f) 071 ax = ~ ~~["j/(T/) - j(1I)J Making similar calculations for au/dx, 8u/dy, and iJ2u/dy2 and substituting in Eq. (5.2) with op/iJx zero, we obtain 2 2xLr - V 11f ' (11)! ',/ (1I) + V2 1I}'(1I) - j(1I)Jl"(71) == 2x _ or j(TJ)I"<',,) + 2/'''(1/) == 0 ~pxp. V e/"'("1) 2 (5.4) The fact that neither x nor y appears explicitly in Eq. (5.4) indicates that the initial assumption that 1fl is a function of TJ alone is correct, that is, Ul is a function of "I alone. The boundary conditions for the solution of Eq. (5.4) are that J(fJ} == 0 !'(71) == 1 I" (11) --+0 for TJ == 0 for 71--+ 00 for 1/ ~ ex> THEORY OF WING SECTIONS 88 Blasius obtained the solution of Eq. (5.4) by an approximate integration in a Taylor series," The solution has not been given in explicit form but is presented in the following table. " 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 !'(.,,) ." - f'{.,,) ." 0 0.0664 0.1328 0.1990 0.2647 2.0 2.2 2.4 2.6 2.8 0.6298 0.6813 0.7290 0.7725 0.8115 4.0 4.2 0.3298 0.3938 0.4563 0.5168 0.5778 3.0 3.2 3.4 3.6 3.8 0.8461 0.8761 0.9018 0.9233 0.9411 5.0 5.2 5.4 5.6 5.8 6.0 I I I 4.4 4.6 4.8 I I !'(.,,) 0.9555 0.9670 0.9759 0.9827 0.9878 0.9916 O.9M3 0.9962 0.9975 0.9984 0.9990 A plot of these values is given in Fig. 41. These values define the velocity distribution in a laminar boundary layer in the absence of a pres­ 1.0 / .8 / ~ ~ ~ /~ V -6 :/ % J .4 JV / .2 ) V -00 FlO. 41. 2 3 4 5 6 7 ~AR; Velocity distribution in the boundary layer over a flat plate according to Blasius. sure gradient. The form of the variables in which this velocity distribution is specified shows that the boundary-layer thickness increases as the square root of the distance from the leading edgeand inversely as the square root of the Reynolds number and that the velocity distribution is similar at all points along a flat plate. In addition to the velocity distribution through the boundary layer, THE EFFECTS OF VISCOSITY 89 the solution of Eq. (5.4) gives a value of fl/('1) at the surface equal to 0.332, that is, 0.332 = /,,(.,,) = d(ujV) = d(ujV) d'1 d[(y/x)v'Rzl For the skin friction, we have .,. = Cf = p,(~~) = 0.332 p,V ~ dy *=-0 X T I L . rnlL V 2 = 0.664 -IT v R% 72P P x } 0.654 = v- /R (5.5) % The total drag coefficient for one side of a. fiat plate is then given by 11% cr = - x CJ 0 r~4~dx = -l~xO~ . l!:- dx z 0 VX pV = !(1.323 x fia.) = 1.328 ~ "\}pV (5.6) The obvious definition of the boundary-layer thickness is the distance from the surface to where the local velocity defect due to viscosity is zero, that is, u/V == 1. Practically, this distance is not 'Yell defined because u/V approaches unity asymptotically. A definition that has frequently been used is that the boundary-layer thickness is the distance from the surface to where u/V = 0.995. The preceding table indicates that, ac­ cording to this definition, o/x 5.3/v'R%. A thickness parameter that is convenient for experimental 'York is the distance from the surface to the point where (U/V)2 = ~2 or o/x'" 2.3/v'llz. For theoretical work it is fre­ quently desirable to obtain a measure of the effect of the boundary layer on the outside flow. The reduced velocities within the boundary layer cause a displacement of the streamlines around the body. This displacement is called the U displacement thickness" of the boundary layer 0* and is calculated from the relation "'J 0* = -1 ~a (V - u)dy 1T 0 or in the general case ~. = Y11 10t' (U - u)dy (5.7) where U = velocity just outside boundary layer For the laminar boundary layer over a flat plate with zero pressure gradient ~* 1.73 x= VR% (5.8) THEORY OF WING 8ECTION8 90 6.6. Momentum Relation. The KarmlLn integral relation27• 111 pro­ vides a method for relating the boundary-layer thickness to the pressure distribution if the skin friction and the velocity distribution through the boundary layer are known or assumed. The relation is useful because good approximations to the skin friction and the velocity distribution can often be obtained from consideration of local conditions. This relation is applicable to both laminar and turbulent boundary layers. The Karman integral relation is based on the principle of mechanics that, if any closed surface is described within a steady stream of fluid, the time rate of in­ crease, within the surface, of momentum in any direction is equal to the sum of the components in that direction of all the forces acting on the fluid. H body forces such as weight are absent, the only forces to be eon­ fI4 sidered are those acting on the surface, dx fix that is, normal forces caused by presu sures and tangential forces caused by _p6# ~'!dx viscous shearing. Consider the equilibrium of the element shown in Fig. 42. The mo­ mentum from left to right entering /':;dx the left face per unit time is FlO. 42. 1"011/_. dy) Forces on an element of the Jo boundary layer. \flu, while the momentum leaving the right face per unit time is [ u(pu dy) + [:xl u(pu dY)] d:c The mass flow per unit time leaving the right face must be the sum of the mass flow entering the left face plus the mass flow entering the upper boundary, The mass flow per unit time through the upper boundary is tx([ PUdy)d:c This mass enters with the velocity U. The momentum entering the upper boundary per unit time is then u(:Xl'pu dY)d:c The total rate of change of momentum from left to right in the element is then mB EFFECTS OF VISCOSITY 91 The forces acting on the element are pa on the left face - [PIS + ~(PIS)dx] on the right face P ::: da: on the upper boundary dx on the lower boundary - T The resultant force is d - - (p8)dx dx do + p -dx dx - T dx which reduces to - {) dp dx dx T dx Equating the resultant force and the rate of change of momentum (:Xl'PU dY)d:x 2 Dividing through by p u(~l'pu dY)d:x = - IS: dx - T dx dx, we obtain r d d f' 2 dx}o u dy - U dx}o u dy 0 dp T = - Pdx - -;; This equation is not very convenient because the values of the integrands do not approach zero at the upper limit of integration. -To avoid this difficulty the integrands are expressed in terms of the variable (U - u). by making the substitution u=U-(U-u) -d l'[ dx o L'[ J2 U-(U-u) dy-Ud dx o ] U-(U-u) dY=--..1!._! 0d pdx p Expanding and collecting terms dU d (0 d t' d t' UIS dx - dx U}o (U - u)dy + U dx}o (U - u)dy - tI.x}o (U.;... u)u dy odp T ==ptb;-p At the outer limit of the boundary layer, Bernoulli's equation is valid; hence + !2 pU2) = dH = 0 dx dx and dp dU dx = - pU dz .!!(p THEORY OF WING SECTIONS 92 Substitution of this relation in the previous equation gives d -dUl' - (U-u)dy-dx o l' dx o (U-u)udy=--T p (5.9) This equation is known as the "!{arman integral relation." In this form the integrals approach zero as the upper limit is approached. This equation is suitable for the experimental determination of the skin friction, although the accuracy of the determination may be affected considerably by the form in which this equation is expressed. Another form of Eq. (5.9) can be obtained by the use of the displace­ ment and momentum thicknesses. The displacement thickness is defined by Eq. (5.7). The momentum thickness is related to the loss of momentum of the air in the boundary layer and is defined as 8 = }o{'(I - ~) 1.£ dy = ~ r U U U 2 }0 (U - 1.£)1.£ dy (5.10) Substitution of tJ· and 8 in Eq. (5.9) gives UlJ* dU +.!!:... U2fJ = i dx dx UtJ. dU dx p + 2U8 dU + tr d8 = ! dx dxp dB +.!!...dU(tJ. +2)=~2 U dx (J pU dx Introducing the shape parameter H = = %plJ! a*If} and the dynamic pressure q dB +(/1 + 2)~dq =..:!.­ 2 dx q d» (5.11) 2q Equation (5.11) is in a convenient form for the calculation of the thick­ ness of the laminar boundary layer when the pressures are not constant, as was assumed in the Blasius solution. 132 In many cases of interest, the pres­ sure gradients are sufficiently small so that the type of velocity distribution through the boundary layer will not differ greatly from that in the bound­ ary layer of a flat plate.' Application of Eq. (5.10) to the Blasius distribu­ tion gives a value of 8/x equal to O.664/~. Consequently the value of H for the Blasius distribution is 1.73/0.664 = 2.605. From Eq. (5.5), T = p,U ~ ~ 0.332- vRs x Substituting the value of 8 T = 0.664 X 0.332 p,U p,U 8 == 0.220 8 THE EFFECTS OF VISCOSITY Substituting the values for Hand dO + dx T 93 in Eq. (5.11) (4.605)(_8_) pU dU = O.220pU 2 dx pU ~pU2 28 By use of the integrating factor 28CJ9·210, this expression becomes iP = 0.4401L i" U 8 •210 dx pUz 9.210j o (5.12) where U z = velocity at limit x If U is known as a function of x, that is, if the pressure distribution is known, Eq. (5.12) permits the boundary-layer thickness to be calculated approximately if the pressure gradients are sufficiently small for the actual distribution of velocity through the boundar)' layer to be approximated by the Blasius distribution. For most wing sections at low and moderate lift coefficients, the pressure gradients satisfy this condition back to the position of minimum pressure. The skin friction may be assumed to be that corresponding to the Blasius velocity distribution for the calculated thickness of the boundary layer. A convenient way of performing these calculations is to find an equivalent length of flat plate'l' 2 which is given by i X. = .,. .. . ~~.-. .. (:. {% U",8.~lOJo lj8.210 dx (5.13) where x, is the length of fiat plate in a stream of velocity Us that will generate the same laminar boundary-layer thickness as actually exists on the body in question at the point z. 5.7. Laminar Separation. It was pointed out in Sec. 5.3 that the laminar boundary layer cannot ordinarily exist for long distances in a region where the pressures are increasing ill the direction of flow, and the extent of the laminar boundary layer under such conditions is limited by separation. The Blasius solution does not, of course, give any information about laminar separation because this solution is premised on the absence of pressure gradients. Calculations of the location of laminar separation require a solution of the boundary-layer equations in more general form. One such attempt was made by Pohlhausen." He assumed that the velocity distribution in the boundary layer could be expressed by a poly­ nomial of the form 0­ Y (Y)2 + a3(Y)3 (Y)4 ~ + a. ~ U = al ~ + at ~ 'U 0 Using plausible assumptions regarding the physical characteristics of the layer, Pohlhausen developed a method for calculating the velocity dis­ tribution in the boundary layer and the position of the separation point. This method has only limited applicability to the calculation of the lam­ inar separation point. 105 Karman and Millikan-" obtained more accurate 94 THEORY OF WING SECTIONS solutions of Eq. (5.2). The solutions are involved and require much cal­ culation, but the results appear to be satisfactory. For most actual 'York on wing sections, the extent of the laminar boundary layer in regions of adverse pressure gradients is too short to require much detailed attention. It is frequently desirable, however, to know the possible extent of the laminar layer as determined by laminar separation. It has been possible to develop a method for applying the I{arman-Millikan solution to the rapid estimation of the laminar separa­ tion point132 which is sufficiently accurate for most applications. More detailed studies may, however, be required for the condition associated with maximum lift. The method for rapidly estimating the position of laminar separation is based on the assumption that it is possible to replace the actual velocity distribution effectively by a simplified velocity distribution consisting of a region of uniform velocity followed by a region of linearly decreasing velocity. The actual pressure distribution up to the point of minimum pressure is used to calculate the equivalent length of fiat plate from Eq. (5.13). Because dp/dx is zero at the point of minimum pressure, the velocity distribution in the boundary layer at this point must approximate the Blasius distribution very closely. The replacement of the actual pres­ sure distribution downstream of the point of minimum "pressure by a linear velocity gradient is usually justified by the relatively small extent of the laminar boundary layer in this region. Consequently, the laminar separa­ tion point may be estimated from positions calculated for a family of such simplified pressure distributions. The position of the laminar separation point was calculated by the Kinnan-Millikan method for a series of pressure distributions of the fol­ lowing type. u - = 1 for 0 < !. < 1 Us - z.­ and u == 1 + F x - X e for 1 < ~ Uo x. - x. where x == distance along surface from leading edge x, == length of flat plate equivalent to length from leading edge to position of minimum pressure U = velocity outside boundary layer at any point x U« == maximum velocity F = velocity gradient (xc/Uo)(dU/dx) (constant for any given case) Irrhe values of U8/ Ue, where U8 is the velocity at the laminar separation POint, were calculated by the Kinnan-Millikan method for a series of values of F, and the results are shown in Fig. 43. In applying this method, the value of F is obtained from calculated values of x,,/Uo and the value of ulj/dx correspondiug to the estimated THE EFFECTS OF VISCOSITY 95 equivalent adverse velocity gradient. The corresponding value of UB/Uo is found from Fig. 43 and is applied to the actual velocity distribution to locate the position of laminar separation. 5.8. Turbulent Flow in Pipes. The earliest information on the nature of turbulent flow was obtained from studies of the flow in pipes. Reynolds found that, if the diameter of the pipe and the velocity were sufficiently 1.00 ~ / / .96 / ~ ~- ./ '/ V I( I .92 .90 / .880 1/ / -.04 -.OS -.16 -.12 -.20 -24 -.28 F I""JG. 43. Velocity ratio for laminar separation as a function of the nondimensional velocity gradient F. small, the flow \'.&8 rectilinear or laminar, as ShO\VD by the behavior mf colored filaments of fluid which did not mix with the surrounding fluid. With larger diameters and greater velocities, the flow became turbulent, as shown by diffusion of the colored filaments. Reynolds found that, if the quantity pl'dlfJ. was less than about 2,000, the flow was always laminar. The upper limit of this Reynolds number for laminar flow was indefinite and depended mainly on the care taken to provide steady conditions and smooth surfaces. Values of the upper limit as much as ten to twenty-five times the lower limit have been found by various observers." With laminar flow, the pressure gradient. along the pipe increases linearly with the velocity; but, with turbulent flow, the gradient increases nearly as the square of the velocity. The velocity distributions across the diameter of the pipe for laminar and turbulent flows are shown in Fig. 44. For turbulent flow, the velocity distribution is more nearly uniform than for laminar flow, and the shearing stresses near the walls are considerably higher. These conditions result from the increased transport of momentum associated with the turbulent motion. It has been observed that the nondimensional velocity distribution THEORY OF WING SECTIONS 96 across the pipe is affected only slightly by the Reynolds number. On the basis of dimensional reasoning similar to that used for the formation of dimensionless coefficients for wings, Blasius" expressed the pressure loss in coefficient form as follows: r_ ~ = Pi - 'P2_ l ~p:;z2 where PI - 112 = pressure drop in a pipe of length l and radius r with a fluid of density p flowing at average velocity u. For smooth pipes or pipes having geometrically similar roughness, the coefficient ~ is a function only of the :JIi!:~:~~~:~= =~~~:;:~:~~::E;::::: in smooth pipes is LAMINAR :11 0.133 ~= ~r : ~£:~::iE:~1::~:~~:~~~:~ the pipe and the rate at which the pressure-loss coefficient changes with Reynolds number. The most im­ portant region in which to study the nature of the phenomena appeared to be in the vicinity of the wall of the pipe where the velocity gradient is high rather than in the middle of the pipe where the velocity is nearly uniform. It appeared reasonable to assume that the velocity distribution near the wall depended on the local quantities p and II- and the skin friction T and was independent of the radius of the pipe. It was assumed, in ap­ proximate agreement with experimental evidence, that the velocity dis­ tribution across the entire pipe could be expressed as a function of y/r alone where y is the distance from the wall of the pipe; that is, the velocity distribution remains similar for various mean velocities and pipe sizes. Equating the pressure loss to the skin friction T\.J8JLENT 44. Velocity distribution across pipe for laminar and turbulent flow. FIG. (PI - Pt)rr! = 2n-lT 2lT PI-P2=-r From Blasius' formula (5.14) THE EFFECTS OF VISCOSITY 97 If it is further assumed that the velocity distribution may be expressed as a power function of y/r, where V = velocity at center of pipe or 3/ 1/ = K p7 -&p.7 4 r- ]/ 1 7/ 7 4: ' 7 4 (r)(~)R 11 where K = a constant factor dependent on ratio of maximum velocity to mean velocity Applying the assumption that the velocity distribution near the wall and hence the skin friction is independent of the radius of the pipe, the ex­ ponent of r in the foregoing expression must be zero, or ~~+ ~n =0 n= ~1 (5.15) The assumption made in the derivation of this equation that the velocity distribution is a function only of local parameters near the wall would indicate that this equation would also apply to the velocity dis­ tribution in a turbulent boundary layer on a flat plate. In this case, r is interpreted as the thickness of the boundary layer and V as the velocity just outside the layer. The validity of the analogy between the flow in pipes and that over a flat plate is shown by the data of reference 19, some of which are presented in Fig. 45. 6.9. Turbulent Skin Friction. It is possible to calculate the skin friction on ~fiat plate with turbulent flow by application of the one-seventh power law [Eq. (5.15)] and the Blasius skin-friction formula [Eq. (5.14)] to the local boundary-layer conditions. In the case of the flow of an incom­ pressible fluid in a pipe, the Reynolds number along the pipe remains constant and the skin friction results in a loss of pressure. In the case of the flat plate, the pressure is constant along the plate and the skin friction causes a deceleration of the fluid particles, resulting in an increase of the thickness of the boundary layer and a corresponding increase of the boundary-layer Reynolds number, At any station along the plate, the skin-friction drag over that portion of the plate upstream of the station can be obtained from the rate of loss of momentum in the boundary layer. It is assumed that, for any station, the velocity distribution in the boundary layer corresponds to that in the pipe and that the local skin friction is the same as that in a pipe at the same Reynolds number. The particles in the boundary layer having an original velocity V up­ KI7$CMJ CMftec eooo 2000 ----- ........ - L.l ..... - ~- L~ ~ 1000 ~~ 1000 ~- ~~,..... ,.... -~ ~ 500 40,,-(:) ~ ~ ~ <:S ti CS II 400 <::i lI\Yv'M -~- - ,...,.... ---- .fOOt"' N '" ...... -- ~ ...... .,.- 1000 _1000' tS I ~ I I " a~ I ~ .... ...--, IIII I I , , ..... ~ ~ II 500 , , , I I!'" I , I '~')CM 8 ''''; It) ~ a ~ ...: ~ ~ ....: a yl'M t\i ~ ---...... l'-' ~ .... ~ ~ 400S? ro:. '" ~ ~ ~ ~ r..:. ,..:.,..:.c::s "....,...., S; C ~ ....... ~ ~ ~ -~,. -- ~ ~ ~ ~ ~ - ..-...-'l ..... _ IIII II1II1 ~ () () ~ ti .... )I - ~680CM II ij -~ ~ I , , ~ -~ ... ~~ ~ ..- x-I25CM - -'-- 500 ~ .... -~ 2000 -~ -"I'" ,..--, - IioIIIOIl CM/sec - 2000 --~ ~ ..--: () ~ K'81$CM CM,&tc 1000 500 8...: ... ~ 1""'""""1 ~oIX·lOCM and X Il 60CM ~ 'to -- - ~ '- ~ K'IOOCM CM/sec g ~ 8 y"CM ...:...:t\i FIG. 46. Logarithmic sua]o dj~"l\mA of tho vptocity in the boundary )nyer as a function 011/ at the sections z = 75, 87.5, 100, and 125 ern from tho leading edgo of the flt,t plate. The solid Imes represent thu one-seventh power law. THE BFFECT8 OF VISCOSITY 99 stream of the plate are decelerated to a velocity u at station z, The rate of loss of momentum at station z is given by the following expression and is equal to the skin-friction drag of that portion of the plate upstream of station x. ' F = p loa u(V - u)dy From the assumption that V=(~)M F i2PV2c5 = but dF dx = If" hence 7 2 dO -r = 72 pV dx A corresponding value of T may be obtained from Eq. (5.14) if the rels­ tion between ii and V is known. A commonly used experimental value of ii/IT is approximately 0.8. Using this value Eq. (5.14) rna)' be written in the form r' = O.0225p~~p.~~o-3-~ TT~ Equating the t\VO expressions for T (5.16) and integrating (;Vy~ z c5 = 0.37 (-tV)%x% ~ c5~ = 7; 0.0225 Solving for a (5.17) Substituting this value of 8 into the expression for the drag F = O.036pV%xG;x)% In coefficient form cr = lL F 7";.P V2X = O.072R:r;-~ (5.18) Equation (5.18) is a formula for the drag coefficient for one side of & smooth flat plate of length x with turbulent flow over the whole surface. Figure 46 shows the skin-friction coefficient for turbulent flow from Eq. (5.18) and for laminar flow from Eq. (5.6). . Nikuradse performed pipe experiments over a range of Reynolds num­ bers much larger than for previous investigations.76 • 7iG These experiments THEORY OF WING SECTIONS 100 showed that the pressure losses departed from values predicted by the Blasius formula above Reynolds numbers of about 50,000, the experi­ mental values decreasing less rapidly than the predicted values as the Reynolds number was increased. As expected from the simplified analysis of the relation between the pressure-loss coefficients and the velocity dis­ tributions, the velocity distributions at the higher Reynolds numbers are approximated better by one-eighth, one-ninth, or smaller powers than .0080 ~ .0070 ~ .0060 .0050 f)()40 <, ~ ~ =--- ~ ~ .-. I!....... - ~.- Turbulent flI=I~-Eq.5.20 £q.5.18./' f'~ "'~ "'i~ ~ ~lt'" ~~ ......... ............. .::::::: ,.... ~LomintT(Eq.5.6) :"r-... 0010 ~ ~ .0007 .0006 "" """ "" ""lIlli, ~~ ~~ fJOO5 "'~ .0004 os .ooaJ ~ ~ ""- .... ,..... 1..5 2 Z53 4 5 6 78906 1.5 2 25.3 4 5 6 78 9d Rx FIG. 46. Laminar and turbulent skin-friction coefficients for one side of a fiat plate. by the one-seventh power, On the basis of an elaborate and extended dimensional analysis, a logarithmic expression was obtained28 • 128 for the velocity distributions corresponding to Nikuradse's values for the pressure losses. q, == 2.5 In 1J + 5.5 (5.19) where u <1>=v* pt,*y 11=­ p. v*={; and is called the U friction velocity." The skin friction on a flat plate may be calculated using this expression by a process using the same general THE EFFECTS OF VISCOSITY 101 principles as that for the derivation of Eq. (5.18).28 The tabular solution to this problem may be approximated by the expression 0.472 cp=---­ (5.20) (log.1l:,;)2.ss Skin-friction coefficients calculated from this formula are shown in Fig. 46. Formulas (5.18) and (5.20) are in convenient form for the calculation of skin friction on flat plates, For application to bodies having arbitrary pressure distributions, it is more convenient to relate the local skin-friction coefficient to a Reynolds number based on the momentum thickness of the boundary layer. For the one-seventh power velocity distribution, such an expression may be obtained from Eq. (5.16). For this case, according to Eq. (5.10), _T _ C/1 L U "2 72P - () 0 · 02~1 (-.!!-)~ ;J ["'0 - . 02r-·lR-~ a, p : (5.21) For the case of the logarithmic velocity distribution, approximate formulas have been ohtuined relating t he local skin-friction coefficient to the boundary-layer Reynolds number Ro• Thut given by Squire and ) . . oung!" is er~ = [5.8HO log (-l.O~5Jlo)f (5.22) T It is known that Eq. (5.21) is valid only for moderately small Reynolds numbers, whereas Eq. (5.22) is a much better approximation- at large Reynolds numbers. Equation (5.22) is, hO\VCVCf, inconvenient for use in connection with the momentum equation (5.11). Interpolation formulas have therefore been developed in the form c, = .ilR,-n (5.23) where the coefficients and exponents nrc selected to correspond with the desired range of Reynolds numbers. For any particular application, the two constants of Eq. (5.23) may be found by making Eq. (5.23) agree with the assumed skin-friction law at t\VO values of R, including the range under consideration. If the subscript 1 corresponds to conditions at the low value of R s and the subscript 2 to conditions at the high value of R" then (T / Pl]2)l log (T jp(]2)2 n-­ - log (R 8,./RIJJ A = c/lRS1" THEORY OF WING SECTIONS 102 The assumed skin-friction law may be Eq. (5.22) or the following formula · 120 suggested by T etervm T 1 (5.24) 2 pU = [ R 2.5 In 2.5(1 _ 5~) + 5.5 J2 The values of cJ corresponding to Eqs. (5.22) and (5.24) are plotted in Fig. 47. 6.10. Calt;ulation of Thickness of the Turbulent Layer. A knowledge of the thickness of the turbulent boundary layer is required for several .007 .006 f){)5 I -~~ ~!"""o .004 ------ ~.OO3 ....... ......... ~I""-~ .... Eq.5.22 ~ Ef~·524 ~"- .. ~ .002 2 s 4 5 6 78 9 f()4 --. ~~ 2 .... 3 ~ 1-1-. 4 .- r- 5 6 78 gl()5 R(J FIG. 47. Local turbulent skin-Iriction coefficient as a function of the boundary-layer Reynolds number. purposes. The profile drag of the wing is, of course, intimately associated with the thickness of the boundary layer at the trailing edge. It is also desirable to know the thickness of the boundary layer at various points on the wing surface in connection with the study of control surfaces, high-lift devices, air intakes, and protuberances. The thickness of the turbulent boundary layer can be calculated from the momentum equation (5.11) if a relation between the local skin-friction coefficient and the dependent variables 8, q, and H is knO\VD. Values of the skin-friction coefficient have been found by assuming that the relation between the boundary-layer Reynolds number R, and the skin-friction coefficient is the .same as the corresponding relation found from pipe experiments. This relation has been plotted in Fig. 47. Although this relation may not be very accurate, .particularly for values of the shape parameter H differing considerably from those found in pipes, such marked differences usually occur only when dqld» is large negatively. In this case Eq. (5.11) shows that the contribution of the skin-friction term to the rate of increase of the boundary-layer thickness dO/dx is relatively small. It may accordingl~ybe assumed for practical calculations that CJ is independent of the value of H. Further simplification of the calculations is obtained by assuming a c, THE EFFECTS OF VISCOSITY 103 skin-friction law of the type indicated by Eq. (5.23). (5.11) may be written in the form de + (H + dx U 2) dU dx (J = A~n (pU)r& e» In this case, Eq. . ' ,;,-." The value of the factor (H + 2) is not greatly affected by the value of H, which usually has values of about 1.4 or 1.5 with extreme variations from about 1.2 to 2.5. Detailed calculations have S110\Vn that excellent agreement can be obtained between experimental and calculated values of 6 if H is assumed to be constant and to have a value"? between 1.4 and 1.6. "~ith H assumed constant, this equation is seen to he of the Bernoulli type, and it can be integrated to give the value of 6 us follows: (~) C ",/c = 1 (UjV)H+2 [(1 + -t: 2Rc " (U)(l/+l) Xo!c V. (u+lH-l ~ dc + (~)"+l( ¥~)rH+2)("+I] 1/<1+..> A (5.25) 6.11. Turbulent Separation. To determine the turbulent separation point, detailed information is needed concerning the velocity distribution in the boundary layer and the effects on the -velocity distribution of such factors as the pressure distribution and the skin friction. Although the value of the shape parameter H can be found for any given velocity dis­ tribution, the value of H in itself does not define the velocity distribution. Thedata collected by Grusehwitz'" and by Tetervin-" show experimentally that the velocity distributions in turbulent boundary layers actually form a one-parameter family of curves and that the velocity distributions can be specified by the value of H. Velocity profiles for turbulent boundary layersl 35 are presented for various values of H in Fig. 48. The value of H increases as the separation point is approached. It is not possible to give an exact value of H corresponding to separation, because the turbulent separation point is not very well defined. The value of H varies so rapidly near the separation 'point, however, that it is not necessary to fix accurately the value of H corresponding to separation. Separation has not been observed for a value of H less than 1.8 and appears definitely to have been observed for a value of H of 2.6. An empirical expression was found 135 for the rate of change of H along the surface in terms of the ratio of the pressure gradient to the skin friction and the shape parameter itself.' The following equation was derived to fit the experimental data: 8 -dH = e4.680 (H dx 2.975) Q2q - 2.03::>(H [8d ,.. - 1.286) ] - - - - qdx (5.26) T The skin friction corresponding to a given value of R6 may be obtained from Fig. 47 or from Eqs. (5.22) or (5.24). Equations (5.11) and (5.26) are THEORY OF WING SECTIONS 104 simultaneous first-order differential equations that can be solved by step­ by-step calculation. It is usually necessary to use such a method, although, 8 7 J/.~9 1.0 ! ~. d .Q r- . ~ .. =­ '"U"'o ~ Q~ l7U r'ii:Pc Cilt't ~ 117' -4 ~ ~~ ~~ ~ , ~~ ~ -. .4 fP'~ .., + ~ ~1( ~ o-Q r4-...~ . ~ ~ -­ ~ r--... ~~ .2 /2 /.6 1.4 .5 ~ ~ s-; N ~ ,r--.. .. ~ r--­ B"'-­ 'I -~ ~ ~~ - 2.2 ~ ~AJ K oht r-- )(~2 ~~ - 0' ~ ~ 2.0 H 1.8' r-­ IOoz ~~ ---.... r---.. ~ --. 0 o ..,~ ... 1> ---v- Q ~~ A It:. ~ ~ 04: v 0 6 ........ .. h. n r .. I\IIl ~A ~ ~~ ~ .6 bh ''''V ft_ .. ~ ..... "U 61. 2 ........... ~ + I 02 D3 ~ N ~~ ~ rw v4 F-- . 2.4 ~5 46 I-- 2.6 q 7 08 ~9 2.8 (0) • varia/ion 01 'rU wi/h H lor various values 01 ~ H ~r--"""--; /.286 1---+--'--;---;r---1 A /.4 I:" .,.....-..--....--.. 0 1--,~,r-,..,..e.,~'7"-:::~~~~--t--t--t--.-,t--t 1.5 a 1.6 0 tr ~~..,4.~~~~~,I'-+---+--t--t-~t-., + l8 x 1.9 1fU.~~~'#7~~-+--t---t---.,t---t---;r---t <J ~.O ~ 2./ ~~~~~t----t--+--"""""'--t----t--'t---t b. 2.2 ~ 2.3 ~~o '---ir---+--t---t--i---t---t---t---t 17 2.4 v 2.5 2.6 -0-2.7 6 ~-+--t---+--t----+--t---+--t--t--1t-"""1-o- 1 2 4 ~8 7 9 (b) - velocity profiles for turbulent botndory layers correspcnding 10 various KJlues of H FIG. 48. Velocity diatributions in turbulent boundary layers. for SOme particular cases, the equations may be integrated directly. The method of calculation is as follows. The values of the variables entering into the computation at the initial station are substituted in the momentum equation (5.11) and the equation for dH/dx (5.26). Values for d8/dx and dH/dx are thus obtained at the initial station. An increment of the length THE EFFECTS or VISCOSITY 105 along the surface of the body x is then chosen and multiplied by d8/dx and dH /dx to give aO and 1::Jl, respectively. These increments of 8 and H are added to the initial values and result in values of 8 and H for the ne-w value of x, The process is repeated until the separation point is reached, as shown by the value of H. For the purposes of the computations the value of H corresponding to separation may be taken as 2.6. The choice of the increments of x is a matter for the judgment of the investigator. AB a general rule, the increment of x should be made small when d8/dx or dH/dx changes rapidly from one value of x to the next. 5.12. Transition from Laminar to Turbulent Flow. Experience has shown that transition from laminar to turbulent flow takes place at a Rey­ nolds number that depends upon the magnitude of the disturbances. For viscous flows at very low Reynolds numbers, as in oil, all disturbances are damped out by the viscosity, and the flow is laminar regardless of the magnitude of any disturbance. .As the Reynolds -number is increased, a condition is reached at which some particular types of disturbances are amplified and eventually cause transition. This value of the Reynolds number is called the U lower critical." Further increase of the Reynolds number causes amplification to occur for a greater variety of disturbances and increases the rate of amplification. Under these circumstances the Reynolds number at which transit.ion occurs depends 011 the magnitude of the disturbances. Transition can he delayed to high values of the Reynolds number only by reducing all disturbances such as stream turbu­ lence, unsteadiness, and surface roughness to a minimum. For ftO\VS in a pipe, the lower value of the critical Reynolds number based on the diameter and tho mean velocity is about 2,300.98 The upper value appears to depend only on the care taken in conducting the experiment, and values twenty times as great as the lower value have been obtained. I t has been found possible 99 • 100. 127,128 to compute the rate of amplification or damping of disturbances of various frequencies and wave lengths. Such computations's" permit. curves of constant amplification coefficient {:ji~* IV to be plotted on coordinates of all· against the Reynolds number R,. = p 1l8*/p. where a = 21rII and l is the wave length. Such a plot is given in Fig. 49. Experimental points obtained by Schubauer and Skramstad'P are also plotted in Fig. 49. The data of Fig. 49 were obtained for a Blasius profile corresponding to flow over a flat plate. It will be noted that there is a Reynolds number below which no disturbance will be amplified. This value of the Reynolds number is the lower critical. The agreement between the experimental and theoretical results is good considering the difficulties of the experimental work and the simplifying assumptions made in the development of the theory. Similar calculations have been made by Schlichting and Ulrich'?' for Pohlhausen's velocity profiles corresponding to various pressure gradients. THEORY OF WING SECTIONS 106 The results are presented in Fig. 50, which is a plot of the lower critical Reynolds number against the parameter X that defines the velocity distri­ bution through the laminar layer. According to the Pohlhausen method, .44r--.....---.....--........----.---.....,.---,----.---,.---,--,-----... I .40 __9110-4 \ .36 \ \ -35 " I -/9 .32 . 28 .24­ a6* .20 ./6 ./2 .08 .04 °0 400 800 /200 16fX) 2000 2400 2800 ~ 3600 4000 4400 R Contours of equal amplification according to Schlichting. V &lUC8 of fji~· i U v (aU valuee to be multiplied by 10-4 ) opposite points are amplifications determined by experiment.. Faired experimental contour of zero amplification shown by broken curves. FIG. 49. the velocity distribution through the layer is given by the following formula and A = p02dU p,dx Figure 5Q shows that decreasing pressures in the direction of flow are favorable to the stability of the laminar boundary layer and that increasing pressures are unfavorable.. The theory has not been developed to such a point that it can be used to predict transition, although it is useful in pointing out the fundamental reasons for the instability of the laminar layer at high Reynolds numbers. The theory does not relate the boundary-layer disturbances to either the THE EFFECTS OF VISCOSITY 107 .surface irregularities or the turbulence of the air stream. The theory also does not relate the transition to the magnitude of the fluctuations of veloc­ ity in the boundary layer. Under these circumstances, it is necessary to t--+--+--+--+--+--J~,--I 8 I p( I V t--~~--+---+I--"'I03jl+--+--+-~~-"""~~ I I I I I -6 -4 -2 I I 'f 1/ 0 2 4 6 8 50. Critical values of lU- as 11 function of the shape parameter '" aceording to Schlichting and Ulrich. I OI FIG. rely on empirical results to predict the location of transition. The empirical data are discussed in Chap. 7. 6.13. Calculation of Profile Drag. In cases where it is known that the. flow is not separated from the surface, the boundary-layer thickness at the trailing edge of a wing section and the profile drag can be calculated from a knowledge of the potential-flow pressure distribution and the location of transition from laminar to turbulent flow, The momentum thickness of the laminar boundary layer at the transition point can be calculated from Eq. (5.12). The initial momentum thickness of the turbulent boundary layer is taken to be the same as that of the laminar layer at the transition point. The thickness of the turbulent boundary layer at the trailing edge can be calculated from Eq. (5.25) . If the pressure at the trailing edge were free-stream static pressure, as in the case of a flat plate, the profile drag would be THEORY OF WING SECTIONS 108 .0/6 .0/4 x Experimenta! meoSU:;menIs ~ R-2.9xlO T R=6.0xlOs ~ - lC/ Theoreticol colcu'r/J8"5 .0/2 - --R=2.9x ----R-6'Ox106 I / .010 c> / , ~ ~ ioo""~ "r;;' /~/ iJl x \ + \\ .006 J J. 'X '" ~' ~-- --.-iIk 1- ,~~ + o .2 -t .-:l~ ~t .002 -.2 C, .4 .8 .6 LO FIG. 51. Comparison of calculated and experimentally measured polars for N.ACA 67,1-215 airfoil. .014 .010 I t .012 '- +~ ~.008 ~.......... ~ ......... , J / /' ¥ !~ "'-- ...~ ...... ~+~- ... .-:. ~ --- -~ ~ + Experimedol .004 measurement - .oO?:s Theoretical Or--'" I--- f - - I--- colcu/olion -.6 -.4 -.2 o .2 Cz. .4 .6 .8 1.0 /2 1.4 Comparison of calculated and experimentally measured polars for NACA ~015 airfoil at R = 5.9 X 10'. FIG. 52. where 8 = sum of momentum thicknesses on upper and lower surfaces In the usual case, the pressure at the trailing edge is not the same as free-stream static pressure, and some means must be found for finding the effective momentum thickness at a point far downstream in the wake THE EFFECTS OF VISCOSITY 109 where the pressure has returned to the free-stream static value. Squire and Y oung'P derived the required relation by setting the value of T in Eq. (5.11) equal to zero in the wake and finding an empirical relation between Hand q/qo- The resulting expression for the profile drag is (8) ctl=2- (U~ - (Jl+fl)/2 c t 11 t (5.27) where the subscript t designates conditions at the trailing edge. The value of H is the value used in calculating the boundary-layer thickness. The agreement to be expected between experimental and calculated profile drag coefficients is indicated by Figs. 51 and 52. The calculated 14 --...r-­ r-­ drag coefficients presented in these .... 8 """'­ --. .............. ~ figures were obtained by a method" r-----. ........... A r---... r--­ fundamentally the same as that pre- ~ 12 r--­ senteel here. ¢ 1.0 5.14. Effect of Mach Number on Skin Friction. Solutions for the velocity distribution through the 080 1 4 2 3 5 laminar boundary layer in corn­ Mach number pressihle flow and for the correspond- FIG. 5~1. Skin-Iriction eoeffir-ienrs, ( ...4) No ing skin friction have been obtained heat transferred to wall. (Ill wall tern perature one-quarter of Iree-streurn temperature, by von Karman and Tsien1ft for the case of the flat plate. The results of the skin-friction calculations are pre­ sented in Fig. 53 for the case of no heat transfer to the plate and for the case of a plate whose absolute temperature is one-quarter of that of the free stream. These results show a moderate decrease in laminar skin friction with increasing Mach number. Although these results indicate that the laminar skin friction increases with beat transfer from the fluid to the plate, this increase is small even for the extremely 10"" plate temper­ atures for which these results were obtained. For the turbulent boundar)' layer, Theodorsen and Regier!" found ex­ perimentally that the skin friction is independent of the Mach number (at least up to a value of 1.69). The experimental data are presented in Fig. 54. These data were obtained by measuring the torques (moments) required to rotate smooth disks in atmospheres of air and Freon 12 (CCI2F2) at various pressures. Data obtained by Keenan and X eumannf for the skin friction in pipes appear to confirm Theodorsen's conclusion for the fully developed turbulent flow. Analytical studies by Leeg6% on the effect of Mach number and heat transfer on the lower critical Reynolds number for the case of the flat plate indicate that increasing th~ Mach number has a destabilizing in­ fluence on laminar flow when there is no heat transfer to the plate. Heat --- THEORY OF WING SECTIONS 110 transfer to the plate has a stabilizing effect, while the opposite is the case for heat transfer from the plate to the fluid, This effect becomes stronger I -1.2 I I I Moch P1essute (in.HgobsJ tvmber Q24toQ62 r-­ 1.8 fo.JO Air .4010 .96 Freon /2 30 t-­ 14 .5J 10/.42 FI1!OII 12 to.77101.69 Fr«Jfl 12 5 I I Symbol 0 -1.4 04­ x D ~ IIlloo.... ~ ~ ~ ........ 100.... """"­ 4.8 5.0 52 ~ KDrrnDns -Ys r--. ........ """­ ~~ 4 ~ KcrnD1s /ominar-f/oW"'/ i" ~ -2.2 fomvlo, C",=.J.871r~ ~, -2·~.6 - futtJu/enI-f/oN formub, /"" Lm CM=O.I46R -~ "'"",,­ -2.0 [,I I 60s 5.4 5.6 a -"'1fI­ M=O.53 ~=l69 ~ ~ ~ ~. ~ 5.8 6.0 6.2 6.4 ....:..; ~ ....... + it-+ 6.6 6.8 -ZO LDgI()R FIG. M. Moment coefficient for disks as function of Reynolds number for several values of Mach number with air and Freon 12 as mediums, Maximum Mach number, 1.69. as the Mach number is increased, and, at moderate supersonic speeds, comparatively small amounts of heat transfer to the plate stabilize the laminar layer to very high values of the Reynolds number. CHAPTER 6 FAMILIES OF WING SECTIONS 6.1. Symbols. Ps resultant pressure coefficient R radius of curvature of surface of modified NAC.A four-digit series symmetrical sections at the point of maximum thickness IT velocity of the free stream a coefficient a mean-line designation; fraction of the chord from leading edge over which loading is uniform at the ideal angle of attack e chord ci. section design lift coefficient c.e/. section moment coefficient about the quarter-chord point d coefficient k1 constant m maximum ordinate of the mean line in fraction of the chord p ehordwise position of 111 r leading-edge radius in fraction of the chord r, leading-edge radius corresponding to thickness ratio t maximum thickness of section in fraction of chord v local velocity over the surface of a symmetrical section at zero lift !:Av increment01local velocity over the surface of a wing section associated 'lith camber Av.. increment of local velocity over the surface of a wing section associated with angle of attack % abscissa. of point on the surfaee of a symmetrical section or a chord line XL abscissa of point on the lower surface of tl. wing section Xu abseisaa of point on the upper surface of a wing section XC abscissa. of point on the mean line YL ordinate of point on the lower surface of a wing section Yu ordinate of point on the upper surface of a wing section Yc ordinate of point on the mean line y, ordinate of point on the surface of a symmetrical section Qi design angle of attack 8 tan-I (dyc/dzc ) T trailing-cdge angle 6.2. Int:"oduction. Until recently the development of wing sections has been almost entirely empirical. Very early tests indicated the desirability of a rounded leading edge and of a sharp trailing edge. The demand for improved wings for early airplanes and the lack of any generally accepted wing theory led to tests of large numbers of wings with shapes gradually improving as the result of experience. The Effie! and early RAF series were outstanding examples of this approach to the problem. 111 THEORY OF WING SECTIONS 112 The gradual development of wing theory tended to isolate the wing­ section problem from the effects of plan form and led to a more systematic experimental approach. The tests made at Gottingen during the First World War contributed much to the development of modem types of wing sections. Up to about the Second World War, most wing sections in common use were derived from more or less direct extensions of the 'york at Gottingen. During this period, many families of wing sections were tested in the laboratories of various countries, but the 'York of the NACA was outstanding. The NACA investigations were further systematized by separation of the effects of camber and thickness distribution, and the experi­ y ./0 °u(xy,Yy) e -./0 'I p!!_.LJ_-_~-=--- ~(XLIYLJ 'Rodius fhrough end of chord (mean line slopeOf a5 % chord) Xy=X-Jf sin9 XL=X"'t sinS Yu=~ +11 cos9 JL 7C -JI cos e Sample calculationA lor derivation 01 the NACA 86.3-818 airfoil (0 -= 1.0) - tan' - ­ ---1----1 sin" FIG. 56. Method of combining mean lines and basic-thickness forms. mental work was performed at higher Reynolds numbers than were generally obtained elsewhere. The wing sections now in common use are either N .~CA sections or have been strongly influenced by the NACA investigations. For this reason, and because the NACA sections form consistent families. detailed attention will be given only to modem NACA wing sections. 8.3. Method of Combining Mean Lines and Thickness Distributions. The cambered wing sections of all NACA families of wing sections con­ sidered here are obtained by combining a mean line and a thickness distribution. The process for combining a mean line and a thickness distribution to obtain the desired cambered wing section is illustrated in Fig. 55. The leading and trailing edges are defined as the forward and FAMILIES OF WING SECTIONS 113 rearward extremities, respectively, of the mean line. The chord line is defined as the straight line connecting the leading and trailing edges. Ordinates of the cambered lying sections are obtained by laying off the thickness distributions perpendicular to the mean lines. The abscissas, ordinates, and slopes of the mean line are designated as Xc, Ye, and tan 8, respectively. If Xu and Yu represent, respectively, the abscissa and ordi­ nate of a typical point of the upper surface of the wing section and y, is the ordinate of the symmetrical thickness distribution at ehordwise position %, the upper-surface coordinates are given by the following relations: Xv = :r - y, sin 9 } yv = Yc + y, cos " (6.1) The corresponding expressions for the lower-surface coordinates are XL = X + Yt sin 8 } YL = Ye - Yt cos 8 (6.1) The center for the leading-edge radius is found by drawing a line through the end of the chord at the leading edge with a slope equal to the slope of the mean line at that point and laying off a distance from the leading edge along this line equal to t.he leading-edge radius. This method of con­ struction causes the cambered wing sections to project slightly forward of the leading-edge point. Beeause the slope at the leading edge is theoreti­ cally infinite for the mean lines having a theoretically finite load at the leading edge, the slope of the radius through the end of the chord for such mean lines is usually taken as t.he slope of the mean line at x/c equals 0.005. This procedure is ~ustified by the manner in which the slope increases to the theoretically infinite value as x/c approaches o. The slope increases slowly until very small values of x/c are reached. Large values of the slope are thus limited to values x/c very close to 0 and may be neglected in practical wing-section design. ­ or The data required to construct some cambered wing sections are pre­ sented in Appendixes I and II, and ordinates for a number of cambered sections are presented in Appendix lll. 6.4:. NACA Four-digit Vmg Sections. a. Thickness Distributions. "'hen the N ACA four-digit wing sections were derived.P it was found that the thickness distributions of efficient wing sections such as the Gottingen 398 and the Clark Y were nearly the same when their camber was removed (mean line straightened) and they were reduced to the same maximum thickness. The thickness distribution for the NAC~4\. four-digit sections was selected to correspond closely to that for these wing sections and is \' given by the following equation: ±'Yt,=_t- (O.29G90·~-O.12600x-O.35160x2+0.28430r-O.l0150zC)(6.2) 0.20 THEORY OF WING SECTIONS 114 where t == maximum thickness expressed as a fraction of the chord The leading-edge radius is r, = 1.1019t2 (6.3) It will be noted from Eqs. (6.2) and (6.3) that the ordinate at any point is directly proportional to the thickness ratio and that the leading-edge radius varies 88 the square of the thickness ratio. Ordinates for thickness ratios of 6, 9, 12, 15, 18, 21, and 24 per cent are given in Appendix I. b. Mean Line«. In order to study systematically the effect of variation of the amount of camber and the shape of the mean line, the shape of the mean lines was expressed analytically as two parabolic arcs tangent at the position of maximum mean-line ordinate. The equations" defining the mean lines were taken to be m Yt: :;: p;. (2px - x 2 ) forward of maximum ordinate and (6.4) Yc = m (1 _ p)t [(1 - 2p) + 2px - r] aft of maximum ordinate where m = maximum ordinate of mean line expressed as fraction of chord p == chordwise position of maximum ordinate It will be noted that the ordinates at all points on the mean line vary directly with the maximum ordinate. Data defining the geometry of mean lines with the maximum ordinate equal to 6 per cent of the chord are presented in Appendix II for chord,vise positions of the maximum ordinate of 20, 30, 40, 50, 60, and 70 per cent of the chord.. c.. Numbering System. The numbering system for NACA wing sections of the four-digit series is based on the section geometry. The first integer indicates the maximum value' of the mean-line ordinate Yc in per cent of the chord. The second integer indicates the distance from the leading edge to the location of the maximum camber in tenths of the chord. The last two integers indicate the section thickness in per cent of the chord.. Thus the NACA 2415 wing section has 2 per cent camber at 0.4 of the chord from the leading edge and is 15 per cent thick. The first two integers taken together define the mean line, for example, the NACA 24 mean line. Symmetrical sections are designated by zeros for the first two integers, as in the case of the NACA 0015 wing section, and are the thickness distributions for the family. d. Approximate Theoretical Characteristic8. Values of (V/V)2, which is equivalent to the low-speed pressure distribution, and values of v/V are presented in Appendix I for the NACA 0006, 0009, 0012, 0015, 0018, 0021, and 0024 wing sections at zero angle of attack. These values were cal­ culated by the method of Sec. 3.6. Values of the velocity increments AIl.IV induced by changing angle of sttaok are also presented for an FAMILIES OF WING SECTIONS 115 additional Iift coefficient of approximately unity. Values of the velocity ratio v/V for intermediate thickness ratios may be obtained approximately by_linear scaling of the velocity Increments obtained from the tabulated values of v/V for the nearest thickness ratio; thus (!!.V.)" = [(~) 1 II 1J ~ + 1 t1 (6.5) Values of the velocity increment ratio ~vc/V may be obtained for inter­ mediate thicknesses by interpolation. The design lift coefficient. Cl i , and the corresponding design angle of attack oa, the moment coefficient Cm.lf) the resultant pressure coefficient P R , and the velocity ratio ~/V for the NAC.L~ &2, 63, 64, 65, 66, and 67 mean lines are presented in Appendix II. These values were calculated by the method of Sec. 3.6. The tabulated values for each mean line may be" assumed to vary linearly with the maximum ordinate Ye; and data for similar mean lines with different amounts of camber, within the usual range, may be obtained simply by scaling the tabulated values. Data for the NACA 22 mean line may thus be obtained simply by multiplying the data for the NACA 62 mean line by the ratio 2:6, and for the X.~CA 44 mean line by multiplying the data for the N.t\CA 64 mean line by the ratio 4:6. Approximate theoretical pressure distributions may be obtained for cambered lying sections from the tabulated data for the thickness forms and mean lines by the method presented in Sec. 4.5. 6.6. NACA Five-digit Wlng Sections. 4. Thickne&8 Distributions. The thickness distributions for the NAC.A~ five-digit wing sections are the same as for the NACA four-digit sections (see Sec. 6.4a). b. Mean. Lines. The results of tests of the N.&-\.CA four-digit series wing sections indicated that the maximum lift coefficient increased as the posi­ tion of maximum camber was shifted either forward or a~t of approximately the mid-chord position. The rearward positions of maximum camber were not of much interest because of large pitching-moment coefficients. Be­ cause the type of mean line used for the NACA four-digit sections was not suitable for extreme forward positions of the maximum camber, a new series of mean lines was developed, and the resulting sections are the NACA five-digit series. . The mean lines are defined 46 by two equations derived so as to produce shapes having progressively decreasing curvatures from the leading edge aft. The curvature decreases to zero at a point slightly aft of the position of maximum camber and remains zero from this point to the trailing edge. The equations for the mean line are Yc = ~kl[x3 - 3mx2 + m2(3 - m)x] from x = 0 to z = ml y.; = %k1m3(l - x) from x = m to oX = c = l~ (6 6) . THEORY OF WING SECTIONS 116 The values of m were determined to give five positions p of maximum camber, namely, 0.000, O.IOc, O.ISe. O.2Oc, and O.25c. Values of k1 were initially calculated to give a design lift coefficient of 0.3. The resulting values of p, m, and k1 are given in the following table. Mean-line designation Position of camber p "" kl 210 0.05 0.10 0.15 0.20 0.25 0.0580 0.1260 0.2025 0.2900 0.3910 361.4 51.64 15.957 6.643 220 230 240 250 3.230 This series of mean lines was later extended" to other design lift coefficients by scaling the ordinates of the mean lines. Data for the mean lines tabulated in the foregoing table are presented in Appendix II. c. Numbering System. The numbering system for wing sections of the N..\ CA five-digit series is based on a combination of theoretical aerodynamic characteristics and geometric characteristics. The first integer indicates the amount of camber in terms of the relative magnitude of the design lift coefficient; the design lift coefficient in tenths is thus three-halves of the first integer. The second and third integers together indicate the distance from the leading edge to the location of the maximum camber; this distance in per cent of the chord is one-half the number represented by these .-integers. The last two integers indicate the section thickness in per cent of the chord. The NACA 23012 wing section thus has a design lift coefficient of O.3 J has its maximum camber at 15 per cent of the chord: and has a thickness ratio of 12 per cent. d. Approximate Theoretical Characteristics. The theoretical aerody­ namic characteristics of the N ACA five-digit series wing sections may be obtained by the same method as that previously described (Sec. 6.4<1) for the NACA four-digit series sections using the data presented in Appen­ dixes I and II. 6.6. Modified NACA Four- and Five-digit Series Wing Sections. Some early modifications of the NACA four-digit series wing section· included thinner nosed and blunter nosed sections which were denoted by the suffixes T and B, respectively. Some sections of this family49 with refiexed mean lines were designated by numbers of the type 2R112 and 2R212, where the first integer indicates the maximum camber in per cent of the chord and the subscripts 1 and 2 indicate small positive and negative moments, respectively. Another series of N ACA five-digit wing sections" is the same as those previously described except that the mean lines are reflexed to produce theoretically zero pitching moment, These sections are distinguished by FAMILIES OF JV[.;V'G SECTIONS 117 .he third integer, which is always 1 instead of o. These modified sections lave been little used and will not be discussed further. More important modifications common to both the NACA four-digit md five-digit series wing sections consisted of systematic variations of the hickness distributions. 1M These modifications are indicated by a suffix -onsisting of a dash and two digits as for the NACA 0012-64 or the 23012-64 sections, These modifications consist essentially of changes of the leading­ edge radius from the normal value [Eq. (6.3)] and changes of the position of maximum thickness from the normal position at 0.3Oc [Eq. (6.2)]. The first integer following the dash indicates the relative magnitude of the leading-edge radius. The normal leading-edge radius is designated by 6 and j, sharp leading edge by o. The leading-edge radius varies as the square of this integer except for values larger than 8, when the variation becomes ]xbitrary.U6 The second integer following the dash indicates the position of maximum thickness in tenths of the chord. The suffix -63 indicates sec­ tions very nearly but not exactly the same as the sections without the suffix. The modified thickness forms are defined by the following two equations: ± y, := aov'X + alX + ll2X2 + a~ ahead of maximum thickness } () 7' :l:Yt==do+dl(1-x)+d2(1-x)~d3(1-x)3 aft of maximum thickness (.) The four coefficients d-o, dt , d~, and ds arc determined from the following conditions: 1. Maximum thickness, t. 2. Position of maximum thickness. In. 3. Ordinate at the trailing edge x = 1, y, = O.Olt. 4. Trailing-edge angle, defined by the following table: m dl/ --:., (x dx 0.2 0.3 0.4 0.5 0.6 = 1) 1.000t 1.170l 1.575t 2.325t 3.500t The four coefficients 00, at, a2, and U3 are determined from the following conditions: 1. Maximum thickness, t. 2. Position of maximum thickness. m, 3. Leading-edge radius, r, == ~2/2. T, = 1.1019(tI/6)2, where 1 is the first integer following the dash in the designation and the value of I does not exceed 8. 4. Radius of curvature R at-the point of maximum thickness R= 1 2d: + 6d3(1 - 11t) 118 THEORY OF WING SECTIONS Data" far some of the modified thickness forms are presented in Appendix I. The data presented may be used together with data for the desired type of mean line to obtain approximate theoretical characteristics by the method presented in Sec. 4.5. The family of N ACA wing sections defined by these equations has been studied extensively by German aerodynamicists, who have applied designa­ tions of the following type to these sections: NACA 1.8 25 14-1.1 30/0.50 where 1.8 = maximum camber in per cent of chord 25 = location of maximum camber in per cent of the chord from the leading edge 14 = maximum thickness in per cent of the chord 1.1 = leading-edge radius parameter, r/t,2 30 = location of maximum thickness in per cent of the chord from the leading edge 0.50 == trailing-edge angle parameter, (l/t)(tan T/2) and t = thickness ratio T = leading-edge radius, fraction of chord T = trailing-edge angle (included angle between the tangents to the upper and lower surfaces at the trailing edge) For wing sections having this designation, the selected values of the leading-edge radius and of the trailing-edge angle are used directly with the other conditions in obtaining the coefficients of Eq. (6.7)~ The selected values for the maximum camber and location of maximum camber are applied to Eq. (6.4) to obtain the corresponding mean line. 6.7. NACA l-series W-mg Sections. The NACA l-eeries wing sec­ tionsG- 113 represent the first attempt to develop sections having desired types of pressure distributions and are the first family of NACA low-drag high-critical-speed wing sections. In order to meet one of the require­ ments for extensive laminar boundary layers, and to minimize the induced velocities, it was desired to locate th~ minimum-pressure point unusually far back on both surfaces and to have a small continuously favorable pressure gradient from the leading edge to the position of minimum pres­ sure. The development of these wing sections (prior to 1939) was ham­ pered by the lack of adequate theory, and difficulties ,,:ere experienced in obtaining the desired pressure distribution over more than a very limited range of lift coefficients. As compared with later sections, the N .~C.A. I-series wing sections are characterized by sOla).lleading-edge radii, com­ paratively large trailing-edge angles, and slightly higher critical speeds for a given thickness ratio. These sections have proved useful for pro­ pellers. The only commonly used sections of this series have the minimum pressure located at 60 per cent of the chord from the leading edge, and data "ill be presented only for these sections. FAMILIES OF WING SECTIONS 119 o. Thickne88 Distributions. Ordinates for thickness distributions with the minimum pressure located at 0.6c and thickness ratios of 6, 9, 12, 15, 18, and 21 per cent are presented in Appendix I. These data are similar in form to those presented for the KACA four-digit series. These sections were not developed from any analytical expression. The ordinate at any station is directly proportional to the thickness ratio, and sections of intermediate thickness may be correctly obtained by scaling the ordinates. b. Mean Lines. The NACA l-series wing sections, as commonly used, are cambered with a mean line of the uniform-load type [Eq. (4.26)]. Data for the mean line are tabulated in Appendix II. Thi, type of mean line was selected because, at the design lift coefficient, it does not change the shape of the pressure distribution of the symmetrical section at zero lift. This mean line also imposes minimum induced velocities for a give~ design lift coefficient. c. Numbering System. The N.A.CA l-series wing sections are designated by a five-digit number as, for example, the N ACA 16-112 section. The first integer represents the series designation. The second integer represents the distance in tenths of the chord from the leading edge to the position of minimum pressure for the symmetrical section at zero lift. The first Dum­ ber following the dash indicates the amount of camber expressed in terms of the design lift coefficient in tenths, and the last t\VO numbers together indicate the thickness in per cent of the chord. The commonly used sections of this family have minimum pressure at 0.6 of the chord from the leading edge and are usually referred to as the Nl\.CA 16-series sections. d. Approximate Theoretical CharacleristiC8. The theoretical aerody­ namic characteristics of the NACA l-series wing sections may be obtained by application of the method of Sec. 4.5 to the data of Appendixes I and II by the same method as that described for the NACA four-digit wing sections (Sec. 6.4d). 6.8. KACA &-series Wmg SectioDS. Successive attempts to design wing sections by approximate theoretical methods led to families of wing sections designated NACA 2- to 5-series sections." Experience with these sections showed that none of the approximate methods tried was sufficiently accurate to show correctly the effect of changes in profile near the leading edge. Wind-tunnel and flight tests of these sections showed that extensive laminar boundary. layers could be maintained at comparatively large valdes of the Reynolds number if the wing surfaces were sufficiently fair and smooth. These tests also provided qualitative information on the effects of the magnitude of the favorable pressure gradient, leading-edge radius, and other shape variables. The data a~ showed that separation of the turbulent boundary layer over the rear of the section, especially with rough surfaces, limited the extent of laminar layer for which the lying sections should be designed. The wing sections of these early families generally showed relatively low maximum lift coefficients and, in many 120 THEORY OF WING SECTIONS cases were designed for a greater extent of laminar flow than is practical. It w~ learned that, although sections designed for an excessive extent of laminar flow gave extremely low drag coefficients near the design lift coefficient when smooth, the drag of such sections became unduly large when rough, particularly at lift coefficients higher than the design value. These families of lying sections are accordingly considered obsolete. The NACA 6-series basic thickness forms were derived by new and improved methods described in Sec. 3.8, in accordance with design criteria established with the objective of obtaining desirable drag, critical Mach number, and maximum-lift characteristics. a. Thick"neB8 Distributions. Data for NACA 6-series thickness dis­ tributions covering a wide range of thickness ratios and positions of minimum pressure are presented in Appendix I. These data are com­ parable with the similar data for wing sections of the NACA four-digit series (Sec. 6.4a) except that ordinates for intermediate thickness ratios may not be correctly obtained by scaling the tabulated ordinates proportional to the thickness ratio. This method of changing the ordinates by a factor w ill, however, produce shapes satisfactorily approximating members of the family if the change of thickness ratio is small. b. Mean Lines. The mean lines commonly used with the NAC_.\. 6-series wing sections produce a uniform chordwise loading from the leading edge to the point x/c = a, and a linearly decreasing load from this point to the trailing edge. Data for NACA mean lines with values of a equal to 0, 0.1, 0.2, 0.3, 0.4, 0.5,0.6, 0.7, 0.8, 0.9, and 1.0 are presented in Appendix II. The ordinates were computed from Eqs. (4.26) and (4.27). The data are presented for a design lift coefficient e" equal to unity. All tabulated values vary directly with the design lift coefficient. Corresponding data for similar mean lines with other design lift coefficients may accordingly be obtained simply by multiplying the tabulated values by the desired design lift coefficient. In order to camber NACA 6-series wing sections, mean lines are usually used having values of a equal to or greater than the distance from the leading edge to the location of minimum pressure for the selected thickness distributions at zero lift. For special purposes, load distributions other than those corresponding to the simple mean lines may be obtained by combining two or more types of mean line having positive or negative values of the design lift coefficient. c. NtUmbering System. The NACA 6-series wing sections are usually designated by a six-digit number together with a statement showing the type of mean line used. For example, in the designation NACA 65,3-218, a == 0.5, the 6 is the series designation. The 5 denotes the chordwise posi­ tion of minimum pressure in tenths of the chord behind the leading edge for the basic symmetrical section at zero lift. The 3 following the comma FAJIILIES OF n'IlVG SECTIONS 121 ives the range of lift coefficient in tenths above and below the design lift oefficient in which favorable pressure gradients exist on both surfaces. 'he 2 following the dash gives the design lift coefficient in tenths. The .LSt two digits indicate the thickness of the lying section in per cent of the horde The designation a = 0.5 shows the type of mean line used. When he mean-line designation is not given, it is understood that the uniform­ oad mean line (a = 1.0) has been used. When the mean line used is obtained by combining more than one mean ine, the design lift coefficient used in the designation is the algebraic sum )f the design lift coefficients of the mean lines used, and the mean lines are Jescribed in the statement following the number as in the following case: {a cs, NAC.A. 65 3-218 = 0.5 = 0.3 } , a = 1.0 ci, = - 0.1 Wing sections having a thickness distribution obtained by linearly increasing or decreasing the ordinates of one of the originally derived thick­ ness distributions are designated as in the following example: N.~CA 65(318)-217 a = 0.5 The significance of all the numbers except those in the parentheses is the same as before. The first number and the lust two numbers enclosed in the parentheses denote, respectively, the low-drag range and the thickness in per cent of the chord of the originally derived thickness distribution. The more recent XACA 6-series wing sections are derived as members of thickness families having a simple relationship between the conformal transformations for 'ling sections of different thickness ratios but having minimum pressure at the same chordwise position. These wing sections are distinguished from the earlier individually derived wing sections by writing the number indicating the Jew-drag range as a subscript, for example, N4~C.A. oor 218 a == 0.5 Ordinates for the basic thickness distributions designated by a subscript are slightly different from those for the corresponding individually derived thickness distributions. As before, if the ordinates of the basic thickness distributions arc changed by a factor, the low-drag range and thickness ratio of the original thickness distribution are enclosed in parentheses as Iollows: X.:\C..\ 65(311)-217 a = 0.5 For wing sections having a thickness ratio less than 12 per cent, the low-drag range is less than 0.1 and the subscript denoting this range is omitted from the designation, thus N .t\C..-\ 65-210 or 122 THEORY OF WING SECTIONS The latter designation indicates that the 11 per cent thick section was obtained by linearly scaling the ordinates of the 10 per cent thick sym­ metrical section. If the design lift coefficient in tenths or the thickness of the wing section in per cent of the chord are not whole integers, the numbers giving these quantities are usually enclosed in parentheses as in the following example: NACA 65(318)-(1.5)(16.5) a == 0.5 Some early experimental wing sections are designated by the insertion of the letter x immediately preceding the dash as in the designation 66,22:-015. Some modifications of the NACA 6-series sections are designated by replacing the dash by a capital letter, thus NACA 641A212 -In this case, the letter indicates both the modified thickness distribution and the type of mean line used to camber the section. Sections designated by the letter A are substantially straight on both surfaces from about 0.8c to the trailing edge. d. Approximate Theoretical Characteri8ticB. Approximate theoretical characteristics may be obtained by applying the methods of Sec. 4.5 to the data of Appendixes I and II as described for the NACA four-digit series sections in Sec. 6.4d. If t\VO or more of the simple mean lines are com­ bined to camber the desired section, data for the resulting mean line may be obtained by algebraic addition of the scaled values for the component mean lines. 8.9. NACA 7-series W-mg Sections. The NACA 7-series wing sections are characterized by a greater extent of possible laminar flow on the lower than on the upper surface. These sections permit low pitching-moment coefficients with moderately high design lift coefficients at the expense of some reduction of maximum lift and critical Mach number. The N ACA 7-series 'ling sections are designated by a number of the following type: NACA 747...\315 The first number 7 indicates the series number. The second number 4 indicates the extent over the upper surface, in tenths of the chord from the le&ding edge, of the region of favorable pressure gradient at the design lift coefficient. The third number 7 indicates the enent over the lower surface, in tenths of the chord from the leading edge, of the region of favorable pressure gradient at the design lift coefficient. The significance of the last group of three numbers is the same as for the NAC..~ 6-series wing sections. The letter A which follows the first three numbers is 8 seria1letter to distinguish different sections having parameters that would FAMILIES OF WING SEC7 1/ONS 123 correspond to the same numerical designation. For example, a second' section having the same extent of favorable pressure gradient over the upper and lower surfaces, the same design lift coefficient, and the same thickness ratio 88 the original lying section but having a different mean­ line combination or thickness distribution would have the serial letter B. Mean lines used for the NACA 7-series sections are obtained by combining t\VO or more of the previously described mean lines. The basic thickness distribution is given a designation similar to those of the final cambered wing sections. For example, the basic thickness distribution for the NACA 747A315 and 747A415 sections is given the designation NACA .747A015 even though minimum pressure occurs at O.4c on both the upper and lower surfaces at zero lift. Data for this thickness distribution are presented in Appendix I. The NACA 747A315 lying section is cambered with the following com­ bination of mean lines: a == 0.4 { a == 0.7 e'i == 0.763 } eli == - 0.463 The NACA 747..\415 wing section is cambered with the following com­ bination of mean lines: a == 0 4 . a == O~7 { a == 1.0 C" == 0.763 } ci, = - 0.463 c.; = 0.100 8.10. Special Combinations of Thickness and Camber. The methods presented for combining thickness distributions and mean lines are suffi­ ciently flexible to permit combining any thickness distribution, regardless of family, with any mean line or combination of mean lines. Approximate theoretical characteristics of such combinations may be obtained by ap­ plication of the method of Sec. 4.5 to the tabulated data. In this manner, it is possible to approximate a"'considerable variety of pressure distributions without deriving new lying sections. CHAPTER 7 EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS 7.1. Symbols. R U V Reynolds number, pVc/p. velocity just outside the boundary layer velocity of the free stream a mean-line designation; fraction of the chord from leading edge over which loading is uniform at the ideal angle of attack ac aerodynamic center c chord Cd section drag coefficient minimum section drag coefficient c/ local skin-friction coefficient c, section lift coefficient ci, design lift coefficient ClanR'I: maximum section lift coefficient section pitching-moment coefficient about the aerodynamic center c../4 . section pitching-moment coefficient about the quarter-chord point k height roughness t time ~ abscissa measured from the leading edge 11 ordinate measured from the chord line Go section angle of attack " angle of flap deflection p. viscosity of air p mass density of air c.r.m. e_. of 7.2. Introduction. The theories presented in Chaps. 3 to 5 permit reasonably accurate calculations to be made of certain characteristics of wing sections, but the simplifying assumptions made in the development of these theories limit their over-all applicability. For instance, the perfect­ fluid theories of Chaps. 3 and 4 permit the pressure distribution to be calcu­ lated," provided that the effects of the boundary-layer flo,v on the pressure distribution are small. Similarly the viscous theories of Chap. 5 permit the calculation of some of the boundary-layer conditions, provided that the pressure distribution is known, Obvious interactions between the viscous and potential flows are the relatively small effective changes of body shape caused by. the displacement thickness of the unseparated boundary layer and the seriously large changes caused by separation. N one of the wing characteristics can be calculated with confidence if the flow is separated 124 EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS 125 over an appreciable part of the surface. Under these circumstances, wing­ section characteristics used for design are obtained experimentally. Wind-tunnel investigations of wing characteristics were made before airplanes were successfully flown and still constitute an important phase of aerodynamic testing. Until recently wing characteristics were usually obtained from tests of models of finite aspect ratio, The development of lying theory led to the concept of wing-section characteristics that were derived from data obtained from tests of finite-aspect-ratio wings, These derived data were then used to predict the characteristics of wings of different plan forms. The systematic investigations in the N'AC..~ variable­ density wind tunnel46, 48, 49 were examples of this type of investigation. This method of testing is hampered by the difficulties of obtaining full-scale values of the Reynolds number and sufficiently low air-stream turbulence to duplicate flight conditions properly without excessive cost for equip­ ment and models. Other difficulties were experienced in properly correcting the data for the support tares and interference effects and in deriving the section characteristics from tests of models necessarily having varied span­ load distributions and tip effects.Go In order to avoid some of these difficulties and to permit testing of models that are large relative to the size of the wind tunnel, two-dimensional testing equipment was built by the X.t\C.i\. The X.:\(~..\ two-dimensional low-turbulence pressure tunneP33 provides facilities for testing wing sections in two-dimensional flowat large Reynolds numbers in an air stream of very )O\V turbulence, approaching that of the atmosphere. The wing-section data presented here were obtained from tests in this tunnel. This tunne}1D has a test section 3 feet wide and 7~~ feet high and is capable of operation at pressures up to 10 atmospheres. The usual models are of 2-foot chord and completely span the 3-foot width of the test section. The lift is measured by integration of pressures representing the reaction on the floor and ceiling of the tunnel. The drag is obtained from wake­ survey measurements, and the pitching moments are measured directly by a balance. "~ing-section characteristics can be obtained from such measurements with a high degree of accuracy. The usual test.s were made over a range of Reynolds numbers from 3 t.o 9 million and at Mach numbers less than about 0.17.· This range of Reynolds numbers covers the range where large-scale effects are usually experienced between the usual low-seale test data and the large-scale flight range. Individual tests have been made to provide some data at lower Reynolds numbers applicable to small personal airplanes and at much larger Reynolds numbers to indicate trends for very large airplanes­ The tunnel turbulence level is very low, of the order of a few hundredths of 1 per cent; and, although it is not definitely known that the remaining turbulence is negligible, the tunnel results appear to correspond closely to 126 THEORY OF WING SECTIONS eo 3 .6 A .. ~ Q ~.) Serles ~.) -V­ e ~ ~ 0 - 0 o ... I'~ I) • • -a r.. .. ~ 2 20 8 16 12 Airfoil thIckness. percent or cbord 0 I) 2 (a) NACA tour- and t l ve-d1s1t series. -6 2 •• ""o • ~ ~ o ••o - -2 ~ .... .I'to. 0 - - - ~l eo o 0.2 > V A. •c :: -c v 6 0.1. " 0.6 to :: ..• ~ • 2 0 II 8 12 16 20 24 Airroil th1ckness, percent or cnord (b) FIG. NACA 6~- aeries. 56. Measured section angles of aero lift for a number of N..t\CA 127 EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS -6 -- Cl ~ -2 /\. o ( .. s::: o ~ () A. .... •., - -. "0 t) ~ • :I 20 1& 8 12 16 20 .Alrtoll thickness. percent of chord (c) NACA 6J~- serIes. eo • "; -6 '-4 rot -4 0 s.. •" '" ....• --JI ~ -4 -2 *'If.~ 5 0 .p 4~ ~ t. 110 ...c A A 0 0 0 -. 4-) A. v V -. " v 0 •• 't:J • ~ •• :I II. 8 12 16 Alrtoil thickne.s. percent (d) IIACA o 20 or chord 65- eeriea. airfoil sections of various thicknesses and cambers. R, 6 X 10'.. 0.2 0.L. " 0.6 A - 0" o ~ 1 a 0 a 0.1 EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS -6 f • .~ ~ ~ It CI I ....o Q .!f -2 taO c C e (> ~ tI .....,o A.. o 0_ ~ ".. - •• - .- ..., • ~ •I 20 1& 8 12 16 20 .Airtoll thlckaess. percent of chord (cJ NACA 6~- serles. eo • ..,.. .... ~ .. -6 ro4 0 •. s. --.ll ~ -4 I;. ~ 0 ...• -2 110 ~4·~ 5 .. c 0 .p 0 A. .. ~ ..:tt n 0 ...... 0 A.. .,;,. V - '" ~ •• i ~ i 20 Ja. 8 12 16 A1rfoil tblckn•••• percent (d) BACA or 20 chord 65- eerie•• airfoil eectiooa or various thicknesees and cambers. R, 6 X lOS. 0 a 0.1 0.2 6 O.L. " 0.6 o A. 24 127 THEORY OF WING SECTIONS 128 those obtained in flight. Application of these wing-section data to the prediction of the characteristics of wings of finite span depends on the adequacy of three-dimensional \V~g theory. These data are not applicable at high speeds where compressibility effects become important. 7.3. Standard Aerodynamic Characteristics. The resultant force on a "ring section can be specified by t"90 components of force perpendicular and parallel to the air stream (the lift and drag, respectively) and by a moment in the plane of these t\VO forces (the pitching moment). These forces are functions of the angle of attack of the section. The standard method of presenting the characteristics of 'ling Sections is by means of plots of the lift, drag, and moment coefficients against angle of attack or, alternately, plots of angle of attack, drag, and moment coefficients against lift coefficient. Plots of wing-section characteristics are presented in Appendix IV for a wide range of shape parameters. On the left-hand side of each plot, the lift coefficient and the moment coefficient about the quarter-chord point are plotted against the angle of attack. On the right-hand side of each plot the drag coefficient and moment coefficient about the aerodynamic center are plotted against the lift coefficient. In most cases, the data indicated in the following table are presented. Surface condition Charneteristie Split flap deflection degrees Reynolds number, millions Left-hand side Lift tift Lift . . . I4Iift..•......... Moment . ~Io~ent . Smooth Rough­ Smooth Rough­ Smooth Smooth 0 0 60 60 0 60 3,6,9 6 6 6 3,6,9 6 0 0 0 3,6,9 6 3,6,9 Right-hand side Drag Drag Moment . . . Smooth Rough" Smooth *O.011-ineh amin carborundum lipread tbinJy to cover 5 to 10 per cent of the area from the leadinc ed«e to 0.0& alung boUa surfaces of a IleCtion wit.h a chOld of 24 inchee. 7.4:. Lift Characteristics. 4. Angle of Zero lift. As indicated in Chap. 4, the angle of zero lift of a wing section is largely determined by the camber. The theory of wing sections provides a means for computing the angle of zero lift from the mean-line data presented in Appendix II. The agreement between the calculated and the experimental angles of zero lift EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS 129 depends on the type of mean line used. Comparison of the theoretical data given in Appendix II 'lith the experimental data of Appendix IV shows that the agreement is good except for the uniform-load type (a = 1) of mean line. The angles of zero lift for this type of mean line are generally closer to 0 degrees than predicted. The experimental values of the angles of zero lift for a number of NACA four.. and five-digit and NACA C)-series wing sections are presented in Fig. 56. The thickness ratio of the wing section appears to have little effect on the angle of zero lift regardless of the type of thickness distribu­ tID -3_ -6 ~ '-t ..c .... o ; II -4 '-'4 o ol(.) MACA FIG. -. C) ~ 8 12 16 Alrton thicknes.. percent (e) Q -. u e o A.. v 6 or v 20 chord 66- .eries. 56. (Cond1ldal) tion or camber. For the NACA four-digit series wing sections, the angles of zero lift are approximately 0.93 of the value given by the theory of thin wing sections. For the NACA 23o-serics wing sections, this factor is approximately 1.08; and for the X.\Cl\ f.)-~(~r ies sections with the uniform­ load type of mean line, this factor is approximately 0.74. b. L~Jt-curve Slope. Lift-curve slopes fOI" a number of NACA four- and five-digit series and KACA fJ-8Cri~ wing ~~·t.jons are plotted against thick­ ness ratio in Fig. 57. These values of the lift-curve slope were measured for a Reynolds number of 6 million at vulues of the lift coefficient approxi­ mately equal to the design lift coefficient of the wing sections. This lift coefficient is approximately in the center of the low-drag range for the NACA f>-Series wing sections. In the range of thickness-ratios from (i to 10 per cent, the NACA four­ and five-digit series and the XAC.-\ 64-series wing sections have values of the lift-curve slope very close to the value given by the theory of thin wing sections (2... per radian, or 0.110 per degree). Variation of Reynolds Dum­ THEORY OF WING SECTIONS 130 P1a~d .~bola 1ndlcat~ roUSb condltl~n .. •, • •ca. •t .12 .- .. ~ .10 ...• r--' . - ... "'8aIootb -- --:t t-- -.iii -. e .06 A "I 0 c- -~ ~1~ ~~ - '\ --. -.: ~ Bogp- 4 1g1t) I 230 (5 4181t) " 8 10 12 14 16 18 Alrtoll thickneas. percent or chore hal -33 at(4 e , --- 00' EI o :J .. ~- ~ E .~ :s ...• t-~ 1 - Berl•• 22 20 NaCA tour- and f'lve-d1s1t aerie•• • .lIa. • r 8lllooth_ .12 ~ J ·'0 - III .Iw-- ~I' -- , -- ·.!t- - I I --~--4 -~ ~-, \..acuab ~O °1.1 eO 00.2 40.~ .08 vo. , , .a 16 18 chord· 20 22 O£ .ACIl " - •• rlea. I t .12 ..I. ~.) ~r ~~ 1'f - -- .... ~ -~ r- - ...•t • .10 °1.1 C 1 .08 e O.~ A o. '9 o. 5 I 12 14 Alr1"o11 thlcknes.. percen1i 10 (b) .. • I I 8 e e , I 8 ~ ........J.~ .... ---, r - 1\ I - T RouP ., aU 0 0.1 I I 10 12 14 Alrtoll. thicknesa. percerat (0) .. ~ vre:-~- - or 16 18 20 22 chord lACA 6l,- .eriea. FrG. 51. Variation of lift-eurve slope 1Ifith arfoil thickness ratio and camber for a 131 EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS na...d "1..0 31;; • ••­ .... -I2t • s. De • ...8'• ,. 4~ J 4t - ~J 1'--1 tt-- po-- 10 1-- -., I-I" 0 08 6 V :I · 064 , • 8' 1"-_ -~ 0 1 10 12 AlrtoU tb1c1cne••• percent or chord (d J MACA 65-•• n ••• 20 .12 22 'r 8IIIooth 11 A 4~ o.~ o. o. I I 8 .....J., \ Roup °'1 E> I· .• ., ~J ...• i• r .ootb I .; • -Jllbo11 indicate rouF condition 14 ~.) ~.) ,~ ... ~ .10 J~ ~ ~.)- t-_ -=i. I rf • .s cl I t E) e A l 0 0.2 • /, - e r'\, - -' to) ~ r--Roup .... ~~ 0.4 I I 6 8 10 Alrtoll 12 <e) number of N.~C.:'... 20 thiem.... percent or chord MACA 66- aer1e•• airfoil sections in both thp smooth and rougb conditions. R, 6 X lOS. 22 THEORY OF WING SECTIONS 132 ber between 3 and 9 million and variation of camber up to O.04c appear to have no systematic effect on the lift-curve slope. The thickness distribu­ tion appears to be the primary variable. For the NACA four- and five­ digit series wing sections, the lift-curve slope decreases with increase of thickness ratio. For the NACA 6-series wing sections, however, the lr---- -- 1'\ 2.8 V) /4-Series (4-Digi1) ~ r--- 2.4 (4-Digit) ) r-, I I I OO-!:aies - - (4-0igil) " .- ~ ~~ I4-Series (4- Digif).~ ~ ~ ',.. r- ..... ~ .. ~ I 230-5eries '/~ -: 7 If ~~ ~ (5-DigiIJ V I I 44-5eries (4-DigifJ --r ~ ~~ ~~ l v-: QoIo." ~ ~ ~/ V~ -o 4r Airfoil wiIh split f/cJJ I , ~ ~ !i4-DigiIJ i'X r ~ r\) r ~ L/ ,r-- 00-5eries~ rl~~ " 44-5eries 24-Series (4-Digif) - I ~ ~ ~j / PkJin >airfoil / ~ ~~ ---: - ~) --fbJgh --5rnodh I J I Symbols with fJogs correspond 10 sinllbfed splil fbp deflected 60- .4 I 00 4 8 I I /2 I I I 16 Airfoillhick~pen;:enI 20 24 of chotrI (a) N ACA four- and five-digit series. 58. Variation of muimum section lilt coefficient with airfoil thickness ratio and camber for .veral NACA airfoil sections with and without simulated split ftaps and standard rough­ ness. B, 6 X loa. FIG. lift-curve slope increases with increase of thickness ratio and with forward movement of the position of minimum pressure on the basic thickness form. The effect of thickness ratio is comparatively small for the NAC.\ 66-series sections. The thick-wing-section theory (Chap. 3) shows that the slope of the lift curve should increase with increasing thickness ratio in .he absence of viscous effects. For wing sections with arbitrary modifica­ tions of shape near the trailing edge, the lift-curve slope appears to decrease with increasing trailing-edge angle. Some KACA6-series wing sections show jogs in the lift curve at the end . EXPERl}.{ENTAL CHARACTERISTICS OF WING SECTIONS 133 of the low-drag range, especially at lo,v Reynolds numbers. This jog be­ comes more pronounced with increase of camber or thickness ratio and with rearward movement of the position of minimum pressure on the basic thickness form. This jog decreases rapidly in severity with increasing Reynolds number, becomes merely a change of lift-curve slope, and is practically nonexistent at a Reynolds number of 9 million for most lying I - ...i "6 - - _~o.' 2.8 - .1,. -'" 0 i , '"'\ -. ..." ~-- ~ L-...... ~ '\ £ // "'0 . 2 - - o I °Il r-- ~'i ~ ,...:3 ... )\, -- :~ ~, ~- ~):;J' \ Y ~ M ~:~ r-, 7" Cl 1 ~ k:-: -_ .I' ~ r-, \.. 0.'J,. ,"F='=" - e ~" ~ I ,. / V~·V -&.-o ~ ...... JII 1/ -- :i'~ .4~ _J V,.) ~ ~ ~"O ~~ .... L:..- ~ ~ ft .2 ~ "00 ~ ......,s:: o o : C&l ~ 1.2 l - - ~ ... 6 .2~ e e L L.. iIl............. ~ ~-- 90.6- e ~~ ~ .~ ..0- ---- ... ~,,~ ..... ~ ,'. ~- ~ ·f ~-\~ ~;; Plain a1rt'o11 -6 ~ \ \\ ~ '"' \\ ~'( -vo.6-4 C~l 8IDOotb \ \.- -6 ~ -0 .2 00 I ltouab I--SJlDbola with 1"l-p correepood ~o 8!'-w..ated- ap11t 1'1ap :Serlect~d 600 o o A1rt'oll W1tb ep11t. ~J.ap ~ 8 12 16 A1rtoll th1ckDe... perceD t or (6) NAC.A 63-series. }"'IG. 58. (Continued) 20 Cbor4 sections that would he considered for practical application. This jog may he a consideration in the selection of wing sections for small low-speed airplanes. An analysis of the flow conditions leading to this jog is pre­ sen ted ill reference 134. The values of the Iift-curve slopes presented are for steady conditions and do not necessarily correspond to the slopes obtained in transient con­ ditions when the boundary layer has insufficient time to develop fully at each lift coefficient. Some experimental resultglOG indicate that variations of the steady value of the lift-curve slope do not result in similar variations of the gust loading, THEORY OF WING SECTIONS 134 c. Mazimum Lift. The variation of maximum lift coefficient with thickness ratio at a Reynolds number of 6 million is shown in Fig, 58 for a considerable number of NACA wing sections. The sections for which data are presented in this figure have a range of thickness ratios from 6 to 24 per cent and cambers up to 4 per cent of the chord. From the data for the NACA four- and five-digit wing sections (Fig. 58a), it appears that the '.2 I °Il 2.8 Ie E ~ .. 0 .... ~~ -.",,~ .~",. 2.0 S 0 _~ 1.6 .... ...... 0 •• i - °Il 1.2 o~ v f' '70.6- ..../ ~ 06=3 - g ~,;~ ('., ~ ~ ::::- ~ ... ..J~• ~ ...~ :, or, , ~ o>~ .,.,~ -1 ~ , " ~ ,JI ,p .......i ~ .4- -- -, ~ 0 .~ v ~ .2 "a .1' t l O - ~~.~ lS' ~.,~ h: :.II ..• 2.4­ 0 -- ~ .. 0.6 .4- -"'" 01 "Y- ~ - ~'V .~ ,," 00---, :~ ... 1~,<:~ A <> ~-- .2- o .1 I--- ~ .8 ~-- Ji>- iI_~ or~ -, <, =­ ~ "9 0.6 .. .4 0 1:1 .2 .1 '00 ~ ............. h : ... -... o~ ~ ...... ',,:.~ ~ ..&- ~ ~. - Airtoll wltn split flap '°1 1 PlAln airfoil -~ ~~: "-oE) ''0 \\ -:'0 \\-90.6.14 °Il -<> ~ ~ -s ~A .4 "ooth I -----R~ 1 ~. • 00 with tlaS8 correspond to ted .pllt flap detlected 600 " .2 .1 -00 8 12 k 16 20 A1rto11 thiCknes., percent or chord (c) NACA 64-eeriea. FIG. 58. (Continued) maximum lift coefficients are the greatest for a thickness ratio of 12 per cent. In general, the rate of change of maximum lift coefficient with thickness ratio appears to be greater for sections having a thickness ratio less than 12 per cent than for the thicker sections. The data for the KACA 6-seri~ sections (Figs. 58b to d) show a rapid increase of maximum lift coefficient with increasing thickness ratio for thickness ratios less than 12 per cent. The optimum thickness ratio for maximum lift coefficient increases with rearward movement of the position of minimum pressure and decreases with increase of camber. For wing sections having thickness ratios of 6 per cent and for wing sections having thickness ratios of 18 or 21 per cent, the maximum lift 2.8.--......-.....-----...--..,.-.....,..-...---~ .......- -........... i 2.4t--t--+---+- 0' i­ ....c:: "'•oo" Airfoll with spllt.flap 2.0 ..---+--f--I---4-~-+-~~ ........~~~~~ '-4 .... 1.6 ~ rot g :: o •• J Plain airfoil 1.2 .8 i t--4---t-~--+-~-+------t-~-.-_Smooth - - + - - + - - - + - - t __ _ flougl1 .4 1---+--+--......- SJlDbola with tlasa correspond to 0 .1aIula~e4 spllt rlap d.flected ·60 I&. 8 12 J.6 20 Airtoil thlCkne.-;, percent or chord (tl) NACA 65-series. FIG. 58. (Continued) 2.8 Cl l no.q- -'" ~ . ~ L.,./ ~ 0" ~ 1.6­ 01,1 O.~..-" fi 1.2 .. ...o OO~ V •• 1IIl'-­ ~ ~ ...a­ .8 .. J;1" .,.,,-" ~ oy-­ ~_-4 .)-- , ~ ~""'" ~ ....~J .............. Alrtol1. wltb apllt flap ~11 ~ -, 'a 0.4 <; -~ .2 ...........-. -l:) 0 ~'\. -o -0­ ~ _J:>­ --.. ..J!\ -,\_0lil '\1: -G ~ ~ .)00­ .).-- --<7' ~, - P1a1n ail'1'o!l -e ~ '\~ ~A Cl l 0 ..... '""'0 .2_ ~ ,-G 0 I .~ »: ~ .,...- () .2-..J,V j 4 .J---. .~ ~ - ­ -V ....~ ) ''T'' ~ ~ .~ ~ .~ v:a ."~~ o~- V) .V "J¥'(~ --" 2.0 ~ ~ 8o ~ ~ .­ c: ~ .2­ bO ­ ~U Slooth - - - - - Roup i-­ I- °0 SJIDbola witb 1"lap correspond to a1aulatecl spIlt t"lap 4.nectecl 60° 8 12 16 20 A1rf'oll th1ckD.... perc.ati or chord (e) NACA 66-eeriea. FIG. 58. (C01&eluded) J&. 135 ~ THEORY OF WING SECTIONS 136 coefficients do not appear to be very much affected by the position of minimum pressure on the basic thickness form. The maximum lift coeffi­ cients of sections of intermediate thickness, however, decrease with rear­ ward movement of the position of minimum pressure. The maximum lift coefficients for the NACA 64-series wing sections cambered for "a design lift coefficient of 0.4 are slightly higher than those for the NACA 44-series 2.0 ..I 1.' ...•.. ...'"' 0 a C) I0 1.2 ... ~ ... .8 ...I .4­ Ii ~ ReJDo1dsD~ '­ Q GJ ~ ..... ~ 1""""' -- r--J ~ -rr .p 6X~~ ...... 7 ,Xl 0 •• ••M o o ., 2Jpe of o-.ber. .8 a 59. Variation or maximum lift coefficient with type of camber lor some NACA 65a-418 airfoil sections from testa in the Langley two-dilnensional low-turbulcnce pressure tunnel. FIG. sections. The NACA 230-series sections, however, have somewhat higher maximum lift coefficients. The maximum lift coefficients of moderately cambered sections increase with increasing camber (Fig. 58). For wing sections of about 18 per cent thickness, the rate of increase of maximum lift coefficient with camber is largest for small cambers. The effect of camber in increasing the section maximum lift coefficient becomes progressively less as the thickness ratio increases above 12 per cent. The variation of maximum lift with type of camber is shown in Fig. 59 for one condition. No systematic data are available for mean lines with values of a less than 0.5. It should be noted, however, that lying sections such as the NACA 230-series with the maximum camber far forward show large increments of maximum lift as compared with symmetrical sections. Wipg sections with the maximum camber far 137 EXPERIAIENTAL CHARACTERISTICS OF lVING SECTIONS forward and with normal thickness ratios stall from the leading edge with large sudden losses of lift. A more desirable gradual stall is obtained when 2.0 .. '----. -.... -a... <, -...... 1.6 ~ - 1.2 ...i .8 ..., .4 . -~ ~R ~ ~ --, ~ ~ t=rQB l;Joo...; ~-!-- ~~ e ~ A ~ k?' "1~ 0 MACA 24-serlea (4 digit) c ~ 0 w4 .... .... 8 0 1.6 ~ ~' .p R G 9.0 x 106 ir ~ .... 1.2 s:: ....0 •• .8 i 1.6 V .p 0 J JlACA e 6.0 Q V '.0 at. Stendard rougbne,s 14-•• rl•• (4 6 )( lOb dl&1t) R G 9.0 x s JV 1.2 I.{/ .r 1&. --A a 6.0 e '.0 A StaDdard l'OUSbDeaa MACA CO-aeries 8 12 16 Alrto1l thickness. percent (a) 106 (4- d1a1t) 6 x 106 20 or chord MACA tour-dlg1 t aeries. 60. Variation of maximum section lift coefficient wid. airfoil thickness ratio at several Reynolds numbers for a number of NACA airfoil sections of different cambers. FlO. the location of maximum camber is farther back, as with the NACA 24-, 44-, and 6-series sections with normal types of camber. The variations of maximum lift coefficient with Reynolds number (scale effect) of a number of XACA wing sections are presented in Fig. 60 for a range of Reynolds numbers from 3 to 9 million. The scale effect for the NACA 24-, 44-, and 23D-series wing sections (Figs. 60a and b) having THEORY OF WING SECTIONS 138 2.0 - ---" ... ~ P) ~ ~ l~ ~ 1.6 ~ 100­ Jl I- e Z_O B I .. 1.2 ~ e I- 4. 0 ..•­ 1! ~. .0 V »c 1.06-/ / ~ '.0 r~.a , x i06­ V ~ Ill"'" ~~ ataDdard ~ ~ ~ ............... .. ~ ~~ .8 ~ ~. ~~ ~ r--...... I ' r-, ~ ~ '"8 \.e ... 0 IfACA 2,O-a.rl•• (5 digit, .4 ... ~ ...s.. 2.0 0 •• I R 1.6 .9.0 x 10' B 6.0 ------. G '.0 ~___JI~-+-M..-~:aiII'~-+-I~~ . . .--I AStan4ard ro\IIbDell8 -t-~t---+--~~--I-~--I----I 6 )( 106 1.2 XACA 4 ~-.erl.8·(4 dlglt) 12 16 20 Airtoil thickness. percent of chord (b) 8 RACA rour- and rl".-dlg1~ eerie•• FIG. 60. (Continued) 24 EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS S,mbols with tlase correspond to 2.0 0'1 =0.4 aad 0.6 I'~ ~ r ::::t~ R ~ r-; e-, ~ ~~ r""" ......... = 0.6 ~ ~~ - c~l 139 ~ -....... ~ ~ ~ ............ .... 106 v 9.0 )( A Standard rouatmeaa e 0.0 e ~ '.0 6 )C 10 6 ~ .8 .." ..• o .. 2.0.---.--......~.-.......-..-......- -.--..-....- ......- ...-..... c .p ~ :: • 9.0 )( 106 e 6.0 0 8 .p ::s e roushn••• 6 )( 106 c 4J ~ •g '.0 4 Stand.ard ~ 1.2 ...o R 1.61--+~-+-~t---4--4.,...g~~:&-.d---4--+--~-I .8 ...... ......__a.--....._ ......-....I_....a- ....... _~.... ! a 1.6 ...-....-........~......- -..-..-.............,....-............-..--......- -. :c R e 9.0 x 106 6.0 1.2.I-----r-t--;--n~-r_;_-=-t_;~'__f::;~'a==:t() 0 ;.0 ~+---t-~ .----+--t----1~~~-+----.\r'_-+--+_-___+_-~¥.ir_+_____1 .8 ..-........_ ......--. o --------I-""""'--..... 12 8 10 --a.-....... -~-01 20 Alrtol1 thickneas, percent or chord (e) NACA 63-aerles. Flo. 60. (Continued) Standard rougbnea 8 6 x 106 THEORY OF WING SECTIONS 140 SJlDbola nth flag e~reapond to et.i 2.0 0Il =0.4 an4 0.6 =0.6 R ~-..............-....o 9.0 1-'--'86.0 x 106 1. 6 t---t""--t--t--+--t---+--~~"""'~rIl'-l.'---I-~ 0 3.0 A t----t---t--+--+--+--+---+-~~.a.----a.--'.....I-..'--I ~ ...e Standard r~e'8 6 x 100 1.6 It ~ 9.0 ,. 1.2 t---t"--t--'t"-~~-+--~.....+-a. l?J 6.0 .....-...a-~;::a.-+:a~(:) 3.0 0 x 106 ~ c . -......- ~ t..e ~ 4D 0 ...A Standard t--t--t-~~~r:::;;....-+--+--+--4---I-~--+---I S rougbne, a 6 x 100 .8 0 ~ ~ .... c 1.6 ... R 0 1'----+--......-.---..-.... <2> 9.0 .p 0 •• -,--~-+--4---+El 1.2 )C 10 6 03.0 Standard I .-a-~t--.......- - I - - - - L 6 rougbne,• 6 x 100 .-t i 6.0 .8 1.6 R e 9.0 x 106 fl 6.0 1.2 t---T--r--t---+-,.f"--+-~-4-~o-~~-I---I 1-.....;;;.--.--.... 0 3.0 A t---t--t--'t--i'#--f--r,,,,+--+--01--4"':::'-+...--1 .8 Standard roughne,s 6 x 111' 0 4 8 12 J.6 hlrrol1 thickness. percent (dl NACA FIG. 20 or chord 64-serles. 60. (Continued) 24 EXPERIMENTAL CHARACTERISTICS OF WI}/G SECTIONS 8J11lbola with flap correspond to 01 1 = 0.6 4 .p !...o t •oo .p ....""... C ...o .p o •• J 1.6 r---,...-r--~r---"Ir---~.----...---~~"-' U =­ .8 "-......_~-.liil- ......_..a..---a._...a.---IIo.--...a._..&........._ - I o 4 8 u ~ ~ 6 Airtoil thlckn•• s. percent at Chord (e) MACA 6S-8erl•• ~ FIG.60 (Continued) 141 THEORY OF WING 8BO'l'ION8 142 ...I o 1.2 ""-......... --a_-a..._"--....... _.-.....~_....... _..a.___._.. , J 1.6 a 9.0 x 106 '.0 . rougme,_ 8 &.0 1.2 .--r---,r-I---,r--o"'L..-;---t-;---,~::t::::::l0 t---+--+--t--­ .8.0 6. Standard =EJ-+-+---t---+-~~c:;;p-~~-~ 6 x 100 4 8 12 16 Air£oU thickne.s. percent (r) IIACA o~ 66- •• rl••• FIG. 60. (Concluded) 20 chord 24 EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS 143 thickness ratios from 12 to 24 per cent is uniformly favorable and nearly independent of the thickness ratio. Increasing the Reynolds number from 3 to 9 million results in an increase of the maximum lift coefficient of approximately 0.15 to 0.20. The scale effect on the NACA 00- and 14­ series sections having thickness ratios of 0.12c and less is very small. The scale-effect data for the NACA 6-series wing sections (Figs. 60c to e) do not show an entirely systematic variation. In general, the scale effect is favorable for these sections. For the NACA 64-series wing sections, the increase of maximum lift coefficient with increase of Reynolds number is generally small for thickness ratios less than 12 per cent, but it is some­ what larger for the thicker sections. The character of the scale effect for the NACA 65- and 66-series wing sections is similar to that for the NACA 64-series, but the trends are not so well defined. The scale effect for the NACA 6-series wing sections cambered for a design lift coefficient of 0.4 or 0.6 is greater than that for these sections with less camber. The data of Fig. 61 show that the maximum lift coefficient for the NACA 63(420)-422 section continues to increase with Reynolds number up to values of at least 26 million. The values of the maximum lift coefficient presented are for steady con­ ditions. It is known that the maximum lift coefficient increases with the rate of change of angle of attack. tOto The significant par~eteris (daoldt) (cjV). Even such low rates of change of angle of attack as those encountered in landing flares produce increases of the maximum lift coefficient. d. Effect of Surface Condition on Lift Characteri8tics. It has long been known that surface roughness, especially near the leading edge, has large effects on the characteristics of lying sections. The maximum lift coeffi­ cient, in particular, is sensitive to leading-edge roughness. The effect on maximum lift coefficient of various degrees of roughness applied to the leading edge of the NACA 63(420)-422 wing section is shown in Fig. 62. The maximum lift coefficient decreases progressively with increasing roughness. For a given surface condition at the leading edge, the max­ imum lift coefficient increases slowly with increasing Reynolds number (Fig. 63). Figure 64 shows that roughness strips located more than about O.2Oc from the leading edge have little effect on the maximum lift coefficient or lift-curve slope. It is desirable to determine the relative effects of leading-edge rough­ ness on various wing sections. In order to make a systematic investigation of this sort, it is necessary to select a standard form of roughness. The standard leading-edge roughness selected by the NACA for 24-inch chord models' consisted of 0.01I-inch carborundum grains applied to the surface of the model at the leading edge over a surface length of 0.0& measured from the leading edge on both surfaces. The grains were thinly spread to cover 5 to 10 per cent of the area. This standard roughness is considerably THEORY OF WING SECTIONS 144 2.8 2.0 1.6 ... ... A .,. 1.~ .:s::: ~ ... .8 C) ; "­ to. ~ 0 () 1'4 ~ 0 II J' I ri C ~..r.,.. r ..."'" . ..,.­ .JP 0 ... -- - ~~ ~ ~"E) -­ PJ '-' -JJ 0 ~ 0 • co j - .4 v j - .8 j J:t -1.2 ~~ -~ :A ~ Il G 6.0 lt0 x 106 a • 0 A 20.0 • 26.0 If r-fF • -1.6 ~ -1' 0 , J6 24­ or attack, Clo ' des FIG. 61. Lift and drag e~acteristiC8 of the NACA 63(420)-422 airfoil at high Reynolds number; TDT tests 228 and 255. Section -8 anel. EXPERIMENTAL CHARACTERISTIC8 OF WING SECTION8 ·032 .02 a - -: -""""" 0 .2 - r--. ---... <, ........... -­ --­ ., - ...-­ -4 zJ" ............... ............ ~ .8 1.0 c .02." "d o R , I 6.0 x 106 0 B 10.0 .01 14.0 V 26.0 6 ~ :~ ~ , e II J '~-.6 20.0 l~j ~ r-, ~ ~~ f!A V 4'1 ~ ~ .~ ...... r: ~~ , v o -1.2 -.8 -.1,. o &eo'lon lUt ooettlclent. FIG. 61. (Concluded) .8 cl 145 THEORY OF WING SECTIONS 146 I I 1 1 I B O.0Q4-1ftOh-sra1n - X roUIbD... ..1.2 - roqJme.. Oft L.S. ~~ O.ooe-1aOh~ • i• .:= : ::. ... - L. L ~~ • O.Oll-hoh-ara b ro~••• OIl .8 ~ + lb.l1ao ~ ~ -..­ CL.. ~~ ~ID r L.L OIl 1,••• ~r elllootil ~ , ,1 .~ j ...•=.. tI OIl ('-­ ~ II 0 / ­.... -) -.8 r Airfoil. DCA ~, c 2' Ie 10' " In. T.ata. m!' 2'1) an4 262 III Chord I I I I -1.' ~ "(WO)~a -al. 0 8 16 01 4e. FIG. 82. Lift characteriatica of a N ACA 63(420)-422 airfoll with varioua at the leadiq edae. -16 -8 I I I ""loa .'taok. .0.. I • dtcreee of roughnea 2.0 ...-01 ~ ~ ~ " ,.. I ... . ~ \. " ----- ­ .. . '0 \\ ...... ~ .~ 1-.. i-a. !~ ~•• M'!It'=•• .8 '-­ -6 .p -.om o o o 8 12 16 20 ~14a""r •• FIG. 63. Effects of Reynolds number on maximum section lift coefIicient 63(420)-422 airfoil with roughened and smooth leading edge. 28 CI of the NACA EXPERIMENTAL CHARACTERISTIC8 OF WING SECTIONS - .. 1.2 o ~ . 1; .8 :: : 44 44 I - - I I I o.~ G IaooUl . a ... ~ rw~ 9Ill I' , 1 J( ~ ...o ......-:.;; ~, ... ...I ~ rc IV LeE. + BoqbM.. atrip OIl .... o r at 0.200 • BoqbM.. atnp at ? I I 1(1Iou. . . . .vip 147 0 I / -.'­ r ~o111 a. 2' x1,p DCA ,,(!a20 J-Ja22 j •• 8 -1.2 ~~ CboN. ~ 111• <r; 255 ~.tl ) 'if ~ ~ 0 8 ""loa aq1e of .tWk. ~ .0. • del M. Lift characteriatice of a NACA 63(420)-422 airfoil with O.Oll-inch-grain rouahnesa at various ehordwiae locatioDL FIG. I .,j .•­ DCA a DCA ~ DCA ~ c .012 .a, ..,r.• .0OS ........ ., tI .., & .~ J "2-215 'It.2-215 '5z-215 4 IlACA "~5 9 IACA 1~ ~ •• J e 67.1-215 r----, >---­ ~ . ~ ..J c a .1 ., Po~1tlon .~ ·5 or 111n1mUlll pressure, ......... ., .7 x/c FIG. 65. Variation of minimum drag coefficient with position of minimum pressure for some NACA 6-seriea airfoils of the same camber and thickness. R, 6 X I ()I. 148 THEORY OF WING SECTIONS more severe than that caused by usual manufacturing irregularities or deterioration in service, but it is considerably less severe than that likely to be encountered in service as a result of accumulation of ice, mud, or damage in military combat. Maximum-lift-coefficient data at a Reynolds number of 6 million for a large number of NACA wing sections are presented in Figs. 58 and 59. The variation of maximum lift coefficient with thickness for the NACA four- and five-digit series wing sections with standard roughness shows the same trends as those for the smooth sections except that the values are considerably reduced for all these sections other than the NACA OO-series of 6 per cent thickness. The values of the maximum lift coefficient for these rough sections with thickness ratios greater than 12 per-cent are sub­ stantially the same for a given thickness. Much less variation of maximum lift coefficient with thickness ratio is shown by the NACA 6-series wing sections in the rough condition than in the smooth condition. The varia­ tion of maximum lift coefficient with camber, however, is about the same for the wing sections with standard roughness as for the smooth sections. The maximum lift coefficients of rough wing sections decrease with rear­ ward movement of the position of minimum pressure; and, as in the smooth condition, the optimum thickness ratio for maximum lift increases with rearward movement of the position of minimum pressure. The NACA 64-series wing sections cambered for a design lift coefficient of 0.4 have maximum lift coefficients consistently higher than the NACA 24-, 44-, and 23o-series sections of comparable thickness when rough, with the exception of the NACA 4412. For normal wing sections, the angle of zero lift is practically unaffected by the standard leading-edge roughness. The results presented in Ap­ pendix IV show that the effect of roughness is to decrease the lift-curve slope for wing sections having thickness ratios of 18 per cent or more. The effect increases with increase of thickness ratio. For lying sections less than 18 per cent thick, tile effect of roughness on the lift-curve slope is relatively small. '1.6. Drag Characteristics. a. Minimum Drag of Smooth: Wing Sections. The value of the minimum drag coefficient for smooth lying sections is mainly a function of the Reynolds number and the relative extent of the laminar boundary layer, and it is moderately affected by thickness ratio and camber. If the extent of the laminar boundary layer is known, the minimum drag coefficient may be calculated with reasonable accuracy by the method presented in Sec. 5.13. The effect on minimum drag of the position of minimum pressure that determines the possible extent of laminar flow is shown in Fig. 65 for some N J.~CA 6-series wing sections. The data show a regular decrease of drag coefficient with rearward movement of minimum pressure. The variation EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS 149 of minimum drag coefficient with Reynolds number for several wing sec­ tions is shown in Fig. 66. The drag coefficient generally decreases with increasing Reynolds number up to Reynolds numbers of the order of 20 million. Above this Reynolds number, the drag coefficient of the NACA 65(Gl)-420 section remains substantially constant up to a Reynolds number of nearly 40 million. The earlier increase of drag coefficient shown by the NACA 66(2%15)-116 section may be caused by surface irregularities because the specimen tested was a practical construction model. It may be noted that the drag coefficient for the NACA 65,-418 lying section at low Reynolds numbers is substantially higher than that of the NACA 0012 section, whereas at high Reynolds numbers the opposite is the case. The higher drag of the NACA 65,-418 wing section at low Reynolds numbers is caused by a relatively extensive region of laminar separation downstream from the point of minimum pressure. This region decreases in size with increasing Reynolds number. These data illustrate the inadequacy of low Reynolds number test data either to estimate the full-scale characteristics or to determine the relative merits of lying sections at flight values of the Reynolds number. The variation of the minimum drag coefficient with camber is shown in Fig. 67 for a number of smooth 18 per cent thick N .A.CA 6-series wing sections. These data show little change of minimum drag coefficient with increase of camber. A considerable amount of systematic data is included in Fig. 68 showing the variation of minimum drag coefficient with thickness ratio for some NACA wing sections ranging in thickness ratio from 6 to 24 per cent of the chord. The minimum drag coefficient is seen to increase with increase of thickness ratio for each series of wing sections. This in­ crease, however, is greater for the XACA four- and five-digit series lying sections (Fig. GSa) than for the NACA 6-series sections (Figs. 686, c, and d). b. Variation oj Profile Drag with Lift Coefficient. Most of the variation of drag with lift for "rings of finite span results from the induced-drag coefficient, which varies approximately as the square of the lift coefficient for a given wing configuration. It is important to keep the induced drag in mind when considering the variation of profile drag with lift, because the variation of the wing drag coefficient with lift coefficient will be largely determined by the induced drag, which is a function of the aspect ratio. At low and moderate lift coefficients where there is no appreciable separation of the flow, the drag is caused almost completely by skin fric­ tion. Under these circumstances, the value of the drag coefficient depends upon the relative extent of the laminar boundary layer and the induced velocities over the surfaces of the section and may be calculated by the method of Sec. 5.13. As the lift coefficient increases, the average square of the velocities over the surfaces increases, resulting in a. small drag increase even though the relative extent of laminar flow was not affected. The THBORY OF WING SECTIONS 150 .05 .05 J.ra (roup 1e b"(UO~ . o !:= t: .010 I .008 J...I .­ i .005 r----.- ~ ~ .... ., - <, .oct. ,.--r--.. -- ,'" r-, ....... , ~ IIICA .,.... ~ I--~ .6 --- 001Z 10-- -- r--... DCA 25021 ----.'- r:-- r--~Iooo. r--- r-- ~t- ~> r----. ~ '5(Je21)-420 .....-1 ~IVIIICA'S,-ua 1/ IIAC& ~ .8 1.0 ...... r---....;.: \ ~ ..... --l-I-1 ~ III ~ ~, l~~ '-0, l'~ .GCI1.. ~ ----.. ~- ) eesae DCA 25021 (a-oup 1etl41ll& edp) A ~( DCA "(2d5)-11& - ' ~'" 2 I I I fN:[ ....... ,.. ~ , 8 10 20 50 ~o' Flo. 66. Variation of minimum d.... coefficient with Reynolds number for several airfoils. together with laminar and turbulent akin-friction coeflicienta for a flat plate. ~ .JI .016 o I I 6,-.. I RACAalr.toU e rl•• - £16­ ~U65.,-818 A 9 ..s ." .004 ~ o •• g ! a o0 .2 .4 .6 .8 1. o De.laB .eotion litt oo.~~lclint. 0li FIG. 67. Variation of section minimum drag coefficient with camber for ee1.-eral NACA 8-:Ieriee airfoil eeetioDa of 18 per cent thickness ratio. R, 6 X 10'. EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS 151 more important effect of lift coefficient is to change the possible extent of laminar flow by moving the minimum-pressure points. In many of the older types of wing section, the forward movement of transition is gradual and the resulting variation of drag with lift coefficient occurs smoothly" The pressure distributions for the XACA 6-series wing sections are such as to cause transition to move forward suddenly at the end of the low-drag range of lift coefficients. A sharp increase of drag coefficient to the value corresponding to a forward location of transition on one of the surfaces results. Such sudden shifts of transition give the typical drag curve for these wing sections with a II sag" or ((bucket" in the low-drag range. The same characteristic is shown to a smaller degree by some of the earlier wing sections such as the NACA 23015 when tested in a low-turbulence air stream. The data presented in Appendix I for the NACA 6-series thickness forms show that the range of lift coefficients for low drag varies markedly with thickness ratio. It has been possible to design wing sections of 12 per cent thickness with a total theoretical low-drag range of lift coefficients of 0.2. This theoretical range increases b)" approximately 0.2 for each 3 per cent increase of thickness ratio. Figure 69 shows that the theoretical extent of the low-drag range is approximately realized at a Reynolds num­ ber of 9 million. Figure 69 also shows a characteristic tendency for the drag to increase to some extent toward the upper end of the low-drag range for moderately cambered wing sections, particularly for the thicker ones. All data for the KACA 6-serics wing sections show a decrease of the extent of the low-drag range "pith increasing Reynolds number. Extrapolation of the rate of decrease observed at Reynolds numbers below 9 million would indicate a vanishingly small low-drag range at flight values of the Reynolds number. Tests of a carefully constructed model of the NAC"'A 65(421)-420 section showed, however, that the rate of reduction of the low-drag range with increasing Reynolds number decreased markedly at Re)·001<18 num­ bers above 9 million (Fig, 70). These data indicate that the extent of the low-drag range for this wing section is reduced to about. one-half the theoretical value at a Reynolds numberof 35 million. The values of the lift coefficient for which low drag is obtained are de­ termined largely by the amount of cumber, The lift coefficient at the center of the low-drag range corresponds approximately to the design lift coefficient of the mean line. The effects on the drag characteristics of various amounts of camber are shown in Fig. 71" Section data indicate that the location of the low-drag range may' be shifted by even such crude camber changes as those caused by small deflections of 8 plain flap," The location of the low-drag range shows SODle variation from that pre­ dicted from the simple theory of thin wing sections. This departure ap­ pears to be a function of the type of the mean line used and the thickness THEORY OF WING SECTIONS 152 SerJ..ea : i}~-dlglt) VI 2~O (5-d1S1t) .016 .... • 012 i. Rough . -"1 . -<:>--1 )-_..A,. ~- --1 e .008 ..... .,p smooth 4.~ ~ I o - .0<:4 .... I .)--"""11" --It­ ~ .~-- '~ .--­ ~ --~.~ ~ ~ c C) .... () ::: G 0 o o (a) 110 NACA four- and rive-d1git series. cd .; .012 co .... I ~ o : RO~J .008 t--­ ...<>---~ .. _.J 1--...... ~ .-r .~-- g ....ee il .004 o - o ~. Salo )th -t­ q 8 12 16 Airfoil thickness. percent (b) NACA 00 00.2 ~ 5--­ ~ or 20 chord A a.It w 0.6 24 63- series. 68. Variation of section minimum drag coefficient with airfoil thickness ratio for several N.~CA airfoil sections of different cambers in both smooth and rough conditions. B, 6 X 1()6. FIG. 153 EXPERIAfENTAL CHARACTERISTICS OF' n'ING SECTIONS A .008 I­ .o~ ~ Smooth ..r (e) MACA 64- aeries • • 012 .008 Jt-­ ... ~ .o~ Roush -e-~ ~-~ ~,)--- >--­ I -­ ir: I ~ -- ...... ~-- Cl ~ '-!\ Smooth 4\ ~ ~ ..-- ~ (d) NACA 65- aerles. A .012 ~.#Q A .008 ~ Rough 1·~-- .o~ ~oo7h 'r .()-~ ).-~ ..... v ,)--­ __ ...4 ~.,~ '~ v - 8 12 4 16 Alrrol1 thickness. percent or (e) NACn 66- series. FIG.6S. (Corduded) c:> 0 0.2 ~ A V o --A 20 Cho~a l 0.4 0.6 THEORY OF WING SECTIONS 154 ratio. The effect of thickness ratio is shown in Fig. 69 from which the center of the low-drag range is seen to shift to higher lift coefficients with increasing thickness ratio. This shift is partly explained by the increase of lift coefficient above the design lift coefficient for the mean line obtained when the velocity increments caused by the mean line are combined with .032 G a .028 e A IfAOA 641 - Ja,12 RACA '42-~5 MCA 6~_~ .ADA 611:4-1;21 :'.~ .. .t .p I o .020 I Z.016 ...a If tJ W v IPI/ ~ ~ 8 co .012 ~ .1 ~ .008 'i:1 ~ ~~ ""'i ~ .~ o -1.' ~V ~V ~ -. ~ ...... ~ ~ -1.2 -.8 . ~ rJy AI.~ -- _.... 0 .... .8 8ectlOil 11ft coetftclent.O& 1.2 1.6 69. Drag characteristics of some NACA 64-series airfoil sections of various thicknesses, cambered to a deei&n lift coefficient of 0.4. R, 9 X 10'. FIG. the velocity distribution for the thickness form according to the first­ approximation method of Sec. 4.5. At the end of the low-drag range, the drag increases rapidly with increase of the lift coefficient. For symmetrical and low-cambered wing sections for which the lift coefficient at the upper end of the low-drag range is moderate, this high rate of increase does not continue (Fig. 71). For highly cambered sections for which the lift at the upper end of the low-drag range is already high, the drag coefficient shows a continued rapid in­ crease. Comparison of data for wing sections cambered with a uniform-load EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS 2 .. ..,; 8 V ~~ o i .:s := '-t i• Ir--B I -.• ... ~ 1 ' - - 1--- V 1J .... ... ~ na. ~e I 0 .­ I I I I I I I I Vppel'lJa1t of low-4rq . . . . ~J1 1JIdt of l_-4ras I - 155 ~- ~ _..... J ~ II - ....~ "':" 1-· o 2 o (a) 8 ~ 20 2JJ 111,.148 . .bel'• • 12 Variation of upper and lower limits of low-drag range with Reynolds number, ( J •x J.o' 2.0 .0 e .028 I I II ~.o • 61.' ~ " 25·0 ~ '5.0 , > .~ ~ J., ',- e~ ~ .008 ~ .,... .-. e -1.2 FIG.70 III . ~ a; rw:... -.8 -.4 0 --- ~r.;) -.~ - .1,. .8 "o~l_ 1U~ coer~~ol.at. o~ 1.2 L' (b) Section drag characteristics at various Reynolds numbers. Variation of low-drag range with Reynolds number for the NAC_4 65(421)-420 airfoil. THEORY OF WING SECTIONS 156 mean line with data for sections cambered to carry the load farther forward shows that the uniform-load mean line is favorable for obtaining low-drag coefficients at high lift coefficients (Fig. 72). Data for many of the wing sections given in Appendix IV show large reductions of drag with increasing Reynolds number at high lift coefficients. • 028 4~ .024 IA tcJ 0 - .p cG) is/ .020 V if? ..... ...o '-t ~ CD .016 0 0 tQ ~ ~ s: 0 .012 ..... .p 0 C) It) .008 l~ HACA 6~-4l8" \ ~r\ ~ ~.jl~ .~ I I I I I CA :U ~. "t:>: ~~ ~. ~ .~ .oa,. .",,: I 65,-218 RACA I '"\ \ \ I 6 53-018,\ i\ .~ d~J ':> ~ ~ 1\ I~ ~ ~ ~~ ~ 1. L - ~ p~ ~.~~ .7 J r 1M .... Li. ~ .... ~\ RACA 6~-618JlACA ).\ 65.,-8J.8- W 0 ...........- ......- ............- ......- ...----......- .....----.....- .....- ............ o 1.2 .8 1.6 -1.2 -.8 -.4 Sectlon ~lrt coefficient. CI FIG.71. Drag characteristics of some NACA. 65-seriea airfoil sections of 18 per cent thickness with various amounts of camber. R, G X 10'. This scale effect is too large to be accounted for by the normal variation of skin friction and appears to be associated with the effect of Reynolds number on the onset of turbulent flow following laminar separation near the leading edge.1M A comparison of the drag characteristics of the XACA 23012 and of three NACA 6-series wing sections is presented in Fig. 73. The drag for the N.~CA 6-series sections is substantially lower than for the NACA 23012 section in the range of lift coefficients corresponding to high-speed flight, and this margin may usually be maintained through the range of lift EXPERIl.{ENTAL CHARACTERISTICS OF rVING SECTIONS 157 coefficients useful for cruising by suitable choice of camber. The NACA 6-series sections show the higher maximum values of the lift-drag ratio. At high values of the lift coefficient, however, the earlier NACA sections .generally have lower drag coefficients than the N ACA 6-series sections. c. Effect oj Surface Irregularities Oil Drag Characteristics. Numerous measurements of the effects of surface irregularities on the characteristics of wings have shown that the condition of the surface is one of the most important variables affecting the drag. Although a large part of the drag increment associated with surface roughness results from a forward move­ ment of transition, substantial drag increments result from surface rough­ ness in the region of turbulent flow." It is accordingly important to maintain smooth surfaces even when extensive laminar flow cannot be expected. The possible gains resulting from smooth surfaces are greater, however, for wing sections such as the XA(~A G-series than for sections where the extent of laminar flow is limited by a forward position of mini­ mum pressure, No accurate method of specifying the surface condition necessary for extensive laminar flow at high Reynolds numbers has been developed, although some general conclusions have been reached, It may be presumed that, for n given Reynolds number and chordwise location, the size of the permissible roughness will vary direct ly with the chord of the lying section. It is known, at one extreme, that the surfaces do not have to be polished or optically smooth. Such polishing or waxing hus shown no improvement in tesu,.3 in the Kf\CA two-dimensional low-turbulence tunnels when ap­ plied to satisfactorily sanded surfaces. Polishing or waxing a surface that is not aerodynamically smooth ,,;U, of course, result in improvement, and such finishes may be of considerable practical value be cause deterioration of the finish may be easily seen and possibly postponed. Large models having chord lengths of 5 to 8 feet tested in the NACA two-dimensional low-turbulence tunnels are usually finished by sanding in the ehordwise direction with Xo, 320 carborundum paper when an aerodynamically smooth surface is desired," Experience has shown the resulting finish to be satisfactory' at flight values of the Reynolds number. Any rougher surfuee texture should be considered as a possible source of transition, although slightly rougher surfaces have appeared to produce satisfactory results in some cases.. l.;oftin6S 8ho\\'00 that small protuberances extending above the general surface level of an otherwise satisfactory surface are more likely to cause transition Ulan are small depressions. Dust particles, for example, are more effective than small scratches in producing transition if the material at the edges of the scratches is not forced above the general surface level. Dust particles adhering to the oil left on wing surfaces by fingerprints may be expected to cause transition at high Reynolds numbers. THEORY OF WING SECTIONS 158 Transition spreads from an individual disturbance with an included angle of about 15 degrees. H • 42 A few scattered specks, especially near the 2.8 .. I I 2.0 I o e DCA 6~-418 D DCA ~- t---- r--- 1.2 65,-418, a ---.. =0.5 - r-. ~~~ .......... -~-- ~. rJ) ~ ~.- I r:' 'r II .8 r ; l~ ~ o I o -rwlr --4 sa •o ..... -.2 J • - .8 7 1 ~...I - -- ~ - ~ " (.;1 '-t ~ S o -.~ ~ I =­ -.4 -16 -8 8tQ~lOD 0 a aDS1. or attack. ~b (10. deg 24 FIo.72. Comparison of the aerodynamic characteristics of the NACA 65,-418 and NAC_~ 66,-418, CI ==.0.5 airfoils. B, 9 X 10'. leading edge, will cause the flow to be largely turbulent. This fact makes necessary an extremely thorough inspection if low drags are to be realized. Specks sufficiently large to cause premature transition can be felt by hand. The inspection procedure used in the N ACA two-dimensional low-turbulence EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS 159 tunnels is to .feel the entire surface by hand, after which the surface is thoroughly wiped with a dry cloth. 0J2 028 o td 0- • G24 ...• 020 . 1i ...o • .~ ~.....'r---.....-_.8 I .2 or-......._ 1.0 .. _ + - - - t - _ t - - - t -......._+---I rI '-4 '-4 8 0 .ou :: . a : .b a0 012 0 If /J 008 AftL "T . o· 0 ....-•.. - 0 4 1; a.o. position 0 ""8 .... ..1 Q -.2 JUOA-'5,~8 -II IIICA 653-418, 0 I -., 011 =0.5 -.060 .267 -·Qq1 I I • -1.6 • zlo .2'5 -1.2- -.8 -.J.. 0 .k "etlon 11ft coeff1cient. FIG. 72. (Ccmcluded) 7/0 I •J 1.2 1.6 z.0 c& It has been noticed that transition caused by individual sharp pro­ tuberances, in contrast to waves, tends to occur at the protuberance. Transition caused by surface waviness appears to move gradually up­ stream toward the wave as the Reynolds number or wave size i~ increased. The height of a small cylindrical protuberance necessary to cause THEORY OF WING SECTIONS 160 transition when located at 5 per cent of the chord with its axis normal to the surface" is shown in Fig. 74. These data were obtained at rather low 2.8 I 1 T II~ e DCA EI lIACA .1IACA """"- I I I I I I I I -l&J.~ ~~ I-- . I Jl x 106 6 6 lIACA 65~~5 9 DCA " 215)~, o .: 1. 6 I !012 9 9 roupn... 9 8taJdaiid ~. Q~ ... I 9 F nJ. L. ~~ e: ....•u ....... I Ih ,1. 2 .p ....... "­ 4"1 , g .... .p o ~ ~. ~ ro IJ' o a ~ ~ II ~ ~t\ J " ." ) o ~ .~ .Art t· 0 ~ --. '9 ~ - ~ ... ,[) "'"'" ~lfl \,.I ....... Ul.Y ~ ,1-- f\ <JsV j3 .~ rv 7 -1 .2 ~ e • ! -.4 .-1. .6 -2J,. -16 seotion -8 8D&1~ 0 of attack. 8 l' "0' dec 24 FIG. 7~. Comparison of the aerodynarnie characteristics of SOD1C N.ACA airfoils from tests in the Langley two-dimensional low-turbulence pressure tunnel. values of the Reynolds number and show a large decrease of the allowable height with increase of Reynolds number. Analysis'S of these data showed that the height of the protuberance that caused transition depended on the shape of the protuberance and on the Reynolds number based on the height of the protuberance and the local EXPERIMENTAL CHARACTERISTICS OF It''lNG SECTIONS 161 velocity at the top of the protuberance. This Reynolds number is plotted against the fineness ratio of the protuberance in Fig, 75 for protuberances located at various chordwise positions on two wing sections. •0,2 I I I t ·02" i• 9 Ie 10' ..... .9 ~"' r-, $-ur; -415 ~ IlACA o • " 6 ~ ,~ StaDel rougbne•• • DCA. '52-415 " DCA 66(215)~ Iu 1 It ~ G DCA B DCA .;: • ... . .-­-­ , ,I 1 9 -_.~ \ \ . Ol~ .I 012 ~ '" ~ 008 , ~ f\ ',-"" ..... ' -v"f..:J ~ ~ . o ..... l~" f\Vp ) ~W , J~ ~v I j /)1 / I/~ V J~ / _llt V ~I I'oT V 1_ 1"1 ~ ~ V 0 o ..... ..;. :-or o ...- ....:. - ...~ ­ .2 ...o Ii CI. 1­ ·m 0 I I .... W ... - v v --.. I -.~ -.~o .2 D ~.o x 1 4 .0 x 10 " 9.0 x 10 I ... ..c. poaIt}On 7./c 7. c o 9.0 " l~ . ­ ., - -­ ... R . ~ ~ ~ - .1 -1.6 r :1 i"'-.~ ~ v ~" ~\ ,1\ 2 o I \/ 1 ~ ~ ~ ~ ~J to !... I .26 -. .265 I I 2 -.105 I I 1 I -1.2 -.8 -.If. 0 -fa. sectIon 11ft coefficient. FIG. .8 c~ 1.2 1.6 2. o 73. (Cuncluded) The effect of Reynolds number on permissible surface roughness' is also indicated in Fig. 76, in which a sharp increase of drag at a Reynolds number of approximately 20 million occurs for the model painted with camouflage lacquer. Experiments with models finished with camouflage paint" in­ 162 THEORY OF WING SECTIONS dicate that it is possible to obtain a matte surface without causing pre­ mature transition but that it is extremely difficult to obtain such a surface sufficiently free from specks without sanding. ~ .050 i .* ~:i I . "',­ ',,­ ~i·* S'" ",I .... ~:.Q20 <, ....... OCto ~ .......... ~ i .010 i °0 , 2 1 5 ---... t"-­ , -- -- -8 , -10 uJ -JDs BaJD014a m.ber. .. FIo. 74. Variation with wing Reynolds number of the miDimum height of a cylindrical protuberance neceusry to cause premature transition. Protuberance has O.035-incb di­ ameter with axis nonnal to the wing surface and is located at 5 per eent chord of a 9O-ineh­ chord symmetrical6-series airfoilaection of 15 per cent thickness and with minimum preseure at 70 per cent chord. Low-drog airfoil numIJer 2 ~ 0.035 in. .20 .035 iii. 015 in. .20 .50 OJ5 in. Low-dtog oirfrJllllllnber I + JC ~ ~+ D x.~ '" ~... e ~ <> ~~ ~ ~ + ~ DiomeIer 0.035 in.. .035 in• .035 in. 035 in. .035 in. .35 .50 V V ~~~ ~ a058 •20 A ~ Diameter a05 0 'q •65 ~ ............... r---~ 1 ~~ 1.0 2.0 sa 40 Projeclion fineness ratio, ~ 50 6.0 lO 75. Variation of boundary-layer Reynolds number factor with projection fineness ratio for two low--drag airfoils. FlO. The magnitude of the favorable pressure gradient appears to have a small effect on the permissible surface roughness for laminar flow. Figure 77 shows that the roughness becomes more important at the extremities of 163 EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS the low-drag range where the favorable pressure gradient is reduced on one surface.' The effect of increasing the Reynolds number for a surface of ... <IlL 12 /6 20 24 28 32 36 40 .. - -­ 44 48 52 56 6OxKJ4 ~ W # ~ ~olei Reynolds number, R (o)SMOQTH CONDITION 4 8 ~ f ~ - OC "'" k70 ~ $ ~ N M R Reynolds nmoe; R (b') LACQUER CAMO..FLAGE UNIMPROVED AFTER ~INTING 76. Variation of drag coefficient with Reynolds number for a 6O-inch-chord model of the NAC.4 e.S(m)-420 airfoil for two surface conditions. FIG. Smooth cond1 tion enamel ca:nourlase wIth all specl:a ~~lt orr with bl&.de S~thetlc ~ .. o .012 ..• .p s:: .: .0G8 t--1--~--+---+---+---I----b~~'./-----4-~~---1 ~ '-t 8o ~ 2 '0 a o ... .p o cI section lUt coefficient. c 1 FIG. 77. Drag characteristics of NACA 65 l 4n )-420 airfoil for two surface conditions. marginal smoothness, which has an effect similar to increasing the surface roughness for a given Reynolds number, is to reduce rapidly the extent of the low-drag range and then ~o increase the minimum drag coefficient THEORY OF WING SECTIONS 164 (Fig. 77). The data of Fig. 77 were especially chosen to show this effect. In most cases, the effect of increasing the Reynolds number when the sur­ faces are rough is to increase the drag over the whole low-drag range. The effect of pressure gradient as shown in Fig. 77 is apparently not very powerful and tends to be evident only in cases of surfaces of marginal smoothness. 17 .;,5 I---~ 0Z8 io--- ~ . .....•" .. o ~ - ~ I I I I I I I Alrf"o11s DCA 26 x lit Chords .... -Test.s . In. 262 O.OI1-J.aoh-sralD rowdal••• on L.L ~ ...a O.OQb-laob-gra1n ;J ~rJJ ... ! l\ ~., l ~ ./ r,. ~ • 016 012 ~. A l o .I ~ '1'D'l' 21jC; aIl4 l'f 10 :J (lao)-!a22 • Io . 020 ia 6, lr!' ~ lit: 008 ~~ r-, ~ r-, r--- L::: :r.A. ~ r-----. ~ ~ "'. ~ ~-V" - ~ t- ,. OD L.B. x O.OO2-1Dah-sraua rousJm••• OIl L. L V If JJ :",A N' ~ t""j; ." ~ ~ ~ rOQBbD... _ 1:7 t. rA..... +She11&o Oft L.E. ~~Ga.oo~ "V Gala. 0 -1.6 FIG. ne88 • -1.2 -.8 -.1&. 0 ... .8 seotlon 11ft ooern.leat. 0,1.2 1.6 2.0 2.4 78. Drag characteristics of a NACA 63(420)-422 airfoil with various degrees of rougb­ at the leading edge. More difficulty is generally encountered in reducing the waviness to permissible values for the maintenance of laminar flow than in obtaining the required surface smoothness. In addition, the specification of the required fairness is more difficult than that of the required smoothness. The problem is not limited to finding the minimum wave size that will cause transition under given conditions because the number of waves and the shape of the waves require consideration. H the wave is sufficiently large to affect the pressure distribution in such a manner that laminar separation is encountered, there is little doubt that such a wave will cause premature transition at all useful Reynolds num­ bers. Relations between the dimensions of a wave and the pressure dis­ tribution may be found by the method suggested by Allen." If the pressure EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS 165 distribution over the wave is known, the criteria for laminar separation discussed in Sec. 5.7 may be applied. The size of the wave required to reverse the pressure gradient increases with the pressure gradient. Large negative pressure gradients would therefore appear to be favorable for + .o" A1rtol1 I DCA 6,lLzO)-422 III 2& x 106 ChoN, ~ 111. Teat. '1' 255 .052 A , I ~ ¥ .012 f ..... ~'" ~~ ~ ~ .008 ..---0 Jloastm... .trip on L.s. " "r-, I ~ ~ r­ '---. """"--. :..­ rv­ '--<~ ~ .... ~ ~ """"­ o -1.6 .1.2 .. 8 ... Il f - ~ . ~ V ..,.. - .... /1 ...-..4' ~ - ../ ~AC.A Jl1. "Jl1 ~~ ~ " ~ '" 4­ " -' 0 .... .8 aeetlan 11ft coe1"l'lClent. FIG. 79. Drag characteristics of a at various chordwise Iocatione, ~ V1 K. I( Ioastme•• atrlp­ at 0.200 _ + Bouatme•• at.rip at 0""0 _ G 8IIooUa 1.2 01. 1.6 2.0 2.... 63(420)-422 airfoil with O.Oll-inch-grain roughness Experimental results have shown tills conclusion to be qualitatively correct. For the t}1JCS of waves usually found on practical-construction wings, the test of rocking a straightedge oyer the surface in a ehordwise direction is a fairly satisfactory criterion." The straightedge should rock smoothly without jarring or clicking. The straightedge test will not show the exis­ tence of waves that leave the surface convex. Tests of a large number of practical-construction models, however, have shown that those models wavy surfaces. THEORY OF WING SECTIONS 166 which passed the straightedge test were sufficiently free from small waves to permit low drags to be obtained at flight values of the Reynolds number. It does not appear feasible to specify construction tolerances on or­ dinates of wings with sufficient accuracy to ensure adequate freedom from dt ~ "' ~' .,. - _a ·.'"...... ~ .. .­ a ...::• o • 02- + •oo ..,.016 S .. . t: ~ OU ~~+ IlACA '5(22')~2 (-at'le4) 0...1 I ,8 IUCA " ( i20)~2Z ~~ ~ _ ~A - -1.2 I I ~~ -F:=o. ~ (/ _.... ~~~ ~ ~ "" a. -.8 ...... 0 26 • .... ltJ .8 1.2 1.6 "otloa l1tt oCMt1'lol_t. . , FIG. SO. Drag characteristics of two NACA 6-...~rics airfoils with O.Oll-ineh-grain roughnese at O.3Oc. waviness. If care is taken to obtain fair surfaces, normal tolerances may be used without causing serious alteration of the drag characteristics. If the wing surface is sufficiently rough to cause transition near the leading edge, large drag increases are to be expected even if the roughness is confined to the region of the leading edge. Figure 78 shows that, although the degree of roughness has some effect, the increment of minimum drag coefficient caused by the smallest roughness capable of producing transi­ tion is nearly as great as that caused by much larger grain roughness when the roughness is confined to the leading edge.! The degree of roughness has a much larger effect on the drag at high lift coefficients. If the roughness is sufficiently large to cause transition at all Reynolds numbers considered, EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS 167 the drag of the wing with roughness only at the leading edge decreases with increasing Reynolds number.' The effect" of fixing transition by means of a roughness strip of car­ borundum of O.011-inch grains is shown in Fig. 79. The minimum drag 32 x 10' 28 ..... 0: • 'Ia ...-3 0 ~ 20 16 12 ! 8 .1' 0"----11.----........-.-....-.........- o .08 ........- - ' - -.......- ... .2k .~ .~ .48 .56 .'4 .72 Sectloo l1tt coetrl~1ent. 01 .56 .'4 .72 o 8ec~1oa lift coet~lcleDt. 01 FIG.81. Comparison of section drag coefficients obtained in flight on various airfoils. Tests of NACA 27-212 and 35-215 sections made on gloves. increases progressively with forward movement of the roughness strip. The effect on the drag at high lift coefficients is not progressive; the drag increases rapidly when the roughness is at the leading edge. Figure 80 shows that the drag coefficients for the X.A.C.-\. 65(223)-422 and 63(420)-422 lying sections' are nearly the same throughout most of the lift range when the extent of the laminar flow is limited to O.30c. These data indicate that, THEORY OF WING SECTIONS 168 for wing sections of the same thickness ratio and camber, the drag coeffi­ cients depend almost entirely on the extent of the laminar boundary layer if no appreciable separation occurs. The variation of minimum drag coefficient with thickness ratio for a number of NACA four-digit, five-digit, and 6-series wing sections with .0/2 I o Surfaces 05 received excepllighllysmded I- 0 Both surfocespointedandfinished10 rear spar -.. 0/2 /6 20 24 28 32 36x/0 6 Rf!I'1OIds number, R FIG. 82. Drag scale effect on l00-inch-ehord praetieal-eonstruction model 65(216)-3(16.5)(approx.) airfoil section. C1 = 0.2 (approx.). or the NACA leading edges rough is shown in Fig. 68. The minimum drag coefficients increase with thickness ratio but are substantially the same for all the sections of equal thickness ratio. The increments of drag coefficient caused by leading-edge roughness are correspondingly greater for the sections having the lower drag coefficients when smooth. The section drag coefficients of several airplane wings have been measured in flight l 60 by the wake-survey method, and a number of practical­ construction wing sections have been tested in the NACA two-dimensional low-turbulence pressure tunnel! at flight values of the Reynolds number. Some of the flight data obtained by Zalovcik l 60 are summarized in Fig. 81. EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS 169 All wings for which data are presented in Fig. 81 were carefully finished to produce smooth surfaces. Care was taken to reduce surface waviness to a minimum for all the sections except the NACA 2415.5, N-22, Republic r ~ .ex»' t ·005 •S f ..s .. i - .our .: ~ 0.910 ( 101 .12• 58 := .010 ,,~ Cd" • I-- +I - ~ .... - ./ !\lrtnIl..,t ! .~ I t .003 <, .002 k' Cd r-, r-, 1Aa1nC" I I .001 1 • 2 2.6511'" ,.... c CondlUon of lIodel 1 0 0.°79 " 0.07~ AJJ recelnd PalMed with gray pr1Jller aurracer C...ourlase p&1nced with almulatad door jolnta • o.O"n II 10 ! I I I I I " Ii I I I 20 IIaJD014. n"","er. B FIG. 83. Variation of the drag coefficient with 1I"Y1101<l8 nUll11JCr for the ~ACA 2301li airfoil IIOCtion together with laminar and turbulent ski ll-fricti on cocffieioute for a flat plate. 8-3, 13, and the XACA 2i-212. Curvature gauge measurements of surface waviness for some of these sections are presented by Zaloveik.P" These data show that the KACA 2-,3-, and 6-sel;es sections that permitted ex­ tensive laminar flow had substantially lower drag coefficients when smooth than did the other sections. Data obtained in the XACA two-dimensional low-turbulence pressure tunnel' for typical practical-construction sections are presented in Figs, 82 THEORY OF WING SECTIONS 170 to 86. Figure 87 presents a comparison of the drag coefficients obtained in this wind tunnel for a model of the NACA 0012 section and in fiight for the same model mounted on an airplane.' For this case, the wind-tunnel and flight data agree to within the experimental error. The wind-tunnel tests of practical-construction wing sections as de­ livered by the manufacturer showed minimum drag coefficients of the order of 0.0070 to 0.0080 in nearly all cases regardless of the type of section .016 ' 1 I I I • ~OIlACA 6~(216)-4l7(~F.' .. bull' ~OIlACA 2 011) \\ \\ ~ l..n----"" ~ o -.2 ~ ~ ~ (.pprox. ~ V" / .. bull t ~ o .8 sect10n 11rt coefficient. c~ Flo. 84. Drag characteristics of the NACA 65.2-417 (approx.) and NACA 23015 (apllrox.) airfoil eectiODB built by praetical-conatruction methods b)~ the same manufacturer. R, 10.23 X lOS. (Figs. 82 to 86). Such values may be regarded as typical for good American construction practice during the Second World War. Finishing the 800­ \ tions to produce smooth surfaces always resulted in substantial drag re­ ductions, although considerable waviness usually remained. None of the sections tested had fair surfaces at the front spar. Unless special care is taken to produce fair surfaces at the front spar, the resulting wave may be expected to cause transition either at the spar location or a short distance behind it. One practical-construction specimen tested with smooth surfaces maintained relatively low drags up to Reynolds numbers of approximately 30 million [NACA 66(2%15)-116 wing section of Fig. 66]. This specimen had no spar fonvard of about 35 per cent chord from the leading edge and no spanwise stiffeners forward of the spar. This type of construction resulted in unusually fair surfaces. Few data are available on the effect of propeller slipstream on transition or wing drag; the data that are available do not show consistent results. 0 11&0& - - a r- 0 l\1 ~ I ~ o ,\ •• 012 ....11• 1.008 66(21S)-U6. lOO-lDch Obor4. 0& 80A 25016. 1OO-lDeb 60N. 0, • :I &!Ii IllaA "C211i )-U' (n1Nllt . .1 J. 0.11) laoh obol'4. 0.19 IIICA 25016 ("bullt .04(:1), l00-1aob chON. 0, • 0.19 \~~\ C) - ~' I\. ~~ o eo . ~ ..,• I~ !.­ ., \ ~ "~ .A.~ .A ~ o & o FIG. o 28 8 85. Scale effect on drag of the NAC..-\. 66(215)-116 and NACA 23016 airfoil sections 88 received. built by practical-eonstruction methods by the same manufacturer and tested •016 I ~ o .012 0 a I D...l. wins. or1g1Dal concU tlon DCA 65__erle. -ins, original coD41~10D_ ~ ~ o0 '" 8 ~~~ ..... 12 "JDOu. ~ 20 D1aber. - 21,. 28 a ~ )C 10' 86. Drag scale effect for a model of the N AC.A... 65-serics airfoil section 18.27 per cent. thick and t.he Davis airfoilllOOtion 18.27 per cent thick, built by practical-construction nletbods by the same manufacturer. c, == 0.46 (approx.) . FIG. 0 u • 012 - .p s: ....•0 ... 0 0 .008 W1ncl tunnel FlIght '-4 ~ 80 ·~.t Z "G ..... r-""U eo .... - ""' ~ • ad,. s :I 0 cI 0 o 8 16 12 Re,nolds number, 20 28 )( lrP R FIG. 87. Comparison of drag coefficients measured in flight and wind tunnel for the NACA 0012 airfoil tteCtiOll at zero ilit. 171 THEORY OF WING SECTIONS 172 This inconsistency may result from variation of lift coefficient, surface con­ dition, air-stream turbulence, propeller advance-diameter ratio, and num­ t A1rtoU ...'1. . Boot. KACA "(2X15)-en.8 !!P. ileA 67.1-(1.,)15 .....\ ..~, 5.98 II ~ n4J._---. -.Uer 'ip ~.012 0" I ~l' 1d.D'-',11DI !'rope11al' ......a • .---..,."~ I.~ ! ....-..; ~r ~ ~~ a-­ ~ o I °8 1 , 5 Dlat. . . tfta 4 ~1 3 2 1 0 oater 11De. tt 88. The effect of propeller operation on section drag coefficient of a fighter-type airplane. from tests of a model on the Langley 19-foot pressure tunnel. CL = 0.10; R, 3.7 X 10'. FIG. ber of blades. Some early British investigation Sl51 , 158. 159 showed that transition occurred at 5 to 10 per cent of the chord from the leading edge in the slipstream. Similar results were indicated by tests in the N..~CA 8-foot high-speed tunnel. 4I Drag measurements in the N.~CA 19-foot pressure tunnels (Fig. 88) indicated that only moderate drag increments EXPERIJIENTAL CHARACTERISTICS OF WING SECTIONS 173 resulted from a windmilling propeller. Although these data are only quali­ tative' because of the difficulty of making wake surveys in the slipstream, they seem to preclude very large drag increments such as would result from a movement of transition to a position close to the leading edge. A I 20 x 106 ........ I~ a 10 Lef't wing .eC1;iOl1}/ in slipatream 1/r---.. . . . f=:-- r- -r--- J.. . . . . V Bight wing .ection out.1~e a11pstream f o Ie .p e, ..s -50 o'-~ ~ .-of • =a .40 e Bight outaide allpatream ". ~ "" .30 :: .20 0 ..s• ..• o CIt t) ~ Jt o 6 .10 wing .ection} til I I I x.ttt wing section 1D al1patre8lll D Power on----- , o Power or~---I .1 ~- V 10-- ~- ~ r> ~> "" ..,Ito... I .2 section 11ft coe1'£lclent. Fro. 89. Flight measurements of transition on a NACA ............ ~ c~ f)()"~rics wing within and outside the slipstream. similar conclusion is indicated by X ..\.C..\ flightdata/ (Fig. 89), which show transition as far back as 20 per cent of the chord in the slipstream. Even fewer data are available on the effects of vibration on transition. Tests in the X . .\ CA 8-foot high-speed tunnel 43 showed negligible effects, but the range of frequencies tested may not have been sufficient.ly wide to represent conditions encountered on airplanes. Some K .~(~.~ Hight data" showed small but consistent rearward movements of transition outside the slipstream when the propellers were feathered. This effect was noticed even when the propeller on the opposite side of the two-engined airplane was feathered, and it was accordingly attributed to vibration or noise. In THEORY OF WING SECTIONS 174 some cases, increases of drag have been noted in wind tunnels when the model or its supports vibrated. A tentative conclusion may be drawn from the meager data that vibration may cause small forward movements of transition on airplane wings, The skin friction associated with a turbulent boundary layer on a smooth surface decreases with increase of Reynolds number, as shown in Chap. 5. This favorable scale effect is not obtained at high Reynolds , , .­ e; .IS~ o .., ~ .s '\ .... ~ ~ .. .5 <, ~ ~ r \ FIG. a 3.IJ J.1 .I., • .0 --- ~~ ~ ~ - ~ - [~'" ~ ~ V'''' ~ ....... ~ \ J.6 U to U ff --..••¥ - 'Wl,,­ ~ ~~ .. ~ ~--- --- f6 - ~-.., ~ •f .3 IIU 1Sl.JJ ~~ ~ , ,;' ~~ .~, .' 0 •• • ~ -... .... ..... ...-­ ~ r-, r-w­ S/J - 0 ~ i!b.a II - U -""1"­ ~ s., 1­ -- ..-.a.IIIl ~r- SI II u 90. Coefficient of resistance of rough and smooth tubes dependent on Reynolds number. numbers on rough surfaces. An explanation of this effect is that the pres­ sure drag of the individual protuberances constituting the roughness con­ tribute to the skin friction when they project through the laminar sublayer, The shapes of the individual protuberances are usually such that the scale effect on the pressure drag should be small. The result at large Reynolds numbers for generally roughened surfaces is that the skin-friction coeffi­ cient is essentially constant. The general nature of the scale effects for rough surfaces has been indicated b)' experiments in pipes." These data (Fig. 90) show that at lo"r Reynolds numbers the pipe-loss coefficient decreases with increasing Rey­ nolds number at the rate expected for laminar flow, When transition occurs, the pipe-loss coefficient increases to the value expected for tur­ bulent flow over smooth surfaces. For very small roughness, the pipe-loss coefficient decreases with increasing Reynolds number along the turbulent skin-friction curve for smooth surfaces until large values of the Reynolds number are reached. For any given size of roughness, however, there ap­ EXPERIMENTAL CHARACTERISPICS OF liTING SECTIONS 175 pears to be a Reynolds number beyond which the coefficient increases slightly to a constant value. The value of the Reynolds number decreases and the constant value of the coefficient increases as the grain size of the roughness increases. At large values of the Reynolds number, the rough­ ness causes large increments of skin friction over the values corresponding to smooth surfaces. On the basis of the pipe experiments and reasoning similar to the fore­ going, von KarmAn139 obtained a formula for the grain size just sufficiently large to affect turbulent skin friction. This formula may be written pl:k=3.<L p. f! V4 Reynolds number based on grain size of roughness and local 'vhere -pUk - ::: velocity outside boundary layer JI. c, = local skin-friction coefficient For a 'ling of approximately 9-foot chord at a speed of 300 feet per second, the limiting grain size is approximately 0.0004 inch and varies only slightly over the surface, increasing toward the trailing edge. The effect of roughness on the drag of wing sections with largely turbu­ lent flow is analogous to the effect in pipes as ShO\\·113 by Fig. 83. At the lower Reynolds numbers, the drag coefficient decreases with increasing Reynolds number at about the rate expected for turbulent flow over smooth surfaces, In this case, the drag remains essentially constant at Reynolds numbers above about 15 million. ...~t a Reynolds number of 70 million the increment of drag caused by the moderate roughness was of the order of one-quarter of the drag that would have been obtained had the favorable scale effect continued. The sensitivity of the turbulent boundary layer to roughness is so great that airplanes should not be expected to show favorable scale effects at large Reynolds numbers unless considerable care is taken to obtain smooth surfaces. It should be noted that, in contrast to the situation with respect to the laminar layer, it is relatively easy to obtain reductions of drag by attention to the surface conditions for turbulent flow. Even though the allowable size of the roughness is very small, each speck of roughness presumably contributes onI)" the drag of itself and does not have any appreciable effect on the skin friction over the surface downstream. Consequently favorable effects may be expected from careful finishing of the general surface of modem high-performance airplanes even though imperfections such as rivets and seams may be present. d. Unconsenxuioe li'ing Sections. The need for low drags in order to obtain long range for large airplanes flying at speeds below the critical Mach number leads to designs having high wing loadings to reduce the wing area and profile drag together with relatively )O\V span loadings to THEORY OF WING SECTIONS 176 avoid high induced drags. These tendencies result in "rings of high aspect ratio that require large spar depths for structural efficiency. The large spar depths require the use of thick root sections. The comparatively high lift coefficients corresponding to the cruise condition for such designs leads to the use of large cambers to obtain low profile drags. Such designs are encouraged by the drag characteristics of modem smooth wing sections ~ .......... .yr r=­ / 4 ~ o • OOO} (l~ cUBit) ",,\~ :~ I 644 v2,o 8 A1rf'oU (a) ~ r-, ,,~ lV ....s ~ v ~~ I I r-, MeA 12 16 perceDt of cho1"4 tblokD.... to~ I (5 cU81t) 20 ao4 tlve-4161\ serle•• 'f4 FIG. 91. Variation of the lift coefficient eorresponding to a drag coefficient of 0.02 with thick­ ness and camber for a number of NACA airfoil sections with roughened leading edges. R. 6 X lOS. which show relatively small increases of drag coefficient with increasing thickness ratio and camber (Sec. 7.00). Unfortunately airplane 'lings are not usually constructed with smooth surfaces, and, in any case, the surfaces cannot be relied upon to stay smooth under all service conditions. The effect of roughening the leading edges of thick wing sections is to cause large increases of the drag coefficient at high lift coefficients. The resulting drag coefficient may be excessive at cruising lift coefficients for heavily loaded high-altitude airplanes. Wing sections that have suitable characteristics when smooth but have exces­ sive drag coefficients when rough at lift coefficients corresponding to cruis­ ing or climbing conditions are called "uneonservative." -The decision as to whether a given lying section is conservative will depend upon the power and wing loadings of the airplane. The decision may be affected by expected service and operating conditions. For ex­ EXPERIMENTAL CHARACTERISTICS OF U·ING SECTIONS 177 ample, the ability of a multiengine airplane to fly with one or more engines inoperative in icing conditions or, in the case of military airplanes, after 1.1, <, 1.2 1!C o / 1.0 / ...• .. ......•a t <~ 8 / o • V o o II • ~ .> -: ~ / .. c ~ ~ ~ ~ 1 00 () .2 A .~ v • V 41 ~ Ct / L ~ ~ V o ,.a o •• s: o NACA 63- (b) i.. ler1~ae .. ........• ~ 1.2 r--- o ...•o ~~ o 1.0 ;.... .. ~ V ~ ...... ~ 5 a e ) 1 .8 ~/ ) ,.. ~) .J" \ 'I\, el t 00 00.1 00.2 .Oe2 c/ 4 ~ <, V / t '" -, -, "----- ".-..:.r /0 ! 0 // / ~ ........6 A VO• 8 16 12 A:'rCol1 thlckne:.s, percent (C ) FIG. i;ACA 64- or ZO 2~ c.hord se rlea. 91. (Continued) suffering damage in combat may be a consid~ration. As an aid in judging whether the sections are conservative, the lift coefficient corresponding to a drag coefficient of 0.02 was determined from the figures of Appendix IV for a number of NACA wing sections with roughened leading edges. The variation of this lift coefficient with thickness ratio ar.d camber' is shown THEORY OF WING SECTIONS 178 in Fig. 91 for a Reynolds number of 6 million. These data show that, in general, the lift coefficient at which the drag coefficient is 0.02 decreases with rearward movement of the position of minimum pressure and with .1.2 ~~ ~ o ~ 1.0 ~~ • ...Ii ef ..-= .8 ~7 t Z o , ~ .6 I / ..V / ~~ ~ V ~l r-, ~ ~ <, o,!. GO ~ ~O.2 AO.~ jO. II .c ...o .:• .~ ~ (4) i MACA 65- .erie. CJ,l - -00 e 0.2 / 40.4 ~ / & :: .8 o • O'J ( / / 7 ---- / ~ i': -e1. ~~ r-, 2\ ~~ \ ~ 0 • n 12 1) 20 2" Alrtol1 thickness. percent or chord (e) FIG. NACA 66- aeries. 91. (Concluded) increased thickness above thickness ratios of about 15 per cent. For wing sections thinner than approximately 18 per cent, the effect of camber is to increase this lift coefficient. For the thicker sections, however, increasing the camber becomes relatively ineffective and may even be harmful in extreme cases. The highest values of this lift coefficient for wing sections having thickness ratios greater than 15 per cent are obtained with the NACA 64-series sections having a design. lift coefficient of 0.4. EXPERIltfENTAL CHARACTERISTICS OF WING SECTIONS 179 7.6. Pitching-moment Characteristics. The variation of the quarter­ chord pitching-moment coefficient at zero angle of attack with thickness SlDa1e •1 n.aAId •pIbola o , aN t or (,tf' dada eel .p,1It ...... aerlea ......­ .... ~ ioS :I ...""I -.1 ,.o - - - r---Ii OOr 41&1t) e :.~ A J,.Ii • 250 (5 41&1t) .. it -.2 t ~ .l.. i -., iI ~ 4 (a) tbl..... 8 AutoU MACA 12 to---.. r---, k 1& 20 peroent or c:bor4 tour- and tlYe-dl81t aerlea • •1 --.. "... i ... .....o t : o -v --& :r­ -.1 V­ °ls. ~~ 00 .0.2 ~ .......... -.2 r- ~ h ""c I .., h 'h_ -·5 ··!JO ~ Io.......~ - ~ ~ ~ ""V­ -.::Ill 4 8 12 Alr1"o11 thiekn•••.• 16 pe~ent \.. -, I...-.­ f'---. Ao.h .0.' ~ -- 20 24 or chord (b) HAC. 63- eerlell. 92. Variation of section quarter-chord pitching-moment coefficient (measured at an angle of attack of zero degrees) with airfoil thickness ratio for several NACA airfoil sections of dift'erent camber. R. 6 X I ()I. FIG. ratio and camber is presented in Fig. 92 for a large number of NACA wing sections," The pitching-moment coefficients of the NACA four- and five­ digit series sections become more negative with decreasing thickness ratio. Comparison of the experimental data in Fig. 92 with the theoretical values obtained from the theory of thin wing sections shows that the absolute THEORY OF WING SECTIONS 180 magnitudes of the pitching-moment coefficients for the NACA four- and five-digit series sections are somewhat less than those indicated by the theory. The pitching-moment coefficients for the NACA 6-series wing .1 - . ~ ......• ~ ,,:.. ~ .A. .A. ..,;,. ..;. ...... - - --- - -.1 o I o ... ~.~ -.2 ~o~ ~ '"-- B 0.1 ---- .~:~-L I-&- i -., ..... 64- oJ-- - 0.2 o.L. " 0.6 H1 ~ ~~ ~ 8 12 16 Alr1'o11 tblClmeaa. percent lIACA - ~ .A 'V "V"'" ~ 4J-- ~ (c) c&l eo '- 20 or cborcl aeries. $lasle t"lasP4 sJIIIbols are £or 6fP siMulated split 1'lap .1 1 .....j t .. ... 0 • • o. 5 ~.aa -. .... ~ '" --- - ..... ~~ C ~ ~ 0 «-: oJ-..... a I -., ,,:.. r«: ~ o~ .~ ~h --* A.. -.1 I0 -.2 .l1De, ~ ~ ~ ......, f'--- -<J ..... L. 8 12 16 20 Airfoil thlckness. percent ol chord (d) 65- aeries. 92. (Continued) MACA FIG. sections show practically no variation with thickness ratio or position of minimum pressure. As indicated by the theory of thin wing sections, in­ creasing the amount of camber causes a nearly uniform negative increase of the pitching-moment coefficient. In general, the absolute magnitude of the quarter-chord pitching-moment coefficients for the K AC.~ 6-series wing sections having mean lines of the type a = 1.0 are approximately three­ EXPERIltlENTAL CHARACTERISTICS 0// WING SECTIONS 181 .1 ~ 0 I .. ~ c; .: 7" "':" "":" ~ A. ~ h­ ~~ -.1 0 :: '-c •a 0 -.2 ~R Nt... ~ ~ .pi c: I :.I -., ~ ~h r«.. r---., .h­ -­rc;r--... -.., "'" --1&.0 ~ ~~ 4 8 12 16 20 AlrtoU thlcJ.:nes.•, percent or chorcl (e) HACA FIG. 66- aeries. 92. (Concluded) .­ •. c: .: V ~ "s... o o -­ V ..• v 65,-618/ I» ­ ~ • ~ o I . V c ,i lL -'1--66(21 «; )-216 • • 2'012 -.02 -.oL I j -.06 = 0.6 -.08 ....10 -.12 -.l~ -.16 1beoretlcal IDoment coefficient tor ~t.e airfoIl . . .n liae about quarter-chord point FIG. 93. Comparison of theoretical and measured pitching-moment coefficients for eome NACA airfoils. R, 6 X l()S. 182 THEORY OF WING SECTIONS quarters of the theoretical values. The pitching-moment coefficients for sections having mean lines of the type a < 1.0 are equal to or slightly more negative than the theoretical values as shO\VD in Fig. 93. Consequently, changing the type of mean line from a = 1.0 to a < 1.0 to reduce the magnitude of the pitching-moment coefficient is relatively ineffective un­ less the value of a is reduced to a small value. The variation of chordwise position of the aerodynamic center at a Reynolds number of 6 million for a large number of NACA wing sections is presented in Fig. 94. From the data presented in Appendix IV, there appears to be no systematic variation of ehordwise position of the aero­ dynamic center with Reynolds number between 3 and 9 million. For the N ACA 24-, 44-, and 23o-series wing sections with thickness ratios ranging from 12 to 24 per cent, the chordwise position of the aerodynamic center is ahead of the quarter-chord point and moves forward with increases of thickness ratio. The data for the NACA 00- and 14-series sections show that the aerodynamic center is at the quarter-chord point. The aero­ dynamic center is aft of the quarter-chord point for the NACA 6-series wing sections and moves rearward with increase of thickness ratio. There ap­ pears to be little systematic variation of the chordwise position of the aero­ dynamic center with camber or position of minimum pressure for these sections. For the thick cambered sections, the chordwise position of the aerodynamic center appears to move forward as the design position of minimum pressure moves back. For wing sections with arbitrary modifica­ tions of shape near the trailing edge, the trailing-edge angle appears to be an important parameter affecting the chord,vise position of the aero­ dynamic center. For such sections, the aerodynamic center moves forward as the trailing-edge angle is increased." EXPERIMENTAL CHARACTERISTICS OF WING SECTIONS 183 .26 .24 ~ HACA r--- r---.­ ~~ 2~- •• rl.a "'" ~~ .22 .2& ~~~ IlACA ~ ~.) -..... re.-. ---­ ---­ _G e ~ a, ~ . JIACA 14-aerl•• . MCA OO-aerle. .22 0 I ~ I I . 8 12 16 Airfoil thieme.a. percent ot (a) r-, 4·) Hl.CA 24... erlea .22 ~ 44-aerl•• 20 ch~rd NACA four- and five-digit series. 94. Variation of section chordwise position of the aerodynamic center "pith airfoil thickness ratio for several NACA airfoil sections of different cambers. R, 6 X 10'. FIG. THEORY OF WING SEC7'IONS 184 SJIIlbo18 with f'lap corre8poDd to ~ c11 ~ ~ . =0.6 ~ ~ ~~ G Cl., =0.4 and 0.6 A . V~ V .-... C11 W ~ 0 ~ -21,­ l~ ~ =0.2 . 4~ ....­ ~~ ....... ~ L..e ~ G ell =0 o 8 12 16 20 Alrl'o11 thicknesa. percent ot chord J,. (b) FIG. NACA 63-series. 94. (Continued) 2JI. EXPERIMENTAL CHARACTERISTICS OF IVING SECTIONS .~ =0.6 Bplbo18 with tlas8 correspond to 01 1 .--t.~ o :::i--~ • •24"-.......­ ......-----......~-------.a..-...a.---.l~~ ~ .28 c.~ .-......~ ~ .~. 01 1 ~., --­ = 0.2 ..",J0) ~ ..",- Cl -­ c·,......... ---­ ----­ Cl 4 8 =0.1 l l o ~..,­ ,...".,..... ~. - =0 12 16 20 Alrroll th1ckness, percent or chord (c) N ACA 6·l-serics. FIG. 94. (Continued) 185 THEORY OF WING SECTIONS 186 8Jmbo18 with nasa cCll'reapord to 0Il .28 =0.6 ~ .~ ··L ~ .26 - ... • • 01 ~ .~ 1 = 0.4 and ~.) 0.6 0 0 1 .28 f:.~ c ..• .2& ... .24 0 .28 0 .p a. e e,....­ a ~ j.:L..-­ .,~ ~Il =0.2 .~ •• ~ ~ ~ CI ~~ to­ e .2& ~ 0Il ~ w 4-) 4·) 8 =0 - -­ .­ v ~ . 12 16 A1r.toU thlolme••• perOeDt (d) NACA 65-series. FIG. •.) .... 94_ (Continued) 20 ~ chord EXPERIlYfENTAL CHARACTERISTICS OF WING SECTIONS .28 l...--J.) \;I s:• .24 c, 1 = 0.4 .p i o o 1.28 ~ 2 : .26 ~ o --. 4r c~l 5 .....24 .p ..• ...:.. v t:\ =0.2 8. .28 ....•• t• .26 a G :.-.--­ ~ ~ r--­ ~ ...;.. ,...­ ct. l =0 4 8 12 16 20 AirtoU thickness. percent or chord (e) NACA 56-series. FIG. 94. (Concluded) 187 CHAPTER 8 mGH-LIFT DEVICES 8.1. Symbols. CD C DOlin CLmax /i.C L DUlX drag coefficient minimum lift coefficient maximum lift coefficient increment of maximum lift coefficient C N normal-force coefficient eN/flap normal-force coefficient based on the area. of the flap CQ volume-flow coefficient, Q/VS E flap-chord ratio, cllc G moment arm, in terms of the chord, of the basic normal force about the quarter­ chord point u ~-Iach number P Q& incremental additional load distribution associated with flap deflection P b" incremental basic load distribution associated with flap deflection Ps incremental load distribution associated with flap deflection Q volume of flow through slot R Reynolds number, pVcJp. S wing area V velocity of the free stream c chord Cd section drag coefficient CI flap chord CA section flap hinge-moment coefficient, h/~pytcr Cl section lift coefficient e'l section flap lift coefficient, ll/>~pVtcl Cl mas maximum section lift coefficient ~Imax increment of maximum section lift coefficient em section pitching-moment coefficient, m/~p lltc2 Cm~/, section pitching-moment coefficient about the quarter-chord point c.1 section pitching-moment coefficient about the quarter-chord point with flap neutral ~ section pitching-moment coefficient about the quarter-chord point with flap deflected ~c". incremental section pitching-moment coefficient about the quarter-chord point Cal section normal-force coefficient with flap neutral Cnt section normal-force coefficient with flap deflected ~a incremental additional normal-force coefficient associated with flap deflection Ca" incremental basic normal-force coefficient associated with flap deflection C-n/a incremental flap normal-force coefficient At:. incremental section normal-force coefficient (cp)J center of pressure of the load on a flap measured from the leading edge of the flap k section flap hinge-moment coefficient (positive in direction of positive flap deflection) 188 HIGH-LIFT DElT/CES l, m n/ a q r v ~I'a x y a ao &to "Yael' "Yb& 8 8/ a a/_ ~cl p T Tn. T.. 189 lift force acting on the flap per unit span section pitching moment incremental flap normal force per unit span dynamic pressure, ~p VI leading-edge radius thickness of wing section local velocity over the surface of a symmetrical section at zero lift increment of local velocity over the surface of a wing section associated with angle of attaek distance parallel to chord distance normal to chord angle of attack section angle of attack increment. of section angle of zero lift ratios of the flap normal force to the section normal force for the incremental additional and the incremental basic normal forces flap deflection deflection of fore flap or vane deflection of main flap of a double-slotted flap configuration increment of flap deflection mass density of air turbulence factor, effective Reynolds number /tcst Reynolds number see Eq. (8.8) 8.2. Introduction. The auxiliary devices discussed in this chapter are essentially movable elements that per­ mit the pilot to change the geometry and aerodynamic characteristics of the wing sections to control the motion of the airplane or to improve the per­ formance in some desired -manner. The desire to imp ove performance by increasing the wing loading while maintaining acceptable landing and take-off speeds led to the developmen t of retrae able devices to improve the maximum lift coefficients of wings without changing the characteristics for the cruising and high-speed flight conditions. Some typical high-lift de­ vices are illustrated in Fig. 95. The aerodynamic characteristics of some typical high-lift devices are presented and discussed in the following sec­ tions. Primary emphasis is given to the capabilities and relative merits SPlIT FLAP C----~:::--=­ ~ EXTERNAl AIRFOIL RAP c: ~ DOUBLE· SLOTTED FlAP" / ~-----:::-::--==LEADING EDGE SlAT FIG. 95. Typical high-liit devices. THEORY OF WING SECTIONS 190 of the various devices. Numerous references to the literature are given to provide design information. •8 I I I I I ., ~ /~ 1<aJ ./ 4 .. CD to > .,.f ~ 0 CD Cf-t ft-t CD D. as .-l fH ~ / V ./ ~, .2 CD 0 ~ / .~ s::CD .L ~" ri.7 ,-.. ~ 10-'" r\ I BxperlaenU.l ~ ~ ~ C ~!J j~ /rfJ !/ r 0 (a) .8 range f'rom 0° to 10°. I I I I I I ., 4fI#' ~~ -:lheoretlcal l ~", -'~ AI ~ ~ ,/ /~ .4 IAl , "L I I y ~ 'Iii' \ I BxperSmental - - ~ _l/~ II .2 - ~~ itf / til ~ ~, .JIll' ...,~ ",~ ~ 0 0 CD ,. ~ ....... ~ "'" V /- ~ s:: ......, "'~ ." ",. r-, ,-~ dOloca I '1heoret1cal /~ ~~ , J~ ~ o 0V .1 .2 ., .0 • flap-chord ratio, E (b) 6 range from 0 0 to 200 • Flo.96. Variation of section flap effectiveness with flap chord ratio for true-airfoll-eontour flaps without exposed overhang balance on a number of airfoil sections; gaps sealed; Cz -= o. 8.3. Plain Flaps. Plain trailing-edge flaps are formed by hinging the rearmost part of-the wing section about a point within the contour. Down­ ward deflections of the trailing edge are called "positive-flap deflections." Deflection of a plain flap with no gap effectively changes the camber of the wing section, and some of the resulting changes of the aerodynamic ehsrae­ Sym­ hoi - 0 +­ X 0 - o -­ V Air-flow characteristics Type of flap Basic airfoil NACA 0009 NACA0015 NACA 23012 NACA 66(2x15)-009 NACA 66-009 NACA- -­ 63,4-4(17.8) approx, ~ 66 (2xl5)-216, a = 0.6 -­ <1 E> -­ ~ -­ b. -­ ~ 17 $) -­ JJ -­ -0 -­ b S> -­ ~ -­ <f Q M R Plain Plain Plain 1.93 1.93 1.60 0.08 0.10 0.11 1.4 X lOS 2.2 X lOS Plain, straight contour 1.93 0.10 1.4 X 10' Plain 1.93 0.11 1.4 X 10' Internally balanced Approach­ ing 1.00 0.17 2.5 X 10' Internally balanced Approach­ ing 1.00 0.18 5.3 X 10' Intemally balanced Approach­ ing 1.00 0.14 6.0 X 10' NACA NACA Piain 64,2-(1.4) (13.5) NACA 65,2-318 approx, NACA 63(420)-521 approx, NACA 66(215)-216 a ,. 0.6 NACA 66(215)-014 NACA 66(215)-216 a == 0.6 NACA NACA 8.0 X IO' Approach­ ing 1.00 0.20 to 0.48 2.8 X lOS Plain 1.93 0.09 1.2 X lOS Plain Approach­ ing 1.00 ..... 6.0 X 10' 0.13 6.0 X 10' 0.13 6.0 X 10' 0.13 6..0 X 1()6 0.14 8.0 X lOG 0.13 6.0 X 106 0.13 6.0 X 106 Internally balanced Internally balanced NAC.A Internally balanced Internally balanced KAC4~ 64,3-1 (15.5) approx. Approach­ ing 1.00 Approach­ ing 1.00 .... Internally balanced 12)-213 745A317 approx. NACA ~3-013 approx. .... 13.0 X loa mg 1.00 6..0 X 10' Plain NACA Approach­ .... 0.14 Plain 65.-421 65 ( Internally balanced Plain 65r415 NACA 65r418 - .,. NACA 66(2x15)-116, a == 0.6 - Intemally balanced Approach­ ing 1.00 Approach­ ing 1.00 Approach­ ing 1.00 Approach­ ing 1.00 Approach­ ing 1.00 Approach­ ing 1.00 A:pproach109 (c) Supplementary information. Fla. 96. (CQTJditded) 191 1.00 to 6.8 X 106 I 0.13 I 6.0 X lQ6 • THEORY OF WING SECTIONS 192 teristics may be calculated from the theory of thin wing sections (Chap. 4) if the flow does not separate from the surface. For most commonly used wing sections, this condition is satisfied reasonably well for flap deflections of not over 10 to 15 degrees. The theory permits calculation of the angle of zero lift, the pitching-moment coefficient, and the ehordwise load dis­ tribution with reasonable accuracy. The flap loads and hinge moments may also be obtained, but the accuracy of these quantities is relatively poor because the effects of viscosity are particularly pronounced over the aft portion of the section. The effectiveness of the flap in increasing the to maximum lift coefficient cannot be (6 in rOdor/s) calculated. ~'Ch I Glauert" calculated the effect of .8 do plain flaps in changing the angle of zero lift, the pitching moment, and dcm e / ' ~ <, the flap hinge moments. The cal­ ~_~4 .6 If d4 r-, I culated effect of flap deflection on the angle of zero lift is shown in Fig. '~ "", dch _ 96, where the flap effectivenessJ l1aol alJ is plotted against the ratio I ~ -, dcz~ .2 of the flap chord to the section chord ~~ ~ -, L.-­ ~ cllc. Numerous experimental points ~ r-, are also shown for flap deflections of 6 .2 .4 c· .8 I.IJ 0 to 10 degrees and 0 to 20 degrees. N -Z- " / I -: '" I'< -, -, ~,..... In general, the flap effectiveness is less than that indicated by the the­ ory, and the discrepancy increases with increased flap deflection for small chord flaps. The calculated ef­ fectiveness of a plain flap in changing the section pitching-moment coeffi­ cient about the quarter-chord point dc./dO is shown in Fig. 97. The calculated values of the rate of change of hinge moment with lift coeffi­ cient dch/dc, and the rate of change of hinge moment with flap deflection dcA/tU are also shown in Fig. 97. For symmetrical wing sections, the theo­ retical pitching-moment and hinge-moment coefficients are 97. Theoretical hinge moment and pitching moment characteristics of plain trailing-edge flaps. FIG. em = a(~) _ (den) + ~ (dC\) lU en. - C, - 0­ dCI The linearity of the theory of thin wing sections permits the applica­ tion of the values obtained from Figs. 96 and 97 to cambered as well as Symmetrical sections. In the case of cambered sections, these values are applied as increments to the corresponding values for the unflapped section. HIGH-LIFT DEVICES 193 Pinkerton" calculated the flap loads that are presented in Fig. 98. The rate of change of the flap lift coefficient with the section lift coefficient dell/dc l and with flap deflection dc,l/dJ are plotted against the ratio of flap chord to section chord. 3.0 2.8 1 I W/Ild dir«:lion ..... L.I 1 ~''''''''''': I \oc 2.6 I 2.4 2.2 ik CII:;;;' r\.. ~ 2.0 " 1.8 ~~ ~I~ a "c, +::;"6 de: I\. 1.6 1.4 '\. 1.2 "r\.-, !f:JL 1,,<6 1.0 ~ .6 .4 .2 ) ~ ooV ./ ~ ~ ~ ~ ~ / V ~ t<~ r\ .2 .3 .4 .5 .6 .7 .8 £, Ratio of flap chord 10 totatchord l_ '\~ -, .9 1.0 FlO. 98. Theoretical variation of flap lift coefficient with section lift coefficient and 1lap deflection. The pressure distribution over a lying section with a deflected plain flap may be calculated by the "ring-section theory of Chap. 3, but such calcula­ tions for a number of flap deflections would be unduly laborious. Some knowledge of the ehordwise load distribution is given by the theory of thin lying sections. The change of load distribution caused by flap deflection can be resolved into t\VO components, one of which is simply the additional load distribution associated with angle of attack. The other component is the difference bet,veen the load distribution on the original and deflected mean lines at their respective ideal angles of attack, These load distribu­ THEORY OF WING SECTIONS 194 tions may be calculated by the theory of thin wing sections (Chap. 4) as explained by Allen,' but the results are not very accurate. For example, the simple theory predicts infinite load concentrations at the leading edge and hinge location. The infinite load concentration at the leading edge may be avoided by use of the additional types of load distribution associ­ ated with angle of attack as obtained from thick-wing-section theory and presented in Appendix I. Allen' obtained empirical load distributions analogous to those associated with changes of the camber that permit calculation of reasonable approximations to the load distributions of flapped 6 - - - R.N. .356 x 10 - - R.N.6.10 x/OS -3 6=50­ 8·:40· 8=30 0 +/ 0 4=20 . . 4=/0 0 ConsIont angle of o~/ack 20%cflop. art" FIG. 99.. Distribution of pressure over a wing section with a plain flap. wing sections. Allen's method is presented in Sec. 8..8. Once the load distribution is obtained, the pressure distribution may be calculated by the method of Sec.. 4.5. Some typical pressure distributions for sections with plain flapsfl are presented in Fig. 99. The lift, drag, and moment characteristics of a typical NACA 6-series wing section with a O.2Oc plain ft ap3 are presented in Fig. 100 for several flap deflections up to 65 degrees. These and other data" show that the angle of maximum lift coefficient with the flap deflected is generally some­ what less than for the plain wing section. Curves of the maximum lift coefficient plotted against flap deflection for two sections" are presented in Fig. 101. Plain flaps of 0.2Oc appear to be capable of producing incre­ ments of the section maximum lift coefficient ranging up to about 1.0 and are more effective when applied to sections with small amounts of camber. Addit~onal data on plain flaps are presented byJacobs," Abbott," and Wenzmger. l 48, 1., 116 HIGH-LIFT DEVICES :~:: ~ e •• .... ... .. i!:: I~:: :i:· i~" . . .•. :::: :::i ;.. *:=! :i:: ;.!:.... :E: :r-: !::: .•. 195 e •••••• ;!~: eo. :e:: :::: ;~!: ~~g . 'mm i ! :::~ m= ~j:~ m: .., . . . 2.8 i::; ;: .... .. ... .... :~~ 2.4 '::: •• : _ • • e o•• ....... _ 2.0 ':i: .. • ••• e .. . ~:.:.. .... m~ : ...•.­ ..... i;;.. .... . : .. ' . :­ ~~:fiI;1.' J~~'J ...... ;i:; . . ~:~: ~:: • , ... .. ••.•••• . .): .;:4 .. (: : :.: : : :. : :.: . .: . :i:~: ~ I~ ~~ .~ ~ '.e :1::.. '::Jf:j .. :::: . .. .. .... f: I(. ~!!'JYI\" ,) :~' •.. ~J~j,~ (iii ..: .. :. i.~ ~j i~ ~ilih~~, :/·r. ~J 'i: II."'·· • Ji ".J:l.!;i!.i.i :;:~.. ~: " ••, i· ~ii:· ~ 'e' '" :;:::: . , ~.::. ':. :.: :' ;~~~ ~~; ~p~ ~~: (. . :::~~'.:' lJ.i!.llI~ 1 i /1: ­ ~ -: ~J~' ).J::: ":. ... ... .. ~:~:'1. 0.4 o -0.4 I~ I - " .. 0 I ::iIJ~;:·~j:tfaj~.-j..jl ',oi: II. 0."/ F' r~1 r t. ~ Jilt:· t-.I P J -; I -. ;.. , - .~ .J:/ 1/' I ( ~ " ..... : If· (;~ ~/t: I Ifill " 0'0 ._; .. ..... .~; .... :: . .. : .. ;;!~ ;=: .. : ..•~: .'-.' ~: ~.~. ~~:~ •. ... -1.2 0 J' (4es) :':-::<::"~_:::~~;;~:;~;~:J':~~~~.g fi '.: ':'. .. ~: -0.8 If. ~- .. -24 •. .~ .:. :·f :" .. ~.• ~ . .:. (. ~." ... ... . ~. . ~~ =r~ - .... ..16 ;;: -8 , 1+_~-+-+-tf-o+""'+-~ ~ I I 6 (4es) ... .... : -- & -i~ i"'~ ~ -15 :i ;.-I: : .. :":. ;&;: ~. •. .. 0 '. ~"l' L '~'I'l; 8 I 16 I .. : 24 32 Section angle of tJtt4ck. «0' U6 100. Aerodynamic characteristics of the NACA 66(215)-216 airfoil section with O.20c sealed plain flap. FlU. THEORY OF WING SECTIONS 196 ....;g; :ii : . ~~:: " . " 0.028 ............ :::!................•............................. : .... ... ! ! ... ·1 I I t ........ i ~ .. ~-' 0.024 !. ~ i I O.ORO I 1 .$1 0.016 t l 0.012 j ! '" :::... :::.::: :::.:: ... :=::.. .,. .... l;" =, "fI :~ :/i/, I : i ~ 0.008 0.004 . 0 ~-O.2 Co» if .! .!l -0.4 ~ § 1 ::~1,15~:~;~f:j;:~: ;~~~ . . :' ::: ~.~~ r:~ ~~\ r .. .. .. ::: ~: : -0.6 ........................: ~ .. r:' , ~ :.' .. . _m: t~ :' : ._ . .. ;~ :.. I I ! i I : : .. ., ..: i i -0.8 -0.8 -0.4 o FIG.. 0.4 0.8 1.2 Section li/t coefficient, c, 100. (Concluded) 1.6 2.0 2.4 HIGH-LIFT DE1'ICEB 197 When the turbulent boundary layer over the aft portion of the wing section is thin and resistant to separation (as when extensive laminar flow is obtained), small deflections of a plain flap do not cause separation but shift the range of lift coefficients for which low drag is obtained.' If the extent of laminar flow is not large or if the flap deflection is sufficiently 2.8 J3 2.~ DCA ~ ..It .. 2.0 ~ s: ~ c... ~ 1.6 3 0 ~ ~ c 1.2 0 ... V J~ ~ ~~ '-4 ...... ~ DCA 65,3-618 ~. 0 0 66(215)-216 ~ ~~~ ....... > ; JI e-Vc?";S -t ..g• 0 .!S •• ~ =­ .k o ~ ~ ~ ~ ~ Plap detlection. 6 . des FIG. 101. Aiaxirnuill lift coefficients for the NACA 65,3-618 and NACA foils fitted with O.20..airfoil-chord plain fiaps. R. 6 X 1(;'. . ()6(215)-~16 air­ large to be of interest as a high-lift device, the flow over the flap separates and large drag increments result. .A. typical variation of drag with lift coefficient for the NAC..\ 23012 wing section with a O.20c plain flap de­ flected the optimum amount at each lift coefficient/ is shown in Fig. 102. 8.4. Split Flaps. The split flap is one of the simplest of the high-lift devices. The usual split flap is formed by deflecting the aft portion of the lower surface about a hinge point on the surface at the forward edge of the deflected portion (see Fig. 95). One variation of the simple split flap has the hinge point located forward of the deflected portion of the surface in THEORY OF WING SECTIONS 198 such a manner as to leave a gap between the deflected flap and the wing surface. In another variation the leading edge of the flap is moved aft as the flap is deflected, either with or without a gap between the flap and the wing surface. Split flaps derive their effectiveness from the large increase of camber produced and, in the case of some of the variations, from the effective increase of wing area. , I I II , J I I l J ~ I I IPlDfnf_~ I" I~~ I I ,,~~~ .04 ~ ~ ~ ~~ ~ .-.~ ~ 'I 1/ r-, ~ ... ~ I J ~ 'Y~ t> U ~ ~ ~ StJditJn /lfl Df)tI/Iit:itlnI,e" FIG. 102. Comparison of profile-drag envelope polars for the NACA 23012 airfoil with O.2Oc plain and split Saps. A ~ The lift and moment characteristicsl for typical NACA 6-series wing sections with O.2Oc split flaps deflected 0, 40, 50, 60, and 70 degrees are pre­ sented in Figs. 103 and 104. Similar data for a flap deflection of 60 degrees are presented for most of the wing sections of Appendix IV. The lift-curve slope with split flaps is higher and the angle of maximum. lift is somewhat lower than for the plain section. There appears to be a tendency for the lift-curve slope for large flap deflections to be less than that for moderate deflections.v 154 Inspection of Figs. 103 and 104 shows that the split flap is more effective in increasing the maximum lift coefficient when applied to the thick than when applied to the thin "ring sections. Figure 58 shows that the maximum lift coefficients of NACA 6-series wing sections with O.20c split flaps de­ flected 60 degrees increase with thickness ratio up to ratios of at least 18 per cent. Figure 58 shows comparatively little variation of maximum lift coefficient for NACA four-digit wing sections with O.2Oc split flaps for thickness ratios greater than 12 per cent, but the maximum lift coefficient HIGH-LIFT DEVICES 199 •.••• "I~r- 2.4 +_+,-,= ............. ~i)1 :.~ • . . AV-Vi e J. , 'J .~~ _. r/ : \ ~ 1 }. 'II II 1, , r,,'/ / "I 0.8 ~-;.-...I--+-f--l~oIltH-II-~-+-I-""""'" r. rl. .. ..J"'; i .1 ~ 0.4 I/r~/ :~~:. Ifl/.: ... ~ - ..'­ .. _ 0 - "'-I-­ 6 "'- .... (deS)",-":';': .. 0 1­ .... .. ;.. " ~r .. J.. ..• ' .., .. ._ . - ... :; ~"--r-~..;""",,,-~~-+--&o-~~~~~~ 'il _. .. .. .­ '. I' . . :.. . ~·:IJ. .J ,.' ... .. .. 1/ .: ,J/J.l 1J. Jl II. I, r- I' ..~ .. " I , J :.'~I-1-'-+-~""""'~oI--i- "'-~- I -M -16 -8 8 0 16 &dioII G1I6k of atIGei, ...,. . . .. ~ ? -0.1 ! ! .! : t i :. i ; ~ ·.i-· I 'l t ~- : I : l 4 l ! I ~~+............;--hIl!=-4--i-~-I-~~~~~-I-~~ .. .i r. ~ -0.2 ; .....-t.. .T-lt-+-.......-+-I.....:..r....... Cj :I r­ .:... '..: .~' .. i. i' : : =-­ .t ! ; !. I &i8~+t+~RmP+4++++tt~~:++:t=d=~~ L f- ; .' A ~ : I -O.8ltttJet!:tt!lt:tJjtt:1:ttttrnm~!eI!~~~tt1 ~ ~ - .... : I-'i~~- ·i:··· ~:!: ..• : •.... l'.:.. 0 .- . . • . • • • . ··1·.;.. -0.4 ...... I:... , ... -0.8 -0.4 : ': ... ::'- o '" '" ... 0.4 :j' i --:;! _.' 0.8 i t ~ ~.,.'-: eo ,-; .T.! IJ· .. II ~. l~ :.' 1.6 Set:tita lift t:«/fit:iDII. c, FIo. 103. Lift and moment characteristics of the NACA 66(215)-216 airfoil O.2Oc split flap. R. 6 X lOS. eectiOD with THEORY OF WING SECTIONS 200 o e 6 V ~/:I ~-t-+--+-+~~ ~ ~~;"""'-'-J-""""""""""~-+-.f--I.--I-~-i ( ) tIl} iJ I i' i7t --.'f "I j - ~-:-. i .. r·· _...... I .. ~ -16 - 8 0 8 ~4 16 Section ",.". of atled. «0. . . o · -j : joii"" 1 • r­ : :...! tin · .iV-. i\ ,,~. N\. .... .. - &ft -, I ~ ! · - r- -0.4 -0.8 ~ : ! - .. l -0.4 1---_ I . .. -~.~ - 0 -­ -- ...,. ~- -­ : --­ .O~ . 0.4 . . ~_. _. 0.8 ~- I­ - ~.- .. Q' ., 1.2 8edioIIli/t eoeffidat. c, t, ; ~o- 1.6 ­ .. _. - -_. .... -:::. -u -. ._­ - 1 ~.~ ~ _. _0. - ··1 .­ -­ ~ ~ .~ 1 .-. -~ ~ 'I I 1-. ~. , ~ ~..J ! .­ ~-.:/ ,. ~ - - - .~ ~:.. : ..... 1-­ _.­ i -2.0 2.4 104. Lilt and moment eharacteristiea of the NACA 651-212 airfoil eectiOD with O.2Oc split flap. R, 6 X 10'. FIG. HIGH-LIFT DEli''lCE8 ~1 decreases for smaller thickness ratios. These results are in essential agree­ ment with those1M of Fig. 105, which shows that, for the NACA 23o-series sections, the maximum lift coefficient generally increases with thickness ratio for large-chord split flaps but decreases for small-chord flaps. 3.2 I . ef· ~ 2.8 [~ 2.4 ~ r---­ --- ~ ::z­ •4,0c ....., .- .30c ~ ~ - .""..... ~ s:I ~ ~ ...... 2.0 j I r----c ~ .IOc C­ ~ 1.6 ~ I - .2Oc i 1 I ..... ...-.. ............. 1.2 .~ ~ ---- -...........­ ~ ~, ~ '-.. ~PIoin airfoil ~ i .. .8 a 12 16 20 24 28 AirIo1II1icknt1ss,prctJnI e (NACA. 230 u,itJs) 32 FIG. 105. Effect of airfoil thiCknC88on maximum lift coefficient of N AC~\ 2.1O-8eriea airfoils 1\ith and without split flaps. The variation of tile increment of maximum lift coefficient resulting from deflection of split flaps of to, 20, 30, and 40 per cent chord 1i4 is shown in Fig. 106. For wing sections of normal thickness ratios, substantially full benefit is obtained for flap deflections of 60 or 70 degrees. The effect of the chord. of split flaps on the increment of maximum lift coefficientl M is shown in Fig. 107. For wings of normal thickness ratio, comparatively little benefit is obtained by increasing the flap chord to more than 20 or 25 per cent of the section chord, although large flap chords are more effec­ tive on thick than on thin sections. Deftection of a split flap produces a bluff body, and large drag incre- THEORY OF WING SECTIONS 202 ments are to be expected. Figure 102 shows that the drag of the NACA 23012 section with a 0.2Oc split flap deflected the optimum amount at each lift coefficient is about the same or less than that for a plain flap.! The pressure distributions over wing sections with highly deflected split ftaps are similar to those for plain flaps (Fig. 99). The pressure distribu­ tions with deflected split flaps may be predicted by the semiempirical method of Allen.' v j// /1/ ....,->-­ 7/AI 1// lb 20 40 60 80 (a)-NACA23012 ARFOIL FIG. 020406080 Flog delleclion,66 deg. (b)-MACA 23021 AIRFOIL r o .20 40 60 80 sx (c)-NACA 23030AIRFOIL 106. Effect of split-flap deflection on increment of maximum lift coefficient for various airfoils and flaps. Data required for the structural design of split flaps are given by Wenzinger and Rogallo.l11 The normal force coefficients and centers of pressure for a 0.2Oc split flap at various deflections are shown in Fig. 108. The data of Wenzinger and Rogallo show that the chordwise position of the forward edge of the flap has only a small effect on the flap loads. The effect of a gap between the wing surface and a split flap with a nominal chord of 0.2Oc hinged at O.SOc from the leading edge of the sec­ tionllO is shown in Fig. 109. The data show that the lCBJ of maximum lift coefficient associated with the gap is considerably greater than that caused by removing the same area from the trailing edge of the flap. Figure 110 summarizes the effect 1," on maximum lift coefficient of moving a O.2Oc split flap toward the trailing edge from its normal position. These data show that the percentage increment of maximum lift coefficient obtained by moving the split flap to the rear is about the same as the percentage increase of wing area measured as projected onto the original HIGH-LIFT DEVICE8 203 chord line. Similar results have been obtained!" for O.3Oc and O.4Oc split flaps. 8.6. Slotted Flaps. G. Description oj Blotted Flaps. Slotted flaps pro­ vide one or more slots between the main portion of the wing section and the 2.0 I 6: 75· fJO° 1.8 1.6 9fJ.)r N.AC.A. - 23OJ() 2102/ V / ~ ~ ~.I ~ 75- /SO· I / / g5*' / V V ~ 1/ /785·7 7/ VI V J,~ I BO D17 [,.......- ..-75- 60· 55P-23012 c u~ .4 r/ I .2 a 20 30 40 50 Flap chotrJ,pen:enI C Flo. 107. Effect of chord of split flap on inerement of maximum lilt coefficient for three airfoil thicknea8eL deflected flap; and they derive their effectiveness from increasing the camber and, in some cases, from increasing the effective chord of the sec­ tion. The slots duct high-energy air from the lower surface to the upper surface and direct this air in such a manner as to delay separation of the flow over the flap by providing bdundary-layer control. The numerous types of slotted flap are classified by their -geometry. Several types are shown in Fig. 111. The primary classification is the number of slots. The single-slotted flap is the simplest and most generally THEORY OF WING SECTIONS 204 used type. Double-slotted flaps have been used to some extent, and multiple-slotted or venetian-blind flaps have been investigated. An im­ portant consideration in the design of slotted flaps, especially of the single- r- i s: ~I c ~ .20~ .SOc Hnge axis l8­ ;::: t 20 c - 40 ~- ~ ,no..... -----...~ i 6. 41\ .............. 60 Ci ~ deg. 0 20 A 40 60 a .. ~ I-"""" 1.20 0-""'"'" ~.80 ",.,- ~ .-a-~ "......- ~ ...... ~ - -- :I r-a \S ---- ~ c:,.,...-- .40 ----- ro--..-~ ,- 0 o 40 .80 1.20 eN 1.60 ~ \ LOO FIo.108. Normal force coefficients and centers of pressure of a O.2Oc split flap at O.8Oc on a NACA 2212 wing. slotted type, is the extent to which the flap moves aft as it is deflected. The movement of the flap may vary from a simple rotation about a fixed point to a combined rotation and translation that moves the leading edge of the flap to the vicinity of the normal trailing-edge position. Rearward movement of the flap requires an extension of the upper surface over some or all of the flap in the retracted position. This extension of the upper surface serves to direct the flow of air through the slot in the proper diree­ HIGH-LIFT DEVICES 205 tion and is called the "lip." The external-airfoil Hap (Fig. Ill) may be considered as a special case of the single-slotted flap with the distinguishing feature that it does not retract within the section. The flow about a wing section with a deflected slotted flap is very complicated, and no adequate theory has been developed to predict the aerodynamic characteristics. Consequently the information required for (0 ) ,...---------c=IO.00"---------"'=---~ (b) (e) Y.IOC 60° 0.20';( \:~ FIG. 109. (4) Details of split flaps with gaps tested on Clark Y wing. design is obtained entirely by empirical methods. Although many experi­ mental data have been accumulated, the large number of configurations possible and the sensitivity of the chacteristies to small changes in the slot configuration make the design problem a difficult one. b. Single-sloUed Flaps. One important parameter in the design of single­ slotted flaps is the chordwise position of the lip. Although completely comparable data are not available for configurations with varied positions of the lip, the maximum lift coefficients appear to increase as the lip position approaches the trailing edge for lying sections of moderate thick­ ness. This effect19• 153 is shown in Fig. 112, where increments of maximum lift coefficient are presented for the NAC£.\. 23012 section with single­ slotted flaps having lips located at O.82ic, O.9OOc, and 1.000c. The con­ THEORY OF WING SECTIONS figuration with the lip located at 0.827c is not exactly comparable with the others because the chord of this flap is 0.2566c as compared with 0.3Ocfor the others. It is apparent, nevertheless, that the increment of the max­ imum lift coefficient is considerably higher for the configuration having the lip at the normal trailing-edge position than for those with a more forward lip location. It is uncertain whether any of the difference between the 2.4 I I I I I I ~No gqJ(remating flt:p Itdlhg edge) I A", ~ 2.0 ..... ......... ---- ----..... ~ ''x 8 . . . ,... I <-s­ 1.6 >-- ~ r-, '''" r..,., I"- Witlrgrp_V (temrNi7g flop Ieodilg edge) C 1'.- ~ ,,~ i'o"' ..... ~ .......... to--............. 1.2 ..~4i AtJfJop/ .8 --~ "--- ~--::"'~ r-- ::::::-::- t::-o--. ....... .~I--"::: oo I ~ ~ I ~ ~ w Reduclial ofchotrJ,percenl flcp chad FIo. 109. (Conduded) (b) Effect on CLmas and on CD at CLmaz of reducing the chord of a 0.2Oc aplit flap. a.. 60 dep'8e8. other two configurations should be attributed to the difference in lip posi­ tion. Envelope polars for these flaps are presented- in Fig. 113. When the lip is located at or near the normal trailing-edge position, the thickness of the flap is necessarily less than that for a more forward location of the lip20 (Fig. 114a). In the case of thin wing sections, especially of the NACA 6-series type, the flap thickness may become too small with a rearward location of the lip to permit favorable slot configurations. Under such conditions, the favorable effect on the maximum lift coefficient of moving the lip toward the normal trailing-edge position may not be realized. CahillJO shows (Fig.. 114b) that the maximum lift coefficients for HIGH-LIFT DEVICES 207 c FIo. 110. Contours of eL..... for various positions of trailing edge of 20 per cent flap. c:::: ~ SlDTTED FLAPWITH LONG LIP c ~----- c ~ Ell1£RNAL-AIAfllIL FLAP , 1"'10. Ill. Several types of slotted flaps. ~ 1.8 ~~~ / 1.6 7 00 \ ", II" / 1.4 I ~ V·" 1"2 ,",I ~ 1.0 ·~1 ~ .6 ~/ / V .2 ,I ~ I ,, .... ~ ~ ,," ...... 1-0-­ ---'" -­ -­ , I ~ V~~ / "'< C I ~ ~ ~ ~ I( ~ " ~ ~ ~. J4 / I I j / 7 V ) "","--. ~ ~ / I ~ /" ,I " j J I V I i .4 ,"l J7 11 .8 i ~ ~, ,I ~ O.30c FOWLER FLAP;GAP-O.0I5c .\ I -s ---­ Q2!J66c..sldled1/q'J 2-h (ref.153) 1 0 ­ _.- Q3Oc..... Fowler l/(pj gcrJ·01Jl5c - ­ Q3Oc••... slolled1/q)"!11 ex/ended I - ­ IlPIgq:J-O.02c ~ » ~ ~ Flopdeflection, 4, deg. Flo. 112. Comparison of increments of seetlon maximum lilt cneffi~ieut for three flaps on a NACA 23012 airfoil. ~ C ~ HIGH-LIFT DEl"ICEB 209 the NACA 65-210 section are essentially the same for lip positions of O.84c, 0.9Oc, and 0.975c. The structural difficulties presented by a long thin lip extension and the mechanism necessary for the corresponding large rear­ ward movement of the flap are such as to discourage the use of rearward locations of the lip unless such configurations result in substantial improve­ ment of the maximum lift coefficient. :> Q30c sIoIfed llop with extended lip; -w---+--+-t--+---+----f gop=O.02c .20~ ~ -- _ ---- \ asoc Fowler Ilop;gop'O.o15c o.2566c s/olted flop 2-h (reference J53) I Plain airfoil 'V I I t I I .4 .8 1.2 1.6 2.0 2.4 2.8 3.2 $«lion liftcoefIJc/enl1cz FIG. 113. Envelope polar curves for three slotted flaps on a NACA 23012 wing section, The effect of flap chord on the increment of maximum lift~· U3 is in­ dicated by Fig. 115. The data presented .for the O.. 25(j(ic and O.4Oc flaps are reasonably comparable in that the shapes of the slots are generally similar. Figure 115 shows that larger increments of maximum lift coeffi­ cient are obtained with the larger chord flap, but the increased effectiveness is small compared with the increase of flap chord, The slightly higher maximum lift coefficients obtainable with large chord flaps do not appear to justify the structural difficulties encountered with such flaps, and flap chords in excess of O.25c to 0.3Oc are seldom used. The maximum lift coefficients obtaineiP" 96. 1&3 with various arrange­ ments of slotted 8ap on NACA 23012, 23021, and 23030 wing sections are plotted in Fig. 116. The flap chord was 25.60 per cent of the section chord in all cases. These data indicate little variation of the maximum lift coeffi­ cient with thickness ratio from 12 to 30 per cent for this type of wing section. A few data3, . are also shown in Fig. 116 for comparable slotted flaps on NACA 6-series wing sections, In this case, the flap chords are 25 or 30 per cent of the section chord. These limited data indicate that, for the NAC.A. 6-8eries sections, the maximum lift coefficients obtainable with ~ ---O.84c ~ o""'" 2.,8 2.,4 Slotled flap I - ~ :;;;...00""" ~~~ -~ ~ J-~~ (.) ~ I '.6 - - O.90c ~ ~ :f § s .6 1.2 5/011«1 flop 2 -~ ~ >---- l---' J .4 ~ gj es c1 (degJ ~ o Ph/no/rfoI1. ---~ e SbHfd fk¥J 1_ _ 45 A '1 ~ SIdled/Iff' 2 _ _ 41.3 SIdled IlqJ 3._ _ 35 I 0 - Sio/led flop 3 FlO. ~ ~ i .8 (0) CONFIGURATICWS "'< .-- - .. _.-: - Reynolds trr!1Iw, R (b) MAXI~ LFT OlTA 114. Variation of maximum eect.ion lift cooflicjent. with .RM)Onoldi' number Cor several slotted flaps on the NACA 66-210 whig 8(wtion. HIGH-LIFT DEVICBS 211 1.4 I' c / D.2566cflap V f / I , ........... .-' -,' - -­ -­ ~- -­ ~' ~o.4Oc flap i. i I I ,, I c-------­ I ./ 'I l "" O.2566c flop I I'J " l' If 00 (0) CONFIGURATIONS D 20 .JO 40 Flop *,~,dtJg. !JO 60 (b) INCREMENTS OFMAXIMUM UFT COEFFICIENT FIG. 115. Effect of flap chord on increments of section maximum lift coefficient for the N ACA 23012 wing section. 3.2 I + --- ---.. § 1 ............., 4 E .~ 2.0 ~ .~ ~ 1.6 - Symbol Kfng section series 0 x ... ~ I 1.2 6 10 , N.A&.A.230N.AC.A. 6 N.A.CA 6 - 14 Flop chord ratio O.2566c -- o.ese 1 i , B.. 22 O.JOc 1 -- I 26 30 section Ihickness rolio, Pc, percent FIG. 116. Maximum lift coefIiciente for various arrangements of slotted flaps. 212 THEORY OF WING SECTIONS slotted flaps on 10 per cent thick sections are appreciably less than those obtainable with thicker sections. The effects on the maximum lift coefficient of some variations of shape of the slot are illustrated'" in Fig. 117. In configurations l-a and l-e, the slot is only slowly converging, if at all, at the end of the lip, and these COD­ [ Skilled flop 1-0 8,640~. 62.~6 Czm.. [ SIoIIedflop I-b 6,6SOdtlg. clmax.· 2.76 l SJolltJd flop t-c 6,.:55 de9. "1mcr.'X. ·2.75 [ SIrJIIedfkJp '-ee 6i 645 t!tJ9. 'f"...•2.il9 l SIdled f/cfJ e-« 6,.:50~. 'ir.. ·2.81 l SbIt«J fltJp2- i 4,·60dtlg. Czm... • 2.674 [ SItJIIt1d fltJp 3-1 t1,.50~ '"'-w. .:2 I SItJIItId fltJp 3-g 6,.55~ c'--.· .60 FIG. 117. l\laximum lift coefficients attainable with various ammgements of slotted flaps on the NACA 23012 wing section. figurations have the lowest maximum lift coefficients of the I-series configurations. A short extension of the lip as in configuration I-b, which makes the slot definitely convergent and directs the air downward toward the flap surface, is effective in increasing the maximum lift coefficient. It may be concluded from these and other data that the slot should be definitely convergent in the vicinity of the lip and shaped to direct the air downward toward the flap. The effects of changing the radius of curvature at the entry to the slot from the lower surface arc shown by configurations I-b and l-c of Fig. 117 for a flap having a comparatively small rearward displacement when deflected. Decreasing the radius of curvature from about 0.0& to O.04c did not produce a significant difference in the maximum HIGH-LIFT DEV"ICES 213 lift coefficient. Other data indicate that this radius of curvature is of little importance when the flap is displaced rearward enough to produce 8. large area for the entry of air into the slot. It is difficult to draw general conclusions about the proper shape of a slotted flap. Figure 117 shows that the highest maximum lift coefficients were obtained with flap 2, which is shaped more like a good wing section than flaps 1 or 3. The difference between the maximum lift coefficients produced by flaps 1 and 2 is small and may be caused by the difference in slot shape and lip extension rather than by the difference in flap shape. FlO. 118. Contours of flap location for 23012 wing eeetion. Q lllar Slotted flap 2-11., &, = 60 degree. on NACA Flap 3, however, appears to be too blunt with a too small radius of curva­ ture on the upper surface aft of the lip in the deflected position. Typical cont.ourst61 of the maximum lift obtainable with various flap positions at one flap deflection are shown in Fig, 118. In' general, the optimum flap position for good flaps at large deflections appears to he that which produces a slot opening of the order of O.. Olc or slightly more and which locates the foremost point of the flap about O.Olc forward of the lip. The maximum lift coefficient, however, is frequently sensitive to the flap position, and the optimum position is best determined by test. A complete set of section eharacterisrics'P for a typical single-slotted flap configuration is shown in Fig. 119. This figure illustrates the charac­ teristic ability of slotted flaps to produce high lift coefficients with com­ paratively small profile drag coefficients. The increment of moment coefficient associated with the use of single­ slotted flaps" is illustrated by Fig. 120. This figure shows the ratio of the increment of the section pitching-moment coefficient" to the increment of the section lift coefficient at an angle of attack of 0 degree for three wing sections and several flaps. The moment coefficients used in this case are ~ o ~ t9 ~ - ~~ 0, d6g. o .20 , --b. ~ 2O--v 30--0 50--1> 60--6 -- d I. _nose - [ ~ I:lJ" / ~ I'V for deflecfed flop II ~ J~v ~ .A~ rr .... J~ ~ lD'"~!o"""'" 10-~1IIr' - o ~~ ~ " - ID- ~ ~ -' ~ ~v l--""" l-=== ~~ I /6 ~ ~ ~~ ~- ~I-""" L...;l~ ~I""'" ~~ lJW-­ -~ IP" o .4 ~v l/~ I...oo"'~ L..--'~ 10<'" 1.2 .8 ..... ~ "J" N'''' ~ 'C l...oo"'~ ....~ ~t,I' .~ ~,;IIiT ;/..". l/~ ~Io""" " 1/ ...... 1.6 20 2.4 2.8 Section lifl coefficienll cl. Path of flap nose for various fiap deflections. lip in per cent airfoil chord c. 81, degrees 0 10 20 30 x y 8.36 5.41 3.83 2.63 3.91 3.63 3.45 3.37 Distances measured from lower edge of If, degrees x y 40 50 60 1.35 0.50 0.12 2.43 1.63 1.48 I FIG. 119. Section aerodynamic characteristics of NACA 23012 airfoil with slotted flap 2-A. 214 HIGH-LIFT DElTICB8 215 based on the total chord with the flap deflected. The ratio plotted appears to be fairly constant at yaIues between about - 0.17 and - 0.21, instead of varying with flap-chord ratio as in the case of plain flaps at small deflections. The normal-force coefficient, pitching-moment coefficient, and center of pressure for the flap of the configuration of Fig. 119 (configuration 2-h of reference 153) are shownlU in Fig. 121. All the coefficients are based on the flap chord. The pitching moments are taken about the quarter-chord point of the flap, and the center of pressure is given in per cent of the flap o -.04 1'--- -.08 "~I" ~ ~ Airfoi/ sections ~- o a 1-0--- o NAG4 23012 / N4CA 2J02I N4G4 65-2/0 ~ /' <3-./2 /V -./6 V •10 ~o() .20 ~ ~/ ~T/le(relcol (plain flop) V 0 ~) V- V i-'- - 0 . .30 .40 .50 .60 F/q) -chotdratio, £ FIG. 120. Variation of ratio of increment of aection pitching-moment eoefIicienf, to increment of section lift coefficient with flap-ehord ratio for several sections with elotted flaps. GO = 0 degrees. chord from the leading edge of the flap. These data are useful in determin­ ing the loads on the flap and the flap linkages, and they show that the normal force coefficients on the flap are less than those for the wing section. c. External-airjoil Flaps. The external-airfoil flap investigated by Wragg: 56GPlatt,838.f86 and Wenzinger161 may be considered as a special case of the single-slotted flap in which the ftap does not retract within the wing section. The maximum lift coefficients'4 obtained at an effective Reynolds number of 8 million for a N ACA 23012 wing section with & O.2Oc external­ airfoil flap of the same section are shown plotted against flap deflection in Fig. 122. These maximum lift coefficients are based on the combined areas of the wing and flap. The values presented were obtained from tests of a ... ~ 0) §I L.E. ~~20 't5 '.-: et i i~ ~t 40 0 ~ -./ ~ ~ j{-.2 I) ~ ~3 .... j'~ 2.0 (j ~8 1.6 O.i ~ ~ ~~ 21·w 1.2 Cb~ ~~~ .8 4~ ~.~ l~ .4 ~ §' °0 .6'8 ~ oJ::. .4 .8 1.2 1.6 2.0 .4 .8 1.2 1.6 2.0 2.4 0 .4 Norma/-force coefficient of-combination,cn .8 1.2 1.6 2.0 0.f =40() (c) is =50 0 I Seolion chnrBcteristiclt of the flap alone or the NACA 23012 airfoil with a 0.25660 slotted flap. (0) FlO. 121. 2.4 0 0.f =30 D (b) 2.4 28 '3 ~ ~ ~ ~ ~ S ~ HIGH-LIFT DEVICES 217 finite span model and were only partly corrected to section data. The corresponding section maximum lift coefficients are judged to be about 4 per cent higher than those presented. If these maximum lift coefficients were based on the chord of the wing section, as is customary for other types of flap, the resulting values would be about the same as those for single­ slotted flaps. Although the extemal-airfoil flap appearedu, 8& to offer some 2.4 ..:­ a/f"" .i / V ./ '/ i-r ~~ ~ \) ~ 1.6 1 ~ C", ~ 1.2 ~ 1§ ~ .§ .8 ~ P,iKJ/~ f 4 o JO 20 XJ Flop defleclion l degrees 40 50 FIG. 122. Variation of maximum section lift coefficient. based on chord of wing section plus flap, with flap deflection for the NACA 23012 wing section with 0.20 c. external-airfoil flap. advantages as a full-span flap, it has not been used extensively because of its failure to show definite advantages over retractable Haps and because its use would probably aggravate the icing hazard. d. Double-slotted /laps. Double-slotted flaps3 (Fig. 123) produce sub­ stantial increments of maximum lift coefficient over that obtainable with single-slotted flaps. The fore flap or vane of the double-slotted flap assists in turning the air downward over the main flap, thus delaying the stall of the flap to higher deftections. It has frequently been found possible to develop flap-vane combinations that can be retracted into the wing section without relative motion between the vane and the flap. Such an arrange­ ment is desirable because the linkage system is much less complicated than when relative motion is required between the flap and vane. 218 THEORY OF WING SECTIONS Investigations by Harris,40 Purser," and Fischel' ! of approximately O.3Oc and 0.4Oc double-s1otted flaps on the NACA 23021 and 23012 wing sections indicated maximum lift coefficients of the order of 3.3 and 3.5 for the two sizes of flap. Typical results for the NACA 23012 section are shown in Figs. 124 and 125. The fore flap and the main flap did not deflect together as a unit for these configurations. Aerodynamic characteristics' are pre­ sented in Fig. 126 for the NACA 65,-118 section with the O.309c double­ slotted flap shown in Fig. 123. For this configuration, the vane and flap .....-------lOCLf----------..I ...----.78/----------1... moved together as a unit up to deflections of 45 degrees. At higher angles, the flap rotated about a pivot and the vane remained fixed. The variation of maximum lift obtained for thin NACA 64-series seetions" with approximately 0.3Oc double-slotted flaps is shown in Fig. 127. The type of fiap used for these tests is illustrated in Fig. 128. The :8ap and vane deflected as a unit. These data. (Fig. 127) show that the maximum lift coefficient decreases rapidly 88 the thickness ratio of the wing section is decreased to values below 10 or 12 per cent. The maximum lift coefficient obtained for the NACA 1410 wing section21 is also plotted to indicate the effect of type of section. The maximum lift coefficient ob­ tained for the NACA 65r118 wing section with a flap deflection of 45 de­ grees (Fig. 126) is also plotted to indicate the effect of larger thickness ratios. The NACA 65,-118 data are not exactly comparable, but the indicated gradual rise of the maximum lift coefficient as the thickness rano is increased to 18 per cent is believed to be representative. The effect of the design position of minimum pressure on the maximum lift coefficients obtained 'lith double-slotted flaps on 10 per cent thick NACA 6-series wing sectionsD is shewn !D Fig. 129. The type of flap used HIGH-LIFT DEVICES ~ I- - ~ /Chad - . 219 ~~'I I ~~ ~'''~~ - v~~ +~ 40'i;~ - ~ I-I-- I-- --~~,... .28 6~~ -~ -- ~ X2 40 50 60 70 0.60 - 40 /.60 1.60 Y2 .05 205 2.05 1.05 '-- .......... / ~ ·'1 11. .24 ~ Of2'd!g. . D ~.20 70~~ ~ .~ ~ ./6 § t·/2 :tl 6fV ~ I V • l~ 1..os -~ .~ ~.04 ~ "'50/ V V tJ ~ ~, A .<.J ) VI P ..-10~ VO ~ l o t /6 i 8 ~ 6'2,deg ".;; ~ \ .\ .\ .V' ~ ~ \J~ v ot'r~ 40 50 60 70 li ~ ; I 0 ~ ~ ~ -8 -.4 o .4 .8 1.2 1.6 2.0 2.4 2.8 , )~ J 3.2 3.6 5ec/lon lifl coefficient, 'i FIG. 124. Aerodynamic eection characteristics of the NACA 23012 airfoil with a O.3Oc double-elotted flap. ala == 25 degrees; %1 :: 0.41; III == 1.72. (Values of %1, 111. %It and 1/1 are given in per cent of airfoil ehord.) HIGH-LIFT DEVICES 221 was similar to that shown in Fig. 128. These data show that the highest maximum lift coefficient was obtained for the NACA 64-series sections and that the maximum lift coefficient decreases rapidly as the minimum-pressure I~ ~ 3.2 11) ~n r-, 45 6 (degJ / ~J V >55 '~ 6~1 ) I I ~ '( / V; .\ 'p ~J I ~ 1ft / /J /35!?o ~ '~ V) / I .\ 1 ~ I--- .A 2.8 ;x 2.4 1/ )' 2.0 , / I, I f lJ 'J0d ~ Vj / I ! III o~ ~ ~ 1.6 .r )'f' I I / I If il ~ ~J ) j I~ V 6 // f I~ j (deg) fI I{ ~l / /1 If t I f 'P~It j .4 ~I / T ,V If ~ I j 7 ) o II J r if ,I , r 1 ~ ,~ ~ .~ ~ Ci) ( ~. 1 -.4 -.8 ~) } ) I JPI ~ 4 -/·?'24 'Ii- V j r! / -/6 J I -8 0 8 /6 2~; Section angle of oIl~ tXo,deg. FIG. 126. NACA 653-118 airfoil section with O.309c double-slotted flap. position moves farther aft. It is thought that the rather large variations shown in Fig. 129 are associated with the fact that the thickness ratio of the sections \\"88 in a critical range, as indicated by Fig. 127. Load data for typical double-slotted flaps are given by Cahill.!! These data show that a disproportionately large part of the load is carried by the vane. Normal-force coefficients for the vane based on the vane chord and the dynamic pressure of the free stream reached a value in excess of 5.0. THEORY OF WING SECTIONS 222 r­ I .028 e -/ V 6 .020 ) ~ V 1,/ .... ...... ro-.. '~ ~ '/ I ~ -.... ,~ 1, ~ ~ r-, ~ ~P-- ~ p.­ ~ '" ........tI. ) ~ V~ J LJ ...-It! ~ v I ~ ~~~ )Jf V - / I 10 )~ - 1;.35 J J (dtJg) r--­ 20 p .". V r-­ ~ ~ r-­ r;JJ1 - / 'I - J ~V ~ J .004 o , ..... f--45 ~ r-, r........ >-­~ ..... ~ o IO "V'll ,..... ~ ~ 20 16-- r---­ .4 t'­ .-..., ~ ~ ~. ~ .­ ..... - -. ~ l...d.. /2 .8 /.6 Seclion lifl txJeffici~Ct FIG. I ~ ~ .......... - fd8g.) ~ ~ ...... ~ ~ ..... r--c:... ~ ~ -.4 - .. ---­ ~ lIo.... .... 65~ ~ -~8 ..D.. A r­ 5~ rd:::: r---.. - 0 'j'U 1,35~l---... I-- ~ -. In 126. (Condtule4) .... 2.0 - --::: ~ 1 r-i l:.,,;R - 2.4 , ... ~f>I 2.8 _d 3.2 ., HIGH-LIFT DEVICES 223 I I ',,..A) N.A.CA. 65., -/18 A!A.aA./4/0~ ) (~ 4 FIG. I l.­ --N.A.C.A.64-series (f _ 8 20 /2 /6 SecIiaIlhickness lf1fio, ~ lpercenf 127. Effect of thickness ratio and type of wing section on maximum lift with double. slotted 1laps. Ftpdud line ~--------- . T 1 ' 5 c - - - - - - - - -.... (o)AIRFOL WITHRAP XI c:: Airfo7chord line~ ,,; r 12· 8~~ FIG. 128. Typical airfoil and flap configuration. THEORY OF WING SECTIONS 224 3.2 I I I I Double slottedflop wiIh fore tip .~ r---...... ......~ 01t75c 2.8 J .>--- 2.4 ~-- r----... ~ ~_J. ---,--- ~) ----4) ~ J... ~ 2.0 i Plain wing section fA ~ 1.6 ~ .~ /1 ~ .............. 11 :~ ~ ............... J 1.2 ~-- p-' ...... )-_1 - ."'4 .8 --Smooth ---- Leading edge rough .4 ':'2 .3 .4 .5 .6 Position ofmimil1lU71 pteSSIIe, % maximum section lift coefficient with position of .7 129. Variation of minimum pressure for 80rne NACA tHeries wing aectiODS of 10 per cent tbiclmees and a design lift coefficient of 0.2 .. R, 6 X 10'.. FIG. FIXED AUXILIARY WING SECTION (AXED SLAT) ~<C-------------LEADING EDGE RETRAC'1MlE SlAT FIG.. 130.. Examples of fixed and retractable slats. 225 HIGH-LIFT DE"VICES 8.6. Leading-edge High-lift Devices. a. Slats. Leading-edge slats are airfoils mounted ahead of the leading edge of the wing in such an attitude as to assist in turning the air around the leading edge at high angles of attack and thus delay leading-edge stalling. They may be either fixed in I.lir--r-.---,r--...,..-...,...-..~-....-..-----.------- ~ 2/.2t----r--t---+-~~-+-_I_______1.~~~-_J___J,.-__1 ~ 10 20 30 40 50 60 'Flop deflection, 6, ,deg. FIG. 131. efficient. Eft"eet of flaps and leading-edge slat on increment of maximum section lift co­ position or retractable (Fig. 130). The fixed leading-edge slat consisting of an auxiliary airfoil mounted ahead of the wing leading edge has been investigated in detail by Weick and Sanders.s" This investigation showed that leading-edge slats of this type with chords varying from 7.5 to 25 per cent of the wing chord and with various sections all produced substantially the same maximum lift coefficient when located in the optimum position for the ratio CLmax2 ( Dmin THEORY OJ! WING SECTIONS 226 The value of this maximum lift coefficient was about 1.64 for the rather low Reynolds numbers of the tests. It is doubtful whether such configura­ tions would experience much beneficial scale effect. The effectiveness of the retractable leading-edge slat107 shown in Fig. 130 in increasing the maximum lift coefficient and the angle of attack for 24 I I I I I I I I I - - ­ Q2566c sidled fk:p ---O.20c - ­ O.2566c --Q2Oc split fkp stxled f~arJB:xJing. sial splif flopmd 1eodiIg. sbf 6 2 .JI"'- 8~ ~ ~ ~ -- _...­ ~--- -~--. ........ ­ ,,/ 4 n. - .......... ~~ '--r--­-... ------......! 4 o 10 20 30 .. ---..IL_ ~~ 40 ~ 50 r-, 60 Fkp deflechOnl 6 , ,deg. FIG. 132. Effect of 1lape and leading-edge slat on angle of attack for maximum lift. maximum lift is shown in Figs. 131 and 132. These data, which were obtained on a K.~CA 23012 lying section, indicate an increment of about 0.5 for the maximum lift coefficient and about 8 degrees for the angle of attack for maximum lift. The increment of maximum lift decreased to about one-half of its value for the unflapped section when either a split or slotted flap was deflected to optimum deflections despite readjustment of the slat to optimum positions with flap- deflected. Weick and Platt!· obtained considerably larger increments of maximum lift coefficient with a special retractable slat (Fig. 133) having a shape providing a rounded entrance into the slot. The increments of maximum lift coefficient with HIGH-LIFT DEVICES 227 this configuration were 0.81 for the unflapped section and 0.45 with a deflected slotted flap. b. Slots. Slots to permit the passage of high-energy air from the lower surface to control the boundary layer on the upper surface are common features of many high-lift devices. The most common application is the slotted flap. When the slot is located near the leading edge, the con­ figuration differs only in detail from the leading-edge slat. Additional slots may be introduced at various chordwise stations. Weick and Bhortal'" made a systematic study of slots on a Clark Y airfoil. The results of this investigation are summarized in Fig. 134. For the unflapped section, the most effective position for a single slot is near the leading edge, and the effectiveness decreases as the slot is moved aft. Multiple slots are relatively ineffective on the plain airfoil unless they include a slot near the FIG. 133. Special retractable slat on Clark Y wing section. leading edge, in which case a total of three slots, all located well forward, is optimum. For the flapped section, the slot located near the leading edge was effective. A single or double slot at the flap changed the type of flap from plain to slotted with- a corresponding increase of the maximum lift coefficient. Load data for the leading-edge slot are given by Harris and Lowry." These data show resultant-force coefficients as large as about 7.5 for that portion of the wing section ahead of a slot near the leading edge. If slots are considered as a fixed high-lift device, the profile drag in the high-speed. flight attitude is an important characteristic. Figure 134 shows that any of the slots investigated cause large increments in the minimum profile drag. This increment increases with the number of slots and de­ creases with rearward movement of the slots. Attempts have been made to maintain low drags with slots open by locating the slots so that there would be no flow through them in the high-speed. condition. lOS Such con­ figurations have failed to improve the ratio of maximum lift to minimum drag over that for the plain wing section. c. Leading-edge Flap8. A leading-edge flap may be formed by bending down the forward portion of the wing section in a manner similar to that in which the trailing edge is deflected in the case of plain flaps. Other types of leading-edge flap are formed by extending a surface downward and forward from the vicinity of the leading edge. As shown in Fig. 135, such flaps may extend smoothly from the upper surface near the leading edge, may be hinged at the center of the leading-edge radius, or may be hinged on the lower surface somewhat aft of the leading edge. Although THEORY OF WING SECTIONS 228 Slot combination CLma x CDmi D CLma'!C. a CDm iD degrees CL max «: =:=::--.-.... 1.291 0.0152 85.0 15 rr: :::--=----.. 1.772 0.0240 73.8 24 c:7~ 1.596 0.0199 SO.3 21 C 7c:=::::..-.. 1.548 0.0188 82.3 19 c: :Jc::::,... 1.440 0.0164 87.8 17 r~~ 1.902 0.0278 68.3 24 7~ 1.881 0.0270 69.7 24 :J~ 1.813 0.0243 74.6 23 ~L7C?c:::=::--..- 1.930 rr: ­ rr: - rL7c=---r~ I i 0.0340 I I 56.8 I 25 I 1.885 0.0319 59.2 24 rC7C7oc:::::::::. -z:':70~ c:7C?c:=::::..-.. 1.885 0.0363 51.9 25 1.850 0.0298 62.1 24 1.692 0.0228 74.2 22 c:7t:==7c::::,... 1.672 0.0214 c: - 1.510 0.0208 7a~ c:7C7oc::::,... II I I 78.2 72.6 I 1.662 I I 0.0258 r I 64.4 I i Ii I II 22 19 I 22 I (a) Multiple fixed slots. FIG. 134. Aerodynamic characteristics of a Clark Y wing with slots and flaps. HIGH-LIFT DEVICES Slot combination CD mi u* CLms x 229 II -----------1----1----1----11----1 I 1.950 0.0152 2.182 ! 128.2 12 0.0240 91.0 19 2.235 0.0278 80.3 20 2.200 0.0340 64.7 21 2.210 0.0270 81.8 20 1.980 0.0164 120.5 12 1.770 0.0164 IOS.O 14 0.0208 117.5 16 2.500 0.0258 96.8 18 2.185 0.0214 102.0 18 0.0243 93.2 19 2.320 0.0319 72.7 20 2.535 0.0363 69.8 20 0.0298 87.3 20 0.0298 68.3 21 I 2.-1-12 I l 2.261 I 2.600 t 2.035 I II i • CD' with flap neutral. man (b) Multiple fixed slots and a slotted flap deflected 45 degrees. FIG. 134. (Concluded) 230 THEORY OF WING SECTIONS none of these devices is as powerful 88 trailing-edge flaps, they may be used full span without mechanical interference with lateral-control de­ vices and they are effective when combined with trailing-edge high-lift devices. Leading-edge flaps reduce the severity of the pressure peak ordinarily associated with high angles of attack and thereby delay separa­ (0) DROOPED LEADING EDGE ] (b) UPPER SURFACE LEADING EDGE FLAP (c)LOWER Sl,eqfACE LEAOINGEDGEFLAP (d) FLAP HINGED ABOUT LEADING EDGE RADIUS FIG. 135. Various types of 1eading-edge flaps. tion. Leading-edge flaps received little consideration until German in­ vestigators became interested in them during the Second '\\"'orld War. Krueger,.,60 Lemme,l3· M, 65 and Koster" showed increments of the maximum lift coefficient of 88 much as 0.7. These increments were, how­ ever, applied to maximum lift coefficients for the plain wings of the order of 0.72. These low maximum lift coefficients resulted from the small leading-edge radii of the wing sections usually used for these investigations and the very low Reynolds numbers of the tests. These investigations in­ dicated that the effectiveness of leading-edge Haps increased with decreasing leading-edge radius. Typical rcsultsH are shown in Fig. 136. The inere­ HIGH-LIFT DEVICES 231 moot of maximum lift coefficient, deLmasJ is plotted against the leading­ edge-radius parameter {r/c)/{t/c)2 where r is the leading-edge radius. The value of this parameter for NACA four- and five-digit wing sections is 1.1. FullmerU investigated t\VO types of leading-edge flap on the NACA 641-012 section at a Reynolds number of 6 million. The chord of the flap was 10 per cent of the section chord, and the configurations corresponded to ........ .6 o Withou/ split fbp .~ .8 ~ I <, """,-re ?J -R=0.57x 10' a Corf(c~fig.135";lh split flop ~ rI ~~ rr R=2.I Xd <, "', r-, n Corf(bJ,fig.l35 - <> - <, lei ~ o Ccd(bJ,"~r.135wifh splitfbp r-, r-, ~ .2 =60 cf Corf(c~fig.135 I \ .2 s ~~ itR=O.72x 106 - ~=a20 (UBM3067) I ~ - Withsplit flop r-R=2.2 x KJS t- r-, ~ /11/ r:,. .6 .8 '" <, 1.0 ~ '" <, "- £4 ~ ~ (~;2 FlO. 136. Effeetiveneea 01 leadi11ll-4M1ge flaps for various wing sections. Increment of lift resulting from leading-edae flap, c, 88 a function of the leading-edge radius coefficient. (r/c}/{t/c)t. .al: b and c of Fig. 135. The increments of maximum lift coefficient and of the angle of attack for maximum lift are shown in Fig. 137. The leading-edge radius parameter for the NAC...\ 641-012 section is 0.72. The maximum increments of Fig. 137 are shown plotted on Fig. 136 for comparison with the German results. The maximum lift coefficient of the plain wing sec­ tion for these tests was 1.42. 8.7. Boundary-layer Cpntrol. The idea of removing the low-energy air of the boundary layer, or of adding kinetic energy to the boundary layer, as a means for increasing the maximum lift has been obvious since the basic mechanism for separation was first understood." The kinetic energy of the layers of air close to the surface may hi increased by removing low­ energy air through suction slots or a porous surface. Another common method is to blow high-energy air through backward-directed slots. The 232 THEORY OF lYING SECTIONS air handled through either the suction or blowing slots may be carried through the interior of the wing and the necessary energy supplied by a blower. Alternately, the pressure difference required may be obtained from the variation of pressure about the wing section itself. This method I!!«JJ:... l.J:Mer 5lIfoce~. '-e4e flop } o t.ower:5lIfoce flop willi a ftrJilhg-edoe flqp lJppei-sufoce leodirJg -e4e flop } £fiper-Sllfoce Ieoditig-edge fbp with Imiling-e4e ftp Fig./35, calle) A- /.'71: /{/. ..,~ conl{b) 8 A -. 4 I - - ---- ~ ~ ~ / -./'" '¥ ~ rA 8 / "'" -, Q,.- .4 ~ Do.. A lJ() ~ ~ / I ~. y" /00 /20 140 /60 l.I!odtY;-edge f/qJ deflecfionl 6,LE!deg. FIG. 137. Variation of the increment of maximum section lift coefficient and the increment of section angle of attack for maximum section lift coefficient with leading-edge flap deflec­ tion. NACA 64t..Q12 airfoil section with leading- and trailing-edge split flaps. R, 6.0 X U)6. has been used successfully only in the case of blowing slots on the upper surface using air taken from the lower surface. Examples of such arrange­ ments are the previously discussed slotted flaps, slots, and leading-edge slats. It is obvious that, if boundary-layer control is applied at sufficiently close intervals along the upper surface of a wing section, separation can be avoided up to very high values of the lift coefficient. The increments of maximum lift coefficient are obtained as an extension of the lift curve to HIGH-LIFT DEVICES 233 higher angles of attack as contrasted with the displacement of the lift curve resulting from deflection of trailing-edge flaps. The problem con­ fronting the designer is to apply sufficient boundary-layer control to obtain the desired values of the maximum lift coefficient without increasing the weight and complexity of the airplane to such an extent as to nullify the gains in performance expected from the higher lift coefficients. These considerations of weight and complexity have prevented extensive use of 6.0 ~ 5.0 / ~4.0 ......" i Theory.. . " .~ .~ ~ ) § .10 .~ ~ (j) f /r / / v l( /' ~. )C / ~/ ~. J ~ I 2.0 V» '--- Experiment _ r--­ CO=Q038 / V)'­ 1/ 1.0 a I V· JV .-20. -/0 0 10 20 Angled ottoclt1t%, deg. 138. Variation of lift coefficient with angle ot attack for a thick wing section with boundary-layer control. FlO. any except the simple types of boundary-layer control such as slotted flaps and leading-edge slots and slats. Successful application of boundary-layer control must delay both tur­ bulent separation over the aft portion of the wing and permanent laminar separation over the forward portion (see Sec. 5.3). The prevention of permanent laminar separation for thin or even moderately thick sections by means of suction slots is a difficult problem involving critically located slots in a region of greatly reduced pressure that requires high-pressure blowers. Nearly all the early investigations avoided this problem by using very thick sections with large leading-edge radii. 102. 101, 104 Using such sec­ tions, Schrenk103 obtained lift coefficients of over 5.0 using a single suction slot (Fig. 138) with a volume-flow coefficient Co of 0.038. The high drag coefficients of such thick wing sections in the high-speed flight condition led to investigations of boundary-layer control on thinner THEORY OF WING BECTIONS sections.II,u,WI Knight and Bamber" obtained maximum lift coefficients-­ of about 3.0 (Fig. 139) for the NACA 84-M wing section with a single backward-opening blowing slot. NACA 84-Mwingsecfion. SloIwidfh=Q667%c Slotof 53. 9 ~ c from leading edge. Inlemo/ pressure /2 q higher than free- sllfJOlTl sialic pressure I ~ , I I I I c, s/olled Ming sediJn JV 2.4 L' / V 7 If I 7 \ \ y \ ./ .~ ,V 8/ FIG. .rl. I) ~ ~ ~ 0 -: / Il o .L ~ ~c, pbiI wing section I I · •• I II'-lfdstnIt/'wing ssdion ~ II .I 7. ~J I / / 1/ Jif 1/ ..4 ~ ~"",. -.-...... ~ ~ 8 K:.~~~~_ ~ -~ ~ ~ 16 24 32 40 48 ~ofolfodr,f4deg. 139. Effect of boundary-layer control by means of a baekward-opening blo'triDc 1101. Much of the more recent work'" lion boundary-layer control has com­ bined suction slots with flaps and leading-edge slots. Quinn" obtained maximum lift coefficients exceeding 4.0 for the NACA 65a-418 section with a double-elotted flap and a single suction slot (Fig. 140). The maximum tift coefficient for the unfIapped section increased rapidly with small HIGH-LIFT DEVICES 235 ~-----.45c----~-a 44 4.0 -V "'--V 0, =65° ~ -" L-----' ~ ~ ~ / l- """'" ~.6 ~ ~~ ~ ~ ~ ~ "6f=4S~ / 0,=0°"",,,,V~ ~ ~ ~ 10-""" ---- ~ ~ ~ 1.6 I( / ...... 1.20 .008 ••016 .024 .032 .040 Flow coefficient, CQ FIG.I40. Configuration and maximum lift characteristics of the NACA 65,-418 wing aection with double-elotted flap and boundary-layer control by suction. R, 6.0 X 10'. 236 THEORY OF WING 8ECTIONS amounts of flow through the slot, but the variation of maximum. lift coeffi­ cient for the Happed section was nearly linear with the flow coefficient up to a value of 0.040. Quinn93 has concluded that the maximum effectiveness of suction slots is nearly reached when the quantity of air removed is equal to that which would pass through the displacement thickness of the boundary layer at the slot location with the local velocity outside the boundary layer. Quinn found that the maximum lift was limited by leading-edge stalling for the 18 per cent thick sections. He used a leading­ L..,....o-" ~ ~V ~ V .01 ).---" ~ .02 .03 .04 FkM coefficienf,CQ FIo. 141. Configuration and maximum lift characteristics for the NACA M 1A212 wiug section with leading-edge slat, double-slotted flap and boundary-layer control. edge slatt 5 to control the leading-edge stalling of the NACA 641A212 section with a suction slot and a double-slotted flap. This combination of high-lift devices gave a maximum lift coefficient of nearly 4.0 for this thin section95 (Fig. 141). 8.8. The Chcrdwise Load Distribution over Flapped W'mg SectiODS. Allen' obtained empirical load distributions that permit calculation of reasonable approximations to the load distributions of "ring sections with deflected plain or split flaps. These load distributions are analogous to those associated with changes of camber in that the ratio Pb&/c~ is inde­ pendent of the angle of attack where Pili is the incremental basic load distribution associated with flap deflection and en., is the basic normal­ force coefficient associated 'lith flap deflection. The total load distribu­ tion is the sum of the incremental basic distribution P"" the incremental additional load distribution Po" and the load distributionof the unflapped section at the same angle of attack. The incremental additional load HIGH-LIFT DEVICE8 237 distribution P a & is taken to be the same as the additional load distribution associated with angle of attack. That is, for a normal additional force coefficient of unity, P = 4 dV a -! GI. V V where the values of dVa/V and vIV are tabulated in Appendix I for various wing sections. Allen also found empirically that the values of P",/~ at any point (x/c)/(I - E) (points ahead of the flap hinge) and [I - (x/c»)/E (points aft of the hinge) were independent of the flap-chord ratio E for any given angular deflection of the flap. Values of P"./c.., are given in Table 6 for both plain and split flaps. The incremental load distribution associated with flap deflection is thus defined in terms of two known types of distribu­ tions. The remaining problem is to determine the magnitudes of these two types of distributions associated with a given flap deflection. The magnitudes of the load distributions are determined from force test data obtained at the desired angle of attack with the flap neutral and deflected. The additional load distribution contributes no moment about the quarter-chord point. The incremental basic distribution is therefore selected to produce the measured increment of moment coefficient between the flap-neutral and flap-deflected conditions. The remainder of the differ­ ence between the normal-force coefficients with the flap neutral and de­ flected is equal to the incremental additional load distribution. The normal-force distribution is assumed to act as shown in Fig. 142. From force tests of the wing section with flap neutral, C1ft1 (quarter-chord pitching-moment coefficient) and ella (normal-force coefficient) correspond­ ing to the normal-force distribution shown in Fig. 142a are obtained. From force tests of the wing section at the same angle of attack with the flap deflected, ems and CftS corresponding to the normal-force distribution shown in Fig. 142b are obtained. Let aem = Cm: - em l , } = en: - C ra., deft (8.1) where em.' and Cfta' are the pitching-moment and normal-force coefficients corresponding to the normal-force distribution for the section with flap neutral when plotted normal to the chord of the section with flap deflected as shown in Fig. 142c. Then ~Cm and .1c,. are the pitching-moment and normal-force coefficients of the incremental normal-force distribution when the incremental distribution is plotted normal to the chord of the section with the flap deflected as ShO"l1 in Fig. 142d. In many cases, the approximation (8.2) THEORY OF WING SBCTIONS 238 is sufficiently exact except when the flap-chord ratio E and the flap deflec­ tion 8 are simultaneously large or when the shape of the mean line is such that the load is large near the trailing edge. In such cases, values of Crat' and Cat' may be obtained graphically from the theoretical load distribution for the unflapped section adjusted to agree with the experimental data. Let Ac.' and Ac.' be the pitching-moment and the normal-force coeffi­ cients of the incremental normal-force distribution plotted normal to the flap-neutral chord as shown in Fig. 142e. Because the incremental basic em,' h=~n=~/====-__ ~_cn_/.....;:' ;;;;:;::::==--.. (a) NORMAL-FORCE DISTRBI1lON (e) DISTRI8UTION SHOWN IN(a) (b) NORMAL-FORCE DISTRIBUTION FOR AIRR)IL WITH FLAPNEUTRAL. FOR AIRFOL WITH FLAPDEF'l.B:T. ED~ f':: ~~ 6Cn " WITH FLAPNORMAL FORCE 0ISTR8UT1ON PLDTTED NORMAL lOFLAP DEFLECTED CHORD. /l(j (d) INCREMENlJ'L NORMAL -FORCE DISTRBITION CMJSED BYFLAP AC~ ~ (e) DISTRI8UTIDN SHOWN'IN (d) PLDnED NORMAL TOFLAP-NElITRAL DEFLECTION. CHORD. FIG. 1.(2. Normal-force distribution and incremental normal-force diBtribution for flaps neutral and deflected. normal-force distribution is responsible for the increment of the quarter­ chord pitching moment, then, if G is the moment arm in terms of the chord of the basic normal force about the quarter-chord point, ac.' == Gc.., t:.e,., == e.., + en., wliere en _, = additional normal-force coefficient associated with flap de­ flection c...,=~' c,,~. } LAc.' = ~c", - (8.3) G The value of G is a function of E and 8. Values of G are given in Table 7. The correlation between the fictitious values of ~c",,' and ACta, and the measured values of de,. and ~CR must be established in order to determine c... and cfteI from force tests. The incremental flap normal-force coefficient is HIGH-LIFT DEVICES I, - nl, 239 (8.4) Ca -gEc ­ where nl, is the incremental flap normal force per unit span. Then J1c.' =z a~ + Ec"" (1 - cos 8) } and &:." = &:., - Ee. ' (1 - cos 6)(% - E) 'I, (8.5) The incremental1lap normal force may be considered as a combination of two components due to the incremental additional and the incremental basic normal-force distributions. Let 1'0, and "rll, be the ratios of the flap normal force to the section normal force for the incremental additional and the incremental basic normal forces, respectively. Then (8.6) or (8.7) The contribution to the flap load of the additional normal-force dis­ tribution is small compared with the basic contribution; and, for the purpose of determining &:." and J1Ca', the following approximation may therefore be used: and Eqs. (8.5) become 4c.' = J1c. &:." /ii;w + En, G (1 - = &:., - E-y", ~' (1 - cos 6)(~ - E) so that At;., ~c.' where cos a) = 4c. + T" ~c.} = 'T'" ~c. (8.8) E(1 - cos 8)("Yb,/G) cos 8)(~ - E)('YbtJ/G) T" = 1 + E(l - T. = 1 + E(l - 1 cos 8)(% - E)('Y6,/G) The values of'T. and 'T.. as determined by Allen are given in Tables 8 and 9. In using this method, the values of J1c. and tic", are obtained from force tests of the flapped and unflapped section at the same angle of attack. From Eq. (8.8)~ values of 11c"" and &:." are obtained using Tables 8 and 9. ~ TABLE 6.-P",/C"6, DISTRIBUTION b. Plain flap at 8 • 20 degrees a. .Plain flap at 8 - S, 10, and 16 degreel 0.05 0.10 0.15 0.20 0.25 0.30 0.31 0.40 B o.~ 0.&0 0.55 0.80 0.65 0.70 0.05 0.10 0.16 0.20 0.25 0.30 0.36 0.40 0.46 0.50 0.66 0.60 --0 -0 -0 -0 -0 -0 --0 -0 ---0 -0 --0 -0 0 0 0 0 0 0 -- -- 0 0.18 0.26 0.39 0.50 0.62 0 0.19 0.27 0.40 0.62 0.64 0.15 0.23 0.34 0.44 0.55 0.16 0.24 0.36 0.46 O.M 0.67 0.17 0.25 0.36 0.47 0.68 0.17 0.25 0.38 0.49 0.60 0.18 0.26 0.39 0.&0 0.62 0.20 0.21 0.22 0.23 0.29 0.30 0.32 0.34 0.42 0." 0.46 0.49 O.M 0.67 0.60 0.63 O.M 0.67 0.70 0.73 0.77 0.25 0.36 0.52 0.67 0.82 0.27 0.39 0.56 0.72 0.88 0.15 0.23 0.34 0.44 O.M 0.55 0.16 0.23 0.34 0.46 0.66 0.16 0.24 0.35 0.46 0.67 0.17 0.26 0.36 0.47 0.68 0 0.17 0.26 0.38 0.49 0.60 0.66 0.80 0.98 1.24 1.74 0.67 0.82 1.00 1.26 1.75 0.69 0.83 1.01 1.27 1.7J 0.71 0.8& 1.03 1.29 1.77 0.73 0.88 1.06 1.3l 1.79 0.75 0.90 1.08 1.34 1.81 0.77 0.93 1.11 1.38 1.84 0.92 1.10 1.30 1.59 2.06 0.98 1.17 1.39 1.67 2.16 1.05 1.25 1.48 1.78 2.29 0.66 0.79 0.97 1.23 1.73 0.66 0.80 0.98 1.24 1.74 0.67 0.82 1.00 1.26 1.75 0.69 0.83 1.01 1.27 1.76 0.71 0.85 1.03 1.29 1.77 0.73 0.88 1.05 1.31 1.79 0.76 0.90 1.08 1.34 1.81 0.70 0.60 0.60 8.74 6.45 4.96 3.86 3.09 2.48 6.04 4.49 3.51 2.72 2.18 1.72 4.83 8.78 2.92 2.23 1.78 1.42 4.40 3.32 2.67 1.97 1.57 1.M 4.01 2. • 2.31 1.77 1.41 1.11 3.71 2.77 2.12 1.84 1.30 1.02 3.&0 3.35 3.23 3.15 3.11 3.06 3.04 2.80 2." 2.39 2.32 2.26 2.22 2.19 2.00 1.90 1.81 1.74 1.69 1.84 1.60 1.63 1.44 1.37 1.32 1.27 1.22 1.19 3.02 2.16 1.57 1.16 1.21 1.14 1.08 1.04 1.00 0.96 0.93 0.91 0.96 0.89 0.86 0.81 0.77 0.7& 0.72 0.70 5.S3 1.08 4.23 3.71 3.33 2.98 4.05 3.&3 2.99 2.63 2.3& 2.00 3.38 2.98 2.60 2.16 1.92 1.73 3.02 2.61 2.19 1.90 1.89 1.60 2.83 2.36 1.97 1.71 1.62 1.35 2.70 2.18 J.81 1.68 1.40 1.24 0.40 (back of 0.30 hinge) 0.20 0.10 0.05 0 1.90 1.40 0.92 0.48 0.32 0 1.34 0.09 0.63 0.34 0.19 0 1.10 0.81 0.&3 0.27 0.16 0 0.98 0.70 0.46 0.24 0.14 0 0.88 0.63 0.41 0.21 0.12 0 0.78 0.67 0.38 0.20 0.11 0 0.73 0.63 0.35 0.18 0.10 0 2.85 2.32 1.93 1.39 1.16 0 1.87 1.63 1.3& 0.98 0.70 0 I.M 1.34 1.11 0.79 0.68 0 1.31 1.18 0.96 0.69 0.&0 1.19 1.04 0.86 0.62 0.4& 0 1.09 0.9& 0.79 0.67 0.40 0 ~/e 0 0.05 0.10 0.20 0.30 0.40 0 0.16 0.22 0.33 0.43 0.54 1-8 (ahead 01 0.60 0.66 hinge) 0.60 0.79 0.70 0.97 0.80 1.23 0.90 1.73 1.00 --- 0.90 0.80 1 - ~/c B 0.16 0.23 0.34 0.45 0.19 0.27 0.40 0.&2 0.89 0.60 0.33 0.17 0.10 0 0.80 0.96 1.15 1.42 1.88 0.81 0.47 0.31 0.18 0.09 0 0.84 1.00 1.19 1.46 1.92 0.62 0.46 0.29 0.13 0.08 0 0.88 1.0j 1.24 1.52 1.98 0.69 0.43 0.28 0.16 0.08 0 0.57 0.42 0.27 0.14 0.08 0 0.&6 0.40 0.26 0.13 0.07 0 0.&3 0.39 0.25 0.13 0.07 0 0.16 0.22 0.33 0.43 a - --0 -0 0 0 0.20 0.29 0.42 0.14 0.67 0.21 0.30 0.44 0.57 0.70 0.22 0.32 0.46 0.60 0.73 0.23 0.34 0.49 0.63 0.77 ~ ~ ~ 0.77 0.93 1.11 1.38 1.84 0.80 0.96 1.15 1.42 1.88 0.8J 1.00 1.19 1.46 1.92 0.88 1.05 1.24 1.52 1.98 0.92 1.10 1.30 1.59 2.06 ~ 2.63 2.05 1.70 1.48 1.30 J.15 2.58 1.96 1.62 1.39 1.22 1.08 2.56 1.88 1.65 1.32 1.17 1.03 2.56 1.83 1.49 1.27 1.12 0.98 2.58 1.78 1.44 1.22 1.08 0.94 2.62 1.75 1.40 1.18 1.04 0.91 1.02 0.88 0.73 0.53 0.38 0 0.96 0.82 0.68 0.50 0.35 0 0.91 0.78 0.65 0.47 0.33 0 0.88 0.73 0.62 0.45 0.31 0 0.83 0.72 0.69 0.43 0.30 0 0.80 0.89 0.56 0.41 0.28 0 ~ s ~ ~ C':S ~ ~ d. Plain or .pUt flap at a- 40 degrees c. Plain flap at' .. 30 deereee - I 0.05 E z/c 1-8 (ahead of hinge) --­ !-.=. ~/c B (bank of hinge) -­ 0 0.05 0.10 0.20 0.30 0.40 0 0.15 0.22 0.33 0.43 O.M 0.&0 0.60 0.70 0.80 0.00 0.66 0.70 OJ.7 1.23 1.67 1.00 O.DO ".rJO 0.80 0.70 0.60 O.SO 0.40 0.30 0.20 0.10 0.05 0 4.61 4.34 4.08 3.82 3.56 3.25 2.89 2.44 1.81 1.53 0 0.16 0.20 0.26 0.30 0.36 0 0.18 0.23 0.34 0.4& 0.56 0 0.16 0.24 0.3& 0.46 0.67 0 0.11 0 0.t7 0.6H O.US 0.67 0.82 I.OU 1.24 1.26 1.68 1.69 :4.12 3.20 2.60 2.70 2.56 2.37 2.20 0.10 -­ 0 0.16 0.23 0.34 0.44 0.&5 0.66 O.RO 3.07 2.89 2.60 2.4.9 2.29 2.0' 1.72 1.27 0.92 0 0.40 0.45 0.60 0.06 0.10 0.16 0.20 0.25 0.30 0.35 0.40 0.19 0.27 0.40 0.52 0.64 0 0.20 0.29 0.42 0.54 0.67 0 0.21 0.30 0.44 0.67 0.70 0.16 0.22 0.33 0.43 0 0.16 0.23 0.34 0.44 0.66 0.16 0.23 0.35 0.46 0.56 0 0.16 0.24 0.36 0.46 0.67 0 0.11 0.26 0.38 0.47 0.38 0.17 0.23 0.39 0.49 0.60 0 0.18 0.26 0.40 0.61 0.82 0 0.19 0.27 0.42 0.62 O.M tt: 0.77 O.!l:l 1.11 1.38 1.74 0.80 O.!J(\ 0.84 1.00 r.rs 1.IH 1.42 1.77 0.89 0.83 1.01 1.26 1.61 0.71 0.86 1.0:1 1.28 1.61 0.73 0.88 1.05 1.30 1.62 0.7& 0.00 1.08 1.33 1.62 0.77 0.93 1.11 1.36 1.63 :;;~ 2.01) 1.8.'1 1.66 1.62 1.40 1.29 2.01 1.17 1.11 0.98 0.82 0.61 0.44 0 - ­ - ­ -­ -­ - ­ -­ - ­ -0 ­ - ­ -­ -­ - ­ -­ -­ -­ -­ 0 0 0 0.1':4 1.01 1.27 1.70 2.36 2.:~7 2.0~ 2.20& 2.0tJ 1.94 1.79 1.88 1.64 1.GS 1.45 1.41 1.03 0.76 0 O.~ 0.25 0.38 0.47 0.68 0.38 0.49 0.60 0 0.18 0.26 0.39 0.60 0.62 0.71 0.145 0.73 0.75 n.AA o.no 1.05 1.0M 1.34 1.73 I.oa 1.2U r.ai 1.71 1.72 2.22 2.15 2.02 1.88 1.74 1.61 2.13 2.01 1.86 1.73 1.60 1.47 2.08 1.90 1.46 1.29 1.34 1.18 1.00 0.74 1.25 1.10 0.93 0.60 0.49 0 1.22 1.09 0.90 0.81 0.86 o I 0O,6'J 0.53 0 l.i6 1.62 1.49 1.37 1.03 0.87 0.65 0.48 0 1.77 1.59 1.45 1.34 1.22 O.M 0.66 0.80 1.46 1.80 0.88 0.79 O.n7 1.22 1.68 1.23 1.&9 0.68 0.82 1.00 1.2& 1.60 1.UO 1.73 1.S2 1.40 1.28 1.17 4.10 4.24 •. 29 4.27 4.14 3.92 2.90 3.01 3.04 3.02 2.91 2.76 2.42 2.52 2.&3 2.48 2.39 2.27 2.12 2.23 2.22 2.18 2.10 1.98 1.90 2.02 2.00 1.96 1.89 1.77 1.88 1.88 1.84 1.81 1.74 1.62 1.81 1.78 1.73 1.70 1.62 1.61 1.76 1.72 1.66 1.69 1.62 1.42 1.06 3.64 3.28 2.82 2.11 1.79 0 2.&6 2.31 1.98 1.48 1.08 0 2.11 1.90 1.62 1.20 0.89 1.83 1.04 1.40 1.0& 0.77 0 1.64 1.47 1.26 0.96 0.89 0 1.50 1.34 1.15 0.87 0.63 0 1.40 1.24 1.07 0.81 0.58 0 1.31 1.17 1.00 0.76 0.93 0.78 0.68 0.41 0 O.OM 0 :::~ -s ~ tlj ~ ~ t'I.) O.M 0 ~ ~ ~ TABLE 8.-P.,Ie"", DISTRmUTIoN.-(Conlinued) I. Plaln or epUt Sap at' - ,. Plain or 8PUtSap at' - 80 deere- • 0.08 ----~ ale 1-8 [ahead of billie) --1 -ale B (back of billie) 0 0.05 0.10 0.20 0.80 0.10 0.16 0.20 0.26 0.30 0.36 0.40 0.06 0.10 0.15 0.20 0.26 eo d..-. 0.30 --- - --- -- - - - - - - - - - - - - - - - - - - - 0 0.18 0.22 0.33 0.43 0 0.18 0.23 0.36 0.46 O.M 0 0.18 0.24 0.34 0.48 0.67 0 0.11 0.25 0.38 0.47 0.&8 0 0.17 0.26 0.38 0.49 O.~ O.M 0 0.18 0.23 0.34 0.44 0.66 0.60 0.80 0.70 0.80 0.90 0.88 0.'19 0.97 1.21 1.M 0.86 0.80 0.98 1.22 1.66 0.68 0.82 1:00 1.23 I .• 0.89 0.83 1.03 1.26 1.67 0.71 1.00 0.90 0.80 0.70 0.80 0.50 3.81 4.18 4.26 4.27 4.21 4.07 2.88 2.90 8.00 3.02 2.97 2.88 2.21 2.46 2.48 2.43 2.38 2.00 2.16 2.20 2.18 2.14 2.06 1.88 1.96 1.98 1.98 1." I.M 1.81 1.77 1.89 0.40 0.30 0.20 0.10 0.05 0 3.88 3.62 3.08 2.34 2.02 0 2.71 2.49 2.16 1.64 1.22 0 2.24 2.M 1.78 1.33 1.00 0 1.96 1.78 1.63 1.18 0.87 0 1.73 1.68 1.37 1.06 0.78 0 1.59 1.44 1.28 0.06 0.70 0 2.80 O.U 1.OS 1.28 1.61 0.60 0.73 0.88 1.08 1.29 1.67 1.77 1.81 1.82 0 0.18 0.28 0.39 0.61 0.82 0 0.19 0.21 0.40 0.62 0.84 0 0.13 0.22 0.33 0.43 0 0.16 0.23 0.34 0.44 O.M 0.73 0.90 1.11 1.32 1.67 0.77 0.93 1.14 1.36 1.67 1.71 1.72 1.72 1.70 1.83 1.37 1.48 1.34 1.18 0.90 0.66 0 0.36 ~ O.~ -0 - O.M 0 0.18 0.23 0.33 0.43 0.68 0 0.18 0.24 0.36 0.48 0.67 0 0.17 0.26 0.38 0.47 0.68 0 0.17 0.23 0.38 0.49 0.80 0 0.18' 0.28 0.39 0.61 0.82 0.88 0.79 0.97 1.20 1.62 0.87 0.80 0.98 1.21 1.63 0.68 0.82 0.99 1.22 1.M 0.89 0.83 1.01 1.24 1.66 0.71 0.86 1.02 1.26 1.M 0.73 0.88 1.05 1.28 1.63 0.76 0.90 1.07 1.29 1.61 0.77 0.93 1.10 1.32 1.80 1.87 1.68 1.M 1.69 1.M 1.48 3.8l 4.M 4.18 2.22 2.38 2.43 2.48 2.46 2.42 1.97 2.08 2.13 2.18 2.17 2.11 1.81 1.88 1.94 1.98 1.96 1.89 1.71 1.78 1.78 1.81 1.79 1.73 1.88 1.88 1.88 1.70 1.87 1.82 1.69 4.27 4.18 2.69 2.80 2.M 3.02 3.01 2.93 1.82 1.69 1.69 1.87 1.62 1.39 1.26 1.09 0.84 0.81 0 4.01 3.71 3.27 2.&1 2.18 0 2.82 2.82 2.29 1.78 1.31 0 2.33 2.11S 1.88 1.43 1.08 0 2.02 1".88 1.83 1.26 0.93 0 1.80 1.86 1.48 1.66 1.62 1.33 1.03 0.78 0 1.M 1.41 . 1.24 0.98 0.71 0 1.46 1.32 1.18 0.90 0.88 0 4.27 1.1~ 0.84 0 0.19 0.27 0.40 0.62 O.M ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1/. - %/c 1-E (ahead or hiDe.) --­ 1 - %/c B (baok of hinge) .. 0.06 E A. Split Sap at a - 30 d. . . . Split flap at 3 • 20 del.... -_. -­ - ­ - ­ -­ -­ -­-­ -­ -­ -­ -­ -­ -­ --­ -­ -­ -­ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.10 0.16 0.20 0.24 0.30 0.36 0.40 0.46 0.60 0.0& 0.10 0.16 0.20 0.21 0.30 0.3& 0.40 0.46 0.80 0.16 0.23 0.34 0." 0.&& 0.18 0.23 0.36 0.46 0.66 0.18 0.24 0.36 0.46 0.67 0.17 0.21 0.17 0.21 0.36 0.47 0.68 0.38 0.49 0.80 0.18 0.28 0.39 0.61 0.82 0.19 0.27 0.40 0.62 O.H 0.20 0.29 0.42 0.66 0.87 0 0.21 0.30 0.'" 0.&7 0.70 O.lS 0.22 0.33 0.'3 0.14.' 0.16 0.23 0.34 0.44 0.11 0.18 0.23 0.81 0.41 0.68 0.18 0.24 0.3& 0.48 0.67 0.17 0.26 0.38 0.47 0.18 0.17 0.28 0.38 0.49 0.80 0.18 0.26 0.39 0.61 0.82 0.19 0.27 0.40 0.62 0.84 0.20 0.29 0.42 0.&1 0.87 0.21 0.30 0." 0.67 0.70 0.71 0.86 1.03 1.29 1.73 0.73 0.88 1.05 1.31 1.73 0.76 0.00 1.08 1.34 1.73 0.77 0.93 1.11 1.38 1.73 0.80 0.90 1.16 1.42 1.74 0.84 1.00 1.19 1.46 1.74 0.68 0.79 0.97 1.23 1.M 0.66 0.80 0.98 1.24 1.86 0.88 0.82 1.00 1.28 1.86 0.69 0.83 1.01 1.27 1.67 0.71 0.86 1.03 1.29 1.88 0.73 0.88 1.06 1.31 1.88 0.76 0.90 1.08 1.34 1.88 0.17 0.93 1.11 1.38 1.89 0.80 0.96 1.16 1.42 1.70 O.M 1.90 1.88 1.90 1.78 1.81 1.46 1.28 1.83 1.84 1.79 1.63 1.48 1.32 1.16 4.&& 4.60 4.49 <&.21 •. 02 3.71 3.15 3.19 3.18 3.02 2.83 2.80 2.83 2.69 2.16 2.14 2.14 2.09 1.91 1.83 1.88 1.98 1.99 1.89 1.88 1.81 1.70 1.57 1.43 1.83 1.80 1.72 1.59 1.48 1.36, 1.78 1.73 2.14 2.32 2.38 2.32 a.19 2.04 1.87 1.73 1.68 1.18 1.48 1.36 1.22 0.99 3.36 2.96 2.38 I.M 2.08. 1.71 1.89 1.48 2.88 1.88 1.84 0.98 0 1.14 1.88 0.9& 0.70 0 1.29 1.12 1.01 0.73 0.13 0 1.21' 1.15 1.05 1.00 0.96 I 0.90 1.08 1.11 1.32 1.19 0. • 0.88 0 0 0.05 0.10 0.20 0.30 0.40 0 0.16 0.22 0.33 0.43 0.14. O.SO 0.60 0.70 0.80 0.90 0.66 0.79 0.97 1.23 1.69 0.88 0.80 0.98 1.24 1.70 0.88 0.82 1.28 1.71 0.89 0.83 1.01 1.27 1.72 1.00 0.90 0.80 0.70 0.60 0.60 4.79 4.84 4.65 4.32 3.04 3.61 3.33 3.36 3.29 3.06 2.77 2.46 2.79 2.83 2.74 2.11 2.27 2.03 2.• 4 2.69 2.•0 2.21 2.00 1.77 2.22 2.28 2.18 1.99 1.80 1.69 2.08 2.09 1.99 1.84 1.86 1.<&8 1.97 1.98 1.81 1.12 1.64 1.36 0.40 0.30 0.20 0.10 0.06 0 3.06 2.80 2.13 1.14. 1.29 0 2.16 1.77 ,1.14. 1.61 1.30 1.23 \ 1.08 0.88 0.77 O.M 0.66 0 0 1.37 1.26 1.18 1.06 0.96 0.69 0.50 0 0.87 0.83 0.t5 0 1.17 0.98 0.81 0.69 0.42 1.83 1.&0 1.08 0.78 0 1.00 0 1.10 0.92 0.76 0.11 0.89 0 1.70 1.M 1.38 1.21 1.04 0.88 0.72 0.52 0.37 0 O.M 0.68 0.49 0.36 0 1.91 1.63 0 2." 2.32 0.81 0 1.92 1.82 1.69 1.14 J.M 1.12 1.41 1.28 1.00 1.19 1.46 1.70 ~ ~ ~ ~ -s o~ ~ ~ C';J I 1.38 1.21 1.09 0.79 0.17 0 O.es 0.49 o I O.M 0.47 0 1.09 0.96 0.81 0.81 0.44 0 ~ THEORY OF WING SECTIONS 244 The incremental basic eft" and additional en., normal-force coefficients are obtained from Eqs. (8.3) and Table 7. The appropriate values of the TABLE 7.-VALUES OF ~Eaeg ~~I 5,10,15 I 30 20 4. -0.474 0.05 0.10 0.15 0.20 0.25 -0.448 -0.423 -0.397 -0.372 -0.347 0.30 0.35 0.40 0.45 0.50 -0.294 -0.268 -0.242 0.55 0.60 0.65 0.70 -0.215 -0.189 -0.163 -0.136 -0.320 G 40 50 60 Plain flaps -0.476 -0.451 -0.428 -0.404 -0.380 -0.477 -0.453 -0.431 -0.408 -0.387 -0.478 -0.455 -0.434 -0.411 -0.392 -0.479 -0.456 -0.435 -0.414 -0.395 -0.4i9 -0.456 -0.435 -0.415 -0.396 -0.357 -0.334 -0.311 -0.288 -0.265 -0.366 -0.346 -0.325 -0.304 -0.283 -0.372 -0.352 -0.332 -0.375 -0.357 -0.339 -0.377 -0.360 -0.342 -0.242 -0.220 b. Split flaps ...... . . . ... ...... 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 ~ ...... ...... ...... I I ...... ...... ...... ...... incremental basic Table 6 as follows: I P", -0.476 -0.452 -0.430 -0.407 -0.385 -0.477 -0.454 -0.432 -0.409 -0.388 -0:478 -0.455 -0.434 -0.411 -0.392 -0.479 -0.456 -0.435 -0.414 ":'0.395 -0.479 -0.456 -0.436 -0.415 -0.396 -0.363 -0.342 -0.320 -0.299 -0.278 -0.367 -0.347 -0.327 -0..372 -0.352 -0.375 -0.357 -0.339 -0.3i7 -0.360 -0.342 -0.30i -0.287 -0.333 I normal-force distribution are obtained by use of The appropriate value of the incremental additional Paa normal-force dis­ ttibution is obtained from the data of Appendix I as follows: (8.10) HIGH-LIFT DEVICES 245 TABLE 8.-VALUES OF T'. ~ 10 0 0.05 0.10 0.15 0.20 0 -0.00 -0.01 -0.01 -0.01 0.25 0.30 0.35 0.40 0.45 -0.01 -0.02 -0.02 -0.03 0.50 0.55 0.60 0.65 0.70 -0.03 -0.04 -0.05 -0.06 -0.07 ~I 40 30 50 60 ~--_.:...--_-_..:.-.._---=------~--_---:.._--- o 0.05 0.10 0.15 0.20 4. ~0.02 0 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 -0.11 -0.12 -0.14 -0.17 I I Plain Haps 0 -0.05 -0.0; -0.09 -0.11 0 -0.09 -0.14 -0.18 0 -0.14 -0.23 -0.30 -0.23 -0.38 -0.14 -0.16 -0.19 -0.21 -0.25 -0.27 -0.32 -0.37 -0.42 -0.46 -0.54 -0.63 -0.71 I 0 -0.21 -0.34 -0.47 -0.59 I I -0.72 -0.85 -0.98 -1.11 -0.28 , I -.. ----------··--0 b. Split --r----O- -0.02 f1~lPS -0.04 -0.05 -0.05 -0.07 -0.10 -0.12 0.25 0.30 0.35 0.40 0.45 -0.06 -0.07 -0.08 -0.10 -0.11 -0.15 -O.li -0.20 -0.23 -0.26 0.•50 -0.13 -O.2H -0.03 ­ -~.09 1--~.14··· -0.14 -0.18 -0.23 -0.2. -0.32 -0.37 -0.42 o ; -0.23 -0.30 -0.38 -0.21 -0.34 -0.47 -0.59 I -0.46 -0.54 -0.63 -0.71 -0.72 -0.85 -0.98 -1.11 t t i1 ! I i The entire incremental normal-force distribution is obtained by adding the incremental basic and additional distributions (8.11) The final load distribution is obtained by adding this incremental dis­ tribution to the load distribution for the unflapped lying section at the same angle of attack (see Sec. 4.5). Approximate hinge-moment coefficients may be obtained by an exten­ sion of this analysis." Such hinge-moment coefficients are nol considered to be reliable because the discrepancies between the predicted and actual THEORY OF WING SECTIONS 246 TABLB ~IIO 9.-VALUB8 20 01' 'F'. 30 40 50 60 1.00 1.06 1.09 1.11 1.12 1.00 1.10 1.15 1.18 1.21 1.00 1.15 1.22 1.28 1.32 1.14 1.14 1.15 1.15 1.23 1.36 1.38 1.39 1.39 a. Plain flaps 0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 1.00 1.00 1.00 1.01 1.01 1.80 1.01 1.02 1.02 1.03 1.00 1.03 1.05 1.06 1.06 1.01 1.01 1.01 1.01 1.01 1.03 1.03 1.03 1.03 1.03 1.07 1.07 1.07 1.08 1.07 1.01 1.01 1.01 1.01 1.00 1.03 1.03 1.03 1.07 1.24 1.25 1.25 b. Split flaps 0 0.05 0.10 0.15 0.20 .... 0.25 0.30 0.35 0.40 ..... .... .... ..... .... .... .... .... 0.45 .... 0.50 .... 1.00 1.01 1.02 1.02 1.03 1.00 1.03 1.05 1.06 1.07 1.00 1.06 1.09 1.11 1.12 1.00 1.10 1.15 1.18 1.21 1.03 1.03 1.03 1.14 1.14 1.15 1.15 1.23 1.24 1.25 1.25 1.03 1.07 1.08 1.08 1.08 1.08 1.03 1.07 1.03 1.00 1.15 1.22 1.28 1.32 1.36 1.38 1.39 1.39 load distributions tend to be appreciable at the trailing edge where the contribution to the hinge-moment coefficient is the largest. In applying this method, it should be remembered that, although the tabulated characteristics are differentiated by flap deflection, the important variable is the degree to which the flap is stalled. The tabulated charac­ teristics for plain flap deflections up to 15 degrees are for unstalled condi­ tions, whereas those for larger flap deflections represent progressively increased separation. Consideration should be given to the selection of the characteristics to be used in order to represent the actual flow condi­ tions properly. CHAPTER 9 EFFECTS OF COMPRESSmILITY AT SUBSONIC SPEEDS 9.1. Symbols. CII pressure coefficient, (p - PflO)/ Jip 1f1 CIIe critical pressure coefficient corresponding to a loeal Maeh number of unity B to~ energy K constant per unit ID888 M }'{ach Dumber, V fa R universal gas constant, e" - c~ 8 croes-eeetional area of a stream tube T absolute temperature V velocity of the free stream component of velocity along the span V. component of velocity normal to span (J speed of sound a lift-curve slope v. b span c, section lift coefficient Cp specific heat at constant pJ'e8SU1'e specific beat at eoostant volume Cr e base of Naperian loprith~ 2.71828 In logarithm to the base e m Jn888 flow per unit area loeal static pressure Po total pressure PflO static pressure of the free stream t time t thickness ratio u component of velocity parallel to the % axis v volume component of velocity parallel to the y axis V component of velocity parallel to the z axis to z; y, Z Cartesian coordinates a angle of attack fJ angle of sweep .., ratio of the specific beats, c./c. ., P. 11" P 1/ v''! - l\ft ratio of the circumference of a eirele to its diameter density of the fluid ID888 9.1. Introduction. The wing-section theory and experimental data presented in the preceding chapters are applicable to conditions where the variation of pressure lR small compared with the absolute pressure. This 247 248 THEORY OF WING SECTIONS condition is well satisfied when the speed is low compared with the speed of sound. At a flight speed of 100 knots at sea level, the impact pressure is only about 34 pounds per square foot as compared with an ambient static pressure of 2,116 pounds per square foot. At 300 knots, the impact pressure increases to approximately 321 pounds per square foot; and, at 600 knots, the impact pressure is 1,494 pounds per square foot. At the higher speeds, the pressure and corresponding 'volume changes are obviously not neg­ ligible. It is to be expected, therefore, that wing-section characteristics at high speeds will not agree with those predicted by incompressible Bow theory or with experimental data obtained at low speeds. Examination of the complete equations of motion89 shows that the parameter determining the effect of speed is the ratio of the speed to the speed of sound. This ratio is called the "Mach number." The physical significance of the speed of sound becomes evident in considering the difference between subsonic and sup­ ersonic flows, The speed of sound is the speed at which pressure im­ pulses are transmitted through the FlO. 143. Significance of Mach angle. air. At SIOlV flight speeds, the pres­ sure impulses caused by motion ~t the wing are transmitted at relatively high speed in all directions and cause the air approaching the wing to change its pressure and velocity gradually. Slow speed flows are accord­ ingly free of discontinuities of pressure and velocity. At supersonic speeds, no pressure impulses can be transmitted ahead of the wing, and the pres­ sure snd velocity of the air remain unaltered until it reaches the immedi­ ate vicinity of the wing. Supersonic ft.ow is thus characterized by discontinuities of pressure and velocity. As shown in Fig. 143, the pressure impulses are propagated in all directions at the speed of sound a while the 'ring moves through the air at the velocity V. The envelope of the pressure impulses is a straight line (for small impulses) which makes a slope sin-1a/V or sin-t M with the direction of motion, where M is the Mach number. This line is called the" Mach line." The air ahead of the Mach line is unaffected by the approaching wing. The present discussion of the characteristics of wing sections in com­ pressible flow is limited to subsonic speeds. The pressure-velocity relations for a stream. tube will be developed, and a brief summary will be presented of compressible flow theory as applied to wing sections. The theory of wing sections in subsonic flow generally consists of the determination of EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS 249 the first-order compressibility effects on the incompressible flow solutions previously developed. The theory is generally limited to speeds at which the velocity of sound is not exceeded in the field of flow. A brief discussion of experimental data will, however, include data in the lower transonic region where the speed of sound is exceeded locally. 9.3. Steady Flow through a Stream Tube. a. Adiabatic Law. The relation among pressure, density, and temperature for a perfect gas is p = pRT where (in consistent units) p = pressure p = density, moles per unit volume T ~ absolute temperature R = universal gas constant = Cp - c. Cp = specific heat per mole at constant pressure Ct, = specific heat per mole at constant volume This equation may be written (9.1) or PIl't r;= 112"2 T2 where the subscripts 1 and 2 refer to any two conditions of/the gas. When the temperature is constant (isothermal changes), the relation between the pressure and the density is obviously P1 PI -=­ 'P2 P2 Except for this special case, pressure and volume changes are accompanied by temperature changes. In the high-speed ftO\VS of primary interest, these changes occur so quickly that individual elements of the gas generally do not lose or gain any appreciable heat. Changes in state of the gas are thus assumed to be adiabatic. In this case, (9.2) where v = ratio of specific heats, cp/cv From Eqs, (9.1) and (9.2), the corresponding temperature relation is E! = (Tl)"Y/{Y-l) (9.3) T2 b. Velocity of Sound. Expressions for the velocity of sound may be derived from consideration of the equations of motion and of continuity. The equation of continuity for a compressible fluid may be developed in a manner parallel to that for Eq. (2.1) and is P2 THEORY OF WING SECTIONS 250 a(pu) ax + iJ(pv) + a(pw) == _ ap ay az at (9.4) For motion in one dimension, this equation reduces to iJ(pu) ap ax== - at or ap au iJp -+p-+uat ax ax ==0 dividing 1;ly p lap + au+~ap = 0 pat (9.5) pax a2: In the concept of a velocity of sound, the disturbances are considered to be small 80 that the changes in density are small compared with the density and the corresponding velocities are small. Retaining only the first powers of small quantities, Eq. (9.5) becomes ! ap + au.... 0 = a In p + au p at ax at ax (9.6) For one-dimensional flow, Eq. (2.3) becomes _ 8p == p a~+ pu8u ax at ax Again retaining only the first power of small quantities and dividing by p, webave au lap at -=-j;8x (9.7) It is known from the equation of state that p is a function of (9.7) can then be written au at == - Idp8p ; dp az == dp a In p. Equation p - dp ax (9.8) Difierentiating Eq. (9.6) with respect to z and Eq. (9.8) with respect to t, and combining, we have h dpatu (9.9) at' == dp8r The solution of this equation is u =f{~ - Ji, t) + ft(X+ Ji,t) These functions represent disturbances traveling in opposite directions with the veloc~ty V dp/dp. This velocity is termed the velocity of sound a. As stated previously, the changes in pressure occur so rapidly that no heat is considered to be conducted to or away from the elements of gas. EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS 251 The relation between the pressure and density is therefore considered to be adiabatic [Eq. (9.2)]. Then fiiP = FYP vdP v-; a= (9.10) For a perfect gas, a == K-v'i' (9.11) where T is the absolute temperature, and the value of K is given in the following table for various English units, and for dry air, -y == 1.4. Units of temperature Units of velocity Feet per second Miles per hour Knots OF. absolute °C. absolute 49.02 33.42 29.02 65.76 . . . 44.84 38.94 In applying the concept of the velocity of sound to various problems, it should be kept in mind that the concept as derived is valid only for small disturbances. Very strong shock waves and blasts should not be expected to propagate at speeds corresponding to the local velocity of sound. c. Bernoulli's Eqootion for Comprem.1Jle Flow. In deriving Bernoulli's equation for incompressible flow [Eq. (2.5)]J the followingexpression was obtained: \ Writing this equation in integral form, we have !2 JT2 + fdpP = E (9.12) Relating the pressure and density by the adiabatic law of Eq. (9.2) and per­ forming the indicated integration, ! 2 V2 + _'1_. ~ 'Y-1p zr E = ! V! + c T 2 " (9.13) This equation can also be derived from thermodynamic reasoning. The thermodynamic reasoning indicates that the quantity E in Eq. (9.13) is a constant along any stream tube for adiabatic changes whether reversible or not. For example, E is constant throughout & stream tube containing a normal shock. For reversible changes, Eqs. (9.2) and (9.13) together with a knowledge of the initial conditions permit the calculation of the variation of pressure and density with velocity. 252 THEORY OF WING SECTIONS d. Cro8s-sectional Areas and PreBSUTe8 in a Stream Tube. Along a stream tube, the equation of continuity for steady flow is pSV = constant where S = cross-sectional area of tube It is interesting to contrast the manner in which the cross-sectional area varies with velocity for compressible and incompressible flow, For the latter case, the area is seen to vary inversely with the velocity. The determination of the variation of area with velocity for the compressible case requires more extensive analysis. Differentiating the equation of con­ tinuity, and dividing by pSV, we obtain ! dB +! dp +.!.. 8dV pdV V =0 (9.14) Differentiating Bernoulli's equation with respect -to the velocity, V or + _'Y_ (.! dp _ 1!.) dp = 0 'Y - 1 p dp v + _1_ (1. a 2 _ 'Y - 1 p or p'- dV 2 a ) dp = 0 p dV a2 dp V+ pdV =0 (9.15) From Eq. (9.14), dS=_~(l+Vdp) dV V pdV Substituting the value of dp/dV from Eq. (9.15), we obtain dB dV =_ §.(lV2) = V a2 ~(1- W) V (9.16) This relation shows that the stream. tube contracts as the velocity increases for Mach numbers less than unity. The area of the stream tube is a minimum for a Mach number of unity and increases with Mach number for supersonic flow. From consideration of Eqs, (9.2), (9.3), and (9.11), it may be seen that the pressures, densities, temperatures, and velocities of sound are con­ nected by the equations (9.17) where the subscript m may denote the condition at any point but, in the follo"ring development, the subscript will denote the condition at the minimum cross-sectional area of the stream tube where the local Mach number equals unity. EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS 253 In order to obtain an expression for the cross-sectional area of the stream tube in terms of the local Mach number M, it is convenient to express Bernoulli's equation (9.13) as ! 2 l\I2 + _1_ 'Y - 1 = ~+ 2 1 am 2( 'Y - 1) a2 (9.18) where a and am represent, respectively, the velocities of sound at any point along the stream tube and at the point of minimum cross-sectional area. From continuity, we can write s; pV 8 = Pma.. 1.0 J .8 I 'fI" ,/ ~ <, r-, ~ / . J .6 s"'~ / .4 -, / r-, r-, "'r-; J .2 / I f ~ <, -....... ---.... ~ o .4 .8 1.2 2.4 2.0 1.6 2.8 3.2 II FIo. 144. Variation of etream-tube area with local Mach number. where Sand S. are, respectively, the cross-sectional areas of the stream tube at any point and at the point of minimum area. Substituting the value of piPa from Eq. (9.17), ~= (:j2/(Y-U (a9 = M (0:) (7+U/(y-U and, from Eq. (9.18), s. _. ( S - l\{ ~ + .1 ) ('Y+U/2<r-t> (~- 1)M2 + 2 (9.19) A plot of 8 m/S against M is given in Fig. 144. The pressure relations along the stream tube may be obtained by sub­ stituting the value of am/a from Eq. (9.18) into Eq. (9.17) p ( "y + 1 pm = (-y - l)M2 )y/(y-t> +2 (9.20) 254 THEORY OF WING SECTIONS where p == pressure at any point along stream tube. It is often more convenient to express the local pressure in terms of the pressure at the stagnation condition, p ( 2 )-r/<..,-I.) Po == ('Y - 1)M2+ 2 (9.21) where Po is the pressure of the fluid at rest. The temperature relations may be obtained from Eqs. (9.17) and (9.18) as T 'Y+ 1 T,. == (y - I)M2 + 2 or To == 1 + y - 1 MI (9.22) T 2 It is interesting to note that there is a limiting velocity which is reached when the gas is expanded indefinitely into a vacuum. Bernoulli's equation (9.13) may be written 111-+ 20.2 1 'Y- 1 (a)' a.. 'Y+l ==2(-y-l) With infinite expansion, the temperature approaches zero and the local velocity of sound a approaches zero. Accordingly V ....... = a. ~'Y + 1 'Y-l (9.23) The local Mach number M, of course, continues to increase indefinitely as the expansion is increased. e. Relations for a N ormol8hock. In deriving the relations for the cross­ sectional areas, pressures, and temperatures existing in a stream tube, the entire process was considered to be adiabatic and reversible. Experience has shown that this condition is satisfied whenever the velocity is increasing in the direction of flow. However, when an attempt is made to decelerate supersonic flows, discontinuities in velocity, pressure, and temperature generally occur. These discontinuities occur in a very short distance along the direction of motion and are termed U shock waves." The velocity and total pressure of the air decrease and the temperature and static pressure increase in going through the shock. The process, although adiabatic, is irreversible and characterized by an increase of entropy. The shock is stationary for a tube of fixed configuration with constant pressures at the inlet and exit. Since the velocity on the upstream face is supersonic, the shock can hardly be considered to propagate at the velocity of sound. Similar discontinuities occur forward of the nose of a blunt body traveling at supersonic speeds. In the case of a simple tube or of the shock im­ EFFECTS OF COMPRES8IBILITY AT SUBSONIC SPEEDS 255 mediately forward of the nose of a blunt body, the plane of the shock is normal to the direction of flow. This type of shock is a fundamental one and is termed a "normal shock," as distinguished from the oblique shocks shown in Fig. 143. Relations for determining the conditions across a normal shock wave from the conditions on one side are derived'" from considerations of mass, momentum, and energy. If the subscripts 1 and 2 denote, respectively, the conditions on opposite sides of the shock and if m is the mass flow per unit area and p is the static pressure, the condition for continuity for a shock stationary with respect to the observer is m The equation of conservation of momentum is PIUl = P2~ (9.24) = P2 - PI = m(ul - U2) (9.25) The equation for conservation of energy is PIUl - PtU2 _!2 m (u~! - Ut!) =~ "Y- 1 (l!: _PI) [.J2 PI (9.26) The solution" for these conditions is P2 P; = (P2/PI) (1' + 1) / (1' - 1) + 1 (p,./Pl) + (1' + 1)/ ()' - 1) (9.27) This equation relates the density and pressure ratios across the shock when This equation corresponds to a change in entropy except in the case where PI equals P2. It can be shown" that P2 must be greater than PI for the change in entropy to be positive. It would be convenient to relate the pressure and density rat.ios across the normal shock to the upstream Mach number. Such an expression may be obtained by expressing Eqs.. (9.24) and (9.25) in terms of the local Mach number and combining to give PI and PI are the upstream conditions. P2 - 1 = PI Substituting the value of 1\'112 = ,. Mt~'Y(l -~) PI/ P2 from 'Y [1 - \ P2 Eq. (9.27) gives " (P2/Pl) -:- 1 , _ <P!lpl) + ("Y 1""" 1 C:)-y - 1) ] <P:. pd (1' + 1);' (1' - 1) 1 (9.28) + From Eq. (9.24), we can write M~ = ~~ ~ l\I 1 P2 P2 (9.29) Values of the ratio Pot/Po. may be calculated from Eqs. (\J.23) , (9.27), (9,,28), and (9.29) where po is the total pressure. This ratio is an index of THEORY OF WING SBCTIONS 256 the useful energy remaining in the air downstream of the shock. Values of Mt/M1, p,./pl, PI/PI, and PoJPo. are plotted against M in Fig. 145. 9.4. First-order Compressibility Effects. a. Glauert-Prandtl rule. The Glauert-Prandtl rule36 relates the lift coefficient or slope of the lift curve of a wing section in compressible flow with that for incompressible flo,v. This relation was derived for the case of small disturbance velocities and 20 5 1.0 16 4 ~ ~ \" .8 <, M2 J2 3 r-, "- ~ .6 I PD2 r-, ,/ <, ~~ ............ ./ 8 2 .4 /t,I" 4 I /' ./ ~ / ~ ~ V ~ " :>< ~ ./ ~ .2 ~ PDI V ?~ V ~ ~ / ~ '" V ~ ~ V 7 ./ ."".,.,.,..., ~~ .c~2 P, /' ---r-, ~ '-. ~ ~ - ~ -.............. v~ r-----.. ~ r----.- ~~ L4 1.8 26 3.4 4.2 M, FlO. 145. Pressure, density, and Mach-number relations for a normal shock. low Mach numbers. These conditions are approximated for thin wing sections with small amounts of camber at low lift coefficients at speeds well below the speed of sound. The Glauert-Prandtl relation is Cl c at; ci, a; 1 vI-M2 -=-=-==~ (9.30) where the subscripts c and i denote, respectively, the compressible and incompressible eases. In deriving this formula, Glauert considered all the velocities over the wing section to be increased by the same factor. Equa­ tion (9.30) may therefore be applied equally well to the moment coefficient. The Glauert-Prandtl rule agrees remarkably well with experimental data, considering the assumptions made in its derivation. A comparison of this rule as applied to the slope of the lift curve for three wing sections of 6, 9, and 12 per cent thickness is given" in Fig. 146. KapianM obtained a first-step improvement of the Glauert-Prandtl rule EFFECTS OF COMPRESSIBILITY AT ~UBSONIC SPEEDS 257 for the lift of an elliptical cylinder and extended it to arbitrary symmetrical profiles, The Kaplan rule is ~~ = I' + 1 ~ t [1'(1' - 1) + ~('Y + 1)(1'2 - 1YJ where Il (9.31) 1 vl-M2 = -_""'/=== ; t = thickness ratio .24.-----..-----or----or----..,.--....--.... .201-----t-------t------+------1,........--+----1 .'61-----+-----+------I----I-oI-~---&.--_t dCt da, .12....------4-----+-­ ~-~~-+-_+__-t----_t .081-----'-------1-----+------;------1 .2 .6 .4 .8 ·ID N FlO. 146. Lift-eurve slope variation with Mach number for the NACA 0012-03. 0009-63. and 0006-63 airfoils. Equation (9.31) approaches the Glauert-Prandtl rule as the thickness ratio approaches zero. It will also be noted that the Kaplan rule shows a "peak dependence on the ratio of the specific heats 1'. The Kaplan rule for wing sections 5, 10, 15, and 20 per cent thick in air together with the Glauert­ Prandtl rule are plotted in Fig. 147. b. Effect oj j{ach number on tile Pressure CoejJicicnt. The Glauert­ Prandtl rule supplies a first approximation to the variation of pressure coefficient with Mach number. C~l Cp , 1 = vI - ]\{2 THBORY OF WING SECTIONS 258 Numerous attempts have been made to obtain more accurate expressions for the variation of pressure with Mach number, notably by Chaplygin,15 Temple and Yarwood, 119 and von Karman and Tsien. 129,14O Garrick and Kaplan" succeeded in unifying these results 88 approximate solutions of It',l 0.20 ~ J5 2.5 rule ./0 1.05 I I I ~Kaplon J I I 1.Pra1dt1-G!aJerf I I Ir' III rule JI 2.0 flI '1 / J 1.5 ~ 1·°0 -----ae ~ 0.4 0.6 0.8 1.0 M1 FIG. 147. Ratio of lifts for compressible and incompressible fluids as function of stream Mach number. the general problem and presented two other approximations. No simple general solution of the problem is known. The I{arman-Tsien relation is widely used in the United States. Ex­ perimental evidence appears to indicate that this relation is as applicable as any of the solutions. This relation is CPM 1 CPt = viI - M2 + [M2/(I + viI - M2)J(C p J 2) (9.32) EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS 259 A plot of the pressure coefficient as a function of the free stream Mach num­ ber AI is given in Fig. 148 for a number of values of the incompressible /.0 .9 .8 1 .~ i· 7 ~ t~ ~ ~ .6 ~ u~ .5 .4 .3 .2 .­ ~ ./ °0 ./ .4 .5 .6 .7 Free-stream Mach 1III11ber,Mo 148. Variation of local preesure coefficient and local Mach number with free stream Mach number according to KArm8.n-Tsien. FIG. pressure coefficient. Also shown in Fig. 148 are curves of the local Mach number ML. A special use for relations such as Eq. (9.32) is the prediction of the stream Mach number at which the velocity of sound is reached locally THEORY OF WING 8ECTIONS 260 over the wing section. Jacobs" applied the Glauert-Prandtl rule to this problem. It is now customary in the United States to use the Karman­ 3.5 ./ .4 ,,5 .6 .7 Free -stream MochnumberlMo FIG. .8 .9 148. (Concluded) Tsien relation [Eq. (9.32)]. This problem may be solved from Fig. 148 by following the curve for the highest incompressible pressure coefficient on the wing section to the line ML -= 1 and reading the corresponding stream EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS 261 Mach number. Figure 148 may be used conveniently when the maximum local pressure coefficient is known experimentally at some subcritical Mach number. If the incompressible pressure coefficient is known theoretically, it is more convenient to use Fig. 149, which gives the corresponding criti­ cal Mach number directly. 1.0 .9 .8 1 \ \ I \ II ~ -, r-, <, ~ ~:-- ~ .4 I I .~ .4 I l6 2.0 LDw-spetJd presstn coefficient, Cp .8 /2 2.4 2.8 Flo. 149. Critical Mach number chart from KarmAn-Tsien relat.ion. 9.6. Flow about Wing Sections at High Speed. Consideration of the effects of compressibility on the flow over wing sections may be conveniently divided into two parts" The first part deals with the characteristics in the range of subcritieal Mach numbers. 111C second part deals with the characteristics at supercritical Maeh numbers. The critical Mach number dividing these t\VO ranges is defined us the stream Mach number at which the local velocity of sound is just attained at any point in the field of flow. a. Flow at Subcritical Moen Numbers. The first-order effects of com­ pressibility on the potential flow at suberitical Mach numbers are well described by the theoretical relations of Sec. 9.4. An indication of the accuracy of the Karman-Tsien relation is given!" in Fig. 15Ob. This figure shows the experimental pressure distribution for the NACA 4412 section at a Mach number of 0.512 and a pressure distribution predicted from the experimental data obtained at a Mach number of 0.191 (Fig. THEORY OF WING SECTIONS 262 -1.6 -1.2 ~- -.8 -A c'P 0 -- ---­ ~ .......... / ........... ~ . r ~ -- ~ -....... ~ ~~ ... r ~ ........ ~ ....... ~~ -­ ~ ~ (b) M=0.5/2 (a) M=0./9/ > ~~ ( .8 --/.6 -­ ~ -1.2 ~ .......~ ./ ' I -.8 f ,.. r 4 -.4 Cp 0 -- ""'­ ....... ....... -I. 6 ~ -I. 2 \ \ .... ~, I r----."",,-: ~ 1', --. rr -=:-.... ~~ 817 ~ <,~...... 'i~ ~ - \ "\ / J "<, ...... 4y-. ~~ ""lr ""'-..... ...... °T .... : 4 Iii ~ ~ ~ "<, \ \ Cpc ~ -'i' ......1'. ~...... I'-o.. ~~ (f) (e) M=Q664 8 ~ (d) M=0.640 "' Cp -. ~, -­I tp; (c) M=0.596 .8 , ~ M:Q690 ; I. 2 -I.2 -. 8".... -.4 t i I......... ~ ~.- ~ -._~ , \ .'" 1Jc ~ ~~ OJ ~- , ....... '~ ...... .. Dotted line in(b)represents application of Kormon-Tsien .......~ .4 - -~ -~ (g) M=O.735 8 .. reloliJn 10 doloin tal x t,pper surface o Lower surface ~ I. 0 20 40 60 Chor~percenf FIG. 80 100 0 20 40 60 80 Chord,percefl' 150. Pressure distribution for the NACA 4412 airfoil. a=: 1°52.5'. 100 EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS 263 15Oa). The accuracy appears to be satisfactory for such applications. The larger area of the pressure diagram at the higher Mach number corre­ sponds to an increased lift coefficient at the same angle of attack. The resulting change of lift-curve slope has been shown (Fig. 146) to agree fairly well with the Glauert-Prandtl rule. The applicability of these theoretical relations is limited to cases where the effects of separation are not marked. If separation is present, the increased pressure gradients associated with higher Mach numbers tend to increase the separation and thus change the whole field of flow, Moreover, the absence of appreciable separation at low speeds is not necessarily a sufficient criterion, because the increased pressure gradients at the higher Mach numbers may cause separation in marginal cases. The theoretical relations may therefore be applied with confidence only to wing sections of normal shape at low lift coefficients. b. Flmo at Supercritical Mach Numbers. The flow at supercritical Mach numbers is partly subsonic and partly supersonic and is characterized by the presence of shocks. No theory dealing with these mixed subsonic and supersonic flows has been developed to the extent of being useful in the prediction of wing characteristics. The almost complete lack of theoret­ ical treatment requires reliance on experimental data at supercriticaI Mach numbers. For the typical case of a wing section operating at a small positive lift coefficient, the velocity of sound is reached first on the upper surface, as is shown in Fig. 150 by the pressure coefficient increasing negatively beyond the critical value CPc_ Figure 150c shows no drastic change in the pressure distribution when the local velocity' of sound is exceeded by a small amount. Many experimental data indicate that drastic changes in the forces on wing sections do not occur until the critical Mach number is exceeded by a small but appreciable margin. There is some doubt as to whether a shock necessarily occurs when the velocity of sound is locally exceeded by a small margin. DC In any case, the losses associated with a shock from very low supersonic velocities are very small, as shown b)" Fig. 145. Such small losses would not be expected to produce drastic changes in the field of flow. A shock occurs when the critical Mach number is exceeded appreciably, as shown by Fig. 150d to g. The shocks act, at least qualitatively, like a normal shock to reduce the velocity to subsonic values. The resulting rather sudden changes in the pressure distribution are shown dotted in Fig. 150. Schlieren pictures of the shocks corresponding to the diagrams of Figs. 150! and g are shown-" in Fig. 151. The positions of the shocks as shown by the photographs correspond closely with the dotted regions of the pressure diagrams. Figure 152 presents another series of schlieren photographs showing the shocks on a NACA 23015 wing section at an THEORY OF WING SECTION8 264 angle of attack of 3 degrees. These pictures.!" which were taken in the NACA rectangular high-speed wind tunnel, show the progressive intensi- FIG. 151. Schlieren photograph of flow for the NACA 4412 f!,irfoil. a = 1°52.5'. fication and rearward movement of the shock with increasing Mach num­ ber. The character of the flow with shock is greatly affected by the inter­ action of the shock and the boundary layer. It is apparent that the shock cannot extend to the surface through the region of Iow velocity in the EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS 265 boundary layer. If the plane of the shock is perpendicular to the flow, the increase of static pressure through the shock will correspond to that shown in Fig. 145. The relatively high pressure downstream of the shock will tend to propagate upstream through the boundary layer. The pres­ M:O.40 M=O.60 M=O.550 M:O.650 FlO. 152. Schlieren photographs of flow. N.:\CA 2:J015 airfoil. NACA rectangular high­ speed wind tunnel. AiI"foil chord, 2 inches. Angle of attack. 3 degrees. sure distribution measured at the surface cannot , therefore, be expected to show a discontinuity. Figure 150 shows that the sharp pressure rise is, in fact, shown as a steep gradient at the surface. Such propagation of the pressure along the boundary layer necessarily leads to pressure gradients across the boundary layer in the vicinity of the shock. This situation is quite different from the usual boundary-layer conditions at low l\Iach numbers. The large adverse pressure gradients associated with the shock always produce a large increase of boundary-layer thickness and frequently cause 266 THEORY OF WING SECTIONS separation, as shown U 4 in Figs. 152 and 153. This separation is an im­ portant or even predominant factor in causing the large force changes occurring at supercritical speeds. The relative importance of the shock losses and the effects of separation on the drag of a typical wing section M=O.750 M:O.70 M:O.821 M=O.80 FIG. 152. (Concluded) are shown-" by Fig. 154. This figure shows the total pressure loss across the wake of the section. These losses are intimately associated with the drag, and, to the first order, the drag coefficient is proportional to the area under the curve. Intense shock has not occurred at the lower Mach num­ ber, and the diagram has the typical shape obtained at lo,v speeds. Intense shoek has occurred at the higher Mach number, and the drag coefficient has obviously increased. This increase may be considered to be of t\VO parts. The very large losses of pressure in the center of the diagram are similar in shape to the lower speed diagram and are attributed to the usual type of loss associated with skin friction and sep RXat,i on. The smaller 267 EFFECTS OF COMPRE88IBILITY AT 8UBSONIC SPEEDS pressure losses extending far out into the stream are attributed to the losses across the shock. It is apparent that the drag rise associated with the usual type of loss is as important as that directly associated with the losses across the shocks. The pressure rise measured'P on the surface of the wing section in the vicinity of the shock is smaller than that corresponding to a normal shock with the losses of total pressure indicated by the wake surveys. The pressure rise correspond­ ing to a normal shock may also be obtained by means of Fig. 145 from the local Mach num­ ber ahead of the shock as indicated by pres­ sure measurements on the surface. In this case, also, the computed pressure rise is greater than that shown by the surface pressures. The com­ plete explanation of this discrepancy is not known, but two factors could tend to reduce l\>1 0.691 the pressure rise on the surface below that FIG. 153. Schlieren photograph of separated flow for rear portion corresponding to a normal shock. of NACA 23015 airfoil. XACA The first factor is the pressure gradient rectangular biJ!h-slx-t'd wind through the boundary layer previously dis­ tunnel. Airfoil ehord, [; inehes. Angle of ut tar-k, (. dt~~rN'~' cussed. The importance of this factor may he inferred from Figs. 152 and 153. The lines in the schlieren photographs correspond to density gradients. Dark Jines are 8ho''''11 in the vicinity of 81: () ~.4- ~ ~ .s-: M Q810 t1 ~ .2­ 15 ~ .708 ./­ .~ Col) § 0 240 200 /60 /20 80 40 0 40 80 /20 160 200 240 Distance across KOte.l perceri chord FIG. 154. '''''ake shape and total pressure defect as influenced by Mach number, 0012 airfoil, a = 0 degrees; NACA rectangular high-speed wind tunnel. ~ACl~. the shock making a small angle with the surface. Although these density gradients may be partly associated with temperature gradients, the sharp­ ness and intensity of these lines are thought to indicate the presence of appreciable pressure gradients. The second factor is the possible reaction of the boundary layer on the shock. The thickening or separation of the boundary' layer tends to pro­ duce oblique shocks, Such shocks would produce some pressure rise ahead THEORY OF WING SECTIONS 268 of the normal shock and would accordingly reduce the intensity of the normal shock, especially close to the surface. The extent to which the normal shock is locally softened in this manner is uncertain, but Figs. 152 and 153 suggest that this mechanism may be of considerable importance. Investigations made by the NACA, by Liepman,· and by Ackeret4 and his associates emphasize the complexity of the interaction of the shock and the boundary layer. These investigations indicate important differences between the shock phenomena for laminar and for turbulent boundary layers. Although exceptions have been noted, laminar boundary layers .4 CD em -·~I .2 .3 .4 .5.6 .8 Mach numbtJr,M FIG. 155.. Force and moment coefficient variation with Mach number. NACA 23015 airfoil. a == 0 degrees; NACA rectangular high-speed wind tunnel. tend to produce multiple shocks of lower intensity, or U lambda-type" shocks, while those associated with turbulent boundary layers tend to resemble intense normal shocks more closely. The understanding of these phenomena awaits further theoretical and experimental investigations. The typical changes of forces experienced at supercritical speeds by a wing section of moderate thickness are shown!" in Fig. 155. The Mach number for force break is seen to be appreciably higher than the critical Mach number. At" Mach numbers higher than that for force break, the lift coefficient decreases, the drag coefficient increases, and the moment coefficient usually increases negatively. 9.6. Experimental Wing Characteristics at High Speeds. The existing three-dimensional lying theory, which is the basis for the concept of section characteristics, is not valid for supercritical speeds where part of the flow is supersonic. Consequently section data cannot be applied quantitatively to the prediction of the characteristics of wings at such speeds. Moreover, present trends for wings designed for efficient operation at supercritical speeds are toward low aspect ratios and large amounts of sweep. As EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS 269 pointed out previously, section data are of doubtful quantitative signif­ icance for such plan forms at any speed. Under these circumstances, the detailed consideration of section characteristics as design data is not warranted. Nevertheless, the characteristics of wings at supercritical speeds de­ pend to some extent on the lying sections employed. The variation of the wing characteristics with wing section appears to be qualitatively similar to the variation of the wing-section characteristics- if the lying plan form is such as to permit any semblance to two-dimensional flow, Consequently, the purpose of this section is to present typical data showing qualitatively the major effects of variation of the wing profile at high subsonic Mach numbers. a. Lift Characteristic«; One of the first and most severe effects of com­ pressibility encountered in flight was the tendency of many airplanes to "tuck under" in high-speed dives. This U tucking under" tendency con­ sisted of a large negative shift in the angle of trim tbgether with a large increase of stability that resisted the efforts of the pilot to trim at the desired positive lift coefficients required for recovery from the dive. The resulting large elevator motions required to recover from the dive, or even to prevent the dive from becoming steeper, corresponded to excessive con­ trol forces, thus leading to the impression that the "~t.ick became frozen." These changes in longitudinal stability limited the maximum safe flying speed for many airplanes. These effects are intimately associated with tile changes in lift charac­ teristics of wing sections at Mach numbers above that for force break. The relatively thick cambered wing sections commonly used during the Second ,Vorld ar experienced a positive shift of the angle of zero lift and a reduction of lift-curve slope. Although ID3ny effects associated with other characteristics of the airplane are present, the change in angle of zero lift directly affects the angle of trim and the reduction of lift-curve slope directly increases the longitudinal stability. The shift in the angle of zero lift is associated with the camber of the wing section. ..~ symmetrical section shows, of course, no change in the angle of zero lift 116 (Fig. 156). The rather large positive shift in the angle of zero lift shown!" in Fig. 157 for the X.~(~.:\ 2409-34 section is typical for cambered sections without reflex. In this case, the change amounts to about 2 degrees for a change of Mach number from 0.80 to only 0.83. The effect of increasing the camber is showu'" by Fig. 158. Doubling the amount of camber causes a similar shift to occur at a lower Mach number. The effects of thickness on the lift characteristics of"symmetrical sec­ tions are shown" in Figs. 159 to 161 for the N..~CA 0006-34, 0008-34, and 0012-34 sections. The 6 per cent thick section shows a large increase of lift coefficient at a given angle of attack with increasing Mach number at "r THEORY OF WING SECTIONS 270 values below about 0.85. At Mach numbers above the force break, the lift coefficient decreases rapidly, but it never goes below the low-speed values, at least at Mach numbers up to 0.95. The 12 per cent thick section, however, fails to show much increase of lift coefficient at subcritical Mach numbers, but it shows a large decrease of lift coefficient at Mach numbers just above the force break. At a Mach number of 0.85, the lift coefficient, c:: ----~ -====­ I/. 4 , ~ ....-:"'~ ~ ~~ ~ ~~ ~~ , t .06 ,, \ -. , I .... ... f~-~ f'... .. ~---~ ---- , ,, -.2 I , ,, f-- -- o ~ :~ , ,, ,, , ,, ,/ l,/ , l " M= ~ .40-.60-.70--.80 ---- +-.83 ---,-- t r V V ~ _......... :,...-' ~ ,~......... ~ -- --.2 I I .- ---<o---_..J ___ -- - -0:::-- .4 I t -- \ .6 1 .B -~_.- - -- .~ to Lifl coefficienfl Ct FIG. 156. Aerodynamic characteristics of the NACA 0009-34 airfoil. and accordingly the lift-curve slope, are only about two-thirds of the low­ speed value. At higher Mach numbers, the lift coefficients again increase. The effects of Mach number on the lift characteristics of the 8 per cent thick section are intermediate. These data show that the thin sections experience force breaks at higher Mach numbers than the thicker sections and that the lift force breaks are much less severe for the thin than for the thicker sections. The effects of thickness for a series of cambered sections are shown" in Figs. 162 to 165. These data indicate to an even greater degree the adverse effect of thickness on the character of the lift force break. Comparison of these data with those for the symmetrical sections (Figs. 159 to 161) shows the adverse effects of camber on the Mach number for lift force break anti on the shift in the angle of zero lift. EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS /2 01/./" V') 8 ~~~ ~;' .-4# 4 ~ ," --- .., ' ~ ~~ "" ,'" , ~ ". ~ ~ ,,~ , ~~ ~:;::; 7'/ "/ ~~ If' ~ .. " V" I , I , ./2 .40-.60-.70--BO-··.8.3 -- --- ./0 I , ,, <, .02 ~ ./ ~ r-- -.2 .. - --- "'~ .- ,. , , , I , I ~. --r--' o J '1 1/'I I j ,I t: I I I I / ') I , I) hI , ~ -~ ~ .~ r- s: --- ---. -- -- ..2 J I " I v , ",,; ~- Ii II /1 -,' i I ,I , , ," -" I 1 I , --., J / I ,, i', -.. , I M= .04 / -:~ ~ :;::;-- 1- ~~ -", .4 .6 .8 LO Lift coefficientI Ct FlG.la7. Aerodynamic cbaractBistics of the NACA 2409-34 airfoil. 271 THEORY OF WING SECTIONS 272 Comparatively few data are available to show the effect of thickness distribution on the characteristics above the force break. The trends indicated by Stack and von Doenhoff116 cannot be considered representa­ tive for properly designed families of "ring sections because of the arbitrary , .-- 4 ",- ,/ ../' ./ , I .- , / - --~ ~~ ~ ~.'" ~~ 9 d ." ~ rt/ : .......... 17 ,.1 ./0 If M= ,I .40-.60-.70--.80 ---- V .08 -, -- /" \ \ ,~ /' -, ! ! ,/ ~' ~- \ , \ ...... ~', .02 ~ ""- f-- .--- - .~ ".,. ... v·/ - ....=:::::: :;;;0.0- ~ ,/ ~ ./ --- -.2 o .2 Lifl FIG. .--.6 .8 T- .4 /.0 coefficien~ c& 158. Aerodynamic characteristics of the NACA 4409-34 airfoil. geometrical manner in which the sections were varied. If the thickness distributions are varied in such a manner as to avoid peaks in the low­ speed pressure distributions, as in the case of the N ACA 6-series sections, the indications are that the ~ffect of thickness distribution is minor com­ pared with the effects of thickness ratio and camber. There is some indica­ tion 1l 6 that the trailing-edge angle should be kept small to avoid adverse effects on the lift characteristics similar to those for thick sections. Figures 166 to 168 show the17b lift and drag characteristics of the NAC..~ 66-210 wing section. The most outstanding feature of these data as com­ EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS .7 .6 1 a." .5 / fdegJ 5 ..... 4 ~ 3 2 ./ ~ ;..­ ~ L----~ ~ ~~ -­ --" v V ~ ~~ jf ~~ ~ _L.-J ~ lla-- ~ I/' / ~ ~~ \~ 1/ r\~ , -...a 1.1 ~~ \I d ~""'" ~~ ,..,. ~ ~ ~~ ,-oc: ~ r<\ o -:1 o .1 .2 .s .5 .4 Moch .6 .7 .8 .9 1.0 ooroe«, M FIG. 159. EfI'e("t of compressibility on the lift of the N ACA OOOH-34 airfoil. .7 .. )V .6 ao Itdegl .5 6 .4 / ~V 1_ ~ V 5 "IIl"'­ ~ ..... 4 -"~ ~ ~ ~ ~ ...... v: l,I ~ I' I" ~, V ~ 'r'( ~ W' L".....-ol 1\ ~ .JIn \ W \ '~ ~ .3 3 ...- ~ ~ j.....­ ~,..- '\ ....-I~ 2 t-­ ~ ~ ./ 0 ~~ ~ ~ ~ ~..- ~~ 1 L.P '""" ~ ~ ~ 0 ./ -I ... w ..... ~ - ~ ~KI" .2 o ./ .2 .3 .4 .5 .6 .7 .8 .9 1.0 Mach number~ M Flo. 160. Effect of compresaibilit:r on the lift of the NAC.A 0008-34 airfoil. 273 .7 .6 .5 .4 .1' .3 .6 ~ ...2 (d':;J .. L­ 6 .6 ~ ~ ~tA - 5 I .~ 4 ~ 3 1.3 Set> :t: .1 I~ ~ § SO J'.. u -:1 c ...... .,~ - I o ~ ~ 1\ ~, rr u... lrl 1/. ~ ~ l't' ~ I\' '] ~ e ./ 1.$I! '" .... ~L. ~ -.2 ~ ...J~ -:~ """'ll'­ A 0 -A -./ 0 ./ .2 .3 .4 .~ .4 .~ .~ l • J Jr,o -.50~""'."'I""-.""2---'''''''.J--''''',4''''''''''''.''''5''''''-.I.6--''''.7 MtJCh numbe'l M FlO. 161. Ettect of compressibility on the lilt of the NACA 0012-34 airloil. .8 .9 1.0 .Mach ntI'IIber 1 M FIG. 162. Eft'eot of compressibility on the lilt ot the NACA 2306 airfoil. .8 .1 I I J J_ J ",JTIII 'Ir­ - .6 .6 - ---~ 4 ~-""'~----I""- .5 .5 ,....- - .. ',....- ... .4 ~ 3 - ~ :~ ~'" ~ 1 ~§ ~ ./ ;:: ~ o I - r- ,....-~ ---I .t L-..-&.--.-...,._-.-II-. ,......-- -- r- o~ 1,....- -./ -.2 - J I I - ~ - .3 .4 .5 .6 .7 .8 .9 1.0 163. EJleot of eompre88ibility on the lift of the NACA 2309 airfoil. 1 I FlO. L~ \.' -,....- ~ ~~ I'\._ \ " - r-- ->-- .... -.­ .J.1~ ,\ J 7 _ ~:~ .i:. J I 'J I IJ I I -,..., -r­ number, M to-­ Jt.-.­ I I Mach Mach IHJfTIber, M FIG. _ I '.-LI I -~ I I J-1'-- I I I I I I I I I I I I-I I I I I I I I I I I I I o .1 .2 .J .4 .5 .6 .7 .8 .,J r.2 -.­ _'-.. ~ - ./ _ \ ~~~ 1 1 . . _ - -:2 -.40 _ _\ ,n .. ,_ 1\ , I""""""" ~ -:3 _ J J:J I I -r-~ ~\ -r- .2Il.-......&.--t--r---Tt-- 0 - -.J -~r- L-_........L,----+---,..---.-- J L----L-.---r---~~ .il · ~:f .2 .... -.-r--.,.... -- 2 n; r---.l-...-.,.-r­ ~ I I I\. ~ ........ ~ -r- .4 ~~ .3 I I LU'-++BH-r-r-n, -~.-, -.­ I er. ->-(deg)-,-- ~_ ~ _ >­ ictJ I I I I .9 /0 164. Effect of compressibility on the lift of the NACA 2312 airloil. THEORY OF WING SECTIONS 276 pared with th~ for the NACA f~ur-digit series secti~ns is the ability of the 6-series seefion to carry large lifts at moderately high angles of attack and high Mach numbers. The lift-curve slope for the NACA 66-210 section at a Mach number of 0.75 is high and substantially constant up to a lift coefficient of about 0.8 (Fig. 168). At the same Mach number, the NACA 2309 section (Fig. 163) shows a reduced lift-curve slope at a lift .8 ~ I(deg~ .7 5 ~ ~ ,......"", r-, r\ ~ .6 4 ",..-. r--., ~, ,~ IV ...... ~ ~ ~ .5 3 ~ 2 , 0 ./ ~~ ,~~ '\ :~~ t-­ .~ 1\\ r\.~ l~ ~ ­ _JIl ~~ - ~ ~ ~~ ~ ~~ l,..:.A ~ -I ;1 ~ ~ 711 ~ I1J ,r~ ~ IfZ If f~ ~f/ '" o l\ ~[~ ~ J P)"'l -./ ~ao4! 'I ~.J ~ rftlCl , ~~ -:2 o .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 Mach IVfTIIJer. M FIo. 165. Effect of compressibility on the lift of the NACA 2315 airfoil. coefficient of only about 0.55 and almost no increase of lift for a change of angle of attack from 4 to 5 degrees. Figure 169 ShO\VS376 the predicted critical Mach numbers and the Mach numbers for force divergence as functions of the lift coefficient for the NACA 66-210 section. The outstanding characteristic of these data is the wide range of lift coefficients over which high Mach numbers for force divergence are realized as compared with the range for high critical Mach numbers. The predicted critical Mach numbers are lower than those for force divergence but approximate the latter values over the range of lift coefficients where the speed of sound is first reached near the location of minimum pressure at the design lift coefficient. It is apparent that critical Mach numbers predicted on the basis of the attainment of the velocity of EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS 277 sound near the leading edge are useless as an indication of the force charac­ teristics of NACA 6-series sections. The general effects of Mach number on the maximum lift coefficients of wing sections may be inferred from the data'19 of Fig. 170. These data were ....- Symbol a 1.2 I - 10- 1.0 0 -6- + -4· -2· x a 0 A a ...SymJol v e- 6- 8· 10· <J O· 2· ~ 4· - ~ ~ .8 ----' .6. - ~ ~ ....-"" ~~~ ..J.~ ..J~ ~ ~ l.A-1 ~ ~ ~ ..J~ ~ ,4- K ~ ~ v ~ ~ I~ ~~ I"'Q ,--. ~L... - iI_ _-._ '-- - - r-- ~ ~ .. '" ~ l\, ~ >-- ~ ~ .4 .... r'\ .s .~ ~ \ 0 ~ ~~ "" ---- -~~ . " ._ _ .. I( ~ ........ , ' .. I "" It- ~ ~ .3 ~ "-t.. ""'- t. f\ \ - , , \ .6 ~ ' .... /4 ,~ , 6 '~ b-A~ .7 .8 .9 Mochnumber;M FIG. 166. The variation of section lift coefficient with ~lach number at various angles of attack for the NACA 66-210 airfoil. obtained from tests in the NACA Langley Hi-foot high-speed wind tunnel of a lying having an aspect ratio of Gand having X.•\CA 23016 sections at the root and NACA 23009 sections at the tip. At Mach numbers above 0.30, the maximum lift coefficient decreases rapidly with Increasing Mach number up to values of about 0.55 where the maximum lift coefficient is approximately 1.0. At higher Mach numbers, the actual maximum lift coefficient decreases more slowly; but the angle of attack, or lift coefficient, at which the lift curve departs radically from its normal slope continues to THEORY OF WING SECTIONS 278 decrease to the limits of the tests. The resulting variation of maximum lift coefficient with Mach number is shown" in Fig. 171. ' The increase of maximum lift coefficient with Mach number at values below 0.30 is associated with the variation of Reynolds number for these .20 ~ Symbol a 0 + I( .18 D 0 A v .16 ~ CI -6· -4· ~ -2" -­ J\ d' e: I , 4· s­ f 8" JO. <I l. -~ "-­.. ~ J~ I Q.' I r, f I .. 1 ~ V in- , (vi l Jl /If -,J T' J ... .06 ~ ~ .02 4~ I-­ ~ ~ l~ ~ ,..,--J ....... ~~ ~ ,.-­ / .5 ,A f~ i ~' ~ A AI ~ '1. I. ~ J I I I til ~ ~t '' I: I J ~! I(' If v IV I ..J \. ~ -' . -­ r.....­ .4 r~ ~ j ....... ~ , .6 -!.... rJ Yltf -~ -.:;;.:::..­ .7 ~ .8 .9 Mach fJII'TJberlM Flo. 167. The variation of section drag coefficient with Mach number at various angles of attack for the NACA 66-210 airfoil. tests. This effect was studied more extensively in the NACA 19-foot­ pressure wind tunnel where the effects of Mach and Reynolds numbers could be separated" for low Mach numbers. These data indicated, at least qualitatively, that the peak maximum lift coefficient is determined by the critical Mach number at maximum lift which, of course, is attained at relatively low free-stream Mach numbers. At subcritical Mach numbers, the maximum lift coefficient is primarily a function of the Reynolds num­ ber.&8 at very low Mach numbers, although the effect of Mach number is 279 EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS l2 .J 1.0 ~ / ~ .6 ~' .~ s 0 .(t= ~-.2 -.4 / /­ / I O· 4· / / if V / I--­ j J J J / ~ 1/ j / " " / I t/ '1'", ( I / ~/ / / I v I J I I I IfJ I ) v / If I I( /1 I 1/ I 1/ J / / j 1/ j 1/ / I 7/ / / II I v I 1/ II If ~ J J I J I I I I I I I I I / 1/ I J J 1/ II II II [7 II 1/ If 1/ If I II ) / / 1/ .4 ~ .2 ~ ,/'" / .8 r: ~ 1\ ~ ~ J J J J I ~ I I J J I I I-I I I I I I I J / I If If If I i O· 4· 0­ 0" tr'1 If If J 0­ 4· 8 If I II / / 1 / I I I J / / If 1/ V / I I II I I I I J 1 I I j j J 1/ II I I 1£ v ~ 1/ ~ 1(" I II V 1/ / I I II 1 I J )V I / j v VV -.8 .400 .600 700 .800 .900 M FIG. 168. The variation of section lift coefficient with angle of attack at various Mach nurn­ hera for the NACA 66-210 airfoil, .300 .500 .9 , I .8 ,, ;' '/ I , ,, / , / !-o-, r::. ~< Lifl divergence ~ F-::-: ~ / I - ~ 0rrJg divergence ~- ~~ "--- ............ / h, , II -.4 -.2 \, 0 .2 .4 .6 169. Critical Mach numbers for the NACA 66-210 airfoil. "" t-, , 5ecIion lif/ coefficien/)c1 FlO. "- \'¥'Y ,1 -.6 " \ V .4 -, \. \\ 1./ .5 -- , ---, "" 1'"-_- .8 THEORY OF WING SECTIONS 280 not negligible. At supercritical Mach numbers, the value of the maximum lift coefficient decreases rapidly with increasing speed, and the effect of Reynolds number is secondary. For the wing tested, the critical Mach 1.6 I M=O.20 ..... <:» .30-". /.4 ~ ~i \ \, \ .50~ IU \ V.. . . \ ' , . 5 , IA ~~ \ .60, ~ r---- ~ loSS" ~ 40~ .45~ 1.2 - 1.0 .:5~ ~ .8 ~ -- ~ ~ 1,.-- ~-- ~ '~ ~1lt .rot} .6 .4 , .2 II ~~ o J ~, -.2 -.4 '" ~ I ~~ , e: ..4 -.6 -/2 FIG. ~ -8 -4 0 4 8 /2 /6 20 24 28 a1deg. 170. Variation of wing lift coefficient 'with angle of attack for several Mach numbers. "-88 number of the order of 0.25 to 0.30 for the flaps-retracted condition and of 0.17 to 0.20 with O.2Oc split flaps deflected 60 degrees. It should be noted that the maximum usable lift coefficient for an air­ plane at high Mach numbers may not be the same as the actual maximum lift coefficient. Referring to Fig. 170, the airplane may be unable to fly on the relatively flat portions of the lift curves above the force breaks, be­ cause of the serious buffeting associated with the partly separated flow and, perhaps also, because of stability and control difficulties. EFFECTS OF COJIPRESSIBILITY AT SUBSONIC SPEEDS 281 At supercritical speeds, changing the camber near the trailing edge, as by small deflections of a flap or aileron, cannot affect the flow over the front part of the section because pressures cannot propagate upstream through the supersonic region. Consequently, the flap effectiveness must be greatly reduced at supercritical Mach numbers. This effect is clearly shown!" by Fig. 172. This loss of flap effectiveness is related to the positive 1.6 1.4 V ~ _ ...... ............. """"'" <, , r-, 1.2 \. \ ~ ~/.O Maxinun lifl coefficient .\ -, \ ~ rsa Lift coefficient ob/oinedot 2° 10 3°0b0ve the angle of ottoc« 01 whichinitiol separation ofthe ' flow from the wing occurred .8 r--- --t, V" " ", " .6 _2 .3 .4 .5 .6 .7 M FIG. 171. Variation of wing maximum lift coefficient with 1\tlach number, shift of the angle of zero lift for cambered sections. The rearward port.ion of a cambered section rna)" be considered to be a special case of a deflected flap. The loss of flap effectiveness at the force break leaves the forward portion of the section at an effectively lower angle of attack, and the resulting change in lift coefficient corresponds to a positive shift of the angle of zero lift. Conversely, a strongly reflexed section would show the opposite effect. It follows that the change in lift characteristics of cam­ bered sections may be reduced by incorporation of reflex or of an upwardly deflected flap. b. Drag Characteristics. The effect of thickness ratio on the drag characteristics of symmetrical wing sections is illustrated" by Figs. 173 to 175. These data show the typical rapid increase of drag coefficient at high Mach numbers. This rapid rise (force hreak) occurs at smaller Mach numbers as the angle of attack and thickness are increased. Similar data THEORY OF n'ING SECTIONS 282 6,0· -p~ M,O.44 -I -0 \,)q" ...: .~ I -S:! :t: ~ -I ~ ~ ~ et -I 0 I "M,073 , 0 , , 50 100 -----lJpper surface o 50 ,00 0 50 /00 Percent chord - - Lower surface FIG. 172. Variation of aileron action with Mach number. tribution results. Symmetrical airfoil, 19 per cent thick, a high-speed wind twme1 .. Two-dimensional pressure dis­ 0 degrees; NACA rectangular == EFJt'ECTB Oil COMPRESSIBILITY A7' SUBSONIC} SPEEJJS .08 J .07 -,; .06 , II J ao !fdeg.l -~ ~ 5 -~ 4 .02 .....;~ ~ 3 2 .0/ 00 If ./ .2 .3 .5 .6 " r./ ry -If. .L .I r- ~ .4 ./ l...c( ....J~ fq' .....I b 91 1/ ~ II 1I I(J [7 II NW ~ ~~~ - .7 .8 1.0 .9 Mach number~M FIG. 173. Effect of compressibility on the drag of the NACA 0006-34 airfoil. ./0 .09 IJ V .08 I 11 J .07 V, } :1 / ~, J J fd':gJ l--' 6 4 00 ./ .2 A ~ l~ 0 l"I ...­ - I • .J ...-J~ J-..... .3 2 .01 t> ~ . . . . lJ.-1V 5 .02 ~ J i1~ I j J~ I -] IT; j ) J iHl lP' 771 j J7)U ~ IJl l...4 .)fJ/ l..J 'r)1f/ ~~ I .3 .4 .5 .6 .7 .8 .9 1.0 Moch~M FIG. 174. Etfe~t of Mmpre..."Sibility on the drag of the NACA 0008-34 airfoil. 283 THEORY OF WIJ.VG SECTIONS 284 ,. .14 J I ./3 0 I I ·fT ./2 ./ / 'Ill I r ./0 ~ ~r-- PI-­ Ii .09 II I J III 1 - t J }I ~! ao J I J 11i 1/ (deg.l .04 6 .03 ""­ 5 II...-. ~~ ~ ~ ~ "'"""'"I ao (deg) ..",- I ~~ .3 2 ~ 0 .0/ / I V~ V ~'~ I( ---.. ...... 4 .02 11f If II ~? I 00 FIG. ./ .2 .3 .4 .5 .6 .7 .8 .9 to Mach ntmber, M 175. Effect of compressibility on the drag of the NACA 0012-34 airfoil. ./2 I J .101 I I I 1 1 I I I I I I I I ·1 I I I I .091 I I I I I I I I ~ .081 I I I I I I I I I I --I ----. ':b. I. .I I. I. I I II ,I I - 111 J ~ f o., .10 I 1 I .08 Ir •f IJF ~ .06 .~ J <.:) :t: § .05 cSB' .04.---r--t--.L..J-.1 ,rhHH-+-+4~A..Jll~mA .,------T--I---1--- I .03 r--T~-.L. I C5~ .05 ~ .04 , .J .2 .J .4 .5 .6 .., .8 .9 ~." a" I V (deg) .02 J -J-2--2-- --- -...... "";,0- -- .r 1-- - . / t' / ~ ~ FIG. 11 j J J jfiJ ...... t: ~J -s ~ 1J ~ 71'1 -s ~ n rn ~ r ~ ~ Ir Iff C .~ ~' ~ ~~ o.::-----~-~----.I .2 .J .4 .5 .6 o ~ t;x, 1/ I ....... ~ ~ ~ .. '-_II ?;] '1 II IJ [J ? ...... r/ ~ JJ lJ 1r _..... 5 4 to Mach fNJmber,M Flo. 176. Effect of compressibility on the drag of the NACA 2306 airfoil. I .03 .01 00 ~ a ~ ~ .()6 ~ , .09 i· O? .~ ~ ~ ~ 'Il 1 \)~ If ttJ , .11 Mach number, M ~ .r .8 .9 1.0 177, F.Jrect of compressibility on the drag of the NACA 2309 airfoil. ~ ~ ~ THEORY OF Ti'ING SECTIONS 286 for the cambered sections, NACA 2306, 2309, 2312, and 2315, are shown lO in Figs. 176 to 179. The effect of camber is to reduce the Mach number at which the drag force break occurs as compared with the data of Figs. 173 to 175. Although the thickness distribution is not the same for these 1 .121-~~---t--t---+---+-+---f----t---+---+---+---t---+--+--+-~-I--I .111--~1--t---t--t---+-+--+--f---+---+---+--+---t--+--+--1-""" , r J ~ J .......... .07 J---.ot---lf--+--+-+--+---+---+--+--+--+--I--+---+--+-+-Io+-fM.~1---I I .~ .~ J ~ • 061---'~~--+--t--+-+--+--+--+----t----t---+--t-~"""'t-'l-fW--+----t--t J a 1 I I J IIItJ ) I J 'JJ V it r~(1 J' ~ .05 ~t---I--t---t---t---t---+--f---+---t--+---+---t---+-tl""-+-lll~""'Il--+--+--t cs «0 If'degl 4 3 2 I -I I ./ .2 I .3 .4 .5 S .7 .~ .~ Moch f'lU77ber,M J4·IG. 178. Effect of compressibility on the drag of the NACA 2312 airfoil. symmetrical and cambered sections, the differences attributable to this fact are very small, at least for the lower thickness ratios. Comparisons of the predicted critical Mach numbers and the Mach numbers for drag force break are presentedf" in Fig. 169 for the NACA 66-210 wing section. For the range of lift coefficients corresponding to high predicted critical speeds, the Mach number for drag force break is about 0.01 to 0.03 greater than the critical. The Mach number for drag force break is intennediate between that for lift force break and the critical EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS 287 value. The slope of the boundary for drag force break is similar to that for the critical Mach number, but high Mach numbers without force break r I, I I~ "l v I f ~ If I I j I I :/ I/~· 1 ) ~ .04 ---....--+-.....--4-(deg} I---I---"--....-..+- 5 j It --+--+--I--~-+---+-;J-~~""""'--"""--'---+---t 4 3 2 i .4 .5 .6 .7 .8 ,9 1.0 Mach nunber, M 179. Effect of compresaibility on the drag of the NACA 2315 airfoil. .010 FlO. / -I .I .2 .3 may be realized over a much wider range of lift coefficient than is indicated by the theoretically predicted critical speed. c. M omen: Characteristics. Figures 157 and 158 illustrate116 the nega­ tive shift of the moment coefficient with increasing Mach number that is THEORY OF WING SECTIONS 288 typical for cambered wing sections. In addition to this shift, there is a tendency at the highest Mach numbers shown for the moment coefficient to increase negatively with increasing lift coefficient as is shown37b more .3 I I I .2 1 9 ,I II 'I ./ P '­ ,.I IiIo.... ~ ~ ~ r-s, ~ ~ r-J ~ ~~ ~ ~ )-a ~~ "I"·T~ I~ ~ ~ \ Symbol 0 + X 0 e -.4 ~\ \ \' \~ _6° -4° -2° -0° \ , v I, t JT II 11 I II ~ I 1 6° 8° E> <1 -~ {I ~\\ 4° v -:5 a \ 2° A I' 6 ~ ~II rI -' V _Je/ r;t) ~~ I I fJO I I t e -;:2 .3 .4 .5 .6 .7 B .9 Mach number, M FIG. ISO. The variation of section moment coefficient with Mach number at various angles of attack for the NACA 66-210 airfoil. clearly for the N ACA 66-210 section in Fig. 180. This rearward shift of the aerodynamic center at high Mach numbers is stabilizing, and the more negative values of the moment coefficient tend to make the airplane trim at lower angles of attack. These moment changes thus add to the pre­ viously discussed effects of the lift coefficient in producing the "tucking under" tendency of airplanes with thick cambered wing sections. EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS 289 The moment characteristics of the NACA 0006-34, 0008-34, 0012-34, 2306, 2309, 2312, and 2315 wing sections are shown" in Figs. 181 to 187. These data indicate that the changes in moment characteristics with Mach / o ..lli ~ _. Cz =QI -./ ./ -­ o Ct 1"'"'04kt... -.....I:t ~- =Q2 ... 'Vll ~~ we Ct =0.3 o , ./ Ct FIG. 181. 1 I -./ ./ 2 3 .4 i '~~ I { =0.5 .5 ~~ hJ I o o "' Ct =Q4 -./ I .6 7 i 8 .9 to Mach number, M Effect of compressibility on the pitching moment of the N ACA 0006-34 airfoil. number are minimized for symmetrical sections, although a rearward shift of the aerodynamic center at high Mach numbers persists, 9.7. Wings for High-speed Applications. As discussed in the previous section, all wing sections experience undesirable variations in their charac­ teristics at high subsonic Mach numbers. Although these undesirable variations can be minimized by the use of thin symmetrical sections, it does not appear to be possible to eliminate them in two-dimensional flow. In three-dimensional flow, however, it is obvious that the shocks cannot extend tothe wing tips without modification. The presence of the shock at THEORY OF WING 8ECTIONS 290 ./ o Ct =QI -./ .1 I I o - T ct. =0.2 t-· I J ~ .1 TTl T .~ ~ § i ~ 1I 1I 0 .,..~ C =Q3 t -- ... 1 .1 o ~ Ct =0.4 -~ -.1 ./ 1 I I 11 I o -.1 FIG. r I ct. =0.5 1 -",~ ~~ .4 .5 .1 .4 .6 l '.0 9 Mach numbe'l M 182. Effect of oompre88ibility on the pitching moment of the NACA 0008-34 airfoil. 0 1 .2 .3 EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS 291 ./ "­ o ~ Ct =0./ -./ ., -..W-O ""'" i'D«l c" =0.2 ~ ~ -./ ./ I - o j T ct. =Q4 -"0 FIG. ~ ct. =Q3 ./ .2 .s .4 .5 .6 ra... I .7 .8 --. ~ .9 1.0 Mach rvmber,M 183. Etlect of compreeaibility on the pitching moment of the NACA 0012-34 airfoil THEORY OF WING SECTIONS 292 o -.1 o -.1 ~ T r-­ I I I I Ct ~~ T I r-.r ~ I I I I J .§ ~ ~ -.~ ~ .A. 1 ~ I ~'""- I I I Ct =0 I 1 I I""--. I...A.. I fI-·1 - I~ ~ =-0./ Ct I o )~ >--. ~ =-0.2 o -.1 ~ I I I .,., ct. =Q/ ~ V ~~ -.-' 0 a j I r r Ct =0.2 0 1 I .L o -.1 1 I ~ Ct I I Ct '-"'" '-c ~ L­ ~ ~ l-ft.. =Q3 .1 rv"" I 1"'-.-.. ~~l4. =0.4 • i. T I ~ Ct I _a. 1~ ~ I I I =Q5 o FIG. ~ I I I I -:.1 .... I I I o -:/ ~~ I I I -:1 -.1 ""'-t ~~ 'T I I I r 1\.e ............. '"' ~ ~~ Ct=Q6 o .1 .2 .3 .4 .5 .6 .7 .8 .9 I,{) Mach ntmberl M 184. Effect of compressibility on the pitching ~oment of the NACA 2306 airfoil. EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS o )~ I I I ....... ITt -./ Ct o =-Q2 I I I I T -./ I .-.'1' """'117' .)~ .- ~ ",,,,,,,,,. ~ CZ=-O./ o I I I ~t lo' t I I I T I Cl =0./ j~ """"'Il~ ~~ LA I I I r I -1 -./ .. ~ ~ ~ ~ ~ ~ ~ c t =0.3 o I I I p--r 1 -./ FIG. ." ct =0.2 o W~ -­ r.ea., ~!r .. I I I 1 1 f -./ o 293 Ct ./ l =Q4 .,... " ~ .4 .5 .6 .7 .8 .9 /.0 Mach numbet; M 185. Effect of conlpressibility 011 the pitching moment of the NACA 2309 airfoil. .2 .J THEORY OF WING SECTIONS 294 o I I I I 1 I -./ )"'" ~ ---.~ Ct=O o I I 1 I .­ ./ N ~ ~ Ct.= 7.1 I I I ~ ~~ I I I c" =Q2 1 I 1 -./ I 1 I .­ ""'lJIIIl I I I -./ FIG. ~ ~ ~ ct =0.3 o o -A. I I I ./ .2 .3 c,,=Q4 .6 .5 Ma:h number. M ., .7 .8 - rw ~~ . ( o L 186. Effect of compressibility on the pitching moment of the NACA 2312 airfoil. EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS· o 295 ". ~ ~ ll( ~ ~ t ;,0 -J C - o -./ Ct ~r ~ -J't'" ~~ ~ "" -­ ct.~a2 o - :.Mil 'L ~ -./ ct - Q3 o ...-III ~ .1 .~ .. .4 .5 ~ ~ Ct=a4. -:/ 0 ".". .6 .T ,8 .9 I. Mach number, M FlO. 187. Effect of compressibility on the pitching moment of the NACA 2315 airfoil. 296 THEORY OF WING SECTIONS the lying tip with the accompanying sudden pressure rise would induce a three-dimensional flow that would tend to reduce the severity of the shock. The existence of such H tip relief" became apparent in early tests of propellers at high tip speeds. The data showed that losses of efficiency did not OCcur until tip Mach numbers were reached that were well in excess of those at which the sections used at the tips suffered large increases of drag in two­ dimensional flow, Corresponding advantages for "rings may be expected if the aspect ratio is sufficiently low, Experimental data have confirmed the advantages of low aspect ratio at high Mach numbers. The theoretical approach for wings of very low aspect ratio is entirely different from that for normal wings of high aspect ratio, and the concept of wing-section characteristics is not applicable. Jones" showed that, for wings with aspect ratios approaching zero, the lift depends on the angle of attack and on the positive rate of increase of span in the direction of the air flow, or db C, = ret dx where C, = local lift coefficient of section perpendicular to direction of air flow a = angle of attack b = span x = distance in the direction of air flow db/dx is positive This simplified theory indicates that C, = 0 for those portions of the wing where db/d,x is zero or negative. This theory shows that the center of lift for a triangular wing with the apex forward is at the center of area. The theory indicates that, for the particular ease of a slender triangular plan form, the lift coefficient and center of pressure are independent of Mach number at both subsonic and supersonic speeds so long as the wing lies well within the Mach cone. The applicability of the theory at a Mach number of 1.75 is shown by Fig. 188. Low-aspect-ratio wings of triangular plan form thus appear to provide one solution to the stability difficulties in the transonic speed range. Another solution to the problems of the transonic speed range is the use of large amounts of sweep. The manner in which sweep is effective is clearly seen from consideration of a wing of infinite span with sideslip (Fig. 189). It is obvious that the spanwise component of velocity V. will produce no forces on the wing, neglecting viscosity, and that the force on the wing will be determined by the normal component of velocity V". The- resulting velocity l' may be transonic or even supersonic, while the normal component V ft is less than the critical speed if the angle of sideslip, or sweep, fJ is sufficiently large. This effect of sweep in avoiding compres­ EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS 297 I .os" "" li <, ' .. ~ ~/ / /v ~ / .02 ) Cm o ) / ~ TheoreIicol (slope ~a02J V - - ./Theorelical ,V .~ -.02 V / .l' V V /V -.04 -.J , /V -2 -I 0 I 2 3 «,dig 188. Test of triangular airfoil in Langley model supersonic tunnel 1\lach number, 1.75; Reynolds number, 1,600,000. FIG. veIOCi/,ry component normal ( 10 wingtVcosjJ) --,----~ 117 streon veJocily..... V Angleof sweea-« fi Spamvise velocity component (VsinjJ) FIG. 189. Velocity components on a swept wing. THEORY OF WING SECTIONS 298 10------------ 80mm - - - - - - - - - - -.. -k:: ~--------------- Profile R-4009 I I 1 / 1/' / ~ol rnetJSUI'1!fTI( 400 <,,/1 (.,'-1' -..tso~ FIG. 190. Wing placed obliquely across a two-dimensional wind tunnel. -qs x D,S ~O o 0 u P 1 - e - /J-'Oj• ,...... cos' 0 -O""'i ~c'tiS'ZlF H=(J FIo. 191. Comparison of pressure distribution over normal and oblique wing. EFFECTS OF COMPRESSIBILITY AT SUBSONIC SPEEDS 299 sibility troubles was independently discovered by German investigators and by JonesU during the Second World War. Application of this principle results in wings with large amounts of sweepback or sweepforward. Such plan forms obviously violate the simple two-dimensional considerations at the root and tip even if the wing has no taper. These violations, however, do not appear to be sufficient to destroy the value of sweep if the wing is of reasonably high aspect ratio, as in­ dicated by tests'" of a wing placed obliquely across a two-dimensional wing tunnel (Fig. 190). Figure 191 shows that the pressure distribution over the center section of the oblique wing is reasonably close to that expected from the simple theory, despite the fact that the tunnel w8JIs violate the assumed end conditions. Experimental studies have shown that swept wings of all useful aspect ratios are effective in postponing and alleviating compressibility troubles. Consequently, the theory of swept wings of arbitrary plan forms is being intensively investigated. 1I • u , lG. U7 It thus appears that the wings of the high-speed airplanes of the future may be quite different from those of comparatively slow subsonic air­ planes. These highly swept low-aspect-ratio "rings, however, have serious disadvantages for the low-speed flight necessary for landing and take-off. Some of these disadvantages are lo\v lift-curve slopes, high angles of stall, and poor stability characteristics. These disadvantages seriously com­ promise the design of airplanes using such plan forms and may force the use of more conventional plan forms with very thin supersonic sections for some applications. In anycase, these disadvantages at slow speeds are expected to dictate the use of conventional plan forms for airplanes de­ signed for such speeds that it is possible toavoid serious compressibility difficulties by the use of thin wing sections. REFERENCES 1. ABBOTr, FRANK T., JR., and TURNER, HAROLD R., JR.: The Effects of Roughness at High Reynolds Numbers on the Lift and Drag Characteristics of Three Thick Airfoils. NACA ACR No. IAH21, 1944 (Wartime Rept. No. L-46). 2. ABBOTT, IRA H., and GREENBERG, llARBy: Tests in the Variable-Density Wind Tunnel of the NACA 23012 Airfoil with Plain and Split Flaps. NACA Rept. No. 661, 1939. 3. ABBOTr, IRA H., VON DOENHOFF, ALBERT E., and STIVERS, LoUIS S.: Summary of Airfoil Data.. NACA Rept. No. 824, 1945. 4. ACKERET, J., FELDMAN, F., and RoTr, N.: Investigations of Compression Shocks and Boundary Layers in Gases Moving at High Speed. NACA TM No. 1113, 1947. 5. AutEN, WILLIAM, JR.: Standard Nomenclature for Airspeeds with Tables and Charts for Use in Calculation of Airspeed. NACA TN No. 1120, 1946. 6. ALLEN, H. JULIAN: Calculation of the Chordwise Load Distribution over Airfoil Sections with Plain, Split or Serially-hinged Trailing Edge Flaps. NACA Rept. No. 634, 1938. 7. ALLEN, H. JULIAN: A Simplified Method for the Calculation of Airfoil Pressure Distribution. NACA TN No. 708, 1939. 8..-\LLEN, H. JULIAN: General Theory of Airfoil Sections Having Arbitrary Shape or Pressure Distribution. NACA ACR No. 3029, 1943. 9..~LLEN, H. JULIAN: Notes on the Efieet of Surface Distortion on the Drag and Critical Mach Number of Airfoils. NACA ACR No. 3129, 1943. 10. ANDERSON, RAYMOND F.: Determination of the Characteristics of Tapered Wings. NACA Rept. No. 572, 1936. 12. BAIBSTOW, L.: Skin Friction. J. Roy. Aeronaut. Soe., January, 1925, pp. 3-23. 13. BAMBER. ~fILLABD J.: Wmd-tunnel Tests on Airfoil Boundary Layer Control Using a Backward-opening Slot. NACA Rept. No. 385, 1931. 14. BLASIUS, H.: Grenzschichten in FlOssigkeiten mit kleiner Reibung. Z. J.fath. PhyBik, vol 66, pp. 1-37, 1908. 15. BL.~SIUS, H.: Ver. deut. lng. ForachungSMft 131, 1911. 16. BOSR.Ul, JOHN: The Determination of Span Load Distribution at High Speeds by the Use of High-speed Wind Tunnel Section Data. NACA. ACR No. 4B22, 1944 (Wartime Rept. No. L-436). 17. BRASLOW, .ALBERT L.: Investigation of Effects of Various Camouflage Paints and Painting Procedures on the Drag Characteristics of an NAC.A 65(.al)-420 a = 1.0 Airfoil Section. NACA CB No. UG17, 1944 (Wartime Rept. No. L-141). 18. BROWN, Cu~nroN E.: Theoretical Lift and Drag of Thin Triangular Wings at Supersonic Speeds. N.ACA TN No. 1183, 1946. 19. BURGERS~ J. 1.1.: The Motion of a Fluid in the Boundary Layer along a Plane Smooth Surface. Proc. First Intern. Congr. AppL. Mech., Delft, 1924, pp. 113-128. 20. CAHILL, JONES F.: Two-dimensional Wind-tunnel Investigation of Four Types of High-lift Flap on an NACA 65-210 Airfoil Section. NACA TN No. 1191, 1947. 21. CAHILL, JONES F., and RACIsz, STANLEY: Wmd-tunnel Development of Optimum Double-elotted-flap Configurations for Seven Thin NAC.o\. Airfoil Sections. NACA RM No. L7B17, 1947, also TN No. 1545. 300 REFERENCES 22. 301 F.: Aerodynamic Data for a Wing Section of the Republic XF-12 Airplane Equipped with a Double Slotted Flap. NACA MR No. L6AOSa, 1946 (Wartime Rept. No. L-544). 23. CAHILL, JONES F.: Summary of Section Data on Trailing-edge High-lift Devices. NACA RM No. LSD09, 1948. 25. CHAPLYGIN, SERGEI: Gas Jets. NACA TM No. 1063, 1944 (from Scientific Mem­ oirs, Moscow University, 1902, pp. 1-121). 26. CIlABTEBS, ALEX C.: Transition between Laminar and Turbulent Flow by Trans­ verse Contamination. NACA TN No. 891, 1943. 27. DRYDEN, H. L., and KUETHE, A. l\{.: Effect of Turbulence in Wind Tunnell\feasure­ menta. NACA Rept. No. 342, 1929. 28. DURAND, W. F.: Aerodynamic Theory, vol. 3, Div. G. "The Mechanics of Viscous Fluids" by L. Prandtl. Verlag Julius Springer, Berlin, or Durand Reprinting Comm., Cal. Inst. of Tech. 29. DURAND, W.F.: Aerodynamic Theory, vol. 3, Div. H. "The Mechanics of Com­ pressible Fluids" by G. I. Taylor and J. W. Maceoll, Verlag Julius Springer, Berlin, or Durand Reprinting Comm., Cal. lost. of Tech. 30. FERRI, ANTONIO: Completed Tabulation in the United States of Tests of 24 Airfoils at High Mach Numbers. (Derived from interrupted work at Guidonia, Italy, in the 1.31- by 1.74-foot High-speed Tunnel) N.ACA ACR No. LSE21, 1945 (War­ time Rept. No. L-143). 31. FISCHEL, JACK, and RIEBE, JOHS 1\1.: "~'ind-tunnel Investigation of a NACA 23021 Airfoil with a O.32-airfoil-chord Double Slotted Flap. N ACA ARR No. IAJ05, 1944 ("·artime Rept. No. L-7). 32. FULLMER, FELlCIEN F., Ja.: Two-dimensional "-:-ind-tunnel Investigation of the NACA 641-012 Airfoil Equipped with Two Types of Leading-edge Flap. NACA TN No. 1277, 1947. 33. FURLONG, G. CHESTER, and FrrzPATRlCK, J.~:MES E.: Effects of Mach Number and Reynolds Number on the Maximum Lift Coefficient of a Wing of NACA. 230­ series Airfoil Sections.. NACA l\IR No. L6F04, 1946, also TN No. 1299. 34. GARRICK, I. E., and KAPLAN, CARL: On the Flow of a Compressible Fluid by the Hodograph ?\Iethod. I. Unification and Extension of Present-day Results. N..4.CA ACR No. 1AC24. 1944, also Rept. No. 789. 35. GLAUERT, H.: Theoretical Relationships for an Aerofoil with Hinged Flap. R. &. AI. No. 1095, British ARC, 1927. 36. GLAUERT,H.: The Effect of Compressibility on the Lift of an Aerofoil. R. & M. No. 1135, British ARC, 1927. 37. GLAUERT, H.: "'The Elements of Aerofoil and Airscrew Theory," Cambridge University Press, London, 1926. 374. GOLDSTEIN, SIDNEY: Low-drag and Suction Airfoils, Eleventh \Vright Brothers Lecture, J. Imt. Aeronaut.. Sei., '·01. 15, 1\0. 4, pp. 189-214, April, 1948. 37b. GRAHAM, DONALD J.: High-speed Tests of an Airfoil Section Cambered to Have Criticall\fach Xumbers Higher than Those A.tt:lmnble with a Uniform-load Mean Line. NAC.~ TX No. 1396, 1947. 38. GJlUSCHWrrz. E.: Die turbulente Reibungsschicht in ebener Stromung bei Druckab­ fall und Druckanstieg, Ing. Arehi», Bd. 11. Heft 3, PP. 321-346, September, 1931. 39. H ..\ BRIS, THOMAS A.: .'\\"ind-tunnel Investigation of an K.t\CA Airfoil with Two Arrangements of a Wid~hord SIQtted Flap. N.ACA TN No. 715, 1939. 40. HARRIS, TIIOM.~S .~., and RECAST, ISIDORE G.: Wind-tunnel Investigation of NACA. 23012, 23021, and 23030 Airfoils Equipped with 4Q-percenkhord Double Slotted Flaps. !\.-\CA Rept. No. 723. 1941. CAHILL, JONES 302 THEORY OF WING SECTIONS 41. HARRIS, "rHOKAS A., and LoWRY, JOHN G.: Pressure Distribution over an NACA 23012 Airfoil with a Fixed Slot and a Slotted Flap. NACA Rept. No. 732, 1942. 42. HOOD, MANLEY J.: The Effects of Some Common Surface Irregularities on Wing Drag. NACA TN No. 695, 1939. 43. HOOD, MANLEY J., and GAYDOS, M. EDWARD: Effects of Propellers and of Vibra­ tion on the Extent of Laminar Flow on the NACA 27-212 Airfoil. NACA ACR, October 1939, (V-lartime Rept. No. L-784). 44. JACOBS, EASTMAN N.: Methods Employed in America for the Experimental Investi­ gation of Aerodynamic Phenomena at High Speeds. NACA Misc. Paper No. 42, 1936. Paper presented at Volta meeting in Italy, Sept. 30 to Oct. 6, 1935. 45. JACOBS, EASTMAN N.: Preliminary Report on Laminar Flow Airfoils and New Methods Adopted for Airfoil and Boundary-layer Investigations. NACA ACR, June, 1939, (Wartime Rept. No. L-345). 400. JACOBS, EASTMAN N., and ABBOTT, IRA H.: Airfoil Section Data Obtained in the N.A.CA Variable-density Tunnel as Affected by Support Interference and Other Corrections. NAC.~ Rept. No. 669, 1939. 46. JACOBS, EASTMAN N., and PINKERTON, RoBERT M.: Tests in the Variable-density Wind Tunnel of Related Airfoils Having the Maximum Camber Unusually Far Forward. NACA Rept. No. 537, 1935. 47. JACOBS, EAsTMAN N., and PINKERTON, RoBERT M.: PressureDistribution over a Symmetrical Airfoil Section with Trailing Edge Flap. NACA Rept. No. 360, 1930. 48. JACOBS, EASTMAN Nc, PINKERTON, ROBERT M., and GREENBERG, HARRY: Tests of Related Forward-camber Airfoils in the Variable-density \Vind Tunnel. NACA Rept. No. 610, 1937. 49. JACOBS, EAsTMAN N., WARD, ]{ENNETH E., and !)INKERTON, RoBERT ~f.: The Characteristics of 78 Related Airfoil Sections from Tests in the Variable-density Wind Tunnel. NACA Rept. No. 460,1932. 50. JONES, RoBERT T.: Correction of the Lifting-line 1"heory for the Effect of the Chord. NACA TN No. 617, 1941. 51. JONES, RoBERT T.: Properties of Low-aspect Ratio Pointed \Vinga at Speeds below and above the Speed of Sound. NACA TN No. 1032, 1946. 52. JONES, RoBERT T.: Wing Plan Forms for High-speed Flight. NACA TN No. 1033, 1946. 53. JONES, RoBERT T.: Thin Oblique Airfoils at Supersonic Speed. NACA TN No. 1107, 1946. 54. KAPLAN, CARL: Effect of Compressibility at High Subsonic Velocities on the Lifting Force Acting on an Elliptic Cylinder. NACA TN No. 1118, 1946. 55. KEENAN, JOSEPH H., and NEUMANN, ERNEST P.: Friction in Pipes at Supersonic and Subsonic Velocities. NACA TN No. 963, 1945. 56. KENNARD, EARLE H.: "Kinetic Theory of Gases," l\lcGraw-Hill Book Company, Inc., New York, 1938. 57. KNIGHT, MONTGOMERY, and BAMBER, MILLARD J.: Wind Tunnel Tests on Airfoil Boundary Layer Control Using a Backward Opening Slot. NACA TN No. 323­ 1929. 58. KOSTER, H.: Messungen an Profile OJ)() 12-0, 5545 mit Spreitz- und Nasanspreiz­ klappe, Deutsche VerSuchsanstalt fm Luftfahrt, UM 1317, July 29, 1944. 59. KaeGER, W.: Systematische Windkanalmessungen an einem LaminarftUgeI mit Nasanklappe, Aerodynamische Versucbsanstalt GOttingen, Fb Nr. 1948, June 13, 1944. 60. KROGER, W.: Wind-tunnel Investigation on a Changed Mustang Profile with REFERENCES Nose Flap. 1177,1947. 61. 62. Force and Pressure Distribution Measurements. 303 NACA. Tl\1 No, "Hydrodynamica," Cambridge University Press, London, 1932. LESTER: The Stability of a Laminar Boundary Layer in a Compressible Fluid. NACA TN No. 1115, 1946. 63. LEMME, H. G.: Kraftmessungen und Drukverteilungmessungen an einem Fliigel mit Knicknase, VorflOgel, Wolbungs- und Spreizklappe, Aerodynamische Ver­ suchsanstalt GOttingen, Fb. Nr. 1676, Oct. 15, 1942. 64. LEMME, H. G.: Kraftmessungen und Druckverteilungmessungen an einem Rech­ teckfliigel mit Spaltknicknase, Wolbungs- und Spreizklappe oder Rollklappe, Aerodynamische Versuchsanstalt Gottingen, Fb. Nr. 1676/2, l\fay 14, 1943 (also NACA Tl\{ No. 1108, 1947). 65. LEMME, H. G.: Kraftmessungen und Druckverteilungmessungen an einem Rech­ teckftiigel mit Doppelknicknase, Aerodynamische Versuchsanstalt Gottingen, Fb. Nr. 1676/3, June 2, 1944. 66. LIEPMAN, HANS WOLFG~"G: The Interaction between Boundary Layer and Shock Waves in Transonic Flow, J. Aeronaut. Sci., vol. 13, No. 12, pp.623-637, Decem­ ber,I946. 6&. LIN, C. C.: On the Stability of Two-dimensional Parallel Flows, Part I, QU4rt. Appl. jfath., vol 3, No.2, pp. 117-142, July, 1945; Part II, voL 3, No.3, pp, 218-234, October 1945; and Part III, vol. 3, No.4, PP. 277-301, Januar.)", 1946. 67. LIPPISCH, A., and BEUSCHAUSEN, 'V.: Pressure Distribution Measurements at High Speed and Oblique Incidence of Flo,,·. NACA Tl\1 No. 1115, 1947. 68. LoFTIN, LAURENCE K., JR.: Effects of Specific Types of Surface Roughness on Boundary-layer Transition. NACA. ~\CR L5J~ 1945 (Wartime Rept. No. L-48). 6&. LoFTIN, LAURENCE K., JR., and COllEN, I{ENNETR B.: Aerodynamic Characteristics of a Number of Modified NACA Four-digit-series Airfoil Sections. NACA RM No. L7122, 1947. 69. LoWRY, JOHN G.: "rind-tunnel Investigation of an NACA 23012 Airfoil with Several Arrangements of Slotted Flap with Extended Lips. NACA TN No. 808, 1941. 70. MUNK, l\IAX !\'1.: The Determination of the Angles of Attack of Zero Lift and Zero Moment, Based on Munk's Integrals. NACA TN No. 122, 1923. 71. l\IUNK, !\{AX M.: Elements of the \Ving Section Theory and of the Wing Theory. NACA Rept. No. 191, 1924. 72. :l\{UNJt, Max 1\1.: Calculation of Span Lift Distribution (Part 2), Aero Digest, vol. 48, No.3, p. 84, Feb. 1, 1945. 73. NAlMA.N, IRVEN: Numerical Evaluation of the e-Integral Occurring in the Theo­ dorsen Arbitrary Airfoil Potential Theory. NACA ARR No. lAD27n., April, 1944 (Wartime Rept. No. L-136). 73a. NAD1AN, IRVEN: Numerical Evaluation by Harmonic Analysis of the e-Funetion of the Theodorsen Arbitrary Potential Theory. NACA ARR No. LSRI8, Septem­ ber, 1945 (Wartime Rept. No. L-153). 14. NEELY, RoBERT H., BOLLECH, THOMAS V., WESTRICK, GERTRUDE C., and GRAHAM, RoBERT R.: Experimental and Calculated Characteristics of Several NACA 44­ series Wings with Aspect Ratios of 8, 10, and 12, and Taper Ratios of 2.5 and 3.5. NACA TN No. 1270, 1947. 75. NIItURADSE, J.: Gesetzmissigkeiten der turbulenten Stromung ill glatten Rohren, ForBCh. Gebiete Ingenieunc., 1932, FOfSChungwjt 356. 700. NIItURADSE, J.: VOrtrige aus dem Gebiete der Aerodynamik, p, 63, Aachen, 1929. 76. NITZBERG, GERALD E.: A Concise Theoretical Method for Profile-drag Calculation. NACA ACR No. 4005,1944. LAMB, HORACE: LEES, 304 77. THEORY OF WING SECTIONS W.: Wind-tunnel Tests of a Wing with a Trailing-edge Auxiliary Airfoil Used as a Flap. NACA TN No. 524, 1935. 78. PANKHURST, R. C.: A Method for the Rapid Evaluation of Glauert's Expressions for the Angle of Zero Lift and the Moment at Zero Lift. R. &; 1\1. No. 1914, British ARC, 1944. 79. PEARSON, E. 0., JR., EVANS, A. J., and 'VEST, F. E.: Effects of Compressibility on the Maximum Lift Characteristics, and Spanwise Load Distribution of a 12-foot­ span Fighter-type Wing of NACA 23O-series Airfoil Sections. NACA ACR No. L5GI0, 1945 (\Y'artime Rept, No. L-51). 80. PEIRCE, B. 0.: U A Short Table of Integrals," Ginn &: Company, Boston, 1929. 80a. PIERCE, II.: On the Dynamic Response of Airplane Wings Due to Gusts. NACA TN No. 1320, 1947. 81. PINKERTON, RoBERT ~'I.: Analytical Determination of the Load on a Trailing Edge Flap. NAC.A TN No. 353, 1930. 82. PINKERTON, ROBERT 1\1.: Calculated and Measured Pressure Distributions over the Midspan Section of the NA.C..A\ 4412 Airfoil. NACA Rept. No. 563, 1936. 83. PLATT, ROBERT C.: Aerodynamic Characteristics of Wings with Cambered Ex­ ternal-airfoil Flaps, I neluding Lateral Control with a Full-span Flap. N ACA ltept. No. 541, 1935. 84:. PLATr, ROBERT C., and ASBOTr, IRA H.: Aerodynamic Characteristics of NACA 23012 and 23021 Airfoils with 2O-percent-chord Extemal-airfoil Flaps of NACA 23012 Section. NACA Rept, No. 573, 1936. 85. PLATT, H.OBERT e., and SHORTAL, JOSEPH A.: Wind-tunnel Investigation of "rings with Ordinary Ailerons and Full-span External Airfoil Flaps. NACA Rept, No. 603, 1937. 86. POHLIIAUSEN, 1\:.: Zur niiherungsweisen Integration der Differentialglcichung der laminaren 'Grenzschicht, Abhandl. aero. Inst. Aachen, 1. Lieferung, 1921. See also z. angelO. ,,"tath. !alech., Btl. 1, Heft 4, pp. 252-258, 1921. 87. PRA.~DTL, 1.1.: Motion of Fluids with Very Little Viscosity. NACA TN No. 452, 1928 (originally presented to the Third International Mathematical Congress, Heidelberg, 19(4). 88. PRANDTL, L.: Applications of Modern Hydrodynamics to Aeronautics. NACA Rept. No. 116, 1921. 89. PRANDTL, L. t and TIETJENS, O. G.: "Applied Hydro-and Aeromechanics,"l\{cGraw­ Hill Book Company, Inc., New York, 1934. 90. PUCKETT, Allen E.: Supersonic Wave Drag of Thin Airfoils, J. Aeronaut. Sci., vol. 13, No.9, pp. 475-484, September, 1946. 91. PURSER, PAUL E., FISCHJ<;L, JACK, and RIEBE, JOHN l.f.: Wind-tunnel Investigation of an N.ACA 23012 Airfoil with a O.30-airfoil-chord Double Slotted Flap, NACA ARR No. 3LI0, 1943 (Wartime Rept. No. L-469). 92. PURSER, PAUL E., and JOHNSON, IIAROLD S.: Effects of Trailing-edge Modifications on Pitching-moment Characteristics of Airfoils. NACA cn No. UI30, 1944 (Wartime Rept. No. L-664). 93. QUINN, JOHN II., JR.: Tests of the NACA. 65,-018 Airfoil Section with Boundary­ layer Control by Suction. NACA CB No. IAHIO, 1944 (Wartime Rept. No. L-2(9). 94. Qt:IN~, JOHN H., JR.: Wind-tunnel Investigation of Boundary-layer Control by Suction on the ~.A.CA 65,-418, a = 1.0 Airfoil Section with a 0.29 Airfoil-chord Double Slotted Flap. NACA TN No. 1071, 1946. 95. ,QUINN, JOHN H .• JR.: Tests of the NACA 641A212 Airfoil Section with a Slat, a Double Slotted Flap, and Boundary-layer Control by Suction. NACA TN No. 1293, 1947. NOYES, RICHARD REFERENCES 305 96. RECANT, I. G.: Wind-tunnel Investigation of an NACA 23030 Airfoil with Various Arrangements of Slotted Flap. NACA TN No. 755, 1940. 97. REID, E. G., and BAMBER, M. J.: Preliminary Investigation on Boundary Layer Control by Means of Suction and Pressure with the U.S.A. 27 Airfoil. NACA TN No. 286, 1928. 98. ScHILLER, L.: Die Entwicklung der laminaren Geschwindigkeitsverteilung und ihr Bedeutung fUr Zihigkeitsmessungen, Z. angew. ltfath. ~'/ech., Bd. 2, Heft 2, pp. 96-106, 1922. See also ForBCh. Gebiete lngenieurw., Heft 248, Aachen III, 3, 1922. 99. ScHLICHTING, H.: Amplitudenverteilung und Energiebilanz der kleinen StOrungen bei der Plattenstromung, Nachr. Ga. Wi8s. GiiUingen, Moth, physik. Klaeee, Bd. 1, 1935. 100 ScHLICHTING, H.: Zur Entstehung der Turbulenz bei der Plattenstromung, Nachr. Ge«: Wiaa. Gottingen, Alath. physik. Klasse, 1933, pp. 181-209. 101. ScHLICHTING, H., and ULRICH, A.: Zur Berechnung des Umschlages Laminar Turbulent, Jahrbuch 1942 der deutschen Luftfahrtforschung, pp. 18-135, R. Olden­ bourg, Munich. 102. ScHRENK, OSKAR: Experiments with a \Ving Model from Which the Boundary Layer Is Removed by Suction. NACA T~I No. 534, 1929. 103. ScHRENK, OSKAR: Experiments with a Wing from \Vhich the Boundary Layer Is Removed by Suction. NACA Tl\1 No. 634, 1931. 104. SCHRENK, OSKAR: Experiments with Suction-type Wings, NACA TM No. 773, 1935. 105. ScHUBAUR, G. B.: Air Flow in the Boundary Layer of an Elliptical Cylinder. NACA Rept. No. 652, 1939. 106. ScHUBAUR, G. B., and SKRAMSTAD, If. 1\:.: Laminar-boundury-layer Oscillations and Transition on a Flat Plate. N.AC.A ACIt, April, 1943 (\Vartinle Rept, No.8). 107. ScIIULDENFllEI, l\{ARVIN J.: \Vind-tunnel Investigation of an NACA 23012 Air­ foil with a Handley Page Slot and Two Flap Arrangements. NAC1\ ARR, Febru­ ary, 1942 (Wartime Rept. No. L-261). lOS. SHERMAN, ALBERT, and HARRIS, ''1''. A.: The Effects of Equal Pressure Fixed Slots on the Characteristics of a Clark-Y Airfoil. :KACA rrN No. 507, 1934. 109. SHElWAN, ALBERT: A Simple l\lcthod of Obtaining Span Load Distribution. NACA TN No. 732, 1939. 109a. SILVERSTEIN, ABE, KATZOFF, SAMUEL, and lIooTMAN, JAMKS: Comparative Flight ar.J Full-scale Wind-tunnel Meusureruents of the :\In.xinluDl Lift of an Airplane. NACA Rept. Ko. 618, 1938. 110. SIVELLS, JAMES C.: Experimental Characteristics of Three \Yinb"S of NACA 64-210 and 65-210 Airfoil Sections with and without 2° \Yashout. !\.~Ci\ 'l"X No. 1422, 1947. Ill. SIVELLS, JAMES C., and KEELY, ROBERT II.: Method of Calculating Wing Char­ acteristics by Lifting-line Theory Using Xonlinear Section Lift Data, K.ACA 1'N No. 1269, 1947. 112. SQUIRE, H. B., and YOUNG, A. D.: The Calculation of the Profile Drag of Aero­ foils. R. & l\{. No. 1838, British AIlC, 1938. 113. STACK, JOHN: Tests of Airfoils Designed to Delay the Compressibility Burble. NACA TN No. 976, 1944. 114. STACK, JOHN: Compressible Flow's in Aeronautics, Eighth Wright Brothers Lec­ ture, J. Aeronaut. s«; vol. 12, No.2, April, 1945. 115. STACK, JOHN, LINDSAY, \V. F., and LI'M'ELL, ROBI-.:RT E.: The Compressibility Burble and the Effects of Compressibility on Pressures and Forces Acting on an Airfoil. NACA Rept. No. 646, .1938. 306 THEORY OF WING SECTIONS 116. STACK, JOHN, and VON DOENHOFF, ALBERT E.: Tests of 16 Related Airfoils at High Speeds. NACA Rept. No. 492, 1934. 117. STEWART, H. J.: Lift of a Delta. Wing at Supersonic Speeds, Quart. Appl. Math., vol. IV, No.3, pp. 246-254, October, 1946. 118. TAm, ITIBO: A Simple Method. of Calculating the Induced Velocity of a Mono­ plane Wing, Aeronaut. Research Inst. Tokyo Imp. Uniu. &pt. 111, vol. IX, p. 3, August, 1934­ 119. TEMPLE, G., and YARWOOD, J.: The Approximate Solution of the Hodograph Equations for Compressible Flow. Rept. No. S.M.E. 3201, RAE, June, 1942. 120. TETERVIN, NEAL: A Method. for the Rapid Estimation of Turbulent Boundary­ layer Thick~ess for Calculating Profile Drag. NACA ACR No. UGl4, July, 1944 (Wartime Rept. No. L-16). 121. THEODORSEN, THEODORE: On the Theory of Wing Sections with Particular Refer­ ence to the Lift Distribution. NACA Rept. No. 383, 1931. 122. THEODORSEN, THEODORE: Theory of Wing Sections of Arbitrary Shape. N.ACA Rept. No. 411, 1931. 123. TuEODORSEN, THEODORE: Airfoil Contour Modification Based 011 e-Curve Method of Calculating Pressure Distribution. NACA ARR No. IAG05, 1944 (Wartime Rept. No. L-135). 124. THEODORSEN, THEODORE: ..~ Condition on the Initial Shock. NACA TN No. 1029, 1946. 125. THEODORSEN, THEODORE, and GARRICK, 1. E.: General Potential Theory of Arbi­ trary 'Ving Sections. NACA Rept. No. 452, 1933. 126. THEODORSEN, THEODORE, and REGIER, ARTHUR: Experiments on Drag of Re­ volving Disks, Cylinders and Streamline Rods at High Speeds. NACA .A.CR No. IAFI6, 1944 (Rept. No. 793). 127. TOLLMIEN, "rIO: The Production of Turbulence. NACA T1"1 No. 609, 1931. 128. TOLLMIEN, W.: General Instability Criterion of Laminar Velocity Distributions, NACA T!\l No. 792, 1936. 129. TSIEN, Hsue-Shen: Two-dimensional Subsonic Flow of Compressible Fluids, J. Aeronaut. s«, vol. 6, No. 10, pp. 399-407, August, 1939. 132. VON DoENHOFF, ALBERT E.: A Method of Rapidly Estimating the Position of the Laminar Separation Point. NACA TN No. 671, 1938. 133. VON DOENHOFF, ALBERT E., and ABBOTr, FRANK T., Ja.: The Langley Two­ dimensional Low.- turbulence Pressure Tunnel. NACA TN No. 1283, 1947. 134. VON DoENHOFFJ ALBERT E., and TETERVIN, NEAL: Investigation of the "aria­ tion of Lift Coefficient with Reynolds Number at a Moderate Angle of Attack on a Low-drag Airfoil. NACA CB, 1942 (Wartime Rept. No. L-661). 135. VON DOENHOFF, A. E., and TETEBVIN, NEAL: Determination of General Relations for the Behavior of Turbulent Boundary Layers. NACA ACR No. 3G13, 1943 (Rept. No. 772). 136. VON KARMAN, T.: tJber laminare und turbulente Reibung, Z. angew. Alath. It/ech., Bd. 1, Heft 4. pp. 233-252, 1921. 137. VON KAmIAN, T.: Gastheoretische Deutung der Reynoldssehen Kennzahl, Abhandl. Aero. Imt. Aachen, Heft 4, 1925, Verlag Julius Springer, Berlin. 138.. VON K.AB.MAN, T.: Mechanical Similitude and Turbulence. NACA Tl\{ No. 611, 1931. 139. VON KA.mUN, T.: Turbulence and Skin Friction, J. Aeronaut. Sci., vol. 1, No.1, pp. 1-20, January, 1934. 140. VON K!RM:!N, T.: Compressibility Effects in Aerodynamics, J. Aeronaut. Sci., vol. 8, No.9, pp, 337-356, July, 1941. REFERENCES 307 141. VON KAmfIN, T., and MILLIKAN, C. B.: On the Theory of Laminar Boundary Layers Involving Separation. NACA Rept. No. 504, 1934. 142. VON KAmfAN, T., and TSIEN, H. B.: Boundary Layer in Compressible Fluids, J. Aeronaut. s«, vol. 5, No.6, April, 1938. 143. WARnELD, CALVIN N. for the NACA Special Subcommittee on the Upper Atmosphere: Tentative Tables for the Upper Atmosphere. NACA TN No. 1200, 1947. 144. WEICK, FRED E., and ILuuus, THOMAS A.: The Aerodynamic Characteristics of a Model Wing Having a Split Flap Deflected Downward and Moved to the Rear. NACA TN No. 422,1932. 145. WEICK, FRED E., and PLATr, RoBERT C.: Wind-tunnel Tests on a Model Wing with Fowler Flap and Specially Developed Leading-edge Slot. NACA TN No. 459, 1933. 146. WEICK, FRED E., and SANDERS, ROBERT: Wind-tunnel Tests of a Wing with Fixed Auxiliary Airfoils Having Various Chords and Profiles. NACA. Rept. No. 472, 1933. 147. 'WEICK, FRED E., and SHORTAL, JOSEPH 1\.: The Effect of Multiple Fixed Slots and a Trailing-edge Flap on the Lift and Drag of a Clark Y.Airfoil. NACA Rept. No. 427, 1932. 148. WENZINGER, CARL J.: Wind-tunnel Investigation of Ordinary and Split Flaps on Airfoils of Different Profile. NACA Rept. No. 554, 1936. 149. WENZINGER, CARL J.: Pressure Distribution over an Airfoil Section with a Flap and Tab. NAC.A Rept. No. 574, 1936. 150. WENZINGER, CARL J.: "rind-tunnel Tests of a Clark Y \Ving Having Split Flaps with Gaps. NACA TN No. 650, 1938. 151. WENZINGER, CARL J.: Pressure Distribution over an NAC.A 23012 Airfoil with an NACA 23012 External-airfoil Flap. NAC.A Rept. No. 614, 1938. 152. WENZINGER, CARL J., and DELANO, JAMES B.: Pressure Distribution over an NACA 23012 Airfoil with a Slotted and a Plain Flap. NACA Rept. No. 633, 1938. 153. 'VENZINGEB, CAllL J., and HARRIS, THOMAS A.: Wind-tunnel Investigation of an NACA. 23012 Airfoil with Various Arrangements of Slotted Flaps. NACA. Rept. No. 664,1939. 154. WENZINOER, CARLJ., and HAlUlIS, THOMAS l\.: Wind-tunnel Investigation of NACA 23012, 23021, and 23030 Airfoils with Various Sizes of Split Flap. NACA Rept. No. 668, 1939. 155. WEN ZINGER, CARL J., and H.O\RRIS, THOMAS A.: Wind-tunnel Investigation of an NACA 23021 Airfoil with Various Arrangements of Slotted Flaps. NAC..~ Rept. No. 677, 1939. 156. WENZINGER, CARL J., and RoGALLO, FRANCIS M.: Resume of Airload Data on Slats and Flaps. NACA TN No. 690, 1939. 1564. WRAGG, C. A.: Wragg Compound Aerofoil. U. S. Air Service, March, 1921, PP4 32-34. 157. YOUNG, A. D.: Note on the Effect of Slipstream on Boundary Layer Flow. Rept. No. B.A. 1404. British R.~E, May, 1937. 158. YOUNG, A. D.: Further Note on the Effect of Slipstream on Boundary Layer Flow. Rept. No. B.A. 1404a, British RAE, October, 1938. 159. YOUNG, A. D., and ~{ORJUS, D. E.: Further Note on the Eff~t of Slipstream on Boundary Layer Flow. Rept. No. B.A. 1404h, British RAE, September, 1939. 160. ZALOVICIt, JOHN A.: Profile Drag Coefficients of Conventional and Low-drag Airfoils as Obtained in Flight. NACA ACR No. 1AE31, 1944 (Wartime Rept. No. L-139). APPENDIXES APPENDIX I BASIC THICKNESS FORMS CoNTEN)'S NACA Designation 0006 0008 0008-34 0009 Page 311 312 313 314 315 316 317 0010 0010-34 0010-35 0010-64 318 0010-65 319 0010-66 0012 320 0012-64 321 322 323 0015 324 0018 325 326 0012-34 0021 327 0024 16-006 16-009 16-012 16-015 . . . . 331 16-018 . 332 328 329 330 16-021 . 333 63-006 . 334 63-009 . 335 63-010 . 63r.Q12. 63r015. 63r018. 6&-021. 336 337 338 339 340 341 63AOO6 342 63A008 63AOIO 631A012 344 ~A015 345 343 346 64-006 . 309 THEORY 0// WING SECTIONS 310 Page 347 348 N ACA Designation 64-008 . M-009 . . . . · . 64-010 . 641-012. 64:r01S . . • · • • • • . . 64:r018. . . . . . . . . 64.-021. . 349 350 351 352 353 . .... 64AOO6 64AOOS 354 65-009 . . 355 356 357 358 359 360 361 66-010 . 65t-012. 363 65r015. . 3M MAOI0 64tAOI2 MsA015 . 66-006 . . 65-008 . . 382 65r018. . 65A21.. 365 . ..... 366 65AOO6 867 368 65AOO8 65AOIO 65 1A012 . 369 370 371 65tA015 . 66-006 . 66-008 . . 66-009 66-010 . . ~~12. . . . . . . . 66r015 . . 66r018. . 372 . 373 374 375 376 66r021 .. 377 378 379 67,1-015 . 747A015 . 380 381 APPENDIX I 311 I•• ,_ -----...... ~ r--- '--- r--- h .8 0006 -MICA \ - .4 .",.- ~ D .4 I x I y (per cent c) __ (pe __ r _OO_Il_t_c}_. o o I (t' /1')2 . 1.0 .8 .6 %C 1'/l' 0.5 1.25 2.5 5.0 ..... 0.947 1.307 1.777 0 0.880 1.117 1.186 1.217 0 0.938 7.5 2.100 2.341 2.673 2.869 2.971 ,_._ I .1J.'./ V \----1 3.992 2.015 1.057 1.364 1.089 1.103 0.984 0.696 1.225 1.212 1.206 1.190 1. t 79 1.107 1.101 1.098 1.091 1.086 0.562 0.478 0.378 0.316 0.272 50 60 70 3.001 2.902 2.647 2.282 1.832 1.162 1.136 1.109 1.086 1.057 1.078 1.066 1.053 1.042 1.028 0.239 0.189 0.152 0.123 O.(l97 80 1.312 1.026 1.013 0.073 ~:~ ;::~ 10 15 20 25 30 40 ~. 90 95 100 __.. _ ;:~_l_J: I~.E. radius: 0.40 per cent c 7'HEORY OF lVING SECTIONS 312 1.6 1.2 r: I -- t--- 100.... ---.. r---. r--- r---. . . . .8 /MCA OOOB .4 L,-.--- I'--- - -", L.--- ~ o .4, .8 .6 1.0 S,C z 11 (V/V)t V/V At'./V 1.263 1.743 2.369 0 0.792 1.103 1.221 1.272 0 0.890 1.050 1.105 1.128 2.900 1.795 1.310 0.971 0.694 7.5 10 15 20 25 2.800 3.121 3.564 3.825 3.961 1.284 1.717 1.272 1.259 1.241 1.133 1.130 1.128 1.122 1.114 0.561 0.479 0.379 0.318 0.273 30 4.001 3.869 3.529 3.043 2.443 1.223 1.186 1.149 1.111 1.080 1.106 1.089 1.072 1.054 1.039 0.239 0.188 0.152 0.121 0.096 1.749 0.965 0.537 0.084 1.034 0.968 0.939 1.017 0.984 0.969 0.071 0.047 0.031 0 (per cent c) (per cent c) 0 0.5 1.25 2.5 5.0 40 50 60 70 80 90 95 100 -_ .._---. 0 ..... ..... . .... L.E. radius: 0.70 per cent e NACA 0008 Basic Thieknees Form 313 .APPENDIX I I. 6 2/_ ---- I. ---..,~ J """-- (~)' o. 1\, I o. ~ i ,.--- I , '--- o o , 0.2 0.4 I ~ ~ f 0.6 08 a 1.0 C % 11 (per cent c) (per cent c) (v/V)! ,,/1" ~'./V 2..089 0 0.917 1.023 1.092 1.137 0 0.958 1.011 1.045 1.066 4.839 1.338 0.966 0.691 0.564 1.162 1.188 1.206 1.217 1.202 1.078 1.090 1.098 1.103 40 2.436 2.996 3.396 3.867 4.000 0.485 0.387 0.326 0.248 0.197 50 60 70 80 90 3..884 3.547 2.987 2.213 1.244 1.185 1.163 1.127 1.067 0.993 95 0.684: 0.080 0.932 0 _.­ 0 1.25 2.5 5.0 7.5 10 15 20 30 100 0 0.756 1.120 1.662 1.096 1.089 1..079 t L062 1.033 0.996 0.965 0 L.E. radius: 0.174 per cent c NACA 0008-34 Basic Thickness Form 0.157 0.128 0.100 0.074 0.047 0.031 0 THEORY OF "K'ING BECTION.r; 314 I.'./I _r _.... r--­ ~ -, I. -r----. ---...... r----. r--...... '\ -- \ IMCA 0009 1 ..­ ~ ~ ...... < :, II % Y (peI: cent c) (per cent c) 0 0.5 1.25 2.5 5.0 0 ..... 1.420 1.961 2.666 .8 .6 %,C (V/V)2 tJ/V lJJJ./V 0 0.750 1.083 1.229 1.299 0 0.866 1.041 1.109 1.140 0.595 1.700 1.283 0.963 0.692 0.560 0.479 0.380 0.318 0.273 i 7.5 10 15 20 25 3.150 3.512 4.009 4.303 4.456 1.310 1.309 1.304 1.275 1.145 1.144 1.142 1.137 1.129 30 4.501 4..352 3.971 3.423 2.748 1.252 1.209 1.170 1.126 1.087 1.119 1.100 1.082 1.061 1.043 0.239 0.188 0.151 0.120 0.095 1.967 1.086 0.605 0.095 1.037 0.984 0.933 0 1.018 0.982 0.966 0 0.070 0.046 0.030 0 40 50 60 70 80 90 95 100 ~~, .. I...E. radius: 0.89 per cent r. NACA 0009 Basic Thieknees Form _---­ 315 APPENDIX I 2.D .­ 1.2 (v)' rf ---- :---.­ , r---­ r--... ---... r---­ ~ ....... .8 NACA 0010 --"\ 1 - ---­ -­ ~~ '--.. "--­ .4, ste X i Y I I 0 0.5 1.25 2.5 5.0 0 0 ..... 0.712 1..061 1.237 1.325 0.844 1.030 1.112 1.151 2.962 1.341 1.341 1.341 1.329 1.309 3..500 3.902 4.455 4.782 4.952 25 ~D./lr 0 1.578 2.178 7.5 10 15 20 l?/V' (r,/V)2 (per cent c) (per eeut c) 1.0 .8 .6 ~ 2.372 1.618 1.255 0.955 0.690 0.. 559 0.479 iO.380 0.318 0.273 1..158 1.158 1.158 1.153 1.144 I" 30 40 5.002 4.837 4.412 3.803 3.053 50 60 70 80 90 95 100 I I 2.187 1.207 0.672 0.105 1.284 1~237 1.190 LI38 1.094 I ! II I 1.040 0.960 0.925 ..... f I I t l I 1.133 1..112 1.091 1.067 1.046 1.020 0.980 0.962 . .... L.E. radius: 1.10 per cent c NACA 0010 Basic Thickness Form 0.239 0.188 0.150 0..119 0.094 0.069 0.045-' I 0=1 0.030 THEORY OF WING SECTIONS 316 i6 - ~ L2 f ""----­ ............ ~ (vY.8 ~ r-, , ['\ NACA 00/0--.14 .4 ~ --­----­ --­- r---. o .4 ~ .8 .6 ~ ~~ 1.0 x/c --x y (per cent c) (per cent c) (u/V)t vv ~Va/V 0 0.944 1.400 2.078 2.611 0 0.892 1.011 1.113 1.167 0 0.944 IJ)05 1.055 1.080 3.857 1.282 0.950 0.688 0.564 40 3.044 3.744 4.244 4.&13 5.000 1.200 1.238 1.256 1.265 .1.253 1.095 1.113 1.121 1.124 1.119 0.486 0.389 0.327 0.249 0.197 50 60 70 80 90 4.856 4.433 3.733 2.767 1.556 1.235 1.205 1.157 1.089 0.990 . 1.111 1.098 1.076 1.044 0.995 0.856 0.100 0.910 0 0.954 I 0 I 0 1.25 2.5 5.0 7.5 10 15 20 30 95 100 i I I 0.159 0.127 0.100 0.073 0.045 I L.E. radius: 0.272 per cent c NACA 0010-34 Basic Thickness Form 0.030 0 317 APPENDIX I 1.6 l2 ~ ,r """.-­ r-, \ NACA 0010"J5 l----"' ~ - r---. ---..... o .4 .2 xlc r---. ............. ~ .8 6 v/V ---~. , ---­ 10 bv./V I __.- -_.i- -~.068- o 0 o 1.25 2.5 5.0 7.5 0.878 1.267 1.844 2.289 0.954 1.032 1.087 0.977 1.016 1.043 1.122 1..059 0.679 0.. 555 10 15 20 30 40 2.667 3.289 3.789 4.478 4.878 1.141 1.172 1.194 1.214 1.229 1.068 1.083 1.093 1.. 102 1.109 0.476 0.382 0.323 0.247 0 . 198 50 5.000 60 70 4.867 4.389 80 90 I 3.500 2.100 1.235 1.240 1.227 1.176 1.046 1.111 1.114 1.108 1..084 1.023 0.. 162 0.131 0.104 0.076 0.048 1.178 0.100 0.920 0.959 j 0.030 0 95 100 I I-----~--- ._._ .. ... ~ I j o o 1.309 0..952 _---~_._-~--_.~._._- L.E. radius: 0.272 per cent c NACA 0010-35 Basic Thickness Form THEORY OF WING SECTIONS 318 1.6 ___ L , ,NA1CAoh,o r- ............. 1""Ia...... r ~I ......... .....--...... ...... :::::: ~ R NACA 0010-64- r--- r---. ~ - ~- o :& .~ 10 15 20 30 40 50 60 70 80 90 95 100 ~ to .6 x/c (v/V)t »rv ~../V 0 1.511 2.044 2.722 3.178 0 1.108 1.245 1.286 1.277 0 1.053 1.116 1.134 1.130 2.324 1.286 0.966 0.690 0.556 3.533 4.056 4.411 4.856 5.000 1.269 1.261 1.248 1.244 1.242 1.127 1.123 1.117 1.116 1.115 0.475 0.377 0.316 0.241 0.193 4.856 4.433 3.733 1.556 1.231 1.211 1.155 1.089 0.980 1.110 1.101 1.074 1.043 0.990 0.155 0.126 0.098 0.072 0.045 0.856 0.100 0.912 0 0.955 0 0.030 0 Y (per cent c) (per cent c) 0 1.25 2.5 5.0 7.5 .4 .....-- 2..767 L.E. radius: 1.10 per cent r. NACA 0010-64 Basic Thickness Form I -- APPENDIX I 319 1.6 12 (v! ~~ ......... f .8 WACA 0010-65 .4 - ---.... ~ """""""" ---­ '-- ....... o "',­\ .2 ~ ~ .8 .6 A- , LO x-/c II x y (lIIV)! (per cent c) (per cent c) , 0 1.25 2.5 5.0 7.5 I i 10 .15 20 30 40 50 60 70 80 _ .. I I I I 0 1.068 2.584 1.295 0.970 0.684 0.551 0 1.140 1.273 1.271 1.252 3.300 3.756 4.089 4.578 4.889 1.236 1.213 1.200 1.196 1.212 1.112 1.101 1.095 1.094 1.101 0.470 0.372 0.312 0.239 0.193 1.229 1.234 1.226 1.173 1.049 1.109 1.111 1.107 1.083 1.024 0.158 0.128 0.103 0.076 0.046 0.915 0 0.957 0 5.000 4.867 4.389 3.500 I 2.100 I 1.178 0.100 ---- l 41Ja / V 1.467 1.967 2.589 2.989 90 _-~., »rv 0 95 100 t I ... _. t I 1 tI j __ ! ._...._.­ 1.128 1.127 1.119 I I I II I '-' L. E. radius: 1.10 pet" cent r NACA 0010-65 Basic Thickness Form 0.029 0 ,~ .. ,­ -.. THEORY OF WING SECTIONS 320 l6 l2 r­ ............... ~ f ---­ '" ..... , , \ NACA0010-66 .4 ~ r---... o --..... ............... - --­ ,.,,--­ .2 .4 x/c z 11 (per cent c) (per cent c) .6 lO (V/V)I v/V 4v./V 0 1.489 2.011 2.656 3.089 0 1.130 1.246 1.286 1.282 0 1.063 1.116 1.134 1.132 2.434 1.289 0.959 0.687 0.554 40 3.400 3.856 4.178 4.678 4.822 1.258 1.225 1.209 1.189 1.178 1.122 1.107 1.100 1.090 1.085 0.471 0.372 0.310 0.236 0.190 SO 60 70 80 90 4.956 0.000 4.889 4.300 2.833 1.184 1.214 1.265 1.278 1.135 1.088 1.102 1.125 1.130 1.065 0.153 0.129 0.104 0.080 0.049 95 100 ------- 1.656 0.100 0.960 0 0.980 0 0.030 0 0 1.25 2.5 5.0 7.5 10 15 20 30 L.E. radius: 1.10 per cent c NACA OOlO-M Basic Thickness Form APPENDIX 1 321 1.6 1.2 - ( .............. I ~ ~- --~ ~ ...... .......... .8 , '\ MACA 0012 ~ i'--­ - -r--- r--­ - ~~ .8 D % 11 (per cent c) (per cent c) 0 0.5 1.25 2.5 5.0 7.5 10 15 20 25 ~ ~ - ..­ 1.0 (fJ/V)1 v/V Avo/V 0 0.640 1.010 1.241 1.378 0 0.800 1.005 1.114 1.174 1.988 1.475 1.199 0.685 4.200 4.683 5.345 5.737 5.941 1.402 1.411 1.411 1.399 1.378 1.184 1.188 1.188 1.183 1.174 0.558 0.479 0.381 0.319 0.273 1.350 1.288 1.228 1.166 1.109 1.162 1.135 1.1OS 1.080 1.053 0.239 0.187 0.149 0.118 0.092 1.022 0.978 0.952 0 0 ..... 1.894 2.615 3.555 30 6.002 40 50 60 70 5.803 5.294 80 90 95 100 2.623 1.044 1.448 0.807 0.956 0.906 0 4.563 3.664 0.126 I~E. radius: 1.58 per cent c NACA 0012 Basic Thickness Form 0.934 I I 0.068 0.044 0.029 0 THEORY OF WING BECTIONS 322 L6 -'--­ V - r---. 7 ~ .......... ~ <, f ~ , ~ r---­ r---.. !'--.. t--......... o .2 % 11 (per cent c) (per cent c) 0 1.25 2.5 0.0 7.5 10 15 20 30 40 50 60 70 80 90 ~5 100 , r\ NACA 0012-34 ~ .4 -x./c .8 .6 I'--....... ~~ 1.0 (v/V)t fJ/V I:.fJ./V 0 1.133 1.680 2.493 3.133 0 0.865 0.997 1.122 1.186 0 0.930 0.999 1.069 1.089 3.154 1.251 0.933 0.683 0.560 3.653 4.493 5.093 5.800 6.000 1.229 1.282 1.310 1.329 1.311 1.109 1.132 1.145 1.153 1.146 0.484 0.389 0.329 0.250 0.198 5.827 5.320 4.480 3.320 1.867 1.284 1.249 1.192 1.112 0.985 1.133 1.11'8 1.092 1.055 0.992 0.158 0.128 0.098 0.071 0.045 1.027 0.120 0.894 0 0.946 0.029 0 0 L.E. radius: 0.391 per cent c NACA 0012-34 Basic Thic1mess Form APPENDIX I 323 L6 ,r: r----. ~ ~ .8 r-, 1\ \1 NACA 00/2-64­ .4 - ----. ---­ - ~ ~ lo-......... ~ l---­ ~ o .4- % 11 (per cent c) (per cent c) .8 .6 xle J,O (V/V)2 »rv ~v./V 0 1.813 2.453 3.267 3.813 0 1.072 1.270 1.330 1.325 0 1.035 1.127 1.153 1.151 2.019 0.952 0.685 0.554 1.322 1.313 40 4.240 4.867 5.293 5.827 6.000 1.150 1.146 1.141 1.139 1.140 0.474 0.372 0.315 0.241 0.199 SO 60 70 80 90 5.827 5.320 4.480 3.320 1.867 1.280 1.244 1.189 1.102 1.131 1.115 1.090 0.993 0.996 0.154 0.126 0.096 0.070 0.044 95 1.027 0.120 0.889 0 0.943 0 1.26 2.0 S.O 7.5 10 15 20 30 100 1.303 1.297 1.300 1.050 0 L.E. radius: 1.582 per cent c NACA 0012-64 Basic Thickness Form 1.236 0.028 0 - THEORY OF WING SECTIONS 324 1.6 I.Z r I ~ --......... i"'-<, ~ I .............. ~ --; ''\ .8 \, NACA 0015 <, o - .-- r --- --- :---~ ---- --- 0 0.5 1.25 2.5 5.0 I--' 1.0 I Av./V - 2.367 3.268 4.443 0 0.546 0.933 1.237 1.450 0 0.739 0.966 1.112 1.204 1.600 1.312 1.112 0.900 0.675 7.5 10 15 20 25 5.250 5.853 6.682 7.172 7.427 1.498 1.520 1.520 1.510 1.484 1.224 1.233 1.233 1.229 1.218 0.557 0.479 0.381 0.320 0.274 30 7.502 7.254 6.617 5.704 4.580 1.450 1.369 1.279 1.206 1.132 1.204 1.170 1.131 1.098 1.064 0.239 0.185 0.146 0.115 0.090 3.279 1.810 1.008 0.158 1.049 0.945 0.872 0 1.024 0.972 0.934 0 0.065 0.041 0.027 0 40 50 60 70 80 90 95 }OO 0 ~V (V/V)t ~ .s z/c I X Y (per cent c) (per cent c) r---. ~ . ..... ..... L.E. radius: 2.48 per cent c NACA 0015 Basic Thickness Form _.- _ .. _~ ...­ 325 APPENDIX I 1.6 1.2 ~ ( .......... I I ~~ <, ~ ......... ......... ~ "­ <, 1\ \, NACA DOl. .4 -: <, - ~ ..---- I"--. .4 () 7.5 10 15 20 25 30 40 50 60 70 80 90 95 100 --­ ---­ ~ ~ .6 r--­ ~ .8 ~ ~ 1.0 (,'1V)2 __v~1 4v./IT 0 0.465 0.857 1.217 1.507 0 0.682 0.926 1.103 1.228 1.342 1.178 1.028 0.861 0.662 6.300 7.024 8..018 8.606 8.912 1.598 1.628 1..633 1.625 0.555 1.592 1.264 1.276 1.278 1.275 1..262 9.003 8.705 7.941 6.845 5.496 1.556 1.453 1.331 1.246 1.153 1.247 1.205 1.154 1.116 1.074 0.238 0.184 0.144 0.113 0.087 1.051 0.933 0.836 0 1..025 0.966 0.914 0 0.063 0.039 0.025 0 v x t (per cent c) (per cent c) 0 0.5 1.25 2.5 5.0 ~ 0 ..... 2.841 3.922 6.332 3.935 2.172 1.210 0.189 I I L.E. radius: 3..56 per cent c NACA 0018 Basic '£hickness Form 0.479 . 0..381 0.320 0.274 THEORY OF 'A,'IING SECTIONS 326 I ~ r-, <, 7 1.2 "' r-, 1 -" 11 (per cent c) (per cent c) 0 0.5 1.25 2.5 5.0 0 r--­ ---.. (V/V)t ~ I \, r--- r---. ~ .8 .6 X/C -, \ L.---­ ~ .4 <, (J(J2/ --- ~ V r-, ...-... () X r-, ~~ MeA .4 r-, ~ 1.0 fJ/V li.v./V 0 0.630 0.887 1.087 1.242 1.167 1.065 0.946 0.818 0.648 6.221 0 0.397 0.787 1.182 1.543 7.5 10 15 20 25 7.350 8.195 9.354 10.040 10.397 1.682 1.734 1.756 1.742 1.706 1.297 1.317 1.325 1.320 1.306 0.550 0.478 0.381 0.320 0.274 30 10.504 10.156 9.265 7.986 6.412 1.664 1.538 1.388 1.284 1.177 1.290 1.240 1.178 1.133 1.085 0.238 0.183 0.142 0.111 0.084 4.591 2.534 1.412 0.221 1.055 0.916 0.801 0 1.027 0.957 0.895 0 0.061 0.037 0.023 0 40 50 60 70 80 90 95 100 ...... 3.315 4~576 L.E. radius: 4.85 per cent c NACA 0021 Basic Thickness Form 327 APPENDIX I rr-«: 7 I r-. ,,~ " r-, <, I r-, '" .s M4CA 0024 . .4 o V --r-, .j. y x (per cent c) (per cent c) - r---. r----.. r----.. ~ x/c L..--- ~ .8 ..8 ~ \ \ t---. 1 ~ I. D (V/V)! V/V 4D./V 3.788 5.229 7.109 0 0.335 0.719 1.130 1.548 0 0.579 0.848 1.063 1.244 1..050 0.964 0.870 0.771 0.632 7.5 10 15 20 25 8.400 9.365 10.691 11.475 11.883 1.748 1.833 1.888 1.871 1.822 1.322 1.354 1.374 1.368 1.350 0.542 0.476 0.383 0.321 0.274 30 50 60 70 12J)04 11.607 10.588 9.127 7.328 1.777 1.631 1.450 1.203 I 1.333 1.277 1..204 1.151 1.097 0.238 0.181 0.140 0.109 0.082 80 90 95 100 5.247 2.896 1.613 0.252 1.065 I I 1.032 0.944 0.879 0 0..059 0.035 0.022 0 0 0.5 1.25 2.5 5.0 0 ......... 40 I 1.325 0.891 0.773 0 j L.E. radius: 6.33 per eent c NACA 0024 Basic 'I'hickness Form THEORY OF WING SECTIONS 328 2.0 1.6 1.2 ---. r: ~ \ .B NACA 16-006 -­ e---' 11 10 15 20 30 40 lit) 60 10 80 90 95 100 ! (fl/V)t v/V 0 0.646 0.903 1.255 1.516 0 1.059 1.085 1.097 1.105 0 1.029 1.042 1.047 1.051 5.471 1.376 0.980 0.689 0.557 1.729 2.067 2.332 2.709 2.927 1.108 1.112 1.116 1.123 1.132 1.053 1.055 1.057 1.060 1.064 0.476 0.379 0.319 0.244 0.196 3.000 2.917 2.635 2.099 1.259 1.137 1.141 1.132 1.104 1.035 1.066 1.068 1.064 1.051 1.017 0.160 0.130 0.104 0.077 0.049 0.707 0.060 0.962 0 0.981 0 0.032 0 (per cent c) (per cent c) 0 1.25 2.5 5.0 7.5 , /.0 .8 % \ \ L.E. radius: 0.176 per cent c NACA 16-006 Basic Thickness Form l1v./V APPENDIX I 329 .. , 1.2 ,­ ........... ~~ .8 \ NM:A 16-00g 1 .4 --- ~- r---­ - ~ .4 (J % 11 Z/c \ .8 .6 """""""­ ~ /.0 (V/V)2 V/V J1v./V 0 0.969 1..354 1.882 2.274 0 1.042 1.109 1.139 1.152 0 1.021 1.053 1.067 1.073 3.644 1.330 0.964 0.684 0.554 10 15 20 30 40 2.593 3.101 3.498 4.063 4.391 ·1.158 1.168 1.177 1.190 1.202 1.076 1.081 1.085 1.091 1.096 0.475 0.378 0.319 0.245 0.197 50 60 70 80 90 4.500 4.376 3.952 3.149 1.888 1.211 1.214 1.206 1.156 1.043 1.100 1.099 1.075 1.022 0.160 0.. 131 0.103 0.076 0.047 95 1.061 0.090 0.939 0 0.969 0 0.030 0 (per cent c) (per cent c) 0 1.25 2.5 5.0 7.5 tOO I 1~106 L.E. radius: 0.396 per cent c NACA 16-009 Basic Thickness Form THEORY OF WING SECTIONS 330 1.6 I-Z T ---- to--"'" ~ ~ ""r\.\ .8 NACA 16-DI2 .4 ~ I'--- - ~ 100-0-.. D !.........-. .t .4 \ t--- t--....... L--- ~ I.D .6 SIC. x y (v/V)t V/V I1v./V 0 1.292 1.805 2.509 3.032 0 1.002 1.109 1.173 1.197 0 1.001 1.053 1.083 1.094 2.624 1.268 0.942 0.677 0.561 10 15 20 30 40 3.457 4.135 4.664 5.417 5.855 1.208 1.223 1.237 1.257 1.271 1.099 1.106 1.112 1.121 1.128 0.473 0.378 0.319 0.245 0.197 50 60 70 80 90 6.000 5.836 5.269 4.199 2.517 1.286 1.293 1.275 1.203 1.051 1.134 1.137 1.129 1.097 1.025 0.161 0.131 0.102 0.075 0.045 95 100 1.415 0.120 0.908 0 0.953 0 0.027 0 (per cent c) (per cent c) 0 1.25 2.5 5.0 7.5 L.E. radius: 0.703 per cent c NACA 16-012 Basic Thickness Form 331 APPENDIX I 1.6 :",.--­ ~ / ~ - --'" -, "'" { \ M4CA /6-0/5 ~ i'---.. D x ~ ---­ - 10 15 20 --­ ---­ ...---­ .8 /.0 0 1.615 2.257 3.137 3.790 0 0.956 1.105 1.200 1.239 0 0.978 1.051 1.095 1.113 0.916 0.668 0.547 4.322 1.256 1.278 1.297 1.327 1.349 1.121 1.130 1.139 1.152 1.161 0.471 0.377 0.318 0.245 0.197 0.161 0.131 0.102 0.074 0.043 40 SO 60 7.500 7.293 1.364 70 6.. 587 80 90 5.248 3.147 1.348 1.254 1.053 1.168 1.172 1.161 1.120 1.026 95 1.768 0.150 0.875 0 0.935 0 100 ~ v/V 6.168 5.830 6.772 7.318 30 ............... (V/V)2 (per cent c) (per cent c) 0 1.25 2.5 5.0 7.5 r--­ ~ v \, 1.374 L.E. radius: 1.100 per cent c NACA 16-015 Basic Thickness Form I ~v./V 2.041 1.209 I 0.025 0 THEORY OF WING SECTIONS 332 - .w - /' ~ - - --­ .......... <, -7 t-, \ I \ .8 V ~ ~ t---- - r--­ ~ .6 I--­ (J % 'Y (per cent c) (per cent c) , \ NACA /S-OI8 .4 ~ ~ L.---­ ~ .8 ~ ~ /.0 (V/V)I fJ/V ~fJ./V 0 1.938 2.708 3.764 4.548 0 0.903 1.092 1.217 1.271 0 0.950 1.045 1.103 1.128 1.744 1.140 0.883 0.657 0.541 5.186 6.202 6.996 8.126 8.782 1.302 1.332 1.357 1.399 1.426 1.141 1.154 1.165 1.183 1.194 0.468 0.376 0.318 0.245 0.198 70 80 90 9.000 8.752 7.904 6.298 3.776 1.447 1.452 1.421 1..306 1.051 1.203 1.205 1.192 1.143 1.025 0.162 0.131 0.102 0.073 0.042 95 100 2.122 0.180 0.837 0 0.915 0 0.024 0 0 1.25 2.5 5.0 7.5 10 15 20 30 40 SO 60 f L.E. radius: 1.584 per cent c NACA 16-018 Basic Thickness Form APPENDIX 1 .8 ~ V - I -/ , ---- --­ --.... <, ~ I. \ ~ NACA /S-(J21 .4 V ---­ ~ r--.-. ...-­ -~ . ---.--.... ~ _l.---' ~ ~ D .4 z 11 (per cent c) (per cent c) 1\ '\ \ zjc .8 B ~ to (V/V)t tJ/V !lv.IV 0 2.261 3.159 4.391 5.306 0 0.826 1.062 1.221 1.295 0 0.909 1.031 1.105 1.138 1.574 1.069 0.828 0.640 0.534 10 15 20 30 40 6.050 7.236 8.162 9.480 10.246 1.342 1.391 1.419 1.474 1.506 1.159 1.179 1.191 1.214 1.227 0.463 0.374 0.317 0.245 0.198 50 60 70 10.500 10.211 9.221 7.348 4.405 1.535 1.536 1.495 1.361 1.039 1.239 1.166 1.019 0.162 0.131 0.102 0.072 0.041 2.476 0.210 0.801 0.895 0 0.023 0 0 1.25 2.5 5.0 7.5 80 90 95 100 I 0 1.239 1.223 ThE. radius: 2.156 per cent c NACA 16-021 Basic Thickness Form THEORY OF WING SECTIONS 334 1.8 ,0 1.2 (f)~ I t ..... I ~CI,,03 ,..,.,.. $Ur"~ ~03 L~ ?? J ."foce r- r--.. ~ ~ ~ .8 NACA t13-D06 ~ ............... D .4 x y (per cent c) (per cent c) '/. S,C 1.0 .8 .6 (f)/V)2 V/V J1v./V 0 0.503 0.609 0.771 1.057 0 0.973 1.050 1.080 1.110 0 0.986 1.025 1.039 1.054 4.483 2.110 1.778 1.399 0.981 5.0 7.5 10 15 20 1.462 1.766 2.010 2.386 2.656 1.130 1.142 1.149 1.159 1.165 1.063 1.069 1.072 1.077 1.079 0.692 0.562 0.484 0.384 0.321 25 30 2.841 2.954 3.000 2.971 2.877 1.170 1.174 1.170 1.164 1.151 1.082 1.084 1.082 1.079 1.073 0.279 0.245 0.218 0.196 0.176 2.723 2.517 2.267 1.982 1.670 1.137 1.118 1.096 1.074 1.046 1.066 1.057 1.047 1.036 1.023 0.158 0.141 0.125 0.111 0.098 1.342 1.008 0.683 0.383 0.138 1.020 0.994 0.965 0.936 0.910 1.010 0.997 0.982 0.967 0.954 0.085 0.073 0.060 0.047 0.032 0 0.886 0.941 0 0 0.5 0.75 1.25 2.5 35 40 -is 50 55 60 65 70 75 80 85 90 95 100 LE. radius: 0.297 per cent c -NACA 63-006 Basic Thickness Form APPENDIX.] I.IJ ro.. 1.2 f/< .8 , c, •. / 33~ I ~ ~ slrfaet!J ~ ~ ~ I ~ r--.08 Lower .•ur'f«:e ~~ V ~ ~ MACA 63-otJ!J .4 [,--­ ~ r--­ " - ­ _L--- ~ - "'""-­ .z a % .4 Y {11/V)2 (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 5.0 7.5 10 15 20 0 0.749 0.906 1.151 1.582 2.196 %/c 0 0.885 i I I 1.002 l.OSI 1.130 lO .6 fJ/V , J1tJ./V 0 0.941 1.001 IJ)25 3.058 1.889 1.063 1.647 1.339 0.961 1.180 1.205 1.221 1.241 1.255 1.08B 1.105 1.114 1.120 0.689 0.560 0.484 0.386 0.324 4.275 1.264 1.124 0.281 4.442 4.500 4.447 4.296 1.269 1.265 1.255 1.235 1.126 1.120 1.111 0.248 0.220 0.196 1.208 1.099 1.175 65 4.056 3.739 3.358 2.928 70 2.458 1..084 1.068 1.051 1.032 0.156 0.140 0.124 0.109 0.095 75 80 1.966 1.471 85 90 0.550 95 0.196 0.984 0.942 0.903 0.868 1.012 0.992 0.971 0.950 0.932 0.082 0.069 0.057 0.044 0.030 0 0.838 0.915 0 25 30 35 40 45 50 55 60 100 2.655 3.024 3.591 3.997 0.990 ! 1.141 1.104 1.065 1.025 1.098 1.125 L.E. radius: 0.631 per cent c N.~CA 63-009 Basic Thickness Form 0.175 THEORY OF WING SECTIONS 336 ttl /' I.e .8 c, .. I 10 , . . , . ..,-'!ot» ro;: ~~~ /r V v I 1'--.10 LOWf1r ~ I .".,~ ~~ ~~ ~ NACA 63-010 .4 -r --­ r---­ t---..­ ~ L----­ I"-­ "'-­ -­ o .4 I ~- /.0 .8 .6 %/C x JI (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 5.0 7.5 10 15 20 25 30 35 40 45 50 55 60 65 70 (,,/V)2 fJ/V 11v./V 0.841 0.978 1.037 1.131 0 0.917 0.989 1.018 1.063 2.775 1.825 1.603 1.316 0.952 1.193 1.223 1.245 1.270 1.285 1.092 1.106 1.116 1.127 1.134 0.687 0.560 0.484 0.386 0.325 4.753 4.938 5.000 4.938 4.766 1.296 1.302 1.299 1.286 1.262 1.138 1.141 1.140 1.134 1.123 0.282 0.248 0.220 0.196 0.175 4.496 4.140 3.715 3.234 2.712 1.231 1.193 1.154 1.113 1.069 1.110 1.092 1.074 1.055 1.034 0.156 0.139 0.123 O.lOS 0.094 1.025 1.012 0.989 0.967 0.945 0.924 0.081 0.069 0.056 0.043 0.030 0.907 0 0 0 0.829 1.004 1.275 1.756 2.440 2.950 3.362 3.994 4.445 I 75 80 85 2.166 1.618 90 95 0.604 0.214 0.979 0.935 0.893 0.853 0 0.822 100 1.088 L.E. radius: 0.770 per cent c NACA *':>.-1)10 Basic Thickness Form APPENDIX I l8 , ~ _C,.. 337 14 Lt:Per surfQCfl ............. I loy ~ ~ £2 <.J (/ .14 Lo",.,- surfoct!!I """ ~ ~ ~ ---­ L.--. I{ .8 ~ , '- ~ NACA 63J-a2 .4 ~ "--..... ~ - r--- ~ - a .P .4 I ~ .6 -----­ --­-- v (vIV)2 "/1' 0 0.985 1.194 1.519 2.102 0 0.750 0.925 1.005 1.129 0 0.866 0.962 1.003 5.0 7.5 10 15 20 2.925 3.542 4.039 4.799 5.342 1.217 1.261 1.294 1.330 1.349 1.103 1.123 1.138 25 5.712 5.930 6.000 5.920 5.704 1.362 1.370 5.370 4.935 4.420 3.840 3.210 2.556 1.902 1.274 0.707 0.250 1.023 0.969 0.920 0.871 0.826 x (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 ---­ I i 0 1.063 I I Jlv.IV 2.336 1.695 1.513 1.266 0.933 1.161 0.682 0.559 0.484 0.387 0.326 1.317 1.167 1.170 1.169 1.161 1.148 0.283 0.249 0.221 0.196 0.174 1.276 1.229 1.181 1.131 1.076 1.130 1.109 1.087 1.063 1.037 0.155 0.137 0.121 0.106 0.091 1.011 0.984 0.959 0.933 0.909 0.079 0.067 0.055 0.042 0.029 0.889 0 1.153 1.366 1.348 II 1.0 .8 %C 0.791 I I L.E. radius: 1.087 per cent c NACA 631-012 Basic Thickness Form THEORY OF WING SECTIONS ~ ,•.22 ~ !IUrlI:It» ......... ( '.I r III I - I (v)' --........ r-, .J1I'" ~ / ~~~~ r-, ~~22 ~ Loww- "'oce . . ~ ~ ~ A ~~............ NACA tJ3z-tJl5 .4­ V ~ -r---- r--- <, t---. L.---- J---­ r--- t--­ ~ a ~,... .8 % 11 1.0 (vlv)! V/V 4v./V 0 1.204 1.462 1.878 2.610 0 0.600 0.822 0.938 1.105 0 0.775 0.907 0.969 1.051 1.918 1.513 1.379 1.182 0.903 5.0 7.5 10 15 20 3.648 4.427 5.055 6.011 6.693 1.244 1.315 1.360 1.415 1.446 1.115 1.147 1.166 1.190 1.202 0.674 0.557 0.484 0.388 0.330 25 30 7.155 7.421 7.500 7.386 7.099 1.467 1.481 1.475 1.446 1.401 1.211 1.217 1.214 1.202 1.184 0.286 0.251 0.222 0.196 0.174 6.665 6.108 5.453 4.721 3.934 1.345 1.281 1.220 1.155 1.085 1.160 1.132 1.105 1.075 1.042 0.153 0.135 0.118 0.102 3.119 2.310 1.541 0.852 0.300 1.019 0.953 0.894 1.009 0.789 0.976 0.946 0.916 0.888 0.076 0.063 0.051 0.039 0.026 0 0.750 0.866 0 (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0.83~ L.E. radius: 1.594 per cent c NACA 63r015 Basic Thickness Form 0.088­ APPENI)[.\'- I eo ~,-.32 L /.6 0,. 1.2 /" / / I,l V' J ~ " r-, ~~ <, ~~ ~ I I ~ ~ NfCA 633-018 -- ~ - r--­ o % <, ~32 LOflllf!lr ""acll I , .............. V I V lIppe". surfloc. <, ( 339 '" I I---­ ~ ZIt: ~ r--­ ~ 1.0 .B 6 (v/1T ) 2 y (per cent e) .(per rent c) r--.. r--- "IV /i,v./V 1.639 0.848 0 0.664 0.837 0.921 i 0 ! i 0.5 0.75 I 1.25 f 2.5 f 5.0 7.5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 I I I I I I 0 0 1.404 1.713 2.217 3.104 0.441 0 ..700 1.065 1.032­ 4.362 5.308 6.068 7.225 1.260 1.360 1.424 1.122 1.166 1.193 1.225 1.244 1.500 8.048 1.547 8.600 8.913 9.000 8.845 8.482 1.579 1.598 1.585 1.550 1.490 1.361 1.258 1.105 0.871 0.663 0.553 0.484 0.390 0.333 1.257 0.289 1.264 0.253 1.259 1.245 1.221 0.223 0.197 0.173 1.188 1.153 1.119 1.082 0.152 0.133 0.115 0.099 1.043 0.084­ 1.004 7.942 7.256 6.455 5.567 4.622 1.252 1.170 1J187 3.650 2.691 1.787 0.985 0.348 1.009 0.933 0.868 0.807 0.753 0.966 0.932 0.898 0.868 0.072 0.059 0.048 0.036 0.024 0 0.712 0.844 0 1.411 1.330 I ~-- L.E. radius: 2.120 per cent c N.~CA 63...018 Basic Thickness Form THEORY OF WING SECTIONS 340 a .z z 71 (per cent C) (per cent C) .4 Z/c .6 (V/V)2 vlV Av.IV 0 1.583 1.937 2.527 3.577 0 0.275 0.564 0.7'5 1.010 0 0.524 0.751 0.851 1.005 1.439 1.236 1.156 1.034 0.842 5.065 6.182 7.080 8.441 1.260 1.394 1.487 1.592 20 9.~O 1.655 1.122 1.181 1.219 1.262 1.286 0.653 0.550 0.484 0.392 0.335 25 10.053 10.412 10.&00 298 9.854 1.698 1.578 1.303 1.312 1.307 1.286 1.256 0.291 0.255 0.225 0.198 0.173 50 9.206 55 60 65 8.390 1.479 1.380 1.281 1.180 1.084 1.216 1.175 1.132 1.086 1.041 0.150 0.130 0.112 0.096 0.081 0.997 0.954 0.916 0.880 0.849 0.068 0.057 2.021 1.113 0.392 0.994 0.911 0.839 0.774 0.721 0..046 0 0.676 0.822 0 0 0.5 0.75 1.25 2.5 5.0 7.5 10 15 30 35 40 45 to. 70 7.441 6.396 &.290 75 4..160 80 85 90 96 100 a.OM 1.721 1.709 1.654 . L.E. radius: 2.6liO per cent c r~ACA 63M}21 Basic 'Thickness Form 0.035 0.023 APPENDIX I 0 1.2. r-l IlL" r: 341 c/ =.01Uppersurface ." I-+-- r---. "".01 lower surrace I .8 I -- r-. NACA 6JAOO6 .4 .....- r--- "--- o % .2 11 1.25 2.5 x/c .6 vlV J1v./I·· 0 0.495 0.595 0.7M 1.045 0 0.900 1.063 1.086 1.112 0 0.949 1.031 1.042 1.055 4.560 1.447 1.134 1.747 1.989 1.142 1.1SO 1.159 1.165 1.065 1.069 1.072 1.077 5.0 7.5 10 15 2.362 20 2.631 25 30 35 4{) 2.820 2.942 2.996 2.985 45 2.914 50 55 60 2.788 1.168 1.079 1.370 0.976 0.693 0.563 0.485 0.383 0.321 1.162 1.151 1.067 1.058 1.049 1.039 1.028 0.. 140 0.126 0.112 0.085 0.072 0.060 0..047 0.033 0 1.169 1.859 75 1.556 1.035 80 1.248 1.010 85 0.939 90 95 0.. 630 0.986 0.96-1 0.322 0.939 1.017 1.005 0.993 0.982 0.969 100 0.013 0 0 2.613 2.396 2.143 2.079 1.794 1.081 1.082 1.081 1.078 1.073 1.170 1..138 1.120 1.100 1.079 1.057 65 70 to .8 (fllV)t (per cent c) (per cent c) 0 0.5 0.75 A- L.E. radius: 0.265 per cent c T.E. radius: 0.014 per cent c NACA 63AOO6 Basic Thickness Forol 0.278 0.24:4 0.217 0.195 0.175 0.158 0.098 THEORY OF WING SECTIONS. 342 l6 , c/ , =.05Upper surface I 0 t ~ ?( -r- ~ ~ ~ '.OS lower surface ............. ~ NACA6.1AOO8 ~ .............. o -- .2. .4 .6 lO .8 X/I! % (V/V)2 »rv AIl./V 0 0.658 0.791 0 0.850 1.034 0 0.922 1.017 11 (per cent c) (per cent c) 1.003 1.080 1.039 1.391 1.132 1.064 3.465 1.961 1.674 1.344 0.967 1.930 2.332 2.656 3.155 3.51& 1.168 1.185 1.198 1.212 1.221 1.081 1.089 1.095 1.101 1.105 0.689 0.562 0.484 0.383 0.322 3.766 3.926 3.995 3.978 3.878 1.227 1.230 1.228 1.219 1.204 1.108 1.109 1.108 1.104 1.097 0.279 0.246 0.218 0.195 0.174 3.705 3.468 3.176 2.837 2.457 1.183 1.159 1.132 1.104 1.073 1.088 1.077 1.064 1.051 1.036 0.156 0.138 0.123 0.109 0.096 90 95 2.055 1.647 1.240 0.833 0.425 1.042 1.010 0.980 0.951 0.919 1.021 1.005 0.990 0.975 0.959 0.083 0.070 0.058 0.045 0.030 100 0.018 0 0 0 0 0.5 0.75 1.25 2.5 5.0 7.5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 L.E. radius: 0.473 per cent c T.E. radius: 0.020 per cent c N ACA 63AOOS Basic Thickness Form APPENDI..Y. I 343 l6 .~". cl =.09 Upper surface .... r/L ~ 1.2 (vY 1fA --..;~ ,...- ~ ~ ~~ ""~ '.09 lower surface .8 NACA6JAOIO ~ - ~ ---..; '--........... o , .6 .8 x/c x 11 (per cent c) (per cent c) 0 0.5 0.75 o ~ 2.5 0.816 0.983 1.250 1.737 5.0 7.5 10 15 20 2.412 2.917 3.324 3.950 4.400 25 4.714 30 4.913 35 40 4.995 4.968 4.837 !I 1.25 45 50 55 60 65 70 75 80 85 90 95 100 I I i I II I i I II , t 1 ! 4.613 4.311 3.943 3.517 3.044 2.545 2.040 1.535 1.030 0.525 0.021 I (V/V)2 I i 1.043 i 1.140 1.068 I 1.200 1.095 1.107 1.116 1.126 1.132 0 i 0.774 0.985 ! I II I II Ii II I 1.225 1.245 1.268 1.282 1.290 1.294 1.291 1.279 1.258 LO &./".. 'II V 0 0.880 0.992 1.021 ~ , 1.136 1.138 1.136 1.131 1.122 I I I I I 0.280 0.247 0.218 0.195 0.174 I I 1.048 1.009 1.02-1 1.004 I! 0.972 0.938 O.9CiO 0.969 0.949 I j 0 0 I I 1.078 1.061 1.042 0.684 0.560 0.483 0.383 1.109 1.094 2.805 1.833 1.594 1.307 0.957 I 1.230 1.196 1.162 1.125 1.086 Ii - i I I , 0.324 0.155 0.137 0.122 O.lOS 0.09-1 1 0.986 L.E. radius: 0.742 per cent c T.E. radius: 0.023 per cent c NACA 63AOI0 Basic Thickness Form I I 0.081 0.068 0.057 0.044 0.030 0 THEORY OF WING SECTIONS 344 16 IJI' cl :./2 Vpp~r surface rl)Lt ~ ~ ~/~ (fit rr .B , ~ "'-o ~ ~ ............ '.It Lower surface ~ NACA6~AOI2 -- ~ ~ ..4 ~ ~Iooo..:. t - ~- .6 .8 x/c z 11 (per cent c) (per cent c) LO (vlv)'J plV i1v./V 2.361 1.701 1.515 0 0 0.5 0.973 1.173 1.492 2.078 0 0.686 0.924 0.985 1.136 0 0.828 0.961 0.992 1.066 0.935 2.895 3.504 3.994 4.747 5.287 1.229 1.265 1.291 1.324 1.344 1.109 1.125 1.136 1.151 1.159 0.679 0.559 0.482 0.384 0.325 25 30 35 5.664: 40 45 5.957 5.792 1.355 1.360 1.357 1.340 1.312 1.164 1.166 1.165 1.158 1.145 0.281 0.248 0.219 0.196 0.174 50 5.517 5.148 4.700 4.186 3.621 1.275 1.234 1.191 1.145 1.098 1.129 1.111 1.091 1.070 0.154 0.136 0.120 0.106 0.092 85 90 1.826 1.051 1.007 0.964 1.025 80 3.026 2.426 1.225 0.925 9S 0.625 100 0.025 0.75 1.25 2.5 5.0 7.5 10 15 20 55 60 65 70 75 "-­ ~ ............ ~ jIlII"""" 5.901 5.995 1.048 1.258 0.880 1.003 0.982 0.962 0.938 0.079 0.066 0.055 0.042 0.029 0 0 0 L.E. radius: 1.071 per cent c T.E. radius: 0.028 per cent c NACA 6&A012 Basic Thickness Form APPENDIX I 345 ,~t:l =.18 Upper surface 1.6 i2 ............... f O~ ~- .................. ~ I 't" ~ V f---- II ~ , r--:::: ~ e-, ~~ \ .8 r/ J8 lower $ur'Fact! I ~ r--... ~ NACA 6:12 AO/5 - r--- ~""""" <, r---.. o .2 .4 I'--- ~ ~~ .,.....- "---~ .6 J.() x/c z 11 (per cent c) (per cent c) (V/V)2 "IV &J./V 0 0.550 0.825 0.882 1.120 0 0.742 0.908 0.939 1.058 1.504 1.370 1.176 0.005 1.257 1.121 1.150 1.167 1.187 1.199 0.669 0.555 0.482 0.384 0.326 1.396 1.206 1.210 1..207 1.198 1•.182 0.282 0.250 0.220 0.196 0.174 6..858 6.387 5.820 5.173 4.468 1.349 1.296 1.237 1.175 1.115 1.161 1.138 1.112 1.084 1..056 0.152 0.134 0.118 0.104 0.090 90 95 3.731 2.991 2.252 1.512 0.772 1.055 1.000 0.950 0.900 0.850 1.027 1.000 0.975 0.949 0.922 0.077 0.064 0.052 0.040 0.028 100 0.032 0 0 0 0 0.5 0.75 1.25 2.5 5.0 7..5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 0 1.203 1.448 1.844 2.579 3.618 4.382 4.997 5.942 6.619 7.091 7.384 7.496 7.435 7.215 1.323 1.361 1.408 1.437 1.455 1.464 1.458 1.435 1.930 I L.E. radius: 1.630 per cent c T.E. radius: 0.037 per cent c NACA ~015 Basic Thickness Form THEORY OF WING SECTIONS 346 1.2 ke,..oe ~ - * ­ ,D ,r..,. ~ I~ ~oe Lower """ot:e r----... r----. .............. ~ .B !MeA M-008 .4 L..­ ~ o .2 .4 ,I .s- .11 1.0 %,C 'II % (per cent c) (per cent c) (vlV)t fJIV &D./1' 0 0.5 0.75 1.25 2.5 0 0.494 0.596 0.754 1.024 0 0.995 1.058 1.085 1.1OS 0 0.997 1.029 1.042 1.053 4.623 2.175 1.780 1.418 0.982 5.0 1.405 1.692 1.928 2.298 2.572 1.119 1.128 1.134 1.146 1.154 1.058 1.062 1.065 1.071 1.074 0.692 0.560 0.483 0.385 0.321 2.772 2.907 2.981 2.995 2.919 1.160 1.1M 1.168 1.171 1.160 1.077 1.079 1.081 1.082 1.077 0.279 0.246 0.220 0.198 0.178 2.775 2.575 2.331 2.050 1.740 1.143 1.124 1.102 1.079 1.054 1.069 1.060 1.050 1.039 1.027 0.1fi8 0.142 0.126 0.112 0.098 1.412 1.072 0.737 0.423 0.157 1.028 1.000 0.970 0.939 0.908 1.014 1.000 0.985 0.969 0.953 0.085 0.072 0.060 0.047 0.031 0 0.876 0.936 0 7.5 10 15 20 25 30 35 40 45 50 55 60 85 70 75 80 85 90 95 100 L.E. radius: 0.256 per cent c NACA 64-006 Basic ThicJmess Form 347 APPENDIX 1 .6 ( /' ~ ~C', •.04 Up~ $urfoce I ~~ ~ ~IU Lower surlOCtfI ~ ~ r r---.. ~ ~ ~ ~ NACA 84-008 .4 ~ r--o x - .... 10--. .2 I .4 y (V/V)2 I 0 0 II 0.658 0.794 1.005 1.365 1.875 /.0 .8 .6 xjc I--­ V/V fJ.V./V 0.912 1.016 1.084 1.127 0 0.955 1.008 1.041 1.062 3.544 l.994 1.686 1.367 0.969 1.152 1.167 1.179 1.195 1.208 1.073 1.080 1.086 1..093 1.099 0.688 0.560 0.480 0.385 0.323 1.217 1.225 1.230 1.235 45 3.704 3.884 3.979 3.992 3.883 1.103 1.107 1.109 1.111 1.105 0.279 0.246 0.220 0.198 0.176 50 55 60 65 70 3.684 3.411 3.081 2.i04 2.291 1.191 1.163 1.133 1.102 1.069 1.091 1.078 1.050 1.034 0.158 0.141 0.125 0.110 0.096 75 80 85 90 95 1.854 1.404 0.961 0.550 0.206 1.033 0.995 0.957 0.918 0.878 I.Q16 0.997 0.978 0.958 0.937 0.083 0.071 0.059 0.046 0.031 0 0.839 0.916 0 (per cent c) . (per cent c) 0 0.5 0.75 1.25 2.5 5.0 7.5 10 15 20 25 30 35 40 I 2..259 2..574 3.069 3.437 1.220 1.064 I I t 100 I L.E. radius: 0.455 per cent c NACA 64-008 Basic Thickness Form THEORY OF WING SECTIONS 348 /.6 { --V ~ ",.- K ~ ~ c,•. 06 UpptIr surftXVI -.l t--""_ ~ ~ ~ T r--:otJ Lower .urf'DCfI ~ T ~ ~ .8 ~ HACA 1U-IJ09 ~ -- ,...- r-----. ......... o .2 x 'Y (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 5.0 7.5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 .4 %/c II--.. I - - to--" .6 .8 to (V/V)2 V/V ~./V 0 0.739 0.892 1.128 1.533 0 0.872 0.990 1.131 0 0.934 0.995 1.037 1.063 3.130 1.905 1.637 1.340 0.963 2.109 2.543 2.898 3.455 3.868 1.166 1.186 1.200 1.221 1.236 1.080 1.089 1.095 1.105 1.112 0.686 0.560 0.479 0.383 0.323 4.170 4.373 4.479 4.490 4.364 1.246 1.255 1.262 1.267 1.246 1.116 1.120 1.123 1.126 1.116 0.281 0.248 0.221 0.198 0.176 4.136 3.826 3.452 3.026 2.561 1.217 1.. 183 1.149 1.112 1.073 1.103 1.088 1.072 1.055 1.036 0.158 0.140 0.125 0.109 0.095 2.069 1.664 1.069 0.611 0.227 1.033 0.992 0.950 0.907 0.865 1.016 0.996 0.975 0.952 0,930 0 0.822 0.907 0.082 0.070 0.057 0.044 0.030 () 1.075~ L.E. radius: 0.579 per cent c NACA M-009 Basic Thickness Form APPENDIX I I. 6 349 I I ve, -.08 ~ I I surf.:. L .--­ ~ /. .. rti: ~ -~ ~ r--.D9 I -~ ~ I~ L...- ""oc. ~ ~ 8 r-, r-, NACA 64-010 " ~ r - ­ ~ I"'---­ ~ - I'---­ _ () z 5.0 7.5 10 15 20 25 30 ~ I.D .8 .6 1/ (v/l')2 v/l" l1v./V 0 0.820 0.989 1.250 1.701 0 0.834 0.962 1.061 1.130 0 0.913 0.981 1.030 1.063 2.815 1.817 1.586 1.313 0.957 2.343 2.826 3.221 3.842 1.181 1.206 1.221 1.245 1.262 1.087 1.098 1.105 1.116 1.123 0.684 0.559 0.480 0.386 0.325 1.129 1.134 1.138 1.140 1.131 0.280 0.246 0.220 0.199 0.176 (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 ~ 4.302 4.639 4..864: 4.980 4.988 45 4.843 1.275 1.286 1.295 1.300 1.279 50 4.586 4.238 3.820 3.345 2.827 1.241 1.201 1.161 1.120 1.080 1.114 1.096 1.077 1.058 1.039 0.158 O.lS9 0.124 0.109 0.095 2.281 1.722 1.176 0.671 0.248 1.036 0.990 0.944 0.900 0.850 1.018 0.995 0.972 0.949 0.922 0.081 0.069 0.057 0.044 0.030 0 0.805 0.897 0 as 40 55 60 65 70 75 80 85 90 95 100 L.E. radius: 0.720 per cent c NACA 64-010 Basic Thickness Form THEORY OF WING SECTIONS 350 I - -~ 1.2 .8 1 I I ,-J2 Upper .8Urface ~~ r~ ~ ~ ~ ~ ~ 1(/ ~2 Lower .".r«11 r/ ~~ l'. "­ MCA 841-tJ12 t-> - r-­ ~ i'--­ t - - . - ~ """"'-­ ~~ ~ 1.0 .8 (J y x II (V/V)2 »rv 0 0.978 1.179 1.490 2;035 0 0.750 0.885 1.020 1.129 0 0.866 0.941 1.010 1.063 5.0 7.5 10 15 20 2.810 3.394 3.871 4.620 5.173 1.204 1.240 1.264 1.296 1.320 1.097 1.114 1.124 1.139 1.149 0.685 0.559 0.482 0.388 0.328 25 30 35 40 45 5.576 5.844 5.978 5..981 5.798 1.338 1.351 1.362 1.372 1.335 1.156 1.162 1.167 1.171 1.156 0.281 0.247 0.221 0.199 0.177 50 55 60 5.480 5.056 4.548 3.974 3.350 1.289 1.243 1.195 1.144 1.091 1.136 1.115 1.093 1.070 1.044 0.158 0.138 0.122 0.103 0.088 2.695 2.029 1.382 0.786 0.288 1.037 0.981 0.928 0.874 0.825 1.018 0.990 0.963 0.935 0.908 0.074 0.063 0.052 0.045 0.028 0 0.775 0.880 (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 65 70 75 80 85 90 95 100 L.E. radius: 1.040 per ce~t c NACA 041-Oi2 Buic Thickness Form I ~Vtl./V 2.379 1.663 1.508 1.271 0.943 II I 0' .-- ~ ~" APPENDIX I 351 .22 LtlWt!r ."(Dce .2 .4 .6 xjc .8 1.0 I X Y 1---_ (v/l'F I tJ/V -----I-----!-----:­ (per cent c) (per cent c) ~o/l' I o o o o 0.5 0.75 1.208 1.456 1.842 2.528 0.670 0.762 0.896 1.113 0.819 0.873 0.947 IJJ55 1.231 1.109 1.284 15 3.504 4.240 4.842 5.785 20 6.480 1.323 1.375 1.410 1.133 1.150 1.172 1.187 0.670 0.559 0.482 0..389 0.326 25 6.985 7.319 7.482 7.473 7.224 1.454 1.470 1.485 1.426 1.198 1.206 1.213 1.218 1.195 0.285 0.250 0.225 0.202 0.179 1.25 2.5 5.0 7.5 10 30 35 40 45 50 70 6.810 6.266 5.620 4.895 4.113 75 80 3.296 2.472 55 60 65 85 1.677 90 95 0.950 0.346 1.434 1.365 1.300 1.233 1.167 1.101 1.033 0.967 0.902 0.841 0.785 1.168 1.140 1.110 1.080 1.049 1.016 0.983 0.950 0.917 0.886 1.939 1.476 1.354 1.188 0.916 i I I I i I i I " 0.158 0.135 0.121 0.105 0.090 0.078 0.065 0.054 0.041 0.031 100 o 0.730 o -----~-----_...:..-----_I~~._I_ -----1 L.E. radius: 1.590 per cent c NACA 64 r015 Basic Thickness Form THEORY OF WING SECTIONS 352 () .4 y x (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 5.0 7.5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 0 1.428 1.720 2.177 3.005 4.186 5.076 5.803 6.942 7.782 8.391 8.789 8.979 8.952 8.630 (V/V)2 0 0.54.6 0.705 0.862 1.079 I I 1.244 1.327 1.380 1.450 1.497 1.535 1.562 1.585 vii' 1.518 J1fJ./l' 0.739 0.840 0.920 1.039 1.646 1.360 1.269 1.128 0.904 1.115 1.152 1.175 1.204 1.224 0.669 0.558 0.486 0.391 0.331 1.239 1.259 1.265 1.232 0.288 0.255 0.228 0.200 0.177 0 1.250 1.600 1.0 .8 .6 Z/c 8.114 7.445 6.658 5.782 4.842 1.354 1.272 1.190 1.109 1.198 1.164 1.128 1.091 1.053 0.154 0.134 0.117 0.102 0.088 3.866 2.888 1.951 1.101 0.400 1.028 0.952 0.879 0.812 0.747 1.014 0.976 0.937 0.901 0.864 0.074 0.063 0.051 0.039 0.027 0 0.695 0.834 0 1.436 i L.E. radius: 2.208 per cent c NAC.A 64r018 Basic Thickness Form APPENDIX I 353 (~)z .4 -I. 1.0 .8 .6 %,C I I (per ::nt C) (per 0 0.5 0.75 1.25 2.5 5.0 7.5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 ~nt C) I () j 1.646 I \ {V!V)2_ 3.485 0 0.462 C.603 0.759 1.010 4.871 5.915 6.769 8.108 9.095 1.248 1.358 1.431 1.527 1.593 9.807 10.269 10.481 10.431 10.030 1.654 I I 1.985 ! 2.517 I t I 1.681 1.712 1.709 1.607 9.404 8.607 7.678 6.649 5.549 4.416 3.287 2.213 1.245 0.449 0 1.117 1.165 1.196 1.236 1.281 1.297 1.308 1.307 1.268 1.228 0.932 0.851 0.778 0.711 1.010 0.965 0.923 0.882 0.844 0.653 0.808 1.020 1.458 ; 1.274 1.203 1.084 0.878 I i 1.262 1.186 1.143 1.099 1.055 1.112 I I 0.680 0.776 0.871 1.005 ~./V _ _-­ .. 0 1.507 1.307 ! ---- 1.406 1.209 I v/lf" I I I 1 I I i ! 1I II I I I I L.E. radius: 2.884 per cent c NACA 64.-021 Basic Thickness Form 0.665 0.557 0.486 0.395 0.335 0.293 0.259 0.232 0.202 0.178 0.155 0.134 0.116 0.099 0.084 0.071 0.059 0.047 0.036 0.022 0 THEORY OF WING SECTIONS 354 0 l2 (vt ..-I P:-rJf II / "I =.02 Upper surface H-- ~02 Lowersurface ............... .8 ........... ~ NACA 64AOO6 .--~ o .2 y x (per cent c) (per cent c) .4 lO .6 x/c (V/V)2 v/V t.Vo/V 0 0.485 0.585 0.739 1.016 0 1.019 1.046 1.076 1.106 0 1.009 1.023 4.688 2.101 1.798 1.037 1.052 0.980 5.0 7.5 10 15 20 1.399 1.684 1.919 2.283 2.557 1.118 1.126 1.132 1.141 1.149 1.057 1.061 1.064 1.072 0.694 0.564 0.482 0.382 0.321 25 2.757 2.896 2.977 2.999 2.945 1.154 1.158 1.162 1.165 1.156 1.074 1.076 1.078 1.079 1.075 0.278 0.246 0.219 0.197 0.177 70 2.825 2.653 2.438 2.188 1.907 1.142 1.125 1.107 1.087 1.066 1.069 1.061 1.052 1.043 1.032 0.159 0.143 0.126 0.112 0.099 75 80 85 90 95 1.602 1.285 0.967 0.649 0.331 1.043 1.018 0.992 0.964 0.935 1.021 1.009 0.996 0.982 0.967 0.087 0.074 0.061 0.047 0.033 100 0.013 0 0 0 0 0.5 0.75 1.25 2.5 30 35 40 45 50 55 60 65 1.068 L.E. radius; 0.246 per cent c T.E. radius: 0.014 per cent c NACA 64A.OO6 Basic Thickness Form 1.422 355 APPENDIX I / Cl = .046 Upper surface / ~ l2 (vt K J I ~r-- ~ r---. ~ ... '.tJ4.6 Lower sarfoce ............. .8 ~ NACA 64,4008 .4 ~ ,........ -.-., r-­ .......... o .2 .6 .8 LO xle x y (per cent c) (per cent c) 0 0.5 0.75 1.25 0 0.646 0.778 0.983 2.5 1.353 (V/V)2 v/V lw./V 0 0.947 1.005 1.068 1.122 0 0.973 1.033 1.059 3.546 1.972 1.697 1.352 0.971 0.692 0.564 0.481 0.382 0.323 1.002 5.0 1.863 7.5 2.245 2.559 3.047 3.414 1.151 1.165 1.176 1.191 1.201 1.073 1..079 1.084 1.091 1.096 3.681 3.866 3.972 3.998 3.921 1.2\>9 1.217 1.221 1.225 1.211 1.100 1.103 1.105 1.107 1.100 3.757 3.524· 1.191 1.167 1.141 1.113 1.084 1.091 1.080 1.068 1.055 1.041 0.158 0.141 0.125 0.111 0.098 1.026 1.010 0.993 0.975 0.956 0.084 0.072 0.059 0.045 0.032 0 0 10 15 20 25 30 35 40 45 50 55 60 65 3.234 70 2.897 2.521 75 2.117 80 85 1.698 90 95 1.278 0.858 0.438 1.053 1.020 0.987 0.951 0.914 100 0.018 0 L.E. radius: 0.439 percent c T.E. radius: 0.020 per cent c NACA 64AOOS Basic Thickness Form I I I 0.279 0.247 0.221 0.198 0.177 THEORY OF WING SECTIONS 356 L6 /C, 0, = .08 Upper surface Ii 12 r t. ~ '-::=: t;( r - ~loo...- ~ """'iiillIII ~ ~~ ~ t', .08 Lower 6urface ~ NACA 64AOIO A --­ ~- r---- I--.... o ~ .6 .4 t - - - ..... ~ ....-­ .B LO x/c % 11 (V/V)2 vtV &J./V 0 0.804 0.969 1.225 1.688 0 0.868 0.952 1.042 1.130 0 0.932 0.976 1.021 1.063 2.868 1.845 1.603 1.300 0.957 2.327 2.905 3.199 3.813 4.272 1.178 1.201 1.217 1.238 1.254 1.085 1.096 1.103 1.113 1.120 0.280 0.248 0.221 0.199 0.177 (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 5.0 7.5 10 15 20 I I II 0.688 0.562 0.480 0.382 0.324 25 4.606 1.266 30 4.&17 25 4.968 4.995 4.894 1.275 1.282 1.268 1.125 1.129 1.132 1.135 1.126 4.684 4.388 4.021 3.597 3.127 1.240 1.208 1.174 1.139 1.102 1.114 1.099 1.084 1.067 1.050 0.158 0.140 0.124 0.109 0.096 1.063 95 2.623 2.103 1.582 1.062 0.541 0.981 0.938 0.893 1.031 1.011 0.990 0.969 0.945 0.083 0.070 0.058 0.044 0.031 100 0.021 0 0 40 45 50 55 60 65 70 75 80 85 90 1.288 1.023 L.E. radius: 0.687 per cent c T .E. radius: O'()?-3 per cent c I II 0 APPENDIX I l6 , 0 # 357 / c1 =./3 Upper surFQ(Y - ~........ ~ ~~~ ~~~ ~ 1.2. L {~ (vt V .8 ~ ........ ~ "J~ lower surFace ~~ ~ I ~ NACA 64,AO/2 ~ -r--- !--- I'""""""" f'-.-. ~ o .2 .4 .6 x/c z I r--. .--­ ........ L.-­ ~ LO ,._­ i l7/ v y (V/V)2 I 0 0.792 0.893 1.25 0 0.961 1.158 1.464 1.006 2.5 I 2.018 1.127 l.Oti2 0.941 5.0 7.5 10 15 1.201 1.000 1.235 1.111 1.257 1.121 20 2.788 3.364 3.839 4.580 5.132 25 30 5.534 5.809 35 40 5.965 5.993 45 5.863 50 55 60 5.605 (per cent c) (per cent c) 0 0.5 0.75 65 70 75 80 85 5.244 4.801 4.289 3.721 I 1.135 1.144 1.324 1.151 1.156 1.160 1.164 1.152 0.281 0.249 0.221 0.199 0.177 1.135 1.118 1.099 1.079 1.057 0.157 0.139 0.123 0.108 0.094 0.873 1.035 1.011 O.98i 0.962 0.934 0.080 0.068 0.056 0.042 0.029 0 0 0 1.34fi 1.289 1.250 1.207 1.164 1.118 0.974 1.263 0.925 0.644 100 0.025 I.U71 1.023 i 1.254 1.308 3.118 2.500 1.882 90 95 ! 2.408 1.720 1.515 1.2..~ 1.354 1.326 I 0 0.890 0.945 1.003 0.681 0.560 0.478 0.383 0.325 1.336 I i ~l'CI/V L.E. radius: 0.994 per cent c "f.E. radius: 0.028 per eent c NAC.A.. 64 1A012 Basic Thickness Form THEORY OF U'ING SECTIONS 358 , L6 /c, = .2/ Upper surface .-- r----- ~~ ~ ~ ,0 l2 / ~ If J K, (vt '/ .8 ~~ ~ ~ ~ I ~~ ~ ".2/ lower surface r-, NACA 64-2 A O/5 I .4 - r--- r--- ~ r '-- r--- - o .2 - .6 ~~ .4 ------- ,...... ~ LO .8 x/c x 11 (per cent c) (per cent c) (VJV)2 vJV .:1t·a / l' 0.5 0.75 1.25 2.5 0 1.193 1.436 1.815 2.508 0 0.678 0.789 0.936 1.110 0 0.823 0.888 0.967 1.054 1.956 1.552 1.404 1.189 0.912 5.0 7.5 3.477 4.202 1.226 1.280 4.799 5.732 6.423 1.314 1.390 1.107 1.131 1.146 1.166 1.179 0.671 0.552 0.478 0.384 0.326 40 45 6.926 1.270 7.463 7.481 7.313 1.413 1.430 1.445 1.458 1.414 1.189 1.196 1.202 1.207 1.189 0.283 0.249 0.222 0.201 0.177 50 55 60 65 70 6.978 6.517 5.956 5.311 4.600 1.364 1.311 1.255 1.198 1.139 1.168 1.145 1.120 1.095 1.067 0.156 0.137 0.121 0.106 0.091 75 80 1.079 1.020 0.961 0.901 0.843 1.039 1.010 0.980 0.949 0.918 0.078 0.065 0.053 0.041 95 3.847 3.084 2.321 1.558 0.795 100 0.032 0 0 0 0 10 15 20 25 30 35 85 90 .1.360 LE. radius: 1.561 per cent c T.E. radius: 0.037 per cent c NACA 64tA015 Basic Thickness Fonn 0.027 APPENDIX I ~,Cls.OJ .I.e 10 359 I I ............. ~r:L r--:Ol Lower""'or:e .8 I I Lt;per surfoce r----.. r-----... -....... r-. I NACA 65-006 .4 --­ ~ D .4 x y 5.0 7.5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 /.0 .8 (V/V)2 v/V ~Va/V 0 0.476 0.574 0.717 0.956 0 1.044 1.055 1.063 1.081 0 1.022 1.027 1.031 1.040 4.815 2.110 1.780 1.390 0.965 1.310 1.589 1.824 2.197 2.482 1.100 1.112 1.120 1.134 1.143 1.049 1.055 1.058 1.065 1.069 0.695 0.560 0.474 0.381 0.322 2.697 2.852 2.952 2.998 2.983 1.149 1.155 1.159 1.163 1.166 1.072 1.075 1.077 1.078 1.080 0.281 0.247 0.220 0..198 0.178 2.900 2.741 2.518 2.246 1.935 1.165 1.145 1.124 1.100 1.073 1.079 1.070 1.060 1.049 1.036 0.160 0.144 0.128 0.114 0.100 1.594 1.233 0.865 0.510 0.195 1.044 1.013 0.981 0.944 0.902 1.022 1.006 0.990 0.972 0.950 0.086 0.074 0.060 0.046 0.031 0 0.858 0.926 0 (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 .6 L.E. radius: 0.240 per cent c NACA 65-006 Basic Thickness Form f'HEORY OF WING SECTIONS 360 ~ 1 , 1/' -.001 lJp~r sur'a~ 0 1.2 ~ I I """'ilIlIiiii ~IH LtItIWr ~1ICtI ~ "'" r-, .1 NACA 85-008 ·4 ~ ~ - r---........ ..... ~ a .4 x 11 (per cent c) (per cent c) Z/c 1.0 .8 .11 (V/V)2 V/V Avo/V 0 0.627 0.756 0.945 1.267 0 0.978 1.010 1.043 1.086 0 0.989 1.005 1.021 1.042 3.695 2.010 1.696 1.340 0.956 5.0 7.5 10 15 20 1.745 2.118 2.432 2.931 3.312 1.125 1.145 1.158 1.178 1.192 1.061 1.070 1.076 1.085 1.092 0.689 0.560 0.477 0.382 0.323 25 30 35 40 45 3.599 3.805 3.938 3.998 3.974 1.203 1.210 1.217 1.222 1.226 1.097 1.100 1.103 1.105 1.107 0.281 0.248 0.221 0.199 0.178 50 55 60 65 70 3.857 3.638 3.337 2.971 2.553 1.222 1.193 1.163 1.130 1.094 1.105 1.092 1.078 1.063 1.046 0.160 0.145 0.128 0.113 0.098 75 80 85 2.096 1.617 1.131 0.664 0.252 1.055 1.014 0.971 0.873 1.027 1.007 0.985 0.961 0.934 0.084 0.072 0.059 0.044 0.031 0 0.817 0.904 0 0 0.5 0.75 1.25 2.5 90 95 100 (i923 L.E. radius: 0.434 per cent c NACA 65-008 Basic Thickness Form APPENDIX I I.G I /e I -.IM 1.2 361 r I I tP/HIr ",'act1 f C\ ,...-r ~ ~~ I'- ~.D6 I -Low.- .ur'ac. f' V ~ .8 /MeA ts5-009 ~ r---- I000o...... o L--- -- r--- I.... - ~ 1.0 .8 .2 x y (per cent c) (per cent c) "' r-, (vlV)2 vjV I1v.IV 0 0.700 0.845 1.058 1.421 0 0.945 0.985 1.037 1.089 0 0.972 0.992 1.018 1.044 3.270 1.962 1.655 1.315 0.950 5.0 7.5 10 15 20 1.961 2.383 2.736 3.299 3.727 1.134 1.159 1.177 1.200 1.216 1.065 1.077 1.085 1.095 1.103 0.687 0.660 0.477 0.382 0..323 25 30 35 40 4.050 4.282 4.431 4.496 4.469 1.229 1.238 1.246 1.252 1.258 1.109 1.113 1.116 1.119 1.122 0.280 0.248 0.220 0.198 0.178 4.336 4.086 3.743 3.328 2.856 1.250 1.220 1.185 1.145 1.103 1.118 1.105 1.089 1.070 1.050 0.160 0.144 0.128 0.111 0.097 2.342 1.805 1.260 0.738 0.280 1.059 1.013 0.963 0.912 0.856 1.029 1.006 0.981 0.955 0.925 0.084 0.071 0.059 0.044 0.030 0 0.797 0.893 0 0.5 0.75 1.25 2.5 45 50 55 60 65 70 75 80 85 90 ss 100 L.E. radius: 0.552 per cent c NACA 65-009 Basic Thickness Form I 0 THEORY OF WING SECTIONS 362 1.4 1.2 .8 I --­ --­ (/ )\ r t --::: :::-­ ~ \08 I J I / ', •.08 IJpptIr 6Ur'ace , ~ ~ LDWf!r Sur~DC4P ~ ""'--, NACA 65-010 ,--............. - r--­ ­­ .z (J y x .4 --­ ~ .s I"'---­ ~~ .8 /. (V/V)2 v/V l:.VA/V 2.5 0 0.772 0.932 1.169 1.574 0 0.911 0.960 1.025 1.085 0 0.954 0.980 1.012 1.042 2.967 1.911 1.614 1.292 0.932 5.0 7.5 10 15 20 2.177 2.647 3.040 3.666 4.143 1.143 1.177 1.197 1.224 1.242 1.069 1.085 1.094 1.106 1.114 0.679 0.558 0.480 0.383 0.321 25 4.503 4.760 4.924 4.996 4.963 1.257 1.268 1.277 1.284 1.290 1.121 1.126 1.130 1.133 1.136 0.280 0.248 0.222 0.199 0.179 70 4.812 4.530 4.146 3.682 3.156 1.284 1.244 1.202 1.158 1.112 1.133 1.115 1.096 1.076 1.055 0.160 0.141 0..126 0.110 0.097 75 80 85 90 95 2.584 1.987 1.385 0.810 0.306 1.062 1.011 0.958 0.903 0.844 1.031 1.005 0.979 0.950 0.919 0.082 0.070 0.058 0.045 0.030 0 0.781 0.884 0 (per cent c) (per cent c) 0 0.5 0.75 1.25 30 35 40 45 50 55 60 65 100 L.E. radius: 0.687 per cent c NACA 66-010 Basic Thickness Form APPENDIX I It---+--t-----i~ 363 NACA 6~-OI2 .41----+---+-~f--__+_-_+_____f-__+_-_+___t-"'"""1 o .2" x y .4 1.0 .8 .6 X/C (t,/lr)~ v/1T &toll" 0 0.923 1.109 1.387 1.875 0 0.848 0.935 1.000 1.082 0 0.921 0.967 ].000 1.040 2.444 1.776 1.465 1.200 0.931 5.0 7.5 10 15 20 2.606 3.172 3.647 4.402 4.975 1.162 1.201 1.232 1.268 1.295 25 30 35 40 5.406 1.316 5.;16 1.332 5.912 5.997 5.949 1.343 1.350 1.357 (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 45 50 55 iI I 1.078 1.096 1.110 1.126 1.1.38 1.147 1.154 1.159 1.162 1.165 0.282 0.251 0.223 0.204 0.188 1.159 1.138 1.115 60 65 70 3.743 1.295 1.243 1.188 1.134 75 80 85 90 95 3.059 2.345 1.630 0.947 0.356 1.073 1.010 0.949 0.884 0.819 1.036 1.005 0.974 0.940 0.905 0 0.748 0.865 100 I 0.702 0.568 0.480 0.389 0.326 5.757 5.412 4.943 4.381 1.343 1 1.065 0.169 0.145 0.127 0.111 0.094 0.074 0.062 0.049 0.038 0.025 1.090 L.E. radius: 1.000 per cent c NACA. 65 1-012 Basic Thickness Form I 0 THEORY OF WING SECTIONS 364 c, •. 22 ~ surl'oce o .2 x 'Y 8 6 1.0 (V/V)2 V/V Av./l 0 1.124 1.356 1.702 2.324 0 0.654 0.817 0.939 1.063 0 0.809 0.904 0.969 1.031 2.038 1.729 1.390 1.156 0.920 5.0 7.5 10 15 20 3.245 3.959 4.555 5.504 6.223 1.184 1.241 1.281 1.336 1.374 1.088 1.114 1.132 1.156 1.172 0.682 0.563 0.487 0.393 0.334 25 6.764 7.152 7.396 7.498 7.427 1.397 1.4:18 1.438 1.452 1.464 1.182 1.191 1.199 1.205 1.210 0.290 0.255 0.227 0.203 0.184 7.168 6.720 6.118 5.403 4.600 1.433 1.369 1.297 1.228 1.151 1.197 1.170 1.139 1.108 1.073 0.160 0.143 0.127 0.109 0.096 3.744 2.858 1.977 1.144 0.428 1.077 1.002 0.924 0.846 0.773 1.038 1.001 0.961 0.920 0.879 0.078 0.068 0.052 0.038 0.026 0 0.697 0.835 0 (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 L.E. radius: 1.505 per cent c X ACA. 65r015 Basic Thickness Form Y APPENDIX I (J .4 x JI (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 0 1.337 1.608 2.014 2.751 365 .s z/c 14 (v/V)~ v/1T j.r:./V 0 0.625 0.702 0.817 1..020 0 0.i91 0.838 0.. 904 1..010 1.746 1.437 1.302 1.123 0.858 5.0 7.5 10 15 20 3..866 4.733 5.457 6.606 7.476 1.192 1.275 1.329 1.402 1.452 1..092 1.129 1.153 1.184 1.205 0.650 0.542 0.474 25 1.488 1.515 1.539 1.. 561 1.578 1.220 45 8.129 8.595 8.886 8.999 8.901 0.285 0.251 0.225 0.203 0.182 50 8..568 55 1.526 1.440 1.353 1.262 1.235 1.200 65 70 8.008 7.267 6.395 5.426 75 4.396 80 30 35 40 60 85 90 95 100 1.170 0.. 385 0.327 1.231 1:241 1.249 1.256 0.157 0.137 0.118 0.104 1.163 1.123 1.082 3.338 2.295 1.319 0.490 1.076 0.985 0.896 0.813 0.730 1.037 0.992 0 0.657 0.811 0.087 0.074 0.062 0.050 0.039 0.026 0.947 0.902 0.854 L.E. radius: 1.96 per cent c NACA 65r 0 18 Basic Thickness Form I 0 THEORY OF ,"lING SECTIONS 366 2.0 1.2 .8 »: 'I I ...--- IV ~ x I ---- [\ /~ ~ ~ r--"\ -: or / v / / \.J/4 I IV a I I \ . / 1 :1 -.-I/. ~ svr'oce I 1.6 I I r: <, LDWtlr ~ '" ~~~ NACA 65. -OZI - --- ---.r--- -- --- ~ y "~ $urf"~ <, {U/V)2 l---' ~ L----- 1.0 .8 f'/l~ (~r~nt~ 1_(p_e_r_~_n_t_~_I_~~~_~~~~~I!~~~~1 o o o o 0.5 0.75 1.25 2.5 1.522 1.838 2..301 3.154 0.514 0.607 0.740 0.960 0.. 717 0.779 0.860 0.980 5.0 4.472 5.498 6.352 7.700 8.720 1.186 1.293 1..371 1..469 1.533 1.137 1.171 1.212 9.487 10.036 10.375 10.499 10.366 1.580 1..621 9.952 9.277 8.390 7.360 6.224 1.633 5.024. 1.073 3.800 2.598 1.484 0.546 0.970 0.872 0.778 0.694 o 0.616 7..5 10 15 20 25 30 35 40 45 50. 55 60 65 70 75 80 85 90 95 100 1.654 1.680 1..700 1.508 1.397 1.286 1.177 1.531 1.333 1.215 1.062 0.838 1.238 0.649 0.544 0.478 0.388 0.330 1.257 1.273 1.286 1.296 1.304 0.289 0.255 0.229 0.206 0.184 1.278 1.228 1.182 1.134 1.085 0.158 0.139 0.120 0.101 0.087 1.036 0.985 0.882 0.833 0.073 0.058 tl.047 0.035 0.020 0.785 o 1.089 0.934 L.E. radius: 2.50 per cent c NACA 65.-021 Basie-Thickness Form APPENDIX I 367 1.6 0 I L2. Cl / =.01 Upper surface ,!, ~ '""""IIIIl ~- r----. __ r-, "a/lower surface I .8 NACA6SAOO6 ~ r--.. .4- () LO .8 .6 x/c % Y ! (per cent c). (per cent c) I (v/1T )2 ! v/V 0 0 0 0.5 0.75 1.25 2.5 0.718 0.981 5.0 1.313 7.5 1.591 10 15 20 25 0..164 O.5ft3 40 45 .2.992 50 2.925 2.793 2.602 2.364 2.087 35 55 60 65 70 75 80 85 1.775 1.029 1.039 1.139 1.145 1.149 1.153 1.157 1.159 1.070 1.072 1.074 1.076 1.077 1.157 1.141 1.07(; 1.131 - II I 4.879 2.145 1.763 1.365 0.966 0.688 0.562 0.480 0.382 0.323 I t I ! ii ! 0.278 0.246 0.219 0.198 0.178 o.isu I 1.124 1.()(i8 1.06U ,I 1.106 1.083 1.052 1.041 0.143 0.127 0.112 0.099 i 1.059 1.032 0.973 0.936 1.029 1.016 1.001 0.986 0.967 0.076 0.061 0.047 0.033 0 0 0 I 95 1.437 l.OS3 0.727 0.370 100 0.013 I 90 0 1.017 1.021 1.049 1.055 1.058 1.063 1.067 1.101 1.112 1.120 1.824 2.194 2.474 2.687 2.842 2.945 2.996 30 1.034 1.043 1.058 1.080 4v./V I -~_._--- i 1.003 L.E. radius: 0.229 per cent c T.E. radius: 0.014 per cent c NACA 65AOO6 Basic Thickness Form OJl87 THEORY OF WING SECTIONS 368 L6 0\ ,'" cl-.05 Upper surface I T ......... .8 t?~5 lower surface r ~ ......... ............... r-, WACA65AOO8 A --- ~ r---- ............ o .4 I I .6 lO ~/c i % Y (per cent c) (per cent c) 0 0.5 0.75 I---... (V/V}2 v/V I Av./V ! 0 0.973 1.001 1.038 1.088 0 0.986 2.5 0 0.615 0.746 0.951 1.303 5.0 7.5 10 15 20 1.749 2.120 2.432 2.926 3.301 1.127 1.145 1.157 1.175 1.186 1.062 1.070 1.076 1.084 1.089 25 3.585 3.791 3.928 3.995 3.988 1.195 1.202 1.207 1.213 1.217 1.093 1.096 1.099 1.101 1.103 0.279 0.247 0.219 0.198 0.178 3.895 3.714 3.456 3.135 2.763 1.214 1.191 1.167 1.108 1.102 1.091 1.080 1.067 1.053 0.161 0.144 0.128 0.112 0.098 95 2.348 1.898 1.430 0.960 0.489 1.076 1.041 1.002 0.961 0.916 1.001 0.980 0.957 100 0.018 0 0 1.25 30 35 40 45 50 55 60 65 70 75 80 85 90 i.iss 1.000 ~ 1.019 1.043 1.037 1.020 L.E. radius: 0.408 per cent c T.E. radius: 0.020 per cent c i I I 3.698 2.010 1.693 1.333 0.954 0.685 0.561 0.479 0.382 0.322 0.086 0.073 0.060 0.046 C NACA 6SAOOS Basic 'rhickness Form APPENDIX I 369 L6 ,e, = ./0 Upper surface I ~ 0 , 0.V- r[ ~ .-. """"""'"' ~ ~ ~ ....... "'lIIIIIIlI "'JO lower surf"ace ~ INAcA 6SA010 - ~~ ~ t--... o % .2 .4 x/c II y (per cent c) (per cent c) o 0 0.5 0.75 1.25 2.5 0.765 0.928 1.183 1.623 5.0 7.5 10 15 20 2.182 2.650 3.040 3.658 4.127 25 35 40 4.483 4.742 4.912 4.995 45 4.983 50 55 4.863 4.632 4.304 3.899 3.432 30 60 65 70 90 95 2.912 2.352 1.771 1.188 0.604 100 0.021 75 80 85 I I I I I ,I I Ii I i II II I I I I r--- ~- - .6 .8 (V/V)2 v/V 0 0.897 0.948 1.010 1.089 0 0.947 0.974 1.148 1.176 1.194 1.218 1.234 1.071 1.247 1.257 1.265 1.272 1.277 1.117 1.121 1.125 1.128 1.130 1.271 1.241 1.208 1.127 1.114 1.099 1.083 1.172 J.O , Av./V I 2.987 1.878 1.619 1.303 0.936 1.084 1.093 I 1.104 1.111 I 0.679 0.559 0.478 0.382 0.323 I I 1.005 1.044 1.133 1.064 1.091 0.999 0.949 0.893 1.045 1.023 0.999 0.974 0.945 0 0 1.047 ~~ L.E. radius: 0.639 per cent c T.E. radius: 0.023 per cent c NACA 65AOIO Basic Thickness Form I ! II 1 I 0.281 0.249 0.222 0.198 0.178 0.161 0.144 0.127 0.111 0.097 0.084 0.071 0.058 OJM5 0.029 0 THEORY OF WING SECTIONS 370 L6 u .-I C, ·./S Upper surface - <. . (/ ,...O~ r( ~ ~ ~ ~ ~~ ~ .8 I ~ .... ~ " ./5 lower surface -, NACA6~AOI2 ~ r'---- o ~ .2 --- -------- ---------- - I--""""'" .6 A 1.0 .8 -x/a I, y (V/V)2 v/V AVa/I' 0 0.913 1.106 1.414 1.942 0 0.824 0.883 0.969 1.081 0 0.908 0.940 0.984 1.040 2.520 1.757 1.543 5.0 7.5 10 15 20 2.614 3.176 3.647 4.392 4.956 1.166 1.204 1.228 1.263 25 30 5.383 5.693 5.897 5.995 5.977 % (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 i.ses 0.914 1.285 1.097 1.108 1.124 1.134 0.672 0.557 0.477 0.382 0.324 1.301 1.313 1.324 1.332 1.338 1.141 1.146 1.151 1.154 1.157 0.281 0.250 0.224 0.198 0.178 5.828 5.544 5.143 4.654 4.091 1.329 1.292 1.251 1.156 1.153 1.137 1.118 1.097 1.075 0.161 0.143 0.126 0.111 0.096 90 95 3.467 2.798 2.106 1.413 0.719 1.104 1.051 0.994 0.936 0.871 1.051 1.025 0.997 0.967 0.933 0.082 0.069 0.057 0.043 0.027 100 0.025 0 0 0 35 40 45 50 55 60 65 70 75 80 85 1.204 1.080 L.E. radius: 0.922 per cent c 1'.E. radius: 0.029 per cent c NACA 65 1A012 Basic Thickness Form APPENDIX I _/~ Cl L6 J,: r ~ ~ -r-, ~ l,...---­ ~ 7 i> .4 ~~ "" ~ 1A fl ~ -r--- r---. L-.­ ~ .2 ~ 11 0 1.131 1..371 1.750 2.412 r---- "'"-­ ~ ~ lO .8 x/c (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 ~ WACA 652 .40 15 ~,.... o ~ ' ... .22 lower surrac« [,....­ ~ I"----........ % =.22Uppersurface '"--. / .8 371 (v/V)% vlV &J./V 0 0.714 0.781 0.891 1.059 0 0,,845 0.884 0.944 1.029 2.048 1.586 1.417 1.195 0.880 1.089 1.115 1.131 1.152 1.166 0.660 0.553 0.476 0.282 0.252 0.227 5.0 7.5 10 15 20 3.255 3.962 4.553 1.187 1.243 5.488 1.328 1.359 25 45 6.734 7.122 7.376 7.496 7.467 1.401 1.416 1.427 1.437 1.176 1.184 1.190 1.195 1.199 50 55 60 65 70 7.269 6..903 6.393 5.772 5.063 1.419 1.368 1.311 1.249 1.186 1.191 1.170 1.145 1.118 1.089 0.161 0.142 0.124 0.109 75 4.282 3.451 2.598 95 0.887 1.123 1..056 0.986 0.913 0.841 1.060 80 85 90 1.028 0.993 0.956 0.917 0.080 0.067 0.055 0.041 0.026 100 0.032 0 0 0 30 35 40 6.198 1.743 1.280 1.383 0.382 0.326 0.204­ 0.181 0,(194 L.E. radius: 1.446 per cent c T.E. radius: 0.038 per cent c ....,.. • ,.... ,. . . . n . . . " ' .. ~ _ ~ :_1 'T." _ THEORY OF WING SECTIONS 372 /.6 /C'i 01 1.2 uprr Surface JO ~- --~, / Lower .urface ~'" .8 NACA 68-006 .4 --­ -~ o .2 % 11 (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 5.0 7.5 10 15 20 , 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 tOO I I I I .4 X/C (V/V)2 1.0 8 6 v/IT Av./V 0 0.461 0.554 0.693 0.918 0 1.052 1.057 1.062 1.071 0 1.257 1.524 1.752 2.119 2.401 1.086 1.098 1.107 1.119 1.128 1.042 2.618 2.782 2.899 2.971 3.000 1.133 1.138 1.142 1.145 1.148 1.064 1.067 1.069 1.070 1.071 2.985 2.925 2.815 2.611 2.316 1.151 1.153 1.155 1.154 1.118 1.073 1.074 1.075 1.074 1.057 0.161 0.145 0.130 0.116 0.102 1.953 1.543 1.107 0.665 0.262 1.081 1.040 0.996 0.948 0.890 1.040 1.020 0.998 0.974 0.943 0.089 0.075 0.061 0.047 0.030 0 0.822 0.907 0 1.026 1.028 1.031 1.035 I I I 1.048 1.052 1.058 1.062 L.E. radius: 0.223 per cent c NACA 66-006lJasic Thickness Form I I I ! / I 4.941 2.500 2.020 1.500 0.967 0.695 0.554 0.474 0.379 0.320 0.278 0.245 0.219 0.197 0.178 373 APPENDIX I '41'!1 ~~,. /0 (;)' I(~ ~ 1~03 LO~ 03 LPPer surfoce .s: """lIIIlIil ~~ -= "' <, MACA 66-D08 -- l...-­ r--- ........ .4 % 11 X/C r-­ ~ ..... .~ 1.0 j {V/V)2 »rv 0 0.610 0.735 0.919 1.219 0 0.968 0 0.984 1.011 1.023 1.038 6.0 7.5 10 15 20 1.673 2.031 2.335 2.826 3.201 1.107 1.128 1.141 1.1SS 1.171 1.052 1.062 1.076 1.082 0.552 0.474 0.379 0.321 25 3.490 30 36 3.709 3.865 40 3.962 46 4.000 1.178 LI86 1.191 1.196 1.201 1.085 1.089 1.091 1.094 1.096 0.278 0.246 0.220 0.198 0.178 1.098 0.161 0.145 0.130 0.115 0.101 (per cent c) (per eent e) 0 0.5 0.75 1.25 2.5 1.023 1.046 1.078 1.068 liO 3.978 55 3.896 80 3.740 3.459 1.205 1.208 1.213 1.202 3.062 1.156 1.075 2.574 2.027 1.447 0.864 0.338 1.103 1.048 0.989 0.926 0.855 1.050 1.024 0.994 0.962 0.925 0 0.768 0.876 6S 70 75 80 85 90 IS 100 i Av./J-· 3.794 2.220 1.825 1.388 0.949 I I 1.099 1.101 1.096 0.689 0.087 I -_1--- ______ L.E. radius: 0.411 per cent c NACA 66-008 aasic Thickness Form 0.073 0.058 0.045 0.029 0 THEORY OF WING SECTIONS 374 zo f.Z ¥,C, •• OS ttJper eurface I' ~ (t)2 ~ V .8 ~ I ~~ ~05 tppM­ .ur'~ ~ "" r-, f44C4 66-oot/ .4 -r---- l"...--' - r-­ '""--­ a .4 x y ~/C ~ --­---­ .8 .6 1.0 wv» fJ/V ~./V 0 0.687 0.824 1.030 1.368 0 0.930 0.999 1.036 1.079 0 0.964 0.999 1.018 1.039 3.352 2.100 1.750 1.340 0.940 5.0 7.5 10 15' 20 1.880 2.283 2.626 3.178 3.601 1.119 1.142 1.159 1.178 1.190 1.058 1.069 1.077 1.085 1.091 0.686 0.552 0.473 0.379 0.323 2S 30 35 40 45 3.927 4.173 4.348 4.457 4.499 1.201 1.210 1.217 1.221 1.228 1.096 1.100 1.103 1.105 1.1OS 0.280 0.246 0.220 0.197 0.178 50 4.475 4.381 4.204 3.882 3.428 1.232 1.237 1.240 1.230 1.172 1.110 1.112 1.114 1.109 1.083 0.161 0.145 0.130 0.116 0.100 2..877 2.263 1.611 0.961 0.374 1.113 1.050 0.985 0.915 0.839 1.055 1.025 0.992 0.957 0.916 0.085 0.071 0.057 0.043 0.028 0 0.747 0.864 0 (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 55 60 65 70 75 80 85 90 95 100 L.E. radius: 0.530 per cent c NACA 66-009 Basic Thickness Form APPENDIX 1 . .........'-~ I. ~ I. I" I 375 I I /~, -.07 ~ Arf~ ~ ~ K -­I I r-.o71.-wr " " c . I ~ f'~ '" r-,"" ,~ "-.­ M-OJ(} .4 L.o-- ....-­ r--- r--.. ,........ .-- I - - r---- ......... I.D .B % 11 (per cent c) (per cent.c) (rJ/V)2 v/IT ~.IV 0 0.5 0.75 1.25 2.5 0 0.759 0.913 1..141 1.616 0 0.896 0.972 1.023 1.078 0 0.947 0.986 1.011 1.038 3.002 2.012 5.0 7.5 2.087 1.125 1.154 1.174 1.198 1.215 1.061 1.074 1.084: 1.095 1.102 0.682 0.551 0.473 0.379 0.322 4.363 1.226 4.6.36 1.236 1.243 1.249 1.255 1.107 1.112 1.115 1.118 1.120 0.279 0.246 0.220 0.198 0.178 1.261 1.265 1.270 1.250 1.190 1.123 1.125 1.127 1.118 1.091 0.161 0.146 0.130 0.114 0.099 to IS 20 25 30 35 40 45 SO 55 60 65 70 75 80 85 90 95 100 2.536 2.917 3.630 4.001 4.&12 4.953 S.OOO 4.971 4.865 4.665 4.302 3.787 1.686 1.296 0.931 3.176 2.494: 1.773 1.054 0.408 1.121 1.059 1.052 1.026 0.979 0.904 0.821 0.989 0.951 0.906 0.085 0.070 0.056 0.043 0.027 0 0.729 0.854 0 L.E. radius: 0.662 per cent c NACA 66--010 Basic Thickness Form THEORY OF WING 376 I. 6 I. 2 8ECTIONl~ I I I _/~, -./Z ~psr l~urFac. --­r: ---,(J IF ~ V< 'r--:IZ ILow."I I ~ -~ ,~ :wr,~ ~ )1I ~ '" /MCA tiIia-O/2 ~ - ~ ~ t---..L... o -4 x (per cent c) 11 5.0 7.5 10 15 20 I I 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 I t--..... I xc 1.0 .8 .6 (V/V)2 V/V s1v./V 0 0.906 1.087 1.358 1.808 0 0.800 0.915 0.980 1.078 0 0.894 0.957 0.990 1.036 2.569 1.847 1.575 1.237 0.913 2.496 3.037 3.496 4:.234 4.801 1.138 1.177 1.204 1.237 1.259 1.067 1.097 1.112 1.122 0.674 0.549 0.473 0.380 0.323 5.238 5.568 5.803 5.947 6.000 1.275 1.287 1.297 1.303 1.311 1.129 1.134 1.139 1.142 1.145 5.965 5.836 5.588 5.139 4.515 1.318 1.323 1.331 1.302 1.221 1.148 1.150 1.154 1.141 1.105 3.767 2.944 2.083 1.234 0.474 1.139 1.053 0.968 0.879 0.788 1.067 1.026 0.984­ 0.938 0.888 0 0.687 0.829 (per 0 0.5 0.75 1.25 2.5 I'--- r--- _I----­ ~ ~ cent c) 1.085 f ! i 0.280 0.246 0.221 0.197 0.176 0.162 0.147 0.132 0.113 0.098 0.084 0.069 0.053 0.040 0.031 I 100 L.E. radius: 0.952 per cent. c NACA 66t-012 Basic Thickness Form I 0 APPENDIX I 377 -., /.~ M l,...-C",•• ' I./pp#Ir :!IU"fOC. 1 ~ 1 ~ 1.1 - 0/ ~J ( I 21#:Per~ ~II NACA ~ ~~ ~~ I , ,~ -r/, i-> --"- 661 -0 15 r--- to--- 1---- t--- o ~ .2 I .4 z c) (per cent Y c) (per cent i X/C .6 ---- ~ r--- "--~ ~ 1.0 .8 I ("'IF): to/I' 0 0.760 0.840 0.929 1.055 0 2.139 0.872 0.916 0.9641.027 1.652 j ..1"./1' 0 0.5 0.75 1.25 2.5 0 1.122 1.343 1.675 2.235 ~.O 7.5 10 15 3.100 3.781 4.358 5.286 20 ~.995 1.288 1.317 1.078 1.099 1.114 1.134 1.148 25 6.543 6.956 7.250 7.430 7.495 1.340 1.356 1.370 1.380 1.391 1.158 1.164 1.170 1.175 1.179 7..450 7.283 6.959 6.372 5.576 1.401 1.411 1.420 1.367 1.260 1.184 1.188 1.192 1.169 1.122 4.632 3.598 2.530 1..489 0.566 1.156 1.053 0.949 0.841 0.744 1.075 0.974 0.920 0.863 0.080 0.065 0.051 0.039 0.025 0.639 0.799 0 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 1.163 1.208 1.242 0 ~~---_ ... -_._......-- ..... _- ­ 1.026 L.E. radius: 1.435 per cent c N ACA 66::-015 Basic Thickness Form j I I I I I ! t I i 1.431 1.172 0.895 0.663 0.047 0.473 0.381 0.322 0.280 0.248 0.222 0..200 0.180 0.163 0.146 0.131 0.113 0.096 7.'HBORY OF WINO .SECTIONS 378 % 11 (per oeot c) (per cent c) (fJ!V)I fJ/V MJ./v 0 0.806 0.857 1.773 1.456 1.312 1.121 0 0.5 0.15 1.25 0 1.323 1.571 1.952 2.5 2.646 1.0 7.5 10 3.690 4:.613 5.210 16 20 8.333 7.las 1.393 1.134: 1.182 1.1SO 25 30 3& 40 4& 7.848 8.348 8.101 8.918 8.998 1.423 1.193 1.4:45 1.202 1.484 1.481 1.210 1.217 1.223 SO IS 60 8.942 8.733 8.323 65 70 7.580 6.197 1.S09 1.522 1.534 1.438 1.302 75 80 5.4:51 8S 90 95 100 0 0.650 0.735 0..850 1.005 1.114 1.234 1.285 1.350 1.496 0.897 1.002 1.074 1.111 1.228 1.234 1.238 LI99 1.141 0..858 0.649 0.645 0.472 0.381 0.323 0.282 0.250 0.223 0.201 0.181 0.163 0.147 0.131 0.114 0.095 0.848 1.172 1..046 0.922 0.803 0.692 0.896 0.832 0.077 0.061 0.048 0.037 0..022 0 0.S87 0.766 0 4.206 2.934 1.714 1.083 1.022 0.9S0 L.E. radius: 1.955 per cent c NACA 66r018 Basic Thickness Form APPENDIX I .4 z JI (per cent c) (per cent c) 0 0 0.5 0.75 1.25 2.5 1.525 1.804 5.0 7.5 10 15 20 25 30 35 40 45 SO 55 60 65 70 75 2.240 3.045 4.269 Z/c (V/V)J 0 0.580 0.635 0.755 0.952 0 0.~1 0.797 0.869 0.976 &1.1 V 1.547 1.314 1.218 1.054 0.828 6.052 7.369 8.376 1.148 1.185 1.208 9.153 9.738 10.1M 10.407 10.500 1.499 1.528 1.551 1.574 1.594 1.224 1.236 1.245 1.255 1.263 .0.251 0.224 0.202 0.183 1.611 1.629 1.648 1.269 1.276 1.284 10.434 10.186 9.692 8.793 7.610 6.251 4.796 3.324 1.069 1.116 0.283 1.508 1.228 1.335 1.155 0.165 0.148 0.132 0.114 0.093 1.084 0.073 1.015 0.944 0.873 0.805 0.058 0.046 0.034 0.020 0.734 0 90 1.924 0.717 0 0.539 100 _IV 0.635 0.542 0.472 0.381 0.324 95 85 1.0 .• 1.143 1.2-&6 1.318 1.405 1.459 5.233 1.176 1.031 0.891 0.763 0.648 80 379 L.E. radius: 2.550 per cent c NACA 66r021 Basic Thickness Form THEORY OF WING 380 ~ECTIONS 2.D I /e, • .J2 1.6 1.2 (v)' 8 (/ e­ t. / ~ ~ I---~ t=-­ ~ - ~ ~ ~ ,~ /MeA t-> I svrFac» i"'"-./2 LorNtN'" .,ftl~ r/ I .4 o ~ I Uppttr -----­ --­ .z x '!J (per cent c) (per cent c) -, 67,1-015 r---~ ., x/c' (V/V)2 -----.. r--..... ~ ~ /.0 .8 6 »rv ~G/V I 0 0.5 0.75 1.25 2.5 5.0 7.5 10 15 ~ 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 I 0 1.167 1.394 1.764 2.395 0 0.650 0.970 1.059 0 0.806 0.985 1.029 1.140 1.068 2.042 1.560 1.370 1.152 0.906 3.245 3.900 4.433 5.283 5.940 1.209 1.100 1.113 1.122 1.134 1.142 0.667 0.548 0.470 0.370 0.312 6.404 6.854 7.155 7.359 7.475 1.318 1.330 1.341 1.851 1.360 1.148 1.153 1.158 1.162 1.166 0.276 0.248 0.221 0.201 0.180 7.497 7.421 7.231 6.905 6.402 1.368 1.375 1.381 1.388 1.390 1.170 1.173 1.175 1.178 1.179 0.160 0.142 0.124 0.111 0.108 5.621 4.540 3.327 2.021 0.788 1.321 1.176 1.018 0.864 0.712 1.149 1.084 0.930 0.844 0.094 0..071 0.060 0..045 0.025 0 0.570 0.755 0 1.239 1.259 1.285 1.304 1.009 L.E.. radius: 1.523 per cent c NACA 67,1-015 Basic Thickness Form II APPENDIX I 2.0 381 r--....,.-...,...-.....--r--.......­ ....... -....-~ ~l ............_ .... I aurtlce =.22 Upper ,r-oIo---+--I-L---­ ......K~ •8 ~ II'( .. . . . I .IJ--;---t----ir--f--+---+--+--J----+_-I ----I--­ '-........r--­ r----I-_.... _ _I---,-­ .2 .4 xlc. 6 ~ .8 1.0 j % 11 i (per cent c) (per cent r) i - - - - - -.._ -­ ~_._.--- (I'll')"! --.. -"--~"--- ~t~CI/l! 1,'/1" - _.._ ­ .. --- --- () 0 () 0.5 0.75 1.199 1.25 1.801 2.462 0.660 0.799 0.942 1.100 0 0.812 0.894 0.971 1.049 2.028 1.680 1.560 1.325 0.990 6.384 1.201 1.259 1.295 1.339 1.369 1.096 1.122 1.138 1.156 1.170 0.695 0.551 0.465 0.383 0.324 6.898 7.253 7.454 7.494 7.316 1.390 1.409 1.42"& 1.435 1.391 1.179 1.187 1.193 1.198 1.179 0.283 0.252 0.224 0.199 0.176 7.003 6.584 6.064 5.449 4.738 1.348 1.306 1.265 1.221 1.178 1.161 1.143 1.125 1.105 1.085 0.156 0.138 0.122 0.108 3.921 3.020 2.086 1.193 0.443 1.115 1.027 0.938 0.852 0.774 1.056 1.013 0.969 0.923 0.880 0 0.703 0.838 1.435 2.5 5.0 7.5 10 15 20 25 30 35 40 3.419 4.143 4.743 5.684 I I I 45 50 55 60 65 70 75 80 85 90 95 100 I i ! i I t I I I ---_._- I I 0.093 0.079 0.065 0.052 O.MO II L.E. radius: 1.544 per cent. c NACA 747A015 Basic Thickness Form 0.028 0.018 APPENDIX II MEAN LINES CONTENTS NACA Designation 62 63 . . . . . Page 383 384 64 65 66 385 386 387 388 389 67 210 . 220 . 390 230 . 240 250 . a91 . 392 393 394: 4==0 4 == 0.1 4 == 0.2 a == 0.3 a -= 0.4 a::s= 0.5 4 == 0.6 a == 0.7 a == 0.8 . . . . . . .... a := 0.8 (modified for use with N ACA 6A-eeries sections) a -= 0.9 a -= 1.0 . 382 395 396 397 398 399 400 401 4<rJ . 403 404 405 APPENDIX II )~ I.0 1 I I~ 383 ~ i""---. r--­ r--- r--..... -.... .... r-, NACA 62 f1ItJt:1nJine ~ .2 D Cl. - ...-­ == 0.90 .4 (Xi == 2.81 x/c 0 ~ Ca e/ . r - - to--... 1.0 .8 .6 =:I - 0.113 I % I/e (per cent c) (per cent c) 0 0 0.726 1,.406 2.625 3.656 0.60000 0.56250 0.52500 0.45000 0.37500 0 0.682 1.031 1.314 1.503 0 0.171 0.258 0.328 0.376 10 4.500 15 5.625 1.651 1.802 20 25 6.000 5.977 5.906 0.30000 0.15000 0 - 0.00938 - 0.01875 1.113 0.413 0.451 0.383 0.318 0.279 5.625 5.156 4.500 3.656 2.625 - 0.03750 0.05625 0.07500 0.09375 0.11250 0.951 0.843 0.741 0.635 0.525 0.238 0.211 0.185 0.159 0.131 90 1.406 95 100 0.727 0 - 0.13125 - 0.14062 - 0.15000 0.377 0.261 0 1.25 2.5 5.0 7.5 30 40 50 60 70 80 I dyc/d.c Pa t:J.vlV '"'" P./4 1.530 1.273 Data for NACA Mean Line 62 I I 0.094 0.065 0 THEORY all It·ING SECTIONS 384 1.0 o I v- -........... <, r--­ ~ ---­ ----r---_ ... ~ NACA 63 ",.", line ~ ~ o .2 e'i x (pe~~ent c) I = .4 0.80 ai '/ X,C = 1.60° --- t--- r--­ .8 .6 Ca e/ 4 ~ 1.0 = - 0.134 I y. PR dYe/dx (per cent c) , .It'jV = PR/4 ---~~~.- 0 0.489 0.958 1.833 2.625 0.40000 0.38333 0.36667 0.33333 0.30000 10 15 3.333 20 5.333 5.833 6.000 0.26667 0.20000 0.13333 0.06667 0 () 1.25 2.5 5.0 7.5 4.500 25 30 50 60 70 80 5.878 5.510 4.898 4.041 2.939 - 00 95 100 1".592 0.827 0 - 0.14694 - 0.15918 - 0.17143 40 I I I I 1 0.02449 0.04898 . 0.07347 0.09796 0.12245 0 0.389 0.553 0.788 0.940 0 0.097 0.138 0.197 0.235 , 1.259 1.233 1.160 0.267 0.305 0.315 0.308 0.290 0.949 0.850 0.762 0.673 0.560 0.237 0.213 0.191 0.168 0.140 0.406 0.291 0 0.102 0.073 1.066 1.220 i I I I i I i Data for NACA Mean Line 63 0 385 APPENDIX II 2.0 to / -- V ----r--­ r--­ --........... ............ ( (J r-, NACA 64 mean line ---- --- - ~ II .2 e'i = 0.76 X ( per cent c o ) .4 ai' = 0.74° I(per cent Yc ) c -------1------ ------- I I d!l</ti.r I I 0 1.25 0.369 2.5 0.726 5.0 1.406 7.52.039 I 10 15 20 25 30 2.625 3.656 4.500 5.156 5.625 40 50 60 70 80 6.000 5.833 5.333 4.500 3.333 90 1.833 0.958 I - o J 95 100 I %C I Data for ­ - .6 C.r/ t ------ =- I! pR I--. 1.0 0.157 ,j--~V-/-V-=-P-R/--4 i --.-- ----- ---.------ ;------_._­ O.3()()OO 0 . 0 0.29062 0.28126 0.26250 0.24375 0.257 0.391 0.546 0.668 _ I 0.064 0.098 0.137 0.167 0.22500 0.18750 0.15000 0.11250 0.07500 O.i48 0.871 0.966 1.030 1.040 I 0 0.03333 0.06667 0.10000 0.13333 0.999 0.910 0.827 0.750 0.635 0.16667 0.18333 0.20000 0.466 0.334 0 X.~C.:\ l\IC1Ul Line 64 'I I I I 1 0.187 0.218 0.242 0.258 0.260 0.250 0.. 228 0 ..207 0.188 0.159 0.117 0.084 0 THEORY OF WING SECTIONS 386 3D 2.0 1.0 ~~ D V ----- -- - ----.. ~ '\ NACA 65 mean . . o ---- .8 e'i == 0.75 'l/c % (per cent c) (per cent c) 0 1.25 2.5 5.0 7.5 10 16 20 25 30 40 50 60 70 80 90 95 100 r--- r--- L.--- ~ 0 Qi a: 00 d1lo/dz ~ 1.0 c.c/4 == - 0.187 PB AvIV c: Pa/4 0.296 0.S85 1.140 1.665 0.24000 0.23400 0.22800 0.21600 0.20400 0 0.205 0.294 0.413 0.502 0 0.051 0.074 0.103 0.126 2.160 3.060 3.840 4.500 5.040 0.19200 0.16800 0.14400 0.12000 0.09600 0.671 0.679 0.760 0.824 0.872 0.14.3 0.170 0.190 0.206 0.218 5.760 6.000 5.760 5.040 3.840 0.04800 0 - 0.04800 - 0.09600 - 0.14400 0.932 0.951 0.932 0.872 0.760 0.233 0.238 0.233 0.218 0.190 2.160 1.140 0 - 0.19200 - 0.21600 - 0.24000 0.571 0.413 0 0.143 0.103 0 Data for NACA Mean Line 65 APPENDIX 11 387 .0 I .IJ tJ ~ V ...---- ~ ---- -- :-.... '" r-, \ NACA 66 I'I'N!'GJ . . - 41!11 .--- ~ ~ .2 D c'- == 0.76 ---- ~ .4 a, == - '/ Z,t: 0.74° t--- ~ 1.0 .8 .6 Ca e/ . == - 0.222 ---~ % 1/1: (per cent c) (per cent c) dy,:!dx 1 PR 4v/V = PR/4 ~._-----_.-...----- 0 1.25 2.5 5.0 7.5 0 0.247 0.490 0.958 1.406 0.20000 0.19583 0.19167 0.18333 0.17500 0 0.135 0.244 0.334 0.408 0.034 0.061 0.084 0.102 20 25 30 1.833 2.625 3.333 3.958 4.500 0.16667 0.15000 0.13333 0.11667 0.10000 0.466 0.557 0.635 0.700 0.750 0.117 0.139 0.159 0.175 0.188 40 50 80 70 80 5.333 5.833 6.000 5.625 4.500 0.06667 0.03333 0 - 0.07500 - 0.15000 0.827 0.910 0.999 1.040 0.966 0.207 0.228 0.250 0.260 0.242 90 2.625 95 1.406 - 0.22500 - 0.26250 - 0.30000 0.748 0.546 0 0.187 0.137 0 10 15 100 0 , Data for NACA Mean Line 66 0 I THEORY OF WING SECTIONS 388 A .W' I -­... ./ oV ~ ~ ~ ~ ~ r-, ~~ \ , NACA 67 nwon line ~ - " ~~ x ..-­ r--...... ~ ./J cu == 0.80 (per cent c) ~ (Xi Yc (per cent c) == - 1.60 0 dyc/dx c.c/. == - 0.266 PB J11J/V == PB/4 0 0.212 0.421 0.827 1.217 0.17-143 0.16837 0.16531 0.15918 0.15306 0 0.137 0.195 0.291 0.356 0 0.034 0.049 0.073 0.089 10 15 20 25 30 1.592 2.296 2.939 3.520 4.041 0.14694 0.13469 0.12245 0.11020 0.09796 0.406 0.483 0.560 0.616 0.673 0.102 0.121 0.140 O.IM 0.168 40 50 60 70 80 4.898 5.510 5.878 6.000 5.333 0.07347 0.04898 0.02449 0 - 0.13333 0.762 0.850 0.949 1.160 1.259 0.191 0.213 0.237 0.290 0.315 90 95 100 3.333 1.833 0 - 0.26667 - 0.33333 - 0.40000 1.066 0.788 0 0 1.25 2.5 5.0 7.5 I I Data for NACA Mean Line 67 I I 0.267 0.197 0 APPENDIX II 389 20 to " ~ \ \ ... ........... ------. o --­ NACA2/0 mean/ina 2.e c o _ ....•.. .2 _----,eli x ) (per rent c __ _,--_.= 2.09° = 0.30 .---. X/c .4 __. a; .. .. __ .... - - -_._.==__- CwL~/. ... _~ y~ e ) ! dlleld..r !I (per cent ll 1.0 .8 .6 O.OO() .~_.--;-_ Pit .. _.-. -_ ..... _-_ .._.-._. i ~v/l/" == PR/4 -- - - - - ~'-_._---.:----....---- -_. 1-"·- ---1--------····­ o t .25 2.5 5.0 7.5 10 15 20 25 30 ! i I ! -to 50 00 70 80 90 95 100 I f 0 0.596 0.928 1.114 1.087 . II - I 0.59613: 0.36236 0.1 S5O-l 0.00018 1.058 0.999 0.940 0.881 0.823 I I 0.705! 0.588 I Q4W 'I 0.353 0.235 0.118 0.059 0 I IJ _ 0.01175 o 0 1.381 1.565 1.221 0.781 0.345 0.391 0.305 0.195 0.626 0.489 O.-lO8 0.348 0.302 0.156 0.122 0.102 0.242 0.198 0.061 0.049 0.040 0.032 0.025 ' Ql00 0.128 0.098 0.065 ! 0.044 0 Data for N ACA :\lean Line 210 0.087 0.075 i O.(}16 0.011 o THEORY OF WING SECTIONS 300 0 1.0 T' r-, I'---.... " - ­ IJ NACA 220 ItI«JI7line .. "" D .4 Cl. == 0.30 x 1/e (per cent c) (per cent c) 0 ai ,I sIC == 1.86 0 dYe/dx .8 .6 1.0 c.c/t -= - 0.010 P. 4v/V == Pa/4 0 0.442 0.793 1.257 1.479 0.39270 0.31541 0.24618 0.13192 0.04994 0 0.822 1.003 0.988 0.900 0 0.206 0.251 0.247 0.225 10 15 20 25 30 1.535 1.463 1.377 1.291 1.205 0.00024: 0.801 0.615 0.465 0.378 0.326 0.200 0.154 0.116 0.095 0.082 40 SO 60 70 80 1.033 0.861 0.689 0.516 0.344 0.253 0.205 0.169 0.135 0.100 0.063 0.051 0.042 0.034 0.025 90 95 100 0.172 0.086 0 0.064 0.040 0 0.016 0.010 0 1.25 2.5 5.0 7.5 - 0.01722 Data for NACA Mean Line 220 APPENDIX II 391 (J I.D 1/r-, -- ............... r---­ ~ NACA 23IJ lin. IIWJt1n ,. ~ II . Cit .2 == 0.30 z (per cent c) 0 (per Yc cent c) .4 a, == sIc 1.650 0.30508 0.26594 0.22929 0.16347 0.10762 10 15 20 25 30 1.701 1.838 1.767 1.656 1.546 0.06174 - 0.00009 - 0.02203 40 50 60 1.325 1.104 0.883 0.662 0.. 442 2.5 5.0 7.5 70 80 90 95 100 0.221 0.110 0 1.0 c.c/t == - 0.014 Pa dyc/dx 0 0.357 0.666 1.155 1.492 1.25 .11 0 0.853 0 0.132 0.168 0.198 0.213 0.859 0.678 0.519 0.419 0.361 0.215 0.170 0.130 0.105 0.090 0.274 0.217 0.. 177 0.144 0.105 0.069 0.054 0.044 0.036 0.026 0.069 0.042 0 0.017 0.011 0 0.528 0.673 0.791 - 0.02208 I AviV == PRI4 Data for NACA l\fean Line 230 THEORY OF WING SECTIONS 392 Zeo -: ~ V o ............. r---. ~ -­ P-............ ~C;l240 m~ o .2 ci, x = 0.30 Ye (per cent c) (per cent c) .4 ai = 'l"e .1 %,C 1.45° dyc/ch .8 .6 Cr../. =- 1.0 0.019 ..lv/II' = PR/4 PR - ------­ 0 0.301 0.572 1.035 1.397 0.25233 0.22877 0.20625 0.16432 0.12653 0 0.377 0.491 0.625 0.718 0 0.094 0.123 0.156 0.180 10 15 20 25 30 1.671 1.991 2.079 2.018 1.890 0.09290 0.03810 - 0.00010 - 0.02169 0.750 0.677 0.566 0.477 0.410 0.188 0.169 0.142 0.119 0.103 40 50 70 80 1.620 1.350 1.080 0.810 0.540 0.304 0.234 0.186 0.150 0.110 0.076 0.059 0.047 0.038 0.028 90 95 100 0.270 0.135 0 0.071 0.047 0 0.018 0.012 0 0 1.25 2.5 5.0 7.5 60 I - 0.02700 Data for NA.CA Mean Line 240 393 APPENDIX II 2.0 1.0 .-1_ --.. r-............ r--­ V a r-­ ~ -~ NACA 250 ",.an "". .!.!...2 c o - .4, .8 .6 1.0 SIC --- ._.. cr, _~~-_.- == 0.30 at = 1.26° - _._--------- ----------. .. X Yc (per cent c) (per cent c) ----~ .... _---­ 1 dYc/dx _ _ _...._ .. _ .. __ -4-..--_ _,_ 0 0.258 0.498 0.922 1.277 0.21472 0.19920 0.18416 0.15562 0.12909 10 15 20 25 30 1.570 1.982 2.199 2.263 2.212 40 50 60 70 80 1.931 1.609 1.287 0.965 0.644 0 1.25 2.5 5.0 7.5 90 95 100 I I 0.322 0.161 0 c.. ~/. =- 0.026 -------------.- f--------.- -­ --­ l~H i ~/::'1~ = P,,/4 I - ­ ­ -­ . --------- ._-­ -_.-~- - _.-...... 0 0.281 0.369 0.477 0.552 0 0.070 ().092 0.119 0.138 0.10458 0.06162 0.02674 - O.OOOOi - 0.01880 0.592 0.624 0.610 0.547 0.470 0.148 0.156 0.153 0.137 0.117 - 0..03218 0.346 0.255 0.197 0.154 0.119 0.087 0.064 0.049 0..038 0.030 0.076 0.051 0 0.019 0.013 0 ! i I i : Data for KAC.-\. Mean Line 250 ! THEORY OF WING SECTIONS 394 2.0 -.............. ~'" ......... <, /.0 ~ ~ r-, ""­ ........ ~ ~ .-. v MACA 0-0 IIIt1t1I'I 'M 2 D Cia sa 1.0 % 1/c (per cent c) (per cent c) 0 0 0.460 0.641 0.964 1.641 0.5 0.75 1.25 2.5 - ~ V CZi =­ 4.56 0 c.c/. t--- t - -........ a: - 0.083 p. 4v/V == PR/4 0.75867 0.69212 0.60715 0.48892 1.990 1.985 1.975 1.950 0.498 0.496 0.494 0.488 1.900 1.850 1.800 1.700 1.600 0.475 0.463 0.450 0.425 0.400 d1lc/d% 5.0 2.693 7.5 10 3.507 4.161 6.124 6.747 0.36561 0.29028 0.23515 0.15508 0.09693 6.114 6.277 6.273 6.130 5.871 0.05156 0.01482 - 0.01554 - 0.04086 - 0.06201 5.516 5.081 4.581 4.032 3.445 - 0.07958 0.09395 0.10539 0.11406 0.12003 0.900 0.800 0.700 0.600 0.250 0.225 0.200 0.175 0.150 75 80 85 2.836 2.217 90 95 1.013 0.467 - 0.12329 0.12371 0.12099 0.11455 0.10301 0.500 0.400 0.300 0.200 0.100 0.125 0.100 0.075 0.050 0.025 ·0 - 0.07958 0 0 Ui 20 25 30 35 40 45 50 55 60 65 70 100 1.604 1.500 1.400 1.300 1.200 1.100 1.000 Data for NACA Mean Line a =0 0.375 0.350 0.325 0.300 0.275 APPENDIX II 395 3JJ - 20 r-; ........ ~'" r-; 1.0 -.......... "'~ -....... '" r-; o NACA a-al mtIQI7 IinII .!.L.e c .2 eli - z' (per cent c) 0 0.5 0.75 1.25 2.5 5.0 7.5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 -r---- r--­ ~ V () 1.0 «i sa 1/e 4.43° dYe/dx (per cent c) 0 0.440 0.616 0.933 1.608 2.689 3.551 4.253 5.261 5.906 0.38235 0.31067 0.25057 0.16087 0.09981 6.282 6.449 0.05281 0.01498 - 0.01617 - 0.04210 - OJ)6373 5.664 5.218 4.706 4.142 c.c/. , a: - to 0.086 PB AviV = Pll/4 1.818 0.455 1.717 1.616 0.429 0.404 1.515 0.379 0.354 I i I ! 0.73441 0.67479 0.59896 0.49366 6.443 6.296 6.029 r--..~ .8 .6 3..541 - 0.08168 0.09637 0.10806 0.11694 0.12307 2.916 2.281 1.652 1.045 0.482 - 0.12644 0.12693 0.12425 0.11781 0.10620 0 - 0.08258 h 11 II ; I I} 1 .j jJ lJ ; ! I ! 1.414 1.313 1.212 1.111 0.328 0.303 0.278 1.010 0.909 0.808 0.707 0.606 0..253 0.227 0.202 0.177 0.152 0.505 0.404 0.303 0.202 0.101 .. 0.126 0.101 0.076 0.050 0.025 0 0 Data for N ACA Mean Line a == 0.1 THEORY OF WING SECTIONS 396 ~ ~ 1.0 ~ ............. o ""' .......... ~" r-; /MeA G-D.2 'e C mtH1f7/ine .2 ~ r - - t--- ~ D .2 .4 ./. r--- t - - - t-- 1.0 .8 .6 %,C e'i -= 1.0 x (per cent c) 0 0.5 0.75 5.0 7.5 10 15 20 I I I I 45 60 65 70 I I 80 85 90 95 100 I =: - 0.094 PR l!JJ/IT == PB/4 0 0.414 0.581 0.882 1 1.530 0.69492 0.64047 0.57135 0.47592 2.583 3.443 4.169 5.317 6.117 0.37661 0.31487 0.26803 0.19373 0.12405 1.667 0.417 6.572 6.777 6.789 6.646 6.373 0.06345 0.02030 - 0.01418 - 0.04246 - 0.06588 1.563 1.459 1.355 1.250 1.146 0.391 0.365 0.339 0.313 0.287 5.994 5.527 4.989 4.396 3.762 - 0.08522 0.10101 0.11359 0.12317 0.12985 1.042 0.938 0.834 0.729 0.625 0.260 0.234 0.208 0.182 0.156 3.102 2.431 1.764 1.119 0.518 - 0.13363 0.13440 0.13186 0.12541 0.11361 0.521 0.417 0.313 0.208 0.104 0.130 Q.I04 0.078 0.052 0.026 0 - 0.08941 0 0 I 75 c. e/ . d1lc/dx (per cent c) 25 30 35 40 50 55 == 4.17° 1/e 1.25 2.5 (Ii Data for NACA Mean Line a = 0.2 397 APPENDIX II ~~ <, I.: '"'~ ......... r-, <, '"' A NACA 4-0.3 <, mean line .­ A ~ -- ~ o ,l .4 :-- r--- r--- .8 %/C e'l (per == 1.0 3.84 (Ii - 0 o 0.5 0.75 1.25 2.5 5.0 7.5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 (per ' __ - . o 0.389 to C.ri. == - 0.106 ~~ I :nt_:~Ldy./~~~ c) ­ _~~~~~= 1'_/4 0.832 1.448 0.65536 0.60524 0.54158 0.45399 2458 3.293 4.008 5.172 6.052 O.363-t-l O.3078U O.26(i21 O.2tr24U 0.15068 6.685 7.072 7.175 7.074 6.816 0.10278 0'-»833 - 0.00205 - OJlfH92 1.319 1.200 6.433 5.949 5.383 4.753 4.076 - 0.()8746 - O.l()567 U.989 - 0.1201-4 - 0.13119 - 0.13901 0.879 0.769 O.fl59 U.275 0.2-1. 0.220 0.192• 0.165 3.368 2.645 1.924 1.224 0.570 - 0.14365 0.14500 0.14279 0.549 0.440 0.330 0.220 0.110 0.137 0.110 0.082 0.055 0.028 o - 0.09907 0 o 0.5-16 - O.0371() 1.538 0.385 1.429 0.357 0.330 fl. 302 1.099 O.1363R 0.12430 j Data for NAC..A.. Mean Line a = 0.3 THEORY OF WING SECTIONS 398 :uJ1_ _-.....-~--,--r---r-.-....,.-.,.--, - r---- r---. .r---",",",-­ o .4 ct; -.: 1.0 3.46° (J[i - x fie (per cent c) (per cent c) 0 0.5 0.75 1.25 2.5 0 0.366 0.514 0.784 1.367 6.0 7.& 10 15 20 "'2.330 z/c c..clt 1.0 .8 .6 -= - 0.121 p. 4v/V - P/f/4 1.429 0.357 dYc/tk 0.61759 0.57105 0.61210 0.43106 3.131 3.824 4.968 5.862 0.34764 0.29671 0.25892 0.20185 0.Ui682 6.&46 7.039 7.343 7.439 7.275 0.11733 0.07988 0.04136 - 0.00721 - 0.05321 1.310 0.327 70 6.929 6.449 5.864 6.199 4.475 - 0.08380 0.10734 0.12567 0.13962 0.14963 1.190 1.071 0.952 0.833 0.714 0.298 0.268 0.238 0.208 0.179 76 80 8& 90 95 3.709 2.922 2.132 1.361 0.636 - 0.15589 0.15837 0.15683 0.lli062 0.13816 0.695 0.476 0.357 0.238 0.119 0.149 0.119 0.089 0.060 0.030 0 - 0.11138 0 0 25 30 35 40 4S 50 86 60 65 100 Data for NACA Mean Line a == 0.4 APPENDIX II 399 r-. <, r-, <, o r-, NACA 1:1=0.5 ",.., lin. ~ ~ r---- t--- ~ () (per :ent C) , (per :nt C) I d~:dz_1 o o 0.5 0.75 1.25 2.5 0.345 0.485 0.735 1.295 0.58195 0.53855 0.48360 0.40815 0.33070 0.28365 0.24890 0.19690 0.15650 5.0 7.5 10 15 2.206 2.970 3.630 4.740 20 5.620 25 30 6.310 6..840 7.215 35 40 7.430 45 7.490 50 7.350 6.965 6.405 5.725 4.955 55 60 65 70 70 4.130 80 85 90 95 3.265 100 t--- t-.... .8 .6 1.0 4IJor - p. P./4 1 I 1.333 0.333 0.12180 0.09000 0.05930 0.02800 - 0.00630 - 0.05305 - 0.09765 - 0.12550 - 0.14570 - 0.16015 2.395 1.535 0.720 - I - 0.16960 0.17435 0.17415 0.16850 0.15565 o II - 0.12660 1.200 I I 1.067 0.933 0..800 I 0.300 0.267 0.233 0.200 0.667 0.533 0.400 0.267 0.133 0.167 0.133 0.100 0.067 0.033 o o Data for NACA Mean Line a =-: 0.5 rHEORY OF WING SECTIONS 400 zo 1"- 1.0 "', <, o r-, NACA 41:0.6 If'IINIn Ii,. o ~ l.---­ ~ r-- r---­ I"""""""'" .4 eli -= 1.0 G, == 2.58 % 1/c (per cent c) (per cent c) 0 0 0.5 0.75 0.325 0.455 0.695 ,I. %,C 0 2.5 1.220 5.0 7.5 10 15 20 2.080 2.805 3.435 4.495 5.345 0.31325 0.26950 0.23730 0.18935 0.15250 25 30 6.035 6.570 6.965 7.235 7.370 0.12125 0.09310 0.06660 0.04060 0.01405 35 40 45 == - t-­ 1.0 0.158 p. dllt:/dz 0.54825 0.50760 0.45615 0.38555 1.25 c.c/4 r--- .8 .4 ~v/V· = PH/4 1.250 0.312 70 7.370 7.220 6.880 6.275 5.505 - 0.01435 0.04700 0.09470 0.14015 0.16595 1.094 0.938 0.273 0.234 75 80 85 90 95 4.630 3.695 2.720 1.755 0.825 - 0.18270 0.19225 0.19515 0.19095 0.17790 0.781 0.625 0.469 0.312 0.156 0.195 0.156 0.117 0.078 0.039 0 0 50 55 60 65 100 0 I - 0.14550 nata for NACA Mean Line 4 == 0.6 APPENDIX II 401 (Ji 20 "­ 1-0 -... '"' -, -, NACA a-47 ",." /WM ~ r-- I ----- ---. ...--- ---­ Cia =-: 1.0 I ·--I~-d I~ I yc Ii 0.305 0.425 0.655 0.51620 0.47795 0.42960 2.5 1.160 0.36325 5.0 7.5 1.955 2.645 0.29545 0.25450 10 15 20 3.240 4.245 5.060 0.22445 0.17995 0.14595 25 30 35 40 45 5.715 0.11740 O.092()() O.On840 OJ)4570 0.02315 50 7.155 7.090 0 - 0.02455 6.900 6.565 6.030 - 0.05185 - 0.08475 - 0.13650 X (per :nt c) 0.5 0.75 1.25 (pe: rent c) y, 6.240 6.635 6.925 7.095 55 60 65 70 75 5.205 - 0.18510 80 85 90 95 4.215 3.140 2.035 0.965 - 100 I 0 /.IJ c...j. -= - 0.179 2.09 0 ai - ~ 0.20855 0.21955 0.21960 0.20725 , - 0.16985 Data for XACA ~ienn t I } 1.176 0.294 i ! t I 1 II I J 0.980 0.784 0.588 0.392 0.196 o Line a -- 0.7 I I I 0.245 0.196 0.147 0.098 0.049 o THEORY OF WING 8ECTION8 402 ..---­ ...., - <, " (J NACA -e/J.8 ",.., n,. .2 a ,-------I.---- .4 e'i -= 1.0 1.64 eli - fie % 2.5 0 0.287 0.404 0.616 1.077 5.0 7.5 10 15 20 1.841 2.483 3.043 3.986 4.748 25 30 5.367 5.863 6.248 6.528 6.709 1.25 35 40 45 50 55 60 65 70 75 80 85 90 95 100 SIC 0 .8 .6 """"""--­ /.0 c../. - - 0.202 p. dl/c/a (per cent e) (per cent e) 0 0.5 0.75 --- r---.... to-""" AviV == PIl/4 0.48535 0.44925 0.40359 0.34104 0.27718 0.23868 0.21(}lj() 0.16892 0.13734 0.11101 0.08775 0.06634 0.04601 0.02613 6.790 6.770 6.644 6.405 6.037 - 0.00620 0.01433 0.03611 0.06010 0.08790 5.514 4.771 3.683 2.435 1.163 - 0.12311 0.18412 0.23921 0.25583 0.24904 0 - 0.20385 1.111 0.278 0.833 0.556 0.278 0.208 0.139 0.069 0 0 Data for NACA MeaD Line a .. 0.8 APPENDIX 11 403 2.0 1.0 "" ,,~ o \ WACA • - O.B (modlfl_d) Mean II". 2 o t-.--­ o ~ - r--- r----. ~ .2 .4 .6 ~ .8 1 0 x/e eli -= 1.0 Gr, -= 'lie % 1.25 2.5 5.0 7.5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 I c.ct. 0 0.281 0.396 0.603 1.055 0.47539 0.44004 0.39531 0.33404 1.803 2.432 2.981 3.903 4.651 0.27149 0.23378 0.20618 . 0.16546 0.13452 5.257 5.742 6.120 6.394 6.571 0.10873 0.08595 0.06498 0.04507 0.02559 6.651 6.631 6.508 6.274 5.913 - 0.00607 0.01404­ 0.03537 0.05887 0.08610 5.401 4.673 3.607 2.452 1.226 - 0.12058 0.18034 0.23430 0.24521 0.24521 0 - 0.24521 :II: 0.219 ------ PIl AfJ/V .. P./4 1.092 0.273 11.~ 0.274 }1.100 0.276 } 1.104 0.276 1.108 1.108 0.277 O.Z17 1.112 1.112 0.840 0.588 0.368 0.278 0.278 0 0 dy./a (per cent e) (per cent c) 0 0.5 0.75 1.400 Data for NACA Mean Line a -= 0.8 (modified) 0..210 0.147 0.092 THBoRY OF WING SECTIONS 404 D I. D -- ., 1\ \ - r--­ ~ NACAa.O,9 meon/i/'ltl L"....--­ ~ ~ IJ t(J ci, == 1.0 x (per cent c) 0 0.5 0.75 1.25 2.5 Gi == 0.90 0 Yc 0 0.269 0.379 0.577 1.008 5.0 7.5 10 15 20 1.720 2.316 2.835 3.707 4.410 0.25786 0.22153 0.19liOO 0.15595 0.12644 2S 4.980 5.435 5.787 6.M5 6.:212 0.10196 0.08047 0.06084: 0.04234 0.02447 40 45 50 55 60 65 70 75 80 85 90 95 100 6.290 6.279 6.178 5.981 5.681 - 0.00678 0.01111 0.02965 0.04938 0.07103 2.984: 1.503 - 0.09583 0.1260:) 0.16727 0.25204: 0.31463 0 - 0.26086 5.265 4.714 3.987 0.225 z= - Pa dyc/dz (per cent c) 0.45482 0.42OM 0.37740 0.31821 30 35 c.c/. ~/V = PR/4 1 1.05:1 0.263 0.526 0.132 0 0 Data for NACA Mean Line (I :II: 0.9 APPENDIX II 405 :J.O .0 .0 D NACA ••/.0 mean /;n. ~ - r--- r--­ ..",..-- ~ - D .4 e'i 1.25 2.5 0.930 5.0 7.5 10 1.580 i 15 20 25 30 35 40 45 65 70 75 80 85 90 95 100 00 c..,J. I J 2.120 2.585 3.365 3.980 4.475 4.860 5.150 5.355 5.475 0.08745 0.00745 0.04925 0.(13225 0.01595 5.515 5.355 5.150 4.860 4.475 3.980 3.365 2.585 11&: - 0.250 PR j 0.23430 0.19995 0.17485 0.13805 0.11030 5.475 , dyc/dx 0.42120 0.38875 0.34770 0.29155 I Ii I - 0 0.01595 I,II 1.580 ! - 0.23430 0 ti i II I I II I If urn i 1( t f; ! I II II - 0.08745 - 0.11030 - 0.13805 = PR/4 Ii I' II I~V/V II :I OJ>3225 I: 0.04925 0.06745 UJ .8 .6 0 ! 0.250 0.350 0.535 50 55 60 a, = Yc (per cent c) X (per cent c) 0 0.5 0.73 = 1.0 sIc I I I i I I j I - 0.17485 I I Data ior N ACA :\IcuD Line Il = 1.0 j I 0.250 APPENDIX III AIRFOIL ORDINATES CoNTENTS NACA Designation Page 408 0010-34 a == 0.8 (modified) e'i II: 0.2 . 0012-64 a == 0.8 (modified) e'i == 0.2 . 1408 1410 1412 2408 2410 . . . . . 2412 2415 . . . . . 408 408 409 409 409 410 410 410 2418 2421 2424 411 ..... 411 411 412 412 4412 4415 ..... 4418 4421 4424 23012 23015 23018 . 23021 23024 63-206 . 63-209 . 63-210 . 63 1-212 63.-412. 63r215 . . 412 413 413 413 414 414 414 415 415 416 416 417 . 417 418 418 63r415. '. . . . . . 63t-615. . . . . . 63,-218. . 63r418. 419 419 63,-618. . 420 420 63r221 . . 63.-421. 421 421 63A21 0 64-108 . . . . 64-110 . 64-206 . . 422 423 422 423 424 64-208 . . . . 64.-209. . . . . 424 406 APPENDIX III NACA Designation 64-210 . 641-112 . . .... 641-212 . 641-412. . 642-215. . 64r415. . . . . . . 64,-218 . . 64r418. 64r618 . . 64 221 407 . 425 426 426 427 427 428 428 429 . 64.-421 .. Page 425 . .... 429 430 430 431 431 432 64A210 64A410 64IA212 . 642.~215 . 65-206 . . 65-209 . . 65-210 . 65-410 . 651-212 . . 65.-212 a == 0.6 . . 65.-412 . 65r215. . . . . . . 65r415. . . . . 65r415a == 0.5 . . 65r 218 . . . . . . . 65,..418. . . . . 65r418 a - 0.5. ­ 65r618. . . . . 65,-618 a == 0.5 . 65...221 . 65r421 . 65.-421 a == 0.5 . 66-206 . . 66-209 . . 66-210 . . 66.-212. 432 433 433 434 434 435 435 436 436 437 437 438 438 439 439 440 440 441 441 442 442 443 66r215. 443 66r415. 444 66r218. 66r418. 66...221. 67,1-215 . 747A315 . 747A415 . . • • • • . . .... 444 445 445 446 446 447 (Stations and ordinates given in NACA 0012-64 a = 0.8 (modified) e'i = 0.2 (Stations and ordinates given in per cent of airfoil chord) per cent of airfoil chord) NACA 0010-34 a = 0.8 (modified) c" == 0.2 _ .._. Upper surface Lower surface Upper surface Ordinate Station Ordinate 0 1.928 2.659 3.623 4.295 0 1.393 2.664 5.177 7.678 - 0 1.686 2.237 2.901 3.323 4.832 6.221 6.974 7.279 10.175 15.161 20.142 30.100 40.054 - 3.640 4.083 4.361 4.678 4.721 49.993 60.038 70.077 80.120 90.091 7.157 6.622 5.662 4.253 2.355 50.007 59.962 69.923 70.880 89.909 - 4.497 4.018 3.296 2.383 1.375 95.050 100.000 1.271 0.120 04.950 100.000 Ordinate Station Ordinate Station 0 0.813 1.324 2.593 5.113 - 0 0.632 0.820 1.186 1.714 0 1.107 2.336 4.823 4.887 0 0.790 1.062 1.608 2.436 7.378 9.875 14.876 19.886 20.017 3.094 3.637 4.523 5.172 5.080 7.622 10.125 15.124 20.114 30.083 - 2.122 2.445 2.961 3.312 3.684 9.825 14.839 5.645 19.858 29.000 39.946 39.955 49.994 (lO.O31 70.ntH 80.100 6.279 6.186 5.735 3.700 40.045 50.006 59.960 OH.H3H 79.900 - 3.721 3.526 3.131 2.5"Ul 1.83U 00.076 95.042 100.000 2.044 1.100 0.100 89.92-1 94.958 100.000 - 1.004 - 0.610 - 0.100 Station 0 0.687 1.176 2.407 tj.015 L.E. radius: 0.272 Slope of rudius through L.E.: 0.095 7.322 (Stations and ordinates given in per cent of airfoil chord) Upper surface Lower surface - 0.781 - 0.120 L.E. radius: 1.582 Slope of radius through L.E.: 0.095 ~ 00 NACA 1408 Station Ordinate Lower surface Station Ordinate 0 1.189 2.418 4.896 7.386 0 1.324 1.862 2.602 3.138 0 1.311 2.582 5.104 7.614 9.883 14.889 19.904 24.926 29.950 3.558 4.171 4.574 4.819 4.939 10.117 15.111 20.096 25.074 30.050 - 2.682 2.953 3.074 3.101 3.063 40.000 50.020 60.034 70.041 SO.03D 4.869 4.502 3.931 3.193 2.305 40.000 49.980 59.966 69.959 79.961 - 2.869 2.556 2.153 1.693 1.193 90.027 95.016 100.000 1.271 0.698 0.084 89.973 94.984 100.000 0 - 1.200 - 1.620 - 2.134 - 2.458 - 0.659 - 0.378 - OJJ84 L.E. radius: 0.70 Slope of radi us through L. E.: 0.05 ~ ~ ~ ~ ~ ~ ~ ~ ~ C ~ NACA 1410 NACA 1412 NACA 2408 (Stations and ordinates given in per cent of airfoil chord) (Stations and ordinates given in (Stations and ordinates given in per cent of airfoil chord) Upper surface Station () Station 0 1./i15 0 t.tuM 2.378 4.845 0 1.ll30 2.398 4.870 7.358 2.297 3.194 3.837 9.854 29.937 4.338 5.062 5.531 5.809 5.940 40.000 50.025 00.042 70.051 80.049 5.836 5.385 4.692 3.804 2.741 90.034 95.021 100.000 1.513 0.832 0.105 24.907 Upper surface Lower surface Upper surface Lower surface .. Ordinate Station Ordinate 1.174 14.861 19.880 _­ Lower surface -- per cent of airfoil chord) 0 1.320 2.002 7.642 - 2.055 2.726 3.157 10.146 15.139 20.120 25.093 30.063 - 3.462 3.844 4.031 4.091 4.064 5.130 I I I 40.000 49.975 I - 3.836 - 3.4:i9 59.058 69.949 - 2.914 - 2.30,1 79.951 - 1.629 89.966 0.901 94.979 , - 0.512 I 100.000 - !- 0.105 L.E. radius: 1.10 Slope of radius through L.E.: 0.05 Ordinate Station 7.330 0 0 l.or)t1 2.733 3.786 4.537 1.:\tI2 2.U22 5.118 5.951 6.486 6.799 6.040 9.82'. 14.833 19.857 24.889 29.925 40.000 50.029 60.051 70.061 80.0SS , 6.803 5.155 7.670 lO.l7G 15.167 20.143 ss.iu 30.075 40.000 49.071 0.267 5.453 4.413 3.178 60.930 79.942 1.753 90.040 1 0.966 95.025 0.126 100.000 89.960 94.975 100.000 I 5fi.940 Ordinate 0 Station Ordinate Station 0 1.372 2.663 5.206 7.273 0 1.3140 1.977 2.829 3.471 0 - 1.8..10 1,12R - 2.491 - 3.318 4.794 - 3.857 2.337 7.727 Ordinate 0 - 1.134 - 1.493 - 1.891 - 2.111 - 4.242 4.733 4.986 5.081 5.064 9.768 14.778 19.809 24.852 29.900 3.987 4.776 5.320 5.677 5.875 10.232 15.222 20.191 25.148 30.100 - 2.237 2.338 2.320 2.239 2.125 - 4.803 4.321 3.675 2.913 2.066 40.000 50.039 70.081 80.078 5.869 5.473 4.820 3.942 2.858 40.000 49.961 59.932 69.919 79.922 - 1.869 1.585 1.264 0.942 0.636 - 1.141 - 0.646 - 0.126 90.054 95.033 100.000 1.575 0.855 0.084 89.946 94.967 100.000 L.'E. radius: 1.58 Slope of radius through L.E.: 0.05 no. 008 ~ ~ t:::J ""-c ~ ""-c '"'-.. ""-c - 0.353 - 0.217 - 0.084 L.E. radius: 0.70 Slope of radius through L.E.: 0.1 .... o co NACA 2410 (Stations and ordinates given in per cent of airfoil chord) Upper eturface Station Lower surface Ordinate Station Ordinate 1.098 2.297 4.742 7.217 0 1.694 2.411 3.420 4.169 0 1.402 2.703 6.258 ' 7.783 9.710 14.722 19.761 24.814 29.875 4.766 5.665 6.276 6.668 6.875 10.290 15.278 20.239 26.186 30.126 40.000 lj().049 60.085 70.102 80.097 90.067 96.041 100.000 6.837 6.366 5.lS8O 4.651 3.296 0 1.816 0.990 0.105 - 0 1.448 1.927 2.482 2.809 - 3.016 - 3.227 - 3.276 - 3.230 - 3.123 - 2.837 - 2.468 - 2.024 -l.Ml - 1.074 - 0.594 - 0.352 40.000 49.951 59.915 69.898 79.903 89.933 94.959 100.000 - 0.105 L.E. radius: 1.10 Slope of radius through L.E.: 0.1 NACA 2412 (Stations and ordinates given in per cent of airfoil chord) Upper surface Lower surface Station Ordinate Station Ordinate 0 1.2a 2.5 6.0 7.5 .. .... 2.15 2.99 4.13 4.96 10 16 20 25 30 6.63 6.61 7.26 7.67 7.88 40 7.80 7.24 6.36 6.18 3.75 so 60 70 80 90 95 100 100 2.08 1.14 (0.13) .. .... 2.5 5.0 7.6 10 15 20 25 30 40 50 60 70 80 90 95 100 100 Upper surface 0 1.66 2.27 3.01 3.46 0 ...... 1.25 2.71 3.71 5.07 6.06 - 3.76 4.10 4.23 4.22 4.12 10 15 20 25 30 6.83 7.97 8.70 9.17 9.38 - 3.80 3.34 2.76 2.14 l.l5O 40 - 0.82 - 0.48 (- 0.13) 0 L,E. radius: 1.58 Slope of rArdius through L.E.: 0.10 • •.o.... Lower surface Station Ordinate Station Ordinate - 0 1.25 NACA 2415 (Stations and ordinates given in per cent of airfoil chord) 2.5 5.0 7.5 50 60 70 80 90 95 100 100 0 1.25 2.5 5.0 7.6 - 0 2.06 2.86 3.84 4.47 10 15 20 25 30 - 4.90 5.42 5.66 5.70 5.62 9.25 8.57 7.50 6.10 4.41 40 50 60 70 80 - 5.26 - 4.67 -3.90 - 3.05 - 2.15 2.45 1.34 (0.16) 90 - 1.17 - 0.68 (- 0.16) 0 ...... 95 100 100 L.E. radius: 2.48 Slope of radius through L.E.: 0.10 ~ ~ ~ e.c: ~ s ~ ~ ~ ~ ~ r,j NACA 2418 NACA 2421 NACA 2424 (Stations and ordinates given in per cent of airfoil chord) (Stations and ordinates given in per cent of airfoil chord) (Stations and ordinates given in Upper surface Station Ordinate 0 1.20 2.5 6.0 7.5 10 I 16 20 2S 30 40 50 60 70 80 90 95 100 100 •• I •••• 3.28 4.45 6.03 7.17 8.05 9.34 10.15 10.65 10.88 10.71 9.89 8.66 7.02 5.08 2.81 1.55 (0.19) ..... ... Lower surface Station Ordinate 0 0 1.25 2.5 5.0 7.5 Station Ordinate Station Ordinate 0 1.25 2.5 5.0 7.5 - 2.45 3.44 4.68 5.48 - 6.03 6.74 7.09 7.18 7.12 - 6.71 5.99 5.04 3.97 2.80 40 100 - 1.63 - 0.87 (- 0.19) 90 95 100 100 0 100 10 15 20 25 30 40 50 60 70 80 90 95 L.E. radius: 3.66 Slope of rndius through L.E.: O.to Lower surface Upper surface 10 15 20 25 30 50 60 70 80 •••• t •• 1.25 2.5 5.0 8.29 7.5 9.28 10.70 11.59 12.15 12.38 10 15 20 25 30 12.16 11.22 9.79 7.94 40 5.74 80 3.18 1.76 (0.22) 90 95 100 100 ....... 50 60 70 Upper surface Lower surface Station Ordinate Station Ordinate 0 3.646 4.965 6.614 7.692 - 0 2.82 4.02 5.51 6.48 0 0.885 2.012 4.380 6.820 0 3.892 5.449 7.502 9.052 - 7.18 8.05 8.52 8.67 8.62 9.300 14.333 19.427 24.555 29.700 10.215 11.888 12.959 13.593 13.874 10.700 - 8.465 15.667 - 9.450 20.573 - 9.959 25.445 - 10.155 30.300 - 10.124 - 8.16 7.31 6.17 4.87 3.44 40.000 50.118 60.203 70.244 80.233 13.606 12.532 10.903 8.824 6.352 40.000 49.882 59.797 69.756 79.767 - 1.88 90.161 95.098 100.000 3.502 1.930 0 3.87 5.21 7.00 per cent of airfoil chord) - 1.06 (- 0.22) 0 L.E. radius: 4.85 Slope of radius through I".E.: 0.10 ..... 0 1.615 2.988 5.620 8.180 - - 9.606 8.644 7.347 5.824 4.130 ~ ~S2 >< ~ ~ ..... 89.839 - 2.280 94.902 - 1.292 0 100.000 L.E. radius: 6.33 Slope of radius through L.E.: 0.10 ~ ..... ..... .... ~ NACA 4412 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Ordinate 0 0 1.25 2.5 2.44 3.39 4.73 5.76 5.0 7.5 10 15 20 25 30 40 50 60 70 80 90 95 100 100 Lower surface Station Ordinate 0 1.25 2.5 5.0 7.5 NACA 4415 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station 0 1.25 2.5 5.0 - 0 1.43 1.95 2.49 2.74 9.41 9.76 10 15 20 25 30 - 2.86 2.88 2.74 2.50 2.26 9.80 9.19 8.14 6.69 4.89 40 50 60 70 80 - 1.80 1.40 1.00 0.65 0.39 60 70 80 2.71 1.47 (0.13) 90 95 100 - 0.22 - 0.16 (- 0.13) 0 90 95 100 100 6.59 7.89 8.80 ..... . 100 IJ.E. radius: 1.58 Slope of radius through L.E.: 0.20 Ordinate Station Ordinate t •••••• - 3.27 6.91 7.5 - .3.71 Station 0 1.25 2.5 5.0 7.5 ....... 3.76 5.00 6.75 8.06 3.98 4.18 4.15 3.98 3.75 10 15 20 25 30 9.11 10.66 11.72 12.40 12.76 11.25 10.53 9.30 7.63 5.55 40 - 3.25 2.72 2.14 1.55 1.03 40 50 60 70 80 12.70 11.85 10.44 8.55 6.22 3.08 90 95 100 100 - 0.57 - 0.36 (- 0.16) 0 90 95 100 100 3.46 20 9.27 1.67 (0.16) II ••••• 50 60 70 80 L.E. radius: 2.48 Slope of radius through L.E.: 0.20 ~ Lower surface Ordinate Station Ordinate - 10.25 10.92 11.25 50 Upper surface 10 15 20 25 30 7.84 40 0 - 1.79 - 2.48 5.74 10 15 30 0 1.25 2.5 5.0 3.07 4.17 7.5 25 Lower surface NACA 4418 (Stations and ordinates given in per cent of airfoil chord) 1.89 (0.19) ••••••• I 0 0 2.11 - 2.99 - 4.06 - 4.67 ~ 10 - 5.06 15 - ~ 1.25 2.5 5.0 7.5 20 25 30 40 50 60 70 ~ 5.49 5.56 5.49 5.26 - 4.70 - 4.02 -3.24 80 - 2.45 - 1.67 90 95 100 100 - 0.93 - 0.55 (- 0.19) 0 L.E. radius: 3.56 Slope of radius through L.E.: 0.20 ~ ~ ~ ~ ~ ~ ~ ~ C ~ NACA 4421 (Stations and ordinates given in per cent of airfoil chord) Upper surface Lower surface Ordinate Station Ordinate Station ..... . - 0 1.25 2.5 5.0 4.45 5.84 7.82 9.24 7.5 0 1.25 2.5 5.0 7.5 0 - 2.42 - 3.48 - 4.78 - 5.62 15 20 10.35 12.04 13.17 10 15 20 - fi.t5 - (}.75 25 13.88 25 - 6.92 30 14.27 30 - 6.76 40 50 60 14.16 13.18 11.60 9.50 6.91 40 - 10 70 80 00 95 100 100 I 3.85 2.11 (0.22) ..... .. 50 60 70 80 90 95 100 100 - 6.98 6.16 5.34 4.40 3.35 2.31 - 1.27 - 0.74 (- 0.22) 0 L. I~. radius: 4.85 Slope of radius through 1.1. E.: 0.20 NACA 4424 NACA 23012 (Stations and ordinates given in (Stations and ordinates given in per cent of airfoil chord) per cent of nirfoil chord) I .ower surface Upper surface Station 1.536 0 3.964 5.624 3.775 6.153 7.942 9.651 8.611 13.674 18.858 24.111 29.401 11.012 13,<)45 14.416 15.287 0 0.530 I5.nOG 50.235 14.47·' 12.674 10.312 00.405 70.487 80.464 7.447 i 4.099 2.240 • It •• •• L.E. radius: 6.33 0 0 1.970 - 3.472 3.4M 6.225 8.847 .io.ooo I ··JU.7G5 59.505 00.513 iO.536 80.680 94.804 lOOJ)(X) I I Station 0 1.25 - 4.656 2.5 - 6.0f>6 - 6.931 5.0 7.5 11.389 - 7.512 16.326 - 8.I09 21.142 - 8.416 25.&~9 - 8.411 30.500 - 8.238 15.738 40.000 90.320 95.196 100.000 Station Ordinate Ordinate Upper surface 10 15 20 25 30 - 7.006 - 0.698 40 - 5.562 Ordinate Station Ordinate ...... 2.67 3.61 4.91 5.80 6.43 7.10 7.53 7.60 7.55 60 7.14 6.41 5.47 - 4.312 70 4.36 - 3.003 80 3.08 - 1.655 90 95 - 0.964 0 Slope or radius through L.E.: 0.20 50 100 100 Lower surface 1.68 0.92 (0.13) ...... 0 1.25 2.5 5.0 7.5 10 15 - 0 1.23 1.71 2.26 2.61 20 - 2.92 - 3.50 - 3.97 25 - 4.28 30 - 4.46 40 50 60 70 80 - 4.48 4.17 3.67 3.00 ~ ~ ~ ~ ~ ....... >< ....... ....... ....... - 2.16 90 95 - 1.23 - 0.70 100 100 (- 0.13) 0 L.E. radius: 1.58 Slope of radius through L. E.: 0.305 ~ ~ ~ NACA 23015 (Stations and ordinates given in per cent 01 airfoil chord) Upper surface Lower surface Station Ordinate Station Ordinate 0 1.20 2.5 5.0 7.5 .... ., 3.84 4.44 5.89 6.90 0 1.25 2.5 6.0 7.5 - 0 1.54 2.25 3.04 3.61 NACA 23018 (Stations and ordinates given in per cent of airfoil chord) Lower surface Upper surface Station 0 1.25 2.5 5.0 7.5 Ordinate ••••• t 10 7.64 10 - 4.09 10 8.83 15 8.52 20 26 30 8.92 15 20 26 15 20 25 30 - 9.86 10.36 10.56 10.OS 40 - 5.92 50 60 70 80 - 40 50 60 70 80 90 95 100 100 9.08 9.06 8.59 7.74 6.61 5.25 3.73 2.04: 1.12 (0.16) .. .... 90 95 100 100 4.84 6.41 6.78 5.96 5.SO 4.81 3.91 2.83 - 1.89 - 0.90 (- 0.16) 0 L.E. radius: 2.48 Slope of radius through L.E.: 0.305 30 40 50 60 70 80 10.04 9.05 7.76 6.18 90 2.39 1.32 (0.19) ss 100 100 • 4.09 6.29 6.92 8.01 4.40 .... ... Station Ordinate 0 1.25 2.5 5.0 - 3.80 7.5 - 4.60 0 - 1.83 - 2.71 NACA 23021 (Stations and ordinates given in per cent of airfoil chord) 0 1.25 2.5 5.0 7.5 Ordinate ••• It •• 4.87 6.14 7.93 9.13 10 15 20 25 30 - 5.22 - 6.18 - 6.86 -7.27 -7.47 10 15 20 25 30 10.03 11.19 11.80 12.05 12.06 40 50 60 70 80 -7.37 - 6.81 - 5.94 - 4.82 - 3.48 40 50 60 70 SO. 11.49 10.40 8.90 7.09 5.05 - 1.94 90 - 1.09 95 (- 0.19) 100 100 2.76 1.53 (0.22) 00 95 100 100 0 L.E. radius: 3.56 Slope of radius through L.E.: 0.305 ~ Lower surface Upper surface Station ~ ..... ....... Station Ordinate 0 1.25 2.5 5.0 7.5 10 15 20 25 30 40 50 60 70 80 90 95 100 100 - 0 2.08 3.14 4.52 5.50 - 6.32 7.51 8.30 8.76 8.95 - 8.83 8.14 7.07 5.72 4.13 - 2.30 - 1.30 (- 0.22) 0 L.E. radius: 4.85 Slope of radius through L.E.: 0.305 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~r;,j APPENDIX III NACA 63-206 (Stations and ordinates given in percent of airfoil chord) NACA23024 (StatiODS and ordinates given in per cent of airfoil chord) Upper surface I Lower surface Station Ordinate Station Ordinate o 415 Upper Surface Station Ordinate Lower Surface Station Ordinate 1 0 0 0 .458 .703 1.197 2.438 .551 .677 .876 1.241 .542 .797 1.303 2.562 4.932 0 -.451 0.?:l7 1.331 3.853 6.601 0 4.017 5.764 8.172 9.844 9.423 15.001 20.253 25.262 30.265 11.049 12.528 13.237 13.535 13.646 10.577 - 7.647 14.999 - 8.852 19.747 - 9.703 24.738 - 10.223 29.735 - 10.454 7.429 9.930 14.934 19.941 1.776 2.189 2.526 3.058 3.451 5.068 7.571 10.070 15.066 20.059 - 1.144 1.341 1.492 1.712 1.859 40.256 50.235 60.202 70.162 80.116 12.928 11.690 10.008 7.988 5.687 39.744 - 10.278 49.766 - 9.482 59.798 - 8.242 69.838 - 6.664 79.884 - 4.803 24.950 29.960 34.970 39.981 44.991 3.736 3.926 4.030 4.042 3.972 25.050 30.040 35.030 40.019 45.009 - 1.946 1.982 1.970 1.900 1.7R2 90.064 95.036 3.115 1.724 89.936 94.964 50.000 55.008 60.015 65.020 70.023 3.826 3.612 3.338 3.012 2.642 50.000 54.992 59.985 64.980 69.977 - 1.620 75.023 80.022 85.019 90.013 95.006 2.237 1.804 1.356 .900 .454 74.927 79.978 84.981 89.987 94.994 100 ...... 0 2.223 3.669 6.147 8.399 100 - 0 3.303 4.432 5.862 6.860 - 2.673 - 1.504 0 L.E. radius: 6.33 Slope of radius through LE.: 0.305 100.000 0 100.000 - .537 - .662 - .869 -- 1.4221 1.196 - .952 - .698 - .447 - .212 - .010 .134 .178 0 L.E. radius: 0.297 Slope of radius through L.E.: 0.0842 THEORY OF WING SECTIONS 416 NACA 63-209 (Stat.ions and ordinates given in per cent of airfoil chord) Upper surface Lower surface Station Ordinate Station 0 0.436 0.680 1.170 2.408 0 0.796 0.973 1.255 1.765 0 0.563 0.820 1.330 2.592 4.897 7.394 9.894 14.901 19.912 2.510 3.077 3.539 4.263 4.792 24.925 29.940 34.956 39.971 44.986 Ordinate NACA 63-210 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Lower surface Ordinate Station Ordinate - - 0 0.696 0:833 1.041 1.393 0 0.430 0.669 1.162 2.398 0 0.876 1.107 1.379 1.939 0 0.570 0.831 1.338 2.602 - 0 0.776 0.967 1.165 1.567 5.103 7.606 10.106 15.099 20.088 - 1.878 2.229 2.505 2.917 3.200 4.886 7.382 9.882 14.890 19.902 2.753 3.372 3.877 4.665 5.240 5.114 7.618 10.118 15.110 20.098 - 2.121 2.524 2.843 3.319 3.648 5.169 5.414 5.530 5.518 5.391 25.075 30.060 35.044 40.029 45.014 - 3.379 3.470 3.470 3.376 3.201 24.917 29.933 34.951 39.968 44.985 5.647 5.910 6.030 6.009 5.861 25.083 30.067 35.049 40.032 45.015 - 3.857 3.966 3.970 3.867 50.000 55.012 60.022 65.029 70.033 5.159 4.834 4.429 3.958 3.430 50.000 54.988 59.978 64.971 69.967 - 2.953 2.644 2.287 1.898 1.486 50.000 55.013 60.024 65.032 70.036 5.599 5.235 4.786 4.264 3.684 50.000 54.987 59.976 64.968 69.964 - 3.393 3.045 2.644 2.204 1.740 75.034 80.032 85.027 90.019 95.009 2.861 2.267 1.663 1.067 0.512 74.966 79.968 84.973 89.981 94.991 - 1.071 0.675 0.317 0.033 0.120 75.038 80.036 85.030 90.021 95.010 3.061 2.414 1.761 1.121 0.530 74.962 79.964 84.970 89.979 94.990 - 1.271 0.822 0.415 0.087 0.102 100.000 I 0 100.000 0 L.E. radius: 0.631 Slope of radius through L.E.: 0.0842 100.000 0 100.000 ~.671 0 L.E. radius: 0.770 Slope of radius through L.E.: 0.0842 APPENDIX III NACA 63 1-212 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Upper surface Station 0 0.336 0.567 1.041 0 1.032 1.260 1.622 2.284 0 0.583 0.843 1.355 2.622 - 0 0.932 1.120 1.408 1.912 4.863 7.358 3.238 3.963 4.554 5.470 6.137 5.137 7.642 10.141 15.132 20.118 - 2.606 3.115 3.520 4.124 4.545 6.606 6.901 7.030 6.991 - 4.816 4.957 4.970 4.849 4.609 9.859 N ACA 63 1-412 (Stations and ordinates given in per cent of airfoil chord) Ordinate Station Ordinate 0 0.417 0.657 1.145 2.378 14.868 19.882 Ordinate 2.257 4.727 7.218 9.718 14.735 :::1 3.544 5.273 7.782 10.282 15.265 20.235 - 25.200 30.160 35.118 - 3.919 4.379 5.063 6.138 6.929 7.499 7.872 8.059 8.062 7.894 50.000 55.016 60.029 65.038 70.043 6.473 6.030 5.491 4.870 4.182 50.000 54.984 59.971 64.962 69.957 - 4.267 3.840 3.349 2.810 2.238 7.576 50.000 55.031 1 7.125 6.562 60.057 65.076 5.899 70. 087 1 5.153 75.045 3.451 2.698 74.955 79.958 84.965 89.975 94.988 - 1.661 1.106 0.601 0.190 0.066 75.089 80.084 i 85.0iO I 90.025 95.012 100.000 1.947 1.224 0.566 0 100.000 0 I 4.344 3.492 2.618 90. 049 1 1.739 0.881 95. 023 1 0 100.000 •• _ . _ _ •• _ _ _ _ ... _ L.E. radius: 1.087 Slope of radius through L.E.: 0.0842 Ordinate - 6.799 I Station 0 0.664 0.933 1.459 2.743 25.100 30.080 35.059 40.038 45.018 29.840 882 34. 39.924 1 44.9H4 . Lower surface 0 1.071 1.320 1.719 2.460 24.900 29.920 34.941 39.962 44.982 80.042 85.035 - Lower surface 417 ...... _ _ •• r 0 0.871 1.040 1.291 1.716 2.280 2.685 2.995 3.446 3.745 - 3.984 45.036 - 3.939 - 3.778 - 3.514 50.000 54.969 59.943 64.9201 69.913 - 3.164 - 2.745 2.278 1 1.i79 I - 1.265 40.076 I I- I- I 74.911 I - 0.764 79.916 t - 0.308 84.930 0.074 89.951 0.329 94.977 0.383 I I ioo.ooo ._ ... -- I __0 _.. .- .._ ...­ I.,.E. radius: 1.087 Slope of radius through L.E.: 0.1 sss THEORY OF WING SECTIONS 418 NACA63r215 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Ordinate Station Ordinate 0 0.601 0.863 1..380 Upper surface Station Lower surface Ordinate Station Ordinate 2..652 - 0 1.150 1..388 1..766 2.420 0 0.300 0.525 0.991 2.198 0 1.287 1.585 2.074 2.964 0 0.700 0.975 1.509 2.802 5.171 7.677 10.177 15.166 20.148 - 3.328 3.999 4.535 5.336 5.895 4.660 7.147 9.647 14.669 19.705 4.264 5.261 6.077 7.3'8 8.279 5.340 7.853 10.353 15.331 20.295 - 3.000 - 3.565 - 4.009 25..125 30.100 35.074 45.023 - 6.259 6.448 6.470 6.315 6.004 24.760 29.800 34.852 39.905 44.955 8.941 9.362 9.559 9.527 9.289 25.250 30.200 35.148 40.095 45.045 - 5.361 5.474 5.439 5.243 4.909 7.768 7.203 6.524 5.751 4.906 50.000 54.981 59.965 64.953 69..947 - 5.562 5.013 4.382 3.691 2.962 50.000­ 55.039 60.070 65.093 70.106 8.871 8.298 7.595 6.780 5.877 50.000 54.961 59.930 64.907 69.894 - 4.459 3.918 3.311 2.660 1.989 4.014 3.106 2..213 1.368 0.616 74.945 79.949 84.957 89.970 94..986 - 2.224 1.513 0.867 0.334 0.016 75.109 80.102 85.085 90.059 95.028 4.907 3.900 2.885 1.884 0.931 74.891 79.898 84.915 89.941 94.972 - 1.327 - 0.716 - 0.193 0.184 0.333 0 0.399 0.637 1.120 2.. 348 0 1.250 1.528 2.792 4..829 7.323 9.823 14.834 19..852 3.960 4.847 5.569 6.682 7.487 24.875 29.900 34.926 39.952 44.977 8.049 50.000 55.019 60.035 65.047 70..053 75.055 SO.051 85.043 90.030 95.014 100.000 Lower surface NACA63r415 (Stations and ordinates given in per cent of airfoil chord) 1.980 8.392 8.530 8.457 8.194 0 40..048 100.000 0 L.E. radius: 1.594 Slope of radius through L.E.: 0.0842 100.000 0 100.000 - 0 1.087 1.306 1.646 2.220 - 4.656 - 5.095 0 L.E. radius: 1.594 Slope of radius through L.E.: 0.1685 APPENDIX III NACA 63r615 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Ordinate Lower surface Station 419 NACA 63a-218 (Stations and ordinates given in per cent of airfoil chord) Upper surface Lower surface Station Station - 0 1.017 1.214 1.517 2.013 0 0.382 0.617 1.096 2.319 0 1.449 1.778 2.319 3.285 0 0.618 0.883 1.404 2.681 - 0 1.349 1.638 2.105 2.913 2.664 3.123 3.476 3.972 4.290 4.796 7.288 9.788 14.801 19.822 4.673 5.728 6.581 7.895 8.842 5.204 7.712 10.212 15.199 20.178 - 4.041 4.880 5.547 6.549 7.250 25.150 30..120 40.057 45.027 - 7.704 7.940 7.970 7.774 7.387 - 6.839 6.161 5.384 4.537 3.650 - 2.754 1..894 1.113 0.467 0.032 Ordinate 0 0.205 0.418 0.866 2.0li0 0 1.317 1.634 2.159 3.129 0 0.795 1.082 1.634 2.950 4.492 6.973 9.473 14.504 19.558 4.560 5.667 6.578 8.010 9.066 5.508 8.027 10.527 20.442 - 24.625 29.700 39.857 44.932 9.830 10.331 10.587 10.598 10.384 25.37S 30.300 35.222 40.143 45.068 - 4.460 4.499 4.407 4.172 3.814 24.850 29.880 34.911 39.943 44.973 9.494 9.884 10.030 9.916 9.577 50.000 55.058 60.105 65.139 70.159 9.974 9.393 8.665 7.809 6.847 50.000 04.942 59.895 64.861 69.841 - 3.356 - 2.823 - 2.239 9.045 8.351 7.526 - 1.015 50.000 55.023 60.042 65.055 70.062 5.594 50.000 64.977 59.958 64.945 69.938 75.163 5.800 74.837 79.847 84.873 89.911 94.958 - 0.430 0.083 0.483 0.704 0.651 75.064 80.059 85.049 90.034 95.016 4.544 3.486 2.459 1.501 0.664 74.936 79.941 84.951 89.966 94.984 34.778 so. 153 85.127 90.089 95.042 100.000 4.693 3.555 2.398 1.245 0 1~.496 100.000 Ordinate Ordinate - 1.629 0 100.000 6.597 0 35.()89 100.000 0 ~_._-- L.E. radius: 1.594 Slope of radius through L.E.: 0.2527 L.E. radius: 2.120 Slope of radius through L.E.: 0.0842 THEORY OF WING SECTIONS 420 NACA 63r418 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Ordinate Lower surface Station NACA 63.-618 (Stations and ordinates given in per cent of airfoil chord) Upper surface Ordinate Station Ordinate Lower surface Station Ordinate 0 0.267 0.487 0.945 2.140 0 1.484 1.833 2.410 3.455 0 0.733 1.013 1.555 2.860 - 0 1.284 1.553 1.982 2.711 0 0.156 0.361 0.797 1.965 0 1.511 1.878 2.491 3.616 0 0.844 1.139 1.703 3.035 - 0 1.211 1.458 1.849 2.500 4.593 7.077 9.577 14.602 19.645 4.975 6.139 7.087 8.560 9.632 5.407 7.923 10.423 15.398 20.355 .;.. - 3.711 4.443 5.019 5.868 6.448 4.393 6.868 9.367 14.404 19.469 5.268 6.542 7.586 9.219 10.418 5.607 8.132 10.633 15.596 20.531 - 3.372 3.998 4.484 5.181 5.642 24.699 29.760 34.823 39.886 44.949 10.385 10.854 11.058 10.986 10.672 25.301 30.240 35.177 40.114 45.054 - 6.805 6.966 6.938 6.702 6.292 24.549 29.640 34.734 39.829 44.919 11.273 11.822 12.086 12.056 11.767 25.451 30.360 35.266 40.171 45.081 - 5.903 5.990 5.906 5.630 5.197 50.000 55.046 60.083 65.110 70.125 10.148 9.446 8.596 7.626 6.564 50.000 54.954 59.917 64.890 69.875 - 5.736 - 5.()66 4.312 3.506 2.676 50.000 55.069 60.125 65.164 70.187 11.251 10.541 9.667 8.655 7.534 50.000 54.931 59.875 64.836 69.813 - 4.633 3.971 3.241 2.475 1.702 75.128 80.119 85.099 90.069 95.032 5.438 4.280 3.130 2.017· 0.978 74.872 79.881 84.901 89.931 94.968 - 1.858 - 1.096 - 0.438 0.051 0.286 75.191 80.178 85.147 90.103 95.048 6.330 5.073 3.800 2.531 1.293 74.809 79.822 84.853 89.897 94.952 - 0.960 - 0.297 0.238 0.571 0.603 100.000 I 0 100.000 0 L.E. radius: 2.120 Slope of radius through L.E.: 0.1685 100.000 0 100.000 0 L.E. radius: 2.120 Slope of radius through L.E.: 0.2527 421 APPENDIX III NACA 63r221 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station NACA 63.-421 (Stations and ordinates given in per cent of airfoil chord) Lower surface Upper surface Ordinate Station Ordinate Station 1.425 0 - 1.527 - 1.861 - 2.414 0.237 0.452 0.902 2.708 - 3.385 o Lower surface Ordinate Station Ordinate 0 0.7fl3 1.048 1.598 2.914 0 - 1.461 - 1.774 - 2.289 3.181 0 0.367 0.600 1.075 2.292 0 1.627 2.001 2.628 3."57 0 0.633 0.900 4.763 7.253 9.753 14.767 19.792 5.375 6.601 7.593 9.111 10.204 5.237 7.747 10.24i 11).233 20.208 - 4.743 5.753 6.559 7.765 8.612 4.5271 5.675 5.473 7.007 I 7.010 7.993 9.506 I 8.097 10.494 14.535 1 9.774 t 15.465 19.5851 10.993 20.415 24.824 29.860 34.897 39.934­ 44.969 10.946 11.383 11.529 11.369 10.949 25.176 30.140 35.103 40.066 45.031 - 9.156 9.439 9.469 9.227 8.i59 24.6491 11.837 29.719 12.352 34.;93 J 12.558 39.8()7: 12.439 4-1.937; 12.044 50.000 55.027 60.048 65.063 70.071 10.309 9.485 8.512 7.426 6.262 8.103 50.000 55.054 11.412 10.580 60.096 9.582 65.126, 70.1-l3 i 8.455 7.232 75.073 5.054 3.849 2.693 1.629 0.708 75.1451 80.135 i 5.947 74.855 2.367 4.643 I 79.865 I - 1.459 85.1·11 3.364 so.067 85.056 90.039 95.018 100.000 0 50.000 54.9i3 59.95~ I- I- - ;.295 I- 6.370 5.366 64.931 i - 69.929\74.927 79.933 l 84.944 1 89.961 94.982 - il 4.318 3.264 2.257 1.347 0.595 0.076 __ .100.000 L_.~ . -.--~ L.E. radius: 2.650 Slope of radius through L. E.: 0.0842 I {i 2.086! 9(!.07~ i I 0 1.661 2.054 2.717 3.925 - 4.U! 5.314 6.029 7.082 7.809 25.351 30.281 35.207 40.133 I 45.OG3 1 - 8.257 8.464 8.438 8.155 7.60-1 5O.0on 7.000 6.200 54.946 I 11 I' J - !- I- 59.9t).J 5.298 6-l.8~: I - 4.335 69.851 3.344 II- 8-4.889 I - O.G72 2.144 , 89.922 I- O.01H l;:;;L..~.l~ l;:~ L_.~·242 1..E. radius: 2.650 Slope of radius through L.E.: O.l6S5 THEORY OF WING SECTIONS 422 NACA 63A210 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Ordinate Lower surface Station NACA 64-108 (Stations and ordinates given in per cent of airfoil chord) Upper surface Ordinate Station Lower surface Ordinate Station Ordinate 0 0.423 0.664 1.151 2.384 0 0.868 1.058 1.367 1.944 0 0.577 0.836 1.349 2.616 - 0 0.756 0.900 l.l25 1.522 0 0.472 0.719 1.215 2.460 0 0.682 0.828 1.058 1.457 0 0.528 0.781 1.285 2.540 - 0 0.632 0.758 0.950 1.271 4.869 7.364 9.863 14.869 19.882 2.769 3.400 3.917 4.729 5.328 5.131 7.636 10.137 15.131 20.118 - 2.047 2.428 2.725 3.167 3.468 4.956 7.455 9.955 14.958 19.962 2.032 2.471 2.832 3.405 3.835 5.044 7.545 10.045 15.042 20.038 - 1.716 2.047 2.316 2.733 3.039 24.898 29.916 34.935 39.955 44.975 5.764 6.060 6.219 6.247 6.151 25.102 30.084 35.065 40.045 45.025 - 3.662 3.764 3.771 3.689 3.523 24.968 29.974 34.980 39.987 44.994 4.152 4.370 4.494 4.528 4.431 25.032 30.026 35.020 40.013 45.006 - 3.256 3.398 3.464 3.456 3.335 49.994 55.012 60.028 65.041 70.052 5.943 5.637 5.245 4.772 4.227 50.006· 54.988 59.972 64.959 69.948 - 3.283 2.985 2.641 2.262 1.861 50.000 55.005 60.010 65.013 70.015 4.236 3.959 3.617 3.219 2.777 50.000 54.995 59.990 64.987 69.985 - 3.132 2.863 2.545 2.189 1.805 75.061 80.074 85.072 90.050 95.026 3.624 2.974 2.254 1.519 0.769 74.939 79.926 84.928 89.950 94.974 - 1.464 1.104 0.812 0.539 0.279 75.016 80.015 85.013 90.010 95.005 2.302 1.802 1.297 0.808 0.364 74.984 79.985 84.987 89.990 94.995 - 1.406 1.006 0.625 0.292 0.048 100.000 0.021 100.000 - 0.021 100.000 L.E. radius: 0.742 T.F. radius: 0.023 Slope of radius through L.E.: 0.095 0 100.000 0 L.E. radius: 0.455 Slope of radius through L.E.: O.()4-2 APPENDIX III NACA 64-110 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station 0 0.465 Ordinate Station Ordinate 1.207 2.450 0 0.844 1.023 1.303 1.793 4.945 7.443 9.944 14.947 19.953 2.500 3.037 3.479 4.178 4.700 24.959 29.967 34.175 39.984 44.992 5.087 0.71~ 50.000 55.007 60.012 65.016 70.019 75.020 80.019 85.016 90.012 95.006 100.000 Lower surface 0 o.sss 0.788 1.293 2.550 D.055 1.557 10.056 1"5.053 20.04:7 25.04:1 30.033 - 0.442 0.524 0.645 0.836 - 2.184 2.613 2.963 3.506 3.904 4.934 7.432 9.933 14.937 19.943 1.719 2.115 2.444 2..970 3.367 5.066 7.568 10.067 15.063 20.057 - 1.087 1.267 1.410 1.624 I.nS 4.191 4.378 4.465 4.452 4.295 24.952 29.961 34.971 39.981 25.048 30.039 35.029 40.019 45'(J09 - 1.877 - 1.935 - 1.951 44.991 3.667 3.879 4.011 4.066 4.014 4.034 3.690 3.284 2.830 2.341 50.000 55.008 ! 60.015 I 65.020 70.023 I 3.878 3.670 3.402 3.080 2.712 50.000 - 1.672 54.992 - 1.480 59.985 - - 1.260 64.980 - 1.020 69.977 - 0.768 50.000 54.993 59.988 64.984 69.981 - 74.980 79.981 84.984 89.988 - 1.833 - 1.324 I 94.994 100.000 - 0..840 - 0.413 I - I Ordinate 0 0.541 0.796 1.302 2.560 5.138 4.786 4.356 0 Ordinate Station 0 0.542 0.664 0.859 1.208 40.016 45.008 2.729 2.120 1.512 0.929 0.406 Station Lower surface 0 0.459 0.704 1.198 2.440 35.025 3.313 Upper surface 0 0.794 0.953 1.195 1.607 5.495 5.524 5.391 3.860 NACA 64-206 (Stations and ordinates given in per cent of airfoil chord) - - 5.350 423 0.090 0______ L.E. radius: 0.720 Slope of radius through L.E.: 0.042 I i 2.307 800024 1.868 85.020 1.410 90.015 1I 0.940 0.473 95.007 75.025 100.000 1 0 0 - 1.924 - 1.824 I 74.975 I I i - 0.517 79.976 - 0.276 0.064 84.980 0.094 89.985 I 94.993l 0.159 Ii 100.000 t 0 LE. radius: 0.256 Slope of radius through L.E.: 0.084 THEORY OF WING SECTIONS 424 NACA 64-208 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Ordinate Lower surface Station NACA 64-209 (Stations and ordinates given in per cent of airfoil chord) Upper surface Ordinate ·Station Ordinate Lower surface Station Ordinate 0 - 0.686 - 0.819 - 1.018 0 0.445 0.688 1.180 2.421 0 0.706 0.862 1.110 1.549 0 0.555 0.812 1.320 2.579 - 0 0.606 0.722 0.896 1.177 0 0.438 0.680 1.172 2.411 0 0.786 0.959 1.232 1.716 0 0.562 0.820 1.328 2.589 4.912 7.410 9.909 14.915 19.924 2.189 2.681 3.089 3.741 4.232 5.088 7.590 10.091 15.085 20.076 - 1.557 1.833 2.055 2.395 2.640 4.901 7.398 9.899 14.905 ~9.915 2.423 2.965 3.413 4.127 4.663 24.935 29.948 39.974 44.988 4.598 4.856 5.009 5.063 4.978 25.065 30.052 35.039 40.026 45.012 - 2.808 2.912 2.949 2.921 2.788 24.927 29.941 34.956 39.971 44.986 , 50.000 55.011 60.020 65.027 70.031 4.787 4.506 4.152 3.733 3.263 50.000 54.989 59.980 64.973 69.969 - 2.581 2.316 2.010 1.673 1.319 50.000 55.012 60.022 65.030 1 70.035 , 75.032 2.749 2.200 1.634 1.067 0.522 74.968 79.969 84.973 89.981 94.990 - 0.959 0.608 0.288 0.033 0.110 34.961 ~.031 85.027 90.019 95.010 100.000 0 100.000 0 I,.E. radius: 0.455 Slope of radius through L.E.: 0.084 - 1.344 5.099 7.602 10.101 15Jl95 20.085 - 1.791 2.117 2.379 2.781 3.071 5.064 5.345 5.509 5.561 5.459 25.073 30.059 35.044 40.029 45.014 - 3.274 3.401 3.449 3.419 5.239 4.921 4.523 4.056 3.533 50.000 54.988 59.978 64.970 69.965 - 3.033 2.731 2.381 . 1.996 1.589 75. 036 1 2.964 2.360 80.035 85.030 ; 1.742 90.021 i 1.128 0.543 95.011 74.964 79.965 84.970 89.979 94.989 - 1.174. I I I 100.000 0 100.000 - 3.269 - 0.;68 - 0.396 - 0.094 0.089 0 L.E. radius: 0.579 Slope of radius through L.E.: 0.084 t APPEl·lDIX III NACA 64-210 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Lower surface Ordinate Station Ordinate 425 NACA 64 1-112 (Stations and ordinates given in per cent of airfoil chord) t· pper surface Station Ordinate o 0 0.431 0.673 1.163 2.401 0 0.867 1.056 1.354 1.884 4.890 7.387 9.887 14.894 19.905 2.656 3.248 3.736 4.514 5.097 2.024 5.110 2.400 7.613 10.113 I - 2.702 15.106 - 3.1~ 20.095 1 - 3.50il 2.967 3.605 4.128 4.956 5.571 24.919 29.934 34.951 39.968 44.985 5.533 5.836 6.010 6.059 5.938 25.081 30.066 i 35.049140.032 45.015 - 3.743 3.Sg-1 3.950 3.917 3.748 6.024 6.330 6.493 6.517 6.346 50.000 55.014 60.025 65.033 70.038 5.689 5.333 4.891 4.375 3.799 50.000 54.987 1 59.975 I 64.967169.962 - 3.483 3.143 2.749 1.855 6.032 5.604 5.084 4.489 3.836 75.040 80.038 85.033 90.024 95.012 3.176 2.518 1.849 1.188 0.564 74.960 I 79.962 , 84.968 89.977 94.988 - 1.386 - O.!2? - O.D03 - 0.15-1 0.068 3.143 2.427 1.718 1.044 0.446 100.000 0 0 0.569 0.827 1.337 2.599 0 0.767 0.916 1.140 1.512 - Ii- I- ! 100.000 1 2.31~ 0 ---- L.E. radius: 0.720 Slope of radius through L.E.: 0.084 1.002 1.213 1.543 2.127 o L.E. radius: 1.040 Slope of radius through I ... I~.: 0.042 'l'HEORY OF WING SECTIONS 426 NACA 64 1-212 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Lower surface Ordinate Station Ordinate NACA 641-412 (Stations and ordinates given in per cent of airfoil chord) Upper surface Lower surface Station Ordinate Station - Ordinate 0.841 1.353 2.618 - 0 0.925 1.105 1.379 1.846 0 0.338 0.569 1.045 2.264: 0 1.064 1.305 1.690 2.393 0 0.662 0.931 1.455 2.736 3.123 3.815 4.386 5.291 5.968 5.132 7.636 10.135 15.128 20.114 - 2.491 2.967 3.352 3.945 4.376 4.738 7.229 9.730 14.745 19.772 3.430 4.231 4.896 5.959 6.760 5..262 7.771 10.270 15.255 20.228 - 2.166 - 2.535 -2.828 - 3.267 - 3.576 24..903 29.921 34 . 941 39.961 44.982 6..470 6.815 7.008 7.052" 6.893 25..()97 30.079 35.. 059 40.039 45.018 - 4.680 4.871 4.948 4.910 4.. 703 24.805 29.842 34.882 39.923 44.963 7.363 7.786 8.037 8.123 7.988 25.195 30.158 35.118 40.077 45.037 - 3.783 - 3.608 50.000 55.016 60.029 65.039 70.045 6.583 fi.151 5.619 5.004 4.322 50.000 54.. 984 59.971 64.961 69.955 - 4.377 3.961 3.477 2.944 2.378 50.000 55.032 60.059 65.078 70.090 7.686 7.246 6.690 6.033 5.293 50.000 54.968 59.941 64.922 69.910 - 75.047 80.045 85.038 90.027 95.013 3.590 2.825 2.054 1.303 0.604 74.953 79.955 84.962 89.973 - 1.800 1.233 0.708 0.269 0.028 75.094 80.089 85.076 90.055 95.027 4.483 3.619 2.722 1.818 0.919 74.906 79.911 84.924 89.945 94..973 - 0.903 - 0.435 - 0.038 1 0.250 0.345 0 0.418 0.659 1.147 2.382 0 1.025 1.245 1.593 2.218 4.868 7.364 9.865 14.872 19.886 100.000 0 0 0~582 94.987 100.000 0 L.E. radius: 1.040 Slope of radius through L.E.: 0.084 100.000 0 100.000 - 0 0.864 1.025 1.262 1.849 - 3..898 - 3.917 - 3.839 3.274 2.866 2.406 1.913 1.405 0 L.E. radius: 1.040 Slope of radius through L.E.: 0.168 APPENDIX III NACA 64t-215 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Lower surface Ordinate Station 427 NACA 64r415 (Stations and ordinates given in per cent of airfoil chord) Upper surface Ordinate Station Ordinate Lower surface Station Ordinate 0 0.399 0.637 1.122 2.353 0 1.254 1.522 1.945 2.710 0 0.601 0.863 1.378 2.641 - 0 1.154 1.382 1.731 2.338 0 0.299 0.526 0.996 2.207 0 1.291 1.579 2.038 2.883 0 0.701 0.974 1.504 2.793 4.836 7.331 9.831 14.840 19.857 3.816 4.661 5.356 6.456 7.274 5.164 7.669 10.169 15.160 20.143 - 3.184 3.813 4.322 5.110 5.682 4.673 7.162 9.662 14.681 19.714 4.121 5.075 5.864 7.122 8.066 5.327 7.838 10.338 15.319 20.286 - 2.857 3.379 3.796 4.430 4.882 24.878 29.901 34.926 39.952 44.977 7.879 8.290 8.512 8.544 8.319 25.122 30.099 35.074 40.048 45.023 - 6.089 6.346 6.452 6.402 6.129 24.756 29.803 34.853 39.904 44.954 8.771 9.260 9.541 9.614 9.414 25.244 30.197 35.147 40.096 45.046 - 5.191 D.372 5.421 5.330 5.034 50.000 55.020 60.036 65.048 70.055 7.913 7.361 6.691 5.925 5.085 50.000 54.980 59.964 64.952 69.945 - 5.707 5.171 4.549 3.865 3.141 50.000 9.016 8.456 7..762 6.954 6.055 50.000 54.960 59.928 64.904 69.889 - 4..604 - 4.076 - 2.834 - 2.167 15.058 4.191 3.267 2.349 1.466 0.662 74.942 79.945 84.954 89.967 94.984 - 2.401 - 1.675 5.084 4.062 3.020 1.982 0.976 74.885 79.891 84.908 89.934 94.968 - 1.504 - 0.878 - 0.328 0.086 0.288 100.000 0 80.055 85.046 90.033 95.016 100.000 0 1 100 .000 - 1..003 - 0.432 - 0.030 0 L.E. radius: 1.590 Slope of radius through L.E.: 0.084 55.040 60.072 65.096 70.111 75.115 so. 109 85.092 90.066 95.032 100.000 0 0 - 1.()91 - 1.299 - 1.610 - 2.139 - 3.478 L.E. radius: 1.590 Slope of radius through L.E.: 0.168 THEORY OF VlING SECTIONS 428 NACA 64r218 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Ordinate Lower surface Station NACA 64,-418 (Stations and ordinates given in per cent of airfoil chord) Upper surface Ordinate Station Lower surface Ordinate Station - Ordinate 0 0.380 0.617 1.099 2.325 0 1.473 1.785 2.279 3.186 0 0.620 0.883 1.401 2.675 - 0 1.373 1.645 2.065 2.814 0 0.263 0.486 0.950 2.152 0 1.508 1.840 2.370 3.357 0 0.737 1.014 1.550 2.848 4.804 7.297 9.797 14.808 19.828 4.497 5.496 6.316 7.612 8.576 5.196 7.703 10.203 15.192 20.172 - 3.865 4.648 5.282 6.266 6.984 4.609 7.095 9.595 14.617 19.657 4.800 5.908 6.823 8.277 9.366 5.391 7.905 10.405 15.383 20.343 - 3.536 4.212 4.755 5.585 6.182 24.853 29.881 34.912 39.942 44.972 9.285 9.760 10.009 10.023 9.725 25.147 30.119 35.088 40.058 45.028 - 7.495 7.816 7.949 7.881 7.535 24.707 29.763 34.823 39.885 44.9-15 10.176 10.730 11.037 11.093 10.820 25.293 30.237 35.177 40.115 45.055 - 6.596 . 6.842 6.917 6.809 6.440 50.000 55.024 9.217 8.540 7.729 6.812 5.814 50.000 54.976 59.957 64.943 69.935 - 7.011 6.350 5.587 4.752 3.870 50.000 55.047 60.086 65.114 70.131 10.320 9.635 8.799 7.841 6.784 50.000 54.953 59.914 64.886 69.869 - 5.908 5.255 4.515 3.;21 2.896 4.760 3.683 2.623 1.617 0.716 74.932 79.936 84:.946 89.962 94.981 - 2.970 2.091 1.277 0.583 0.084­ 7~.135 5.654 4.477 3.294 2.132 1.030 74.865 79.873 84.892 89.923 94.963 - 2.074 1.293 0.602 0.064 0.234 60.043 65.057 70.065 75.068 so.064 85.054 90.038 95.019 100.000 0 100.000 0 L.E. radius: 2.208 Slope of radius through L.E.: 0.084 80.127 85.108 I 90.0771 95.037 100.000 I 0 , 100.000 0 - 1.308 - 1.560 - 1.942 - 2.613 0 L.E. radius: 2.208 Slope of radius through L.E.: 0.168 APPENDIX III NACA 64r618 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station 429 N ACA Mc-221 (Stations and ordinates given in per cent of airfoil chord) Lower surface L pper surface Ordinate Station Ordinate Station Lower surface I Ordinate Station Ordinate 0 0.150 0.359 0.805 1.982 0 1.534 1.885 2.452 3.518 0 0.850 1.141 1.695 3.018 - 0 1.234 1.465 1.810 2.402 0 0.362 0.596 1.075 2.297 0 1.690 2.049 2.618 3.665 0 0.638 0.904 1.425 2.703 4.417 6.895 9.395 14.427 19.486 5.093 6.312 7.322 8.937 10.153 5.583 8.105 10.605 15.573 20.514 - 3.197 3.768 4.220 4.899 5.377 4.772 7.264 9.763 14.776 19.799 5.182 6.334­ 7.282 8.778 9.889 5.228 7.736 10.237 15.224 20.201 - 24.560 29.645 34.735 39.827 44.917 11.065 11.698 12.065 12.163 11.915 25.440 30.355 35.265 40.173 45.083 - 5.695 5.866 5.885 5.737 5.345 24.829 29.861 34.897 39.933 10.701 11.240 11.510 11.502 11.125 25.171 30.139 35.103 40.067 45.032 - 50.000 55.071 60.129 65.171 70.196 11.423 10.730 9.870 8.870 7.754 50.000 54.929 59.871 64.829 69.804 - 4.805 4.160 3.444 2.690 1.922 50.000 55.027 60.050 65.065 70.0i5 10.507 9.702 8.749 7.679 6.521 54.9731- 50.000 59.950 64.935 69.925 - 8.301 7.512 - 6.607 - 5.619 - 4.577 75.203 80.191 85.161 90.115 95.056 6.544 5.270 3.963 2.646 1.344 74.797 79.809 84.839 89.885 94.944 - 1.174 - 0.494 0.075 0.456 0.552 75.077 80.073 85.001 90.044 95.021 74.923 79.927 - 3.520 - 2.490 M.939 . 89.956 94.979 - 0.727 100.000 0 100.000 !- _. __ 0 .. .-._-­ L.E. radius: 2.208 Slope of radius through L.E.: 0.253 4-1. 008 100.000 -- --~ i 1 5.310 4J)82 2.885 1.761 0.765 I 0 Il~.OOO 1 L.E. radius: 2.884 Slope of radius through 0 - 1.590 - 1.909 - 2.404 - 3.293 4.550 5.486 6.248 7.432 -8.~7 8.911 9.296 9.450 9.360 8.935 - 1.539 - 0.133 0 I~.I~.: O.OSl - THEORY 0/;' WING SECTION8 430 NACA 64r421 (Stations and ordinates given in per cent of airfoil chord) Upper surface Ordinate Station Lower surface NACA 64A210 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Ordinate Station 0 0.424 0.665 1.153 2.387 Ordinate - Lower surface - Station Ordinate 0 0.856 1.044 1.342 1.895 0 0.576 0.835 1.347 2.613 0 - 0.744 - 0 0.227 0.445 0.903 2.096 0 1.723 2.101 2.707 3.834 0 0.773 1.055 1.597 2.904 - 0 1.523 1.821 2.279 3.090 4.545 7.028 9.528 14.553 19.599 5.482 6.744 7.786 9.442 10.678 5.455 7.972 10.472 15.447 20.401 - 4.218 5.048 5.718 6.750 7.494 4.874 7.369 9.868 14.874 19.885 . 2.685 3.288 3.792 4.592 5.200 5.126 7.631 10.132 15.126 20.115 - 1.963 24.657 29.723 34.794 39.865 44.936 11.591 12.209 12.539 12.572 12.220 25.343 30.277 35.206 40.135 45.064 - 8.011 8.321 8.419 8.288 7.840 24.900 29.917 34.935 39.955 44.975 5.656 5.984 6.192 6.274 6.208 25.100 30.083 35.065 40.045 45.025 - 3.554 - 3.688 - 3.744 11.610 10.797 9.819 8.708 7.491 50.000 54.945 59.901 64.869 69.850 - 7.198 6.417 5.535 4.588 3.603 49.994 55.012 60.028 65.042 70.054 6.014 5.714 5.323 4.852 4.310 50.006 54.988 59.972 64.958 69.946 6.203 4.876 3.556 2.276 1.079 74.846 79.855 84.878 89.913 94.958 - 2.623 1.692 0.864 0.208 0.185 75.063 80.076 85.074 90.052 95.027 3.702 3.037 2.301 1.551 0.785 74.937 - 1.542 1.167 79.924 84.926 I - 0.859 89.948 - 0.571 94.974 - 0.295 100.000 0.021 50.000 55.055 60.099 65.131 70.150 75.154 80.145 85.122 90.087 95.042 100.000 I 0 100.000 0 L.E. radius: 2.884 Slope of radius through L.E.: 0.168 - 0.886 - 1.100 - 1.473 - 2.316 - 2.600 - 3.030 - 3.340 - 3.716 - 3.080 I 354 1 -- a.3.062 - 2.719 - 2.342 - 1.944 I- 100.000 - - 0.021 LE. radius: 0.687 T.E. radius: 0.023 Slope of radius through L.E.: 0.095 I APPENDIX 111 NACA 64A4:10 (Stations and ordinates given in per cent of airfoil chord) Upper surface I i 0 0.350 : 0.582 1.059 2.276 NACA 64 1A212 (Stations and ordinates given in per cent of airfoil chord) Lower surface Station Ordinate Station Ordinate 431 Upper surface Station Lower surface Ordinate Station Ordinate 0 0.902 1.112 1.451 2.095 0 0.650 0.918 1.441 2.724 - 0 0.678 0.796 0.969 1.251 3.034 3.865 4.380 5.366 6.126 5.251 7.770 10.263 1'5.252 20.230 - 1.592 1.919 1.996 2.244 2.406 4.849 7.343 9.842 14.849 19.862 3.145 3.846 4.432 5.358 6.060 5.151 7.657 10.158 15.151 20.138 6.705 7.131 7.414 7.552 7.522 25.200 30.166 35.129 40.090 45.050 - 2.499 2.537 2.518 2.436 2.266 24.880 29.900 34.922 39.946 44.970 6.584 6.956 7.189 7.272 7.177 25.120 / - 4.482 30.100 - 4..660 35.078 f - 4.741 40.054 4.714 45.030 4.549 49.989 . 7.344 55.025 ' 7.040 60.057 . 6.624 65.085 j a.l06 70.108 J 5.490 50.011 2.O'M 49.993 55.015 60.034 65.050 70.064 6.935 6.570 6.103 5.544 4.903 50.007154.985 59.966 i 64.950 69.936 - 75.075 80.090 74.925 I - 2.037 79.910 - 1.563 84.912 - 1.159 89.938 - 0.771 94.968 - 0.398 4.749 7.230 9.737 14.748 19.770 24.800 29.834 34.871 39.910 44.950 ; ; : . I I 4.780 54.975 59.943 64.915 69.892 3.967 3.018 2.038 1.028 74.874 79.849 84.852 89.896 94.947 100.000 1 0.021 100.000 75.126 SO.151 85.148 90.104 95.053 J i ; j j 1~ 0 0.409 0.648 II 1.135 2.365 0 1.013 1.233 1.580 2.225 - 1.736 1.418 1.086 0.760 - 0.460 0.229 0.132 0.076 0.048 90.062 95.032 4.197 3.433 2.601 1.751 0.888' - 0.021 100.000 0.025 L.E. radius: 0.687 T.E. radius: 0.023 Slope of radius through L.E.: 0.190 85.088 - I 0 0.591 ·0.852 I 1.365 2.635 I - I- 0 0.901 1.075 1.338 1.803 - 2.423 - 2.874 3.240 3.796 - 4.200 1I ­ II- 1- I IO?OOO I- 4.275 3.918 3.499 3.034 2.537 0.025 L.E. radius: 0.994 r.s, radius: 0.028 Slope of radius through L.E.: 0.095 THEORY OF WING SECTIONS 432 NACA 64JA215 (Stations and ordinates given in per cent of airfoil chord) L pper surface Lower surface NACA 65-206 (Stations and ordinates given in per cent of airfoil chord) Upper surface Ordinate Station Ordinate Station 0 1.243 1.509 1.930 2.713 0 0.612 0.876 1.393 2.667 0 1.131 1.351 1.688 2.291 0 0.460 0.706 1.200 2.444 4.811 7.304 9.802 14.811 19.827 3.833 4.683 5.391 6.510 7.351 5.189 7.696 10.198 15.189 20.173 - 3.111 - 3.711 - 4.199 24.849 29.875 34.903 39.933 44.963 7.975 8.417 8.686 8.766 8.627 25.151 30.125 35.()97 40.067 45.037 49.992 55.018 60.042 65.063 70.079 8.308 7.843 7.258 6.566 5.782 50.008 54.982 59.958 64.937 69.921 75'()93 80.111 85.109 90.076 95.039 4.926 4.017 3.039 2.046 1.039 74.907 79.889 84.891 89.924 94.961 100.000 0.032 100.000 Station 0 0.388 0.624 1.107 2.333 I Lower surface - Ordinate Station Ordinate 0 0.524 0.642 0.822 1.140 0 0.540 0.794 1.300 2.556 0 0.424 0.502 0.608 0.768 - 5.491 4.939 7.437 9.936 14.939 19.945 1..625 2.012 2.340 2.869 3.277 5.061 7.563 10.064 15.061 20.055 - 5.873 6.121 6.238 6.208 5.999 24.953 29.962 34.971 39.981 44.990 3.592 3.824 3.982 4.069 4.078 25.047 30.038 35.029 40.019 45.010 - 5.648 5.191 4.654 4.056 3.416 50.000 55.009 60.016 65.022 70.<Y>-6 4.003 3.836 3.589 3.276 2.907 50.000 54.991 59.984 64.978 69.974 - - 2.766 2.147 1.597 1.066 0.549 75.028 80.027 850024 90.018 2.489 2.029 1.538 1.027 1 95'(x)9 ! 0.511 74.972 79.973 84.976 - 0.699 - 0.437 - 0.192 0.007 0.121 - - 4.948 i - 0.032 L.E. radius: 1.561 T.E. radius: 0.037 Slope of radius through L.E.: 0.095 100.000 t 0 89.982 94.991 I 100.()()() - - - 0.993 - 1.164 - i.306 - 1.523 - 1.685 - 1.802 - 1.880 - 1.922 - 1.927 - 1.888 1.797 1.646 1.447 1.216 0.963 0 L.E. radius: 0.240 Slope of radius through l..E.: 0.084 433 APPENDIX III NACA 65-209 (Stations and ordinates given in per cent of airfoil chord) Upper surface Ordinate Station Lower surface Station Ordinate 0 0.441 0.684 1.177 2.417 0 0.748 0.912 1.162 1.605 0 0.559 0.816 1.323 2.583 - 0 0.648 0.772 0.948 1.233 4.908 7.405 °2.275 2.805 3.251 3.971 5.092 7.595 10.096 15.091 20.082 - 25.071 30.058 35.044 40.029 45.014 - ~.904 14.909 19.918 4.522 24.929 4.944 29.942 5.254 5.4:61 34.956 39.971 44.986 I 5.567 5.564 I 5.439 50.000 5.181 013 55. 1 4.814 60.024 4.358 65.033 3..828 70.039 3.237 75.041 2.601 80.040 1.933 035 85. 90.026 1 1.255 0.596 95.013 100.000 I 0 Upper surface Station Lower surface I Ordinate Station Ordinate o 0.435 0.678 1.169 2.408 0 0.819 0.999 1.273 1.757 - 0 0.719 0.859 1.059 1.385 1.643 1.957 2.217 2.625 2.930 4.898 7.394 9.894 14.899 19.909 2.491 3.069 3.555 4.338 4.938 5.102 7.606 10.106 15.101 20.091 - 1.859 2.221 2.521 2.992 3.346 3.154 3.310 3.401 3.425 3.374 24.921 30.0641- 3.788 25.079 - 3.607 34.951 5.397 5.732 5.954 39.9()s 6.067 40.032 t - 3.~ 6.058 45.016 50.000 I - 3.233 I 54.9871 - 2.991 ~9.976 I - 2.672 ti4 .967 - 2.298 69.961 - 1.884 74.959 79.960 84.965 89.974 987 94. 1 100.000 NACA 65-210 (Stations and ordinates given in pel" cent of airfoil chord) 1.447 1.009 0.587 0.221 0.036 0 L.E. radius: 0.552 Slope of radius through L.E.: O.OSl 29.936 44.984 I i 5.915 5.625 5.217 4.712 4wl28 75.045 8l!.044 , 85 90.028 95.01-l, 3.479 2.783 2.057 1.327 0.622 1\ Jl38I' II- 35.~~ - I- 3.894 - 3.868 I 50.000 i 55.014 OO.tl27 j (\5.03() 70.043 I 0 i 0.565 0.822 1.331 I 2.592 I 50.000 I 54.986 SU.973 I 64.96! 69.951 i - 3.709 3.435 3.075 2.652 2.184 74.955 Ii - 79.9~6 I - 1.689 1.191 0.711 0.293 0.010 I- 1° I II! !- 1- 1 8-1.062 I 89. 972 94.986 I! I_~:~~J_~ i 100.000 - 0__ 1..E. radius: 0.687 Slope of radius through L.E.: 0.084­ THEORY OF WING SECTIONS 434 NACA 65-410 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Lower surface Ordinate Station NACA 651-212 (Stations and ordinates given in per cent of airfoil chord) Upper surface Lower surface Ordinate Station Ordinate Ordinate Station 0 0.661 0.781 0.944 1.191 0 0.423 0.664 1.154 2.391 0 0.970 1.176 1.491 2.058 0 0.577 0.836 1.346 2.609 0 0.372 0.607 1.089 2.318 0 0.861 1.061 1.372 1.935 0 0.628 0.893 1.411 2.682 4.797 7.289 9.788 14.798 19.817 2.800 3.487 4.067 5.006 5.731 5.203 7.711 10.212 15.202 20.183 - 1.536 -1.791 - 1.999 - 2.314 - 2.547 4.878 7.373 9.873 14.879 19.890 2.919 3.593 4.162 5.073 5.770 5.122 7.627 10.127 15.121 20.110 - 24.843 29.872 6.290 6.702 6.983 7.138 7.153 25.157 30.128 35.()97 2.710 2.814 2.863 2.854 2.773 24.906 29.923 34.942 39.961 44.981 6.300 6.687 6.942 7.068 7.044 25.094 30.077 35.058 40.039 45.019 - 4.510 - 4.743 45.032 - 7.018 6.720 50.000 54.971 59.947 64.927 69.915 - 2.606 2.340 2.004 1.621 1.211 50.000 55.017 60.032 65.043 70.050 6.860 6.507 6.014 5.411 4.715 50.000 54.983 59.968 64.957 69.950 74.910 79.912 84.924 89.943 94.971 - 0..792 75.053 80.052 85.045 90.033 95.017 3.954 3.140 2.302 1.463 0.672 74.947 79.948 84.955 89.967 94.983 34.903 39.936 44.968 50.000 55.029 60.053 65.073 70.085 75.090 80.088 85.076 90.057 95.029 100.000 6.288 5.741 5.099 4.372 3.577 2.129 1.842 0.937 0 40.064 100.000 - - 0.393 - 0.037 0.226 0.327 0 L.E. radius: 0.687 Slope of radius through L.E.: 0.168 100.000 I 0 100.000 0 - 0.870 - 1.036 - 1.277 - 1.686 2.287 2.745 3.128 3.727 4.178 - 4.882 - 4.926 - 4.854 - 4.654 - 4.317 I1 - 3. 872 - 3.351 - 2.771 - 2.164 1.548 0.956 0.429 - 0.040 0 L.E. radius: 1.000 Slope of radius through L.E.: 0.084 1 APPENDIX III NACA 65.-212 (J == 0.6 (Stations and ordinates given in per cent of airfoil chord) Upper surface Lower surface 435 NACA 65t-412 (Stations and ordinates given in per cent of airfoil chord) Upper surface Lower surface Station Ordinate Station Ordinate Station Ordinate Station Ordinate 0 0.399 0.638 1.124 2.356 0 0.982 1.194 1.520 2.113 4.837 7.329 9.827 14.833 19.848 0 0.601 0.862 1.376 2.644 - 0 0.852 1.012 1.242 1.625 3.017 3.728 4.330 5.298 6.042 5.163 7.671 10.173 15.167 20.152 - 2.185 2.606 2.956 3.500 3.904 24.869 29.894 34.921 39.951 44.983 6.611 7.029 7.304 7.444 7.423 25.131 30.106 35.079 40.049 45.017 - 4.197 4.401 4.518 4.550 4.475 50.017 65.051 60.094 65.123 70.124 7.231 6.856 6.318 5.634 4.842 49.983 54.949 59.906 64.877 69.876 - 4.283 3.968 3.566 3.124 2.640 75.112 80.090 85.064 90.036 95.013 3.983 3.082 2.173 1.297 0.521 74.888 79.910 84.936 89.964 94.987 - 2.131 1.604 1.085 0.595 0.191 100.000 0 1100.000 0 L.E.. radius: 1.000 Slope of radius through L.E.: 0.110 0 0.347 0.580 1.059 2.28.1 0 1.010 1.236 1.588 2.234 0 0.653 0.920 1.441 2.717 - 0 0.810 0.956 1.160 1.490 4.757 7.247 9.746 14.757 19.781 3.227 4.010 4.672 5.741 6.562 5.243 7.753 10.254 15.243 20.219 - 1.963 2.314 2.604 3.049 3.378 24.811 29.846 34.884 39.923 44.962 7.~93 7.658 7.971 8.139 8.139 25.189 30.154 35.116 40.077 45.038 - 3.613 3.770 3.851 3.855 3.759 50.000 55.035 60.064 65.086 70.101 I 7.963 7.602 7.085 6.440 5.686 50.000 54.965 59.936 64.914 69.899 - 3.. 551 3.222 2.801 2.320 1.798 75.107 I 80.103 85.090 , 90.066 95.033 4.847 3.935 2.974 1.979 0.986 74.893 79.897 84.. 910 89.934 94.967 - 1.267 - 0.751 - 0.282 0.089 0.278 I 100.000 0 100.000 _. -- 0 .-.-._.­ I~.E. radius: 1.000 Slope of radius through L, E.. : 0.168 THEORY OF WING SECTIONS 436 NACA 65r215 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Ordinate Lower surface Station NACA 65r415 (Stations and ordinates given in per cent of airfoil chord) Upper surface Ordinate Station Lower surface Ordinate Stati~n Ordinate 0 0.406 0.645 1.132 2.365 0 1.170 1.422 1.805 2..506 0 0.594 0.855 1.368 2.635 - 0 1.070 1.282 1.591 2.134 0 0.313 0.542 1.016 2.231 0 1.208 1.480 1.900 2.680 0 0.687 0.958 1.484 2.769 4.848 7.342 9.841 14.848 19.863 3.557 4.380 5.069 6.175 7.018 5.152 7.658 10.159 . 15.152 20.137 - 2.925 3.532 4.035 4.829 5.426 4.697 7.184 9.682 14.697 19.726 3.863 4.794 5.578 6.842 7.809 5.303 7.816 10.318 15.303 20.274 - 3.510 - 4.150 - 4.625 24.882 29.904 34.927 39.952 44.976 7.658 8.123 8.426 8.569 8.522 25.118 30.096 35.073 40.048 45.024 - 5.868 6.179 6.366 6.427 6.332 24.764 29.807 34.854 39.903 44.953 8.550 9.093 9.455 9.639 9.617 25.236 30.193 35.146 40.097 45.047 - 4.970 5.205 5.335 5.355 5.237 50.000 55.021 60.039 65.053 70.062 8.271 7.815 7.189 6.433 5.572 50.000 54.979 59.961 64.947 69.938 - 6.065 5.625 5.047 4.373 3.628 50.000 55.043 , 60.079 I 65.106 ~ 70.124 i 9.374 8.910 8.260 7.. 462 6.542 50.000 I 54.957 59.921 64.894 69.876 - 4.962 4.530 3.976 3.342 2.654 75.065 80.063 85.055 90.040 95.020 4.638 3.653 2.649 1.660 0.744 74.935 79.937 84.945 89.960 94.980 - 2.848 2.()61 1.303 0.626 0.112 75.131 II 80.126 1 85.109\ 90.080 j 95.040 I 5.532 4.447 2.175 1.058 74.869 79.874 84.891 89.920 ( 94.960 - 1.952 1.263 0.628 0.107 0.206 0 1100.000 I 100.000 0 100.000 0 L.E. radius: 1.505 Slope of radius through L.E.: 0.084 I I 100.000 I 3.320 I - 0 1.008 1.200 1.472 1.936 - 2.599 - 3.098 0 L.E. radius: 1.505 Slope of radius through L.E.: 0.168 APPENDIX III NACA 65r415 (J == 0.5 (Stations and ordinates given in per cent of airfoil chord) NACA 65.-218 (Stations and ordinates given in per cent of airfoil chord) Upper surface Lower surface Lower surface Ordinate Station Ordinate Station Station Ordinate Station Ordinate 4.099 5.122 5.985 7.383 8.459 5.426 7.946 10.451 15.432 20.389 - 2.335 - 2.746 -.3.081 - 3.591 - 3.963 24.671 29.743 34.825 39.916 45.019 9.280 9.883 10.280 10.470 10.423 25.329 30.257 35.175 40.084 44.981 - 4.232 - 4.411 I- - 4.508 - 4.526 - 4.431 50.152 10.106 55.262 9.501 60.307 8.672 65.314 7.684 70.294 , 6.573 49.848 54.738 59.693 64.686 69.706 - 4.226 - 3.929 - 3.548 75.253 80.199 74.747 79.801 84.863 89.923 94.973 - 90.077 95.027 100.000 , -_._­ ),.~~. 0 100.000 - 2.609 1 radius: 1.505 Slope of radius through L.E.: 0.233 0 0.612 0.875 1.390 2.660 4.819 7.311 9.809 14.818 19.835 4.178 5.153 5.971 7.276 8.270 5.181 7.689 10.191 15.182 20.165 - 24.858 9.023 9.566 9.916 10.070 9.996 25.142 30.116 I- 40.058 45.028 !- I I J 75.077 1 80.074 I 85.(163 f 90. 046 1 95.023 I 2.083 1.545 1.014 0.527 0.139 0 0 1.382 1.673 2.116 2.932 50.000 55.026 60.047 65.063 70.073 -3.1~ - 0 0.388 0.625 1.110 2.340 29.884_ 34.912 39.942 44.972 100.000 ; ... ~ 85.13i 5.387 4.157 2.930 1.755 0.715 - - I 0.755 1.036 1.573 2.874 0 0.957 1.132 1.3i7 1.776 0 - 1.282 - 1.533 - I I I 1.902 - 2.560 - g.546 - 35.0881 - 4.305 4.937 5.930 6.676 7.233 7.622 7.856 7.928 7.806 !- 9.671 9.103 8.338 7.425 6.398 50.000 7.4ti5 54.974 f - 6.913 59.953 j - 6.196 U4.937 : - 5.365 69.927 5.290 4.133 2.967 1.835 0.805 I 0 1 I I - 4.454 74.923 : 79.920 84.93i 89.95·1 94.977 100 .000 . _ _ _· " _ " ' ·_ _ .W __ ". _ _ _ _ ' - 4.574 7.054 9.549 14.568 19.611 0 - 0 1.233 1.520 1.965 2.812 ~ 0 0.245 0.464 0.927 2.126 ~ - Upper surface 437 3.500 2.Ml 1..621 0.801 0.173 0 . . . . . . _ _ _ - . . . , _... _ L.E. radius: 1.96 Slope of radius through L.E.: 0.084 THEORY OF WING SECTIONS 438 NACA 65a-418 (Stations and ordinates given in per cent of airfoil chord) Lower surface Upper surface Station Ordinate Station Ordinate NACA 65,-418 a == 0.5 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station 0 0.278 0.503 0.973 2.181 0 1.418 1.729 2.209 3.104 0 0.722 0.997 1.527 2.819 4.639 7.123 9.619 14.636 19.671 4.481 5.566 6.478 7.942 9.061 5.361 7.877 10.381 15.364 20.329 - 3.217 3.870 4.410 5.250 5.877 24.716 29.768 34.825 39.884 44.943 9.914 10.536 10.944.__ 11.140 11.091 25.284 30.232 35.175 40.116 45.057 - 6.334 6.648 6.824 6.856 6.711 50.000 55.051 60.094 65.126 70.146 10.774 10.198 9.408 8.454 7.368 50.000 M.949 59.906 64.874 69.854 - 6.362, 5.818 5.124 4.334 3.480 75.154 80.147 85.127 90.092 95.046 6.183 4.927 3.638 2.3liO 1.120 74.846 79.853 84.873 89.908 94.954 - 2.803 1.743 0.946 0.282 0.144 100.000 J~.E. 0 100.~ __ 0 - 1.218 - 1.449 -1.781 - 2.360 0 radius: 1.96 Slope of radius through L.E.: 0.168 Lower surface Ordinate Station Ordinate 0 0.197 0.411 0.868 2.057 0 1.440 1.766 2.271 3.233 0 0.803 1.089 1.632 2.943 - 0 1.164 1.378 1.683 2.197 4.493 6.966 9.459 14.481 19.533 4.715 5.891 6.882 8.482 9.709 5.507 8.034 10.5041 15.519 20.467 - 2.951 3.515 3.978 4.690 5.213 24.604 29.691 34.789 39.899 45.022 10.643 11.325 11.770 11.970 11.897 25.396 30.309 35.211 40.101 44.978 - 5.595 5.853 5.998 6.026 5.905 50.182 55.313 60.364 65.372 70.347 11.506 10.788 9.820 8.674 7.397 49.818 54.687 59.636 64.628 69.653 - S.626 5.216 4.696 4.094 3.433 75.298 80.232 85.159 90.089 95.030 6.038 4.636 3.247 1.930 0.777 74.702 79.768 84.841 89.911 94.970 100.000 0 - 1.331 - 0.702 - 0.201 _-_ 100.000 -_... - 2.734 - 2.024 0 ... - L.E. radius: 1.96 Slope of radius through Js; E.: 0.233 APPENDIX III NACA 65.-618 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Ordinate Lower surface Station Ordinate 0 1.146 1.356 1.651 2.152 NACA 65,-618 a = 0.5 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station 0 0.172 0.385 0.839 2.026 0 1.446 1.776 2.293 3.268 0 0.828 1.115 1.661 2.974 4.462 6.936 9.431 14.455 19.506 4.776 5.971 6.978 8.602 9.848 5.538 8.064 10.569 15.545 20.494 24.574 29.652 34.738 39.826 44.915 10.803 11.504 11.972 12.210 12.186 25.426 30.348 35.262 40.174 45.085 - 5.433 5.672 5.792 5.784 5.616 50.000 65.077 60.141 65.189 70.219 11.877 11.293 10.479 9.482 8.338 50.000 54.923 59.859 64.811 69.781 - 5.259 4.723 4.063 3.302 2.506 75.230 80.220 85.189 90.138 7.075 5.719 4.306 2.863 1.433 74.770 79.780 84.811 89.862 94.932 - 1.705 - 0.943 - 0.268 0.239 0.463 95..068 100.000 0 100.000 - - 2.880 - 3.427 - 3.876 - 4.564 - 5.072 0 L.E. radius: 1.96 Slope of radius through L.E.: 0.253 439 Lower surface Ordinate Station Ordinate II 0 0 0.941 . - 1.055 1.244 - 1.239 1.811 - 1.493 3.154 - 1.895 0 0.059 0.256 0.689 1.846 0 1.469 1.821 2.375 3.449 4.248 6.706 9.194 14.225 19.301 5.115 6.448 7.575 9.404 10.815 5.752 8.294 10.806 15.775 20.699 24.407 29.537 11.893 12.687 13.209 13.456 13.395 25.593 - 4.321 30.463 . - 4.. 479 35.3161 - 4.551 40.151 - 4.540 44.966 - 4.407 12.974 12.173 11.090 9.806 8.374 49.727 54.532 59.454 64.443 69.481 34.684 39.849 45.034 50.273 55.468 60..546 65.557 70.519 75..445 so.347 85.239 90.133 95. 046 1 100.000 I i ! I I - 2..469 - 2.884 - 3.219 - 3.716 - 4.071 - 4.154 - 3.815 - 3.404 - 2.936 - 2.428 6.851 i 74.555 . - 1.895 5.279 I 79.653 t - 1.361 3.720 84.761 1 - 0.846 2..233 89.867 - 0.. 391 0.920 94.954 - 0.055 I 1 0 I lOO.()~:lO________~ _~ ___ L.E. radius: 1.96 Slope of radius through L. e.: 0.349 THEORY OF WING SECTIONS 440 NACA 65r 221 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Ordinate Lower surface Station Station 0 0.247 0.468 0.933 2.135 0 1.567 1.902 2.402 3.335 0 0.628 0.892 1.410 2.684 - 0 1.467 1.762 2.188 2.963 4.791 7.280 9.778 14.787 19.808 4.783 5.918 6.865 8.370 9.514 5.209 7.720 10.222 15.213 20.192 - 24.834 29.865 34.898 39.932 44.967 10.381 11.007 11.404 11.570 11.461 25.166 30.135 35.102 40.068 45.033 - 50.000 55.030 60.054 65.072 70.084 11.055 10.372 9.461 8.390 7.195 75.088 80.084 85.072 90.052 95.026 5.918 4.595 3.270 2.000 0.861 I 0 Upper surface Ordinate 0 0.372 0.608 1.090 2.314 100.000 NACA 65r421 (Stations and ordinates given in per cent of airfoil chord) Ordinate Lower surface Station Ordinate 0 1.601 1.956 2.493 3.505 0 0.753 1.032 1.567 2.865 - 0 1.401 1.676 2.065 2.761 4.151 5.070 5.831 7.024 7.922 4. 582 1 5.085 7.062 6.329 9.557 7.371 14.575 9.034 19.616 10.304 5.417 7.938 10.443 15.425 20.384 - 3.821 4.633 5.303 6.342 7.120 8.591 9.063 9.344 9.428 9.271 24.668 29.729 34.796 39.865 44.934 11.271 11.976 12.433 12.640 12.556 25.332 30.271 35.204 40.135 45.066 - 7.691 8.088 8.313 · 8.356 8.176 50.000 ~54.970 ~ 59.946 64.928 69.916 - 8.849 8.182 7.319 6.330 5.251 50.000 55.059 60.108 65.145 70.168 12.158 11.467 10.531 9.419 8.166 50.000 54.941 59.892 64.855 69.832 - 7.746 7.087 6.247 5.299 4.278 74.912 79.916 84.928 89.948 94.974 4.128 3.003 1.924 0.966 0.229 75.176 80.167 85.143 90.104 95.051 6.811 5.388 3.940 2.514 1.176 74.824 79.833 84.857 - 3.231 - 2.204 - 1.248 100.000 - 0 L.E. radius: 2.50 Slope of radius through L.E.: 0.084 I 100.000 0 89.896\- 0.446 1 94.949 100.000 I. 0.088 , 0 L.E. radius: 2.50 Slope of radius through L.E.: 0.168 1 APPENDIX III NACA65r421 a = 0.5 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Lower surface 0 o 0.155 0.363 0.813 1.992 1.620 1.991 2.553 3.631 0.845 1.137 1.687 3.008 - 1.344 1.603 1.965 2.595 4.414 6.880 9.371 14.395 19.455 5.315 6.651 7.773 9.572 10.951 5.586 8.120 10.629 15.605 20.545 - 3.551 4.275 4.869 5.780 6.455 24.538 29.639 34.754 39.882 45.026 12.000 25.462 12.765 30.361 13.258 1 35.246 40.118 13.470 13.362 44.974 - 6.952 7.293 7.486 7.526 7.370 50.211 55.362 60.421 65.428 70.398 12.890 12.056 10.942 9.637 8.193 49.789 54.638 59.579 64.572 69.602 - 7.010 6.484 5.818 5.057 4.229 75.340 80.264 85.181 6.664 5.09i 3.550 74.660 79.736 84.819 89.900 94.966 - 3.360 2.485 1.634 0.867 0.257 95.034 100.000 NACA66-206 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Lower surface Ordinate Station Ordinate Ordinate Station Ordinate o 90.IOO 441 I 2.095 0.833 0 1 1100.000: 0 0 L.E. radius: 2.50 Slope of radius through L.E.: 0.233 0 0.461 0.707 2.447 0 0.509 0.622 0.798 1.102 0 0.539 0.793 1.298 2.553 - 0 0.409 0.482 0.584 0.730 4.941 7.439 9.939 14.942 19.947 1.572 1.947 2.268 2.791 3.196 5.059 7.561 10.061 15.058 20.053 - 0.940 1.099 1.234 1.445 1.604 3.513 3.754 3.929 4.042 4.095 25.046 30.038 1.202 24.954 29.962 :H.971 39.981 44.990 1 I I 50.000 t - 1.723 - 1.810 35.029 - 1.869 40.019 f - 1.900 1.905 45.010 1- 4.088 55. 009 1 4.020 60.018 I 3.886 65.026 I 3.641 70.031 , 3.288 I 75.034 I I SOo0341 85.031 90.023 t 95.012 I 100.000 I 2.848 2.339 1.780 1.182 0.578 l) 50.000 54.991 59.982 64.974 69.969 , - 1.882 1.830 t - 1.744­ ! - 1.581 t - 1.344 Ii 74.966 i - 1.058 84.969 89.977 I 94.988 - 0.434 - 0.148 0.054 I 7909661 - 0.747 ! 1 100 .000 0 L.E. radius: 0.223 Slope of radius through L.E.: 0.084 THEORY OF WING SECTIONS NACA 66-209 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Ordinate Lower surface Station Ordinate Upper surface Station Lower surface - Ordinate Station Ordinate 0 0 0 0.442 0.686 1.179 2.420 0.735 0.892 1.135 1.552 0.558 0.814 1.321 2.580 - 0.635 0.752 0.921 1.180 0 0.436 0.679 1.171 2.412 0 0.806 0.980 1.245 1.699 4.912 7.409 14.912 19.921 2.194 2.705 3.141 3.850 4.396 5.088 7.591 10.092 15.088 20.079 - 1.562 1.857 2.107 2.504 2.804 4.902 7.399 9.898 14.903 19.912 2.401 2.958 3.432 4.202 4.796 588 2. 5.098 7.601 10.102 15.097 20.088 24.931 29.944 34.957 39.971 44.986 4.821 5.145 5.378 5.528 5.594 25.069 30.056 35.043 40.029 45.014 - 3.031 3.201 3.318 3.386 3.404 24.924 29.937 34.952 39.968 5.257 5.608 5.862 6.024 6.095 25.076 30.063 35.048 40.032 45.016 50.000 55.014 60.027 65.038 70.046 5.578 5.476 5.275 4.912 4.400 50.000 54.986 59.973 64.962 69.954 - 3.372 3.286 3.133 2.852 2.456 50.000 55.016 60.030 65.042 70.051 6.074 5.960 5.736 5.332 4.759 50.000 54.984 59.970 64.958 69.949 75.050 3.772 3.058 2.283 1.477 0.690 74.950 79.950 84.9&6 89.965 94.982 - 1.982 1.466 0.937 0.443 0.058 75.056 80.055 85.049 90.037 95.019 4.071 3.289 2.445 1.570 0.724 74.944 79.945 84.951 89.963 94.981 9.~ SO.05O 85.044 90.034 95.018 100.000 0 0 NACA 66-210 (Stations and ordinates given in per cent of airfoil chord) 100.000 L.E. radius: 0.530 Slope of radius through L.E.: 0 0.084 44.984­ 100.000 0 0 0.564 0.821 1.329 100.000 0 - 0.706 - 0.840 1- - 1.031 1.327 - 1.769 - 2.110 - 2.389 - 2.856 - 3.204 1 - 310467! - 3.664 - 3.802 - 3.882 =~:I - 3.770 - 3.594 1 - 3.272 - 2.815 1 - 2.281 - 1.697 1.099 0.536 0.092 0 L.E. radius: 0.662 Slope of radius through L.E.: O.~ I APPENDIX III NACA~-212 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Lower surface Ordinate Station Ordinate NACA66r215 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station 0 0.424 0.666 1.156 2.395 0 0.953 1.154 1.462 1.991 0 0.576 0.834 1.344 2.605 - 0 0.853 1.014 1.248 1.619 0 0.406 0.646 1.134 2.370 4.883 7.379 9.878 14.883 19.894 2.809 3.459 4.011 4.905 5.596 5.117 7.621 10.122 15.117 20.106 - 2.177 2.611 2.977 3.559 4.004 4.855 7.349 90848 14.854 190868 24.908 29.925 34.943 39.962 44.981 6.132 6.539 6.833 7.018 7.095 25.092 30.075 35.057 40.038 45.019 - 50.000 &5.019 60.036 65.051 70.061 7.068 6.931 6.659 6.169 5.487 50.000 54-.981 59.964 64.949 69.939 75.066 80.065 85.057 90.043 95.022 4.661 3.739 2.755 1.750 0.189 74.934 79.935 84.943 89.957 94.978 100.000 0 100.000 443 Ordinate Lower surface Station Ordinate 0 1.168 1.409 1.778 2.417 0 0.594 0.854 1.366 2.630 3.413 4.202 4.872 5.957 6.790 5.145 7.651 10.152 15.146 20.132 4.342 4.595 4:.773 4.876 4.905 7.437 24.886 29.906 1 7.9Z1 34.929 ( 8.280 39.952 I 8.501 44.976 8.590 25.114 30.094 35.071 40.048 - 4.862 4.741 4.517 4.109 3.543 50.000 55.023 6O.0f5 65.063 70.075 8.553 8.378 8.030 7.402 6.547 50.000 54.977 59.955 - 2.871 2.147 1.409 0.716 0.157 75.081 80.079 85.070 90.052 95.026 5.526 4.393 3.202 2.005 0.881 0 L.E. radius: 0.952 Slo~ of radius through L.E.: 0.084 f 1 j 100.000 I 0 - f - 2.781 3.354 3.838 4.611 5.198 - 5.647 5.983 6.220 6.359 6.400 69.925 - 6.347 6.188 5.888 5.342 4.603 74.919 79.921 84.930 89.948 94.974 - 3.736 2.801 1.856 0.971 0.249 45.024 64.937 100.000 - 0 1.068 1.269 1.564 2.045 0 L.E. radius: 1.435 Slope of radius through L.E.: 0.084 THEORY OF WING SECTIONS 444 NACA 66r415 (Stations and ordinates given in per cent of airfoil chord) Upper surface Ordinate Station Ordinate Station 0 0.314 0.544 1.019 2.241 I 4.711 7.199 9.696 14.709 19.736 24.771 29.812 34.SS7 39.904 44.952 Lower surface I Upper surface Station Lower surface Ordinate Station Ordinate 0 1.206 1.467 1.873 2.592 0 0.686 0.956 1.481 2.759 - 0 1.006 1.187 1.445 1.848 0 0.389 0.628 1..115 2.. 346 0 1.368 1.636 2.054 2.828 0 0.611 0.872 1.385 2.654 3.718 4.617 5.381 6.624 7.581 5.289 7.801 10.304 15.291 20.264 - 2.454 2.921 3.313 3.. 932 4.397 4.827 7.320 9.818 14.825 19.841 4.002 4.933 5.724 7.004 7.982 5.173 7.680 10.182 15.175 20.159 - 3.370 4.085 4.690 5.658 6.390 8.329 8.897 9.309 9.571 9.685 25.229 30.188 - 4.749 5.009 5.189 5.287 5.305 24.863 29.887 34.914 39.942 44.971 8.742 9.31,' 9.731 9.989 10.093 25.137 30.113 35.086 40.058 45.. 029 - 6.952 7.373 7.671 7.847 7.903 50.000 55.. 028 60.054 65.075 70.()89 10.045 9.828 9.394 8.610 7.568 50.000 54.972 59..946 64.925 69.911 - 7.839 7.638 7.252 6.550 5.624 75.095 6.345 5.001 3.606 2.230 0.961 74.905 79.907 84.919 89.940 94.970 - 4.555 - 3.409 - 2.260 . 35.143 40.096 45.048 50.000 55.046 60.090 65.126 70.150 9.656 9.473 9.100 8.431 7.518 50.000 - 5.244 54.954 59.910 64..874 69.850 - 5.093 4.816 4.311 3.630 75. 162 1 80.159 I 85.139 t 90.104 95.. ~'i3 6.419 5.187 3.872 2.519 1.196 74.838 79.841 84.861 89.896 94.947 - 2.839 2.003 1.180 0.451 0.068 100.000 NAC.A 66.-218 (Stations and ordinates given in per cent of airfoil chord) 0 100.000 0 LE. radius: 1.. 435 Slope of radius through L.E.: 0.168 80.093 85.. 081 90.060 95.030 100.. 000 0 100.000 0 - 1.268 - 1.496 - 1.840 - 2.. 456 i 96 - 1.1 1 - 0.329 0 L.E. radius: 1.955 Slope of radius through L.E.: 0.084 I THEORY OF WING SECTIONS 446 NACA 67,1-215 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station Lower surface NACA 747A315 (Stations and ordinates given in per cent of airfoil chord) Upper surface Ordiilate Station Ordinate Station Lower surface - Ordinate Station Ordinate 0 0.402 0.642 1.128 2.361 0 1.213 1.460 1.867 2.577 0 0.598 0.858 1.372 2.639 - 0 1.113 i.320 1.663 2.205 0 0.. 229 0.449 0.911 2.109 0 1.305 1.599 2..065 2..935 0.771 1.051 1.589 2.891 - 1.927 4.848 7.344 9.845 14.854 19.869 3.557 4.321 4.947 5.954 6.735 5.152 7.656 10.155 15.146 20.131 - 2.925 3.473 3.913 4.808 5.143 4.564 7.053 9.558 14.599 19.668 4.264 5.286 6.140 7.497 8.503 5.436 7.947 10.442 15.401 20.332 - 2.518 2.952 3.304 1 3.843 4.247 24.887 29.908 34.930 39.953 44.976 7.348 7.825 8.185 8.430 8.570 25.113 30.092 35.070 40.047. 45.024 - 5.558 5.881 6.125 6.288 6.380 24.758 29.867 35.001 40.200 45.375 9.242 9.731 9.982 9.962 9.572 25.242 30.133 34.999 39.800 44.625 - 4.546 4.773 4.926 5.020 5.040 50.000 55.024 60.047 65..068 70.086 8.600 8.516 8.302 7.935 7.373 50.000 54.976 59.953 64.932 69.914 - 6.394: 6.326 6.160 5.875 5.429 50.447 55.463 60.435 65.366 70.241 8.964 8.206 7.324 6.365 5.354 49.553 54.537 59.565 64.634 69.759 - 5.014 I 4.930 I 4.772 1 4.509 75.098 80.100 85.092 90.071 95.037 6.515 5.335 74.902 79.900 84 ..908 89.929 94.963 - 4.725 3..743 2.653 1.. 503 0.471 75.130 80.073 85.038 90.016 4.336 3.295 2.257 1.289 95.004: 0.481 74.870 79.927 84.962 89.984 94.996 100.000 3..999 2.537 1.103 0 100.000 0 100.000 0 0 I 100.000 -­ L.E. radius: 1.523 Slope of radius through L.E.: 0.084 0 - 1.031 - 1.207 - 1.473 - 4.110 I - 3.502 - 2.743 - 1.91sl - 1.097 - 0.405 0 - .._---,_.­ L.E. radius: 1.544 Slope of radius through L.E.: 0.232 445 APPENDIX III NACA 66,-418 (Stations and ordinates given in per cent of airfoil chord) Upper surface Station 0 1.205 1.412 1.719 2.256 0 0.372 0.610 1.095 2.323 0 1.570 1.869 2.342 3.226 0 0.628 0.890 1.405 2.677 - 3.042 3.651 4.163 4.977 5.589 4.800 7.291 9.788 14.797 19.815 4.580 5.653 6.565 8.039 9.170 5.200 7.709 10.212 15.203 20.185 - 6.053 - 6.399 24..840 4.306 5.347 6.231 7.669 8.773 9.f133 10.287 34.829 ! 10.759 I 885 11.059 39. 1 11.188 44.943 I I 75.191 80.185 85.162 90.120 95.000 100.000 I - Ordinate Station Ordinate - 4.656 7.140 9.636 14.651 19.683 I Lower surface Station 0 1.405 1.692 2.147 3.000 50.000 55.056 ftO.l07 65.149 70.178 Upper surface Ordinate Station Ordinate 0 0.280 0.509 0.981 2.194 24.726 29.775 Lower surface NACA66r221 (Stations and ordinates given in per cent of airfoil chord) 11.148 10.923 10.464 9.639 8.539 7.238 5.794 4.276 2.744 1.275 0 0 ~720 0.991 1.519 2.806 5.344 7.860 10.364: 15.349 20.317 40.11~ - 6.i;~ f 45.051 1- 6.80~ 10.047 10..709 29.869 11.183 34.900 11..478 39.933 44.967. 11.595 50.000 54.944 59.893 M.SSI 69.822 50.000 55.032 60.063 65.087 70.103 25.274 30.225 35.171 - 6.6.39 - I- 6.736 6.543 6.183 5.519 4.651 74.809 . - 3.658 79.8151 - 2.610 84.838 - 1.584 89.880 - O.6i6 94.940 - 0.011 100.000 0 L.E. radius: 1.955 Slope of radius through L.E.: 0.168 J 75.109 so. 106 85.092 90.067 95. 034 1 100.000 t i 11.537 11.281 10.763 9.823 8.581 7.145 5.591 3.996 2.440 1.032 0 - 0 1.470 1.729 2.128 2.8M - 3.948 - 4.805 - 5.531 6.693 - 7.578 I- 1- 25.160 f - 8.257 8.765 30.131 35.100 I - 9.123 40.06719.~ 45..033 - 9.405 5O.0~ I- 9.331 54.968 ! - 9.091 59.937l - 8.621 64.913 69.897 - 7.763 I - 6.637 1- 74.891 5.355 79.894 ; - 3.999 84.908 2.f>50 I- 89.933 i- 1.406 94.966 , - 0.400 lOO.ooo 1 0 L.E. radius: 2.550 Slope of radius through L.E.: 0.084 APPENDIX III 447 NACA 747A415 (Stations and ordinates given in per cent of airfoil chord) Lower surface Upper surface Station Station Ordinate Ordinate 0 0.183 0.398 0.852 2.041 0 1.318 1.622 2.106 3.016 0 0.817 1.102 1.648 2.959 4.487 6.972 9.476 14.521 19.598 4.411 5.488 6.390 7.827 8.897 5.513 8.028 10.524 15.479 20.402 - 3.501 - 3.845 24.698 29.818 34.964 40.176 45.364 9.687 10.216 10.497 10.499 10.121 25.302 30.182 35.036 39.824 44.636 - 50.447 55.474 60.454 65.393 70.273 9.516 8.753 7.859 6.878 5.838 49.553 54.526 59.546 64.607 69.727 - 4.462 4.381 4.235 3.992 - 3.622 75.164 80.107 4.783 3.692 2.592 1.546 0.639 85..066 90.037 95.015 ioo.ooo 0 f t - 0 0.994 1.160 1.406 1.822 - 2.349 - 2.730 - 3.038 4.095 4.286 4.411 4.485 4.~93 II I- 74.836 79.893 84.934 89.963 94.985\100.000 3.053 2.344 1.578 0.838 0.247 0 L.E. radius: 1.544 Slope of radius through L.E.: 0.274 APPENDIX IV AERODYNAMIC CHARACTERISTICS OF WING SECTIONS C.aNTENTS NACA Designation 0006 . ()()()9 . . • . . Page 452 . 0016-34: ..... 0010-35 . . . . .. 0016-34: a == 0.8 (modified) eli = 0.2 . . 0012 . . . . . . . . . . . . . . 0012-64 0012-64 a 1408 . == 0.8 1410 (modified) C'l, == 0.2 . . . 462 464 466 468 470 472 1412 2408 474 476 478 2410 2412 2415 2418 2421 454 456 458 460 480 482 . . 484 2424 4412 486 488 4415 490 4418 4421 4424 23012 23015 23018 23021 23024 63-006 492 494 496 498 500 502 504 506 508 510 . . .. . . 63-009 . 63-206 . 63-209 . . 63-210 . 631-012 . . 631-212 . 520 6&.-412 . 522 524 63r015.. 63r215 . . . 63r415. 63z-615. . 512 .514 516 518 . 526 528 530 THEORY OF WING SECTIONS 450 Page 532 534 NACA Designation 63r018. 63r218. 63r418. 63,-618. 536 538 540 63.-021 . . 542 544 546 634-221 . . 6&-421. 63AOI0 63A210 64-108 64-110 64-206 64:-208 64:-209 548 ..... 64-()(M). . • • • • . • 64-()()9 • . . . . . .. . . . .. 558 560 562 liM 64-210 . . 64.~12. . . . . . . . 566 568 570 64t-112. . Mt-212 . . 64.-412 . . 64r015. 64r215 . 64r415 . 64r018 . 572 574 576 578 580 582 584 586 588 590 592 64r218 . . 64r418. 64r618 . . 64r021 . . 64.c-221. 64r421. MAOIO 594 596 li98 600 602 604 606 608 610 612 614 616 64A210 MA410 64 1A212 . . 64,A215 . 85-()()6. . ..... . . . . . 65--009 . 66-206 . . . 65-209 . 65-210 . . 65-410 . 6&-012. 65t-212 65s-212 II == 0.6 . ~-412. 550 652 554 556 . . . . . 6Sr015. . . . . . . . 65r215. . . . . . . . 66r415. . . . . . 65.t-410 II :II: 0.5 • . . 618 620 622 624 626 628 630 APPENDIX IV NACA Designation 65r018 . 65,-218 . 65r418. . . . . 65r418 a - 0.5 . 65.-618. . . . . 65r618 a :=II 0.5 . 65.-021 . 65.-221 . 65r421 . 65r421 a - 0.5 . 66-006 . . . 66-009 . . ~206 •.. 66-209 . . 66-210 . . 66r012. 66,-212. 66:-015. 602-215. 66r415. 66,-018. . 66,-218. . 66,.418. 66c-021 .. 66r221. 67,1-215 . . 747A315. 147A415. 451 Page 632 634 636 638 640 642 644 646 648 650 652 8M 656 6li8 660 662 664 666 668 670 672 674 676 678 680 682 684 686 THEORY OF WING SECTIONS 452 3. 6 ,­ 3. "!:II S 2. 4­ 2J) I,..'" a; ., ~J~ I .6 'I. 1'\'5 , 1'4 .. ~ ~ i 16 '~ I~ .8 ~ P\.~ 16 I~ .i 't.;; t [7 •4­ rtt 0 • I III 1, A 1(/ L._ fI) 0 • /r ,A ~ ~r 'J. r\1 r­ I ) oR ~ tJ -!!I'=' iiiiL...;' ~ I~ .J. ~r ~ R , r­ ~ j I 4 I"'~ A fULI'-I, 4 .~ 8 ~-:3 ~ j) 10 .4 I J ftI) r e -.1 Wl J I :\ !l -/..2 , 1'\ .~ -.~ -.5 .....1'lI -16 -8 0' Section D!'19. NACA 0006 Wing 0 8 /6 Dffack, «0, dsq . SectiOI' 32 453 APPENDIX IV . ~ 0 ~ ,, i"""oo...... I , .032 "~I _4l"I -0 .e .4 .6 • I.O.t128 .8 .tJ24 J .IJ2(J 1 ~ II r .DI6 \ 1/'; , \ , I). J r­ _...... P } V i\. I(~ ) ~ JJtJB I J I 1/ \ Aid ~~ ~~ ~p lP" L&I ~ ."'t"" -.....rt; ~r"c I .... IPi.fd~ lt1~ ~--. '""'Pc ~,..... - (J -­ 1­ -_.­..... ;.J I.­ 1-.: *f. G.e. p08ition 8 fJ_ ~ o 3.lh"'~~250 0 R ao ~t~~250 90 ~~250 00 ~ 6D ~ __ Standard roughneu 3 -. -.5 -1.6 -1.2 -.8 Q2(Jc eil1vJa1'«l8plt flap defected ISO· : ,,6.0 • &0 S~" rouq/1ne$s -.4 -1111 - 0 .4 St!!Cfb7 lift COtI'f';cient, ·c, N ~C.A 0006 Win« Section (Continued) .8 /.2 ~~ ~I- 1.6 THEORY OF WING SECTIONS 454 3.8 3.2 _.... ......, - 24 ~.~ 8f/ J IJ 1.6 fJ r­ ~ . , ~. "Pi."\ ~ 7 ... r IJ L2 /; .8 J. .1J V ....,j~ ~ r ~i ~~ , :8 U rJ 'II .4­ ~, ~ !8 rl.. .­ ~ A L­ - ~ ,r er 1 I~ ~~. i'I~ ~ :> l r v J .... ~~ ,j 1/ iJ II' -.4 • rc ~J 'K il:l fl 1~1I\ ~~ "I ~ iJ ~ -.8 '\II~J I.J~ 11 . . . rv low. e _ 1-7) , ~1 -ce Ir4 ." l \. , \.. -1.6 .-.5 -2.IJ. -~ -18 -8 0 8 /6 Section ongIe of Q/~ . . deg NACA 0009 Wing Section 24 455 APPENDIX IV .IJ3, .2 ~~ ..- ~ ..• --. -, I \ '\ .e .8 .IJ .4 .D3I / " -_. "'. 1.0 .-- * .024 II ~ \ \ CI· oeo ....... .§ ~ .016 ..... •e , In iQ~ ~.OI2 ~ t ~ ,.. \ \ \. • "\ \. ~, \ en ~ - .. ... II ~ ( lfS V jV / ¥ d ,.,. t> ...... ~ r-~ l.) J I~ ~ V ........ ..Mol! .... u.. 1- - .004 o :1 J 1/ '- I\, " ~ .008 J ~ 'l\. I"\kl ~ ""~ ~....,~ ~ I 7 01 I\. ~ & I D z- ~ ~ I t ~ ... '" .. I"lIo.. !- --""" ..... I """" t f I t I i , f 2 l 1 G.c. ppsifion R->--,. *~~I o 30-'0_ .250 0 I I a 6.01 I .2500 I I :J c 9.0~ 250 .005 6 6.0 ~ Standard roughness . O.zOe simotatea split flop deflected 60· v 6.0 4 I 5. StJ"do).d' 6.0 p Iial I I I I I ·1.2 -.8 I ..4 I I ~ks rout} J I I I I I I I I I t I I I 0 .4 Section liff coefficient, e, NACA 0009 ·,\ing Section (Conlinued) .8 1.2 /4 THEORY OF WING SECTIONS 456 2.4 2.0 1.8 I I.. 1.2 , 0.1 .. ~ I 0 'C.I ..: :I 1 ... -0.1 5 J J ··i IA·· l" +. ;ilJ ..•! i·· I ~ 0.4 ! ! ···To. j l···· . } .. -r : 1":."~ 0.' f-t~~~~-. . ..' .. ..­ - i-: ~. J. ···f·· I I i: ... o I.. _ •••• -'1--1---; . . . ­ 1- . l I;"': 'j'" ... ~I 1'. +. 10: t ~ _i,~ j""~'+ .~.-\-.. ••! -0.8 '1"+ .·r··· I '='0.3 -0.4 ;"~~~ _: ..·t·. ':'.'I":If-..'1--" : R ., • •'..saN -- ... ::!~:< "ill:;!', ... ..... ~ " - _....._. 1-' •••. . 0.200 81atlated 8pll' .~: ... flap detlaoted 600 _... - ~ f- ••• ; .•••.• ;.. ~ R t I: v, .l~ l~k:'" .+ . .!... ·t·· '" .+.. .-!-. -1- +.. . -24 I -16 -8 o tV 8 •. .. :~ ... ". . . eo .,. ; .. !j .. ~...:.~~~ .. r;.. :: .: .. . ... ~ - ti'-+ ....... 18 SeetioD ....1. olaUac:k. «0.... NACA 0010-34 'Wmg Section . ­ ••• -1.1 - -1.6 .. I'~,{::~;i:;:.,···i::·';g:::.~r ,..' .....:: \,-: r'~' ~•• ~ICT~,7,O.x l;i~i:. ~"1" -~. "' ,0 : :1::· uP 1iM I'OI"! ~. '0' .a;~.. . ~~. ._~ t'" _ •.••••• ~~. . _ . .. .. -0.2 _.~ .••• ... +.. ,.,.. a .·l-,' :.' ..: .~i ._~J I -J - .'i-' -i~-' i. o. ~ "F ~ ']. : .. T .~S'c--!' ,.. ~ -i-... i -~ ~ ..4~: .L..~~ e 3.1 0 -0.4 II +..+ ! lJ 1 ..lH ~t·_···~· : . .. . 0.8 II I :11 I "J"" .. j. . ~ j ~. ·~·t ........- ......'... 457 APPENDIX IV : ..... . '" ' ~~I"'::" - . ... c.a 0.024 ~. 0.020 0.018 J .' ~ I--~- _. --...- _ - t--- . - -_... •• -I'I t--•.••. 6.0 106 : I ·l·- f- •.• ~ 1-1'" .... _. .1 f t-- ... _.- - _. .,... "1-' .- : .~ : 'i --1 i : ~ i . '_1- Jf;{ .1: ~~"!' ~~. --+-+-t-:. _f.-._ .. i "T" j I"'"' ., ._-- .. ~. !!"':. r: . rr -to : f t t--. f-. ~t-- &.0 • 0 3.0 I . . .1 '" •'.'.' c, SectioD lift coefficieDt. NACA 0010-34 Wing Section (Continued) i:~: :' I. a.c. pos1 tloD R i Q : _.+-~-:-t-~~~I-'+~'" --.: --r' ". _. :.-i -; + .. .•-~t-t--L : _. .•.•_~, .- 1 .! iQ ~ v .. - ::-:~~... I ·t- ~'r- 1-' . j ·If··· . -. .1- -I'" ·f " J. 1ii~ i.'~" .- - ...-t~-+-+-"""-+-~+-l"'-+-+-+-. 1-10-1--+-+-:' 0 ~..o x -0.2 .. _, . __ :--t· .-1 fo-_ •.•.• "i"- i -"i+_...._: !J -~-~_ .._1 ... ..-t. . I ~~~~~: _ __,.~~: . ! 1 ·\1···. .!..- ..... ....- ~- -f- _. _.... . -.,- .~:. "-I-'~""""-+-"+--i-I--t o .__ ····f··· 106 .~. '~-r- .. I ....,.., ~~ ..,. ~•. _ I. 6.0 x ---,.- - - - .._-;:-.. ••. '" : :~ ~~dard ~\IIba••• -~--y- _·t--+~· !..- - , Ii: ~+ 0.004 It .-t-- --~ _! ..•. ...~ ...J- ._~.t-- •• 0.008 -:~~.J. .. , : i ~.~.- .. ~- ....,.- ··4··· 0.012 _.-"'~""""" I t--f:-~""~'!=II~~~&-i""""".~~·~··r=Ji:~'~~-rl ............-+--....-+........ ......-t~---t-o!·D •• a .8 ~~.;,;,.. ~. _. -+-I--I~""""'~<> 9.0 x .- 8 t! _. .f ~""~.....: x/c 106 .245 .245 .24' "'"1,''' .:,. 1. "lIe .018· .001= .055 "'1" I" THEORY OF WING SECTIONS 458 2.8 -, ~~ ~1tf:r ,:i~~li:,f ~ '.... :;. ! ~I:;' r;!~lr·:T :.... '1:· I •. I· 2.4 2.0 e: iw is 0.1 0.8 1 ~ ~ ..; I ~8 -0.1 .2 J 0.4 0 ~ .•... •.••. .. .;;.~ ~~ of: ._.. ~.1 1. L; •• - ... ~. --.If;~ ._ . . . , I -.~. ':f" 1-' ••• .- - ~ -0.8 _. T •. ~_ Ill.. "':""""- .. r1I""- ~l!n J !". ·'1 "1'" .- ~- ~- f-:'! ..- .~~. ... .• _'0 -0.4 -1.6 ••• ••• . - ••.•. ~ .6xl~ ,::,":'1::' -24 l::r";J::';,:, I. 1...1 . ,..16 -8 ,~.~~, .... :1 ~::.•... "-1-:" '!. _. o. \:;;:r" -" - foo··· .- 0- -8 16 SectioD aaale of attack, ~, dee. NACA 0010-35 Wing Section ~ ••• =-::­ ~;~ ~ . "-r'- _.­ ...... _. ••• •• ~-l~ ~?~ - _.. - .. _.~; '~T'~' ... ~~ A ... - 0.200 alaulated .p11t t·~~·~- ~-. ·tlap deflected 600 · \.. R .-L ;•.._.;.. 1 - ••... ~ ~~ ..~.:..; '.0 SUDdU4 0 _ - ~~.: :~:- _ ; ....t.... ~ .. - .... - ~~":'~~ "',.~~-~~..... -1.2 R 9.0 x 10' 6.0 . .. .. - .- .-.. ... r-.-.. •••• - r I.. _ :~ .; - :~ ••- -•• ··1· . , \... . ~ • j~ '1;' : r' .;~p I~I"'!":'" ;..; 8' i I ; . ~F. i • .. .... ~~. ~!·I··· . , -- '. ,. -: :;~,-. .r, ··~;1-.d. --.::-1Ii~ WI~~~.~I ..ifII"T' l i·· ··l·· +. i··I~~·· - ;~I""" IW"O '. . a ,. I '.,'" .::: ·;i.·: "~J. ~." ;~.IJJ ::- .. ~I .. ... .. .­ .. '" -I' . . . -::- .~ '::'"1: • • •• . • .-. ···I·]f;·: "'j ~.. .... .1-:.... ; .~~._.- r.- ":i' -.. '- .t. - :,-,"," . - - "-1 '" .. -0.3 _~- I ..~ ·~l·~ ~ ~ 'j -0.4 •a I - - - _.... .. - I ~ I . .. .., .... l" Stan4a~t.rouabne.. 6 x 10­ I I " •...•..•• •.•... ......, ._. -0.2 ..•.."'. :..... ~._-_ .._~.,~ .. L.... ic: 0 ;;l: :~:m': ~i;: .::' .::l...~:~ ':::, ,'...,; :::.1,;; ::' :':':~ ~~, ~ :~. .:.I~ ~~ ~~ 4 0.200 al.w.ate4 ap!it tlap 4e,leote4 60 ::: 96 x 10 6 - 1.2 I ~ ... 1.8 I··I~· ._" . .. .- . . .- .:~. ·:·0 .. ~.. . .:: 24 ... - ~'S 32 APPENDIX IV . . ~;': . ::~ ~~L;:, ::. . . O.osl :::: :.:. :::. ~!~: 0.024 ~ ~~ ..... ... -' - .....-­ . -::t-, .' -i" .. . ! I I a'j" - 1-- '" t:: ·8ell f... 0.016 .. 0.012 ,-.op 'a ••• •..­ .., ~ ... -'•.•­ _. '-1­ .. __ ~~' . _a._~_~ +.....-.. ~ CM ••• ~ '" .~ ";" •••• "l'" .I. ,. • I • . ···1 ...j.... i· I ~... 1 t f .•_~ _... ~ .•J ..; I~_.. .•••_! ! .a t.., I .. 8 0.020 f ..... :...:'" .. 13 a i : _..­ •••..•­ .•• ­ ••. ••• ..a ­ .• ....!.., ..; .2 !;;:;:::;:> ..; ,..! ~ r-:; - :.-.:. 'j . 1. ................. 459 ! / u fI~ .. ._.. . ••• _ . 'a_ _• ~Jl~~ Tf .•.. - ••.•..•... '. f-." 0.004 _ _•. _ ~'.-z.'.'; _ ~r! .DiD IT ! ' .-1' ."'! ~ -.-~ ,., :jiI': \. " 0.008 -1,,­ .a+~ 'J.i~ v~ M,! -'fA i ~L . « 1 ) : '.'"fa.. -1-- :..1,­ ~ i ._.. ~;~ ..­ -­ -~.~,~ ..~. t: ~_._ .. :: :,J..... ·i...· " ';-I-~ . 1i1 .. - ... .. .:.. ... ·~Rg.... a ... ..~ ... '.-­ "'1 .. -t- -: I r -:_ ':.. § --:­ - a :..;. I:::' .. .......·.....~...-.I···· ~ 'j­ •• ~:I-.!-;.: ~- ._a. ····i I ii : ~- -0.3 •• -f­ . .~ ··1 ::-,.,....,"". I ; poa"1tion ­ 1 Ie . • • 0. x/c R .~- -t~.0 ".'1 es.o I· ~~...... -I--I-+-~f-+-+-r--t-+-i~:.......- t-+-.... jD .0 : .~ !-..:L: -­ It--···.· '-'":" . ~ . "\" I ~., ll!-.. .• ­ .', i...;.:-. .- ··t;;·-~ - :::.' .... ;.- - ~ :.1': .. . - l : l•.­-t--;.+-...... o .. I ; _1.4!~ i I< 6 10 .205 .205 • 2 0 5..; .075­ 02 7 .. .O~O • I ~ iI i ~~~--"I......... -1.6 - .­ -1.2 ••• -0.8 .• ·1· ~. l t , .i· i 0.4 0 -0.4 SeetiOD lift coefilCieDt, c, NACA 0010-36 Wing Section (Continued) 0.8 1.2 1.6 THEORY OF WING SECTIONS 460 2.8 :t······ '" '.. t~· ;:g gg :~:~ ~ ...., .. I 2.4 2.0 -T" ..! .. ; : .. -·i- r-;~:~. -; ..i i 1.8 .. , " t 0.1 ~ Cf , 0 ...8J fa B J 1.2 0.8 0.4 0 -g -0.1 --0.4 -0.2 -0.8 -0.3 -1.2 -OA -1.8 i ~ . ~. ; . !... : i i . APPENDIX IV 461 0.082 0.028 0.024 ;: ..; '~i'" ~. .- _. ···1··· ..L~ .... .. j. ~... Il ~r, , ..~_. I.. 'l! . ~.. ... g~ ·0° x 10 I • i I I ~ .. ...;.. ... j.. .po' i·,. -~+.. -! o. • .~L ·t· • :.- -+-+:-+-..,.~-+-~~-~I I I ' t· ':"j'- ... . ······l··· 1-....+->..:""'- ... - "1-- 'f' -;--·1·····-j- ...J,}...+. -t"'+-'':;f ... 0.018 ...... -·1,.. -+ ... ...-"- --·I\·r: '1'~ . . '1'" ""1"" .},.. -1'" -+ -l' .:)~ , J' I.lL I! .J' II ::=u 8 ·S .... 0.020 . I • ... .... .! .. i- '1"-' .. 03.0 : . ..4-...A8t~~ l~usJm .. ·t~· '''T' as ~ ~ a .9 J 0.012 0.008 0.004 . "~ .. "-i" 0 I 0 ..: I JE r' ; .... " .-1'- ._(.. ····r·I -''1, ' - ~.!"I. .. i, . ~ '" ··t···· ....~_ .. "-r - .- . ;.. . .- -t. .~ ...r i r;..,1.... ·1·· ·. · LL I ~.. too M.. 1 i.- .. ~._.~ i I . . .1 ... ..1 .. . t-- '. . .:- .....- . !t,... ~.: ···i···· f V ;' r V ~ Jj "', ir k· . Ie i '~~I' ~ .. _~~..Cl~ !-{W- i !... I· ·..ii i I ci II . I j ... ,i., I I: ··i • 'f ! : I !.... i··· ".. . -;. -. f -,- ;..- rt1 ~! I .. "'''' . . _.. ! 'r c: •e 0 ~ ~ f •I .. ·f..·· : -0.3 .1.. :"i i -1.8 : ~ i .: : i t t i I -1.2 -0.8 '! I .f i i' ,1-­ ~ + .. ! -! .,• I .. - i - ... i t-··· . ·l·· I ~ i: i.. i i I ; t I .. 1. .. ,J .,.. 1 ....t.. .J. _...J.._ t ~OA 0 0.4 0.8 SectioD lift coefrlCieDt, eI e NACA 0010-34 'Viog Section eli i i i a = 0.8 (modified,) i ., ', i , I i ... = 'JI : ·! J I ~ -,;- I ; i !.... L i "'i , ! ! i ~~ L...... .:.... L.~,,_....... ... ~ - L ";0.1 -0.2 I :o~ . ·1 I~:· j . . ·0 .,.8 .i. :r:TI~,l·· I' ',;.' ! i' --JIMI .... ~,.-~ !,' I ...;:~ ..... - I~ I i. t j ~ ~-I_."" I i"·r-'""r't~.... i t ... _. --I I I . = 0.2 (Continued) I 1.2 1.8 THEORY OF WING SECTIONS 462 2.8 2.4 2.0 : 1.6 i e 0.1 ~ 8 .::: ~I-._. ~ c; 8 j ) ~ .~~ ._.~ ; :~ r-- ..I~ ·fal····~ Jr". i .... : ~~ i' .., .. '. •..... ~r: .. , j. ~ .j.. It e- 9.0 x 10 6 &.0 =-~~-- Ij~ -_. . . . _- .~~) J '.06.0 ~~ - - - ~ rousbDe.·6 0 --0.1 -0.4 ....is • 8 -0.2 -0.8 -0.3 -1.2 -0.4 -1.6 .a .;a~; ~rhi. St_ciaM s ..: 11 + ; .1". !_J .. _! .. .L: ~ ...+... I ··'·--1··· : .+ 1l~ ~ ._...~~._t·~.._.~-!./:; r . ·-r 'i ,.i : ! := 0 : : 0.8 0.4 ~ " ~ lli 1.2 i iJ ;~. x 10 a I :I -24 -16 -8 0 8 16 Section aaale of ~ttack, GO J deg. NACA 0012 Wing Section 24 32 APPENDIX IV 0.032 463 :::;P: .~....:I:::: ::~:1:;:: :::;i:::; ::::-1::': ;:::1;::; :::1' ..... .. .. r- .. .. ,( .. :...+- ...-r... ~~ I + ! ..: ... -. .. .. ..~. ~ ~~ i--rt : '. : : 0.028 ~ : I : .... :. I .. .. ... :,.. ~ .,;e. 11~ : ~mrm; =LI:-: ~+ ~1~ ~q. .. '~I +: I ~ i 0.024 ;, "';'''1 ~ ..~ ,. : I , .. ··r··· . ... ..; I T·· '1 '-r ; i i !.. ..-.:.~ - ~~I~ t·!· i • -i-.. i . ! " , , ---- ~i. •.. ~ "i~~ .. .+. ;.' --:-- II"~ "!-~ l j ...... ~-- t. .... ij.... ··1···· ..~ T' ~ I .t ... 'j " 0.004 ; r- 'i ! , _. 0 '--:- : ! IS 8 -0.2 .... :....- - .. ... ~ , - .~ ! .. : -1.6 i J. -. ~ I ., ~ -1.2 I t .. i i : ....... T· .. I , .1 ~ ~ ..... : .. ~ ~ ! : .. ; !' ..' : ... ~ <> D 0 ~ 1_•0 '.0 ,·1 .. i,-·J 0 x 1~ t ~ ; : . zlo "l/0 ~~ o o. o .250 ···1 ...;- 1· I ·i .. .~ .. r: ~ , t ,~ ; .. - :Y"" ..L. , ••-c • poal'loa ! R I I r '. 0 0.4 SeetioD tift c:oeff"JcieIlt, -0.4 : ; "'IIIlV i I ; : .. L .' ... - ' -! ~ .. - ,A -- - -- I- -! "1 ..I.... ..LJ J I i ... ~ : --- tj?~ ip. ~ .. .. t i_ '" ."-'" ~L . .~ : : .. .. ~~ .....: ~ I, .' .. .: ~ i "'1'" ~f" ~ --- .r•. 10 .. .. .. ~ : -'! ; :/ II ~~ ~ ~ .: .! I ~r : i -0.8 .&••• "f ..... -". ... f ; -0.3 . - t ,i t·· j i 8 ~ .... I '" .. I.... : I , : ~ - ·r: ~ .... ·l,·· ~-a- ; ! .. ,i I ~ ; -0.1 . !... f.a II .g : I ..·i.... i ! "j'" ! .' :- Nei ..... ... . .+_. .. ~-i··l~· t ~ ~~ i ! ! ... l'i ~. -s .. ~ ! -."• : ~":" : 0.008 : I_~. 1 s ."tI .. 1-+. ..:: .. 1 ; . "-r- . : i I i H~ 'It: ~.o X 106 m .0 8 0.020 ...1.:- .. ! ! .... ., 83.0 ,,i ! i .- 4Stua4al'd rouabn·" -t-o _.-i-~ 6.0 x 106 ! . 8 0.016 "'r'" ··-1·" .~I~: .1... too ~ : : ::-:~: ! ..1 I ~.+ .....i. ~ ._,.... ~.~II ';q "....: . i -1'" f 0.012 , I . I ... L. ... .. J" '. ... ~~ ~ ~~i"'T'" ,-' ~ ~ ~~ l ! t. - : 'i - i .. f- ; /1 ~()F' -T" ~, ! I R ····f _·t·· ·-t- ~i'! 'r'- .. ~I .. -.L ; -.--- ~ .. ·-t' ; "I ; "i' c, NACA 0012 Wing Section (Continued) 0.8 1.2 ~ : 1.6 THEORY OF WING SECTIONS 464 2.8 ~;+ _.~:.~+- : :(. "'j"';": ~.:. :::~:;~~:;:: ::" ;~:: ': 2.4 . J.. 1.6 1.2 .... " .I ··t··· . -.- . . -..-- .~:- :I 2.0 ;;;; m; :':":' :: 0 200 a1alatfad .plit . ~tlap c1en.ote4 60° ... -.. -I~ : B ....t. . ' ., V 6 X luv ~ :.~ ~~. ._\... vStan4ard6I'O\I8bD· · · : I" : "{: 6 x 10 :~~ -l·, . ";;! . ; I .'; . ! 'J: . ·-t· ··t···· , _·t· . . : ····1·· I {. -··1-1.~r·· . -i....: : . .+.. ·~··f-· ····l··~; ~~.. .I. +.. I I -J- .. \7;. r ........ -._ .." . ~~ :Mt\;" f ,..· ~~~ IL~~. _ _..­ . !a~ ·1·\ r -'i _J1.'·· · .]. ---··-i··· · ': . ... j .. : ; "':1'" ..J._ .-4--....·.. >­..: ' -.1....1·. "I:' I. . 'ff' . . ··1···· _._..._...- .. -f·· I I :,.. f +. --j--··+..·.. I . .I ' ~. . I . '1 rT­ b!,i·" -. _­ _ . f - • • - •••..•..- ...sa 0.8 "1" . :-\.- ...... ·"r"-:~..J··: -_.. . _- ~~_. +~ . -i:.. - .._>-~::. "­ ~ .!l ! .~ t: 0.1 8 .••. 1. 0.4 ~ = ~ IE 0 " C.» ..: 8 :g '; ...8a .I... 0 cZ -0.1 -0.4 -0.2 -0.8 i.; .... ii • -1.2 .1'r. .J. -1.6 ° 6.9.°0 0 0 : • I .; R 6 x 10 , 3.0 ,.+.. . It.. ··!.. +. +·· . ·1 +-1" ·.. 1 !.. .... l~-.:---:.:..- '''i~~~ ~. 1,. ~. ~')I' t---t-....,..-.~! -24 : 1.. ~. .: .~ .' -OA ... I -+. ~~.' -0.3 ...'_ ~ ~~ ... ... ~ i..J'" . 1 .. l........ !I .. .r" _ "'f" IJ~" ·~·l··· ....•,,:- ... l' "'1'" A 'S~r:8. : 'i"lt;(li V..... i.' ·:l.... ~' .... !.. L !-: ,6.0 x l~ ~. 'f if '~r-~ .~f" -I "- _.. ... . . . :t1~. . i···· .. " .. ,... 'fr ;.... j.. #. ·1...... ...1­ I --~t·j ......... I..· i' .. ; \ ... ...~ ...• :.-!­ -t· .... --j .. I.. ·..1 .. .: u I ~ ~ + ···1..··:..+···1(;..· .:..-­ . 1~~:.t~:I·· ~ .. .. • .. : -16 j -4­ .~.... !.. i ... ·..1.. ·....... .i ·1 i--;Y -;"·T!H ...... l-j­ "I '0.200 .1.alated ~!. i ."lIt flap detlected 60U ~ 106 :.1 .. t j . . i +. l., , j. i. ~~__;~~i-+-~~-+-~ ·1..··' .,... , . !.. i I..+·I! I ill..I·'..· . I.. ..+.. j...:.f.. v6 -8 0 8 16 Section aaale of attack, 0"0, del. NACA 0012-64 Wing Section 24 32 465 . APPE.VDIX IV 0.032 0.028 0.024 "~""+--+--+-1--t-- _.. J' ..: 0.020 8 13 is 8 r.. : .. .. r.-- ~·1.··~ f~ ···i·:I -i.ir- ••• 0.018 -a .. sa .2 0.012 ri 0.008 - .•.. _...,.;: i~~~~1h':~~ .. '::' ..•• ".. -~ :. ~ ".. ~ ~~ ~~~~.: . ~. ._..- -- .... ..+. 0.004 ~ a :·...I~'-"j~~~+_··_..· ." 0 - "'{ ~ ·--t· ._~~. -~ ; --t·- -~'._. . !.- ~ .,..... --. ',::1......J-"I d:-; ::. -.:. ··~--r-·.i- ... __ H ' __.+;: r: I ~. ~= f .,..~_I ::. ~. ~~_.- ~ ..-..- .. -- '-'" 10"-1-" f-!-=~ •••••••.••. '" " .......­ ~-l. ._.~ ... .-. ... I"""" i i.._t-_ .~ 0 .r_ '" _._ . '1·1· c.J . . ­ ,..- -+.. -':'~=~ll:: ­ -~t·~ a.c. poalt1on ..: -0.1 i ·u E "C -J.- .~. ~~: ~.- ~~~~ ~ ':~~~~- ..-..0. x/c 7/0 -0.2 .y ~ ~ :I -0.3 -1.6 -1.2 -0.8 -0.4 o 0.4 Section lilt coetracieDt, t: I NACA 0012-64 'Ying Section (Continued) 0.8 1.2 1.6 THEORY OF WING SECTIONS 466 . 0.032 0.028 .. t .. ~I'~ ... f z ......0 i 0.020 tg 0.016 II 0.012 j :··. ....­ '....., .... : "";7:" _..- _. "':4 ... •..~~ .....~~ r::'~: ~~:~I:- ...•.~ •.;:~ :';'I~t ~~~~~:~.~ ::: ~ _. ;.~~ ~ ~ ~ foe·· . . . . . :':-1-' ••••••. ~ ...••••.•--ro' _. :~.- ~~ t - •• I-:_~ •.4 ••• ;': ..... ; " j ..: 11 -0.1 t.. J - . . . - - .-1- . _. .. .-. .. -.. J--. ~T ~::: ;: r·.. . ~··· .. •. :!. ..: 11 ';"'7 ::- :w· 1-::' •••••- •••. ••• ~~::.: : ._. : :'1':;'~~' 'i!J. IIQ;~ .....• _:~.. .. I .. . . I ...! . ... +. ···1·_···· l··· . .....J.. ... ' ·1... 1 . i •. i ~ 1"-' ... -. .~.... ·1 1 _.•.. . ).. rJ.f!. t:~. ~1iI1 ~.-- f vtI +_.. J...I W ~~r;::=-~. ... ·..t·· .i: r L ... 1.... -_ _... .... ., =- ..... ::. :::: ....... .•• ~4::;i::..:·~· I~···r,- .~ -~- .~~ •••. :. ;.;,:",_~. ::':i .~:; .... -+:. -1 j' . ·:·lJ _. ... r,"J;~' . .... - ". =- ~ .;: r: T" ·1-:; ..... - .. ;.: ... ... -= ~'. '-."11-.. ~-.-\--of­ +. ~.. ;. ,.. •. _. _. • .:~ .;; . ._....: I ) , -­ - ~~ ,'" ~ .I...; .6 St::~~... ----'j"" _........... : .. \.! '._.. ..::i~.~: ~~.......... ·T··..:- r' Y:i 10' x ~~-::fo-.' ·-·a : ; ~:. :'" .~ .:;.:-... .•••• 0 <> .0 ~ .... .; I" ·~f· :~.: .;..~ ~ 1\ +j: i:g .. '.... :. :::: .• 0.004 .1 ." .:-1-... :... . r\b'I.' 0.008 Ii\ - 'JI~' : ~ ,.; ~Jat . l·IA·· , .. .­ ,~... -. .. .. Io:-fo­. ._.••.•••.•..••• : --r:::.~::: .... .•... ,. .. .r./ 0.024 "'" .. -r- ;.. ~ .. I··· i ! I i ... ] -0.2 .•. ._....! 1 .. , -0.8 -1.6 '.~~ .0- :::. .•..• - I ..• ~~ ••• -'1 . + "1"': -1.2 -0.4 • 1 'f .. J •••..... ....._. -0.8 ~_:. ··f ~ 0.4 0 SectioD 6ft coefftcieat, NACA 0012-64 Wing SectiOD, (I == 0.8 (modified), Cia == 0.2 0.8 C, 1.2 1.8 467 APPENDIX IV 2.8 :0.200 R· 2.4 '16 x ~ 2.0 !-=t J..' :·:I···J· ··r:: ~l~r:':"f 8"lAt. ai.ude4 flap derlected 60 ~ 106 · VStandard rougbne•• 6 x 106 "1 ."f J: : i .~ ._~- : .;.._ ........' ; _~.._._ . ; I.. . Ir~ ~'lf" i . ~ : .; . . .- : ~. ! ; : ~ ~7- ; .. [I:. ... \:... ··+-t+ · .-t-: .._~ : : ~ \ ; ffi_~.rk~-\. i-t:. : ~.+ , oj._~.+.... - .•. : .. . : • .. i : 1.8 .._ 1.2 ~ .. ~ ...:. ~~----+--i-f--o-+""'~' • ~-+-...ioo-f-";'--f-~I-/·~··i ---- ~ i 0.1 . ~)": ! o!; 0.8 1 ~ .. -t-. ; !: ~!: !.. .~ i . .' ~. ~ '\ ~ t:' ~J". : .: ~-1 r' ..... ~ -t- T .~ ~ ; 0.4 ~ 4t. 0 I" ...I. t... .1 J 0 .. 0.1 -0.4­ -0.2 -0.8 -0.3 -1.2 -OA -1.6 a j -16 -8 o 8 16 SectioD aDale of atteck, «0' dec. CI == NACA 0012-64 \Ving Section. 0.8 (modified) J e'i = 0.2 (Continu«l) 24 32 TllEORY OF WING SECTIONS 468 3.2 2.8 1.Ji i" ,JY I~ I.B f ~l 'lL l'\ ~. IF HI~1 B· I 2 I' A ,.II Iff Jr II 8 1/1 Jr B 1 / 'f t~ I 4­ a ~) ill 'I 1.' -:/ J ,. ~ ~ ~ I~b lIS ~, (f I~ ~o--J , .8 I~ ~~ "-~ 'J~ :u - - ~ ~ I .2 1. .8 l rv t -5 -20 -32 .. i\1 ..... t~4 _. J. . I'll 3 ~3 I~ -:~ j I­ if--:2 :~ s: ,., IV ~~ ~u ~ ,, J' '<l .i' /19. w~ ~ ~ d P 11' -" l,( I -8 0 8 /6 Sectlcn tMVs t1ttac~ "0' d~9 NACA 1408 Wing Section -/6 of 24 3Z 469 APPE"NDIX 11' ~ .03tJ.r-r-r-.,......,......,~--r-......-...... -r--­ _ , '\ .8 , , , 'rJ ,, to T . ,e J I "'"' A Jr:t I ·I.o16t--t--t-t-+--+-t-t--+-I-+'Ift-II-+-~--f--+-4~-i--I-f-4--I~~~~~ , ~ ! r2.. . . . . . . \ \ If II , ,.J '/ ) J II ~\ 1\ ~ ,\. \... ~~~~~ ~ s & ~.(J(J(J.-+-+--t-........--t--+-...pDlII~~-+-t-+-..--I--+--I--+--4-~f--Il~~~..-..+--f--+--+--{ -1.2 -:8 0 .4 .8 coel'fic/ent, Cz NACA 1408 'Yin~ Section (Continued) -.4 S~;on lift 1.2 1.8 THEORY OF WING SEOTIONS 470 bIG 11' ~ J" 2.0 \, .J. I~ 'rb r /.6 ., r'R I~ J , r bfJ 'I ~ , 1:1 1 \ ] J o ,," I~ )4 -0 I~ n ~ , .., ,~ r ~ I.J' I' JI r -.4 .. r I.U I. -.1 ~ -~ ~r I I .. 14 or Jf g IQ II.. ;lA ] ., ~~ .C " J , ~ .. .. ~ j~ ,r• XJ II -8 j G I: ): .~ If::l -1.2 , ~;.t ~ ,. I.oJt ~ lIJ '-~ • I- I;' -1.6 -s -20 -32 -24 -/6. -8 0 8 ,.6 SecIion of aI1Dck, «" deq NACA 1410 Wmg Section CntIe 24 471 APPENDIX IV ,, , .4 *.6 I I .8 ~ I , ;> ~ I ....... ~ J , -\j I I \ J .......- +-W"Wo-+--l n ~ .OI6.......- +--+-.....,..-+--+-........ --t--+-+--t~-+-+-4--+--+-+-I!-+-+--++-l....-+ ... a I J J ,, , , ( IP B'.OI2.......-+-+-+---.......-+-....-.+--+-"lIlil-+--t-+-+...-+--+-+--t-+-tI~+-+-+-~~...-. Y/~ ...... ~ ~ s """ _" ,~ r1 v \.) J: .008t-+-+-+-+-+-P-+~~a.-+-+-+-......-+-+-+-.-+-+~~..,......-t-+-+--+-+...... -./ t--+-+--+-+-t--t--+-0f-4-t--+-+-4~-+-+-4~-+-+-lJ-+"-'+--I-"""'-+-o1-~""""'" i ~ ,'" .i-.2t--t--+-t-t--t-it-+-~""""'~I--+-~~~~"""""'--I-~--I--t--+--+-..f---f.-+--I ~ 0 ~3 :0.10 .:r:$~;; o~ ...... -. 1--+--+--+-1-+............ ~ <> 9.0 4 ~ ~ -.4 -.8 .~~~~~~~~~~~ .24 7 .OO7~~-+-~I--+..-.+--+·· I Staodord rOUtJhneSs O.2OC simulafed· spk.f flop defkH:fed 60­ 6.0 I ~ ~Z -1.2 .~ ·St;'~ ~~t-+-t-~I-+-+-o+-f -.4 0 .4 Section lift coeffldenf, c1 NACA 1410 Wing Section (Continued) .8 1.2 1.6 THEORY OF WING SECTIONS 472 3.6 rr­ - - 3.2 - r-­ ~ 2.4 LY r\ 1..1 l I\., II' .J t'1" 2.0 rJ ,., 1(/ :L dJf 1.6 '(I J.~ ~ n #I Jf I­ Jr ~ IllS II I r ,,..­Id J .N. 11 tfi 'I' IA) ~r:: J 1 '.. -1.2 ll't ~ d ~ -.B - r Tr\­ I" ~~ j' v iI.( J' -.4 J ~ IIJ - r'Q 1\ J' 1~ .1'1 ~., .41 i' IJ 'I i\:> , II U , JI!..~ ~rl\ ~ II .4 ,~ .J' l' .8 4~ ~ 1(/ I., l2 \ r ,,~ ,~ ~ ~ '1~ " ~"Vo n ... y .. [D .ft:l 1. ~ IV -1.8 -20. -32 -24 -16 -8 . 0 8 Section angle of oftocltj NACA i412 ",Ving Section ao, deg If} 24 APPE.VDIX I1T .4 * , 473 .8 .8 I t 1\ , J il \ I~ I J 4 \ \ IT \ t ) v If I.E } Jrr J _'I 1 i { "­ ) -. ... "V -. t i __ it - .-­ f ~ I I \i '10-.­ 2 .i-· u ~ • 0 u ..... 3 i i .4 -~.6 -/2 I I I . I I I I I I I t I I I I I ( I t I -.8 -.4 0 ·.4 Section lift coefficient, N-~CA 1412 \\~g Section (Coiltin1ted) Cz .8 lZ /.6 THEORY OF WING SECTIONS 474 1.8 2.4 2.0 1.8 1.2 ~ i :R 0.8 ~ ';J 0.1 , ~ 0 " ..: -0.1 II -0 Ii8 8 0.4 a .. .1 rZ 0 -0.4 ... -0.2 -0.8 :8 -0.8 ..1.2 -0.4 ..1.8 CI • I -18 -8 0 -8 18 8ectioJi ....le of attack, «0, .... NACA 2408 Wing Section APPENDIX IV 0.032 ........ .. ~ .. , : ... ...... ... .... ....... ~~; :;r: .... _. .... .. .­ : .. ; .. ~:-T:~ ~'··t~~ :~4~- :..1 ;~f-: H~':~ :V= i i +~ ... ! 475 '':: '" r ~ , : I : I .­ -.... "t... i" .,. F· ·_:t··· 0.028 '1 ! . ,.­ .. , .-+. ~ ... ....j".. f:t: "',.-6. ~ ~. I : .j' T" : 'J i .'1 -1 _. ... ~ ..- ... ..·1···· .4'.­ ~ .. ~... .. If ~8\ ,. ·..1-·· .. ··~l·_· ! -. I ,i 0.024 : I I ' " . .. :. ... ... ..-1-. .. ... ··-F·· .. - .. _.. .... T··~ . It : "·t~·. I i , ! . •..:..1_ ... . .. ... ... _.. .. ...:;.- ~~ .... ~~ . ;­ 09.0 x 10 6 f D6.0 0.020 0'.1 ··-t+ ?t'~ '­ _. '-fa­ .. .. ASUadard roupm••• : ~ '1 .. .. 1 ~F, 6~O x 106 1 ~':'I-~ .- ... 1- • .. ., : : -. ... I I :­ I II f0.016 . ,­ : .­ .:l ..... .. ._ . -. -. ,- J.... '-J­ .­ I 'r U-~ I I r 4-: : - " I' "','" ~~*. <• L·f·· or' " . " ., :: ~ .I I .ar II .s J ­ ~ _.~. . ­ ~ ; ./ ;< ~ 0.012 ., ." .. ., .. .. .. .. " _.., .. - : .... ... .... 0.004 .. .. '" ., ., o '1: ;,' - --~- -0.1 _.. ..: ~ -0.8 " .. .. ; ~ .. tr : .. ~ . .. o. J . - .. - .. ,­ ~ .;.~ ~- 10_ --- ,­ ~. . '.­... ..- .. ~- .... .. ~:- - ... . ~. ... . - ..­ .. .­ . ·_.a.... rv-! ·-f···· .. .L- :J... ; ····~f· i ... -- .- --; . .­ ... •.•. f-. .. ~ -0.8 ~I--. ···h I '" '1"­ -- 1- • I ! f -0.4 -~- -­ ...... -.. ~. -~ - . •• .1 _. 1 .. : ! a.c. poaltlOD ;:fc 7/C R ....x " ~"'! !' I "j' , .~ .ill j ..~ 'I ':" I ..·t i iI - ...1 -' ... .... 1 a I. _..~:t·· ... .­ ," .. ..;.~ ~- i .j t I .. .J.. .. -j- ... .1.... ... L.. ..J .. .•. _. ... .­ : : j .. j... -~. I , : .+: _.J._ .-!-­ i j­ i -­ I"":,,. 'f -J~ .~ ~~ i' - ._. ·-t·i .~ ~ ..-... .. :1& _.- ~-~~~ ... -.. . ~ ~ ,; f . ., , t····~ ·it ~ :, .. ~ .­ .­ :._'­ .. ._. ­ -1.2 'Or • • ~~- I .. .. .. • I."",. • .;!~ ~,:L • ~·t,;t ~.. f-- _. ... - '.__ . . . - ~ ..;,; - ~~ .. .. - 'j --t· -. ~i" ~t· 'i:­ .. ··~lt r - ~ .jJ/­ ~'1 ! l­ ... ..- J " _ .~ ~~t- u, ~ .-:.~.. .... .... : .. ._. ....:f-: I~ .... .... .. -1.8 .. .. ~ ... ,­ : ! ..- . ...;. : . , .. .- ..: : .-. ~ !-I-' -­ ~ : . .: .. .­ '. ..... -0.1 .;~I; : .. ,­ . .. '. ,..·I~ :·;!1iIii~: ... .. "­ .. ~ : .."/-: .... .... .. .~ : .. .. : -+- . .. -~ .. e.,· : : <. n : ... ­ '::Q '.\ .... i:!: .. .... ; ... ~ '" " 0.008 ,­ - .. .. : :: : : :. .- : -.- ..... .... .. .... : ..#­ v· " , I!<>Z·O 106 .~7 .0 .O~ .0 •2lI 7· .o~ - .­ :-. I,~0'.1 .247 . -:-:t-f. .:tt_i....l-4...J.-l-}·.~ . .L~ .i J f-~ .' 0 t- . . ... ... i -=-­ .1. I 0.4 SeetioD lift coefrlCieDt, c, NACA 2408 \\-ing Section (ConlilLued) I I 0.8 I I I 1.2 ~ J 1.8 THEORY OF WING SECTIONS 476 .. 2.8 f- 0.200 ... flap detleoted R 2.4 6 x 1 •; ! ~.£~: ! ! I ! t ! ....~rx t·: ! : 2.0 :l.6 ~ ~ I· i : ,- ... i ! 1.2 ... ,i I e.t 0.8 0.1 ¢: ~ E " 0 '.J ..: 8 :§ I ; -0.1 ua i i ! i i ~ I' I 0 -0.8 -1.2 -. -I i ;V i I ~ -1.6 -24 _ ~ ...... . -16 ! -8 ! .... i " i. .i ... • ! ; ~~ I j" I .... ~ '" ; ! I ..... , , o.~Oo t I ~- . ',i_ .. . : I ! i' . ....i -:... .J at.a1ate4 split t'lap den.ted &00 t I .' . y" .. t o 8 0" R -,'x·1 I" , , 16 SectioD aDale of attack, «0 t cleg. NACA 2410 Wing Section - . :"s ! " I ,·6 - .- ~ .. -.. ~ L...- i .... .. : :.~.... 10' I i ! : -I I ,.~~ t -0.4 1 1- s ;\ ~ i ~.o x .0 03.0 6~~ugbne.s 6.0 x 10 : v -,;, .Ai.~ ; ; ~ 1 : :4~1 • • :. -0.4 . i : .f Ybl' I ... t .. I 1 vr. : ~ I ' V r\ bl f't a ....j. ·l·:r:· · . i..~ .. - i ; ; .. '~i , I· I ... 1' l i'~ iJ i ! I cb : I -0.8 i I .~ i 8 -0.2 == ..a :it ! 0.4 = .; , I 1l l' : ; : 1\. , ,~~1.1 I, t t t. 1 VStandar~rOUghneas ~ i t~ 106 v6 x ~ , i 0 ' : .~.... I i .. I... ·i... J .....:.J... I al~la~ed'sEltt 0 r ..;..... : i ...;...­ i· ; I "1' 24 I j--' + tf 32 477 APPENDIX II' 0.028 0.024 ~ (oJ ..: Ii ·u 0.020 Su e .,... tal 0.016 ~ .. r! a .9 0.012 0.008 0.004 - .- _.­ . ...... ; : ac1 0 -- It " ~ 8 -0.1 I :§ ::::u 8 a u I I-- e ~ ~j ..:f-·~·J .,T" -. ~ 1'-' t "­ 'j' ..l t ! 1 f ... ; ; .I ---f I I &.0. po.1tt_ ~ . ~ -­ : -0.3 o· : ~ i ~ -1.6 -1.2 i .J. O JI xjc -0.2 .a 0 ! z .•••••• .~ 7/0 : ; -i ! -0.8 .•. -0.4 o 0.4 Section lift coefIJCieDt. c, NACA 2410 "-ing Section (Continued) 0.8 1.2 1.6 THEORY OF WING SECTIONS 478 3.fJ 28 2.4 2.0 ~N 1.6 I e ir ~, 1.2 }~ ., ~V IA I ~I ..4~ ~~ -~ L,\ \\ '" rl , \ ~ ~ '~ ~ I 17 tr J r I ~ a .r 1: a ,"t • 7 '~ ..... It" J .... ~~ ,~ ~ t.' -.4 r"", , ~. ;J J )/ -.8 t" J r L K:~ -/.2 -I.8 -5 ~ -24- -/6 r8 St-ction anti" o attach,8 ~01 de9 /6 Df' N.\CA 2412 Wing Section 24 :J2. APPENDIX IV 479 .036 :'0 V ~ -.2 --­ -r-- --­ --- ---­ ./J32 I ~ 0 .2 .4 aTe 1.0 .8 .6 .D2~ ~ ~ I -­ , \ ~ , I ~ \ 0/6 , I I\. 1\ ~ ~ 1/ d ",~ f ' ....... / r/ ~ .... ~=:IiJ / // I,P:/I ./ "'-. ~ ./JOB r V ~ ~).... ~ -­ ~;t: ~ ~ -..,. -..­ Q () J ,/ ~ l"'I '""'"'-.... I I ~ me J ~ ~ I j ~ l-'l I I I i I t .L. I I i i D04 0 "'­ - - ~ .A. A. .,. u­ -./ ; ! (i 'W' .i -2 .\! ~ CI'.c. R ~e arao«: -239 ~ o ...... o 89 .. _~ ~ 2~7 " -~ I a 5.7 _,...._.243 .. 5.7 _~ ~ f posItitJn FIfe .006 ~0D4 -.0/8 StD/'Jtbrd rDtJ9"~s -4 -~ -/8 -:8 -.4 0 .4 St!cIion Oft coeffICit!ht, e, NACA 2412 Wing Section (Continued) 8 1.2 1.6 THEORY OF WING SECTIONS 480 3. 6 2 2.S 2.4 2. 0 ..J~ I.6 j~ ( ~ \ \ ~ IAV ~rJ 2 IMV iW fj,. J<" .. ", \.r\ ~ \ ~y ~'1. ~~~ ..,) \ \ ~ r J 4 J r If j c -" 1/ 1:.., ] 1/ l.", J .4 1 'I "":'{ ~ ~~t 'I j .8 1# .. i' .... I' \ t c :/ -l 2 ~S flJ -L.6 -20. -'32 -24 -16 -8 0 8 /6 Section angle of attocJr, CllO, deg N.AC.A 2415 Wing Section 24 APPENDIX IV 481 ~u j V ~ ,........ r--- r--­ ~- -.2o r---. r---... I"""---. _L---­ ~ ~ .2 .4 .qc .8 .8 1.0 .D', .01. ~ J I .024 ~ I ~ J T J ~.o20 I II If .. i I ~ ~r\,. -, ~ """""''''- ..-­ ~~ b ""ttl [V' ~~ t ' ~~ L-t, ..,; - .., .. """",:r -­ ~ 1ft.. ~ V ~ ~ If ~ ~ I ) \ c A J V II J " ~ I 1~ If if j v 1)01' 1.<'/ V I ) )..J lI"/~ ~~ -­ -'~""V ioiiOl:I_ 1­ - .. - ~ ! .1 .2 d.e. position R -.ziff o 30.""" o 6.0 ~~ ~ .246 .3 () 9.0 6 6.0 ~~ ~ I-~ ~ 013 013 Sfandard r ~ .246 '.~ -I.z -.8 -.4 0 .4 Section lift coeffic;~nt, ~, NACA 2415 "·ing Section (Continued) JJ . 1.2 1$ THEORY OF WING SECTIONS 482 2.8 2.4 a J.8 ~~L 2 ~ Jr.,. -o )... \ r ~~ , I({.~~ ~\I' t~ 8 J ~, r ~ .I 0 ~ 4I:~ I .~ .... & c I ... J .. 0 "'"'-.,. L,.... , -.1 J. t"'-.2 .~ 8 ~ " -/.2 1-:4 -I. 6 .,..e-:3 ti -:5 -c4 "' -o~ , ~~ ) .J' , I~ I' 1:,,­ '""'l .... } l-eo J ~ !~ <' \ ~ q Ii c t -8 0 8 /6 0f'ItJ/tt of aft7Jt:k, . . d~t} NACA 2418 W'mg Section -/6 Sectbn 3Z 483 APPENDIX IV .­ r--- r---. ~ V" r-; .......... -.2 D - l--­ ~ r----... r---... I--­ ~ .­ ~ A­ .4 ..2 10" .8 .G J(c ~ j L r ,. ~ J ~ ~ \ x u C( ~ " -, ,, -q r\. ~ ~ l'r ,~ ~~ .... ""'~ ..,~ ~ ...,j ~ 7/ v "' b... iJ j~ ,~ ......... ~ "­ ~ ..... :t:r--..~ ~~ 7~ J:/ jl'-;~ ~~ ~~v 1Ioo....c.... ........ ~~ I~ J;I( l/ ~ t II / ~ ~ I I :tl5....... ~ t 0 I~ "'Iloo.... '" I.A,. A: .. ... r­ "I;l , - i_ I ./ t 2 o 2hJr­ -.239 , I .' ••~ POSitIOn .3 *f~ -.044­ D 511-­ - .24Z -./J23 089 __ '- .24/ -./116 6 58 _ ",,- ~ STt:tndDrd ~ .tIII -./} .4 titlctiaJ .4 '/1, 0cOtIfficitmlj '1 NACA MIS W'mg Section (ConMwd> .8 /.6 484 THEORY OF IVING SECTIONS 3. 6 3.Z 2.8 4- z.0 t6 ~~~ fj~ a ...... c l/I/ 8 JJI JI~I A~ 0 ~ ., r\.. Jr l:: ~ 1/1 u J Il 0 II ) J~ ,,,- ~ -./ ......... Ij ...... ~ ~ Jf "v 0 ~~:a 'I) hJ ~ ....~ ~ J...J ~::\. ~'" I.2 J' -. 4 -~ I:::~ -, I J 8 JrJ ... ~ I. ~ ,b 1 -I. 2 JI If A' ,rv " A~ 1-( -l 6 -.s s -2­ - Z -24 -ItS -8 0 8 /6 Section angle of aftock, "I, de9 NACA 2421 Wing Section 24 32 APPENDIX III 485 -.0_ :2 ,/' 0 --­ ---- r---... r---... r----. "'-­ -.2 0 .--­ ~ ~ .2 .4­ .D3Z ~ .8 .6 ~c ~ 1.0 ••• , '( "lJ .D24 \ \ ~ "' !\ ~O ,• 016­ I 012 IF .. " -" ,~ i. ~ ~ Y ~ r: / ~ ~ ~ ...... ......, ~ I 4 ~ ~ ~ r-, ...... r' J d I ~ I .i J / ...... ,~ :0-:"-. 008 J '", .. " ~~ ,II II , L ( J 1 I J j ~ ~:r lh~ p;:~"" l.a ~~-' .... ~.r v~ __ 0 f_ .... ; l.O_ -­ - ...,. - - .­ Q -./ .... -.z D.C. position ~Cf3rc -.032 -.011 R " 2.9JC/O­ - .2.31 "5:.0 .239 -.s ~a..$1. .24/ 65.9 Sfondord rou5 ",,,~ -.0/8 s -.4­ -",5 -1.6 -12 -.8 , -.4 S~ction 0 .4 lift coer'icienlj c, NACA 2421 "·ing Section "Conti-n1Iedj .8 l2 1.6 THEORY OF WING SECTIONS 486 3.6 «e 2.8 2.4 2.0 1.6 e ~ rt: r\ ~~ ~ ~ ~ ~~ ""~~ /.2 10-.~ .~ .Q t: If) AJ .8 ~~ ~'- ~ (.) 'I .... ./ ~ .6 I~ .4 1V IA~ ...,;; Col 0 ~ ,.--.~ 1'­ JV IB o Jr,... r ~~ "'t '­ Li""'o rv ~ ~p -J ", -."'1 " IA~ Jrt ~~~ -.8 1.'( I} ci"[ \ lh' t--- r\. '/ ~ -24 -/6 -8 0 8 ~ction DI'IqIe of OflD'-;';' NACA 2424 Wing Section «" deq /6 32 APPENDIX IV 487 .2 .036 V'. r-, 0 - r---­ ........ ~ - 10-...... .4 .2 ~ ';c r--.......... ~~ .6 ~ ~ ~~ .D3~ lO .8 .02IJ I i ...... j I~ , 1~ \ ~ j C !\ "... \ \ ·i.o16 ~ ) ~ '""" ~ ! ~, \ r\..~ r"2 ~ 11 J /' II \. 1/ "'i'-, N~ ~ ""~ 1""'0 """'" & 1.aB .~ i"""\l~ ~""'" ~ ........ Itr .A ~ ""'" I ~. P 1/1 I ./ Jrf ,/~ ~?--.. ~~ ..... :K; ~~ 1'P ~PJ' ~ ~ .... ~ - (I) i ! - "II. .. , ... ~ I~ ~ ", - :IIA .... ~ .. ..... .;WI' I d -' -2 zw[ a.e position R o 29)(/0* ~ .2/8 a 5.Dt-_ ~ 228 0900 __ .il3/ 459 __ ~ St ~ .~ ~ .5 -/$ -/.P -:8 ~ i 007 019 0'4 rl~fJt'.ss __ 4 0 .4 Section lift coefficient, .8 ~, NACA 2424 Wing Section (CMltinued) I.e 1.6 THEORY OF WING SECTIONS 488 3.0 .32 UJ ~'f'{. I/' J1 2.-r 7 f I ) II 2.~ ~ '4 II L6 JJ.ltl:n~ ]. .4rn I "1. I-~~ JIWl~ ~~ e v ....... .~ ~ ..... .. J If 0 \J 0 ~ ~ , l , J 'I h- I ~ \J ..... -.3 ~ 'Ph.. " [; -.8 ~4 ~ 1~ .,~, ...... 1!!5 ...... 45 II I. 1):f~V ~ l't ~ -1.2 t I I , J ....... .~ -:2 ~ 't.;. r J QI ~ ....4 , I I'r: .... 1 1) (~ 'C 0 " J) I 1/ ...­~ ~~ """"Il~ , ~ A II .J '­ lV I If , .8 ;: .1 (~ ~r I L2 tl ~r;x." ... ... ~ JT I'"" -1.6 -.5 -16 -8 0 8 Section ongle of attocJr, ~ NACA 4412 Wmg Sec..-tion deg /6 24 APPE.VDIX IV 489 .tJ» V - ~ t--. I'---. I'---. '" \ ~ .OJZ \. \ \ .2 .4 .6 * .8 LO .De. J I I ,I ~ J 1. II J 1"\ , ) I\. 1'-' " """ ~ ~~ ~ ~"" ~ )'f' ~ """'I; :....J."" r--.....n It'I!lIt: """""" ~ ~ J Jlo I II" 'I P7 e II' I V Jb r-, ~~ ~ 'I I ~ ~ -­ ". ~ :..... ., ~r- i I o , ~ -./ lr-ot. o - .. - ~ h~ - ! 1 G.C. POSItion . R ~ryJe o3.0xlo- ->­ - .245 ~D68 1 .246 -.05/ 247 -.04/ 66.0.1 J f Stondt:1rdr~ss UZOc simulated "$plif flap deflected 6fr 06.0. ' 09.0; v8.0 t -1.2 -.8 -.4 0 .4 Section lift caefficif!lTf, c, N AC.A 4412 ,"·ing Section (Continued) .8 1.2 l6 THEORY OF ,"'ING SECTIONS 490 3.6 3.2 8 ~~ " If ~ ~ .~ , 2.4 l~ ~ ./ ~ 20 .l I ) 1.6 ~ I ~ ~ I ..... "-", r\. "") ~" ~ j'" ~~ ~~ Jr I 1/ ~ ) ~'J ] [J 8 ~~ J~ 1V lJ 1/ ~V I ! (, 0 J, / II ~ ., J / ~ 0 II ,J .~ ~ -., .. " r­ J r\. 1 ~ :8 I If [ 'l J " ,J ... -.. ~ """lIl~' II ~ 'I;: 1-.3 t, ·4 -I. ...... .6 -IS -8 0 8 /6 $«:lion ~ ~ at~ ~.g NACA 4416 Wing Section APPENDIX IV 491 .2 .txJI V 1= r--... ~ ~ ~ .. t-----... ~ 1\, ., I II ,, I ~ ~ -.2 0 .4 2 _... «Ie •8 .6 IISII J.IJ• ~'? A b II II J JJ20 rY J 0/8 J. '\ ~ ...... ~~ f--"'~ 1: ~ ~ r"<",,- ....... tJ... - u - ...... flY I J/ I v ~ VI/ lid 1..1 'J ~ ()~ ~' ....... -'I .,.".,... I ./ "­ ~ [ll. , Y I - 12 rO I I \ j I ..... -..~ lIJ .... ,... ~ .",.,. 0 ~ ./ I­ 1_1­ _ - ) "_ - - .. ~~~ ~ 2 1~ .qcft .Q.~ 8u-.3 e 9.0 6 )-4 -.5 I I PO$/fion R 1Ic o 3O~0241 -a066 D fJ.O .2-1I-.lU{j 9 -I.P ~8 .245-:lJ.IO 6./J' Sfondord rDUtJlv!ess 020C simulated split flQ) deflecfed &.,. 6.0 -.4 0 .4 .8 Sttctiolt ,,1t c.HicitJnt, Li NACA 4416 Wing Section (Continued) 1.2 1.6 THEORY OF WING SECTIONS 492 3.6 3.t2 2..(J 7 2 ., I )~ rr ., l ~ b ~f I ..., f J 'I J 'I l6 ...J~ ~'--l ~~ J II Q ] I.2 1~ ~ ~ II J' :tip ,lo I ./ I c r] ~ 'II ~ II( .: J , ~ 0 'Ii II ... II I J II , 0 c , 'I J II ~ -.1 ~ .. I. ~ .~ ~2 \ -.8 .. I.J~~ ~ ~ ..... h -..3 In, -l 2 f'l'~ t I - ~U I .~ \J , j I J 0 j. " -.4 'U Jr J '0::: r-~~ ~ J~ 'I 8 ~~) ~ -.4­ ",,5 f;L.. "'-- I - -~ ~ -l6 -24 -/6 -8 0 8 /6 Section Dngle of oftocJr, QO, deg ~.AC..\ 441S Whlg Section 24 32 APPENDIX IV .2 I ~ ~ --~ ~ r--. r----.... ~ -.2 0 493 -,, L­ ~ :Z ~ I '"' ~I .6 .4 .e~ I 10 .8 O~ * .024 I ".020 I - v ~ ~r 11/ j 1/ \ " -~ ... '" r-, I "­ i (;;~ I v: ~ ~ l.A ~~ If I~ r.::. -~ I 1 ~~ Lb ~ ~~ ~.~ )...-...< ~~., ........ ~ .::~ "'"-L...t' .004 i i ~ If' ..-L.-l11"" I I I o ; 1 , a -­ / - ...., f ! 1­ ..... - - ... - -f"'t ~ (J.C. paS; iion :1 *~vfc 030)(10_t-L240 -.060 0601 I -Z42· -.O~ 09.01 .Z4 2J -.03/ Standard rovglr1ess t>.20c SImulated SPlit f~op deflected Ii, 6.0 J 3 -l6 V6:fl I I I I I I I I ( 4. -1.2 ...8 I t I I I I I I I I I I I I eo: I I I t I I I I I I I I 1 I I -.4 0 .4 Section lift coefficient, c, NACA 4418 \\'illg Section (Coltti",ued) 8 l2 /.6 THEORY OF WING SECTIONS 494 3.6 ~2 2.8 ,a IF ~ )., ..( 2.4 I j r; 2.0 J 1/ J7 \ 1/ 1.6 ,~ I ,~ .b. ~r / l2 1/ fS' J ~f'.J I ~ ~ ~ J~ I ~ Jr } 1 ~ 'I jr 1>1 J II I lP II ~ ~J [ il' JI ): .s. ~~ ~1I' ~ ~ -.4 u~~ if' II I -./ ~ ;r~ ~ ~ -- ...... .- ] .,:-~ I )1] 'V , ~~ J ~~ ~ -.8 ,, ~R rl"" J IS' -L2 -1.6 -.5 -2.t!.32 -24 -16 -8 0 8 Secf:"on engle of olftx:lt, «., deg NACA 4421 '\\Ting Section 16 24 32 495 APPENDIX IV , .6 .8 I .. N.fi-+--Io-+--+-~I--+--t-+-I 1.0 A II IA ~020 1--I.....- I -.......~........-I--60-~~-+-OO+.......--t-........ ..foo-........--I-........-+-f--t~..............,...1'-1-1 \ ...... .i .\i ~% § J ,....J \ I I ~I I-t-+-+--+-...... ~-+--+-......--Plr-+--+­......- +--+--t-.......-t-~+-t--t--r-M:-t-t-t--t p , \ 'I J ,. II r ." o. to/2 t-+-+--f-I-+..Jltr~~-I--I-I~-4-~;;;::;t3~p.q.--I--+-1H--f--+-t«I~!R-+-+-I .~ ..... ~.tKJ8 tt±t±tttt±±m_lm~ttttjjjj -1.2 -.8 -.4 0 .4 Section /iff coefficient, t;z N ACA 4421 '\\Ting Section (ContinllM) .8 l2 is THEORY OF lVING SECTIONS ·496 3.6 3.2 2.8 1..- } \ If' ~ V I 2.4 '7 If I ~ J 2.0 v J It J l6 II J ... ~ I?\.. I jD ir 1 (1 ~ 4' I o o ... .....-~ ~, " I .4 ~ ~~ II ./l.& _:.I'll 't8 S tr/ If 1 IJ. ~V If~ I I IV If! I I' , il J( i , -.4 -, Ir ~ '~~ -i, I· ,f •.8 IN ~ ~~ "" ~ .1.2 ~ --.I: ~ D-o ... ~~ -.4 -1.6 ..,.5 -2-D -3Z -24 -16 ~ .. v ~ -8 0 8 /6 Sec/ion angle of affodt, do, de9. NACA 4424 Wing Section 24 .32 APPEJ.VDI4Y 1l'" -.2 ~ --- ,..--­ V ;Ii­ -.to .036 r---. ............... ............... ---- --­ ~.2 ~ <, , ~ I ~ ""T" .()jZ I , I .8 .4­ " 1.0· 1Ja1 .8 * .02':­ 497 :l I ---­ b ~ ~ , J ~ IJ 4~ .01: J V ~~ ~ Cl f\. 012 ~ Jl ,~ I !A ~ ......... ~ ~,~ ---e ""-­ .­ 'O~ "'IJ_ --A ~ _.,0 --­ -V t. ~~ / , .¥VV In If ""'1..",.000 ~ ...... .q ~ "" u a ~ ~) ~ ~~ ", - -­ .~ -. - - -./ ~ ". ".. i is -.2 ~ IU -.c. posdion ..035 *$~ ~ 4» o3.0x/O· - .224 0 (,) - 1o;:;J o~O -.3 .230 .016 239 -.005 standard rciut;ihness I 09.0 b8.0 aZlJC ~ rluloted ! oJi! flOO deflecf;r;KJ .tJ 'Ie I- 96..0 .. ...S 7.6 -1.2 -.8 -.4 0 .4 Section lift coefficient, c,. NACA 4424 \Ving Section (Continued) .8 l2 lfJ THEORY OF WING SECTIONS 498 .6 '.2 2.8 2.4­ 68~ . •. , l~ . . . J... 6 I JV • '#) ~ ~ .. L~ .2 ~,. II f ~ A I ~~~ j ~~ I ..8 jt ~r ""'-l~ Y ,I JI II .. 4 j( I j. 0 .,........ J r IA -;J/ 1.. r\. ,.., J.~ ~ . ~tI ,~ ] -:8 • . • 7 ~J I ' .r. 1.2 #I r"" -.5 ,~~ -=32 -24 -16 ~ 0 8 /6 Set:7itllt tlhq/fI Df t¥fftlCk, "" deg NACA 23012 Willa Section 2.f. 3J. APPENDIX IV 499 .1138, V ~o r--­ ,,:2 0 --­ ---- --­ - ~ ~ r--- .1»1 ~ ~ .2 .6 .4­ .8 1.0 * 02./. ~ 1 ..­ I I I , I 1 A r ( J tJ , 4/8 ~ ~ e If /IV J v 1\ - \ ~ (i N~ Co"" ~~ .IJ08 V ~ c~ 1/ .ifV' ""'-~ ..Ao~ ~ I'.. V 1/ 'f'., "',...... ~ ~ ~r ~ ~~ tv ~ rv tI:"- ~.I"'lI t ­ ~ I II 1/ 1 l ~ .012 / P ~ ;\ 10 I J J \ \ "Ij ;> JV tv V '~ ~ ,.", ~ 0 - ..,. - - .. .. , .... .­ - i ""'" .... ~ A -.1 ~ .. .• I _41!1 .o:position R *E :0 3Oxlt? ~ .241 n 6.0_~ ~ .24/ 088 .241 - -'" °4 6.0-~ ~ -~ l-­ YlC .036 .035 .004 S'tCl11dt1rd I roui/7i1ess -.4 ~5 -/.8 -/.2 -.8 -:.4 0 .4 Sedioli lift coefficient, c, NACA 23012 ,\Ving Section (Contin'fled) .8 1.2 /.6 THEORY OP WING SECTIONS 500 3.6 3.2 Z8 2.4 2.0 .-.'\ d:r"1, 1.6 , t. ~ . 11)"" ~ Jf , v:" 1 ~)i( )( 1M ~~ I "~ Ii ~~ 1 '1 I ~ i J .f J o f r-t o ~ :~ .... ~).",. r'\;~ j ~~~ ., If J -.4 J ...:' .~ .!t 2 sCb -.8 )-.3 -l2 -.4 -1.6 -~ -2.0 -32 If " " J J 3 I j; II t \.J~ -24 -/6 -8 0 8 Section angle o'oIfodt, NACA 23015 Wing Section «en deg. /8 24 32 APPE~VDIX 501 lIT .2 .IJ36 V ­ ~ ---- r-- r--- r--- --­ .~ ~ .2 0 A ;;t(c ~ --- .s .6 ~ .-­ 4 , 1.0'" ~ I II .024 ~ \ \ 020 ~ \ \ I , , ~~ " I r- I "­ 9 , ~ "z, , - ~ , (1 ~'" .-0"" ",If" ""-~ """1: ...... ~~ .... J Jrl P J~ IS) /' J:V ,/ ~I'-. ~ ~~ iI II I rA'l P" ..,~ . I ~ '\ ~ ~ J J \ , t.- f ,I ~, ",~ n..11"'""'" CT_ ~ ~ I\.' .tXJ4 o ~ r-" ... I­ , - .... -./ c5 t I ! , 11 (I.C. : JX?Sifion ~~~ oso oes 2.6" IUt- .23/ 06.0 r-~.~ .239 '0 <>a!1I-~~ A6"() ~ ~ ~ .243 021 Standard rOughnes,.s I I -.5 -1.2 ~8 ~4 0 .4 Sl!lcfion lift coefficienf, N.ACA 23015 ~·!ng c,.8 Section (C!J.""i-nued) 1.2 1.6 THEORY OF WING SECTIONS 5OZ' 3.6 3.2 2.8 2.4 2.0 1.6 ~\ IA~a ~ e .~~ ~ ........ '2 ~, .....& ~ 't.: q, 8 ,v ~) .8 ~, ~ } c~ 0 1/ 1 ~ L , ~ ) .\. ,~) ~v ~ -.8 '(~ a .... ~ , J.' IA ........ K-.3 ......... J ••4 .& (J •.2 '1-.,. ,J ~ ~ ).. ,.F ~ .... .4 s, ~~ ~ I .~ (j ~ II, ~ ~ 0 i"1 "1 A t.'I .1.2 ~ ~ ,.,'s ... -.4 .1.6 -.5 -2.03 - Z .24 -' ~r -18 -8 0 8 /6 Section angle of" ofloclf, «OJ deg. NACA 23018 Wing Secticn 24 32 APPENDIX IV 503 .2 .MB: -e. V .Iii n K -I'--­ ~ ---- ............ --­ --- ---­ ~~ .u. ~ ~ .z .4 .024 .6 * 1.0" .8 ~ \ ,.. \ I ,I 020 ) ~ ;\ \ \ ~ ~ J J I . ~ ,,[\ \ \ 7 " n ~ I\. ~ ~ "­ .... l~ r--..; ~ ....... .A.. -r '" - -r-"" - :/ ~ V V ;d ~ J!.,{ v. ~ -~ i0" L.A ~ ,-'" :I.ooo~ ~ .... If j I J r.[ ~ ~ r-, "'~"" "F~lt1 o C II j ~)' f\. R. " .tl ~ I 1\ ~ .. c. pt?SitiOl? II -­ *~~ 03.1 Xl4 .;..,.... 236 06.0 ­ 08.9 66.0 -.5 -I.6 -1.2 -.8. ~ 1--- ~4 .24/ 243 01:;, 017 001 5'kT1dord ~ S l 0 .4 Section lift coer'icierlt, e, N ACA 23018 ""'ing Section (Continued) .8 1.2 l6 THEORY OF WING SECTIONS 504 3.8 3.2 2.8 2. 0 I. 6 ...... ~~ I,f e ~~ ~v a .ti Ig ~r I. 4 J; 0 ,,~ 'I. -./ .(~ 'l ~r I&: ,.;;; 0 "'lll~ 11Ji'" h 8 ~ .1 t-o ~';J s ~ !t: ".-:: ~i 1/ ......... c..:: ,I J~ t6 1.2 -'r-"I ........ ~ -.4 ... . rJ j ~ • 1 J , ...... )....h -, IA Jr III ~,. 0 -. 8 jF IJ J' -'\ -I. 2 'I \ u'lrs - -I. ~ -.5 i'J - -/6 -8 0 8 ~c1io" enq/I! fJf attock, "0) deg NACA 23021 "ring Section /6 24­ APPEl\~DIX 505 IV 2, - .tI"u r: 10 ~ --.2 o ---. ..... ~ ---­ r---... .z -. .4­ .024 --­ ~ ~~ ............... ~ ~ JRZ ~ ~) .8 .6 1.0 ~- t 1 , ~ .020 J ~ IT ( ... I~ , .\ ~ ~ ~ ~ ~ ~[\.. ~, I ,I. 1\ 1\ v ~ I\.. b.. ~ ~ ~ -"­ II ~'f ~ ,A. If ~ iCo-_ ~ ....... [/ ~~ ~~ ""' - ~iJ' I'Iol'" ­ ""-""" 1'1P- ~ -} ~ j fl v 10 I( I )~ ~;:JAlV 17r1 [.,1 ~~ ~h I 7 j r\. ~ ~ 'V ... ~ ~~ ~~ I-.. ./JIU . '-­ o - - - "'­ ~ ~ a 5.9_~ (> 8.9 _"- A" 5.9 -­ ... :­ ..... 'V l....c "V' •.c. posifiut .lJ26 '*E~' 238 B o saao: - .223 ~ .....­ 234 ­ .072 .:-'.lXJ8 Standard rouf/hness I -5 -1.6 -1.2 -.8 :~ACA -.4 0 .4 S«:fion 11ft coefficient, c, 2:>021 "-i:J'; 3cnion (Conlinued) .8 1.2 -e THEORY OF WING 8ECTION8 506 3.6 2.8 2.4 2.0 1.6 A j:£') rv.~ ~ IlJr ~iiJ~ ~ :..... JU la' J, ~ i/J ..)k 1 111 I' ~\. Jr ~ I J' o , 0 •.4 ~ -.8 -.2 ~ ........ J' ~r 1!J::! ~~ A If) .I' I.(~ ~ 4 ~ IQ A r' }-.3 ...... 4~ ttl-. I ...... .ic cK ~ If ""[~ T (~/ J. .......-a: -LZ ~ .., -.4 -1.6 -.5 -2.Q -32 .24 -16 . -8 0 8 SecflOl7 angle of o1l«Jr, «it deg NACA 23024 Wmg Section /8 24 32 APPENDIX IV .2 v~ """'iiiiiioI; - """-- •.2 0 • ---- - ~ V .... .2 507 ~ ~ ~ t"----. .11:: ~ I.O·N~ .8 .8 .4 r---...... ~ .JJ~ life 02_ ~ ~O \ 0 , , \ .0/2 , ~ ~ t1 ~~ f"""ooolil\. r\ V i'""'"'I:r ~ J)(J8 """l~ J r/ ~'O '1"\ ~ t1 """'~ ~ ......... II 100" 1'\10 " f' r\. b I v ~ ~ ~\ -c I ~ c .0/6 I J V Ij VlA. ..,., .J'" V ~ Ir"i ~v l/ ~~ .....,ooi~ . / ?~ ~ ~7 c- r~ ;,..Jr"I -- ..VV5# -- .... '" ".,. w... ,..,~ ~ v "'t ~ -./ c; J. ~ -.2 ~ 8\oJ ..... a.c. PQSlllon *~we R 03.0-/0-- - .2/Z .102 05.9 .223 065 08.9 23/ 048 65.9 Sfondord r~ss -.3 I _.4 .... -1.2 -.8 ~4 0 .4 Secfion /ilf coefficient, .8 C, NACA 23024 Winlt Section (Continued) 1.2 £6 THEORY OF WING SECTIOJ.VS 508 3.6 2.8 20 it~ ~ i- 1.6 ~ I~ - ?'3" l r ~!J 1.2 e IN ........ .i A .~ r 8 ~ e" j [l~ ~ :.:::: J I;l~ 4 ~ .6 ~ ~ j CI) jf Itl ~r " I~ J'l ~ ~ Ih ,, ,I fT: ,''ll , -./ ~"'" J v~ ~·-.2 .!! Iln. -. 8 , ~ -..;: ~f..J-.3 i ~ g P 4 I rc:; c I ~4 ~~ ~ I"~ ~~ ~ A f ~ II 0 0 r 1i!~ Ii -~( lP~ ru -l. t n; K ~ ~ 1\ , ~ -I. 2 I \ j ~ ~ -.4 -:5 -I. 6 -2­ ~ -32 -24 -/6 -8 0 8 1& Secfi:m angle 01" o/facle, «., cMJg NACA 63-006 Wing Section 24 APPENDIX IV 509 .2 .ON i.ooo""""" .fJJ2 i""""lllo.. \ I ", " III 1 ~~'""'" I I .2 .6 .4 .8 * LO .tJN I ~ ".020 J> i1 I , I 1 ~ 1\ ~ ~ lJar\... ['\. r""'!)P-'Ip;I ~ ~fh a 0 ~ ~~ I-~~ "'rvJl'. ~~ k I_~ ~ lrf'lZ' ~r- U~ ~J I .004 ~ t - .... ~ a 1­ - - j -./ I i f tl; .... a.C, pcsit,'on * E-. R t I ~ o 3.0UO­ - -lJ 258 48 Q 6.0 ~.258 -.029 e 9.0 I I 258 -.033 " Stondard rouqlY1ess ! 4 6.0 al'Oc simulafed split flap def/ecf~d 61r I I v I ea ,,-­ v 6.0 -~ Standard rouqhn~ss' I t , f -.5 -1.6 I t -:8 -.if. Section 0 .~ lift coefficient, c, NACA 63-006 'Ying Section (Continued) .8 1.2 1.6 THEORY OF WING SECTIONS 510 32 2.8 2.4 . 2.0 ~ r~ j L6 v 11 ~ ~ j' II JJ j II IJ! 11 II' ~, , Id dl o , 4-./ V .~ A I IT I~ LLI­ o 4~ Sf ,. .~ I ~r'\ JI " -.4 Q ~f 10 I ). p 1IIIIf\ ~ c: ~ ~ y I' ,,~ ~iCJ , r-; ... r9 • M~ -.8 ~'f I.­ ,. it 1.1 L -.4 J.6 24 M 8 0 8 Secfion ongIe of DlfDdr, ... NACA 63-009 Wmg Section de9 ~ 24 32' APPENDIX IV _. 511 .. .D36 ~..,. r---. '---­ -" ,, \ ... ~ -'0 .2 A .../ \ 'ti' .8 .6 4» I .Me 1.0 ~ .01: It.. , " IA 1\ ,., ~/Z (\ ~~ ~ ~ ~ Il\.. ~ ./JOIJ / ~ l. . . . &.~ ~~ D.r'<~ ~~ ~~ .". J ... ...r­ --­ . ~ I "'­ - -./ ~~~ ~~ r,s-rv ~ 1I:l~ ~ ~ 1/ ~'" ... ~r-~ii ~n ~ ~ J c.. -­ B ~ -...... l... ,... r- ,.. , .....' .i-.2 .~ ~ R §-...,~ :3.0JtIOL... a6!' I I 09.0. Af1.0 ... Q __ ~ -.4 -.s ~J.6 -1.2 -.8 ~~~ r-1: .1,~ .258 .(}()1 .2S~ s 514 .. UI;' s_~ /8 ~c .,gp di 'lticfet. I' ~". I I. rQf9;7n,SS -.4 0 .8 .. .4 StK:tion liff coeffICitN7f, C, N ACA 63-009 'Ving Section (Continued) 1.2 1 512 THEORY OF WING SECTIONS 3. c 2. 8 .4- ~ 2..., I.J~ II ." I .2 ,J 10...... ~ :~ ::::., , ~ J .6 0 lei II l jr\. ( I - 1. I! \ u; • ~B. J ~ rr .r'\ "-~ jt v .4 5 1ft " I iP 1/ III ~ IJ lih. ~I l- .A II It -.1 !_~ J .4 ~~ ... ~ ~~ I .8 '"(J ~ r ) a ~ ... 11 J cr ~ v ~ ~ ~ ~ ~p IQ~ ..J , ""=' c: -.2 .~ o~ -~ ·w ~ 'I...: " ~-.3 C """\~ <....t: ~ " ~ 1.2 L 1\ 'to­ ~ ~ v 1-.4 1.6 -.5 2.t'J -32 I t ! -Z4 -/6 -8 0 8 /6 Section onqIe of attock, «" deg NACA 63-206 Wing Section 24 JZ APPE1VDIX IV 513 .2 10 .1)_ .----~ ~ \ \ I \ \ -.2 0 .2 .4 Jt/c .6 , I .m l I 1.0'" .8 .OP4 ~ .fJ20 ~ ~ I J 1'­ If \ .J ,.. " ~"" r--.)-. IW" ~~ ~.J:.., \;~ ....... •004 ~~ - .. [flo. IA ~"'" ~~ ·"! ..... N a n ."­ I:f~ - ... ..­ -­ ,­ ill­ -./ I I Ci I R o J.O~/()6 a 6.Q o 9.0 6 6.0 ztcg"c G.1l position ~"251 .005 =t=..- .254 ~ ~ :250 I 0 -.011 Sttntk!rd rDuqhness D.20t: simvlated Spilt flap deflecfed 60 v 6.0 I­ I I I V 6.0 I I I I I I -1.2 I I I -.8 4~tandord rouqhness I , I I I I I I I I , I I I I I I I -.~ 0 .4 S«#OI1 /if, coefncien/i c, NACA 63-:::06 "-ing Section (Continued) .8 .I.~ /.6 THEORY OF WING SECTIONS 514 3.2 2.8 2.4 I 2.0 J .~ IY, J" '8 l6 ..4 ] jiJ .~ Id II ~ ~~ .8 J ,I i[ ~ J T ~ ] .-J .. P 1/1 Y 0 - 4r I ~ t d '..... IJ' .. .41. !A • ~ ~ I .. 4 1 If ~ IA "'0~ ". .... 11;1 ,1 \ z -J.6 -.5 -Z!!32 -l ""f,;1 -: -.4 ( I 'I 8 ~1 IC~ 1J ] , l:) " lib ~ J . .. Cl) If ....... l2 4-.1 ~ J "IT .. ~ o ~ ~J'l L -24 -/6 -8 0 8 Section angle DF offocJr, ... deg NACA 63-209 Wing Section /6 24 32 APPENDIX IV 515 .2 .JJ3Ii -~ - ~ ---- IlliIiI,;, I---. ..... I \ \ , -.2 0 .z I 1. 0 .tlZS .8 .6 .4 , .-t I * 024­ 'D 016 , 012 J V l~ ~ ..C ..... If lIlI,,­ ~~ ""'IIll~ ~ IU'" r­ ~ .. ~ f.{ '/~ J\~I"" r-­ lSi J ~P V .z~ ' ~ V \ .()()B ~ I 4> J. ~ I~ ~V ~ j;a::::: ~~-- .JJ:.. :;;r-­ ~~ .J!tJ ~. -----­ ... V 1­ .. ,.. I- ... -­ r­ ...... r­ -.1 . J ...:' ~ \1 ~ ... -~ 1_.«:. posi io ~ ~ -." R ~~gKf~ '" I ...0/8 .~~ ~8 ~,:260 .263 ~~ ¢9.0 ~2 176.0 SfondcJrd roughness a2~k--si.~10~ f ·;pJ·f ~~ p ~ 'fer-fee 6rr -.4 ~~~ I -.5 -l2 ~8 ~4 0 4 Section lift coefficient, Cz NACA 63-209 Wing Section (Continued) .8 l2 l6 THEORY OF WING SECTIOIVS 516 3.6 2 28 4 P u ~ If! l\ ", ~v I~ I. 6 jJ , r r 18 ]t /I z 1 ./ ~ ~ .v ~ I '(;1 , J j I ~~ lJ r r l. ~ IIJ ~ia. .8 1_­ .J 1­ .4­ Qa 0 V"1'Do .... V9 "CJII~ t~ e l .... j-j -/.2 -.4 -I.6 ....5 -z-32 !'[]II ~~ " S }~ ~ ~ JIJ II , ~ ~c J' ",",'I .t u ~-.2 ~ r Q '( 1-. 1 11 J 0 ~ ~~ ~ .4 [ 0 r , I ) d J. ~ /I .8 ..... ~ .... ~ II v~ • •• .d -b ~ -16 -8 0 8 S«-tion QngM of offock; «" deg NACA 63-210 Wing Section /8 24 32 517 APPE~YDIX Il~ .2 .0361 O~ -r---. ~ ~ -, \. .2 0 ........ ----I{ , I ,, .DR 'V' .8 .6 .4 .2 1.0 .Ie '!Ie ~ ._­ cu 16 , ,- .J ~ ~ 12 "h ~ ~ I \ c ~ l: \..~ '" 01/ ip tr /KS' Ol..... a...A ~~ ~ ..... .0 04­ '/ '/'irl V ..I. ~~ ~~ p, I'-~ ~~ j 11. I t ~ ~ ~ ")Jo... "'S / I ~ I~ ." J .","" ~v ~t:;: ~r~F ~ .- -..... 0 --rolL.. I­ :-.. L..­ "­ IAo.olI ~ "'" ~ ~ - "'"~ ,.. r­ ,.. ~ -./ c! J --! nile e.e, position =t= o 3.0:../0_~ 264 R • a 6.0 .26/ (> 9.0 ~ .26/ -..;; -.O.JI -.os 7 -.D3J 6.0 '-" Sftndord rou~ss o.ZOc Simulated spli.,. flop deflected 60 A 9 ~5 -1.& 6.0 -~~ -­ -­ I " 8.0 -~~ ... Stondord rOUf/l'lnl?. . -:.4­ -1.2 e:8 -:4­ 0 .~ S«fioh Nfl coefficient. t't NACA 63-210 'Ying Section (Continued) .8 1.2 I. 8 THEORY OF WING SECTIONS 518 3.6 32 2.8 Z,4 , JI. 20 I IJ' I I J II" ~I t, II IS jl' 1.6 .4~ I¥ r3 ..: 1.2 ~f ]J ~ ~ Q) () \J .1 0 J, 4-··' & ., '- ~ U Llr, ~ o ..a -.8 ~ -.4 -1.6 -.5 ILC~ -Z4 ~-- .. ~ t{ Ie '-I •t J ~ -,.., t o I I -1.2 '--r p J j-3 JE ~ ~~ ~~ ~ {I ~ u ..... ~ ~, ~ \. i .~ I ...., J , ~, ~ it ~r -.4 - II J \ ~~ II 7J ~ ~~- II ,r .4 , -. '" ~~ ). I ~;i -0 ~ :::: , J IJI lr- .s ~~ r~ ~ I .~ 7 , ~~ I 'lID IJ-I -8 0 8 16 Section ong/~ oF t3tfOClt: «" <leg. NACA 631-012 Wmg Section .16 24 .J2 519 APPENDIX IV " , ., If. .8 ,6 * , O2O~-+-+--4---o-lI-+~~+-tI-+-+-+-t-+-t--+-+-tt-T-t--t-t--t--+--t--t-t-t"""'-r-1 14 ~ I.o/~ i \ 1\ ~t-+-+-~+-II-+ , ............. ~\ 'il.OIZ ~ 1 .....+-'t-+-+-+-+-ir--t--+-+-t-A-+-+-......t-1t-t--t-1 I v \ ..\ , r 'I l I~ 19 J " p J ~ ........-+--+-~~-o--t-+-t~I»--t-t-t-t--t-~~m--+-t--t-iil;'f-~~r--t"-r; jQ08'~~-+-+--P~I"'I"'f-t-t-t-+-+-+-t-~~~T-t--t--t-1 lIo. _ :­ - r­ ~ ,­ .. , _.I........-+-+-~I-+-+--t-+-I~-t-+-t-+--t--+-+-1t-1"'-+--t-t--t--+--.-""t-t-1-r--r--1 ci -1..2 -4 ~4 a A section lift coerncient, ~ NACA 631-012 Wing Section (Continued) B I.Z THEORY OF lVING SECTIONS 520 3.6 3.2 2.8 2.4 / ;1 / j 20 ), ~~ ~ JJ' "'it.. " 'II ~~ 1.6 7 !If ~~ ~" :JI JJ ~~ ]~ J! a .r 19 CI i' I. J~ If l"t ... r] I'd 1/ 1'1.. i. , 'F' I d -.~ IA".­ ~ 'A\I ~ ~ 4Il ~ • -.8 .... -. ,. I I -.I u ).rJ ~I 10 J J ,,~ Y \ J ~ 'I ~ I 'P'"' N f I ""< v \ , I J. J J o '""'l~ 1\ \ IJI I( t l ~, I­ ~~ U ./ ~ ~ 1// I ~ Llj~~ t t tel -1.2 I ! I t ! -1.6 t i ,I -.5 -2 0 -.12 -24 -/6 -8 0 0" NACA 63!-'?!2 Wing Section Secfion 0I'"ItJIe olfacJc. i 8 "It I{; deg 24 )2 521 APPEJ.VDIX IlT .2 O~ .0'6 - ~ ~ - ........... ~ r--- r--. ""-­ .D32 ~w- \ \ ", -.20 .4 .2 .6 ~ .lJ28 1.0 .8 I .DE~ .,.. u - ~ I \ t r in If I J I u \ 2 U l- ,,~ " ~ ..... r- f. V Sf ~ ~~ "'~ '8 J ,, "\ II ~lf ~~ ~ ,/ n p J 'v ,..., J.'( ~l J )l $.~ ~'f ;:~ .1Il~ ~l6. ~ r"QI ~~ / > 1-10. ~ ~ •004 I ..... ~ 1 ~ ~ '" .­ ,.. lo"f - ,­ t -.1 .... ~ 1- r­ .... A. ~ I i t .,2 I I i ! I c 6.0 '.3 o ~ 9,0 6.0 ~.26J ;.263 I -.034 -.029, t Standard rouqhness 020c simvloted Sf'it flOp deflecfed 6;;­ v 6.0 : I I I I I V ~O Sfandard rotJqhness: .T 1. I I I -.5 -1.6 t i a.c. postttoa t R -.................... ~/c -.046 vJt: ! o 3.0,,10· 1.264 -1.2 -.8 -+I ~ I iii Ti I I I I 1 I J I I I • , I I t I I I I I I I I f i I I f I -.4 0 .4 Section lift coefficient, c, NACA 631-212 \ring Section (Continued) .8 I.e /.6 522 THEORY OF WING SECTIONS ~6 Z Z8 I~ I,. ..a 2. Jr , IF ~ Jf ~ 'II 1# 1.6 ]r g,­ , I ., ~ 'to.; " .r t 18 I II ~ 0 j 0 , t 1;iI!l ... -./ -.4 ~-.2 ~ ~ a I"r " ~ .... ~~ 1.2 -.4 1.6 -:.5 »» -~- ­ ~f' 1ft -.3 - .. - ... - ~ It ~, J '"' f' .. r'1 -.8 .i ~ - 1­ • ~ -~ l J~ I I .4 g ~ JI I' J 'I ~ 1\ I .A Jr 'l,I ~r .8 11 It ~1 1)" 1.2 lfii~ j " IQ!\ ilJ .~ l-b Id 18 .1 ~ J .4 " <P. ~IV .... j -32 -24 -8 0 8 /6 St!!cfion angle attock, «'0,deg NACA 6.3:-412 Wmg Section -/6 0' 24 3Z 523 APPENDIX IV ./De----------­ .2 I c J II ~ \ I ). , ~ II J 1/ ,.. I IJ~) J/ v v:s ~V T \ ILl Lr -./I--I--..4--+-o&--t-+:=I=:.t-......--+--I-f-t--+-+--t---l........-+--t-t-+-t--t-~t-+--t--t-'t'-t J I I I I I I I I t I I I I1 I I I -u I.Z l6 524 THEORY OF WING SECTIONS 3.6 3.2 2.8 - 24 'i :J 1/ I\. j 1/ 20 J.~ '8 ~ I\, J.l" "'0 J.6 jJ 0 -~ ~"';:J J ~~ IJ ,v {~ ~r'i IJ V ~ 1J 1// 8 ./ , o ~ A .... J ~ 7 ,r , U .~ ~ -'" Id ! U .J. - '" In rt j J r)P tJ 41 4 ~b ~ 1 ~ '\I J-.2 ,....lA.~ ~ j -J ~ \.. ~ -..... t3 r ,~ JJ t"o~ -­ 1/ JI it r\:: ~ J;I IIJ n ~.-I G I ~J ~ ) ~ ~ .".". ~I .~ C~ \ "'hf ~ If' 6 -.5 -e 0 -32 -24 -8 0 8 /6 Section anqIe attoclr. «., deq N.ACA 63t-015 Wing Section -/6 .0' J2 525 .036 ·2 - r--- r--- r--- ~ r <; .........--. l.-­ ~ . 1'"----_ .1»2 ~ ~. '\ , I ,I l'4 .2 ./JIB /.0 .8 .6 .4 * .024 us·020 ~ , , I rt \ ,,~ , -9 ~ ) \ ~ I\..... ~~ ~ ~, ~ ~ ·'l'lJ I~ / , v: " - --.: ~:\ ~~~ N .. ..... - ~ IIt1 - .- .­ '0'" - It j 7r:J ~ ~ ~~ VI~ r ~~ [~~ ~ ,. ~ V ...... ~ ~ .()()4 o II V· .JV V ,,~ ~"\ ~~ -; .J IZ\ ~~ II } i\ ,~ .~ I ~ - .... '= .. ~ - V" I -./ J 'D.C. posiTion '* R ~_'-_ ~ Jic o .20~_ .270 -.032 o 6.0 ..27? -.027 o 9,0 l - .27/ -.034 dO A O.20e v 6.0 IT ~ Sfondord rOUtJfr1ess simvIo1~d split flap deflt!cfed . 6.0 ~~H Standard ~ss l I t ~ '":5 -1.6 i -1.2 -.8 60· • 1 I I 0 .4 Section Jift coefficient, -.4 .8 c, NACA 632-015 "·ing Section (Continuod) 1.2 1.6 THEORY OF WING SECTIONS 526 6 2 2.8 .... 19 ., 1J .4­ OC M , I 2.0 , I ~ j ~ 1.6 ~ 'I ~"Jt.. ~~ Ii ~~ ,J.! ~ -tl :~~ I J 1.2 ~ I~r If) r J .8 , I ~ ~ 0 I Z fd r II I !~ rtt".. f ~ .­ 'A~l. 1"'):11 II II .JI ~"tt. Jr :Q \ ~ 41 ~ -.8 I,.{ .. - J' ~ 1.2 - ~ • -.4 " r .)1 ~ ~I -./ • j~ ~~ c - ~ Jf J" .4 o IA IJ· II ., ;l 'VI.,.. ""-~ ... ) n L6 I\'~) 1.6 -.5 2.0 -32 -24 -/6 -8 Secfirn 0 0I1f/Ie of oflacJr, NACA 63t-215 Wing Section 8 «0, detJ /6 24 .12 APPENDIX IV 527 .•­ 10 / ' ~ .... -r--. <, I'---. .......... ---- -----poo-o, -~ ~ .. 0 .2 .6 .4­ .,. .. I 1.0 .8 ~t- .024 I I ,• , -.2 '---­ J. 6. II 'I l 1\ r1.1J20 \ -..: J .§ \. ~.DI6 "' , \i ~ r., J i I \ w.. I"' " ~ ~ ~ ~~ ~ ~ ~ ~ f't\ l.OOB "" h II '/ ~ r\. r: l/ ~~ ~., I""....-~ ~~.,. ~ ~i' ~F=:::bli ~"" ~t3. vlJ \U~' I'"' 1­ ~P' - ~ .004 o - r-" ~ ~ -J d I t 4C. ptJSIIion * R o 3.fhfO· o 6.0 e iQ 90 6.0 »'c. 1.269-.052 -.OH i=t= rrStaniiord 267 rDl.lf'/hnllss .266 -.lJ2(J ~ Q2Ot: simulafecf $,plH ;,:g t;/tp dt!I'Mded 0<" Sf~ r fdl~1 n '$S ...... ~ _ . I -:5 1.6 -1.2 -~ t I I I -.4 .8 .4­ 0 S«:fion 11ft coe'lia.nf, C', NACA 632-215 Wing Section (Continued) 1.2 -. -rv: ,,) I Or " ,,.., 1'i/rJ) IA~~ "~ L1 THEORY OF WING SECTIONS 528 .3.6 3.2 2.8 , y~ lEz 2.4 .4 7 7 , II' J 1\ ," 2.0 b A J " '/I l6 ~~ ~ro ~ J i' n s: ~~ 1/1 L~ }[ II Ilj I~~ IJ rt i' j~ p J JJ II ~r III A ~J I(J lti 'i o I o J t ""'tt-. -./ .4 J-.2 -.8 ...,'" ~~ IJt. ...... ~r ru ..... ~ I l~ ! r p rl-4=r I .... ~ J.r ~~ .1 ....,)-....,> ~~ r.. ' I~ c 'I 1b I ~ \n. ~ -r'­ [ ] U Voofq. -. v -1.2 ~P 'fIT rP -l6 -.5 -2­0 -.32 -24 -/6 -8 0 8 SectiOn orqle of attack, ~, deg NACA 632415 Wing Section /6 32 APPENDIX IV 529 .2 .Q~ V ~ - r--, .......... - - r---. r--- r---.. ~ ~ ~I\ \ --.... .o,z .. \ .2 .4 * .024 '"'\ 1.o.Qn .8 .6 19 1o" I\. ~ ,I ~ , \ ~ ~ ", '-l~ ~ i'\ r-, ~~ 1/ ~!."" f'..~ ..,~ ~~ I ~ 1/ ,~ ~ ~ r If ) I .J '­ p,," I( ~;1 ~~ JT~ ifl.j~ .............. ~~ ....... ~ J~ ...... ~ ~ ~ ';1' ...... I~t' ""!I:l ro-r" .004 o -. - - .­ -~ .-­ - I i- I­ :-­ ~ A ~ a.e position ~C$-.C43 lie .264 03.0-. 'U 06.0 .204 -.039 09.0 .262 -.036 ~ 6.0 Stondord rouqhness QZOc slmuloted split flap deflected 60· v 6.0 1 5fo~ ';Jrd r: v 6.0 ess J l'W -.,5 -l6 -I.Z -.8 -.4 :J .4 Section lift coefficient, .8 e, NACA 63t-415 Wing Section (Continued) /.2 1.6 THEORY OF WING SECTIONS 530 3.6 3.2 ~ Z8 ~ J1 , '( ~ ~~ 2.4 Jt} \ 11/ A 'fI 2.0 1 J II i II 1.8 J ~~ ~ J I~ ~ru) ~~ ~, I 1 il roJ" ," ~a J p ~jI J II cf 'f1 , I /~ .4 I r o I: I ~J :I a II -.~ I e. 1/4 {~ ,ft.' ~ 1 - I - I J' 1-.2 ~' fJ JfI rt -./ '?II~ .~ ~ fJ iq -.8 4 h1 -1.2 ~ L~ ) t llol ~n J' ~ ,pIi1 IY -/8 -.5 -2.0 -32 -24 -8 0 8 /6 of aHaclr, Rot dt!!tJ NACA 6.1t-6Ui Wing Section -/6 Secflon 0f79Ie 32 APPENDIX IV 531 .oj. 2 ~ '--­ - r--­ r---. ",.,- ~ I--~ ~~ \ \ \ -.2 0 - , , A:HJ .2 .4 qc .6 1.0· .8 ~ .024 ~ I ~"tI.o20 1 T I I 1 1 r1 6 ~.O/6 \ C) 4b .8 \. It ~" .~ I~ ~' ... IJ ~ I\.... v ~ IN I"" "Zlloo..... ~ ~ ~ 10'" ~ In "!'o.. I'tl ~t'-c: ~ u ~ .txJ8 \-l~~ 1 P !"I ) A I pr I \ ~ t t·O/2 ~ ~/ /~ ~'/ IIJ. t. v f'~ ---,.... ~~ ~ ~ _ ..J.1Ji" ' ­ _1II1II ~'"" ~~ ..-~ .004 o -./ ... cs lo~ ~"" II""'" IV" r­ ~I '\01""" """"l IV' r- .... r­ ru ~~ ~ I­ I D.C. positiDt1. • ~C ~ 1I/e o .10'S=_.266 -.037 o 6.0 .E66 -.043 R .___ o A ~O ~ .266 -.040 6.0 ~ Standard rOCKjhness O.2Oc sinrvIofed splH fl", deflected 6lr v 6.0 176.0 _ ~p-- SIa?dord r~s -.5 -1.6 -/.2 -.8 -:4 ·0 .4 section liFt coef'ficient; c, NAC.A 63t-615 Wing Section (Continued) .8 /.2 1.6 THEORY OF WING SEOTIONS 532 ... 28 t't:. IV 2.4 ,, f J , IP . . 2.0 ,. " j ~f A ,1" 1.6 J JIll A fI ~ JJ ~ III I J. o -.1 , 4-.2 ..... " Hr 1r 0 III rJ ..... liio..i: II ~I ~ I ~ V .J ~ .4 ~~ I\.. "" d ~4 l\4 8 r 4' ~~ rv-~ - ~~ .i/ rl.... j I~ ~b HI 1:3 ~tl ..... ~:i: rq~ 1;r Ie 2 II­ J' IJ .1 D~ .J ~ IJ rtJ~ .... - .A ~~ ,'1 ~ "" ,. J~ ~r A ~. I'IfJ ~~ ~ ... [~r -/. 6 -.s so ~2. -32 -24 -/6 -8 0 8 /6 Section ont}le attoc~ a;" de9 NACA 631-OJ.8 0" w lng Section 24 JZ APPE.YDIX IV 533 .2, .4J' ..-----­ -: 10 <, -~ ~ ~ ........... .......-.... r---_ ----­ ~ .t1J~ ~I " I \1 N .2 .4 .III .8 .tJ "!c· 1.0 ~ , I I \ ~20 :t­.. .i , ~ ~ .\i ~.o16 1 j ."., f.olz C \. \ I' , \. ~ ~ " "' c 1'­ ~ M~" " J;.008 J .u ~ I Y .)'1 ~ hr [:£' ~y ~CJ ~ """"'U"'I ~ -. ~ ~ c~ ~~ ~~ lI'rtT I I ",. :\ "" -.... a rJ J~ ,....rtr I"'I'l ~~ ...,;; I' v: .JItt~ 01"", "" "- ~, .§ ~ ,J If - J:V - ,.. r-'lJ" "='-~ ... --' "' -... IV ~ ... -­ ~ -­ ... I Cl.~. R position o .lfPU .277 -.lJ05 *JiV~ 06.0 I - .27/ -.0/9 .27. -.OaJ o 9.0 ~ V 6.0 standard rOUt;Jhness I aa ~ simulated splif "/op deflected 60 6.0 . , i I' I I I I I SftYldtrd r~-= v.ea i I -:5 -1.6 I -1.2 -.8 -.4 0 .4 Section lift coefficient, c, NAC.-\ 63 J-018 "-ing ~.·tion tCoutinued) .8 I.Z 1.6 THEORY OF WING 8ECTIONS 534 3.6 3.2 2.8 ~ jl1 I~ 2.4 ,11 l 17' I 20 iJ II J II 1.6 ~I JltII'v­ IA ,,~ ~ ,V ri ~ --;J IJ N~ IJ u iG ...!tI6..a ,.« A II 1C1 ~ II I/J l~ U o .., ~. I~'" III V J" ~ I..J­ i1J iI II III ~r 1'1~ 0 ...r ,~ , II -./ . 1-.2 ~r 1 -.4 ~ ~ -J t "-'8 ~r ~~ ~ /( I.J -8 ~ I ~\p -1.6 l/ ~ M::'JI b. [iii Id ". ~ -.5 rvolon v -1.2 Q ....u iS-.4 -- - ru-- ~ -.8 ...... ~ -.3 ~ I(' - -­ ~,. l' ./ i.4 o ~ -2.0 -32 -24 -/6 -B 0 8 /6 Section CIMJIe of' attock, «0, dey NACA 63a-218 Wing Section 24 32 535 A.PPENDIX IV .2 r io <, ---- ~- ~ ~ r---. r----.... ~ ~ "'--­ ./J82 . ~~ , I .. I \; --.2 o ~ .6 .4­ .2 lII;~ .tJa .8 1.0 ;021­ l , ".., \ 9 I ~ Q J '/ \ II -~ Q ~ ~ ~ "~ 't ~ ... -""'1""'00­~ A A c I IJJII FIt4 JV L..".,;~ .~ 'I' 1/ ~V ~~ ~[I'.... ~ IIilI.. f"" I V r\.. ~ " ,0 IiJ r.R K. rtD .,~ -"'"~ -- - .. ~- ~~ .... ~"" n...~ ~DJI ~ o o R tt- 2.91(-,0· o 6.0 e ~ 9.0 6:0 6.0 V 6.0 ~ ~ o.2(Jc V ... r­ Ilol"'"" G.C. -1.2 -.8 - '¥" position ;<0 lie :21.3-.050 .272 -.047 .271 ~04~ Stont:I01 -d ro s simulated o$fIII.·t ~fed _I ! - po- po- I 1° I • I I so: I Stondord rouqhne8$ I -.5 -1.6 r.­ I -.4 0 .4 Secfion lift coefficient, e, Nla.CA 633-218 Wing Section (Continued) .8 1.2 1.6 THEORY OF WING SECTIONS 536 3.6 3.Z - 2.8 :rI J' IP ire 2.4 ~·t 1/1 f, J~ 20 " 11 JJ III l l6 II ,r I... A V v. II ~ J.{ ~l Id I/J ~!l , ,J " n ~, tl, CI (, to[ ~"- ~~ -./ .& 9 'II [ ~~ 11 ~ o """'rQry., .A ~r )' Ih 'frJ IJr, .t -~ ~~ rar'b)" fJ 1I J. o ~ I J" -.4 , ~l=d ..... - -- ) 4;> II In L J ~~, -.8 1: .... it I I 0'" IVll~ \.~ ~V -1.2 9 I.- ~IO.. .- V(7 -1.6 -.5 -20 -32 -24 -/6 -8 0 8 Section anqle of attack, «" deg NACA 6::1:1418 Wing Section /6 24 32 APPENDIX IV 537 2 .1)34 10 V f'.... ~ -~ - ~ ~~ r----.. r---­ ......... ~ -\. JJ41 \ • \ I -.''-0 .U. ,o!I .2 .4 .024 .8 .6 life '­ 1.0 I ... rt \ , - , 1 , \ \ II , ~ \ 6 , I I I ... ~ If:V ~ J IT /1-/1 Il' p / i\. ~l h Q :J ~~ 2 1~ I /­ '" ~-. ~ ~, ~tr ~i"Il Mil ~~ ~ V ,~~ ~~ -....."-. ~t-- ~ ~. I ;0 ~r" 1ii!I_ "l> ~".. ~-""" 1L. ...&;;J~ . ., A -­ ; - ./ .2 G.e. .iooca:.. . o R o 6.0 .3 <> gO A 6.0 ~ ~ position *t1/!c 272 .27/ .272 -.()S2 -.05~ -.051 standard roughness f~ dellected 60 a20c simult1fed·splif 96.0 .5 -1.6 <f~dr V 6.0 .4 -1.2 -lJ -.4 0 _.L.. - .4 Section lift coefficient, c NACA f);J~18 \Vlng Section (Continued) .8 1.2 1.6 538 THEORY OF WING 8ECTION8 3.6 ..s 2.8 I{ ~ r IlJ ~ II \ J 11' ~ i~ 2.0 ~f IJJ ,r A /.6 ... ...~ A" ~ .rt ,,.... ~ JJ" ~ " 1'" rlJ ./ II II j~~ JJ W , .. 4-.2 . -. , . .. P _J ll~ ~ W~ ..- J I , Iv .8 ..:' ~ - -I. 2 u "- 4r 1 :'" IP .IN iP .(1 ~ -3 . o II' J ]J I1.PJ -./ j! '1 .... '. ~r IJ o ~ ~- ~. -~ a.: .... .... i -- laC J :, .r 4 ".. M"': 'J Id~ tV J 8 ,.,~~ ;JD ~" -.4 -I.6 -.5 -2.tJ -32 L...1I!koJ IV' f ,e -24 -/6 -8 0 8 /6 S«::fitJl? onfJ'e cdtac/(. cr.. de9 NACA 63.-618 Wing Section D' 24 32 APPENDIX IV 539 16 .,e ~ J J p II r ~ .OI_Ijj~Io--o+--+--+-~~-f--+-+-t-+r-f--+-""""t-4-+--+-~~--f--t-+-t--t"-t--t-+-....-...., " j' 8 I~ 1 t--I-+-t--+-Pwir-+-"'+--t-t-t~a-If-t-t-+--H,.....+-t-t f.OI2'1-t-+-+-+-.....-+Iilf-..... .1... u If 1r1 ~.~ -.~ -1.6 -1.2 -.8 -.4 0 .4 .8 Sect-ion lift COtI'ficJ.,f, ~, NACA 63,..618 Wing Section (Continued) 1.2 /.8 THEORY OF WING SECTIONS 540 3.2 28 ...4~ £ 1.4 2.4 r. , .,.4" ~ II ~., I 2.0 L I jJ ,I JI 1.6 IJ V A ~~ ~~ .d J'rtl I(f J~ , ~ A.1 III o 1!f 10 tl " I~ Ii'"' M~ tl J"" ~N ~iIIal.. p e! -:2 Ii J. r: "'~ -.8 v~ jiJ .i T .~ 8 A. --""Il:lIl~l{ 4 ~ ....."" ~ -.3 ~~ l5 ~ I~ l­ --.~ •• < .' r ~ C -.4 - II' r -./ I'"r'~ j~ IJ rJ QUr 0 :n ~" ~~ J" (/ ,,~ ,~ -- t~ ~~~ IJ ,JI I, -,.. -1.2 .'i:trI. ~ r--; ~~ ~1 v'V" I..J~ o ..... c: :~ -.4 -1.6 -.5 -2.0 ~ -32 -24 -/6 -B 0 8 /6 Section angle of' offack, «0' deg N..~CA ~.-()21 Win.. SecLion 24 32 APPENDIX IV 541 .2 .036 V r-, ~ r---. r-----... ~ ~~ r---.. r---- .... , t " .4 .2 ~t! .012 r-r L---­ ~ I .6 I " .B I .028 1.0 .024 J. ~.()20 1\ ~ \ ·1 .~ ~.OI6 i~ "oo , " 0 ~ 1\ 8' is .012 .§. -...;; ~ J 1/ , J 1.1 I~ ) o ~.008 I li( "~ f'~ ~L\ ~ ~ .. --~j"loo' , )~ ~ ~ J~ i/J~ ~ ~ C~a. ~, LJ' - ""-­ "I""""'( ""'- -"'~v b ;~ ~- :r .004 o -- \:I I '" ", ~ - "W \l~ -./ ! d I I i "'to:' ~ -.2 ~\i ~ Cb o ~ -.3 a.o. posJlion ~ dO·! ~c o 3.0x/~ _ .276 oo 9.0 ~ t\ 6..0 - .273 .273 lie 0 ~OOS I -.001 j ~ Standard rOUtJhness a20c simulated $pit flap deflected 6f)O ~ v 6.0 ~ -.4 V 6.0 ~ 10­ ~ Standard rouqhnesB I -.5 -l6 I -l2 -.8 -. I 1 -.4 0 .4 Section liff coefficient, c, NACA 53e-021 ~Fing Section «('on!'fnued) .8 1.2 1.6 542 THEORY OF WING 8ECTIONS 3.6 3.2 2.8 n d J 2.4 I~ ~ r 0 r 1. ~ \ -v 1 1J tI Iw j l/I J I. G ... 18 ~ ~rt " 't;. ttl 11 5 ~,. II ~r r .J.r- If, II OQI- 't • ,.... rq ~f/ •• 111'-4 1-1 ...... ~N ~h... IV! r V , -./ e.. -.2 -. ~ I~ lrl 0 ~ JJ/ , ~I a J' " ~~ ~ ~~ ~ ~ 1. "I: ~ r\ 'l 8 ... ,.. .A ,~. /1/ IIJ -I A .~ ~'" ~~ J1 II - 8 - "~ • ., IJJ OI~ -1.2 ~ J ~ ~~ ~...-!"U ""' ... IJ' ~ -/.6 -.5 -zn -32 -24 -/6 -8 Secfion CInfJIe 8 0 0' Of/DCI<. «., tMq NACA 63r221 Wing Section /8 24 J2 APPENDIX IV 543 .2 olr r-, -.2 0 .J»I - r----... r----.­ r----. ........ _L.---' ~ r---....-.. .4 .2 .6 ~ .: I--....... .I/JI L.----' "'---rT " " I \ .8 • /.0" 024 j fJ20 t 1 7 , , \ .016 , l '\ \ ~ " .0/2 r J 1\ IY 9 V !\ ~ ~~ ~~ ~ .008 ,r I , III ,~ ., ~ I ) ,l.J ,,~ ~ ~ ~,... t: ~ IfJ v: .-.~ , 1"'lllIll~ ~ .... ,­ ~ ~ lr. _ -­ - " ""­ ..rl~ ~~ ~ ~ -... ~ , - - ro-" -­ v .... -J -«i i­..~ ~ fJ.cpos.~ :S! ..a () .... o ~~n--2~~'~ e &2" ... i J .?10 -.0/1 ~.269 -.03;1 6.0 ~ Stat . . d ~ t12Q1: sinuIo"!d ~ flap dI!IJ~ ~u 6.0 ~~ A - 6.­ Sfandard~.b V40 -:..,. I I I -I: -1.6 -1.2 -.8 -.4 0 .eI section lilf coefficitlnt. 4 NACA 63c-221 Wing Section (Continued) .8 1.2 ~ rr lP THEORY OF WING 8ECTIONS 544 3.2 2.8 1:1 ~ tod 2.4 JJ iIf ~ I~ ~., 2.0 fI J~ IJI ~, 1.6 10 1r ~ ;>0~ :/J LA '"V :JII rl l~ I.., d Ifl Jrf II LJ l.Jr 'M n J" ~ ~I l~ 4V 1. n J ~r.I -./ -.4 fDlb I.' I /I JJ ,. ~ /,I ~I ~ ~~ ,f! I ~~ P ~ ~ ~ ~'"' .~ o "e'" r -/.6 -.5 -2.0 -32 10 rb La -1.2 -.4 4· tit, .~ -.3 r I'z'o -.8 ~~ ~ .......... -J ~ eJ'-.2 , ~~ lift ~I 0 \A ~S lbP ~ PI o .J[ !"1( r'CJ ~ ~"'- ~ M~ ~~ . "VI' ~ -24 -/6 -8 St!cfion tngIe 0'o." attock,8«0, dfJ9 /6 NACA 63~21 Wing Section 24 32 APPENDIX IV 10 -: r-, - ..­ 10'" - t---...... 545 .fJII --..... ............... ~ ~~ ~ r----. "'--­ r-I , I .4 - , I ~ .2 J»Z .6 lIt/c .8 .02' 1.0 .024 j ",.020 ....... . I I ~ .~ '1 ~.OI6 1 Cb 8 - 0\ "' t·O/2 , , i\ 1­ Q ~ \. ~ I ~ "­ \ iQ ~~ ~ ..,~ "'!1-. ~b jp ~ r:r --~ ~~ ~ \. \. ..4 ~~, ~~ ~ ~~~ -~ ~ - n~ ~ ~p 1.~ v .0fH a - t" '"""' -./ q ! R o o 9.0 ~ 6. 0 V f7 -1.2 tLc. ppsifion orIitl~ -.O2~ 3.0)(10' ~-:275 o 6.0 -.5 -l6 _ .2~ .e» -oso -.02, l' ' Sfandard aeoe simulo 'e4f SP1·f (I" ~f~cft1d~ 6.0 6.0 -.8 ~AC.A riJ<JiT7nes~ t C _, I ~o I tc. n( ard roue,h,e :$ -.4 0 .4 .8 Sec tion lift coe''ficien.'f; e, 63.-421 ~·in;; Section (Continued) l2 1.6 546 THEORY OF WING SECTIONS 3.6 3.2 2.8 2.4 0.200 aSa1l1aU4 ~t nap clerleoted 2.0 • 10' V , • V ataadard n . . . . . 7 , .106 ~ 1.6 ~~ ~ I ~7 .J. ~ 7 .~:pc '1/ A I tJ I ~s II ~ ./ 0 .~ ...... u .4 ~ 41 I ,f -, ". I-Ii f7' 'I I o r g ..: .1 &-.3 ~ ~~ ~ -.5 -2.0 -92 -/6 JI01lFDe:l 6.0 x l · ~ I­ .... v ' V I"VI \ 16 \ tlap detlected 6 _ II) lP V 106 ~, 4r -1.6 x '.0 ataadud ~ 0.200 .~u1.ted apllt -.4 9.0 6.0 1 4. ~!f ~ -/.2 ~ ~. y~ & IT II' I~J -.8 e 0 ~ - R A j"­ ~~ .\) s: ~ -.2 ~1 0 II -.4 Y .! " ( [JIll V ~ ~ r 1. j, 11.._ /I ~ -./ ~r ]~ ~ fr ~ I ~ c~ I~ I.,....ooioo""'" ~ 6 10 ------­ I -8 I I I 0 Sec fion angle of offocle. NACA 63AOIO Wing Section 8 do, deg /6 24 32 547 APPENDIX IV .2 .OM .., ~ ~ ~o ............... ........ .2 .4­ Z/c ~~ .8 .6 /.0 .tIN .024 • • Z·O.0 •• •020 D .. &.0 ~ 10' ............. ........'­ ~ r-, -.,.. I' ~ N l\ ~~ ,)) ... ~ ~'f' 'I ,J "1/ IQ'~ i ~~ O'"~~ ~ ~V tal"\. .004 II '1J-~ ~ ~. - - ~ ~ J ~~ VLO ~ ~ i-""'" in­ ~~ r-cJroo... ! ~ ~ ~p V IJ ,.. j I~ ~~i\. ~ ~ .~ 1\ 'lb ....... ----. "­ ~•• o- l. ~ )I -r-- ~Ioo- - 1 ~ ~ 0 1.taDdUd oA air"""" - o - ... .-. .-. . ~ ~ - ! .. t -.1 o • \I !-2 .t' . ~ ~ t ~-.3 ~ j -.4 ff>- f o. I Q, o , I o Co R .rIc 3.0xloe-rQ250 6.0 .2Sg o o 9.0 f­ - -~- --f- f-I­ -­ position 0 vI~ .­ I-- -­ .014 .254 -.003 Standard roughness --~- 6.0 V G20c simulated spIlt flop deflecfed 60­ 6.0 I I I I I I I 1 j V 6.0 ..-~.- I -~.~ i -.8 -.4 - -­ 1--..­ Standard roughness I -.5 -1.6 -­ ~ f- _. 0 .4 Secf/on lift coeffi"cimf, C, .8 NAPA 63AOIO Wing Section (Continued) 1.2 1.6 THEORY OF WING SECTIONS 548 J.O· 3.2 2.8 2.4 ~--+--+-........... O~200 • Saulate4 qli' nap a.n.ote4 600 R V V 6 106 _ x --...i-t---t-+--+--+-~~I----;.--+-f--f--+---~ at&D4ard rousbD••• i-t--f-+-+-+--+-~!---+-4---I--4-4-~ lD ........,:t--f--lt~-;--+--+-~r----;.-+-f--f-+-+---4 I i 6 x 106 - 2.0 ). \ lPI7tz ! I ; : I ! I I : : I 1.6 I' c; "'['.2 .~ r 1. I r I I~ I I I ~I~ j I.')r(: i! j ~ ; i I t .\) ~ \:.1 o .8 v , ~ t: .~ .1 ...... .4 \I c'l o -.4 -1.6 -.5 -2.0 -32 , i i I -24 ~6 -8 a Section onqle of" aftock, NACA 63A210 Wing Seet10n 8 "0' ffl deg I I 24 32 549 APPENDIX IV ~ 0 t-> ~ ~ r--- r--- r--- t--.. .03Z~+-+-+--+--+-+--+--+--+---1 7: o NAca tiaA21U \\ mg t;eCtiOll (Continued) THEORY OF WING SECTIONS 2.8 2.4 2.0 I" l.­ II /.6 .JJ. "7 d ~ J .4 If' 1 , r ~L\A J ./ . , 8 r R :, f: !f 0 :.1 r .. ~ ~ J t JII l' ·1 -./ -.4 t-....­ III ~~ D tin. tJ '. ~ ~ .... ~. ~ ,~ -g~ ... 4' ~ ~. ~ o J "If 1-!cJ ~.4 ~ I~ kJ_ ".~ .., l II ll( \: ~ -.5 -2.0 -.3Z -24 -/6 -8 0 8 /6 Section 01"I91e of attock, «0, deg NACA 64-006 Wing Section 24 ..32 551 APPENDIX IV , I '{ .6 .4 .,, ..:­ .i ~ ~ .016 ..... --+-+--+-+-I~-I--+-~.....-+-+--I-I--I-+-.......-+-~I--+--+--+-......I -+-+-+-.......-t r 1 , ~ r .01.2.......-+--+-+-I-+-I-.....+-l....... j I ...-I\o-+-~~-r--+-~I-+-+.......+ -lt-+-+--+-+--I-+..... II ..,; ~.aA9~-+-~~-+-I-+.......~........-+-+-t--+-~..-+-f--If-f-+-~~......-+-+--t-~ -./I--4i-+.........-+-~~-t--+-+---1Io--oo+-+--+--+- ........--I--+--+-~I-+............-.-t-t--+-t--+-+--t ~ J -lZ -.8 -.4 0 .4 Section lift coefIlcient, cJ NACA 64-006 Wing Section (Continued) .8 I.Z 1.6 552 THEORY OF WING SECTIONS 36 3.2 2.8 2.4 2.0 'I'l J; '\ , 1.6 A V ~7 A IJI ~~ 'R H ..... BI j~ ~, II "{,I ~J IJJ~ III ~ i IJ • J. r g I,.tr r II 4 " IT ,A. ill .4 J ~~I ~ f -./ ~ i. ~b J I~ A "'" 'PI ,.......~ j -';",", ~ ,A ~ I ~L -.8 , • :fI ,f ~~ J -.2 -~ 1l '? -.4 "-~ ~r U n ...J '­ A ~v "'" J'" ,. ',. I • ~ '­ ~ -.S -20 -32 -24 -/6 -B 0 8 /6 Section cnqle of a/tacit, tJC•• de9 NACA fl4-OO9 Wing Section 24 JZ 553 APPENDIX 11' ~ J-I-+~4-I--I-+...-4-~-P-I~"""'-I--I--+--+--+-+--+-+-I~--+--+-+-t--t-+--r-t--t { ~ t"'l ~ .01G1--I-+-6--+-.............................-..t--+........-I-+-+--f--+-+-~+-I~--t--+-.........-t--f--+---+-""'t--t If Cb I 8 ~ \ I J I ~ .012 J--I.-I-oI--+-f,.,I.+-I-I-~~-+-+-+-+-..f---ooI~-+-~-+-+-+.,+-~t-#+-+-+-+--t I ) // .~ ~ r. J.V J.0Q9~4-1--I-~~~D-fIild--+-+----+-+-+-+-+-+~~~~-+--t--l~-+-it--i -:5.......~~......""""-I~~~ .............--'--a.......a....-.............--'-.-..-"'--II......I._ _......-'--"'--Io.-.t­ -,,4 0 .4 .8 -.8 1.2 -l6 -l? Section lift coefficient;. c1 ___ 1.6 THEORY OF WING SECTIONS 554 .3.2 2.8 2.4 2. 0 g~ 1.1. 'r\.L I. 6 ." f1 Jf Il' II l' ; I~\ I 8 J ~ \ 1 If 0 ~ -~ )...,t> ~,. f I lr fJ , ~J ~ J ~ !JP ~=t If r'1 r-.. ~~ / 4 o " jt 2 ~) til 4r ,~ I . 11 -.1 -. ~ a4 IT , 1I \ I_~ 1­ J -.2 I'~ -.8 ~ \ -;3 l. ~ -- U iJ .,~ ~J j I J ~ ~ , -J.r.z ~ () \J \, .... ~ c § -.4 -I. 6 ~ -.5 ..2.0 -32 -24 -/6 -8 0 8 /6 Section orJ91e 01 attack, "0, de9 NACA 64-108 Wing Section 24 .JZ 555 APPBNDIX IV .D36 ~o ~ ~ -- - I--­ \ \ - .D» ) I " / ~i .2 .4 .8 .6 ~/C .lin 1.0 . .024 r I. J ~ I , '/ \ '\. I. IA \ ~ , ,~ J ,.' \.. l"-. ~ '~'1 o ~ '-"';r ~ ~- <~ I ./ r ~, 0-.-" r/r ~~ 1\.\ ~ 5l:~ ~~~ ~~t'\ G~ '-R_ I0Io.0" .004 ~- o 1& ~ ~ -. I 06.0 to 9.0 :66.0 t o.c. pos tion R o .J.Ox'0. Z/C .259 10-1-- 256 vic -029 .0/4 .029 ~:Z55 Sfandard rol.XJhr1ess D.2iJ ~ simulott;d. splif '!CP t:(e'.1t!lfted 60­ v6.0 . "'\.~ ~6.0 I I " I I I I 5 -1.6 TI -I.Z -.8 I I I • fondprdr~s I I I I I I .4 Section lift coefficient, c, -;I I 0 NACA. 64-108 Wing Section (Continued) .8 I.Z 1.6 556 THEORY OF WING SECTIONS 3.2 2.8 2.4 r, 2.0 ~~~ Vl;l If' , .J. 1.6 J 'I " ~, J,l • V J 1I Iff ~ ~J II l~ o III JJ III r rh o I~~ II ~ -./ t , " J' -.4 ~ 1'\ 1 II ~ .J-.2 r ... ~ ~~ I l\.. lfo IlIa -.8 v y " 'lA ~,. -lZ "'" ~, .f , ) ~ - 1S1. ~> hll I rir ~~ ....... lIJ ]r 1/1 Inl) ­ .....lr ~ ./ t7 1\7 - I I , ~~ l. -1.6 -.5 -2.0 -.32 -24 -/6 -8 0 8 /0 Secfion-0179Ie of attach, «"~ NACA 64-110 Wing Section 24 .32 APPENDIX IV 557 .2 .oJ6 ~ -- ~ r---.. I---.. r--~ ------. r\ J \. \ .2 .4 z/c .6 J).J. I ~ I I .6z• .8 1.0 .024 c1.DZO P If IA ~ I~ 1/ 0 1\ \. :\.~ ~f\... v J g,~ '\ $) ~ ,~ l: ~tI ~~ ..A Q~ .J1A ~ ~ t::!t..... N", ~~ III V< IV I$.> ~ _Ifl. ~v ,dgJfO :::i:~ 'WII~ ..., tJ ~ ,,~ r' r" ~"t .004 , r \ ,\ o ~ LA. ~TJ ~.,.~ "" ~ D ...... Ii-. .-. ~- I­ - ,-./ j" pos til In D.C. *=tf'/C 0.3.0)(/0-. .260 -.0/3 06.0 .260 -.0/8 '09.0 .261 -.022 1':'8.0 _' I ~tan(Jord roughr7ess GZOc simUl ¢fG! &PIit fkp<{t!f1erfed v 6.0 f7"6.~ 7 I I I 1,1 I I 6J.)0 I Sfold"Jrd royghness I I -;5 -l6 I -1.2 -.8 -.4 0 .4 Secfion lift coefficient, NACA 64-110 \\ ing Section c, (Continlleu~) .8 l2 L6 558 THEORY OF WING SECTIONS 16 3.2 ~8 2.4 20 £6 IT \a­ IJ .......l2 .§ I ~ 't... ~.e \I ..... 'to.. I JJ ,-, "­ ...... .1 "~......4­ " \) ~ 0 r ..... 0 -.1 !­ J-.2 \ I~ -24 -/6 -8 Section ong1e 0 8 of attock, "01 dsg NACA *'4-206 Wing Section /6 24 .JZ APPENDIX IV 559 .tJJI .2 ~o ---­ ~ ,, , ~ .2 .4 .6 .zlc -,, / .6 '" .8 .141 / 1.0 024 f1.020 T I I~ ~ ""' .016 QI a {.ali! ~ \ \ J 1/ JP iTII ~ j l,6 I\. ~~ I)., ~ ~ .".008 ~ ...... 1/ ~ V }., .--1'1 V In~ .tl~ .\ r ~"'" ~~ ~~ p~ ~--" ....... l(~ .004 tJ1. o -­ .AI ~ - .­ ~ e .A.. ,.... -./ ~ ,­ ~ $ D.C. poSition j vic. -.020 .z/c .25S O.J.O~ '(J4 06.0 254 .011 .253 "09.0 -.020 :-6.0 I Sfandard roughness QZtJ sim~1of1!!d .spit flop deflected 60 v 6.0 v 6.., .~~.'J II~I s I-fondord r. --:.5 -io -LZ -.8 -.4 0 .4 Section lift coeFficient, c, NACA 64-206 Wing Section (Continuc:d) .8 1.2 £6 THEORY OF WING SECTIONS 5uO 3.2 28 2.4 2.9 ..C!~ 'rli IfJ 1.0 '- i'i ~ r ~ I ;.:J.Z ~ Cb .d ~I"':I 4r I , I ~ ~ J :..:: J~. I e ~ .4 ~ II') n I P i' j' "Q, ,~ r~ • JI • n ,. 0 . ~r t _.- ...... ~I , I 41" -.4 .- ~ ~ -.8 .....' ~ ~ .J ".~..- .i ~ I I .. ~~ ~~ ~ I ~, .: -.2 v fAlA I a .8 o ~~~ I .S:! ./ ~ t ~. -~ -.3 -1.2 8 ...... 1-. 4 -1.6 -.5 -2.0 ~ -32 -/6 -8 0 Secfion cnqle of -attack, NACA 64-20~ Wing Ser.tion 8 /6 «Ot deq 24 APPENDIX IV 561 .2 .lJJ6 --­ , ..",..­ ~ ~~ fro ----­ \ \. \ I , I' -~ 'Vi .2 .4 .IRIJ .8 .6 ;ric 4: 1.0 , J. I I \ \ I ~ \ ~ r ~~ 6.... A -,'" A. ,~~ ~~ l'_ "" .......... :J::m-.­ ............... ~:l.. JI ~ AI!!lo - - 1 / ell' / 'c(/~ ~~ ,/ ~~ ~'1 R~"'" II"\.. :lI: a ~ J wa ..... - - - ~ A. .A. -./ q ~. " ac. position R ---­ ~C 1l11C 3.011=>- .c57 .256 -.1J05 -.005 oo 6.0 o 90 g60 ~ 257 -.007 ~St~~~~~ I _ Q20c SJ""ulo~et{.Sf"if ~Jap flef/eef ed V1Z Stdnd~d ~Ol~~~ ., I I I I -.5 -1.6 -L2 -.8 I so: I' t I I I I I I I f I I I 0 .4 Section lift coefficient, t:1 -.4 NACA 64-208 Wing Section (Continued) .8 1.2 £6 THEORY OF WING SECTIONS 562 3.6 3.2 2.8 2.4 2.0 ~ ~~ 7 1.6 r "L '7 J , ~ " ~ ~ J -e I ~ J -./ , 1-.2 .. r t7 IJ J ~ ~ ~. Jr III ~JI ~~ J, U~, l II ~ Dl~ ~q , o , ~~ ~ ~ I If J a ~n. ~II ~ tJ I JI .r . I , 1.1 -.4 ,. I~ '"rot ~U to ~ ¥M1 -.8 v 1-9 ""O...I:Z - .~ ~.~ yv ~l , I ~'» -3 o(oJ l~ -.4 -1.2 I ~- \9 -l6 ~ -.5 -2. 0 -32 -24 -/6 -8 0 SeCtion on(]le of' attock, NACA 64-209 Wing Section 8 "Ot de9 /6 24 32 APPENDIX IV 563 .2 .JJJI "..-- --- ~ ~ ~~ ~ \ .. I \ .4 .2 .6 *' 024 ~ I ./JJ2 I .oa 1,,0 .8 ~ " 1 ~- i )' ~ I \ ~.OJ6 •uo I , .. t J ~ \.012 I ~u \ (~ ~ ~.OOB I ~ ~ '" I v ~'-e. '-.).. ~ 1 ~ ~ h.._ \~ roo",~ r - .-..:II -- ~~ ~L. V ~) :1 1/ l/ .,TJ ~~~'" L,; ,I!~ J' w J V " ~ Iff -~ I~ A~ ~ I 0 ~ l J 1 - IIfMl c.lU .004 o 10", - .- -.I J 2 o ~ 3.0JdO·__ ~58 o 6.0 I t' 3 *$.!Ie c. p'osifion o 9.0 A 6.0 _, t -.0/5 .259 -:.029 .26/ -.041 Standard rOugJr1ess ,O.2Oc simulafed plif flop dt!flecffHI 60 ; ~o 4 Jfal~ar!d~Ju~e~~ r. I I I 5 -1.6 -1.2 -.8 0 -.4 Section NACA 64-~ng .8 .4 liff coe'ficient, c, Wing Seetin", (Cnnb:nued) l2 1.6 THEORY OF WING SECTIONS 564 3.6 3.2 2.8 2.4 w: 2.0 ~ ~'7 , .1/' , 1fT 1.6 ~7 ~ ~ IflU> A~' V ~ ~ If/ 1~J\ ~~ l" U ~~ J,~ ~ I .J. IT • J v: i" J' I o o ~ rrr IJ u 0 r 4~ ~ ~J .1 I d "- ~, /' ~.,.,. I -./ !­ J-.Z rq..A , J ,p ~- 1 I , '\" Jr ~~o -.8 - I ~ I • -.4 ~ \ .. V" ~ , \ -1.2 I -1.6 I ~S -2.0 -32 ,I -24 -/6 -8 0 8 /6 Section onq1e of attack, «OJ deq NACA 64-210 Wing Section 24 J2 APPENDIX IV 565 .z .DJ4 -r--­ ~ l.---- l.--­ - ~o r---...... t--- ~~ A .z/c ~ I .O~6 .8 .6 .oR I \ .2 ­, \ ..~ 1.0 .024 I' ~ .].OcO b I I ~ I 't \ \ r r\ ~~ l;t ~rs: ~~ '­ "'~ P--. / I if ""' ~ V ~Jlt.A """,0' ~ ,~ o " PV Vr:f ~l/ LI ~~~ ..... ~;r ~~ r-s.,[9 ["'i7 ,..,/1 I' P V ~ 0 I lr~ qiloo.. .004 o - - ~ -. I t a,c: pbs tion R_ .rIc yle 03.0­ '0 . .259 -.016 ~~ .259 .258 06.0 090 /:),6.0 -.016 -.011 Staoaora roughness 1. afp::t.·sp'if Q2a $Jmu flap deflected 60" v 6.0 I V6.~'} ~f 1 -I 1 -I.Z -.8 , 'I I I I I I I 1I 5 -L6 J I f Standard rouqhness -.4 I I I I I I I t J I 0 .4 Section lift coefficient, NACA 64-'210 '\~ing Section I I I c, (rnTlti11.1I1'rl) .8 l2 £6 THEORY OF WING SECTIONS 566 3.8 3.2 2.8 ~ j 2.0 J. ~ !J'l~ ~ 1.6 ~ J" r \ ~17 ~ J I ~-~ ~ J I( ~ J l ,4" ..J~ I br:M !I J ~.4 ~ ~ o -./ ~ r-l !B r i. 'I lJ .a ~r 1 "'1~ r ba.l. o r :d !.if 1"""9'''' ( -.4 )J -.8 I; ,, , ~~ 11 \ 1\\ . JJ ~ 4) 1." t1~ JJ J ./ ""IIii~ I» IC:Jti Il:i '-11-0 J( ~- ~, r'Yl~ 't , ,1IAI.' 1\ -/. 2 I ~ bltj ~ -l 6 -:5 -20 -32 -24 -te · -8 Secfion 0 8 <TI9!e 0; aftacli, "01deg NACA 641-012 Wing Section 16 24 APPENDIX IV 567 .IM ~ ~ --­ -­ - ---­ - ~ ~- ~ , " .4­ .2 0 .6 6/e .lJJ5 ~T ~ , I I I ~ RI 10 .8 > :> ,, J .6 1. r I j" , l ~ ~ l 1. c ~ I I \ ~ , \ r\ I d9 l I J , 1\ .... I"' ~ ~ i~ J '1 I R " ~'-. ~ ...... _ l:r~ ~ "'--n. 1:",­ ~t'\ ~'r i,J; ru- 'J /~ /J '/ /' J III /~SI / "' "­'t1 ,,-l~ ~ ~ J)' J I f'­ '\. ;J S r~ J~ ~V -~ lD~f' ........... ~""'[ v~ Q~ ~ ~ _~1' .004 o 'Al-L - t"\ ~ .­ -­ '" '" ,--"-,, iW' ......-.. -- - ~ I *1 '/;.c a.c. position o 3.0Jtltr' R o 6.0 ~ 3 "0 9.0 6.0 · 6 a20c v 6.01 ~ f 5 .259 ~ .26~ t I -1.2 -.8 . '29 .017 -.OOc ~ Standard rOUlJhness simulated split flop deflecfed so: V 6.0­ ~ t -1.6 _ .256 S-l~nJ~d~~eSs ". II I I -.4 0 .4 .8 Sec tion lift coefficient, c, NACA 641-012 Wing ~p.ction (Continued) 1.2 l6 THEORY OF WING SECTIONS 568 3.6 .3.2 2.8 24 ~ J' ~ , 20 \ ( ~ d A ~7 l' 1.6 \ \ 7 I~ jill 'l /) , 'CV IJ I(J 1" 1J-' , o ../ ~1 s " rust... ~ A II o J. -tltj AI :'1 I -~ ~b~ S I -.4 r r ~ ~ AI. ~ if" I I -.... ~ } J-.2 ~. n 1m. ,~ a :tI II ~ ~ """"'1;:1 I J .~ ......4 '" .I ~~ ~I U .1 '4 if" I .I I l -.8 I i'QI . d q (I fy ~- l -l2 l c{ .",,~ I t;1 i~~ -t« ,-.5 -2­ 0 -32 -/6 -8 0 Sec-f,on angle of 8 otrocn, "., deg NACA 641-112 Wing Section /6 24 3c APPENDIX IV 569 .oN .2, ~ ~o r'---. - ~ r---- r--- r---_. - ----­ ~ ~ .()~ ~~ -, , I 'N .4 .2 .6 .8 .IN. ~ LO '* I n .024 I ~ ~ 1 'I fi v, , Q ~ , \ <:f\ ~ ~ l/:a r-, I~I'-.. yo -~ V " . . . ta / ~~~ ~~~ -­ Ilk -­ I-. I"i7 ,.­ . - I --.. 10 R -03.0_10 $ a.c. poslf,on .ric .265 265 267 JIlt: -.0/3 -.0/7 o so -.039 A 6u.J I SfoncJard rouqhness O~Oc Simulated sflit 'lap deflected 60 96­ 060 .3 V 60 4 5~o:'~rd ~~~ I I 1°f I i If JI t I -I 1 I I I I I I 5 -1.6 A ~ ~~ .004 ~ It jQ: ~- ~:t:: ~~ - )­ If/ '/ 1/ ~ l;C 'J""" t:L~ ..­ fJ '/ ,,~ s IJ dV II~ II ) I'\. ~" "'l..: 1/ 1/ ~ ~'" i'l'f I'~ o J J r\ b '\ J D J I "\ , ~ I -1.2 -.8 -4 0 .4 Section /ift coeffIcient, c, NACA 641-112 'Ving Section (Continued) .8 li' l6 THEORY OF WING SECTIONS 570 .16 2.8 2.4 )',\ If fJ L.. IJ r r\z ~~ If) lr ~;r~ • ~5' III 1V J A ~ , 'I ~ o ~. 14 J ,II Jl 1.1 r 'II J I ~ J [9 " 1~ ~ ~'- t ~v. 9 InlV"l -8 ........:: .~ ~ t ~~ " II' J.'D -.4 R.. I 0 I,.. I \. I Ill" II J' U ~ I ~ JlI' '{/ K 10 ~~~ rt JJ II' ~ If) -./ '1 ~ ~r 1.6 o 1\ .. \ ~~ 20 "'L.4 , .~~ IV" I'Q.. Iv...... _ 'I IUt -1.2 1'-' .DIb }. \ -1.6 -.5 -2.0 -3Z -24 -/6 -8 0 8 /6 Section angle of attock. a •• deq NACA 641-212 W"mg Section 24 .12 APPENDIX IV V-­ ~ ~o r----... - r-- r--- t--­ ~ ...-­ -~ I - - ~ " 571 .4. ~ f I .oR fl I \1 -,.2 0 N .2 .4 .8 .6 .z:/c 1.0 4r. 6 I ~ , J I t. ~.o20 I r- , 1 '~ I 1\ \ 6 J 'I III 1\ ]. ~ ~~ ~h. 1/ V f"'w ,~ ~- ~!' -~ ""'~ M: . . l.....oo'' '" ~ ,~ r"~ s: IIi> ~ V v..I"" ~ ...l!~ I): , ~~ ~"j ~~ ]~ P ~p: -­ ... 1­ II' " ." - "'--" r­ "'IIi J ,'( v \ fa. ,,/Ik ( I[ . J Co \. Jif 1/ J n 1 'V' ~ ~ ~ - r­ .1 '.2 D.C. poSition R o .3.OxI Z/C >- 262 ~-.01:3 yIc: o() 9.0 .262 I - .262 -.024 -.oZ4 t 61J J 6.0 OZ "'6..0 Sfondord roughnesS simulated split flop deflected 6lr I I I I I I J I I V.6..o ~~I- Sfandard roughness A .4 I- I I I I I :5 -J.6 -1.02 -.8 rr I T I I I I 111 I I I I I I I 11 I I -.4 0 .4 Section lift coefficienf, c a NACA 641-212 Wing Section (Continued) .8 1.2 L6 THEORY OF WING SECTIONS 572 3.Z 2.8 , d 2.4 F d ,J i~ \, 2.0 ~ v , r )" r '6 --,.. ~~I ~ v ~ J' II iLl I .~.4 jt o I -./ ~I_.Z ~~ t' J' ,...,,.~ l ~r ~~ III r & -.8 ~ ",-pI r I ~'-) ~f 'of' II A 'I 4: I ~ ~~ Jr '­ .• \ 1. • -.4 ,~ ~ I ~ ~ J It: R 0 I' IJI , ~ , ~I J I ~ ,~ II 1 'to.;. ~ P I ./ ...... Jr J' II ,r , ~ J~ ~;rJ .~ ~~ y.",-a.. fVo­ lQ. -.5 -2.0 -.3Z -24 -/6 -8 ... 0 - - ..I~lb. ~, 'Y - 8 Section Dnq/e of attodr, NACA 641-412 Wmg Section ~,deg /6 24 32 5'73 APPE}lDIX IV .2 ~o m6 ~ ----­ - ,..... r--­ ~ - ""'--­ --.... ........ .11M ~ \ \ , \. \. .2 .4 . tk/c LO.J1a .8 .6 .lJ24 1 ~.020 I I ~ ,II , 11 ,, 1 \. ~, ~ ~ :l. "L\-. ~~ .......... """"" r-, ~ ~~ -p­ JJ :/ ~ c .1Y" I IP if) If J:1V I 1\ \ I 0 I~ ) Dl J I.. J II l .A. 0 ~ ~Id ,., ~ ~ ~rf./ ~IJ re I'~ 1"'11 ~~ ~ .... ~~ ....4~ "-'1;, .004 o " - :A.. ... ~ - -./ ~ f.J' L1£. B z/c .266 0.3.0" rJ 06.0 f pbsJflOn y/e -.05.3 -.046 -.034 09.1.1 .266 :267 ~6.0 Sfandord~5 aeo ~ s~(J/~ Sf'if ~/OJ? '{e'.Jeffed 60 v6.0 S'ta~ ~ t . hr:es's £76," ,0UC1~ t I I I -.s I -l2 ~8 -~ 0 I I I I I I I .4 ~ S«:fion lift coefficient, c1 NACA 641-412 Wing Section (Continued) 1.2 1.6 THEORY OF WING SECTIONS 574 - 3.6 2.8 .. I~ 2.4 '7 J It' 1\ ~ ..J { A~ 2.0 ~ 0 ", J." ., J.~ II .At1t1 ),1 IfJ' III IJ ~ ~ o 0 " 4• t~ J" 7r .J. ....­ r. 1""'C • ~'1 j ~ -.2 r'O~ 4J .... " 14 ... ~ !­ J Iq~ rq~ II) ~ -.If. ......",..". ..... ·LJI JI I­ II -.1 .. a IJ t"I.-.... i{ u 1t 19 'I '\. ..fJt. , II /J u '1 ~ (J ~.4- \ ~F ~ ,v •/ ., ..£1.- f1o'J IIJ l' IJ c . ~ W 1.6 J I M -.8 4' ~~ n IA ... ~, , ~":7 J ~, "h. lrf ·wr -.5 -2.0 -32 -24 -/6 -8 0 8 Section t:Jn<;Ile of" attock, «0, det:J NACA 64s-015 Wmg Section /6 24 32 575 APPENDIX IV .z .tJI/J ~ --- ...--­ <; r---. -,,2 ­ - ----­ ',­ ! _-­... ~ ~,. ~ , I N .4 .2 0 """"'-- I"'"--­ I/t: .6 .fJ28 8 1.0 !\. .024 ~ ~ ~.020 I ~ 'I , \ •~\ ~ , J r'. ~ '~ ~ ~~ ,,-' ~~ i'""\ilI"to ~~ ..... 0- -c i'J'1..I--"' -- -... ..... ~ ~ )jJ ~v .-.~ ~ 1.J!l~ v~ .. 12~ ~~ ""-­ ~ " """ "' '" .004 $ ~ ~.",0:1 I'-.~ ltf ') 17 ) (~ If~ IT II! f\. .A.. rl~ ..... If \ h ~ I II r . ~ , r\ ~. J !J rot ~ \ r- J , \ - ",.. .... -- - - .....- _-.10.­ '.I Z 4.~. o - 3.DxI~ 08.0 ~ o 9.0 A 6.0, ~ -.4 .5 II -.8 -.007 -.006 .267 -.007 Sfandtrd roughness D.20e Slinular~t( '¥J'{f ('Of ~flecfed 60­ 6.0 V 6.0 -1.2 Aosiflon .r:/Ct. ylc .268 .268 I 1 , I I I' I Standard rOU4'~ := -.4 0 .4 Secfion liff coefficknt, cL NACA 64,-015 Wing Section (Con:inued) .8 1.2 1.6 576 THEORY OF WING SECTIONS 3.6 3.Z Z8 - I,( I / l\. 9 '1 / ~P'[\ , In l 2.0 ,v f) '7 ,r 1\ ,­A 1.6 '7 .-..II:! rFJ ;JIlII ~ .A IJ :r Ifl IJ Iii. Ie .1 'I )J:~ -.4 ..J~ ~ Jl ...... ­ " ..Iv ~ - J It&. F l:1~ l-l II A. ~~J -.8 ""' 1\ [ \. ~ "p -.5 ~~ " IJ ""'ll\.. t! -.2 ~- 1M ~ ,~ IA I -.1 " r\ If I rJ ~'1 ~ d JI I' "f/ 0 ;v ~r l' I) o .A ~~ J. 1/1 II ,, IJ'P ~~ '(J ~~ ~ ~ r~ ~oC1 II V' "V" ~~ ~~ v 'W' ~ -2.0 -3Z -24 -/6 -8 0 8 Secfion angle of a/tacit, «0) ae9 NACA 61t-215 Wing Section /6 24 32 APPENDIX IV 677 .2 .DJ6 ~o V r--- I"--- ~- - ~ ~I...-. .4 .2 r--.... r---. "-­ .1»2 ~ ~ ~~ " .6 ~C I , I '~ .8 1-\ .IJN 1.0 j I .024 I ro , 11 r \ r; \ \ II ~ ~ \ " " 1:,,­ ""~ ~~ ,t-'t ..... / ~"""-. r:I ~~ ~v v- v: v­ A ,. --­ .004 - V rrT~ iJ1.~ 0. ~Ri ""'II Irf /~ ~w r"-J" ~~ ~~ o .J. J / ~~ I v f6 / / ~ ~ D J J v \. "'" '.G- , IT II if I '­..,. '~ I 1 .4~ ... ~A I \ \ \ l1' n 1/ J \ ~ - T !In , - ""'" -­ ,..... -./ a.c. position 1 o 30xlO~ o 6.0 *$3MC .267 .266 .265 -:045 -.038 -.0/4 A 6.0 ~ Standard roughness O.2Oc simulated split flap ckflec;ed 6V· v 6.0 I I I ,I I t 1 V 6.0 londord roughness <> 9.0 ~ t -.s -l6 -l2 -.8 -.4 0 .4 .8 Sec tion liff coe/ttcent, {'1 NACA 642-215 "·ing Section (Continued) I 1.2 1.6 THEORY OF WING SECTIONS 578 ..J6 I- .3.2 28 lit a: \ V 1.-4 'of' 2.4 1 Id 7 1f' t I~ I\.. . f ., 2.0 J f \ \1 ,r l6 I ~ JJ r 0 \ -J -. JI I~ D } /I ~ ~ r Jf "1~ ~IQ ~ A ~- 1 I~ -.8 r 1'\ ---.. " II III , ,, ~ ~ ,JI ~ .1 0 ., ,:> \ ~~ II o ~~ J~ ] [ , r\\ :l .A 1I~ j 4 ,,~ I~ ~~~ '1 ,I t· 8 ~~ rvl ~ J I ... ~ I ........ II. '­ ~~ ~'l j ~I ..,~\ :I r ~ ~ lH y" -1.2 t2. ~- .­ In Ij ... IV -/.8 ,,:5 -2.0 -JZ -16 -8 0 Section onqle of 8 attock, "" deg NACA 642-415 Wmg Section ~ 24 .32 APPENDIX IV 579 21 .(1M r ""--­ l---"'""" - r--­ r--- r--­ ~ ~ 100.­ ~r- ~ \ .. ~ .(I,Jz ~ \ .2 .4 ~/c .RtJ .8 .6 1.0 024 ~ 1\ \ b \ 1\ ~ r\. ~I\. ill , ~ r\ ~~ r\ ,~ l~ ~ ~~ 19 J II 11\ 1/ " l~ r-, ........ ~ r-, "0' ....... ~ f'.. H:~re ,... po­ 6"'~ "I ,. J J liP I ,...1. VII ~ 1/ /~ ~J.~ J~ 'j. / lIJ V ~~ ~ ... T ~ I{.l' v-.e::::t;~ '""""'!!IL rV .004 -~ ,­ .-\J ~ o - .. -- I " Cl.C. .6 ~JlI ~IO~ a 6.~tl= 3 o 9.0 8.0 A * .264 265 t lo- l.­ !lie -.010 -.051 r- 264 to- L..L position. -:040 standard rouqhness Q20c simulated split 96.0 176.0 4 - .­ .­ ~. I I r I Sraoaard "0f deflected 6 I : t rouq"'ness S -/.2 -.8 -.4 0 .4 S6etion lift coefficient, NACA 64J-415 Wing Section .8 ~ (Continued) l2 £6 THEORY OF WING SECTIONS 580 6 2 28 '} ~ 7 4 12 11 7 ~7 Y ~1 0 ~ I J , I.6 \ b , J f A -- ~"' ~ Iff 2 ~ A U ioo-- J~ I~ o ~, , I 4~ I II "IT r J II A III -./ ~, 4 I !­ rQl;ll:l ~ A