AD-760 100

DEVELOPMENT OF AN EMPIRICALLY BASEDCOMPUTER PROGRAM TO PREDICT THEAERODYNAMFC CHARACTERISTICS OFAIRCRAFT. VOLUME I. EMPIRICAL METHODS

Roy To Schemensky

General Dyn amic s

Prepared for

Air Force Flight Dynamics Laboratory

November 197 3

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Convair Aerospace Division UnclassifiedGeneral Dynamics CorporationFort Worth, Texas

3. REPORT TITLE

Development of an Empirically Based Computer Program to Predict theAerodynamic Characteristics of AircraftVolume I: Empirical Methods

4. DESCRIPTIVE NOTIES (T•p of repof and lnCluivwe da1tc)

Final Report - December 1972 through October 1973•. AUTHOI(SI (FitSf rs.mi. ddlf InlfI@I, Iast Rm.t )

Roy T. Schemensky

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Approved for public release; Distribution Unlimited

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AFFDL/FXM1S. ABSTRACT

This report (Volume I, Emperical Methods) presents the methods, equa-tions, and substantiating data for an empirically based computer pro-gram for the rapid and accurate evaluation of the aerodynamic characteristics of large aircraft (bombers, tankers, and transports) from takeofthrough landing and through the subsonic, transonic, and supersonicspeed regimes. The program calculates lift, moment, and drag character.istics at both low- aud high-lift conditions, including the effects ofground proximity during landing and takeoff. The input requires the

configuration geometry and the aerodynamic conditions for which solu-tions are desired. The program includes the capability of analyzingboth fixed-wing and variable-sweep-wing configurations as well as the

aerodynamic characteristics of the most recent supercritical wing

designs. The accuracy of the program is verified through comparisons

of the predicted results with experimental lata for several configura-

tions. Details of the input and output for this program along with a

FORTRAN source deck listing and sample problem are contained in VolumeII, Program User Guide. Although this .'rogren was developed to handt t"hebomber, tanker, transport class of aircraft, it is also applicable to fighter typeaircraft without maneuver devices.

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K E O D FL IN K A •IN K a . .. tLIN K C_OLA

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Aerodynamic CharacteristicsDrag PredictionLift PredictionMoment PredLctionHigh-Lift PredictionComputer ProgramTransport AircraftDrag Due to LiftVortex Lift

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DEVELOPMENT OF AN EMPIRICALLY BASEDCOMPUTER PROGRAM TO PREDICT THE

AERODYNAMIC CHARACTERISTICS OF AIRCRAFT

Volume I Empirical Methods

R. T. SCHEMENSK Y

I

I Approved for public release; distribution unlimited.

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FOREWORD

This report was prepared by the Convair Aerospace Divisionof General Dynamics Corporation, Fort Worth, Texas, for the AirForce Flight Dynamics Laboratory under Contract F33615-73-C-3043, Project 147601. The work reported here was performed inthe period December 1972 through October 1973.

The results of this work are documented in two volumes.Volume I presents the methodology developed in this study;Volume II contains the user manual for a computer programwhich automates these methods.

The work was accomplished under the direction ofMr. J. Kenneth Johnson of the Air Force Flight DynamicsLaboratory (FXM). The author wishes to acknowledge thevaluable assistance of Mr. Eugene L. Crosthwait, ConvairAerospace Division, in the development of the empiricalmethods for the wing lift and wave drag characteristics.

This technical report has been reviewed and is approved.

PIhif6 . AntonatosChief, Flight Mechanics DivisionAir Force Flight Dynamics Laboratory

i

I " JiiP

_f-- -- ,

ABSTRACT

This report (Volume I, Empirical Methods) presents themethods, equations, and substantiating data for an empiricallybased computer program for the rapid and accurate evaluationof the aerodynamic characteristics of large aircraft (bombers,tankers, and transports) from takeoff through landing andthrough the subsonic, transonic, and supersonic speed regimes.The program calculates lift, moment, and drag characteristicsat both low- and high-lift conditions, including the effectsof ground proximity during landing and takeoff. The inputrequires the configuration geometry and the aerodynamic condi-tions for which solutions are desired. The program includesthe capability of analyzing both fixed-wing and variable-sweep-wing configurations as well as the aerodynamic characteristicsof the most recent supercritical wing designs. The accuracy ofthe program is verified through comparisons of the predictedresults with experimental data for several configurations.Details of the input and output for this program along witha F(JRTRAN source deck listing and sample problem are containedin Volume I1, Program lUser Guide. Although +tj .. a..developed to handle the bomber, tanker, transport class of air-craft, it is also applicable to fighter type aircraft withoutmaneuver devices.

IJ IlI

TP3LE OF CONTENTS

Page

1. INTRODUCTION 1

2. GEOMETRY 3

2.1 Component Geometry 3

2I.1. Body Geometry 32.1.2 Airfoil Surface Geometry 4

2.1.3 Wing Geometry 5

2.2 Variable-Sweep Configuration 7

2.3 Airfoil Section Geometry Dg

3. MINIMUM DRAG 15

3.1 Friction, Form, and Interference Drag 15

3.1.1 Friction Drag 163.1.2 Form Factors 173.1.3 Interference Factors 26

3.2 Camber Drag 29

3.3 Base Drag 29

3.4 Wave Drag 30

3.4.1 Wing Wave Drag 303.4.2 Body Wave Drag 323.4.3 Nacelle Wave Drag 34

3.5 Fuselage Aft-End Unsweep Drag 35

3.6 Miscellaneous Drag Items 35

iv

TABLE OF CONTENTS (Cont'd)

tage

4. DRAG DUE TO LIFT 37

4.1 Subsonic Polar Prediction Below Polar Break 37

4.2 Supersonic Polar Prediction Below Polar Break 43

4.3 Transonic Polar Prediction 46

4.4 Subsonic Polar Prediction Above Polar Break 48

4.5 Supersonic Polar Prediction Above Polar Break 57

5. CRITICAL MACH NUMBER AND DRAG RISE 61

5.1 Pressure Coefficients Calculations 62

5.2 Critical Mach Number Calculation fromCrestline Pressure 64

5.3 Drag Rise 65

6. LTVT 73

6.1 Wing Lift-Curve Slope 73

6.2 Supercritical-Wing Lift-Curve Slope 77

6.3 Tail Lift-Curve Slope 79

6.3.1 Downwash at the Tail 806.3.2 Dynamic Pressure at the Tail. 81

6.4 Body Lift-Curve Slope 82

6.5 Angle of Attack at Zero Lift 82

6.6 Nonlinear Lift of High-Aspect-Ratio Wings 85

6.7 Nonlinear Lift of Low-Aspect-Ratio Wings 86

6.7.1 Potential-Flow Lift 876.7.2 Leading-Edge Vortex Lift 876.7.3 Tip Vortex Lift 91

V

TABLE OF CONTENTS (Cont'd)

Page

6.8 Nonlinear Lift of Cranked Wings 93

6.9 Maximum Lift Coefficient 100

6.9.1 High-Aspect-Ratio Method 1006.9.2 Low-Aspect-Ratio Method 1066.9.3 Tail Lift Contribution to CLMAX 106

7. 1DMENT III

7.1 Zero Lift Moment iII

7.2 Aerodynamic Center 113

7.2.1 Aerodynamic Center of Forebody 1lo7.2.2 Aerodynamic Center of Wing

(Trapezoidal, Single Panel) 1157.2.3 Aerodynamic Center of Wing (Cranked

or Double Delta) 1227.2.4 Aerodynamic Center of Wing-Lift

Carryover on Body 124

7.3 Effect of Trim Deflection 124

8. HIGH-LIFT AERODYNAMICS 129

8.1 Lift of High-Lift Devices 129

8.2 Drag of High-Lift Devices 133

8.3 Moment of High-Lift Devices 135

8.4 Aerodynamic Characteristics of Two-DimensionalHigh-Lift Devices 138

8.5 Ground Effect 148

9. DATA COMPARISONS 163

9.1 Systematic Wing Study 163

9.2 Cranked Wing Study 164

vi

Ji"

TABLE OF CONTENTS (Cont'd)

9.3 C-141A Flight Test Data 164

9.4 High Lift Configurations 165

"10. CONCLUSIONS AND RECOMMENDATIONS 197

REFERENCES 199

v

I I..

+i

N vii

LIST OF FIGURES

Figure Page

I Variation of Leading-Edge Radius withThickness Ratio 13

2 Turbulent Skin Friction Coefficient on anAdiabatic Flap (White-Christoph) 18

3 Skin Friction on an Adiabatic Flat Plate,XTR = 0.1 19

4 Skin Friction on an Adiabatic Flat Plate,XTR = 0.2 20

5 Skin Friction on an Adiabatic Flat Plate,XTR = 0.3 21

6 Skin Friction on an Adiabatic Flat Plate,XTR = 0.4 22

0 _r .)L i c1 ±t L o~l n UInaI d i C-1' Li1 tLJL C ',- I C2 tZ Id a-

XTR = 0.5 23

8 Supercritical Wing Compressibility Factor 25

9 Wing-Body Correlation Factor for SubsonicMinimum Drag 27

10 Lifting Surface Correlation Factor forSubsonic Minimum Drag 28

11 Transonic Wave Drag of Parabolic Noses 33

12 Effect of Fuselage Upsweep on Drag 36

13 Lift and Speed Regions ior Calculation ofDrag Due to Lift 37

14 Body Effects orn Wing Span Efficiency 39

15 Effect of Reynolds Number on Leading-EdgeSuction for Blunt Uncambered Airfoils 40

16 Leading-Edge Suction foLi Sharp Airfoils 42

viii

I LIST OF FIGURES (Cont'd)

FigureI 17 Effect of Reynolds Number and Thickness onthe Leading-Edge Suction Factor 42

18 Effect of Wing Planform on Leading-EdgeSuction 44

19 Lift Coefficient for Minimum Profile Drag -

NACA Camber 45

20 Lift Coefficient for Minimum Profile Drag -

Conical Camber 45tr

21 Transonic Suction Factor 47

22 Airfoil Leading-Edge Shape Effect on aPB° 50

23 Conical-Camber Effect on CLpB 52

24 Reynolds Number Effect on C, 53LýDB

25 Correlation Lift Coefficient for CLDB

26 Weissenger Theoretical Wing Planformj Efficiency Factor 55

27 Separation Drag Factor 56

28 Supersonic Polar-Break Lift Factors 58

29 Mach Critical Prediction Chart 66

30 Prediction of Fuselage Critical Mach Number 67

31 Correlation of Critical Mach Number for Con-ventional Wings 68

. 32 Transonic Drag Buildup 69

33 .ypical Lift-Curve Slopes 78

34 Factor Used in Determination of Body Lift-Curve Slope 83

ix

-n' itam....... M'• .

•1.

LIST OF FIGURES (Cont'd)

Figure

35 Camber Factor for Zero-Lift Angle of Attack b4

36 Variation of Vortex-Lift Constant withPlanform Parameters 88

37 Variation of Vortex Breakdown Factor withAspect Ratio and Angle of Attack 89

38 Experimental Variation of Leading-EdgeSuction Parameter R with Angle of Attackor Round-Leading-Edge Delta Wings 90

39 Sketch of Possible Flow Patterns 92

40 Comparison of Program Results with Experi-mental Data, A = 630 94

41. Comparison of Program Results with Experi-mental Data, A= 58u 95

42 Comparison of Program Results with Experi-mental Data, 4t= 530 96

43 Comparison of Program Results with Experi-mental Data, A= 450 97

44 Construction of Nonlinear Lift Curve forCranked Wings 98

45 Definition of Cranked-Wing Planform forCalculation of Angle of Attack in NonlinearRange 99

46 Parameters Used in Calculation of Angle ofAttack in Nonlinear Lift Range 99

47 Factors for Determining Subsonic MaximumLift 102

48 Factors for Determining Mach Number Cor-rection for Maximum Lift 103

X

LIST OF FIGURES (Cont'd)

Figure Pae49 Section Maximum Lift for Uncambered Air-

foils 104

50 Effect of Airfoil Camber on Maximum Lift 105

51 Angle-of-Attack Increment for SubsonicMaximum Lift of High-Aspect-Ratio Wings 105

52 Subsonic Maximum Lift of Low-Aspect-RatioWings (XT < 35%) 107

53 Subsonic Maximum Lift of Low-Aspect-Ratio

Wings (XT > 35%) 107

54 Mach Number Correction for Low-Aspect-RatioMaximum Lift 108iI

55 Angle of Attack for Subsonic Maximum Liftof Low-Aspect-Ratio Wings 109

56 Mach Number Correction for Angle of Attack 109

57 Effect of Linear Twist on the Wing Zero-LiftPitching Moment 112

58 Wing-Body Geometry 114

59 Supersonic Center of Pressure of Ogivewith Cylindrical Afterbody 116

60 Wing Aerodynamic-Center Position 117

61 Transonic Wing Aerodynamic-Center Location 120

62 Geometry for Cranked-Wing Aerodynamic-CenterPrediction 123

S63 Parameter Used in Accounting for Wing-LiftCarryover on the Body 125

64 Aerodynamic-Center Locations for Lift Carry-over onto Body at Supersonic Speeds 126

Xi

LIST OF FIGURES (Cont'd)

Figure Page

65 Incremental Effect of Flaps 130

66 Flap Chord Factor 131

67 Flap Span Lift Factor 132

68 Slat Span LifE- Factor 134

69 Flap Span Drag Factor 136

70 Flap-Induced Drag Factors 137

71 Span Effect on Moments 139

72 Span Effect on Moments of Sweptback Wings 140

73 Theoretical Lifting Effectiveness ofTrailing-Edge Flaps 141

74 Turning Efficiency of Plain Trailing-Edge

Flaps 142

75 Turning Efficiency of Single-Slotted Flaps 142

76 Turning Efficiency of Double-Slotted Flaps 143

77 Turning Efficiency of Trlp1.Sl-n--tted Flaps 1431

78 Principle of Superposition Theory andExtended Slotted-Flap Geometry 145

79 Maximum-Lift Correlation Factor for Trailing-Edge Flaps 147

80 Flap-Angle Correlation Factor Versus FlapDeflection 147

81 Leading-Edge-Flap Maximum-Lift Effectiveness 149

82 Maximum-Lift Efficiency for Leading-EdgeDevices 149

xii

LIST OF FIGURES (Cont'd)

Figure Page

83 Leading-Edge-Device Deflection-Angle Cor-rection Factor 149

84 Flap Center-of-Pressure Location as Given

by Thin-Airfoil Theory 150

85 Moment Correlation Factor Versus Flap Chord 150

86 Two-Dimersional Drag Increment Due to PlkinFlaps 151

87 Two-Dimcnsional Drag Increment Due toSingle-Slotted Flaps 152

88 Drag Increment for Double-Slotted FlapsVersus Lift Increment at a - 00 153

89 Prandtl's Interfereace Coefficient 155

90 Factor Accounting for Finite Span in GroundEffect 156

91 Parameter Accounting for Variation inLongitudinal Velocity with Ground Height 157

92 Parameter Accounting for Ground Effect onLift Due to Trailing Vortices 158

93 Parameter Accounting for Ground Effect onLift Due to Bound Vortices 159

94 Effect of Flap Deflectionion thE GroundInfluence on Lift 160

95 Effective Wing Span in the Presence of theGround 162

96 Effective Flap Span in the Presence of theGround 162

t. xiii

LIST OF FIGURES (Cont'd)

Figure Pae

97 Comparison of Predicted Drag Versus Liftand Test Data 166

98 Comparison of Predicted Lift Versus Angle ofAttack and Test Data 177

99 Lift-Curve Slope as a Function of Mach Number 188

100 Drag-Due-to-Lift Factor as a Function of MachNumber 189

101 Pitching Moment-Curve Slope as a Function ofMach Number 190

102 Minimum Drag Coefficient as a Function ofMach Number 191

103 Correlation of Predicted and Flight Test DragPolar 192

104 Drag at Constant Lift Coefficient Versus MachNumber 195

105 Comparison of Predictions with Test Data,Single-Slotted Flap 196

xiv

I

LIST OF SYMBOLS

A Area

AR Aspect ratio

b Wing span

CD Drag coefficient

CIDL Drag due to lift

CDR Drag rise

Cf Skin friction coefficient

Cf/C Flap chord to wing chord ratio

CL Lift coefficient

CLd Wing design lift coefficient

SC'd Section camber lift coefficient

CLDB Value of CL for flow separation

C P Pressure coefficient

CLpB Value of CL where drag polar ceases to be parabolic

CR Root chord

INC©a- geUowt•ical chord

CL Lift-curve slope

d Body diameter

e Net polar span efficiency

e. Wing-body polar span efficiency

ew Theoretical wing-alone polar span efficiency

i Incidence angle

K Polar shape factor

xv

ILIST OF SYMBOLS (Cont'd)

ILength

M Mach number

MAC Mean geometrical chord

NCR Critical Mach number

R Leading-edge suction factor

RN Reynolds number

S Surface area

tic Wing thickness-to-chord ratio

Longitudinal location

Y Lateral location

Z Vertical location

Angle of attack

ACL Polar displacement at minimum drag'C~o Lift displacement at Oe - 0

CLo

'Ay Airfoil leading-edge sharpness parameter

A Sweep angle

A Taper ratio

OTE Airfoil section trailing-edge angle

Deflection

Efficiency factor

Subscripts

BT Boattail

c/4 Quarter-chordxvi

LIST OF SYMBOLS (Cont'd)

Subscripts (Continued)

c/2 Hid-chord

CG Center of gravity

e Effective value or exposed

EXP Exposed

f Flap

HT Horizontal tail

i Inboard

LE Leading edge

LER Airfoil leading-edge radius

MAX Maximum value

MIN Minimum value

N Nose or number of wing panels

o Zero lift or outboard

PB Value at CLPB

PLAN Pianform

REF Reference

S Slat or airfoil surface

s Exposed airfoil surface

t Tail surface

TE trailing edge

WE Wing-Body

Wet Wetted area

xvit

1. INTRODUCTION

The need for a computer program for performing rapid andaccurate evaluation of the aerodynamic characteristics of anaircraft is evident during evaluation of a preliminary designconfiguration when both time and geometry definition arelimited. A quick-response program is also needed for performingpreliminary design trade-off studies. This report presentsdocumentation for an empirically based computer program thatwill predict, with minimum aircraft geometry input requirements,the aerodynamic characteristics of large aircraft (bombers,tankers, and transports) from takeoff and landing through thesubsonic, transonic, and supersonic speed regimes.

Some of the methods contained in the AeroModule program(Reference 1) developed at General Dynamics' Convair AerospaceDivision, Fort Worth Operation, are utilized in the Large-Aircraft Aerodynamic Prediction Program. The AeroModule pro-gram was developed to proV1 '• id e f a. A-g - U8- . Lues-or use

in a computerized aircraft design synthesis program. TheLarge-Aircraft program has extended the AeroModule methods toinclude moment calculations along with improving the methodLused to predict lift and drag.

Input to the Large Aircraft program requires the configu-ration geometry and the aerodynamic conditions for whichi solu-tions are desired. Aircraft geometry is represented by aseries of component bodies and airfoil surfaces. The geometryinput requirements are minimized by u-S cf internally calculatedvalues where possible. Component wetted areas can either becalculated internally from other input or input directly,as desired. The program includes the capability of analyzingboth fixed-wing and variable-sweep-wing configurations alongwith either conventional or supercritical wing designs.

The output from the Large Aircraft program consist oftabulated CL, CD, Cm, and angle-of-attack predictions for agiven Mach number and altitude or Reynolds number condition.In addition, 4 breakdown is given of each drag item such asdrag rise, wave, friction, base, trim, camber, etc. Also,other aerodynamic parameters such as lift-curve slope, CLmax,polar-shape factors, wing-body aerodynamic center, etc., arelisted. The output format is designed to provide the programuser with a quick scan of the significant aerodynamic para-meters along with an in-depth lock.

1

. . . . . ---.-.--------

The methods, equations, and substantiating data for theLarge Aircraft program are presented in the following sections.Details of the input and output of this program along with aFORTRAN source deck listing and sample problems are containedin Volune II, Program User Guide.

Although this program was devpeoped to handle the bomber, tanker,transport class of aircraft, it is also applicable to fighter type air-craft without maneuver devices. The program is referred to throughoutthe text as the Large Aircraft Program.

2p

2

L !__

2. GEOMETRY

The Large-Aircraft Aerodynamic Prediction Program requiresa minimum of input data since most of the geometric parametersused in the aerodynamic methods are calculated internally withinthe program. Some geometric parameters such as wetted areas andmean geometric chords can be either generated internally by theprogram or accepted as input data. The conventions and equationsused by the Large Aircraft program to determine the geometricparameters used in aerodynamic calculations are described in thissection.

2.1 Component Geometry

The basic aircraft geometry is represented by a series ofcomponents. The fuselage, canopy, and stores are represented bya series of bodies; the nacelles are represented by another seriesof bodies (opennosed); and the wing, tail surfaces, pylons andventrals are represented by a series of sirgle-panel airfoil sur-faces. For cranked or complex wing planforms the wing can alsobe represented by a series of interconnected surface panels.

the geometric parameters for variable-wing-sweep configurations.

I |! 2.1.1 Body Geometry

The minimum geometry input requirements for the body compo-nents are length. width, height, nose lengthand boattail length;in addition, for open-nosed bodies, the inlet and exit area mustbe specified. If the maximum cross-sectional area of the compo-nent, AMAX, is not input, the value is calculated by

SAMAX = 4 (width x height) (2-1)

Also, if wetted area for the component is not input, a value isthen calculated. The wetted area for close-nosed bodies is

I determined by

A+T 42.8/ + 2-BT AMAX2(2-2)

+ 4 IN d/BT) If/'A (2-2)

3

For open-nosed bodies, wetted area is determined by

T- 2 .5N(I + KmLET ) + 2.51BT(I + AAXITAWET [4(Aryirh AB' A~ k

+ 4(1 - / BT) A4AX (2-3)

2.1.2 Airfoil Surface Geometry

The input required to define the planform for the airfoil

surface components that represent surfaces other than the main

wing are exposed taper ratio, exposed root chord, and the leading-and trailing-edge sweep angles.

The aspect ratio and the exposed area of each componentare calculated by the equations

(i-AS) 4 (2-4)TT777 'ta-( - f-nnL/i-n. l

r 11 A)2 ARS 255EXP - (CR)s (1 )4) 25

If the component wetted area and mean geometric chord are notinput, they are calculated by the equations

F1 (CR) 1 + 1," (2-6)

SWET = SEXP L2+.1843(t/c)s + 1.5268(t/c)s 2

- . 8395(t/C)s3j (2-7)

The wetted area is essentially twice the exposed area with asmall factor to account for thickness.

For trim calculations the location of the hortiontal tailquarter-chord point on the MAC is calculated from

4

XHT blýT F1+2ART~ tan(kLE)H + H +(XE(289- 1 1+-AT I +T (XEHT (28

where bHT is the exposed area span of the horizontal tail calcu-lated from bHT - V ARHT ' SEXP The moment arm of the tail atany angle of attack is computed from

T - XH cs (.-.) (2-9)

where

X (T ZCG)2 + (XHT - XCG)I

and

(ZHT - ZCG)S-CL •arctan (XHT -

XCG and ZCG are the longitudinal and vertical locations of themoment reference point and ZHT is the vertical location of thequa4-ter--huLd point on the horizontal tail MAC.

2.1.3 Win& Geometra

If the main wing is defined as one panel, the aspect ratio,taper ratio, leading-edge sweep, planform area, and the locationof the wing relative to the fuselage are required input to definethe planform geometry. The geometry for a complex wing planformis represented as a series of panels, as shown below in the sketch.

fIXLE, YW & CRW ARE DEFINEDFOR EACH PANEL

3-

y 2

YW(l) PANEL #1- N FUSELAGE

- INTERSECTION

XLE(l)

5

The total wing is defined by up to ten panels. The leading-edge

location and chord length of each panel edge is specified along

with the average section camber and thickness of each panel.

Average values of thickness and camber are computed by the root-

mean-square equations

tic S• EX-P

IN(C-1d) 2i~iC~d • •(2-11)

where SEXP =• Si is the sum of the exposed areas of all thepanels. Certain aerodynamic calculations, such as wing wavedrag and lift-curve slope, require the use of an "equivalent"trapezoidal wing that approximates the planform of the arbitrary

-wing. The sweep angle of the equivalent wing is obtained byarea-weighting the sweep according to the equations

NS(tanALE) i Si

(taiAE) - i (2-12)

SEXP

N2 (cosAc 4 ) Si

(co•/ye4 ) (2-13)

SEXP

N (cosAc/2) iSi

(cOsAc/ 2 )e = Xl (2-14)SEXP

N•itan-A&TE) i Si

(tarE) e = ((2-15)

S EXP

6

If the wing panel wetted areas and mean geometric chordare not input, they are calculated by use of equations similarto Equations 2-6 and 2-7.

For a multiple-panel wing, if the planform area is notinput, a value is calculated by summing the panel exposed areasand adding the area obtained by extending the innermost panelto the centerline of the aircraft. In the case of a wing whoseinnermost panel represents a strake with a large leading-edgesweep angle, extending this panel to the centerline of the air-craft would result in an extremely large planform area. Inthis case the value of the theoretical planform area of the wing,ignoring the strake, should be input. The aspect ratio is de-fined as

AR - b2 /SpLAN (2-16)

Lift and drag parameters are calculated by use of the aspectratio defined with the wing planform area, and the final resultsare then referenced to the reference area, SREF, which is input.In most typical cases, SpTv.A. equals SREF.

2.2 Varlable-Sweep Configuration

IThe planform for a variable-sweep configuration is definedby a trapezoidal movable panel and an optional glove panel, asshown in the sketch below.*

ULE(3) ~

XLE(2)

D XPIVOTXLE(l.) YPIVOT

_XAPEX ,

7

The procedure first defines the coordinates of Points 1, 2,3, and 4 from the input. (The input planform area is equal totwice the area enclosed by these four points.) When the movablepanel is rotated about the pivot point, the resulting geometry isas sketched below. The coordinates of Points 2, 3, 5, and 6 are

2

3

A

5 --4

then determined. The planform area is calculated as twice thearea enclosed by these four points. Since the tip chord is assumedstreamwise at the forward reference sweep, the distance from Point2 to the centerline is the semi-span, b/2. The aspect ratio isthus defined as

AR -2 Q~LN(2 6

The taper ratio is calculated as

A - CT/CR (2-17)

where CT and CR are as defined in the following sketch.

8t

A B AREA A-

AREA B

/,/

_ - H~t7 CR~b/ _

The mean aerodynamic chord (MAC) and wetted area of the

outboard panel are calculated by use of the tip chord, CT, andthe chord, CRX2 . The (MAC)calculated is expressed for the out-board panel as

(MAC)calculated 3 RX2( 1 + (2-18)IAJ

where •' - CT/CRX2. The wetted area (Awet)calcul ted is com-puted using twice the exposed area of the panel and the thick-ness correction to wetted area expressed by Equation 2-7. Thesecalculated values for MAC and Awet are compared with the optionalinput values at the forward reference sweep and the aft reference

~ ,jivA 44-1i~ra * T~ - hF *-I- I I a A .,r"A 4 4-.. 1.a A4 .,,

might occur for non-trapezoidal planforms, the input values areused and the incremental differences between the two are used forinterpolation purposes in the calculation of MAC and Awet atintermediate sweep angles. The equations are as follows:

MAC MACcalc. +•MAC 1 + (AMAC2-AMAC-) (2-19)

Awet Awetcaic + 'Awetl + ('Aet2"wettA-Al (2-20)

9

where

6MAC - MACcal - MACinput

6Awet - Awetcalc. - Awetinput

and where the subscript I refers to the forward reference sweep

position, and the subscript 2 refers to the aft reference sweeppo si tion. ,

The maximum-thickness sweep angle,A(t/C)max, used in fric-

tion drag calculations, is calculated from the quarter-chordsweep of the panel and the inputA(t/c)max at the forward andaft reference sweep positions. The equation is

A(t/c)m, -Ac/4 +4-i + (4 42-44) l (2-21)

where

'A- (Ak/4)calc. - (At/c)max)input

The streamwise camber and thickness of the outboard panelat a given sweep are calculated by

. C '/C '(CL)alc. (CLdref (C'/C)ref (2-22)

(t/C)calc" C I//ef"• •r (2-23)

wherer rsrka cosA.

C'/C " 0.5 LE + c E (2-24)cOBWE-Ac/2) + c-OsGc/2-ATE)J

Equation 2-24 is the relationship between the chord perpendicularto the mid-chord sweep, C', and the streamwise chord, C. For avariable-sweep wing, C' remains constant so that the camber andthickness perpendicular to the mid-panel sweep also remainconstant. Finally, the outboard panel thickness is compared withthe aft reference sweep input value and, if the calculated andinput values differ, the input value is used for interpolation

10

r_" -I

pu'rposes in the calculation of t/c at intermediate sweep angles.The equation is as follows:

(tic) - (t/C)calc" + (/c) - (2-25)

where (lt/c) - (t/C)input - (t/c)calc' at aft sweep.

The variation of wing twist with sweep can be calculatedfrom

t- arctan ( AZ (2-26)C" tip

where AZ is the vertical position of the leading edge (assumingthe wing is twisted about the trailing edge) and C is thestreamwise chord at the tip in the swept position. The tipdisplacement is calculated at the forward reference sweep posi-tion through the equation

(AZ)tip - (CRA) tant

The streamwise chord at the tip is calculated from

Ctip -CR - (b/2)(taALE - tanATE)

The tip displacement is assumed to be independent of sweep.

The variation of wing incidence with sweep is calculatedI fromif iref ( - tanAALE" tartATEl) COSAALE

(2-27)

where .A-LE -ALE -ALE1 . In calculating the variations of twistand incidence with wing sweep, it is assumed that the wing pivotis perpeiidicular to the wing chord plane.

2.3 Airfoil Section Geometry

Several airfoil section parameters are used in the aerodyna-mic predictions. These parameters are generated internally in theprogram for the NACA 6-series and 4-digit airfoil sections alongwith biconvex and supercritical airfoil sections. The proceduredetermines the leading-edge radius as a function of thickness

[ 11

ratio, t/c, for these airfoils, as shown in Figure 1. Thedistance of the position of maximum thickness from the leadingedge, Xt/cmax, is listed in Table I. A leading-edge sharpness'parameter, ky, expressed as

,&y - A(t/c) (2-28)

is defined for uncambered airfoils, where A is a function of theairfoil leading-edge geometry (shown plotted in Figure 2.2.1-8of Reference 2). The trailing-edge angle of the upper surfaceof the airfoil is computed from

TE - BS(t/c) + C(Ced) (2-29)

A, B, C values used in the Large Aircraft program are listed inTable I.

If the airfoil section cannot be approximated by one of thesections contained within the Large Aircraft program, the usercan input geometry to define any arbitrary airfoil section.

Two examples of the designation for a six-series airfoil aregivenl by-y

64--210 and64A210

The 6 for the first digit indicates a 6-series airfoil. Thesecond .ligit (4) designates the chordwise location (in tenths)of the nmnimum pressure for the basic symmetric airfoil at zerolift. The third digit (2) designates the camber design liftcoefficient (in tenths). The last two digits (10) designatetheV airfoil thickness (in percent). The letrer A appearing insome 6-digit. series designations indicates that a modifiedthickness and camber distribution is used.

An example of the designation for a 4-digit airfoil is given

by:

0012-34

where the 12 designates the thickness (in percent chord), the 4designates the position of maximum thickness (in tenths), andthe 3 designates the leading-edge radius (3 designates 1/4 normal,6 designates normal, and 9 designates 9X normal leading-edgeradius).

12

SUPERCRITICAL

3 64- OXXS~63-OXX

66- OXX

2 65-OXX

0

0 48 12 16 .20 24

THICKNESS RATIO, t/c (Z CHORD)

OOXX-6X

2 263AOXX

-,z 64AOXX

Fi~gr~e TVaication o Leading-Edg -e Radius with

& -.......... A013 "

Table 1

TABULATED AIRFOIL SECTION PARAMETERS

Airfoil Type X(t/c)max A B C

63-series .35 22.0 34.6 14.864-series .375 21.7 38.4 14.865-series .41 19.2 46.4 14.866-series .44 18.35 60.2 14.863A .37 22.0 57.5 14.0564A .39 21.2 59.5 14.0565A .42 19.2 66.5 14.05Supercritical .3471 27.0 30 40.0Biconvex .50 11.75 95.0 0.0OOXX-62 .2 24.0 50.0 13.8

-63 .3 24.0 63.0 13.8-64 .4 22.0 82.8 13.8-65 .5 20.0 113.0 13.8-66 .6 20.0 153.0 13.8-33 .3 19.0 63.0 13.8-34 .4 17.0 82.8 13.8-35 .5 15.0 113.0 13.8-93 .3 29.0 63.0 13.8-94 .4 27.0 82.8 13.8-95 .5 25.0 113.0 13.8

14 I

3. MINIMUM DRAG

The drag of an aircraft can be represented as the sum ofminimum drag, plus drag due to lift, plus drag due to trim. Thedrag bookkeeping system used in the Large Aircraft program hasminimum drag comprised of the drag items that are "assumed" tobe independent of lift, such as frictkon, form, interference,wave, base, camber, roughness and proturberance. Drag due tolift is comprised of the drag items that vary with lift, suchas induced drag, profile drag increment due to lift, and flowseparation drag. Transonic drag rise, which varies with lift,is separated for bookkeeping purposes into an increment addedto minimum drag and an increment added to drag due to lift. Incases where the fuselage has an upswept aft end, the incrementin fuselage drag between an upswept fuselage and a symmetricalfuselage is tabulated in the program output as a function oflift. The drag buildup does not include incremental drag con-tributions due to propulsion installation such as spillage drag,bleed, nozzle effects, etc. In many thrust-drag accounting sys-tems the propulsion related drag increments are included in thepropulsion force buildup since these drag increments vary withpower setting. If a horizontal tail is present on the configu-ration, the untrimmed lift and drag is computed for a zero taildeflection condition. The effect of horizontal tail deflectionfor trim is determined by computing the lift and drag incrementrelative to the zero tail setting.

The methods used to determine each of the minimum drag

contributions and the fuselage aft-end upsweep drag are de-scribed in the following subsections. Drag rise, drag due tolift, and trim drag are discussed in Sections 5, 4, and 7,respectively.

3.1 Friction,_ Form, and Interference Drag

A large part of the subsonic minimum drag is comprised of

the sum of friction, form, and interference drag of all theaircraft components. The drag of each component is computed as

CD - (Cf Awet FF • IF (3-1)SREF~

where Cf is the compressible flat-plate skin-friction coeffi-cient, Awet is the component wetted area, and FF and IF are thecomponent form and interference factors.

15

3.1.1 Friction Drag

The flat-plate, compressible, turbulent, skin-friction coef-ficient is determined from the general equation given in Refer-ence 3,

Cf •Cf(RN * F2 ) (3-2)

where Fl and F2 are functions of the freestream Mach number andwall temperature. The incompressible skin-friction coefficient,Cfj, is evaluated at the equivalent Reynolds number, RNL F2.White and Christoph (Reference 3) developed expressions for thetransformation functions Fl and F2 along with a more accurateexplicit equation, bhsed on Prandtl/Schlichting type relations,for computing the i'icompressible, turbulent,flat-plate frictioncoefficient (Cfi) with the following results:

F1I . t- f-

F2 - tl+n f

0. 430

(li og 1 0 RNL) 2 5 6

For an adiabatic wall condition, t and f are given by

t - Tm/Taw + r [--! M 2

f - I + 0.044 r M 2 t

Using a recovery factor r - 0.89 and a viscosity power-law expo-nent n - 0.67, recommended in Reference 3, results in the follow-ing expression for Cf:

Cf - t f2 0.430 (33)

where

t - [1 + 0.178 N

f - 1 + 0.03916 MW2 t

The Reynolds number, RNL, is based either on component length ora admissible-surface roughness, whichever produces a smaller

value of Reynoldb number, as follows:

16

(RN/ft) • LRNL - minimum (3-4)

RN ~K1 . (L/K)I' 0 4 8 9 (3)

where

RN/ft is determined from standard atmospheric tables oris input.L is the characteristic length of the component.K is the admissible surfa..e roughness and is an inputquantity.

and,

K1 - 37.587 + 4.615M + 2.949M2 + 4.132M3

For mixed laminar-turbulent flow, transition location isspecified for the upper and lower surfaces of the wing. For thelaminar portion of the flow, the Blasius skin-friction relation

Cf - l.328IJ[R-7 (Cf-5Xr EcffiLamiutarI whe-re rcf/cf - (1 +4 r. 1,);~256N 2 .12 4. .,.d pA toth46_ n

tion point. At the transition point, Xr, the laminar momentumthickness is matched by an iterative process to a turbulent-momentum thickness, which begins some fictitious distance, AX,ahead of transition. The skin-friction coefficient for theturbulent part of the flow is calculated from Equation 3-3,where the Reynolds number is calculated from

RNL - (AX + L - Xr)0(RN/ft) (3-6)

i i- 1 univin ef rc T.-4 0-, f-v-•-*v 4 #-4 ̂ .v 4 a 64 .- % ,I 1. ^4 -- 1..

Cf = + L-Xr) (3-7)=CfTurb.(37

Calculated values of Cf versus RN are presented in Figures 2through 7 for mixed laminar-turbuient flow.

3.1.2 Form Factors

The component form factors, FF, account for the increasedskin friction caused by the supervelocities of the flow over

17

to

-- -------------

A J.iJA

7T T ý-f-1 -T

144-

2- 7LI-F T CD

44-

14kAA TýL -

7u 00

4+

H A

cli4'A T

-AA-t I Iff if]-L4 T-Týil IEý-V-I=fi iý7 - -4

7- .............. 00

5 04

4

j4F

4

-+4 CN

14

Lr)eq h-r-4#,4 CN -Cl)-----------00

r-4

-A

fit I

777

cn0 0C) 18C!

C:

- 4 1-r7-- -t7rr

________17__

-71

* .. ~ .3

I cl

1-4 7If77 YJ, v j I

ii7' z<~a TI.-

I%,

I-- 1T

44

- ~~ ~ ~ 7 o --.---1

00

_ _ C4

-4

1--4.. -

--- ~,-- - f-j,~~ -

0 4- ~ -- - 7-J ** *

4 -- . - -. -; C

-4D 4 -4~ - C -- - C- ~ 20

......... .

-.- -14

4___

7 ". 7 7:_

-~ ~~ ...--.

IPIN

Ir-7 -7 - ----.- --

1-I21

:-: rr-, -F=T -T

ý4 Tf-1 71

17;'': .. CýLTt-- -PA - -,- -- -,, -- -,:-,

4+

C14

L

L

-j

7 IT - j-ý ITT-] 006- T-T-

Jý- IR-77 -T

4-

4;2 44 f

F , --------------ff I - 71 Pt- It4

L) + +

E- ILI

gn -4

mv-v

7- 4-144E 4341 co

=V-- 4T- A-........... .......5-A- tf ft

z f If

HC) %0

2-

-4 C-i-4- Aj

A

CA C149- -4

so ZO

IT. - co-4 4 10-P, T

L 7 Y-1 T I - --4-411 r.+-;4--4 Itili - - - -------- 04 Nt3- IL ki

-7- 4

2-

C*4

4-ý =t:tA

Ln V) C14C) Cý 4-4

22

- - - - - - - - -0

t-4:1 t-H4

it -1

- - - -

11 it II I [I II

Ik

0 4 423

the body or surface and the boundary-layer separation at thetrailing edge. The form factor for the "body" component iscomputed as

FF - I + 60/FR3 + 0.0025 FR (3-8)

where

Component Length

vWidth x Height

For "nacelle" components, the form factor is given by

FF 1 I + 0.35/FR (3-9)

Equations 3-8 and 3-9 were obtained from the Convair AerospaceHandbook (Reference 4) and also appear in the DATCOM (Reference 2).

The airfoil form factors depend upon airfoil type and stream-

wise thickness ratio. For 6-series airfoils, the form factor isgiven by

FF I + 1.44(t/c) + 2(tic)2 (3-10)

For 4-digit airfoils, the form factor is given by, 2

FF I + 1.68(t/c) + 3(t/c) (3-11)

For biconvex airfoils, the form factor is given by

FF - 1 + 1.2(t/c) + 100(t/c) (3-12)

And for supercritical airfoils, the form factor is given by

FF - 1 + KlCld + 1.44(t/c) + 2(t/c)2 (3-13)

The factor KICId in Equation 3-13 is an empirical relationshipwhich shifts the 6-series form-factor equation to account forthe increased supervelocities caused by the supercritical-section design camber CId. The factor KI (derived f :om experi-mental data) is shown plotted in Figure 8 as a function of theMach number relative to the wing Mach critical. Equations 3-10and 3-11 were obtained from informal discussions with NASA/LRCpersonnel; Equation 3-12 appears in both the DATCOM and theConvair Handbook.

24

00

'.4*0413

C14)

r,'-4

cc1*

.9.4

. -".4

25)

3.1.3 Interference Factors

The component interference factors, IF, account for themutual interference between components. For the fuselage,theinterference factor is given by

IF - RW_ B (3-14)

where RW.B is shown plotted in Figure 9 as a function of fuse-lage Reynolds number and Mach. For other bodies such as stores,canopies, landing gear fairings,and engine nacelles, the inter-ference factor would be an input factor based on experimentalexperiences with similar configurations. The Convair AerospaceHandbook (Reference 4) recommends using

IF - 1.0 for nacelles and stores mounted out ofthe local velocity field of the wing

IF - 1.25 for stores mounted symmetrically onthe wing tip

IF 1.3 for nacelles and stores if mounted inmoderate proximity of the wing

IF - 1.5 for nacelles and stores mounted flushto the wing or fuselage.

The interference factor for the main wing is computed as

IFRLS R (3-15)

where RW.B is the wing-body interference factor presented inEquation 3-14, and RLS is the lifting surface interferencefctor pre......i. Fi4gure 1, F'nr supercritical wings thewing interference factor is set equal to one. Other airfoilsurfaces such as horizontal or vertical tails use an interfer-ence factor determined by

IF - RLS *Hf (3-16)

where Hf is the hinge factor obtained from input (use Hf - 1.0for an all movable surface, 1.1 if the surface has a flap forcontrol). The factors RW.B and RLS are plotted in Reference 3and also appear in the DATCOM.

26

L,

V I

00oYUU

v.4

I;Q

v-4

000

r-f (n

27-

J 4 - "-

1.2

RL

1.0 -12

.8 - - ,..------

l~-oo - - - i--

.6.4 .5 .6 .7 .8 .9 1.0

COS A(•c) ma

Figure 10 Lifting Surface Correlation Factor ForSubsonic Minimum Drag

28

3.2 Camber Drag

The minimum drag contribution of the wing twist and camberis related to the lift coefficient of the polar displacement,

ACL, by the equation

CDCAMBER e e K&C L 2 (3-17)

This increment is called camber drag and represents a drag incre-DMI Te spn eficincre

ment between minimum profile drag and CDMIN" The span efficiencyvalue, e, is related to the induced drag factor, K, by the equa-tion

Ie = •.RV r ARK

If, for some reason, e -> 1, an alternate equation, obtained fromReference 4, is used:

2.SEXPOSEDCDcM~BER - 0.7(ACL) S (3-18)SREF

3.3 Base Drag

Data presented in Reference 5 were used to establish equa-tions from which the base drag of bodies could be determined.The trends of these data show three different phases: (1) agradual rise of CDBase at transonic speeds up to M - 1, (2) arelatively constant drag level supersonically up to about M -1.8, and (3) a steadily decreasing value of drag above M - 1.8.The resulting empirical equation-s S.r. Cien as

(0.1 + 0.1222M8) Bas ý%SRef

CDBa = 0. 2 2 2 2 SBase/SRefl.0 <M < 1.8 (3-19)

1. 4 2 SBase/SRef)/(3.15 + M2 ), M >/ 1.8

29

3.4 Wave-Dra

Supersonic wave drag is determined on the basis of a compo-nent buildup for which simplified shapes are assumed. Threebasic simplified shapes are used to represent the airplane:bodies, .nacelles, and wings. The component buildup assumesthat the total wave drag is the sum of the isolated wave dragof each component and does not allow for the mutual interfer-ence between components. However, the component-buildup methoddoes give wave-drag results comparable to average configurationswhich have some degree of favorable and unfavorable interference.

3.4.1 Wing Wave Drag

The technique used to estimate wing wave drag evolved froma method that applies transonic similarity theory to straightwings. Data correlations at Mach I were performed on a largenumber of unswept wing configurations with blunt and sharpleading-edge airfoils. For the Large Aircraft program, these

results were represented by an analytical function conmmon toboth types. The equations were then modified for M > 1 to pro-duce a peak value at low supersonic . -e-As and then to decreaseat high Mach numbers to values predicted by straight-wing lineartheory for equivalent two-dimensional configurations. Finally,sweep effects were included. The resulting semi-empirical equa-tions are presented below:

2KtKwKcKb

CD 3

Kb' K -(F r l/Ak ....) + @ .41L1+(l+ 1-,- 3#e,

+ -KKwKcK 3 (3-20)

fiKb w 3 .8 ( F 0)m + AR 2/ie 3 l+Koq' 5 + _ _1 +A) FO 8 1+3a ]

where

CD - CD/(t/c)

Kt - airfoil thickness distribution factor

-2 c 2 4 t +)

30c

Kb, Kw, airfoil factors

Kb - 1.0, Kw - 1.2 for double-wedge sectionsKb - 1.069, Kw - 1.0 for curved-type sections

Xt - location of section maximum y ordinate

ro - section leading-edge radius

Kc - airfoil camber factor

1 + 2-(hlt)22

h - section camber (maximum y ordinate)+1/2 2

doSALE + -A-7 (tan2LE - tan 2ATE)Kp = - ... 1 2

1 + (;)--2(tanALE + tanATE)

(1+2r9)FO - 0.3 + 0.7 Kp

- 81/(t/c) 1 / 3 = v/i-7/(t/c)1/3

1+ X2 (2-X) 3

2 (Ulim/$) at > >

1+ X2(2-X) 1 at2 _

Z = cOSALE +LcOSATE

anid whr LAR : I mhtwnAei t valueoand where ARe is the strait-wing AR having the me ue of

CD/(t/c)5/3 at M = 1.0 as AR, where A - AR(t/c)

The value of ARe is determinqd by solving the followingequation by use of an iterative method:

1_ _ __ _ 2_ __ _ __3+__3_332 _3 3.33 K

S+A~e3 2__ + 1 +1 AR3 2_

AI) -33)AeAR AR3

31

The term im represents the approximate value of t8 ac whichCD/(tic)5 / 3 willf maximize, provided the body is essentiallycylindrical where the wing is attached. If the body is aria-ruled, the peak value of CD/(t/c)573 may or may not be closely

approximated.

3.4.2 Body Wave Drag

The fuselage body wave drag is computed by dividing thebody into two parts, consisting of a simplified pointed nose anda simplified boattail. That is,

CDWoY CDpN AMAX + CDpBT AMAXE (3-21)BOY N SREF SREF

Nosewave drag, CDPN, is determined from Linnell's empiricalequation

1.2 + 1.15( 'f2l1)(fm 2 +1) CDP. 1 + (3-22)I + 19pv

for the supersonic wave drag of parabolic noses (Reference 6).For Mach numbers between 1.2 and 1.0, the nose wave drag isdetermined from the curves of Figure 11, which were derived fromthe transonic drag rise of ogive noses, as presented in FigureIII.B.10-9 of Reference 4, and using Equation 3-22 as a supersoniclimit. The nose fineness ratio, fN, is calculated from the noselength. fN, and the maximum cross-sectional area, AMAX, as

)HBoattail wave drag, CDpBT, is determined as a function of

boattail fineness ratio (fB,.base diameter to maximum diameter(dB/d), and Mach number. This is done by computing CDPBT atfive values of (dB/d) and interpolating to the desired value.The general form of these equations is given below:

32A ___ _ _ _ _

. IN

MACHVE FRO. FI .. Il.B 0-7 Rf

-1

0 & FI4. 8, Ref._6 12

'N S I

.Figure i~Transonic Wave Drag o aaoi oe

For I/fN 1 and dB/d - dB(I

For d

CDPBT(l) + (I)./fB+A2(1).(•/fB) 2 +A3(I) ( 3ifB)3fB2

(3-23)For fN >

CDPBT(I) f2 A(l)(fN) (3-24)

The polynominal coefficients of Equations 3-23 and 3- .'determined from a leasz-square fit of Fig. III.B.10-9of Refer-ence 4 for ogive boattails, are tabulated below:

IdB/d Ao Al A2 A3 A4

I 0 1.165 -0.5112 -0.5372 0.3964 0.5132 0.4 1.067 -1.709 1.6632 -0.686 0.33523 0.6 0.7346 -1.4618 1.5795 -0.6542 0.198

-- - - ------ -

5 1.0 0 0 0 0 0

3.4.3 Nacelle Wave Drag

The nacelle wave drag is calculated by a method simiiar to

that used for the fuselage:

'CNac•. + CD-) (3-25)"CDrNac (CDoN + DBT -REF

The equation used to calculate CDN for open-nose bodies is

CDoN I [(i- rIN/R)/fNJ' 5 /] (3-26)

where

• INLET 2 AIN T

R ? A/r

This equation is a curve fit of Figure III.B.10-6 of Referenpe 4.

34

I.

3.5 Fuselage Aft-End Up.weep Drag

The main parameters affecting the fuselage aft-end upsweepdrag are the upsweep angle and the crossflow drag coefficientof the rear fuselage sections in the local flow, including modi-fication by wing downwash. Data in Reference 7 indicate that theafterbody drag increases rapidly with upsweep angle 1, but decreaseswith increasing fuselage angle of attack. The curves in Figure 12,obtained from the data presented in Reference 7, are used to predictaft-end drag as a function of angle of attack.

3.6 Miscellaneous Ddpam Items

In the preliminary design stage of aircraft drag estimaticn,the drag due to surface irregularities such as gaps and mismatches,

fasteners, small protuberances,and leakage due to pressurizationare estimated by adding e miscellaneous drag increment which wouldbe some percentage of the total friction, form, and interferencedrags. The miscellaneous drag varies between 10 and 20 percent ofthe total friction, formand interference drags for typical air-craft. The Large Aircraft program computes miscellaneous drag byuse otapercentage factor specified as input to the program.

35

I I I.

00

-1-CEl)

000

4-41 '1-

.1 - C4

00-r4

Aauao No aav)0(j

~~36

4. DRAG WIE TO LIFT

The Large Aircraft program predicts drag due to lift by oneof several methods, depending on the aerodynamic conditions atwhich a solution is desired. The various regions are illustratedin Figure 13; they are discussed in the following subsections inthe numerical order shown in the figure.

DB , 6

7

CL6

55•

"ICR 2

1 3 4

MCRo MLU ML2

MACH N. MBER

Figure 13 Lift and Speed Regions for Calculation ofDrag Due to Lift

4.1 Subsonic Polar Prediction below Polar Break

Region 1 is bounded by the critical Mach number and by theCL at which the polar break occurs (CL ) In this subsonic,low-lift region, the drag due to liLt 91n'be determined from

4. 37

I-'

CDL K(CL -.CL)2 (4-1)

where the drag due to lift factor, K, is predicted from

K s- -R + R (4-2)CLa, WAR eo

In this equation, a leading-edge suction parameter R, is used torelate K to the lower bound of drag, l/2TAR, for full leading-edge suction (R-1.0) and to the upper bound of drag, I/CLa, forzero leading-edge suction. Body effects are accounted for inEquation 4-2 by computing eo, shown plotted in Figure 14 as afunction of taper ratio and body-diameter-to-span ratio (d/b).

The correlation of leading-edge suction on induced drag wasfirst developed by Frost (Reference 8) and was later extendedfor additional planform effects and high•'r subsonic Mach numbers

.(Reference 9). A study by NASA (Reference 10) showed that air-foil camber, conical camber, sharp leading edges, leading-edgeflaps, Reynolds number, and sweep have significant effects onthe suction parameter. H. John (Reference 11) improved thecorrelation of R for plane wings at low Reynolds number by in-cluding airfoil thickness along with ldng-edge radius.

The procedure followed in the Large Aircraft program todetermine R is as follows:

1. Using the leading-edge radius and the leading-edgesweep for each wing panel, determine fl from theequation

RNLER xO3 cotALE ;l-M 2 cOsAlE; ALE ) 200

.q = (4-3)

RNLER X10 (5 -6 5 1 1tLE) ýl.M2COs7LLE; ALE <(200

(ALE in radians)

The switch from the cotangent term is made to preventflfrom going to infinity as sweep approaches zero.

The value of fl is then used to read RT from Figure15, which is a plot of leading-edge suction for thin,round-nose, uncambered airfoils developed in Refer-ence 9.

38

j

di

" - -- - - _ -"F _. , -I,-.....-• . •

rj =- ==.-=1 i - I -. . ' ' . : , , . " • •

I J - •' . •, •*'*** = .1 .." "

4 " • ' I . I , , .,i • ' , i ' i . i ' I , " " , -, '. • . ! . I - .I ,

I-..7-

i .. . . . .. .- .

[ .. 1" " 0 -. "f -- -.

oe:, . . . . . ..

l ~~~~0 25'; :"

1.8

.47

I . 5, i: ' '

S0 . -. . 2 3 4 • ' 5

BODY DIAMEThR TO SPAN RATIO

".•- : .. . .... .. 7T .,- .~ 7 ]

Fiur 1 BdyEffects on Wing Span Efficiency! i eol T FT. .7T

-4

•....~ . . .

r4.

00

0

0-4#t-l

r-4 4-

W-41

'-.4

'-.4W 00

40- -dC>/

2. Using the leading-edge sweep of the wing panel,determine RMN from Figure 16. The plot ofFigure 16 was obtained from the results reportedin Reference 10 for sharp-leading-edge wings. Theleading-edge-suction value for sharp-leading-edgewings is independent of Reynolds and Mac4 numbers.

3. If the value of RT determined from step (1) is lessthan the value of RMoN determined from step (2), setRT equal to RMIN-

4. Using the wing panel thickness and leading-edgeradius Reynolds number, determine a thickness cor-rection to leading-edge suction, RT, from Figure17. Figure 17 was determined from the data presentedin Reference 11. The increment in suction parameter

ART it then added to the value of RT determined fromstep (3).

5. Determine the effect of either section camber orconical wing camber on the R factor from

R4 - RT + (0. 8 2 4-RT)-(Cr w - /. (4- 4). AA X 'Id'-Lcon'

if Ri > 0.874, Rt - 0.874.

Correlation with experimentai data and the resultsof Reference 10 indicate that R does not decreaseas much for low Reynolds number when the wing iscambered compared to an uncambered wing.

The accompanying sketch CAMBEREDshows the relative effect R AI

that Equation 4-4 willpredict for camberedwings.

/ WING

RNLER

6. Obtain the effective R for the composite wingfrom a span-weighted average of the individualRi for each panel as follows:

41

0.44

0.21

0. 1 a i- - 6

10 20E0G4.5 60 70 80 f

A LE DEG

Figure 16 Leading-Edge Suction for Sharp Airfoils

0.3 _ _'

. -T -4

1 1 -j10.1 IIr

t/c 0,6

.!7 I

RNLER

Figure 17 Effect of Reynolds Number and Thickness on

the Leading-Edge Suction Factorj 42

N

Ri Y(i+l) - YW(i)- R(b/2 - d/2)i-il

where YW(i+l) - YW(i) is *the width of the ith panel.

7. Using Figure 18 and the parameter AR •/cosALE, calcu-late the effect of wing planform on R. The increment ARdue to planform is then added to IT, determined from step(6), to obtain the final R value used in Equation 4-2.

The polar displacement, ACL, is related to the lift coeffi-cient for minimum profile drag, CLoPT, by the equation

( AR K)CLOPT

The lift coefficient for minimum profile drag is related to thecamber, twist, and asymmetry of the configuration. Figures 19and 20 present data (Reference 13) for the effect of NACA camberand coni-. -cal ca 'ern C0-F u .LtLalU WL418b Thelimited amount of data avIalable for correlation indicate that

0.75CLoPT = 0. 5 19 5 (CLd) (4-5)

where CLd is the wing section design lift coefficient.

4.2 Supersonic Polar Prediction below Polar Break

The drag polar in the supersonic region beyond the secondlimit Mach number below polar break (Region 2) is predicted byEquation 4-1, where

CK = - e+ (4-6)

This equation is similar to Equation 4-2 for the subsonic induced-

drag factor except for the use of R, which is a transonic leading-edge suction factor. For Mach numbers greater than Mach critical,the suction factor is predicted from

R Ro/(l + n4M + (nAM) 2 ) (4-7)

43

IV;

.08

AR

..o6

.o4

.02

00 2 46 8 10

AR ACSALE

Figure 18 Effect of Winrg Planforto on Lead ng-EdgeSuction

44

4'4

Ref. 13

.6

.4

CLOPT

.2

0f

0 .01 .02 03 .04 .. 0 . .06f/c

Figure 19 Lift Coefficient for Minimum Profile DragNACA Camber

DESIGN MACH - 1,00.8 b/2 RAY LINE.6

Ref. 13

.4CLoPT

.. 2

00 .I .2 .3 .4 .5 .6

Conical Camber -

Figure 20 Lift Coefficient for Minimum P.otile Drag -

Conical Camber

45

where Ro - leading-edge auction factor at the criticalMach number at zero lift, MA°

n - 12(coSALE) 1 . 6

AM- M- cMCR

The variation of (R/Ro) for Mach numbers greater than MCR° isshown in Figure 21. This method of predicting polar shape factorproduces a continuous decrease in leading-edge suction so thatin the limit as the Mach number approaches the sonic leading-edgecondition,the polar shape approaches (1/CL.).

The supersonic polar displacement for NACA camber is calcu-

lated from

KCLd(O.25-0.225ýcOtALE); ýcotALE 4 111

1CL = (4-8)0o; tcotAL ý 1.11

and the supersonic polar displacement for conical camber iscalculated from

CLcon(O'lll'O'I~cOtALE); ecotA[LE Z/ 1.11

ACL - (4-9)

These equations were obtained from a simple curve fit of thedata presented in Figures 94 through 97 of Reference 13.

4.3 Transonic Polar Prediction

In the transonic region bounded by Mach critical (MCR) andthe first limit Mach number (MLI) (Region 3), the induced dragis computed by adding drag rise to the basic polar:

CDL K(C - CL)2 + CDRCL (4-10)

The basic polar shape is calculated up to MCim in the same manneras described for Region 1. Beyond MCRo, the 6asic polar does notchange. An incremental drag-rise term (CDRCL) is calculated as a

46

t~d Iit4

rij

'-I~~ 7. ~1 77I't ...

2.1 I~lj-4

1 1771 T

Ir-4

1-H1- i!IL~ 1tI2. YTI It fl dý

''447,t~ 7

L:i C1

t 47E1w

function of lift and is added to the basic polar to determinethe total drag due to lift. A complete description of thetechniques used to determine the drag-rise increment is given

in Subsection 5.3.

In the transoniz region above the first limit Mach number,

MLI (Region 4), the drag-rise term in Equation 4-10 becomes lessaccurate. Therefore, the drag polar in Region 4 is calculated

by interpolation between the polar shape factors calculated inRegions 3 and 2. The equations for K and ,&CL are given by i

M-MLI

K= KI+ (K1 2 - KLI) ML2-MLI (4-1)

A CL ACL + (ACL (4-12)

where ML1 < H < 2 The polar shape factors, KLl and CLLI,are determined from a least-square curve fit of the polar shapecomputed by Equation 4-10 at MLI. The polar shape factors KL2and CLL2 are computed from Equations 4-6 and 4-8 at ML2.Me Liimit Mach numbers are determined from

MEL = M'c + 0.05

If M-Ll < 0.95, then MLI 0.95

If MLI ;> 1.00, then MI 1.0

ML2 = MLI + 0.15

1 4.4 Subsonic Polar Prediction above Polar Break

The polar region between the polai-b.beak lift coefficient

(CLPB) and the initial-stall lift coefficient (CLDB) is theregion in which leading-edge separation and reattachment occurs,causing the polar to deviate from a parabolic shape (Region 5in Figure 13). Whet-her or not this region exists (i.e., the flowreattaches after separation and allows the wing to reach a higherlift coefficient before final separation occurs at the trailingedge) is de~ermine3 by the type of airfoil, the Reynolds number, andthe leading-edge wini sweep. For thin wings, lnw Reywolds numbers,or highly 3wept wings, the values of CLp and CLDB are equal.

48

4: l l n lm l m

Leading-edge sharpness is a measure of the type of separationlikely to occur. Blunt, thick airfoils generally exhibit trailing-edge separation, wbile very thin airfoils exhibit leading-edgeseparation. Airfoils of moderate thickness are likely to separateand reattach at the leading edge, followed by trailing-edge sepa-ration (stall) at higher lift coefficients. Associated with theleading-edge separation and reattachment is a loss in leading-edgesuction, which produces an increase in drag due to lift. AboveCLDB, the flow separates completely along the wing and the dragincreases more rapidly.

The prediction method in the Large Aircraft program utilizesthe sharpness parameter of the airfoil, Ay, as defined in Equation2-28. IfAy is less than or: equal to 1.65, leading-edge flowseparation is assumed to occur. Also, if the leading-edge sweepis greater than or equal to 50 degrees, it is assumed that lead-ing-edge separation occurs. If Ay is greater than 2.05, a lead-ing-edgc separation and reattachment occurs, followed by a trail-ing-edge separation. For values of &y between 1.65 and 2.05, atransition region exists in which the behavior varies betweenthe condition of full leading-edge separation at Ay = 1.65 andfu1l leading-edge flow reattachment at Ay - 2.05.

Beccuse the polar-break lift coefficient is a function ofmany variable--z, it has proved to be a difficult quantity topredict. Data correlagions at subsonic and transonic speedsperformed during the development of the Large Aircraft programindicate that the polar-break point can better be determinedwith angle of attack as the parameter rather than lift coeffi-cient. These correlations resulted in a method that determinesthe angle of attack at poier break as a function of Mach number,Ay, sweep angle, and wing camber. Consequently, the polar break,CL, is calcu].ated as

CLpB CL " (CXPBO + A&PB) (4-13)

where (aPBO/cOAc/4) is shown plotted in Figure 22 as a functionof Ay and M cOsAc/ 4 . The term &aPB accounts for section camberani is deteruined from

&I A idA0PB = (12.05-4.1 M cosAc/4)( Acosc/4)

which was derived principally from correlating the experimentaldata in Reference 1U. lor wings with conical camber, an incre-ment in CLFB is obtained from Figure 23.

49

000

Q' cr () L ) 0.0 r- P

AA7J

U,1

4-I

Q)

C4.

*00

'4-4 -

E 00

500

The drag due to lift is expressed in Region 5 as

CDL K(CL-ACL) 2 + K' (CL-CLpB) 2 (4-14)

CLPB < CL < CLDB

where K' 0.5!8/f.7 (Reference 1).

The upper boui.dary of Region 5 represents the lift coeffi-cient at which trailing-edge separation occurs. It is predictedas

CLDB CLpB I+ f RN-ý (E-i) - 6 CL]ORN

where

0 ; Ay •< 1. 65

T- (Ay 1.65)/0.4; 1. 6 5 <&y< 2.05

1.0 Ay > 2.05

and (8CL/ORN) and 6CL are shown plotted in Figures 24 and 25.

The drag polar above CLDB (Region 6 in Figure 13) increasessharply from the subsonic attached-flow condition. The polarprediction for lift coefficients above CLDB is determined by amodification of the empirical method presented in Reference 13whereby

2CDDB + KDCL + ACDB (4-15)

CL > CL.,B1

where CDDB is the predicted lift dependent portion of profile

drag at CLDB, KDCL 2 is the theoretical induced drag, and ACDB

is a correlated separation drag increment obtained as a functionof CLDB- The lift dependent profile drag at CLDB is given by

"KKuCC 2 . 2 (4-16)CDDB K(CLDBn-CL)+'(CLDB-CLpB) -KDCLDB

where the theoretical induced drag factor, KD, is predicted from

KD '(4-17)

where e' is a modification of the clabsical theoretical

51

I

T f

Derived from Fig. 71, Ref. 13

M cosAc/4

.5 0.5

4 0.6

ACLpB 0. 7'COSZAcI4 3

.2

i .1

0 .1 .,2 .3 .4 -. 5 -

CONICAL CAMBER .CLcoN

Figure 23 Conical Camber Effect on CLpB

52

.16"

NOTE: CLDB is assumed to beconstant f rRc •>, 4xlO?

.12-09CL~ 10- 6

ORNRef. I..08

.04

0 L

0 .2 .4 .6 .8 1.0MACH NUMBER

Figuxe 24 Reynolds Number Effect on CLDB

* 1 . . .

.12

.086CL •,•ii:

S04

0(1, .2 .4 .6 .8 1.0

MACH NUMBER

Figure 25 Correl.a.tion Lift Coefficient for CLDB

I, 53

drag-due-to-lift factor I/7rAR to account for nonelliptical span

loadings and body effects. The factor e' is calculated from

el es [1 - (dlb)2] (4-18)

where the wing planform efficiency factor e'w is as plotted inFigure 26 as a function of taper ratio, sweep, and aspect ratio.The data of Figure 26 were obtained from a Weissinger lifting-line solution presented in Reference 13.

The drag above CLDp represents the separation drag componentwhen major separation e fects are present. Simon et al. (Refer-ence 13) measured this drag relative to the profile drag atthe drag-break lift coefficient and present correlated curvesOf LACDB versus CL, CLB, and Mach number. The ACDDB data werecurve fitted and result in the equati.on

BDB + B CLLDB CL> CLDB (4-19)

The factor KB is shown plotted in Figure 27. The program doesnot vary KB with Mach number since the drag-rise term is includedin the polar buildup above CLDB-

For convencional wings the polar predicted by Equation -4-15is continued uF through CLMAX. For low-aspect-ratio or crankedwings (see Secticn 6) that develop vortex lift at the higherHit coefficients, the zero-suction drag polar predicted by

CDL = CL tan a (4-20)

is compared against the drag polar predicted by Equation 4-15and the drag due to lift is set equal to whichever is lowest.This produces a drag po .-lar as shown. in, te Sketch below. TheCL-a variation for vortex lift is predicted by the methodsgiven in Section 6.

54

0.96,

0.88

0Taper Ratio 0.250

ii 1.00

~ 0.92'~ '.Taper Ratio 0

0,ii j c-lI I 1 I t

0.926

.0 10 2 . 0 5060 .00

555

- -- - - --

M -0..6

1.6 Derived from Fig. 81, Ref. 13

".•o 1.2

0 •

p0.8

o.4

0.

2 3 .4 5 6 .8

CLDB

1.3

S~1.21.2 NASA Data, Ref. 12

KB(KB) 6xI0 6

1Ai

MCAIR Data, Ref. 13

1.00 1 2 3 4 5 6 7

REYNOLDS NUMBER RN-C

Figure 27 Separation Drag Factor

56

CL tan

CLDB K 4 - -B

CL

.. .

I °CD

4.5 Supersonic Polar Prediction Above Polar Break

Supersonic Region 7 above polar break is predicted by thesemi-empirical method developed in Reference 13. The equationfor predicting supersonic drag due to lift above polar break isgiven by

SCDL. (K-K')(CLP(-rCL)+K'(

where K and ACL are the polar parameters in the low-lift regiondiscussed in Subsection 4.2.

1The polar-break lift coefficient, CLPB, is correlated as afunction of sweep, aspect ratio, camber, etc. in Reference 13.A curve fit of the data in Reference 13 results in the following

* ~equations forCLB

CLPB CLS 1+l.25(CLS9 +aCLS9 -CLSI)(PcOtJLE-.I)+0.5CId (4-22)

* where the factors CLS1, CLS 9 , and LCLS9 are shown plotted in Figure28 as a function of aspect ratio.

57

.8

CLS 1

.7

.6 CLS 9

CLS9

.5

.4

.3

20 30 40 50 60 70ALE DEC.

.1

&CLS 9

-.2

0 1 2 3 4 5 6AR

Figure 28 Supersonic Polar Break Lift Factors

58

The polar shape factor, K', above CLpB is computed inReference 13 as

CK' ,H (4-23)CL~

where

1.1; ARtanALE 3.5

H - (4-24)1.I + 0.1(ARtanA•LE - 3.5); AR tanj4LE >3.5

59 N

.3

5. CRITICAL MACH NUMBERAND DRAG RISE

The drag-divergence Mach nuniber or Mach critical, is definedas the Mach number at which a rapid drag rise intercepts the sub-sonic trend in drag. The British method of predicting the criticalMach number for two-dimensional airfoils (Reference l11 appears tobe the most accurate empirical method available. The Britishmethod uses the Sinnott "crest criteria",where the low-speedpressure at the airfoil crest is related to the drag-divergenceMach number.

The Large Aircraft program uses a method analogous to theBritish two-dimensional critical Mz.ch number prediction procedurein order to predict the critical Kich number for a finite-aspect-ratio swept wing. The critical Mach number is defined as thevalue oF freestream Mach number which produces a local supersonicflow measured normal to the sweep of the isobar at the crest.The local Mach number normal to the crest isobar is taken to be1.02 for conventional airfoils and 1.05 for supercritical airfoilsin order to define freestream critical Mach number. The soniccondition at the crest can be predicted by means of a simpleeauation in which the incompressible pressure at the crest ofthe airfoil and compressibility factors are used.

The value of 1.02 local Mach number for the weak shock atcrest condition for drag rise used in Sinnort's transonic air-foil theory (Reference 15) was established empirically. Refer-ence 14 shows that this method should predict MCR to within+0.015 for the majority of conventional two-dimensional airfoils.However, as shown in Reference 14, with "peaky" airfoils (as inthe supercritical airfoil) the onset of rapid drag rise may beaelayad ,,ntil the shock is substantially downstream of the crest.The predicted value of .MCR based on a local Mach of 1.02 at thecrest may thus be conservative by more than 0.02, and a localMach of 1.05 is necessary to achieve good correlation.

The following subsections discuss the methods used topredict the pressure distribution around an airfoil and todetermine MCR from the pressure at the crest along with themethod used to estimate the drag rise above MCp.

60

-.. . . .. . - .. i

5.1 Pressure Coefficient Calculations

The incompressible, inviscid pressure distribution aroundthe airfoil is first defined, from which the pressure at thecrest can be determined. The method of Weber (Reference 16)was used for the pressure distribution calculations. Thismethod requires the airfoil surface coordinates to be deter-mined at the chordwise locations defined by

X() - I(i + cos -W) where 0- ) -1. N (5-1)

N may be any integer, but, in this program, N is set equal to 32.

The Weber formula is essentially a second-order lineartheory whereby the pressures are determined from multiplicationof the matrix of thickness and camber ordinates of the airfoilby a matrix of constants given in Reference 16.

The formula for the incompressible pressure distributionon an infinite sheared wing was obtained from the incompressibleform of Equation 93 in Reference 16, resulting in

FI - Cp 2 1 1

i+ S Ms-

x(tcosyL [I + S( 1 )(X)cosA. 4- S(4)(X)cos

+ sin (Y COS_.l + 211

-cos [S(l)(X)sinA S(•)• sinA j

+ sin cv siAIA + S5)- -X( )+~ ~~t sicj 0o se i

+ sin2.ACos 2o l - S(2)( 1 (5))x2 (5-2)

61osA

61

For p orn the uppex surface the + is used, and for the lower-surface Cp the - is used. Also,

N-I

S(() Z S (2 ) ZA

N-Is(2(x) s(

S() N.-i (3), Z +S(3)s(")(x Z t' -+' SN.) VP2s(4((x) + ss(2,,v'Tsc

u(4 )(-I N-i

-N-i (5)

Cfr -I- - - Si.t

Tables of the S~ t) matrices of constants are given in Reference16 Zt. and Z., are the thickness and camber distributionat the control pt•nt,4, given by

zt. = + Yt

and

Zs, •t(yu - yzL )

where y, and yz are the upper-and lower-surface ordinatesdefined at the control point liL given by

X(/) (+ Cos* N

62

5.2 Critical Mach Number Calculation fromCre tline Pressure

The procedure followed to determine MCR for swept wings issimilar to the procedure outlined in Reference 14 to predictthe MCR for two-dimensional sections. In the procedure, Equa-tion 5-2 is used to compute the pressure distribution around

the airfoil at a sweep angle determined from

arcos(cosAc/ 2)n; Ac/2 < 400

arcos [(c2SAl 2 ) f+'76604n]';Ac. 2 > 400 (5-3)

whereAc/2 is the wing mid-chord sweep at the semi-span of thewing and the factor n is determined from

ARn 7 i+AR

The sweep angle A represents an effective isobar sweep at themid-span region of the wing as affected by the root and tipregions of the wing. The procedure used to determine NCRbased on the crest pressures is as follows:

1. Determine a chordwise incompressible, inviscidpressure distribution for an angle of incidence(a). integrate the pressure distribution tonhtain f-hp lift- r'nffie-iPnt (-r..

2. Determine the chordwise position of the crestfor each x, the crest being defined as thepoint at which the airfoil surface is tangentialto the undisturbed freestream direction (9 =a).

3. Determine the incompressible pressure coeffi-cient at the crest (CPcrest).

4. Use Cpcrest to determine MCR from the relation

CPcrest P/ P/L) ( 1+0. 2MCR2COS2A) 3 -(4)

S1C2/ lN~ 2COSA0. 7MCF 2/]-.,t•osA

where (P/Pt) is the ratio of local static pres-sure to freestream stagnation pressure asdetermined from

63

-/P L + O. 2M12] 3.

where HI i- the local Mach number normal, to theisobar sweepA Aat the crest of the airfoil. MHis set equal to 1.02 for conventional airfoilsand 1.05 for "peaky" or supercritical airfoils.Equation 5-4 uses a Prandtl-Glauert compressi-bility factor to correct the incompressiblepressure coefficient for Mach number ratherthan the Karman-Tsien factor used in Reference14. References 17 and 18 recommend using thePraadtl-Clauert factor instead of the Karman-Tsien factor in the M•{ prediction method forhighly cambered airfoils or general airfoils athigh-lift coefficients. The relationship deter-mined by Equation 5-4 is plotted in Figure 29.

5. Use the Prandtl-Gleuert compressibility factor(4D) evaluated at NCR to obtain the lift coef-.ficient CLD from

CLD - CLi/#D

6. Repeat Steps I through 5 for a set of incidencesin order to obtaitn a drag-rise boundary from theset of points (CLD, MCR).

Th ..... -c... ,•1, iuu~tmb predicted by the above six stepsis prevented from exceeding the critical Mach number of thefuselage alone (shown plotted in Figure 30). For aircraftthat are not area-ruled, where the isobars are allowed tounsweep at the wing-fuselage juncture, the method would tendto overpredict MCR when the value approaches the fuselage MCR.Vie prediction-versus-test MCR correlation shown in Figure 31is thus applied for conventional-wing predictions.

5.3 Drag Rise

For Mach numbers less than MCR the drag increases slowlywith increasing Mach number. This drag component is known ascomprestible drag, or drag creep. Methodology for estimatingthis Component of drag for conventional or supercritical wingswas included in the subsonic drag buildup in Subsection 3.1.For Mach numbers greater than MCR, drag rise begins and in.-creases rapidly with Mach. Figure 32 illustrates the dragbookkeeping system followed in the Large Aircraft program

64

S. . 1 6 \ . . . . . . . . . • . . . ... . . ... . . . ... ..

* ~~-1.6 --

: 1.4... -.. ... ........-......... •..............-.. L d .. . .

-1.2 --

-1.0

-0.8.LOCAL MACH

C/= 1.05cos2A \

•-0.6

1.02: \

-U.4j

"-0.2

0 -f... . ! - - - - - -

3.5 1.f 0,7 13 0.9 3.0MCR cosA

Figure 29 Mach Critical Pr `,ýtion Chart

65 ..

1.0

.9 t

.8 7('Mcr) B0 dy.

.6

0 2 4 6 '8 10 12 14FINENESS RATIO, (1/d)Bofdy

Figure 30 Prediction of Fuselage Critical Math Number

66

I'4

00

ClC

CN c

-r4

-r® 0

Cf) 00E-4-

67~

MLI 1.o0 M2

~'!

Ii

CL 0 .3

D T

c,''FRICTION

HCRQ(MACH NUMBER

Figure 32 Transonic Drag Buildup

68

whereby beyond Mach 1.0 the drag rise and the interference plusform drag are replaced by wave drag. The drag rise is separatedinto two components, drag rise due to lifting surfaces and dragrise due to all orher components on the aircraft. The drag risedue to lifting surfaces is represented by

CDRL - PL (M-MCR) 2 (5-5)

where

PL - 25 , (t/c + 2 f/c)-(cosAc/ 2 )3

The factor PL is a function of the wing section thickness, tic,maximum ordinate of the camber, f/c, and the midchord sweep,

Atc/2. The total drag rise, at zei'o lift, due to lifting andnon-lifting components is determined by

CDR= a2(M-MCRo) 2 + a3(M-Mc,-) 3 (5-6)

where d2 and a3 are defined to produce a continuous zero-liftdrag curve between MCpo and Mach 1.0. The drag rise is curve-fitted to begin at MCR with zero slope and end at Macb 1.0,matching the value and slope of the wave-drag cucve. Thecoefficients a2 and a3 are calculated from

3 (CD - CD ) -(1 - MCR )CD/W, F&I o Wi

a9 m (I- MCRo)2

(I - MCRO)GDW1 - 2(CDWI - CDF&I)

3 (1-

where CDwI, CDF&i,and CDwI represent the Mach 1.0 value of thewave drag, form plus interference drag, 4.nd tLe slope of thewave drag, respectively. The non..lifting contribution to thedrag rise is assumed to begin at MCRo and not vary with lift;using Equations 5-5 and 5-6, an equation for total drag riseat any lift condition can be determined from

2 2 3CrYR - PL(M-MCR) + (82-PL)(M-MCRo) + a3(M-MCRo) (5-7)

69

I

The change in MCR with lift causes the subsonic drag polarto increase after MCR (see Figure 32). For bookkeeping, thedrag rise is separated into a minimum drag contribution and acontribution due to lift. The increment to the minimam drag is

determined from Equation 5-6, and the increment at lift isdetermined by

C DRcL CDR CDR° (5-8)

I

U

70|

6. LIFT

The untrimmed lift of an aircraft can be represented bythe equation

CL = CL., (0(- aLo)

For moderate to high aspect ratios and moderate sweeps, thelift equation is linear with @m) so that the lift-curve slope(CLK) is constant. The total lift-curve slope of the aircraftis given by

CL" = (CLwB + (CLT + (CLQB (6-I)

which is the sum of the wing (including body carry-over effects),horizontal tail, and the forward portion of the fuselage.

The following subsections describe the methods by whichCL, andOC1L, are calculated along with the method of calculatingthe lift in the nonlinear range up to CLMAx.

6.1 Wing Lift-Curve Slope

The value of (CL.,)WB is predicted by the use of severalrather involved semi-empirical equations. These equations weredeveloped to predict wing lift-curve slope as a continuous ex-pression in the subsonic, transonic, and supersonic regions.The value of (CLO)WB is expressed as

(CL.)W-B (CL.) Basic* Kt Kb SREF (6-2)

where (CLqL)Basic is the wing-alone CL& with no thickness effects.The factors Kt and Kb account for the effect of airfoil-thicknessplus camber and fuselage interference, respectively. The equationfor (CLg-)Basic was evolved from the Polhamus (Reference 19) equa-tion for trapezoidal wing

,a 0S(! )AR/5 7.3CL= (6-3)

a0 ao0 2 21 ARi•-• ~(L-•)+ (LO)+1- -(McosEtc2 2oA•

c/2 2cosAc/2

71

for subsonic flow and the linear- theory level of

4/57.3CL 4/53 (6-4)

for supersonic flow at the leading edge (14> I/cOSkE).

To extend the Polhamus equation for use with non-trapezo-dalwing planforms, Spencer (Reference 20) replaces cosAc/2 with theeffective cosine mid-chord sweep determined by Equation 2-13.

When Equation 6-3 results were compared with subsonicexperimental data, it was deduced that better agreement wouldbe achieved if the predicted peak CL-K were to occur at M 4 1.0(for moderate and high AR) and if the rate of increase in CL,with increasing M were larger. Consequently, Equation 6-3 wasaltered to

AR/57.3/Basic r.3 ,/3w

1+V1+Ll-((cosAC/ 2)e (•*) "I2 co,. i2 e

(6-5)

where the sectional lift-curve slope ao equals 2kr, and M* isthe limiting M for the application of Equation 6-5. M* is afunction of AR and Ac/2 and is defined by

r 121* " MO* + (i-•M *) l '-(cosAc/2)eJ (6-6)

where

MO* - (10 + 0.91AR3 )/(10 + AR 3

In effect, M* is the Mach number at which the rate of increasein CL. with M begins to decrease. Note that, at M-0, Equations6-3 and 6-5 are identical. However, it was also found desirable(for improved correlation) to limit (cOSc/2)e to the range0.94 >, co5Ac/2 ' 0. Thus, in applying Equation 6-5, sweep anglesof less than 20 degrees are treated as having a value of cosAc/2 I0.94.

72

For thin wings, experimental levels of CL.(characteristicallyreach a peak at speeds somewhat greater than M*. With low sweepand moderate to high AR, the peak occurs below M-I, while with highsweep and/or low AR, the peak may be at or above sonic speeds. Atspeeds well above sonic, CLa then decreases with increasing Mand, when the leading-edge becomes supersonic (M > co-ALE- ), thelevel ap roaches the two-dimensional-theory level of CL, -4/(57.3 ýM2 -I). To emulate these trends, Equation 6-5, was modi-fied by a factor term and an adder term, each to be applied onlyat M > M*. These new terms are included in the modified equa-tion:

1/57.3(CL.)Basic * ; M 14* (6-7)

where CIeo is defined as (CL•)Basic at M1 M* in radians, and

= -M* El + (M*I~M)YJ1+9AR ( 2 - 2/3 rA 23+?rAR

and

Z -+ AR2

3 I (-AR 'AR l)coS.Ac 2/3CL~xo 'Le.oe

Wings having thick airfoils undergo a degradation in CLbeginning at M > M*. The level of CLd< versus M dips, usuallyreaching a minimum at M < 1.0, and then recovers to a secondpeak level at M > 1.0. To account for this phenomenon, the basicCLa equations have been modified by a factor, Kt, as defined by

Kt= 1- L4 - 6 2j "I ; M2 M3 (6-8)

1.0 MI M >, M3

where V' ,7 2, MI, M2, M3 are as defined by the equationsbelow:

73

V9A•(t/c) /(cos.A c• e

where

Atc) - t/c - (tic)xim

1(t/c),rim 312 (6-9)4. 4 AR(coSAc/2)e

The equation for 1 is applicable for

R t/c)(coscl2) 0.0

e

0f A RA-,,t/c•^ 0. 1

if A0 07, use 0.07e

If AR A(t/c)/(cos.Aci2) > 0.10, use 0.10e

M-HI

(M2 -M1)e

M-M2

Mi = 1-2(t/c)(cos 4/2) + (Cd)- (6-10)4+AR

M2 - M1 + t/c

Note: 0 •• MI <M*

If M1 )7 M*, use '41 M*

M3 - 1.0 + t/c

The derivation of Equation 6-8 is based on the data trends andanalyses of Reference 21 and on other limited data (e.g., Refer-ences 22 and 23). It should be noted that a CLQ< "bucket" is

74

predicted only if the wing streamwise airfoil t/c ez-m Žeds thelimit thickness defined by Equation 6-9, The limit-thicknessboundary was established from the statistical boundaries pre-sented in Reference 24.

Another factor in the wing CLaprediction equation (Eq. 6-1),

is the fuselage interference factor, defined by

Kb = (I + d/b)(l - d/b)f (6-11)

where

16+3AR2f2

8+5AR2

b wing total spand = body total width at wing junction

The factor Kb accounts for the change in wing lift due to thebody segment which enshrouds the wing and to the wing-inducedlift on that body segment. Based on semi-empirical derivationspresented in Reference 25, Kb is independent (to the first order)of 11. It is noted thaLe tUtal lift of a wing/body configura-tion is derived by adding the body-alone lift to that obtainedfor the wing-alone as modified by the factor Kb-

Application of Equations 6-2, 6-5, 6-7, 6-8, and 6-11,for the prediction of (CL,)WB will yield trends as sketchedin Figure 33. It is noted that the technique is stronglydependent on the value of (tic) in the transonic speed regimeif (tic) exceeds the limiting value defined in Equation 6-9.

Substantiation of the CT.L prediction technique describprlabove in the form of comparisons with a wide range of experi-mental data is presented in References I and 26. Thederivation of CLa for Mach numbersgreater than M*, presented inoriginal Reference 26, relied heavily on transonic-bump test data,which characteristically produces a trend such as shown inFigure 33. The new derivation (Equation 6-7) relies on sting-mountedtest data, which produces a less abrupt transition in(dCL.•/dM) in the transonic region. N

6.2 Supercritical Wing Lift-Curve Slope

The wing lift-curve slope technique described above inSection 6.1 would underpredict CL,,, when compared againstsupercritical wing data. In a study of the factors affecting

75

t/c<

STING-MOUNTED 4.4AR(cosA,/ 2 ) e 3/2

(CL =)W-B B

TESTI _ _SDATA "• •

bukt -iiu aue i

-------- / LOW AR312

I I

1.0 2-0 3.oMACH NUMBER

MI+M3NOTE: f- is approximately

the location Of CL.bucket minimum value. t/, > 1

4.4'AR(cosAi2). 3/2

(CL )w-B~ j•

M2 i -M

1.0 2.0 3.0MACH NUMBER

Figure 33 Typical Lift-Curve Slopes

S~76

lift, it was concluded that the reason CLwas being under-predicted is due primarily to the thickness correctionfactor, Kt (Equation 6-8), and the incompressible sectionallift-curve slope, ao (Equation 6-3).

For supercritical wings, the onset of the lift diver-gence Mach number, MI, is delayed to a higher value as com-pared to conventional wings. For supercritical wings, thefactors MI, M2 , and S in Equation 6-8 are modified asfollows:

(Ml)Scw = (Mj)CONV + 0.09

(M2 )S CW (M2 )cO1V + 0.045

(M3 - M1)scw

(S)SCW "(Y)CONV' - (3-MI)CONV

This modification delays the thickness correction factor toa higher Mach number and also decreases the extent by whichKt is reduced at Mg.

Supercritical wings have a higher sectional lift-curveslope compared with conventional wings. The program uses

a. 1.174 t/c

for supercritical wings in place of ao/29T= I for conventional

6.3 Tail Lift-Curve Slope

The lift-curve slope of the horizontal tail can beestimated from the equation

(CLOT (CLOI()TKWc1B)+KB(W)1 (- ST (-13

where (CL1)T is the exposed-area lift-curve estimate for thetail; KW(B) and KB(W) are the Pitts, Nelson, and Kaattari body-lift carry-over factors (Reference 27); 9/dco( is the downwashgradient; qt/q•. is the dynamic pressure ratio; and ST is the

77

exposed tail area. The exposed-area lift-curve for the tail isestimated by use of the exposed planform of the tail and themethod described in Section 6.1.

6.3.1 Downwash at the Tail

An empirical method of estimating the low-speed downwashgradient behind straight-tapered wings is given in the DATCOM by

lz) 4 .4 1.*19 (6-14)(5R, = KAKAYKH(coS-Ic/4) •(~4

Tne factors KA, Ki, and KH are wing aspect ratio, wing taper

ratio, and horizontal-tail-location factors, respectively,determined from

KA = 1/AR - l/(l+AR 1.)

K= 1O-3A

and

Kh=(.-ht/b)KH=(2jt'/b) I13

where AR and 9ýare the wing aspect and taper ratios, respectively,ht is the height of the tail relative to the wing chord plane and

(t' is the distance between the exposed MAC of the wing and theexposed MAC of the tail. At higher speeds the effect of compres-sibility on downwash is approximated by

6 = ( ) (CLg)M (6 -15)

where (CL)o and (CLx)M are the wing lift-curve slopes at lowspeed (14=0.) and at the appropriate Mach number, respectively.

78

6.3.2 Dynamic Pressure at the Tail

The method for estLwating the dynamic-pressure qt/q., atOie tail is based on the DATCOM method which relates thed.'namic-pressure ratio to the drag coefficient of the wing.The steps involved in determining the dynamic pressure at somedistance aft of the wing root chord, outlined in Section 4.4.1of the DATCOM report, are as foilowp

1. Calculate the half-width of' the wing wake by

Zw ( 6.8 C0.68 CD 0'+ 0.15) (6-16)

where x is the longitudil:: l distance measuredfrom the wing-root-chord trailing edge, Z. isthe half-width of the wake at any position x,and CDo is the wing zero-lift drag coefficient.

2. Calculate the downwash la the plane of symmetryat the vortex sheet by

4E = A-R (CL'- (6-17)

3. Determine the vertical distance Z from thevortex sheet to the quarter-chord point ofthe MAC of the horizontal tail by

Z = x tan (Y+ E - 2) (6-18)

where Y = tan- 1 (ht/At).

4. Determine the dynamic-pressure-loss ratio atthe wake center by

(A q 2.42(CD°)1/2 . (6-19)

5. Determine the dynamic-pressure-loss ratio forpoints not on the wake centerline by

79

In

s () C(6-20)

q.0 2f Z

6. Determine the dynamic pressure ratio at anarbitrary distance x aft of the wing-root-chord trailing edge by

q 1 -t (6-21)q. q.0

6.4 Body Lift-Curve Slope

As shown in Reference 28, the linearized lift-curve slopefor a body can be expressed as

1/3) k -F--- (6-22)

where L is the body length, LN is the forebody length, F is thebody cross-sectional area, and kI is a linear-potential lift-curve-slope parameter. The factor kI (a function of body widthb, body height, h, and the pevimeter of an elliptical body withequal area, p) is determined from the curve given in Figure 34,Wxncu is taitii .LU.. Refe-ent e 20'.

6.5 Angle of Attack at Zero Lift

The angle of attack at zero lift, 0 Lo, is determined from

CILo = (CILO)cAMBER + (aLO)TWIST + (cLO)INCIDENCE (6-23)

The effect of camber, CLd onaoLo is calculated from

0 a Lo(a'Lo) CAMER (jd ) CLd (6-24)

where (OOLo/OCj/) is shown plotted in Figure 35, which is obtainedfrom two-dimens lonal data.

The increment in aLo due to wing twist, r, is calculatedfrom

(LO)WlST -- - (aa--)r (6-25)

80

*-10

.08

.06

.04-

.02

0 2 4 6bP 8 10 12 14

Figui'e 34 Factor Used in Determination of Body Lift- Li81

0

O d ,-3 .08-.

•-4 cosAc -/4 1

-6

-7.6 .8 .9

M cosAc/4 -

Figure 35 Camber Factor for Zero-LiftAngle of Attack

82

whereLN/at = 0.093 - O.0OC571-A.+ 0.5761,A - 0.2645,f and[ Ap- tan- ( taUIc/4/,6) (deg) The equation for (3c<Lo/,3CI) was

obtained from a curve fit of the parametric data reported byGilman and Burdges (Reference 29) for wings with linear-element

twinst.

The angle of attack in the program is measured relativeto the wing root-chord reference plane. For variable-sweepconfigurations the angle of attack for any sweep position ismeasured from the wing chord plane in the forward sweep posi-tion. The increment in OeLo due to wing and horizontal tailincidence is calculated from

(CLB) (iw) + (CL,) (iw-it)(oL)BODY TAIL

INCIDENCE CT,

+ (iw-d wREF) (6-26)

where

(CL,,)Rnnv = (C14OS o + (• KB(W) (C1,.0

When M > 1, the contribution of camber aicd twist to aLois set equal to zero and only the incidence effect is continuedsupersonically.

6.6 Nonlinear Lift of High-Aspect-Ratio Wings

The lift characteristicsof a high-aspect-ratio wingis illustrated in the sketch. CLA high aspect ratio is defined CLMAXas ARŽARLOW, where ARLOW isdefined in Section 6.9 (Equa- _A&MAXtion 6-37). The lift varies CL-linearly with angle of attackup to CLs, after which thelift variation becomes non-linear.

MAX

83

The angle of attack for a specified lift coefficient iscalculated from

W CL + AOi (6-27)CLd, +&Lo

where

(0; CL Z CLS or M > 1.0

Ac = (CLmaxCLs) A AO•max; CLS - CL !:' CLmax (6-28)

2(CL._CL) 2 " e,,,+ 50; CL > C%,m

Lmax-maxand CLs = CL,,(0&- Oei.o - 2i-W•max)

The prediction of CLmax arid 40<max for high-aspect-ratio wingsis discussed in Section 6.9.

6.7 Nonlinear Lift of Low-Aspect-Ratio Wings

The subsonic charac-teristic of a low-aspect- CLrato winp is illustrated L.E.in the sketch. The total - VORTEXlift is equal to the po- j LVtentiall lift plus the vor- I TIPtex-induced lift from the I VORTEX

leading edge and tin of POTENTIALthe wing. The equation

MAX

84

for lift can be expressed as

CL CLp + CLV + CLTV (6-29)

The method of predicting each of these terms is discussed inthe following subsections.

6.7.1 Potential-Flow Lift

The potential-flow lift L determined fromCLp = Kp sin bicos o(+ CLo (6-30)

where Kp is th; lift-curve slope given by small-disturbancepotential-flow lifting-surface theory, and the trigonometricterms account for the leading-edge separation effects (Refer-ence 30). The value of Kp is the lift-curve slope (CLO), con-verted to radian% obtained from Equation 6-1. The factor CLois the lift at zero angle of attack predicted by

CLo -CL. OCLo

6.7.2 Leading-Edge Vortex Lift

The leading-edge vortex lift is determined from

2CLV = (-R) •FV KV sin2OcosO (6-31)

In this equation, developed in Reyerence 31, the sharp-leading-edge suction analogy of Polhamus is modifie. to account forround-leading-edge and vortex-breakdown effects.

In Figure 36 (taken from Reference 36) the theoreticalvariation of the vortex lift factor KV with aspect ratio andcutout factor is shown. The factor FVB (shown in Figure 37)is a vortex-breakdown factor, which was obtained from theratio of experimental data to theoretical for sharp-leading-edge delta wings. The factor R in Equation 6-31 is a lead-ing-edge suction parameter (Reference 33). For a sharpleading edge, the suction parameter is near zero; for arounded leading edge, the suction parameter is near unityat low alphas. The variation of R versus o/_ is shown inFigure 38 as a function of thickness ratio.

85

Ref. 32, Fig. 3

a-0.5_______ ___

4 - ~~0. 25 . .

.25 ..

5 vo

VL "-'IZLE

a-./-tan ALE

0 . I. 3* 4 .

.4 4flcOtALE/(1..a)

Figure 36 Variation of Vortex-Lift Constant with Planform Parameters

I.

Sd

enenM b

E 0

0 O" 0 ,.

s1d Lfl

C.) 0 W

CO

/ A A

P4-

7) 87in,

Figure 37 Variation of Vortex Breakdown Factor with Aspect Ratio andAngle of Attack87I

00

C0 CJ C)

* -:) Cl) )

.<- <

to tn Uuu

ce<4<<<

C) b

0 ý 0 r--,4 -4J

,a w :

co

J..~ .,Q'

C) 04 1

S-,4

- ,4)

00 4

4-z0

6.7.3 Tip-Vortex Lift

For wings having a tip chord greater than zero, a tip ,ortexforms that induces an additional lift contribution on the w-ng.At low angles of attack the flow around the wing leading edge andtip is attached, and a vortex sheet is formed at the trailing edge(Figure 39a). At: slightly higher angles of attack (Figure 39b),the flow possibly will make the turn around the leading edge ofthe wing without separating, but the flow around the tip separates.In this stage, the flow forms a vortex sheet consisting of a hori-zontal part originating from the trailing edge and two verticalsheets attached to it originating from the two sides of the wing.Kuchemann (Reference 34) noted that a spanwise cross-sectionthrough the vortex sheet has the same shape as that obtainedbehind a wing with end plates. The height of the "end-platevortex" or tip vortex is approximated by

SCT 1 (6-32)/b 2 Z AR

where CT is the tip chord. With the height of the tip vortexknown, the incremental tip-vortex lift can be expressed as

C Z-I¶ :-Z I K1 sin a (6-33)

-L-v V xZ+ IiZ-(xT)2

where

Z = 2 cosAc/ 2 /AR

x = 1.0014-1.969(h/b)+3.0021(h/b) 22.0072(h/b) 3 (6-34)

Equation 6-33 was derived (Reference 34) by modifying t[ic Helmboldlift Pquation, where the effect of the end-plates are expressedas a factor 1/x to the aspect ratio. Equation 6-34 is a curvefit of the end-plate effect shown in Figure III.A.4-1 of Refer-ence 4.

It can be seen from Equation 6-32 that the end-plate effectbecomes smaller with increasing aspect ratio. This explainswhy the end-plate effect of the tip vortex has rarely beennoticed for wings of moderate and large aspect ratios,although it always existed. The tip-vortex method is shown

89

V.0

39b moderate angle of attack

39a low angle of attack

Figure 39 Sketch of Possible Flow Patterns

90

in Reference 34 to give gcod agreement with experimental, datafor unswept rectangular wings ranging in aspect ratio from 0.5to 2.0. It is stated in Reference 34 that the end-plateanalogy can be used for straight or swept wings.

The nonlinear lift calculated by Equation 6-29 is l.mt edby the maximum lift coefficient, CLtpax, of the wing. The valueof CLmax and the angle at which maximum lift occurs,o(max, arepredicted by the low-aspect-ratio method given in Section 6.9.The leading-edge vortex is then limited by the condition

CLV <CLMAX - (CLp) - (CLTV)M (6-35)

where (CLp)MAx and (CLTv)MAX are the values calculated for thepotential lift and the tip-vortex lift at the maximum-lift angleof attack. If (CLV)MAX '5 (C)MAX, it is assumed that theleading-edge and tip vortices are too weak to add much lift tothe wing, and the high-aspect-ratio method discussed in Section6,6 is then used to predict the lift up to stall.

Results of applying the nonlinear lift predictionr procedureare shown Wn Figures 40 through 43. The data were taken fromReference 35, which reports on a test of a series of clippeddelta wings. The program results are shown as the solid linesfor the complete lift and as dashed lines for the initial valueof the lift-curve slope, CL.,. The results, in general, aregood and indicate a substantial amount of the lift is due to thevortices.

6.8 Nonlinear Lift of Cranked Wings

The method available in the Large Aircraft program forpredicting the subsonic lift variation of cranked wings isbased on the technique presented in Reference 9. This methodassumes that as the outboard panel of a cranked wing experiencesstall, the inboard panel still continuEs to lift. Thisbehavior is believed to be caused by the influence of theleading-edge vortex of the inboard panel. Consequently, theflow field is similar to that of a low-aspect-ratio deltawing with leading-edge separation.

The method employs the results of a data correlationaccomplished to provide a technique for determining thenonlinear lift of double-delta wings. It is hypothesizedthat the nonlinear lift of a cranked wing should be similar

91

'7'

C. c'q

p.p

0

Prj C'

Ent

C C- -4

0

z

I'Do -~ -001

4U r-4

6--,

'--

"ccc

4.

<r: NI. 00 .- 'C4Vj0

Figure 40 Comparison of Program Results with Expert- .ittlmental Data, A ~63o0.

110-4 I

0V

_ . •. . , . r.. . . .: ! c n r. ,,,,, - .,,•

* ,.. .

-. .. 030' .

t,'i-' 0 0"-

o\- .. . ..

li 0

"~~~~00 -, C) l . ',: •

-,0*

:*:.Figure 41 Comparison of Program Results with Experj-: - - -, -

CI Imental~C DaaA 8

903

o,

00 't 000. . . .0.i -- . . ... . o ',- -- i

.*. '.',,--':,

- . " : n" . i*: . . .... .

- . .c J-.--.-

r4.

Go 004 0'C'r- - CNC14 C%4r4 r-4

S.0

'•. ........i .;

:W44

o 0 ,-

04 A: ''CC" '•'cC:: " .t:

.n .D I. C)%0C4 0-t cC I Ao

P,4

I.i.

'. C-4 r a 'I - )

> .*I 0% O0C1 -I

00-cc< '

4 - .- M. .n0

040

Figure 42 C;omparison of Program Results with Expert-

mental Data, A= 530

94

<I-

L

44 00 '1 gi

C cJC'4 Cl-

-44

.,..2-.''

'r 'N,, 04 ,, C',4 €'4 -4

E --

O - It - .1

-. : . ' .7 i

En 00 C It 0 %0r N 0

lo 'n

. ..

I.

4 CD

Fi ur 4 Cop ri o ofP orm eut it xei

ItI 00I CA I -'-

K!

40>

Cr) % rCNJC%0 - r C)

A N C44 4 -4

CA.

c')o

0 -4 C4C)-t ---

- ~ ~ ~ ~~~~~~~~O Fiur metlDtA-5 eutDwt ei

to that of a double-delta wing. The nonlinear-lift curveconstruction technique for cranked wings is shown in Figure44.

CL

CIDB ACL

S- Nonlinear Lift Based ontDel~a Wing Correlation

A CL

OeStall

Figure 44 Construction of Nonlinear Lift Curve forCranked Wings

Except for some slight refairing of the data correlationcurves to account for aspect ratios of less than 1.0, the methodemployed in the program is essentially the same as that presentedin Reference 9. The nonlinear angle of attack for a given CL iscalculated from

[CL Ai (6-36)

S=C L . ( (

where

CL x is the linear lift-curve slope (includingoutboard panel)

Ai is the aspect ratio of -he inboard panel,defined as shown in Figure 45.

96

y

Sj/2b1Aj Si

13b

b/ 2

Figure 45 Definition of Cranked-Wing Planformfor Calculation of Angle of Attackin Nonlinear Lift Range

I-.jta nAr v,

.008

a

.006 1 .6

n In.004 -*1.4

.002 81.2

0 1.00 to0 20 30 40 50 60 70

LEADING-EDGE SWEEP, ALE (DEGREES)

Figure 46 Parameters Used in Calculation of Angle ofAttack in Nonline~~r Lift Range

97

'1B is the nondimensional spanwise ordinatefor the break point in the cranked wing

a,n are correlation constants (shown inFigure 461 derived by modifying theReference 9 method.

In the program it is assumed that a stall is the angle ofattack corresponding to CLDB, as shown in Figure 13 of Section 4.The accuracy of the nonlinear angle-of-attack prediction tech-nique is, of course, strongly dependent on knowing O(stall.

6.9 Maximum-Lift Coefficient

The method used in the Large Aircraft program to estimatethe maximum lift and angle of attack for maximum lift is basedon the DATCOM method for both the low- and high-aspect-ratiowings (Reference 2). The maximum lift of high-aspect-ratiowings at subsonic speeds is directly related to the maximumlift of the wing airfoil sections. The wing planform shape isa secondary influence on the maximum lift obtainable. However,for low-aspect-ratio wings, the wing planform is the primaryeffect on maximum liftwhile sectional characteristics aresecoauldary. The program uses the criteria established in theDATCOM method by the equation

3ARLo= (6-37)

(C1+l)cos/,iJwhere C1 is a function of taper ratio, as given in Figure 54.If AR >ARLOW, the high-aspect-ratio method is used, andif AR • ARLOW, the low-aspect-ratio method is used. These twomethods are described in the following subsections.

6.9.1 High-Aspect-Ratio Method

The DATCOM method is an empirically derived method basedon experimental correlations of high-aspect-ratio, untwisted,and constant-section wings. The equations for maximum liftand the angle of maximum lift are as follows:

CLMAX CLMAX )CMAX +ACLMAX (6-38)

98

CLMAX0 CMAX " + O(Lo +'AOeMAX (6-39)

The first term in Equation 6-38 is the maximum lift coefficientat M-0.2; the second term is the lift increment to be added forMach numbers greater than 0.2.

The factor (CLMAX/CIMAX) is computed by a curve fit of thecurves in Figure 4 .1. 3 . 4 -14a in the DATCOM given by

C LMAXCLMAx = A - B &ky' (6-40)

where

0; Ay < 1.4

Sy ' =y-1 .4; 1 .4 .<• & y < 2 .5

y Ž 2.5

and the terms A and B are plotted in Figure 47 as a functionU4 Uweep. (kTheLn y is defined by Equation 2-16.)

The increment to CLMAX due to Mach number is computed froma curve fit of the curves of Figure 4.1.3.4-15 in the DATCODfgiven by

ALE.CLMAX = C + (D-C)(L--0) (6-41)

where the terms C and D are plotted in Figure 48 as a functionof Ay and Mach number.

The section maximum lift coefficient at M-=0.2, CyMAX iscomputed from

=tMAX (Cf MAX) Base + Cd %AX(6-42)

where (CJ2MX)Base and 4 CIMAX are shown plotted in Figures49 and 50 as a function of the sharpness factor, maximum-thick-ness location, and camber.

99

1.3

1.2 1.2

A BA -- _

1.1

1.0 B-- B .4

0.9 0

0 10 20 30 40 50 60

ALE DEGREES

Figure 47 Factors for Determining Subsonic Maximum Lift

IoK

1. 0 DERIVED FROM

FIG. 4.1.3.4-15Ref. 2

C 0.6

0

M--0.4

-o.5 "-'

-1.00 1 2 3 4 5 6

AY

0.5

1.0

0.4 J-M--0.6

-0.5

0 1 2 3 4 5 6

Figure 48 Factors for Determining Mach Number Correction for Maximum Lift

t .)

Lrr)

CV-)

CO.*j P-4~ -

54-4

.14

ca

U rj

000

r-4-

C'T

P20

0.6: f/

A~h~x 7.Chord

0.4

0.2

0 _

0 1 23 4 5 6AY

Fi gure 9;0 Effect of Airfoil rl4mh,-r nn Mnvi ~mrf L.Tft

12 - -I.

- - ) 0.2

lo 20 so -0 .1 6

84~ ALEFOrm)

Figure 51 Angle-of-Attack Increment for Subsonic M{aximumLift of High-Aspect-Platio Wings

103

The angle-of-attack increment for maximum lift, O(MAX,is obtained from Figure 51, which is taken from Section 4.1.3.4in the DATCOM.

6.9.2 Low-Aspect-Ratio Method

The empirical equations in DATCOM for estimating subsonicmaximum lift and angle of attack for untwisted low-aspect-ratiowings are

CLMAX = (CLMAX)Base + .ACLMAX (6-43)

0(MAX = (PMAX) Base +,AaMAX (6-44)

The base value of CLMAX is obtained from Figure 52 if theposition of maximum airfoil thickness, XT, is forward of the35-percent chord point, and from Figure 53 if XT is aft ofthe 35-percent chord. The values of ACLMAX, Cl, and C2 areobtained from Figure 54, the base 0 MAX from Figure 55, andthe value of A:o(MAX from Figure 56. (Figures 52 through 56are taken from Section 4.1.3.4 of the DATCOM.)

6.9.3 Tail-Lift Contribution to CLMAX

Because the horizontal tail usually has a smaller aspectratio compared to the wing and is in a downwash field thatcounteracts the effect of angle of attack to some extent, itis assumed that the tail does not stall before the wing. Thelift generated by the tail at the angle of attack of wing stallis added to the wing maximum lift coefficient to obtain theconfiguration maximum lift coefficient. The configurationmaximum lift is given by

CLMAX = (CLMAX) + (4 CLMAx) (6-45)

Wing Ta il

where

(ACLM) = (CLK) • 57.3 sino~max(coscxmax) 2Tail

T

The term (CLQ,)T is the lift-curve slope of the horizontal tailas determined in Section 6.3.

104

I' -- -- ~-. -~ - --• . •..o --.--- .. -

__.75

1. 0

(°L)" 2 --

112.

SLOW ASPECT RATIO

rlBORD EIR LINE, ASPECTt / RATIO

.4

0 .4 .8 1.2 1.01 2.0 2.4 2.8 82 .6 4,0 4.4

AR(CI + 1) • cOSALE

Figure 52 Subsonic Maximum Lift of Low-Aspect-Ratio Wings (XT < 35%)

1.4 -2 - - - - - - -

.50

". 5, 1\1 ., I _,[ I -

(CL).1.2.) !,, ,

.-- /U/flR LIMI'I-OF.

LoW-A-PECT-RA'IOI _/ BORDLERLINE ASPECT

LO ..PC AI RATIO

.4 .8 1.2 1. 2.0 2.4 2.8 8.2 3.8 c 4.0 4.

AR(ci + I) - cosALE

Figure 53 Subsonic Maximum Lift of Low-Aspect-Ratio Wings (XT > 35%)

105

.01.5---

----1

H0. 1.0,, '4 ' ..- -0-----*1 - ..-=.---------

a .2 .4 .9 .8 1.0 0 .2 .4 .0 .8 1.0

X, TAPER RATIO X. TAPER RATIO

.4, "-' "

-,2 - - - -

A CLmnSK

0 21-

(C 2 + 1) AR tanALE

Figure 54 Mach Number Correction for Low-Aspect-Ratio Maximum Lift

106

|0

---- UER LIMIT OF

(Gg0 - - - '- LOW-ASPECT-RATIO

,d RANGE

LOW A.SPECT RATIO F0- ,_..BORDERLINE ASPECT

RA TIO

0 .4 .8 1.2 21 . 2.4 2.8 3.9 8.6 4.0 4.4AR(C1 + 1) - cos ALE

Figure 55 Angle of Attack for Subsonic Maximum Lift of Low-Aspect-Ratio WingMaximum Lift

(d j

10(C2 + I)AR tanA+4

Figure 56 Mach Number Correction for Angle of Attack

0107

I,=

7. MWMENT

The moment of a wing-body-tail configuration can be repre-sented by the equation

Cm - C% + (n- Xac CRe A-

Mo CR- - CLWB - CLTAIL XHT/C (7-i)

where n is the chordwise distance to the moment reference pointmeasured in exposed wing root chord (CRe), Xac/CRe is theaerodynamic center location relative to CRe, and the last termrepresents the moment contribution of the tail lift times th;moment armAHT, determirned from Equation (2-9).

The following subsections discuss the method used to predictthe elements in Equation 7-1 along with a method of determiningtrimmed lift curves and drag polars.

7.1 Zero-Lift Momentql U_' _-• . _ ___ a .. . .

ý... ,mLetU U. Pu.LedLeLing the zero-lift pitching moment tora wing-body configuration considers only the effect of the wingon Cm. and does not include the effect of an assymmetricalfulse1=;ge or the effect of stores and nacelles located near thewiý,g. However, the Cmo prediction method in the Large Aircraftprogram can be adjusted by input so as to match the test dataCmo on a similar configuration.

The subsonic zero-lift pitching moment for wings withlinear twist , up to the critical Mach number, is given in theW 4J. t 1 ancb J (c ,)Ct.U IW)• + 5.9(t/c)M5

MO = nc)'t'=O + (t -k _554_ (7-2)

where (Cmo•-0 is the Cmo of an untwisted wing and (ACmo/r) isthe change in wing zero-lift pitching moment due to a unitchange In wing twist, 1• The parameter (ACm/r-) was obtainedfrom lifting-line theory and is shown in Figure 57.

The Cm% of an untwisted wing is obtained from

AR+2cos3c/ 4 ( )SECT (7-3)

108

&

" I

(per deg)I

QUARTERJCHORD SWEEP ANGLE, A~p, (deg)

.032.

7-

6Ylv dog)

QUARTER-CHORD SWEEP ANGLE. A C14 (deg) 6

-. 032

m0 0

At, .024A5PfEj' RATIO7'(per deg)

0

QUARTER-CHORD SWEEP ANGLE, A,/4 (deg)50 0Figur~e 57 Effect Of Linear lW~ nteWn eo.LfPitching MOment Twsontl WgZeOLf

'109

Iwhere (C-mo)STCT is the average section pitching moment coef-ficient deteriuned by averaging the section Cmo for each wingpanel, using

VA

i E-l\ •/ iSi

(C'mo)SECT = N

i=1

where (Cm/Cld) is the theoretical pitching moment divided bythe section design lift of the airfoil camber line obtainedfrom Table 4.1.1-D in the DATCOM.

7.2 Aerodynamic Center

The aerodynamic center location of a wing-body configu-ration is given in the DATCOM as

(Xa) (CXac )W(B)~ (Cl,)W(B)+ -d B(W (C (W)

CRe (CT .)__+(CT .)W...+(rT )-" """ • " "• " oLU'-""B(W)

(7-4)

where the Xac/CRe terms are the chordwise distances measuredin exposed wing-root chords from the apex of the exposedwing to the aerodynamic center, positive aft. The sub-scripts N, W(B), and B(W) refer to the lift and aerodynamiccenter contribution of the forebody, exposed wing, and thewing-lift carryover on the body, respectively.

7.2.1 Aerodynamic Center of Forebody

The subsonic location of the aerodynamic center forforebodies with ogive nosecones is approximated in theDATCOM as

(x ac) = 0.54) LN+I. 6 (XLE-LN) (75)

ýCR e/N CRe

Figure58 defines the geometric parameters, LN, XLF, and CRe.

The supersonic forebody aerodynamic center is obtained from

110

I

b/2

d/2

LXN~

XLE

L

Figure 58 Wing-Body Geometry

*1'11

(X 8a \ - XLE / XCP (7-6)kCRe/ k /

where the term XCP/)B is obtained from Figure59 (Figure 4.2.2.1-23a in DATCOM).

7.2.2 Aerodynamic Center of Wing (Trapezoidal, Single Panel)

The aerodynamic center of the exposed wing is determinedfrom the DATCOM charts presented in Figures60a through 60f.These charts are valid for subsonic Mach numbers less than Machcritical and supersonic Mach numbers greater than 1.2. Fortransonic conditions, the data presented in the DATCOM in termsof transonic similarity parameters (Figures61a through61d) areused to determine the a-.rodynamic center position.

The procedure for obtaining aerodynamic center can besummarized as:

For M, MCR; (Xac teXac) (7-7)

For MCR+.05 ; M 7 MCR,

(ac (Xac + f(ac (X' M<I R(78(~) CRe CRe - (CRjj .05

For Vl+(tlc)2/3 M N >MCR + .05,

(Xac)(Xac)(-9

For 1. MC •Z(tcei

For 1.2 M 14N/)

CR CRe CRe (- N 2 1ý+(t)2 . (7-10)

112

Ref. 2, FIGURE 4.2.2.1-23a

SUPERSONIC SPEEDS

Xc-P• .4, .4- - -" -"-• • .-- -

(fraction of .3--body length) __N

.22.0 --. . .

N 'A• ifF d d-

- -=

Figure 59 Supersonic Center of Pressure of Ogivewith Cylindrical Afterbody

113

Ref. 2, FIGUpE 4.1.4.2-22

SUBSONIC AND SUPERSONIC SPEEDS

1.0- 11117 -0!I

ARtanALEJ.

SUBONI 6USOI

0.6-

0 T1A 1__ _ 2_

0 TTNAN ATANALENA E

Figure~TA 60 Win Aeodnaicce Positio

114

1.4

T1NA2 1 0 0'ý

I,,-

Figur 60 (ot)

51

¶J

(e) - --0.5- _ ARtanALE

2.0 ARtanALE _1 711111*- I _ 11

- -i ~

•1 °.2 6_-2- - - '

UN WE TT - 4--3

.6 -22-

116 1 11

-1 i U B S O N I C . - - I .S U E O N IC0 PON=1

0TAN ALt 1 TAN ALL•• TAN A LE TAN ALL.•

20 AraN ALE I A L•

FIlue 60(_n7 d

I16ý

Ref. 2, FIGURE4.1.4,2-25

. 02

(t )2/3__

AARtanlAE ARtaAL-tnL - 6...... AtaIAE.8aLE ,--LS, J6 -- 61•.,

/ ~~AR t a nA

5

41--- - 4 - E-S44. 4

.6 f 3t 33SI ... ..1.. 2 2•_

_ 2 12

(a) t ,01, ,,0 1 2 3 0 2 3 0 1 2 3 0

ARtanA LE-ARtanAlE RtanALE

. .8 4--- __ .. .4

S---- 4- ...... 4

3 3."6v NS WEPT T.E.~ -- 3--PT

.2- le0

(b) X 0.2 O--• I j • --0o 1 3 0 2 1 2 3 0 1 2 3

AR = AR (t/c) 1 1 3

Figure 61 Transonic Wing Aerodynamic-Center Location117

"---• ... N.. ... 11A I

VV( - V-- I V -- 2

1.4 - .-4 -

ARtanALE

1. tan ARtanALE

,,-: 4 _4.•-: ARtanALE

' •4_

3. 2_ 2.6 'SWEPT T.E. NSWE " T.E. ; ',,.--

O.L.4_

N

RETNGLR IGh

0-H--

-4 -2 3-1 2 3 -0 1 2 0 1 2 3

TR AR (/)/

(d) 1.()

(~.2

Figure 61 (Cont'd)8i

For M •1.2,

, •. ~CRe/ IkCReI(-i

where (Xac/CRe)' is read from Figure60 and (Xac/CRe)" is readfrom Figure 61.

7.2.3 Aerodynamic Center of Wing (Cranked orDouble Delta)

The prediction of the a.c. location of cranked or double-delta wings is taken from the method developed at Convair Aero-space as reported in Reference 10. The non-straight-taperedwing is divided into two panels, with each panel having conven-tional, straight-tapered geometry. The individual lift-curveslope and a.c. are estimated for each panel, using the techniquedescribed above for the trapezoidal wing and treating each con-structed panel as an isolated wing. The individual lift anda.c. location for each constructed panel are then combined, usingan inboard-outboard weighted-area relationship

Xa Xac)(CS +Xc(~~~Xac (CRe/iCq i)i

CRe (CL.)iSi + (CLo)oSo (7-12)

where the outboard wing a.c. is referenced to the inboard rootchord length given by

fXT.\ C~o .AY (tanALE) + (taLE)i0o CRe CRe CR

The geometry for the inboard-outboard panel arrangement isillustrated in Figure 62.

119

_____________

AL~i

CRe

yA- ALEOCRe

I I

(b 2) 0

Figure 62 Geometry for Cranked Wing Aerodynamic-CenterPrediction

120

7.2.4 Aerodynamic Center of Wing-LiftCarryover on Body

The location of the a.c. due to the wing-lift carryover on thebody is determined by use of the DATCOM method. For j8ARe > 4 thesubsonic a.c. location is obtained from

(CRe + L tnc/4 ' f(d/b) (7-13)

elB(W) 2e

where ARe is the exposed-wing aspect ratio and the factor f(d/b) isshown plotted in Figure 63. For flARe < 4 the a.c. location isdetermined from

xac) Xac) PA(XaN +_____c) - CRo 4 (cR (7-14)

B(W) E(CeJ R

where (Xac/CRe)' is the a.c. location determined from Equation 7-13and (XaciCRe)" is the theoretical location for 8ARe=O determinedfrom tl.e equation

( =c) i ARe(l+Xe) tanALE (7-15)

where Xe is the exposed-wiLg taper ratio. Equation 7-15 is limitedto values less than or equal to 0.5. For supersonic conditions thea.c. location is estimated from Figure 64. For transonic conditionsthe a.c. location is determined by linear interpolation of the a.c.values determined at the critical Mach number and Mach i.I.

For complex wine planforms the equivalent wing-sweep values areused in the subsonic and supersonic a.c. location methods.

7.3 Effect of Trim Deflection

The effect of trim deflection can be estimated by predicting theincremental change in lift, drag, and moment due to tail deflectionalong constant angles of attack. Tle total wing-body-tail lift,drag, and moment can be represented by

CL = CLwB + CL t(a- tot + CL, 6 HT (7-16)

121

Ref. 2, FIGURE 432.1-35

.32--

.28-

.24-

f(d/b) .201 __ __

.16 -. . .,

0L0 .1.2 .3 .4 .5

body dianieter/wing span

Figu-e 63 Parameter Used in Accounting for Wing-LiftCarryover on the Body

122

Ref. 2, FIGURE 4.3.2.1- 37

(L) WITH AFTERBODY ..S2.0"- • .... .

CO iýL-- - - - - -- - - - - - -- - fCOT A,_ ••

0 .,

re8(W)

1.2" -

- - - I

.4 2 .48'04 .8 1.2 1.6 2.0 2.4 2.8

EFFECTIVE RATIO OF BODY DIAMETER TO ROOT CHORD, c

Figure 64 Aerodynamic-Center Locations for Lift Carry-over onto Body at Supersonic Speeds

123

CD - CDMIN + (CDL) + (CDL) (7-17)

TOTAL il

Cm - Cmo + (dC-) _ CLa t(Cf aot) + CL a HT IHT/E (7-18)

where the induced drag of the tail is predicted by use of a tail-induced drag factor Kt times the square of the lift generated bythe tail, i.e.

(CL) =Kt CIJat(a-clot) + CLa HT (7-19)HT

From Equations 7-16 and 7-17, the incremental change in lift anddrag at constant ce due to a trim deflection can be determined by

ACL CL6 "HT (7-20)

ACD Kt [CL t(C-G t)CL," tH)J 2 - Kt CL (a- ot)] 2 (7-21)

Equation 7-21 can be reduced to

ACD--aHT 2 + b HT(G -dot) (7-22)

where a = KtCL6 2

b = 2 KtCL. t CL6

The factor CL, for an all-movable horizontal tail is predictedfrom

CL, (CL',)t KW(B) qt/q. (7-23)

where (CL') is the exposed-area lift-curve estimate for the tail, andKW is the surface lift in the presence of the body factor. Theinduced drag factor for the tail is determined by the leading-edgesuction method described in Section 4, where

1-Rt Rt SREFKt 3K()(~ + R EF(7-24)Kt 5 wARHT (SEX)HT

124

The Large Aircraft program can predict the lift, moment,and drag for a fixed tail setting or the program can solve for•~HT to trim out the moment. The tail deflection required fortrim is obtained by setting Equation 7-18 equal to zero andsolving for 811T*

I

125

II

8. HIGH-LIFT SYSTEM AERODYNAMICS

The empirical methods for predicting lift, drag, andmoment of an airplane with flap and slats deployed consistof adding the incremental effects of the high-lift systemto the clean airplane aerodynamics. Figure 65 illustratesthe manner in which the incremental effects of a flap canbe applied to the clean-wing aerodynamics. The following

techniques for estimating these increments were derivedfrom DATCOM and References 36 and 37.

8.1 Lift of High-Lift Devices

The untrimmed equation for lift can be expressed as

CL = CLo + CL(,-(O<-b) (8-1)

The term iol varies with c/ according to

S0; 0C "(Omax - 2O(max

/ m\0 ` 2 (8-2)

(• 2ZO~ma • ;O> (o• max- 2 0daW )

(k. 2AOYmax )11. awhere

CLmax-CLo

(>/max CL 0 +AOýmax

The clean-wingAckmax determined in Section 6.9.1 is also usedfor the case with flaps. The increment in CLo and CLmax caused

k, 4--,;1 ,.• r,1,, _ __1 - -l~ ..- "r.. -1o, -• -. -- - -,W"A

6CLo ACo (CL') KcKb (8-3)

ACLax = AC/m (- KcKb cos.Ac/ 2 (8-4)

where Kc and Kb are shown plotted in Figures 66 and 67. The

factor CL.,/Ce.L , determined from the Polhamus lift equation

CL .' AR

2+ 4+S=~~ (csc/ (8-5)

\Plan/ cosJ,/2

126

CL (CLMAx) CL

•CLMAX FLAPSS (CLM~x) CLEAN

cx £xN CAX Cm Am ~ ] -

ACLoC

l CMAX + Cm •_ACo

SHIFT IN POLAR ACLACLDUE TO FLAP

Figure 65 Incremental Effect of Flaps

127

1.0

.8 Ref. 2

.6

.4

.2

0o .2 .4 .6Cf/C .

1.8

1.6 (ods)C

1.41

1.2

.3

1.2 .

0 2 4 6 810AR

Figure 66 Flap Chord Factor

128

.:..Ref. 21

- - VALUEf OF Kb.-.

0- bf!

I. ~1 0.-..

.8-

Kb

1....

.2

0 .2 .4 bf_.6.8 1.

b

Figure 67 Flap Span Lift Factor

129

converts the sectional values ACyo and 'Cjmax to the three-

dimensional case. The sectional values are either obtained

from input to the program or generated internally for certaintypes of high-lift system. The method of generating the sectionvalues is discussed in Section 8.4.

The increment in lift at zero angle of attack is approxi-mately zero when a slat is deflected. Thle slat acts to delayseparation from the wing leading edge and thus allows higherangles of attack and, consequently, higher values of maximumlift before the wing stalls. An estimate of the increase inmaximum lift of a slat is represented by

)LiCl~nax =A• 2max(•-•-) KS - cosq .LE ' -6)

where the partial span effect, KS, is shown plotted in Figure 68

8.2 Drag of High-Lift Devices

The iintrimmed eauation for drag can be expressed by

CD = CDmin + K(CL- ACL)2 (8-)

Total

The drag-due-to-lift factor, K, with high-lift devices can beestimated by

SRefK -- KClean <Z-1

where Kclean is the clean-airplane polar factor and Splan isthe new planform area of the wing if the flap extends thechord if the wing. The drag due to lift factor remainsunchanged until ' '> C'max - 2- A Cmax, after which Kapproaches the zero-suction value given by CDL = CL tanO'max-

Total minimum drag for the high-lift configuration isexpressed as

CDmin CDmin + CDL.G.+ CDFlaps+ CDslats+CDI (8-8)

Total Clean

where increments are summed for the landing gear and the profile

130

IVALUE OF' KsjEFNVPSNO, R-5-6213

F~~ .:j-KS~

-4..

~ o bs /b/2 ... 10.-1.0 L-7- -I-

1-I7i- -7

.6i~

I( 7-7----

.4-

'1 ~7- J7Y

0 .t 2 .L4~ b 11.6.8 1.0

Figure 68 Slat Span Lift Factor

131

drag caused by the flaps and slats. The increment in profiledrag of the flaps and slats can be estimated from sectionaldrag data, using

cDFap ACdf cosHL• Kd (8-9)

/ACDSlat =ACds cOsJ LE Kd (8-10)

where Kd is a partial-span factor, shown plotted in Figure 69.The method for generating section values if given in Section 8.4.

Deflection of a flap produces an increase in lift atzero angle of attack which in turn produces an induced draggiven by

2(AC")2

CDI - KaKf VAR (8-11)

where K. and Kf, shown plotted in Figure "70, are factors whichaccount for the non-elliptical span loading of partial-spanflaps.

The deflection ofa_ fla-p incrcaaes thne cambI-er of tCheairfoil. In Referezice37, thin-airfoil camber theory is usedto relate the displacement of a polar with flaps to the liftincrement of the flap at zero angle of attack by the equation

ACL = (ACL)clean + (LCL)Due to

Wing Flap

where W )(CLo)(ACL)• 0 = ti-,ARK Flap (8-12)

Flap 1+1.16 l6~)(.5-Cf/C)

8.3 Moment of High-Lift Devices

The pitching moment increment caused by a flap on a sweptwing iA represented by

CM Cm Km+TR tanA.A+s (8-13)Ra=o t2 - 0

where ACm• and AC/ are the sectional change in moment and03

,0

132

EFFECTS OF FLAP SPAN ON DRAG

. d VIU .0

1.110

A-0I25:

1.05

1.0 1.0

.6

.4 R f. 3 , Fi . 1

0 .2 .4 .6 .8 1.0b i/b /,,2

Figure 69 Flap Span Drag Factor

133

ACLO2

CDI Kf KA, 7rAR DERIVED FROMREF. 36,1 FIGS. 18,19& 20

4 0

.2~

3

S... - F L A P C U T -O U T0.2 bcIb!0.1

0 1 -

o .2 .. 6i.u0(bc + bf)/b

I 0.2 bc/b

0.15.KA -

20.I.

S-10.05 ,

0 -

0 .4 .8 1.2 1.6 2.0 2.4AR 27T

Figure 70 Flap-Induced Drag Factors

*134.

lift at zero alpha due to flap deflection. The partial-spanfactors Km and KSW are shown plotted in Figures 71 and 72, andthe method for generating sectiotal data is given in Section 8.4.

8.4 Aerodynamic Characteristics of Two-DimensionalHigh-Lift Devices

The lift effectiveness of plain trailing-edge flaps canbe estimated from thin-airfoil theory. The rate of change oflift with flap deflection at a constant angle of attack isgiven by

C4 -2 + sin GfJ (8-14)

wherecos Of 1 I - 2(Cf/C')

and C' is the extended chord.

This equation is plotted in Figure73 as a function of flap-chord ratio. The theory considers only a bent flat plate anddoes not include effects of thickness or large deflectionangles. The effects are accounted for in References 38and 39 by empirical flap efficiency factors. as shown inFigures 74 through 77. The lift of a plain flap may now beexpressed as

•CIo '=?p CI f (8-15)

where

'p is the plain-flap efficiency factor from Figure 74,depending on the flap deflection angle, Sf,plus the included angle of the flap trailingedge, OTE"

Cy is the rate of change of lift with flap deflectionat constant angle of attack from Equation 8-14.

jf is the flap deflection angle in radians.

This procedure is extended to slotted flaps with Fowler motionby evaluating Cj at a flap-chord ratio based on the extendedchord. For double- or triple-slotted Fowler flap segments thelift increment is obtained by summing the incremental liftincrements for each flap segment. The result is

135

1.47 17 1

Ref 36, Fig. 14i

1.2-

Km

.6

.44

I VALUEKm I, OF Km

.211

00 .2 .4 .6 .8 1.0

bf /b

Figure 71 Span Effect on Moments

136

.07 ---- - -

Ref. 36, Fig. 27(a) L.

.0 .. ....

.05

Ksw

.02 I _ \

0 bfbf/b

'Figure 72 Span Effect on Moments of Sweptback Wings

137

IIL

rf

4-i

0 1 0.2 0.3 0.4 . 0.FLAP CHORD RATIO, e~

Figure 73 Theoretical Lifting Effectiveness of Trailing-Edge Flaps

138

0.4 -F 4

11-17 7-1.--

0. .. ... . .. 1 <

010 1.0 20 30 40 5060 70

"f 40TE

Figure 74 Turning Efficiency of Plain Trailing-Edge Flaps

---t - 0 NACA 23012 AIRFOIL. ..... NACA 23012 AIRFOIL

4ýr 0NACA 65-210 AIRFOIL7 47-1 CONVAJR AIRFOIL

A-j 1 NACA 64-2 13 AIRFOIL .. :

I __ - f 7 NACA 4412 AIRFOIL

0.6 j 0 rtTK-~ V'04 4

-7-e

- ..:.~ .....

010 20 30 40 50 60 70 80

8~~TE

Figure 75 Turning Efficiency of Single-Slotted Flaps

139

-~~fF[I TIT] TFtTj jfTfho.1.. ..-.. ~ c 0 =0.240.

4.7 44 _i

I ttI

10 f0 30 50607'2'

0.2 7- -.' ý . I 11 , . , . -. --ý .. ... ... . .. T

Fi'gure 7 6 Turning Efficiency of Double-Slotted FlapsI

1.0- .~

.... A 64 .213 AIRFOI

.44T4P~vY~6 4 -6-0.4 f4 f2~ it

II

Fiur 7 Trnn Ef~iecyofTipe-lot&Flp

2

I,.-.-1 - .

= = ITsI1 ci b (8-16)

where

i is a subscript that indicates the Ist, 2nd, 3rd flapsegment of the slotted flap

I is the number of slots or segments in the flap system

is the slotted-flap efficiency factor from Figures75, 76, or 77 for the ith flap segment

Cesi is the lift effectiveness for the ith flap segment

f is the flap deflection of the ith flap segment.

The method of summation and the geometry definition required toevaluate Equation S-16 is shown in Figure 78.

The effects of leading-edge high-lift devices on the winglift at zero angle of attack is estimated from thin-airfoil

(CS) = 2(sin OLE - OLE) (8-17)

LE

where

cos OLE = 1 - 2(CLE/C')

l .....n c tratng-edge flaps, the deflection of a nose flap causesa loss in lift at zero angle of attack. The increment in lift,

A CI is

S Cc° = (C LE) ' LE (8-18)

where 6 LE is the leading-edge flap deflection angle in radians,

positive nose down.

The two-dimensional maximum lift increment,4 C/max, due to atrailing-edge plain-flap deflection is given in Reference 38 as

KS'mx-- (8-19)'AC/max =KT K8 ACI 0 A (819

141

C,

TT

1 2 2

f33

3~ 33~~

r 7I TE

flf

I f¢

I? f3•---- f T

Slotted-Flap Geometry

142

+t

and, similarly, for single, double- or triple-slotted flaps the

S~equation is

Kc KT Ž (ACi10 )( ACI 0 ) (8-20'

where(A• C is the predicted lift increment determ-ined

i for the ith flap segment from Equation 8-18

KT and K6 are empirical factors developed from experi-mental data shown plotted in Figures 79 and 80

AC/a is the theoretical relationship between *1LAC 4--o A.RGmax and 4CJa,,0 given in Reference 39 as

Cmax Jf + nIsin4(X+of)/sin2(X-ofl ) -

( c 2 -a.") .-- f + sin~f Of tan X/2L -j

where cosX 2(Xs/c')-l and cosOf = l-2(Cf/c'). This equationrelates the theoretical maximum-lift increment to the chord ofthe flap and the position of separation, Xs/c', on the airfoil.The choice of the separation point Xs/c' to determine the maximum-lift ratio from Equation 8-21 depends on the leading-edgeconfiguration. For clean leading-edge airfoils, the point offlow separation is assumed at the leading edge, Xs/C' = 0.For airfoils with leading-edge high-lift devices, the pointof flow separation is assumed to be at the knee of the leading-edge device, Xs/c' = CLE/c'.

The two-dimensional increment in maximum-lift coefficientof leading-edge devices is predicted in Reference 38 as follows

ZCmax) LE = C max 5 " LE (8-22)

where, according to thin-airfoil theory,

C LE)A -- 2 sin OLECý6LEW

and cosQLE (I- 2 CLE/c'W The (CISLE)max is presented in Figure 81.The factors "? a •nd " are empirical factors, introduced

in Reference 38, to correlate A CVimax)LE with available test

143

t2.

FIG. 7

KT

0 .o4 .08 .12 .16 .20LEADING EDGE RADVIS/TIIICKNESS RATIO

Figure 719 Maximum Lift Correlation Factor for Trailit-g-Edge Flaps

REF: 38FIG. 9

1.8 -7..

1.4

jI

1.oI

0 20 40 60 80FLAP DEFLECTION -DEG.

Figure 80 Flap Angle Correlation Factor Ver3uL, FlapDeflection

144

dala on airfoils with trailing-edge flaps (Figures 82 and 83).The rmximum-1ift efficiency factor, 7max, depends on the type ofleading-edge device and on the ratio of leading-edge radius tomaximum airfoil thickness; ?JS is an efficiency factor that accountsfor large leading-edge-flap deflections.

In the case of the two-dimensional moment increment, themethodology for predicting the pitching moment is developed parallelto the methods used for estimating the lift increment, which extendthin-airfoil theory to cover multiple-slotted flaps with extendablechords. The trailing-edge-flap pitching-moment increment at zeroangle of attack is given in Reference 38 as

Z'Cm'=° 0" C° CI (8-23)

where

is the predicted lift increment for eithertrailing-edge or leading-edge devices

is the theoretical center-of-pressure locationfrom thin-airfoil theory (Figure 84)

K11 is an empirical factor developed from experi-m'!nta], data (Figure 85)

In the case of the profile drag increment, flap drag incre-ments at XL =0 for plain and single-slotted flaps are obtained fromFigures 86 and 87. These figures were obtained from Section 6.1.7in the RATCOM. For double- and triple-slotted flaps, Figure 88 isused to obtain the C0=0 drag increment.

8.5 Ground Effect

During takeoff and landing when the clean airplane is closeto the ground, the ground proximity produces an increase in thelift-curve slope, a decrease in drag, and reduction of nose-uppitching mcment. However, some high-lift configurations may showa loss in lift-curve slope due to ground effect.

145

-1 17-4

1.2.

aim LifJ fetvns

2 . .v 7SA S-~_i

LE 1-2CCRVED

.4 KTEE 7"f .. L...- .dFA --

0 040.2 0. 0.4~ 0.5 02

~~~Figure 821aiu itEfcec o Leading-Edge Devices,.i

SLT<2 _, )(REG R~ rG

f~A I~ EDGE INGLAP

0 0.0 0.0 30. 401 0.2

Figure 83 LMaximum Ldft DEvficiencyflerton A ngle de D2evioFctor

21.

1.2 -0.2-

--iFýADING EDGE FLAP

-AL

--. 71.F DI- ED E LA. ...

0.46 ' - 0-0, 12__ _

F C

Cf~

Figure 84 Flap Center of Pressure Location as givenby Thin Airfoil Theory

2.4

- REF: 38-FIG. 10

2.0

SLOTTED'FLAPS

1.6k

-ZPLAIN

FLAP

.2Cf/C .

Figure 85 Moment Correlation Factor Versus Flap Chord

P'LAIN FJ.A'S' 10,_ DEGREEFS.

20,

.24 70.3

- -65y - - --- So-I

60

IV . POINTS .APE USEDTO CONSTRUCT CURY S

A --- V0

4-55

40I

Figure 86 Iwo-Dirnensional Drag Licrement Due to Plain

Flaps

1 48

S.2

I-:SINGLE-SLOTTED FLAPS .

S20.._ - - - .. .I

I / -

-. 20A 30/

.16-

.12 POCINTS ARE USED_.TO CONiS~TRUCTCUVES-

_0 40

.04 30~ii

0 L

0 .0 .1 .2 .32.4

CfC

Figure 87 Two-Dimensional Drag Increment Due toSingle-Slotted Flaps

149

-. ' DOUBLE-SLOTTED FLAPS,65

.1 .- . ........ 5

Ref. 37 data Cf /c- - - - Ref. 38 data .

.12 -I

AlCd

.084

0..8 1.2 1.6 2.0 2.4 2.8 3.2.Act~

0

Figure 88 Drag increment for Double-Slotted FlapsVersus Lift Increment at a - 0

150

A theoretical approach for estimating ground effects wouldemploy an image-vortex system to represent the ground plane.The DATCOM uses a semi-empirical method to estimate the increasein induced upwash at the wing due to the reflected trailingvortices and the change in lift at the wing due to the reflectedflap ground effects. The change in wing-body lift moment anddrag due to ground effect at a constant angle of attack is givenby

A CL= (ACL)wB + (ACL)IIT (8-24)

ACxQ 7 " - ) (ACL) (CL)HTHT/ (8-25)

UC CL2 dCL2 rTCL (ACD ( ~R CD~Wig (R)-26)

Alone

where C0, r and T are ground-effect parameters, shown plotted inFigures B9, 90, and 91. The term C is the minimumdrag contribution of the wing plus the ar g due to lift of theWing i free air*. The i.•crtueeit in lift on the wing due toground effect is calculated from

0L)wB 9."12+14.2/(I+A),3 CLwB(CL)w.) WB 'GOAR LW WB

+ L CLWB 5( ) 42 (4CL)Flap (8-27

where CuBiSthe wi-ng-aloue 'Lift at angle of attack, and U-,(zL/Lo), and A (ACL)Flap are shown plotted in Figures 92, 93,and We. The increment: in lift on the horizontal tail due toground effect is calculated from

(ACL)HT = (AC) CLt / (1- bE (8-28)

where &6 is the change in downwash at the tail given by

eff (H11 - H)2

beff + (HH + H)2

where • is the dcwnwash at the tail in free air, H is the height

151

.7 -/- 4 CHORD- - -

4 - b42-

o7 GROUND

h/2 b/2

.2

0 0.4 .8 1.2 1.6 2.0

hb/2

Figure 89 Prandtl's Interference Coefficient

152

"Ira II

1.0 -

\I....\ 2 ii

4- - - h-b- .7c.6- ""- +.75cOU.9 (h. 7 ,b/1 CHO

(/2_b/2

.75I I/

.4.

73 -- /2 - -7ý

.5 .t o .62 .n Spa 1.2Go.

Effect

153

0I

h

ST 57.3 z

Iii

6

T \

T\

0 .2 .4 .6 .8 1..

Figure 91 Parameter Accounting for Variation inLongitudinal Velocity with Ground Height

154

1.46- 3/ COR

1.4)

1.2 M2 -- b/

-. 75 b/22.5

/ 12Or - _

.6-AxJ

155

.4- -- - -_ _--

.2-

rL -1 57.3 CL

15

0 .28 ~ 1.6 2.0 2.4

Cr

Figure 93 Parameter Accounting for Ground Effect onLift Due to Bound Vortices

156

0- -- ii.R

III

----------. 08- __

12 -v0 2. 6.8 1.0

Cr /'Cr I

Figure 94 Effect of Flap Deflection on the GroundInfluence on Lift

157

of the V/4 of the wing above the ground, HH is the height of the"5/4 of the horizontal tail above the ground, and beff is thleeffective wing span, defined as

CLwo + (ACLO)Flap (8-30)beff (-0

SCLwo 1 (4CLo) Flap

bIw bf,

where CLwo is the wing lift in free air without flaps,

!Lo)Flap is the change in. lift due to the flap, and

ibI. ,' ''b'l

bw =cb b 'b

The ratio (bw/b) is given in Figure 95 as a Linction of taper ratio,and (bj/b') is given in Figure 96, as a function of flap span.

The in'rements in lift, momenLoand drag du1e to ground effect

are calculated at each angle of attack. These effects are thenadded to the free-air calculations; so that a trimmed condition inground effect can be calcolated.

158

'IY bV454

.6

.6 ~J " I12 3 4 5 6

Figure 95 Effective Wing Span in the Presence of the

Ground

bfV .4.S

T-I0 .2 .4 I, 6.8 1.0 .

bf

Figure 96 Effective Flap Span in the Presence of the

Ground

1594

9. DATA COMPARISONS

This section presents comparisor.s between predictions usingthe Large Aircraft program and test data from various sources.

9.1 Sy stematic Wing Study

Comparison of the predicted lift curves and drag polarswith the systematic series of wing reported in Reference 12are presented in Figures 97 and 98. Reference 12 reported onthe tests of 11 wing-body models covering systematic variationsof sweep, thickness-to-chord ratio, position of maximum thick-ness, camber, and aspect ratio. The study was conducted overaMach range from 0.23 to 0.94. The predictions were made usingthe geometry presented in Reference 12. The wetted areas ofthe wing and body were computed internally by the Large Air-craft program and the friction drag solution was for fully tur-bulent flow.

The predicted drag polar shapes up to lift coefficientsslightlv greater than the senaration lift- coeffinient rnmpsclose to matching the experimental data in most cases. Thepredicted minimum drag is quite ciose to experijmental levelsfor most cases except for some test data below 0.3 Mach number.Transition grit was fixed on the leading-edge of the model butits possible that at the low Reynolds and Mach numbers the flowwasn't fully tripped causing the test minimum drags to be Low.The predicted polars in the transonic region tended to have toomuch drag at the higher lift coefficients probably because thedrag rise contribution was increasing too fast.

The predicted lift versus alpha curves are in good agree-ment with experimental data at the low lift coefficients. How-ever, the agreement is poor in the high lift regions, when thepredicted maximum lift coefficients don't agree with test data.At the low speed, low Reynolds number condition CLMAX was under-predicted especially for the high sweep and the thin wing con-ditions. The thick wing lift curve slope was underpredictedat the highest Mach number because the predicted transonicthickness correction became too small.

160

9.2 Cranked Wing Study

Comparison of the predicted lift curve slope, drag-due-to-lift factor, minimum drag and pitching moment slope with testdata for three of the wing models reported in Reference 40 areshown in Figures 99 through 102. Each wing was planer andwas mounted separately on a cylindrical body of revolutionwhich had a Sears-Haack nose. Model I was a 59 degree lead-ing-edge triangular planform and Model 2 and 3 had two straightline leading edge segments of different sweep angles (referredto as "cranked" planforms). The three models were designed tohave the same exposed span and exposed area.

The predictions of the lift curve, drag due-to-liftmini-mum drag and moment curve for the Model 1 wing agreed closelywith the experimental data except for the moment prediction atMach 2.94 and the minimum drag at the transonic conditions andat Mach 2.94. The predicted lift curve slope for Models 2 and3 matched the experimental data at the low subsonic Mach number,but the program did not predict the large increase in lift-curveslope at transonic Mach numbers and the program overpredicted

ItLie t-curve bslope- fo b- - - saupersonia LIL.,• LCI Z. ILI; SUsLfLic

and transonic predictions of drag due-to-lift, for Model. 2 and3, agrees well with experimental data, but the supersonic valuesare underpredicted because the lift curve slopes were overpre-dicted. The predicted pitching moment curve slope was in fairagreement with the test data for Models I and 2 up to a Machnumber of J.6; beyond 1.6 Mach the pitching moment was over-predicted. The minimumn drag prediction for Model 2 was infair agreement with experimental data except in the transonicregion. The minimuml drag predictions for Model 3 were on theaverage 14 percent higher than the test results.

9.3 C-141A Flight Test Data iComparison of the C-141A flight test drag polars with full-

scale corrected wind-tunnel polars and Large Aircraft programresults are shown in Figures 103(a), (b), and (c) for M=0.7,0.75, and 0.775. These results indicate that the Large Air-craft program predictions are in good agreement with the flightdata points at the higher values of CL and lower values ofMach number. Comparison of the flight test versus predicteddrag-rise characteristics are presented in Figure 104 as dragvariation with Mach number ior constant lift coefficients.These results indicate that the reason why the drag polar wasunderpredicted at the higher Mach num' rs and lower lift co-efficients was due primarily to overprec±2ctiig the drag riseMach number, Mcr.

161Im

The C-141A flight test data shown in Figures 103 and 104were obtained from Reference 41. The flight test data inReference 42 are corrected to an equivalent rigid-aircraftcondition trimmed at a c.g. location of 0.25 mean aerodynamicchord at a Reynolds number of 55 x 10 6 /MAC. The correctedwind tunnel polars shown in Figure 103 were obtained fromReference 42 where fully-corrected model test data was extra-poled to full-scale Reynolds number. The predictions of theLarge Aircraft program were made using the full scale C-141Ageometry presented in Reference 41. The program prgdictionsare trimmed at a 0.25 MAC c.g. location and 55 x 10 '/MACReynolds number condition. The program prediction used a7 count (0.0007) miscellaneous drag increment in the dragbuildup to represent the roughness drag. The 7 count rough-ness increment was tho same as used in Reference 42 to scalemodel data to full scale.

9.4 Hiýh Lift Configurations

Figure 105 showq the lift, drag, and pitching momentpredictions for a wing-oJdy cowzfiguration compare - h testdata for a clean wing and a partial ind full-span .,Angle-slotted flap (Reference 43). The prt-gram results are in goodagreement with the clean wing lift momnent and drag. The incre-ment in lift due to flap deflection is underpredicted atCL^AX, but the relative predicted differences between thefull-span and partial-span flaps were similar to the testdata. The increment in moment due to flap deflection wasunderpredicted for both the full-span and partial-span cases,while the drag increment for flap deflection was overpredictedon!- .... -r f.-ll-span u• 1on.. .

1652

0- o

CD ' . ~ 0i

~N-~~r-cOc

-r4

C)n

.. I0

N4 W

4-4 4

w ,r-4

ý-4 CO

0 -

a, 4

(I-0o

.&-) 163

'-4

(I,

0 00

Li 0

C-4-

4z0000

1.0

Ii1-

'Q41

C." ~ 70

D ao7H D06

V0 0

Ic,

c'.0

4-''1 I . L~r ) ~'.*

0(U *~ C.

~4

Cý

CO) r- c 'J

Ccu)r r- OC 00 o .

ih- 0 [3 I L-4 C)-

U-)4

IIr

00

16

C4

-cC4

w0 -4U; ~ '

0 Cl

c~00

040

cnn

0 CD

- C

4 - 4

I•

C)

01

0 O

o

II~

C)

U

V)

41.

0 000

168

ENN

Ell

0C

0 0--0

CC

0 0

00

(12r

C9C

169

C 'zt

00

0o-

col

-no

0 C D 0 0

z -4cnc

4J .

U 04

00

170)

C-4~

Ln L ýo 0 a n m 0

000

II

-1- 0

00

4J4

00

- -4

CC

-4 a

171

Ci

'COj 04 r-. ()ot 0

Cli

00

E. C7 ~

-44II 1

~ 1-r4

.~172

C14

CJIY' Olt 0- Lr C) 0 c')

oO

C)

00

c~ti

I~~i 19

11(Y0

ric2

'17

pol pppwmwo

I7'

0- r- 00

aa

00

0r 0

13114

&c-1 -1

v-4

-... i C .C L~n r-- o flfQ) C14 in '0 t-4 r- c-4 00

eo tst r - CO 0O

oKo

- Is-

:3:

00

00

-t4

C) *-175

I- r-ir, c

0 0 ,

LAL

w4 0 0

..... ..

* bo

4-i

-S)%o t

176,

~ ~ ~tn r-4oo .V;77V ~~ ~ 1 LeC'Jlý ýlo 7

lhT '';J;::. .jIy

zi t

I -CI 1 I1

1-4 : ... t i t N .

1771

xb

II~~, -r-,t-- r-tII~. r- . co oo i

00

12D

4J.

U3

0 03

0.46J

CrCh

'0o

040

178

.Cfr-c-Lw otmfl

oo

000

00

v-4,

17

1%D

c! ' HC%00. r- 00 00 a

vii

.I i00

00 46

"0

00

II4

4V

08

I* q.4 .0. -q %I -- c

Ci-4; ýOýO 1 -

.r40

4C'J

~t00

ot

4,

4P

I'.

r- rIoD -444-

00

4'1

000

C>)

v,-

C) C:

CI

rC-,

4 182

'P Cl - 0 00 04

00

(n

4-iU-

0'

00

C,1

183

(2)

CC

UC) 0 B

oCC>

o Bo

r-

I1I

-* LI

0 Test.06©nL - Predicted

-C)

.02

(a) Model

i

0 .4 .8 1.2 1.6 2.0 2.4 2.8

0Q

.06

0CCL

.012

(b) Model 20-:

0 .4 .8 1.2 1.6 2.0 2.4 2.8

.06

0 G

.04 CL•.

.02

(c) Model 3

0 0.4 .8 1.2 1.6 2.0 2.4 2.8

tMACH NUMBER

Figure 99 Lift-Curve Slope ap a Function of Mach Number

185

.6 0 TestI

- rsredicted

.4

K 0

.2

(a) Model 1

0 .4 .8 1.2 1.6 2.0 2.4 2.8

.6

.4

(b) Model 2 .2III0.

.4 .8 1. 16 2.0 2.4 2.8

.6

.2

(c) Model 30__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

0 .4 .8 1.2 1.6 2.0 2.4 2.8MACH NUMBER

Figure 100 Drag Due-To-Lift Factor as a Functionof Mach Number

186

-. 3o0 Test-- Predictedp O__ G

O CL

-. I

(a) Model I0 -

0 .4 .8 1.2 1.6 2.0 2.4 2.8

-. 3[

- omCCI I

L

(b) Model 20 .4 .8 1.2 1.6 2.0 2.4 2.8

-. 3 _'

-. 2

09C L_OCL

(c) Model 301

0 .4 .8 1.2 1.6 2.0 2.4 2.8MACH NUMBER

F.•gure 101- Ftcih.\.is Mmient-Curve' .V1,ope as a Functionoft H~lalu.h huibe 1

187

0 0.016

CDDmin

000

.012 0 (C)

(a) Model 1

0 .4 .8 1.2 1.6 2.0 2,4 2.8

* Olb

.016

raDin ••...._..

.012 0 0

.008

(b) Model 2ng r ]

' i i-

i -- - --- - - - - -n

0 .4 .8 1.2 1.6 2.0 2.4 2.8

.01 02 M

(c) Model 3

0 .4 .8 1.2 1.6 2.0 2.4 2.8MHAH NUMBER

Figure 102 Minimum Drag Coefficient as a Function

of Mach Number~188

- 444

~ tI 14ý

It-

,~. 4444 -

0.50 :j

0.40 77ý T_ 7.

LTRI F.. hYest Data

HI I

.. ... ... ...

(.3) "IF0~O

_77. ;7.1 71t

I 4-r

CL

0.40 ~-

4 Flight Tcst Data

.ýi4 f flrore ModetlDt

0.30 Program Prediction

191Figur 10 -! Cotne

0.190

0.60 lo

'~ -t-.--L

0.50~ :±- --- 4

0.40t I. . 47

0%30 j l'-:P + * Corrected Model Data-- t Program. Prediction

0.20; 4- .-. .... r

13 *t.,t L i9.]

0.10

0.0160 0.0200 0.0240 0.0280 0.0320

C D

Wc M u0.775i

Figure 103 - Concluded

191

ii

C.- 141A0 2 8

.026 C

I ..

024

CD

.022

.020

31Flight

Test Data

.o 1 6 -Large Aircraft.016

-- -"Program Results

0.6 0,7 0.8MACH NUMBER

Figure 104 Drag at Constant Lift Coefficient Versus Mach Number

192

r4 4 /r-. /

4 H -4 -4 ý

4J

-4

0 OD~ 00

4J4

41 -U

GJ 4

4ý

0))

r- 0

0 .4-4

Fa,

r4.

-4 -4C.) 9)-w0 N c 4'044

on0 0 Cz go.44

A40193I

10. CONCLUSIONS AND RECOMMENDATIONS

The Large Aircraft Aerodynamic Prediction Program provides theaerodynamicist with a quick-response capability for evaluating the

aerodynamic characteristics of arbitrary large aircraft or perform.ing design trade studies. The Large Aircraft program offers a sig-nificant improvement over hand calculatLons based on handbookmethods because:

1. The program can consider more complex relationshipsbetween geometric and aerodynamic parameters thatmay be neglected or not feasible in hand calculations.

2. Typical hand calculations require a long series ofintermediate calculations and chart lookups thatrequire a great deal of time to perform and areprone to error.

3. The program always performs the calculations in aconsistent manner, whereas hand-calculation resultsmay vary between individuals doing the same cal-culations.

Before using the Large Aircraft program to analyze an arbitraryconfiguration, the user should familiarize himself with the methodsand operation of the program. If a similar-type configuration withtest data is available, it should b.ý evaluated first with the programso as to establish limits for credible lift, moment, and drag pre-dictions for those types of configurations.

The modular construction of the Large Aircraft program willallow subroutines to be added or replaced to incorporate new and/or improved aerodynamic prediction procedures as they become avail-able. Future improvements to the Large Aircraft program shouldextend the program to better handle fighter aircraft by Including buffetpredictions, transonic maneuvering devices, higher Mach numbers, etc.

194

_________I ___________,______ . . .. __ _._,

REFERENCES

1. Schemensky, R. T., Preliminary Design Aircraft Lift andDrag Analysis Procedure, AeroModule V, General Dynamics'Convair Aerospace Division Report ERR-FW-1363, December1972.

2. USAF Stability and Control DATCOM, Air Force Flight DynamicsLaboratory, October 1960 (Revised August 1968).

3. White, F. M., and Christoph, G. H., A Simple New Analysisof Compressible Turbulent Two-Dimensional Skin FrictionUnder Arbitrary Conditions, AFFDL-TR-70-133, February 1971.

4. Aerospace Handbook, Second Edition (C. W. Smith, ed.),General Dynamics' Convair Aerospace Division Report FZA-381-II, October 1972.

5. Hoerner, S. F., Fluid-Dynamic Drag (Published by the author,Midland Park, New Jersey, 1965).

6. Linnell, R. D., Similarity Rule Estimation Methods III,Flow Around Cones and Parabolic Noses, General Dynamics'Convair Aerospace Division Report M-'A-1059, 10 August1955.

7. Peake, D. J., "Three-Dimensional Flow Separations on UpsweptRear Fuselages," Canadian Aeronautics and Space Journal,Vol. 15 (December 1969), p. 399.

8. Frost, R. C., and Rutherford, R. G., "Subsonic Wind SpanEfficiency," AIAA Journal, Vol. 1, No. 4 (April 1963).

9. Benepe, D. B., et al., Aerodynamic Characteristics of Non-Straight-Taper Wings, AFFDL-TR-66-73, October 1966.

10. Henderson, W. P., Studies of Various Factors AffectingDrag Due to Lift at Subsonic Speeds, NASA TN D-3584,October 1966.

11. John, H., On Induced-Drag Calculation of Uncambered andUntwisted Wings as a Function of the Reynolds Number,Boeing Company Report B-819.41, March 1969 (Originallypublished by VFX, for GDLR Aerodynamics Colliquium,Berlin, October 1968).

195

REFERENCES (Continued)

12. Ray, E. J., and Taylor, R. T., "Buffet and Static AarodynamicCharacteristics of a Systematic Series of Winds DeterminedFrom a Subsonic Wind-Tunnel Study, NASA TI-; D-5805, June1970.

13. Simon, W. E., et al., Prediction of Aircraft Drag Due toLift, AFFDL-TR-71-84, July 1971.

14. A Method of Estimating Drag-Rise Mach Number for Two-Dimen-sional Aerofoil Sections, Royal Aero. Soc. Transonic DataMemorandum 6407, July 1964.

15. Sinnott, C. S., "Theoretical Prediction -f the TransonicCharacteristics of Airfoils," Journal of the AerospaceSciences, Vol. 29, No. 3 (March 1962).

16. Method for Predicting the Pressure Distribution on SweptWings With Subsonic Attached Flow, Royal Aero. Soc. Tran-sonic Data Memorandum 6312, December 1963.

17. Miller, B. D., Notes on Wing Design Methods forcritical Airfoils, General Dynamics' Convair AerospaceDivision Report ARM-061, October 1971.

18. Drag-Rise Mach Number of Aerofoils Having a SpecifiedForm of Upper-Surface Pressure Distribution: Chartsand Comments on Design, Royal Arer. Soc. TransonicData Memorandum 67009, January 1967.

19. Lowry, J. G., and Polhamus, E. C., A Method for Pre-

Angles of Attack in Incompressible Flow, NACA TN 3911,1957.

20. Spencer, B., Jr., A Simplified Method for EstimatingSubsonic Lift-Curve Slope at Low Angles of Attack forIrregular Planform Wings, NASA TM X-525, May 1961.

21. Polhamus, E. C., Summary of Results Obtained by Tran-sonic-Bump Method on Effects of Planform and Thicknesson Lift and Drag Characteristics-of Wings at TransonicSpeeds, NACA RM L51H30, 30 November 1951.

22. Nelson, W. H., and McDevitt, J. B., The TransonicCharacteristics of 17 Rectangular, Synetrical WingModels of Varying Aspect Ratio and Thickness, NACA RMA-51AT 10 May 1951.

196

k/

REFERENCES (Cont' d)

23. Tinling, B. E., and Kolk, W. R., The Effects of MachNumber and Reynolds Number on the Aerodynamic Charac-teristics of Several 12-Percent-Thick Winas Havirn35 Degrees of Sweepback and Various Amounts of Camber,NACA R• A50K27, 23 February 1951.

24. Donlan, C. J., and Well, J., Characteristics of SweptWings at High b~.2cds, NACA RN L52A15, January 1952.

25. Hoerner, S. F., The Lift of a Body-Wing Combination,Wright-Patterson Air Force Base Air Documents DivisionReport F-TR-1187-ND, September 1948.

26. Schemensky, R. T., Braymen, W. W., and Crosthwait, E. L.,Aerodynamic Configuration Analysis Procedure, AeroPtxdule,Version IA, General Dynamics' Fort Worth Division ReportERR-FW-931, 31 December -1969.

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