A-AlOl 6 75 AUBURN UNIV ALA ENGINEERING EXPERIMENT STATION

F/ 20/4

'AN AERODYNAMIC ANALYSIS OF DEFORMED WINGS IN SUBSONIC AND SUPER--ETC

(VI

A17 PR 81 J E BURKHALTER, J M ABERNATHY DAAG29-78-G-0036

UNCLASSIFIED ARO-15666.4-AE ML

411

ABURN UNIVERSITYENGINEERING EXPERIMENT STATION

U AEROSPACE ENGI NEERING

1 AN AERODYNAMIC ANALYSIS WO DiFORMED WINGS

3 IN SUBSONIC AND SUPERSONIC FLOW

FINAL REPORT

Contract No. DMAG29-78_G-0036

January 1980-April 1981

lumi~rn.U.S. Arrq Research OfficeResarch Triangle Park, N.C. 27709

Ic

AI-

I

AN AERODYNAMIC ANALYSIS OF DEFORMED WINGS

IN SUBSONIC AND SUPERSONIC FLOW

by

John E. Burkhalter

John M. AbernathyMilton E. Vaughn, Jr

FINAL REPORT

Contract No. DAAG29-78-G-0036 CJanuary 1980-April 1981

II

Department of Aerospace Engineering

Auburn University, Alabama 36849

i for

U.S. Army Research Office

Research Triangle Park, N.C. 27709IiAiproy'nn for puliuc releas;

iii

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1 1.

Unclassifiped

SECURITY CLASSIFICATION OF THIS PAGE (Wheni OM. B,,L0t0d)

REPORT DOCUMENTATION PAGE BEFOR DCOKLEI FR1. REPORT NUMOIN -12, 3VT ACCESSION No. /I NEC IET, C ALOG MUM§CfR

4. TITLE (and Subtitle) - ia,~~t

An_ Aerodynamic Analysis of Deformed Wings in, Jan' 180- April P817Subsonic and Supersonic 1iw 4- tM

7- A nt)urimltr S. CONTRACT ON GRANT NUM6E

John M4. kbernathy DAAG9-1-003617

I - Milton E./Vaughn, JrL______________I S PE~o~a~o~~tt11N ~AME AND ADDRESS 10. PNOGRAM ELEMENT. PNOJECT, TASK

u ANEA & WONK UNIT NUM9ENSEngineering Experiment StationAuburn UniversityAuburn, Alabama 36849 ______________

11. CONTROLLING OFFICE NAME AND ADDRESS _L..TZX

U . S. Army Research Office pi08P. 0. Box 12211 UE ~P~EResearch Triangle Park, NC 27709 200_____________

1. MONITORING AGENCY NAME II ADORESS(If different from Confrollind Office) 15. SECURITY CLASS. (of this eport)

Same./~ _______ unclassified150. DECLASSIFICATION/OOWNGRADING

SCHEDULE

16. DISTRIUUTION STATEMENT (of this Report)

Approved for public release; distribution unlimited.

17. OISTRISPUTION STATEMENT (of the abstract en~tered In Block 20, It dLacLmo)

Same

3It. SUPPLEMENTARY NOTES

The findings in this report are not to be construed as an officialDepartment of the Army position, unless so designated by other authorized3 documents.

It. KEY WORDS (Continue on reverse side If necessaryinmd Identify by block number)

Aerodynamic Wing Loading, Elevon Loading, Hinge Moments, Deformed Wings,

Wing-Elevon Gap Effects, Subsonic and Supersonic Flow

20..AArSTRAC r (Continue an reverse siede If necessary and identify by block number)

"The aerodynamic loading for deformed wings with elevons in both sub-sonic and supersonic flow is considered. The solution procedure falls intothe potential flow category with appropriate restrictions. For subsonic

flow, r lifting surface Kernel function formulation is used in which thelocal pressure loading for both wing and elevon is determined simultaneouslyin a ,emi-closed summation manner. Cases under study included gaps betweenwing and elevon in addition to arbitrary wing-elevon deformations. ResulLs.-o.ig

DD AN7 1473 EDITION OF I'NOV 65 IS ODSOLETE Unclassi f iedI :73 ~~~SECURITY CLASSipir*Ty-..nr ,v emn ~-,. ..a a.-nw

20. ABSTRACT (CONT)

I for all cases compared very well with experimental data. Experimental datataken in a low speed wind tunnel is also presented for a cropped delta wingand rectangular elevon in which the wing-elevon gap was the primary testvariable. For supersonic flow, 3-D supersonic theory forms the basis forthe solution procedure. Deformations are accounted for with the use ofdoublet paneling added to the basic 3-D solution. Results agree very wellwith existing experimental data. The gapped elevon and thick wing trailingedge problem is also addressed with satisfactory results.

Accession For

NTIS GIA&InTTC V.3

'" Distribiitlion/

I Avail,blilty Codes

wAvai and/oriDist Special

I

I

I

!

ABSTRACT

The aerodynamic loading for deformed wings with elevons in both

subsonic and supersonic flow is considered. The solution procedure falls

into the potential flow category with appropriate restrictions. For

subsonic flow, a lifting surface Kernel function formulation is used in

which the local pressure loading for both wing and elevon is determined

simultaneously in a semi-closed summation manner. Cases under study

included gaps between wing and elevon in addition to arbitrary wing-

elevon deformations. Results for all cases compared very well with

experimental data. Experimental data taken in a low speed wind tunnel is

also presented for a cropped delta wing and rectangular elevon in which

the wing-elevon gap was the primary test variable. For supersonic flow,

3-D supersonic theory forms the basis for the solution procedure.

Deformations are accounted for with the use of doublet paneling added to

the basic 3-D solution. Results agree very well with existing experi-

mental data. The gapped elevon and thick wing trailing edge problem is

also addressed with satisfactory results.

I PREFACE

The work as reported in this document represents the completion ofthe various tasks as prescribed in Contract No. DAAG29-78-G-0036, forthe Array Missile Command, Huntsville, Alabama. This research ismonitored through the U.S. Army Research Office, Research Triangle Park,North Carolina, and is the result of the work done during the periodJanuary 1980-April 1981. The program manager with the Army MissileCommand was Dr. Donald J. Spring.

Some of the experimental work presented in Chapter III and ChapterIV was supported by Auburn University Engineering Experiment Station.

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ACKNOWLEDGMENTS

The authors gratefully acknowledge the advice and support ofDr. Donald Spring of the Army Missile Command and Dr. Robert E.Singleton of the Army Research Office in the performance of thisresearch study. The typing assistance of Mrs. Marjorie McGee andthe computer programming expertise of Miss Pamela F. Gills arealso greatly appreciated. The experimental work was performedin the Aerospace Engineering Departmental wind tunnel at AuburnUniversity and was sponsored by the Engineering Experiment Station.

I

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TABLE OF CONTENTS

LIST OF FIGURES ............................ v

LIST OF SYMBOLS. ......................... viii

I. INTRODUCTION................... . . ... . ... .. .. .. .....

Subsonic Flow................... . . ... . . .. .. .. .....

Supersonic Flow ......................... 3

II. SUBSONIC FLOW-THEORY. ...................... 7

Fundamental Equations ...................... 7

Loading Function for Wing-Gap-Elevon Configuration .. .. ... 10

Total Aerodynamic Forces and Moments. .. ........... 14

Elevon Hinge Moments .. .................... 19

Non-Planar Wings .. ...................... 22

Interference Effects .. .................... 23

Order of Polynominals. .................... 25

III. EXPERIMENTAL MODEL AND TESTS -SUBSONIC 27

Model. ............................ 27

Test Procedures. ....................... 27

IV. RESULTS - SUBSONIC FLOW. ................... 30

V. CONCLUSIONS - SUBSONIC FLOW .. ................ 53

VI. SUPERSONIC FLOW - THEORY. ................... 55

Fundamental Equations. .................... 55

Lift and Pitching Moment for Wing Elevon Combinations . .. 58

Wing Downwash on Elevon. ................... 62

Wing Trailing Edge Blocking of Elevon .. ........... 68

jTABLE OF CONTENTS (CONT)

VII. SUPERSONIC FLOW - RESULTS. .. ................. 74

IPlanar Wings. .. ......................... 74

Deformed Wings. .. ....................... 90

Elevon Loading .. .. ..................... 103

VIII. REFERENCES .. .. ....................... 107

IAPPENDIX A. .. ........................... 110

APPENDIX B. .. ........................... 172

SUBSONIC COMPUTER PROGRAM USERS GUIDE .. .. ......... 173

SUPERSONIC COMPUTER PROGRAM USERS GUIDE .. .. ........ 179

Ii

I1I LIST OF FIGURES

Figure No. Title Page

1 1 Wing and Subpanel. Coordinates. .............. 9

2 Wing Geometry and Aerodynamic Forces ... .......... .. 12

3 Chordwise Loading Functions ..... ............... .. 13

4 Control Point Location and Corresponding Matrix forWing-Gap-Elevon Configuration ..... .............. .. 15

5 Experimental Model Details ..... ............... .. 28

6 Comparison of Two Theories for a Swept and TaperedWing of Aspect Ratio 2.828 ..... ............... .. 31

7 Comparison of Two Theories for a Rectangular Wing ofAspect Ratio 2 ........ ..................... .. 32

8 Leading Edge Suction on a Highly Swept Cropped DeltaWing with a Fuselage ...... .................. .. 33

9 Chordwise Loading with a Deflected Elevon at=0.5 .......... ......................... .. 34

10 Lift and Hinge Moments for a Wing-ElevonConfiguration ......... ...................... .. 35

11 Aerodynamic Characterirtics for a GappedConfiguration . . . . . . . . . .. . . . . . . . . . .. . 37

12 Gap Effects on Chordwise Load Distribution ........ .. 39

13 Elevon Hinge Moments for Several Hinge LineLocations .......... ........................ .. 40

14 Characteristics of a Wing-Fuselage-Elevon Model for

HL /LCE = 0.016 and /cE = 0.004 .... ............. ... 42

15 Characteristics of a Wing-Fuselage-Elevon Model for

XHL/cE = 0.016 and /cE = 0.497 .... ............. ... 44

16 Hinge Moments on a Wing-Fuselage-Elevon Model for' XHL C = 0.254 ........ ..................... .. 46

V

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LIST OF FIGURES (CONT)

Figure No. Title Page

17 Hinge Moments of a Wing-Fuselage-Elevon Model for

XHL/CE = 0.507 ......... ..................... .47

18 Characteristics of the NACA 747A315 Airfoil Section. 50

19 Characteristics of a Finite Wing that is Both

Chambered and Twisted ....... .................. . 51

20 Schematic of Wing-Elevon Coordinate System and WingTrailing Vortices ........ .................... . 64

21 Schematic of the Wing-Elevon-Trailing Vortex Filament

Configuration ......... ...................... . 66

22 Dimensional Planform and Sectional Sketch of anExperimental Wing-Elevon Configuration, Ref. 1 ..... ... 69

23 Schlieren Photographs of the Blunt Wing Trailing Edge

and Elevon Leading Edge, M=1.94 .... ............. ... 72

24 Dimensional Sketch of Four Wing Planform Shapes Under

Study ........... .......................... .75

25 Pressure Distribution on a Flat Trapezoidal Wing;

M = 2.01, a = 4.0 ........ .................... . 77

26 Sectional Normal Force and Pitching Moment for a Flat

Trapezoidal Wing; M = 2.01, a = 4.0 ... .......... . 79

27 Pressure Distribution on a Flat Delta Wing; M = 2.01,

a= 4.0 .......... ......................... ... 80

28 Sectional Normal Force and Pitching Moment for a Flat

Delta Wing; M = 2.01, a = 4.0 ..... .............. .82

29 Pressure Distribution on a Flat Tapered Wing;M = 1.61, a = 2.0 ........ .................... . 83

30 Sectional Normal Force and Pitching Moment for a Flat

Tapered Wing; M = 1.61, a = 2.0 .... ............. ... 85

31 Typical Multiple Isobars on a Rectangular Wing

Planform .......... ........................ . 86

32 Typical Multiple Isobars on a Delta Wing Planform. . . . 87

vi

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LIST OF FIGURES (CONT)

Figure No. Title Page

33 Typical Muitiple Isobars on a Cropped DeltaWing Planform ......... ...................... . 88

34 Typical Multiple Isobars on a Trapezodial Wing

Planform ......... ........................ .. 89

35 Schematic of Three Deformed Wing Shapes ........... .. 91

36 Pressure Distribution on a Cambered Delta Wing;M = 2.01, a = 6.0 ........ .................... . 92

37 Sectional Normal Force and Pitching Moment for aCambered Delta Wing; M = 2.01, a = 6.0 .. ......... .. 94

38 Pressure Distribution on a Cambered and Twisted Delta

Wing; M = 2.01, a = 6.0 ...... ................. .. 95

39 Sectional Normal Force and Pitching Moment for aCambered and Twisted Wing; M = 2.01, u = 6.0 ... ...... 97

40 Pressure Distribution on a Warped Trapezoidal Wing;M = 2.01, x = 6.0 ........ .................... . 98

41 Sectional Normal Force and Pitching Moment for a WarpedTrapezoidal Wing; M = 2.01; -Y = 6.0 ... ........... . .100

42 Elevon Normal Force and Hinge Moment Coefficientsversus Wing Angle of Attack, 6 = -100, Gap = 0.0 . . . .104

43 Theoretical Normal Force and Hinge Moment Coefficientsversus Wing Angle of Attack for Gap = 0.0 and 4.0",

-10= .......... ........................ 106

I

vii

A- 1

II

LIST OF SYMBOLS

Symbol Definition

a Nondimensional body radius

AR Aspect ratio

b Wing span

BL Wing loading coefficients

B Wing loading coefficientsnm

B Elevon loading coefficientspq

c Mean aerodynamic chord, reference length

c Local wing chord*

c Root chord of elevon*

CTE Tip chord of elevon*

C Local wing chord**

C Local chord of wing**~W

C E Local chord of elevon**

CR Root chord**

j CT Tip chord**

c c Section pitching moment coefficient

CL Lift coefficient

CM, CM Pitching moment coefficient about wing root chord

y leading edge

CH Hinge moment coefficient

C Pressure coefficient (p-p,,)fq.

AC pPressure loading coefficient (pressure difference betweenupper and lower surface)

*Subsonic flow

**Supersonic flow viii

U

LIST OF SYMBOLS (CONT)

Symbol Definition

HL Hinge line of elevon

I Integral of the chordwise term of the pressure loadingfunction

TE

LE

AK Value of Kernel function integral over a subpanel

L Lift force on the wing

M Moment produced by the wing

M.o Freestream Mach number

AP Pressure differential across the wing

P(Xoy o) Pressure loading parameter from 3-D supersonic flow theory

q Freestream dynamic pressure

rb Dimensional body radius

S Wing planform area, reference area

tTE Wing trailing edge thickness

V Freestream velocity

VLE Leading edge vortex

w Non-dimensional perturbation velocity W/V, parallel to2-axis

x,y,z Cartesian coordinates

NH Hinge line location along x-axis

'LE Distance from y-axis to leading edge of wing

XoY o Integration variables

ix

1 7-

LIST OF SYMBOLS (CONT)

Symbol Definition

Greek Symbols

a Angle of attack

Mach flow parameter,

Gap width between wing trailing edge and elevon leading edge

y Non-dimensional vorticity F/V.

6Elevon angle relative to wing (trailing edge down is positive)

A Taper ratio, (tip chord)/(root chord)

A Wing leading edge sweep angle

Nondimensional perturbation velocity potential VV

e Nondimensional spanwise variable

e Local deformed wing slope at a control pointC

a Local deformed wing slope relative to the chord line

T Positive surface slope of the aft portion of the wing

Eq Non-dimensional chordwise and spanwise variables, respectively

Subscripts

E Elevon

H Hinge characteristics

HL Hinge line

LE Leading edge

nm Point, or constanLs associated with the wing

n Nor.ial

pq Points or constants associated with the elevon

o Denotes wing coordinate variables

T Total

x

Ft4LIST OF SYMBOLS (CONT)

Symbo Definition

I Subscripts

TE Trailing edge

C Freestream conditions

w Wing

x,y,z Coordinatu axes

i,j,k Unit vectors

FWD Forward

AFT Aft

I xi

I

I. INTRODUCTION

Subsonic Flow

It has been observed that failures associated with the control

surfaces have occurred on certain missile configurations. It is believed

that these failures have occurred because of adverse loading on the

control surface due to large deformations in the wing surfaces. A pre-

liminary study was completed which attempted to identify the aerodynam-

ic changes which occurred because of these deformations. The present

study addresses the problem in more detail including various wing-elevon-

gap planform configurations, subsonic and supersonic Mach numbers, moderate

wing and elevon deformations of an arbitrary nature, and blunt trailing

edges for the wing.

In recent years, major efforts have been made to model multi-element

airfoil configurations (wing-elevon combinations) and to predict resulting

loads and moments. Most of the research efforts concerning these multi-

element lifting surfaces have been two-dimensional analyses as typified

by the early work oi Glauert2 ,3 in 1924 and 1927 and more recently by the

4,5work of Halsey. However, the case of the wing, air gap, control

surface combination has generally been neglected as pointed out by Ashley6

who also expressed the need for such an analysis.

Extension of analyses to include three-dimensional effects or finite

wings has generally employed vortex lattice formulations or some form of

7 8

constant pressure paneling such as that of Woodward and Lan. .The Lan

approach has received considerable attention of late since the leading

1I!

.1!

2

edge suction terms and Kutta condition at the trailing edge are properly

accounted for in the analysis. DeJarnette 9 extended the "strip" approach

of Lan to provide for a "continuous" loading in the spanwise direction.

Aside from the Lan-DeJarnette formulation, general vortex lattice

approaches suffer from traditional shortcomings such as large computer

storage and long run times even on today's high-speed computers. Formu-

lation of the problem using the Kernel function approach in its usual form

10,11as typified by Cunningham has been generally neglected but offers the

potential for increased accuracy, lower computer run times and less8 9

storage than does the vortex lattice approach. The Lan -DeJarnette

analysis requires comparable run times and storage and offers a viable

alternative to the solution of the problem at least for single element

airfoils. Extension to multi-element airfoils, as far as is known, has

not been completed.

For the most part, previous theoretical analyses have been restricted

to planar lifting surfaces with no camber or twist. Wing-elevon con-

figurations generally have been sealed-6ap cases. However, White and

Landahl12,13 have developed ;, procedure requiring the method of matched

asymptotic expansions which is used to determine the load distribution

when a gap exists between the wing and elevon in two-dimensional flow.

Both the vortex lattice and Kernel function approaches have enjoyed

certain successes but certainly the vortex lattice has been more exten-

sively used. There are certain problemts which, though amenable to vortex

lattice solutions, are more easily handled by the present lifting surface

techniques; namely, that of wing irregularities such as twist, camber,

and arbitrary deformations. Certainly camber and twist have been

! ...I

I ,,-- " .. . i ._,,. ..,. ,

I

addressed in previous efforts, but analyses of moderate "arbitrary"

deformations of a lifting surface have not been found in the literature.

The present work is concerned with the wing-elevon problem for

configurations with moderate wing deformations such as camber, twist,

and general deformations due to high wing loading or distortion of the

surfaces due to aerodynamic heating. Lifting surface theory is employed

with the solution following the procedure established by Purvis14 and

Burkhalter, et al. 1 5 The loading function over the multiple lifting

surface is defined, and the Kernel function is then integrated over the

surfaces. For this work, the gap distances are considered to be small

enough that vortex rollup is assumed negligible.

Comparison is made with experimental data on several general con-

figurations. For the gap case, data was obtained from low speed wind

tunnel tests with gap distance as a primary test variable. Data for

several elevon deflection angles and hinge line locations are shown for

each configuration at numerous angles of attack. Experimental data is

lacking for wing load deformations at subsonic speeds but comparisons

are made with uiings which have t:-d,...te camber anl twist.

Supersonic 1low

For supersonic flow there are several approaches which address the

problem of predicting loads cn a planar wing. These methods range from

two-dimensional shock ezpansi,,n theory to three-dimensional wing theory

to vortex lattice theory. Some approaches are useful for variably swept

wings while others are only applicable to wings with straignt leading

edges. The basic elements of the major supersonic flow theories will be

summarized in the following paragraphs.

I

4

Shock-expansion theory is two-dimensionally restricted to attached

shocks but it is general and can be used for thick and thin 2-D bodies

alike. However, because the results are difficult to express in concise

aLialytical form, shock-expansion theory is mainly used for obtaining

numerical solutions for diamond shaped airfoils. If, however, one

assumes a thin wing in steady flow at a small angle of attack; i.e.,

small perturbations; then the approximate relations for weak shocks and

expansions may be used. This thin airfoil theory results in a simple

analytical expression for calculating the pressure coefficient.

Although these two-dimensional theories are valuable, they are not

sufficient to correctly model three-dimensional flow. Two of the first

three-dimensional theories were point source theory and line source

theory. These theories were adapted to thin supersonic airfoils with

straight leading edges and various types of symmetrical crossections.

In each theory the velocity potential for the wing is obtained by

analytically integrating over the forecone from the point in question.

Either subsonic or supersonic leading edges may be handled in this manner

as well as delta wings with supersonic leading and trailing edges. These

theories, however, do not allow for the interaction between the upper

and lower surfaces of the wing which would result from subsonic leading

edges as well as from subsonic wing tips.

Thin wings with straight subsonic leading edges may be handled by

conical flow theory. As the flow properties are constant along rays

emanating from the vertex of the wing the flow is two-dimensional. The

Tschaplygin transformation is used to transform the small perturbation

equation in polar coordinates into the two-dimensional Laplace equation.

f!I

! I,

5

Complex variable techniques developed for incompressible flow are then

j used to obtain an analytical solution for the pressure coefficients.

This technique has also been extended to wings with straignt supersonic

leading edges and also to wings with straight subsonic and supersonic

leading edges combined.

All of these techniques are basically inadequate in predicting

loads for a deformed wing; a deformed wing in this case meaning moderate

arbitrary deformations which include camber and twist. Most are amenable

to general planform shapes but are usually restricted to straight leading

and trailing edges and do not include fuselage effects. A more recent

attempt to include curved leading edges was completed by Carlson1 6 in

which a vortex lattice scheme was used to model the wing planform. Al-

though resulting computations were highly oscillatory, a "smoothing"

routine introduced by Carlson produced acceptable results for thin planar17

wings. Another recent formulation of the problem utilizing supersonic

line sources has also produced good results on curved planform shapes

and does not require any post data manipulation.

The solution technique used in the present research is fundamentally

based on three-dimensional supersonic flow theory as outlined in Refs. 18,

15 and 19, but is modified to account for wing deformations. Potential

flow is still assumed so that solutions to the potential equations may

be added. .The deformed wing is "overlaid" with a doublet paneling sheet

whose strength everywhere is determined through matrix inversion tech-

niques and the doublet solution is then added to the basic three-dimensional

supersonic solution. This is accomplished by subdividing the wing planform

into very small panels over which the pressure is assumed constaLit.

! ~. -.-- -

!6Integration is completed over each subpanel in closed form and results

may then be written in summation notation. The vorticity paneling

produces perturbation velocities which account for deformations in the wing

surface.

t

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II. SU3SONIC FLOW-THEORY

Fundamental Equations

The appropriate equations and solution procedure for the Kernel

function formulation used in the subsonic analysis are outlined in this

section. The equations are well known (Refs. 20 and 21) and the solution

procedure is developed in detail in Ref. 14.

Compressibility is included in the potential equation which is

written as

XYZ (x ,y) [ (x-xO) 1 dyi *(x,y,z) = 0 0 (-o+z 1 + 0x ° dd °

4"rf J (Yy )2+Z2 I+2 y_2S 0 (x-x )2+ (y-y )z+ 2zZ

0 0J

The downwash at an arbitrary point (x,y) in the z=O plane due to an

infinitesimal area (dx° dy ) of a lifting surface is

0 0

w(x,y,O) = L _J 2 1+ 0 dx dy (I).... (y-Y)2 (X-xo) 2+ 2(y_yo)

2 J

where E and q are dimensionless :hordwise and spanwise variables, respec-

tively. These variables are made non-dimensional by the definitions:

x-x (y)LE , (2)

c(y) b/2

The functional form of the pressure loading coefficient is assumed as

N 1 MnA C(n) B sin(2m+l)e (3)

n=0 m=0

7I!

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where rl = cos 0 and N and M represent an arbitrary number of chordwise

and spanwise control points, respectively. It is readily seen that this

form produces the required leading edge square root singularity, satisfies

the Kutta condition at the trailing edge, and has slender wing behavior

at the wing tips. A discussion of the logic behind this assumed form of

the pressure loading is presented in Reference 14.

Over a sufficiently small subpanel of the wing planform, AC isP

essentially constant and may be taken outside the integral of eq. (1).

The resultin , expression may be evaluated in closed form to give

Aw(x,y) = P [K(x 2 ,Y 2 ) -K(x2,.y) -K(x ,y2 ) +K(x1 ,Y1 )] (4)811

where, as an example, K(xly I) is evaluated in the form

K(x y ) (x-x1) + -x1 )2 + 32(y-yl)2

(Y-Yl)

)2y ay1 ) )+ log [ (y-y I ) + /(x--xl)2 + 2(y-y)2 ] (5)

e -l1 1

The other K's of eq. (4) are evaluated by making the appropriate sub-

stitution of the subscripted variables in eq. (5). Also in eq. (4),

AC (E,rI) is evaluated at the centroid of the subpanel. The wing andP

subpanel are illustrated in Fig. i.

For small angles-of-attack, the boundary condition imposed is the

requirement of no flow through the wing; i.e.,

w(x,y) + sin a(x,y) = 0 (6)

L[

x

-0'~ dy0

X~vy dx

CT

8=7T/2 8=0

j Figure 1. Wing and Subpaiiel Coordinates.

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The total downwash at any control point due to the entire wing is found

by summing the effect of eac. subpanel of the surfacc and Eq. (6) becomes

XYAw(x ,y) + sin a(x.,y.) 0 (7)S

where (x.,y.) are the control points, and are located according to a1j

cosine distribution 9

c(y.)x= XLE (y) + - _ (I - cos -$1) , i = 1,2,.. .N

LE1 2 N+1

"(1 - Cos "i) I = 1,2,.. .M (8)yj 2 M+l

The set of simultaneous, linear equations for the NxM unknown loading

coefficients, B 's, defined by substituting eq. (3) into eq. (4) andnm

the result into eq. (7), may now be solved, andthe loading at any point

on the wing is defined by Eq. (3)

Loading Function for Wing-Gap-Elevon Configuration

The above analysis is for a wing configuration only. When a control

surface (e]e-.on) is added to the wing, the load distribution will of

course change. Vurther changes in this distribution occur when a gap is

introduced between the wing ani elevon. In the present work, it is assumed

that the elevon span is equal to the wing span, and the elevon hinge

line is perpendicular to the wing center!ine. With these stipulations, the

solution procedure for the basic wing is paralleled except for the loading

function.

First, for the sealed-gap case with a deflected elevon, a two-function

jformulation using eq. (3) to describe the pressure distribution is used.

The nondimensional chordwise variable, ', of eq. (3) is redefined and isI

i __ _ _ _ __ _

11

considered separately for the wing and elevon. For the wing and elevon,

respectively, it becomes

X-XL (Y) X-X E(Y)

XXL = , - E (9)

W (Y) + cE(Y) ' F cF(Y)

This geometry is shown in Fig. 2. With these definitions, it is easily

verified that the leading edge singlarity is satisfied on both the wing

and elevon, a:d the Kutta condition is met at the elevon trailing edge.

These singularities agree with usual assumed loading function with con-

trol surfaces pointed out by Landahl in Reference 13. At the wing

trailing edge, the Kutta condition is not satisfied, nor is the usual

singularity obtained, but rather a finite value. This loading is used to

allow a closed form solution to the sectional lift integral. Error due

to this assumption is considered to be on the order of error due to the

assumption of an infinite loading, while the advantages of a closed form

solution are considerable. This (chordwise) distribution is shown sche-

matically in Fig. 3a.

Whmun a gap exists between the wing and elevon, the loading function

over the wing and elevon is defined in order to create the effect of two

wings. The leading edge singularity and Kutta condition must be satis-

fied on each surface. Equation (3) produces this distribution by

redefining the chord-wise variables illustrated in Fig. 2 as

LEw(Y ) XXLEE(Y)

y ) c E (y) (10)

for the wing and elevon, respectively. This loading is shown in Fig. 3b.

12

y TI

I z

W P

1 Figure 2. Wing Geometry and Aerodynamic Forces.

13

p

a) Sealed-Gap Case TE w/LEV

b) Gap Case TE w LE E

Figure 3. Chordwise Loading Functions.

I

14

After the load distribution has been defined, eq. (7) with eqs. (3)

and (4) is used to produce a set of simultaneous, linear equations for

the unknown coefficients for both the wing and elevon. The typical

control point location and corresponding matrix for the unknown coef-

ficients are shown in Fig. 4. The control points are located using eq.

(8) separately for both the wing and elevon. The summation is performed

over the entire surface of the wing and elevon, and for the control

points on the elevon, the boundary condition becomes

w(xiY.) + sin[=(x.,y 0 (11)

It should be noted that the loadings for the -ing and elevon are

solved simultaneously, therefore, iteration of the interference between

the surfa s iminated. The pressure ai any point on the wing or

elevon is given by eq. (3), with the appropriate definition of CW and

E (eq. (9) or (10)), and the appropriate values of the B nm's (wing or

elevon).

Total Aerodynamic Forces and Moments

From the wing-elevon configuration, shown in Fig. 1, tve total lift

is given by

y=b/2 x=x TEJ AP dx dy (12)

y=-b/2 X=XLE

IIII

I 15

10y ,nI

1+

2±+

++ 6+I 1+ 156 179+ 119+E

14 + 16± 18 + 20 +

oing Elevon 1 B

IWing wing

Wing I El evonon on B pqCs-6

Elevon I Elevon p

Figure 4. Control Point Location and CorrespondingI Matrix for Wing-Gap-Elevon Configuration.

16

Separating eq. (12) into the contributions for the wing and elevo 1 , and

using the definition for the pressure coefficient, the above equation

becomes

b/2 x TEWxTE

L = q f J AC dxW CPd d (13)

-b/2 x LEW X LEE

It is convenient to define the sectional lift coefficient as

TE

cc z f AC Pdx (14)

LE

Considering the sectional lift for the entire configuration to be

cc z(Y)0 cc k (y 0 ) + er ' (yo) (15)

the lift becomes

b /2

L = q ccZ(y0 ) dy (16)

j -b/2

wherexTE W TE E

cc (y) AC dx + AC dx (7kL 1 P E (17

E

Combining eqs. (9) and (10), the dimensionless chordwise variable can

be defined by

x L E W ( Y ) X L E E ( )( 8

WTE

I

where

i cW ,gap case

C cW + cE, gap-sealed case

With these definitions, the sectional lift becomes

TEW 1

cc (Y°) =f AC ( W q)c T d[w + f ACP ( E'n)cE dEE (19)

0 0

Using eq. (3) for the pressure distribution, the integration of

eq. (19) may be performed as in Reference 15 to give

NW Mccz(Yo n nmsin(2m+l)O

n=O m=0

NE M

+ I B sin(2q+1)0 (20)

p=O q=O pq

where B represents the pressure loading coefficients obtained fromnm

eq. (11) corresponding to the wing loading, and B represents thePq

coefficients of the elevon loading. Also, in eq. (20) for the wing

* 0= T + T 2 +1 i 1(2T - 1) (21)4 TEW TEW 2 w

and

n 2 n - TE - TEW ) , nl,2,...NW (22)

and for the elevon

I0 = T (23)

(23

and

Ip = 2p-) p-i p = 1,2,...N (24)p 2(p+l) IpI'E

Note that for the gap case, TE = 1, and eqs. (21) and (22) reduce to

eqs. (23) and (24) with n instead of p.

I 1

18

Returning to eq. (16) and using the dimensionless spanwise variable,

n, of eq. (2), the total lift is

1

L = q f cck(9) dri (25)

-1

With the definition of the lift coefficient, and for a symmetric wing,

eq. (25) becomes

1

C f cc 0) d n (26)L S 9(

0

Similar manipulation of the force equations leads to the total moment

about the root chord leading edge (of the wing) as

12 f c (7

CM =- c 2c () dn (27)

CM Sc

0

with the sectional pitching moment coefficient given by

NW

c c (Ir) = XLE [ . i .in(2m+l)

+ C B~W nm+ c Tri' l nmB ;n(2m+l)O

T n=O m= +

N -

+ xLE Ip B sin(2q+l',j

E p 0 q0 p

NE M+ cE X ! I p + Bpq sin(2q+l)O (28)I p0 q0O

where the first two terms are seen to be the moment due to the wing, and

the last two are the moment due to the elevon.

I , III - H ii i - : ii lilli i ii i i

- :1 ...- ~ ~ ii j* * .

19

For the wing alone, the integration of the lift and moment equations

is performed analytically in closed form. However, because of the

discontinuity resulting from the addition of an elevon, eqs. (26) and

(27) must be integrated numerically when a gap exists or the elevon is

deflected.

In defining the lift and moment coefficients for the entire configur-

ation, the reference area, S, is the total surface area of the wing and

elevon combined, excluding the gap area. Similarly, the reference length

in the moment coefficient equation is the mean aerodynamic chord, c, of

the wing and elevon, without considering gap distance.

Elevon Hinge Moments

When an elevon is included in the configuration, elevon hinge moments

are important in determining control forces for the vehicle. For a

straight elevon leading edge with the hinge line located at the leading

edge, the total elevon hinge moment is

1

MH = qb f c2c (q)dn (29)

& -1

where the sectional hinge moment coefficient is defined as

2 (q) = c dE (30)

"'c 0

Writing eq. (29) in terms of the moment coefficient and for a symmetric

Iwing, it becomes1

CH 2 f c (T) dn (31)

S E E mH

EE 0

II

I '20

Considering both the wing and elevon contributions of eq. (27) separately,

j it is noted that eq. (30) is the elevon moment about the elevon leading

edge, the solution of which is given by the last term of eq. (28), or

NEM

C2 C cE I p+ B sin(2q+l)0 (32)p=O q=O Pq

Substituting eq. (32) into eq. (31), the integration may be evaluated in

closed form (Reference 14) to obtain

2c N c -c N M2R NE RE-T E NE M

CH = E B I + E B I G (33)SHE- Bp1 Ip+l - Y YS p0 SEc E p=0 q=O pq p+l q

where

T/2

G = f sin 20 sin(2q+l)0 dO (34)

0

For the elevon hinge moments, the reference area, SE , is the total

area of the elevon. Also, the reference length, cE is the mean aero-

dynamic chord of the elevon.

Thus far, the analysi, has bein limite-d to a hinge line at the elevon

leading edge. If the elevon lift ir also .nown, the lift Liid moment

produce a couple at the elevon leading edge. 1he moment about any chord-

wise station-may be found by summi:ig moments at that point. This gives

MHL NLE I LE XHt (35)

I

II

i21

where XHL is the chordwise location of the hinge line measured from the

elevon leading edge. Dividing by q SECE gives the equation in coefficient

form

x

CH CH + CLEe :F1, (36)

where CH is given by Eqs. (33) and (34).

The total lift for the elevon is

i

C _ cc (f) d, (37)

0

with the sectional lift of the elevon given by

1

cc (r)= c fAG (FEl)d E (38)ceEn E CPE E

0

It is readily seen that this is the second integral in eq. (19) and

the solution is given by the last term of eq. (20); thus

E Mcc (q) = Y I B sin(2q+l)0 (39)!1 , E p=O q=O p p

where I is given by eqs. (23) and (24).

For the elevon, the lift may be obtained in closed form. Substi-

tuting eq. (39) into eq. (37), and integrating gives

NE

CL I B 1 (40)L 2S E pO p

where, in the loading coefficients B pq only the q=0 term appears.

Thus, using eqs. (40) and (33) in eq. (36), the elevon hinge moment

about the hinge line is obtained in closed form. The only requirement

L" IIII ,, ,, , .. . ... , - _ _ -,9 . . .

22

for the analysis is that the hinge line be perpendicular to the wing

root chord.

Non-Planar Wings

Lifting surfaces which do not lie entirely in the x-y plane present

a different view to the flow field than the planar wing previously con-

sidered. To analyze non-planar wings, it is necessary to consider the

angle that each control point and each grid element makes with the x-y

plane. Non-planar effects of primary interest include wing twist, cam-

bered airfoils, and arbitrary chordwise and spanwise deformations.

For the case of w. ig twist, at any given spanwise location on the

wing, the chord line lies in a plane that differs from the initial x-y

plane by the twist angle, a., at that station. This angle may be in-

cluded in the boundary condition such that eq. (6) becomes:

w(xiY.) + sin [t(xi,Y) + TW (xiyj)] = 0 (41)

The boundary condition for the elevon, eq. (11), may be rewritten in a

similar manner.

For cambered or arbitrarily deformed airfoils, the chord line may

be replaced by the mean ca ,f er line. The downwash produced by each grid

element is proportional to the angle which that element makes with the

x-axis, or

Aw(x.,y.) Aw(x.,y.)cos(Y. (42)

where ti is the angle relative to the x-axis and is given by

ic, =tan (dzj (43)dxX.

I23

This is the veh city comp~aent in the z-Oirection; however, each control

point also has a z.-Loordinate. The boundary condition is for no flow

through the wing, or that the normal component of velocity to the wing

be zero. The angle that th, control point makes with the x-axis is

given by

= tan-l(d-z. (44)

C

Thus the component of velocity normal to the wing is

Aw(x.,y.) = Aw(x.,y.)cosu (45)i J 1 i c

It should be noted that the boundary condition of the deformed wing is

satisfied on the x-axis, although the coordinate of the mean line lies

off the axis; however, this is equivalent to the planar wing approxi-

mation essential to thin airfoil theory. The actual normalwash is found

by combining equations (42) and (45)

Aw(xiyj) = Aw(xiyj)cos( - c' ) (4W)

By defining a function z(x) many variations and amplitudes f camber

and wing deformation in the chordwise direction may be obtained. Com-

binations of twist and camber or deformation may also be analyzed. How-

j ever, the condition of no flow separation still applies at all points.

Interference-Effects

j In the present analysis, interference effects, such as fuselagc per-

turbations and leading edge suction, are also included. An infinite

line doublet is used to simulate fuselage interference effects on the

3 induced velocity over the wing. The potential equation for the line

I

I

I

doublet may be differentiated to determine the velocity component normal

to the plane of the wing. To calculate the lift and moment of a fuselage

in the presence of a wing, an image wing inside the fuselage is employed.

The sectional characteristics of the actual wing are integrated over the

image wing span to determine the lift and moment. Both of these principles

are well known and the theoretical development can be found in Reference

22.

Leading edge suction, which makes a significant contribution to the

lift of highly swept wings at large angles of attack, has also been

incorporated into the preseat theory. Purvis (Ref. 23) has shown that,

for an assumed pressure distribution in the form of eq. (3), and for an

elliptic spanwise lift distribution, a general, closed form solution for

the leading edge suction is

C 2

CL = (CL sina - A cos.) cosA (47)LVLE Lp -TR cosA

where CL represents the potential flow lift coefficient, which is givenp

by eq. (26). Also, from Reference 23, the moment may be written as

2.212072 C C b/2

C sin,__- ----P cos:i x [v 77 -+ sin - (q)]dvMVLE c b cos. s AR f LB

0 (48)Using the definition of 7 in eq. (2) gives

1.10603C C I

CVL -P sinij - A cosl IXL[rI'i2 + sin-()]d (49)MVLE c cosx rAR f LE

0IoFor a constant leading edge sweep angle, this integral may be evaluated

in closed form. The coordinate of the leading edge is given by

x LE ri cosA!L

|

I2)

therefore, eq. (A ) becomes

.106036C C(tanA) si L - f T ,- -( , d- LP ( )[sinAt- -- cosCA] [n2/ + r sin-(n)Idy

-'IL cosA - ,AR

o io)

Integration produces

C0.65151 L (tan')

C 0.65151 [C sint - Cosa]( A-- (51)

'%LE - L ~ TAR cosA

Since the current gap analysis assumes small gaps such that there is

no vortex rollup, interference effects will be confined to the fuselage

and leading edge suction. These correction terms are necessary to allow

a comparison between theory and experiment in many instances; therefore,

by including them, a more versatile analysis is obtained.

Order of Polynomials

Before proceeding, comments on the effect of polynomial order on

the solution is in order. The unknown constants, B in eq. (3) arenm

really coefficients for the various terms in the polynomial describing the

pressure loa: on t!o t:in-. It has been suggested that if th. polynomial

order is inc-kea-ed to a large number of terms (i.e., , 50) that the sec-

tional coefficients such as cc will tend to "oscillate" in the spanwisc

direction. For the formulation used in the present analysis, no oscil-

lations are observable at least up to 50 terms in the equation. However,

oscillations do indeed occur if terms on the order of 80 or more are

used in the pressure loading. It does not appear that this is due to

"numerical" instability as would be the case if numerical integration

were used, but rather due to overkill. That is, the pressure loading

function is rather general and for most wings only a few terais (' 20) are

1

I

I

1 26required for a solution If more terms are used, it is analogous to

I fitting a 50th degree polynomial to a linear curve; oscillations are the

inevitable results.

I

III. EXPERIMENTAL MODEL AND TESTS - SUBSONIC

Model

To verify the gap effects and subsequent elevon loading, an exper-

imental model shown in Fig. 5 was fabricated and tested in a low speed24

wind tunnel. The baseline configuration consisted of a thin, low-

aspect ratio, cropped delta wing with rectangular elevons. These

surfaces were symmetrical about both the chord line and the fuselage

centerline. The fuselage was included to facilitate mounting of the

hinge moment balance mechanism. An ogive nosecone and tailcone were

also included. The fuselage and wing tips were designed to allow for a

two-inch translation in the elevon mounting position, which could be used

to vary the gap distance and/or the elevon hinge line location. The

model was floor-mounted to an external six-component pyramidal balance.

Test Procedures

The model was tested at a dynamic pressure of 3.5 inches of water

corresponding to a speed of approximately 126 feet per second and a

Reynolds number of 0.76 million per foot.

Data was obtained for an angle of attack range from -19 to +5

degrees in 2-de~ree increments. The negative range was used to minimize

strut interference effects on the elevons. Several elevon deflection

angles (00 to 200), gap distances (0 to 75% c E) and hinge line locations

(0 to 50% cE ) were tested.

27

Ai

28

C.14

00

00

C-4

79

000

-4-

p i

0 V)*

C

CC

IN=

I

IV. RESULTS - SUBSONIC FLOW

Results for a wing configuration are compared to other well known

theories in Figs. 6 and 7. Agreemenc is good for all cases considered.

In obtaining results using the present theory, the wing subpanel mesh of

constant loading was 45 by 45 (cho,:dwise by spanwise), and the control

point grid was 4 by 4. The total CPU time required was 45 seconds on an

IBM 3031 compared to 1 minute for the method of Lan on the Honeywell

635.8 Storage requirements for the present method are negligible on a

large machine. This mesh size may be rediced such that total CPU time

is reduced to 20 seconds, with a change in accuracy of aproximately 5

percent.

A comparison of theory and experiment for the "base" configuration of

the wing-fuselage model is presented in Fig. 8. Both linear theory and

leading edge suction are included. The importance of the nonlinearity

introduced by leading edge suction in the high angle of attack region is

* easily realized trom this figure.

The theoretical pressure distribution for a deflected elevon is com-

pared with experimental data (from Ref. 25) in Fig. 9. From this figure,

it is seen that the assumed functional form for the chordwise pressuke

distribution for the sealed-gap case represents an adequate model of the

actual flow. Total lift and hinge moments are presented in Fig. 10

where agreement is good, particularly in the low angle of attack regime.

The analysis developed for the general case of deformed wings,

elevons, gap, and fuselage is used to predict the lift, pitching moment,

30

!I

31

-- Lan's theory (Ref. 8)

6.0 -Present theory

4.0

ACP

rad -.

2.0 -

0.0 1

0.0 0.2 0.4 0.6 0.8 1.0

0.8ccj

cc L

0.4

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Figure 6. Comparison of Two Theories for aSwept and Tapered Wing of Aspect

Ratio 2.828.

II

32

8.0

-Present Theory

6.0 DeJarnette (Ref. 9)

AC 4.0

rad

2.0-

0.00.0 0.2 0.4 0.6 0.8 1.0

1.2

0.8

C L 0.4

0.0

0.0 0.2 0.4 0.6 0.8 1.0

Figure 7. Comparison of Two Theories for a Rectangular

Wing of Aspect Ratio 2.

33

0.2 --- Linear theory

_________Leadii,- edge suction

0.0 - 0 Experiment (Ref. 24)

-0.2 2

CL -04

-0.6 7.

1-1-0.8

-10

-20 -16 -12 -8 -4 0 4a (degrees)

0.6

0.2

0.0

-20 -16 -12 -8 -4 0 4a (degrees)

Figure 8. Leading Edge Suction on a Highly SweptCropped Delta Wing with a Fuselage.

34

-1.2 LEE

6-10* 00

- 0.8

p 1

-0.4 1 3 1

0.

0.00.0 0.2 0.4 0.6 0.8 1.0

1.2Theory

Experiment (Ref. 25)6=00 6=001

0.8

0.4

AC ~ ~~El6-0*4p

341 3

1 -0.8

0.0 0.2 0.4 0.6 0.8 1.0

I Figure 9. Chordwise Loading with a DeflectedElevon at nl 0.5.

35

0.8 - Theory

Experiment (Ref. 25) 6-0 o

0.6 6 =0*

0.4 r 6 -40 5=-40

0.2CL

0.0

-0.2

-0.4

-0.6

-0.8

-1.0 i i-10 -8 -6 -4 -2 0 2 4 6 8 10

a (degrees)

0.12

0.08

0.04 - 3

H 0 0S0.0

-0.04 - 6=00

-0.08 1 a * l I I l

-10 -8 -6 -4 -2 0 2 4 6 8 10a (degrees)

Figure 10. Lift and Hinge Moments for a

Wing-Elevon Configuration.

I

l36

and hinge moment slopes for the model tested in the low speed tunnel and

is shown as a function of gap distance in Fig. 11.

Certainly the most interesting trend with respect to the gap is that,

for all cases, hinge moments increase slightly as gap distance increases.

It is also noted from Fig. 11 that the coefficient slopes vary more

noticeably with changes in elevon deflection while the changes with angle

of attack are essentially negligible. Figure 12 illustrates the theoret-

ical spanwise pressure distribution over the wing and elevon for the

present cropped delta wing configuration. This figure confirms the

predicted increase in the loading over the elevon, and shows the slight

loading change with respect to angle of attack. This loading form is in12

agreement with the two-dimensional results obtained by White, et al, as

well as the present experimental data. In Fig. 13, the hinge moments are

presented for several hinge line locations on the elevon. It is seen

here that there is a chordwise point, in this case between 15 and 20 per-

cent of the elevon mean aerodynamic chord, where the hinge moments are

independent of gap distance. It is noted that this point does not coin-

cide with CH 0, which gi"-s the elevon ar, dynamc center. This aspect

could be significont in the design of variible geometry wings where gap

distances are subject to change. Also the minimum value of the hinge

moment, which corresponds to minimum control forces, occurs for the sealed-

gap case.

The slopes of the pitching moments curves illustrated in Fig. 11 are

seen to be essentially constant in nature. Since the moment is referenced

to a fixed point on the wing, the increase in the moment arm with increas-

ing gap coupled with increased elevon loading might tend to suggest an

- J

37

- -- Gap Theory

0.05 Sealed-Gap TheoryCL _ Experiment

L 0.04 0 0 0 0

0.03 !

0.0 0.2 0.4 0.6 0.8 1.0

ICE

-0.020

CM -0.028

- 0 . 0 3 6 , , !J0.0 0.2 0.4 0.6 0.8 1.0

E/E E

0.01

C1. 0.0

0 0 0 o

--0.01 , I i I0.0 0.2 0.4 0.6 0.8 1.0

E/C E

Figure 11. Aerodynamic Characteristics for aGapped Configuration.

i

I

38

0.03

CL6 0.02

0 .01 . I . I

0.0 0.2 0.4 0.6 0.8 1.0/EE

-0.012

CC 5 -0.020

-0.0280.0 0.2 0.4 0.6 0.8 1.0

F-/C

0.0

I 1 -0.01 0

-0.02 I .

0.0 0.2 0.4 0.6 0.8 1.0

Figure 11. concluded

77 '

39

0.7

0.6 -CE=0.0

-- E/E E=0.505

0.5

TE w/LEE

AC 0.4P o,=5O

0.3 - 60*

0.2

0.1

0.00.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.6 -TEW! LE E

0.5 -

AC 0.4 II0.3 -65

0.2

0.1 7>0.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Figure 12. Gap Effects on Chordwise LoadDistribution.

I

40

0.04

I/ IE 0.0

0.02 /C-E 0.122

CI 0.497

CH 0.0

CH600Theory

Experiment

-0 -0 0 HL /ZE 0.0160 XHL/-d E 0.254

-0 04 A XHL/E 0.5070.0 0. 0.2 0.3 0.4 0.5

XHL / E

0.04

0 .02 XHL/E0 .507

A

CH 0.0 XHL/CE=0.254

-0 .0 4 0.0.0 0.2 0.4 0.6 0.8 1.0

Figure 13. Elevon Hinge Moments for SeveralHinge Line Locations.

I41

increase in uhe noment. However, from Fig. 12 it can be seen that the

change in pressire distribution mere than offsets these factors. Yet,

as the gap become-s large, the theoretical pressure distribution must

approach that of two separate wings, without interference. The moment

referred to a fixed point on the wing must become infinitely large as

the gap distance approaches infinity.

For large gaps, it was found experimentally (Figs. 14 and 15) that

there is a considerable increase in the total lift and moment for an

elevon deflection of -20'. The decrease in the forces becomes larger

with increasing gap distance. However, elevon hinge moments did not

reflect this trend, indicating that it is due more to the gap rather than

to flow separation over the elevon. In any case, it is apparently a

viscous phenomena and cannot be modeled properly using potential flow

techniques.

The experimental data for different hinge lines is presented in

Figs. 16 and 17. Theoretically, there are no changes in the total forces

and moments with a change in the hinge line. This result also appeared

in the experimental data despite the fact that, for a deflected elevon, the

elevon leading edge no longer lies in the plane of the wing. From the

data in Fig. 16, it is seen that the hinge moments at XHL/CE - .254, or

approximately the quarter chord of the elevon, are quite small and show

little variation with angle-of-attack or elevon deflection. This in-

dicates that this point is near the aerodynamic center for the elevon.

However, the slopes change with a change in gap distance, thus the aero-

dynamic center must also be gap dependent, as previously shown on Fig. 13.

I

I'42

-Theory

Experiment

0.4- 0 6 = 00

0 =+5 °

0 .2 - 0 - 0 o 6

0+ 5 0

0.0 -0- 100

-0.4 -L

-0.6

-0.

-1.0

-1.2

-1.4

-20 -16 -12 -8 -4 0 4az (degrees)

Figure 14. Characteristics of a Wing-Fuselage-ElevonModel for XHL/ E = 0.016 and E/-c = 0.004.

LE

I

11 43

0.8

0.6 0

0.4

0.2

0.2 -- 0

0.4 E

0.2 08 -4 4p

0.1

o.300200

0.0

-0.1

-20 -16 -12 -8 -a~ (degrees)

Figure 14. concluded

I44

- Theory

Experiment

0.4 0 6= 00

0 6 -+50 6+50

0.2 0 6 = -100 1 6-00

06 =-200

0.0 -6=-1o1

-0.2 -

-0.4

C LL 0/

-0.6 -

o/-0.80

-1.0

-1.2 /

-1.2,-1.4 p I I I I

-20 -16 -12 -8 -4 0 4

a (degrees)

Figure 15. Characteristics of a Wing-Fuselage-Elevon

Model for 0.016 and E ff 0.497.

0.6

0.4

C4

0

0.4 0 C

8=20

0.30

0.2 n~0~~~~0

C10

0.0 3 t3

-0 -16 -12 -8 -4 0

Figure 15. concluded

4 Theory

Experiment

0.2 0 3 0 ° E/Ew 0.004

r +=50 E

0.1 O -i0 ° 6=-109

c 600a

0.0 ;

-0.1 i I a

-20 -16 -12 -8 -4 0 4a (degrees)

0.2

6/- =0.134

0.1 -

CH /-- 00

0.0

-0.1 I I -

-20 -16 -12 -8 -4 0 4a (degrees)

0.2

/CE =0.261

4000 00 .

0.0

; d =-+5

-20 -16 -12 -8 -4 0 4A (degrees)

Figure 16. Hinge Moments on a Wing-Fuselage-

Elevon Model for x HL/Z E= 0.254.

I

I47

0.1

E/E=0.009

6 -+50

o0.0 0 0: : : :

C-0.2

I pzo-20 -16 -12 -8 -4 0 4

cc (degrees)

0.1 E/zE=0.130

0.0 i 0 E l 0 0

-0.1 C0 0 0 0 0 0 0

-0.2 0 0=

i0-20 -16 -12 -8 -4 0 4

a (degrees)

0.1 /E=0.264-=+5o

0.0 0"=00 o c i 6=0:CH - D-

0.1 0 0 0 0 0 0 0 0

-0.2-20 -16 -12 -8 -4 0 4

(degrees)

Figure 17. Hinge Moments of a Wing-Fuselage_Elevon Model for 'HNL/EE 0.507.

I48

Results for arbitrarily deformed airfoils are difficult to find in

the literature; therefore, perhaps the best basis for comparison is two-

dimensional thin airfoil theory. Table 1 illustrates theoretical results

for the two arbitrary ne-an lines shown. The present theory, extended

to two-dimensional results and thin airfoil theory show good agreement.

For cambered airfoils, experimental data may be obtained from Ref. 26.

Results are presented for the NACA 747A315 laminar flow airfoil in Fig. 18.

The current theory predicts experimental data very well. Again, the thin

airfoil and present theory are closely matched. This result should be

expected since, foc two-dimensional flow, the present theory would become

thin airfoil theory as the limit of the wing subpanel areas with a constant

ACp approached zero.

For a finite wing, Fig. 19 compares theoretical predictions with

experimental data (Ref. 27) at several Mach numbers for a wing that is both

cambercd (N-, C\ . -1.0 meer. .mtei I Tne) and f-rsted. It should be noted

that the £.pcr Lrtal model was tapered and t'e maximum camber varied with

the spanwise c)ordinate. In the theoretical model the wing was replaced

by the mean camber line at each 20 percent of lthe ,.pan and this distribu-

j tion remained constant for that section. Also, a linear spanwise

distribution of twist was assumed, while there was a slight variation

from this experimentally. Nontheless, agreement of theory and experiment

is very good-over the entire experimental Mach number range..I1IIII

49

.- c

o 0r, C

CNC

it It

IL C_

cr 0

U r~q

OWN=

[I50

PrV nSil t L I ,:-v

1.0 -- Thin airfoil L.,eory

O Experiment

0.8

0.6

0.4CL

0.2

0.0

41

-0.2

-0.4 .

-6 -4 -2 0 2 4 6 8:t (deg)

10.2

0.0

-0.2

-6 -4 -2 0 2 4 6 8

i (deg)

Figure 18. Characteristics of the NACA 747A315

Airfoil Section.

I

. .. I iF

I,

0.5 M =.6

0. - Theory

0.3

0 Experimient0.2

0.1

CL 0.0

-0.1

-0.2

-0.3

-0.4

-0.5 'I

-10 -8 -6 -4 -2 0 2 4 6 8 1001 (degrees)

0.6 -

0.4 -

0.2

L 0.0 -

-0.2 -

-0.4 0

i-0.6F I I I - I I t I

-10 -8 -6 -4 -2 0 2 4 6 8 10a (degrees)

Figure 19. Characteristics of a Finite Wing that is BothCambered and Twisted.

II I

f 52

Theory

0.08

0 Experiment

0.06

CL a . ---

(per deg)

0.04

0.020.0 0.2 0.4 0.6 0.8 1.0

M

60

40000a.c.

(percent c)

20

0 L0.0 0.2 0.4 0.6 0.8 1.0

Figure 19. concluded

i

I

V. CONCLUSIONS - SU33ONIC FLOW

From the preceeding results, it is seen that the present theory

produces good results in the inviscid flow regime. Configurations

ranging from a basic wing to varying combinations of wings, elevons, gap,

and fuselage may be modeled as well as moderate wing and/or elevon defor-

mations. Since compressibility effects are included, reasonable results

up to large (subsonic) Mach numbers are generally obtained.

For the finite gap case, the theory provides an excellent means for

obtaining early design characteristics in a short time for small angles-

of-attack and elevon deflections. However, it is shown that for large

elevon deflections there was a considerable decrease in total lift and

moment characteristics when a large gap was present. This appears to be

a viscous phenomena and therefore cannot be modeled with potential flow

techniques. Vortex rollup, which must be included for very large gaps,

has also been neglected in the present theory. Results for cambered,

twisted, and arbitrarily deformed wings may also be obtained easily with

the present theory. Results are good as long as the requirements of

potential flow are met on the surface.

Several advantages of the current method occur as a consequence of

the assumption of a constant AC over a sufficiently small subpanel of theP

wing. The wing can be divided into many subsections, yet only a relative-

ly small matrix must be inverted to obtain the load distribution. This

assumption also allows AC to be taken outside the downwash integral andP

yields a closedi form solution to the equation. As a result, this method

53

1 54

requires considerably smaller computational times and storage space than

many methods, such as vortex lattice, since they usually require a

matrix equivalent in size to the number of wing subpanels.

II

( i. .,, ... .._ . -,'i--'-' .. " -. f

I

VI. SUPERSONIC FLOW - THEORY

Fundamental Equations

The theoretical approach for the supersonic Mach number regime is a

potential flow formulation in which solutions are superimposed. The

source solution introduced by Evvard 28 forms the basis of 3-D supersonic

theory used in the present analysis and vorticity paneling is added to

account for wing deformations. The vorticity paneling, as previously

introduced, is actually a continuously distributed doublet panel which

satisfies the supersonic potential equation

1fY(x y) 01x-C(xYZ) i So ___°'° (x-x=) fdx dy (52)21y = 0 (Y-Yo0 )2+Z2 V(xXo)2_ 2 (y_yo)2_ 2Z2J 0 0

As for subsonic flow, the vorticity distribution I(x ,y ) may be replaced0 0

by the pressure loading coefficient AC p(xy ). The downwash equation

is obtained by differentiation in the standard manner and when evaluated

in the z=O plane (planar flow) becomes

1 AC (x ,y ) (x-x

_T f p 0 0 dx dy (53)

S 0 (yy 0)2 v(X-Xo)2-2(y-yo)2 0 0

where the area S is that contained on the wing surface in the upstream0

running Mach cone emanating from the field point (x,y). Although the

vorticity paneling is formulated for planar flow, it may be used to

account for non-planar velocity components if the slope of the nonplanar

j surface is properly taken into account. A more detailed discussion of

how this is done is explained later.

55

IIJ

56

The assumed pressure loading cousists of the basic three-dimensional

supersonic flow term plus the perturbation term fron the vorticity panel-

ing and is then written as

N MAC (xy o ) = P(xoYo)Sin(a49o ) + P(X n 0 BL m co-2 (54)

p 0 0 0Om0=0

The first expression on the right-hand side is the 3-D loading term

modified to account for local wing deformations and is valid for both

subsonic and supersonic leading edges. The second term uses the 3-D15

pressure functions P(xo,Y o ) as a weighting function which, when multi-

plied by the unknown coefficient BL , and the series ql cos -- - accounts

L 2

for small deformations in the wing surface in the upstream Mach cone from

the point (x,y). The order of the terms in the second expression seldom

exceeds 3 or 4 and for planar wings, the entire second term is unnecessary.

Increasing the polynomial order for larger wing deformations or higher

angles of attack does not seem to provide better agreement with experi-

mental data.

The form of the assumed pressure Icading term to account for per-

turbation velocities was chosen as a polynomial in the spanwise direction

and a trigometric or Fourier series type function in the chordwise

direction. No strong justification for these assumptions can be given

except that in the spanwise direction the 3-D loading term should produce

acceptable results, even for deformed wings, and corrections, which the

polynomial provides, should be small. Subsequent good agreement with

experimental data seem to justify this assumption. In the chordwise

direction where more severe changes occur in the pressure loading, a

trigometric correction, which is somewhat more general than theI

II

57

polynomial, was chosen with subsequent good agreement with experimental

data.

As in subsonic flow, the AC term is removed from the integral inP

eq. (53) and the resultant expression is integrated in closed form over

a small subpanel. Results of this integration over various shaped

subelements is presented in Ref. 15.

It is convenient to define the value of this integral as AK so that

the total downwash at a point (x,y) becomes

Iw(x,y) = (AK)i[ACp(Xo5 (y o ) 55())

i~l 1 p 0 ~(55)

where I is the total number of subpanels in the region (cone) of integra-

tion. The boundary condition is

w(x,y) + sin (t+6 C) = 0 (56)C

where e is the local deformation angle at a control point. Rearrangingc

and combining equations (54), (55) and (56) yields

I(AK).[P(x ,y ) sin (ct+6 )]. +

i=l 0 0 1

I N M,y ) B , r C o s 2" .

i=l n= m=O 2

47r- sin(+0 ) (57)

C

where L = n + mN.

Body perturbation effects on the wing loading are accounted for by

assuming that the body is cylindrically shaped. The cylinder is gen-

erated with an infinite line doublet whose axis is in the z-direction

perpendicular to the wing planform. The results of adding the body to the

.-

I56

analysis simpiy adds another term to the right-hand side of eq. (57) which,

after a little manipulation, b~ecomes

4fra2 sina

a(a+n)2 (58)

where a is the nondimensional body radius define-d by a = r b/(b/2).

Equations (57) and (58) are combined and rearranged as

I N M

(AK). [P(x ,y ) Y Y BL rlm co =

i=I 1 0 o0 um=O L 2

S4-f 4T a 2sina

- sin (ot+O) + 4 asn)2

c 3(a+-1) 2

I

+ (AK)i[P(xoy ) sin(c+0 )]. (59)i=I 0 1

The above equation represents a set of N simultareous linear alge-

braic equations and is conveniently arranged in matri- form for a

Gaussian reduction solution for the unknown coefficients B . Once the

I unknown coefficients are determined, the localized loading is obtained

from equation (54).

Lift and Pitching Moment for Wing Elevon Combinations

In the preceding section, the basic equations were presented for

finding the resulting pressure distribution on a wing or elevon in super-

Isonic flow. For the wing alone, these equations may be applied directly

since there is no upstream influence due to the elevon. After the

j pressure loading coefficients in eq. (54) are determined, the pressure

coefficient may be computed at any point on the wing surface and appro-

priate subsequent integration produces the lift.

II

I

1 59

For the elevon, a similar procedure is followed with the exception

that the elevon is flying in the wake of the wing and therefore the down-

wash produced by the wing and the elevon must be considered. The mechanism

used to compute this downwash is discussed in another section.

The lift for the wing-elevon configuration is the sum of that pro-

duced by the wing and that produced by the elevon. It is convenient to

define nondimensiona], chordwise and spanwise variables for the wing as

w = (X-xLE)/Cw (60)

I and

T = y/(b/2) (61)

For the elevon, using the same coordinate system, these definitions

become

g = (X-CR-6)/cE (62)

and

nE = y/(b/2) (63)

With these definitions, integration of the pressure distribution

over the wing for the lift becomesn=l w=l

CL b f (ACP Cw d'w)dqw (64)w w

S=0wand for the elevon

C b f f ( CEC d )dnE (65)L E 2EEl

n=-l E=0

IAI

I n

60

It is convenient to define sectional properties as

C =1w

(cc) w = Cw J ACe dCw (66)

w=0

and

E =1

(cck)= CE f ACPE d E (67)

E =0

from which we get the total lift coefficient

C= f [(cc ) + (cc )E]dn (68)CL Sk kE

n=0

Because of the nature of the pressure loading function, equation (54),

equations (66), (67) and (68) must be integrated numerically.

A similar analysis leads to the moment equation about the y axis

(see Figs. 1 and 2) as

Myf APw x dA + f APE xdA dyw(69)

A AE

For the first integral in eq. (69), we note that

x = XLE + C w (70)

and

dA = dxdy = (Cwd )(b dri) (71)

andb

xLE = y tan A = b n tanA (72)

61

Using these definitions, the first of these integrals over the wing can

be written as

= qb f {ntanA[C f ACp d w ] + Cw2 fACp wdw}dn (73)

0 0 0

In a similar manner, the second of these integrals become

rf=i I I

12 = b f Rw+)[C E d + C E Ac d}d (74)

ri=O 0 0

The moment coefficient about the y axis Ls defined as

CM = M /(qSc)Y

so that

71=i 1

CMy = f 2 ntanA(ccZ)w + cw 2 ACe wd wy Sc Jf w w

T1=0 0

[(CR + )(cc) + C 2 f ACPE dCE ]}dn (75)

0

In order to evaluate eqs. (66), (67), (68), and (75), we must first

determine the constants, BL in eq. (54) for both the wing and elevon.

Since the elevon solution cannot affect the wing solution (zone of silence),

then the wing loading is solved independently of any elevon considerations

through a Gaussian reduction of eq. (59).

For the elevon solution, we must consider the downwash produced by

that portionof the wing in the forecone of some arbitrary point on the

I

1 ll lmh -.. .k. '

I

62

elevon. This downwash from the wing on the elevon will enter the elevon

solution through an additional term in the boundary conditions. That

is, for the elevon

(AK)i [ACe((x , )] = -- [ww + sin(a+6)] (76)

i=lE

where w is the downwash produced by the wing at a particular control

point on the elevon. Evaluation of w is considered in the next section.w

Wing Downwash on Elevon

To evaluate the downwash from the wing on the elevon, a direct

approach was first considered. That is, integration was to be carried

out over the forecone from a point on the elevon surface. However, because

of the gap between wing and elevon, this did not seem to be a feasible

approach, especially in light of its complexity. Thus another approach was

considered and used.

Since the wing solution is completed first, sectional properties on

the wing may be computed at any spanwise location. Thus the wing is

divided into equally spaced "strips" along the span and the sectional lift

is computed for each strip. It is assumed that a horseshoe vortex is

attached to each strip in a lattice manner whose strength is determined by

~the sectional lift coefficient. That is, from the Kutta-Jiukowski theorem,

the lift per unit span for some ith section isI(AL/An)i = pVrj = 2qF i (77)

Swhere

I F' = VII ~ i i

II

I

63

Placement of these vortices are shown in Fig. 20. Note that in the

figure, the bound portions are shown attached to the quarter chord and

the trailing legs trail b Iiind the wing to infinity.

If the wing is symmetrically loaded, the lift may be written as

1

L = qb f (ccR)w dnw (78)

0

so that for a strip

AL = qb f (cc) w d w

Dividing by An and equating to eq. (77) yields

ri=(i-f (cc ) dnI

n 2

If it is assumed tuat over th, strip the term cc ds constant, then the

integration may be c.;mpleted in closed form.

At the junction between ay' two strips, the resulting trailing fila-

ment strength is the difference between adjacent filaments and may be

written as

F1 c - (79)FRi - i+ I kcc i (zci+ 1

The starting point of these filaments is assumed to be the wing

trailing edge (for lack of a better assumption). Fortunately, the down-

wash induced from all trailing filaments is small in comparison to the

I!

I

64

19y

CR

'4 Trailing

I- Vortices

Wingf

LL/

I CR E Lt Elevon

I; iI *1* i

Figure 20. Schematic of Wing-Elevon CoordinateSystem and Wing Trailing Vortices.

III

I 65

free stream components and thus an error in the starting point of the

filament does not significantly influence the final elevon loading.

Obviously, though, the filaments must start somewhere on the wing. From

From a planform view of the wing-elevon trailing filament configuration,

it may be shown that if a particular field point (x,y,z) is in the region

of influence of a particular filament starting at (x,yr), then

x > (xr+8!y.-rl) (80)

2 y 2 2

where r y + z . That is, if eq. (80) is satisfied, then the field

point (x,y,z) "feels" the downwash from the trailing filament (x ,YF).

To compute the velocity induced at some field point (x,y,z), the

Biot-Savart Law is used as an approximation. Although this theorem is

actually only valid for subsonic incompressible flow, it will be suf-

ficient for the present analysis if proper account is taken for the

region of influence (zone of silence).

In vector form, the Biot-Savart relation governing the velocity

induced at some point by a vortex filament is

V(x,y,z) = - _ rd (81)4 r

In this expression, r is the vector from thu field point to an

arbitrary point on the filament, dZ is a vectored differential length

along the filament, r is the magnitude of the vvxior r, and Y' is the

filament strength. Referring to Fig. 21, these quantities may be

written as

dT = (i cosO + k sin0 F)d (82)

66

zA

Figue 2. Scemaic o th Win-ElvonTrailingVorte Filaent Cnfiguatenn

L aIlvo

FilIonIXYZ

67

and

rt(x -x+L coseF)i + (y.-y)j + (LosinO1 - z)k (83)

Equations (82) and (83) are introduced into eq. (81) so that the

integrals for the three velocity components are

Z=L0

V - f (Ylsin0F/Dm)dZ (84)

Z=0Z=L

0

V = C ((xlsinO. + z cosO )/Dm)dZ (85)y 4v i

Z=0

and&=L

0

V =- f (y cos6F/Dm)d (86)z 47r

Z=0

where

XI X xF-x, Yl = YF-Y' (87)

A =2xI cosO F 2 z sinO (88)

B (x 12 + Y + z2) (89)

D =(L 2 + AL + B)3/2 (90)

m 0 u

and

2 2L +Ao=_____ A(91)

(4B-A' 2 +AL +BI

I

,68

These integrals may be evaluated in closed form with the results

V = -Fr(yIsinO )I/(4TT) (92)

Vy = F(x1sinO F + z cosO )I/(4 T) (93)

and

V = F(y1coser)I/(47T) (94)

The induced velocity from these filaments on the elevon is of

primary concern and more specifically the velocity induced normal to the

elevon. In vector form, a unit normal to the elevon is

n = -i sin6+k cos6 (95)

so that

V = Vn = - V sins-V costS (96)n x z

or

V = -FY1l[Cos(O +6)]/(4"T) (97)

Wing Trailing Edge Blocki;,- of Eleven

To complete the theoretical analysis, the wing thickness and blunt

wing trailing edge and the subsequent effect on the levon loading must

be considered in addition to the gap between the wing trailing edg,. and

the elevon leading edge. The configuration being used in this analy.sis

is shown in Figure 22. The flow just aft ot the bMunt wing trailing

edge is highly turbulent as expected and details de.scribing the sub-

sequent flow field is fundamentally grounded in solutions of the Navier

Stokes equations. The implication then is that viscous elfects domini

the solution as was suggested in Ref. However, this conclusion is

not necessarily true. It is clearly recognized that viscous eftects

II

69

y

4.413 .70n 611

77.23301.00

5.174

5.181 -t*

a) Planform View

St ea IsH inetI '

BIw alikie

Figure 2-. 1) 1m'nis eIna I Plani t~rtn ,ind S~t i oil I I t (IallFcxpcr tment i I Wing-1-: evon Coni t i gtirat ion, et I. I.

I-di2

I

70

immediately behind the wing trailing edge are very important, but it is

believed that the dominant effects over most of the elevon can be pre-

dicted by potential flow methods. Justification of this conclusion is

presented in a later section.

As depicted in Fig. 22, the flow is assumed to be tangent to the

surface of the aft portion of the wing for all angles of attack. The

streamlines thus emanating from the upper and lower aft wing surface

impinge on the elevon surface as shown in Fig. 22, forming a "blanked out"

region bounded by the blunt wing trailing edge, the upper and lower

streamlines and a section of the elevon chord line. in this blanked

region, details of the flow field are not well known. Thus for purposes

of this analysis, it is assumed that this wake pressure coefficient is

zero such that essentially free stream conditions exist.

From Fig. 22 the location of the aft impingement point is

(x C )tan - 12xAF T .C R . . + (98)

tan' - tan'

and the forward location i, ,

(x -(,R t;n +- LXFW = CR + .. . .. . , 4 (99)

tan + tan' 1"

where )T is the positive sortace slopt of the alt portion of the wing,

That is, for the schematic in F g. 22, is about .'. and ' is about

-10'.

Equations (98) and (99) are valid lor 0.0. If the elevon deflec-

tion angle is positive, then the forward location is determined from

eq. (98) and the aft location from eq. (99). It is obvious from Fig. 22

i.i

I

71

that there are elevon angles where the leading edge of the elevon lies

entirely within the wake region with some portion of the aft section out-

side the wake. Also, for very small elevon deflection angles, the entire

elevon is immersed in the wing wake and thus is completely unloaded.

In order to justify the use of eqs. (98) and (99) to form a blanked-

out region (Newtonian approach) a wind tunnel experiment was designed to

study this flow field. A two-dimensional model consisting of a wing with

a blunt trailing edge and a diamond shaped elevon was constructed (see

Fig. 22) and tested at a Mach number of 1.94 in a 4" by 4" test section.

The total pressure was about 30 psia with a freestream unit Reynolds

number of 1.6 x 106. Selected schlieren photographs of the results of

these tests are shown in Fig. 23. In the upper left where 'i = 0.0' and

6 = 0.0', the elevon lies entirely within the wake of the wing as is

clearly seen. The important thing to note here, though, is the well de-

fined streamline emanating trom the top skiriace of the wing and impinging

on the elevon boundary layer near the apex ot the elevon midclord. In

the upper right wherte L= . and - - th( leading e,,ge of the

elevon is about 4' iutsidt, ;i, wing,, w; k, rt-gion it , pparent lv t ill

inside the wing l'indarv liver. Fhn ,lit i njitgt rert ;,,lilt d Il it ,IItj er

elevon surface is st ill >1 %, . i; i l l bi lt t , , rw.i' 1,,,int , ,iTA be

clearly located.

In the lower left, whit-re -4. 1-4 a st -- 4) , the at i :p I g et n

point is located easily wiile Lilt wL I vo le.d Ing edge remains submere.,d in

the wing wake. In the lower right where = -4.14' and -10.(0 ' , the

results are incGonclusive. The overall results of these tests indicate

arIvr] et

I

Fig. 23. Sc'il1ieren Photo-raphs of the 1 iunt Wing Trailing Edze and

Eilevon Leading Edge, M=1.94.

I

1 73

[ that some sort of blanking region is formed which is not totally inde-

pendent of angle of attack and is more strongly dependent on the elevon

deflection angle than that predicted by eqs. (98) and (99). Nevertheless,

these equations do provide "bounds" for the mathematical analysis and, as

will be shown later, provide good agreement with experimental data.

I

I

I1

II

VII. SUPERSONIC FLOW- RESULTS

Comparison with experimental data of the theoretical analysis as

previously presented is divided into two major categories: (1) planar

wings and (2) deformed wings. In many cases, the planform shapes chosen

were dictated by available experimental data. In some cases, experimental

data was available for both deformed and undeformed wings with the same

planform shape. Four of the planform shapes analyzed are shown in Fig. 24.

Planar Wings

Results of the pressure loading and sectional lift and pitching

moment for three of these configurations are shown in Figs. 25-30 for

selected Mach numbers and low angles of attack. More complete data summarv

results for these configurations are presented in Appendix A. In these

figures the theoretical solution for the pressure loading is basic three-

dimensional supersonic theory as presented in Rets. I' and 18. No

vorticity paneling Is required. The double summat ion tcrms in eq. (54) are

omitted and only the P(xoy o ) sin (i) term is used Lk, comput,, the pressure.

In most cases, the sectional normal forces agiree wl1 with the exper-

imental data especially on the inboard sections ot the wings. It is

observed that the worst agreement is near the wing tips a, one would

expect. This deviation at the tips is not as detrimental as might first

appear since most of the configuration loading is concentrated on the

inboard sections.

1 74

I

i I

1 75

700 77.230

2. 74748 5.79162

HighlySwept

Delta Wing Cropped 1.37844Delta

Wing

1.2488

LowI] i ghAspect ApctRatio iRatio0.19048

! Trapezoidal Wing TrapzoidalI Wing ~

1.1.0

Fig. 24. Dimensional Sketch of Four Wing Planform Shapes Under Study.

'1.

I

76

The sectional moment coefficients about the y-axis shows the same

kind of agreement as does the normal forces. In each case the dis-

crepancies are not severe and the location of the sectional aerodynamic

center is predicted quite well.

The experimental data for these plots (including those in Appendix

A) were obtained from Refs. 29 and 30 and was integrated numerically to

obtain the sectional normal forces and pitching moments. Integration to

produce the normal forces agreed quite well with that tabulated in the

original reports; however, the sectional pitching moment data as pub-

lished in Refs. 29 and 30 was not reproducible by numerical means. Con-

sequently, these experimental data as presented in Figs. 25-30 (and

Appendix A) for the sectional coefficients was obtained by numerical

integration of the pressures and is not that which is "tabulated" in

Refs. 29 and 30.

For these planar configurations, the theory is a linear theory and

thus tabulation of the total forces and moments is sufficient to illus-

trate overall agreement. Table 2 is a comparison of the lift curve slope

and moment curve slope for the three planar configurations under study.

as found by numerical integration of the curves in Figs. 26, 28 and 30.

From Figs. 25-30 and those in Appendix A, it is difficult to picture

exactly how the sectional loading profiles are spread over the wing.

Consequently, Figs. 31-34 were produced showing the isobars and the large

influence produced by the wing tips. Note also the supersonic leading edge

behavior for the low aspect ratio trapezoidal wing and rectangular wing.

It is apparent from Figs. 25-30 and from Table 2 that the linear

theory is excellent for most cases considered. The disagreement near the

! I

I77

o FLAT TRAPEZOIDAL WING C FLAT TRAPEZOIDAL WINGO ALPHA = 4.0000 ALPHA 1.0000

LAMOA 25.7917 LAMDR = 25.7917MACH NO. = 2.0100 MACH NO. = 2.0100CR = 1.7324 CR 1.732U

o CT = 1.2488 CT = 1.2488Y/IB/2) = 0.1000 m Y/(B/2) 0.55000 In

(CD

CD CD CD0" 0 0

o 0

T0.o0 0.25 0.50 0. 75 1.00 0 .. L) o. 0LJ 0.7 0 1.00PSI PSI

o FLAT rHAPEZOIDAL WING FLFT 'HEZ TIL .L>N,

O ALPH(l 4.0006 0" HL PH* - 41,[fULAMOP 25.7917 -LnM[f 2.5.7911MACH NO. = 2.01fQr MACH N0. = 2. rI, L0CR = 1.7324 CR ( ].7324

o CT = 1.2488 C CT I 1.2I8HYI!82) 0.3500 1/ (m/21 OlT

0 0

__. C n C) C CDCD

CCD

D 0n

CD CD

Th.oo 0'.25 0'.SO 0'. 75 1'.00 .OO 0 .2S 0.-0 0. 7, 1.00PSI PSI

Figure 25. Pressure Distribution on a Flat Trapezoidal Wing; M = 2.01,c= 4.0.I.

II '

Il

78

0 FLAT TRAPEZOIDAL WING 3 FLAT TRAPEZOIDAL WING

o ALPHA = 4.0000 0- RLFHA = '.DDDLAMOR = 25.7917 LAMOR = 25.7917MACH NO. = 2.0100 MACH NO. = 2.0100CR = 1.7324 CR = 1.7324

CD CT = 1.2488 CT = 1.2488! Y/(8/21 = 0.7700 m Y(B 2= 0.9700

0 C

o

a- ooo 0 d

o 0

PSI PS I

o FLAT TRAPEZOIOAL WING

o ALPHA = 4.0000LAMOR = 25.7917MACH NO. 2.0100CR = 1.7324o CT : 1.2488

PSIPS

ALPHA2 = 0.800

Figure =5 25.7917e

I0

I0 0 i

c 0 0'. 2S 0 .50 0'.7S 1'.00PSI

Figure 25. Continued

II

1 79D FLAT TRRPEZO]DRL WING

] LPHA = 4.0000LRMOR = 25.7917MACH NO. = 2.0100i CR = 1.7324

o CT = 1.2488

C -

C)D

o.25 0.50 0.75 1.00Y/ (B/2)

0

FLRT IHOPEZODORL WING0)

ALPHA = 4.0000LRTMR = 25.7917MACH NO. = 2.0100CR = ].7324CT = 1.2488

CL

LI

'0.00 0.25 0.50 0.75 1.00Y/ (B/2)

Figure 26. Sectional Normal Force and Pitching Moment for a FlatTrapezoidal Wing; M = 2.01, a = 4.0.

I

80

o FLAT DELTA WING c FLAT DELTA WING

O ALPHA 4.0000 HLFrIR = 4.0000LAMOR 70.0000 LRMOD = 70.0000MRCH NO. 2.0100 MACH NO. = 2.0100CR = 2.7475 CR = 2.7475

C CT = 0.0000 CT = 0.0000,/(B/2) = 0.1000 m Y/(B/2) = 0.4000

Dci o

o c C)

C " C)

3 (90 -

C) C)o ci

To 0.2 0.- 7Y -o.D .2 .s 7 .o'.00 o0.25 0.5 0.75 1.00 .50 0.75

PSI PSI

o FLAT DELTA WING c FLAT DELTA WING

CO" ALPHA = 4.0000 Co- ALPHA = q.0000LAMDR = 70.0000 LRMOR 70.0c00MACH NO. = 2.0100 MACH NO. = 2.0100CR = 2.7475 CR = 2.7475

o CT = 0.0000 cm CT = 0.0000S"/18/2) 0.2500 m '/ tB.rj 0.5000

m 0 c-i(: (D

o c

. . . . . . .y . . . . F -C -)-

CC)

C3 C

I(

- C)

Th.oo 0.25 0.50 0.75 1.00 00hf) 025 l% 0. 505PS I PS I

Figure 27. Pressure Distribution on a Flat Delta Win~v,; N 2.01, 4 .0.

l1

AN AFRODYNAMIC ANALYSIS OF DEFORMED WINGS IN SUBSONIC AND SUPER-ETC(U)APR 81 J E BURKHALTER. J M ABERNATHY OAAG29-78-G-0036

UNCLASSIFIED ARO-15666.4-A-E NL

23 fflfflfllfllflfflfEIIIIIIIIIIIIuEIEIIIIIIEEEIllllllumllllluEEEIIEEIIEEIIEEIIIIIIIIIIIIIEEEElllllllllIE

81

o FLAT DELTA WING 0 FLAT DELTA WING

8-ALPHA = 4.0000 VLPti =47.0000LAMOR 70.0000 LORH 70.0000MACH NO. = 2.0100 MACH NO. ( 2.0100CH = 2.7q75 CR = 2.7t7SCl = 0,0000 CT = 0.0000

/ T'B/2) - 0.6000 0 Y/ (B/2) 0.8000

00

0

0

O0 O. 2 0 .SO

'. 7 !.O0

O0 0 -. 25

0.50 O.75 1. 0 0

P S0

PSI

PS

~FLT DELT WING

QLP R = 70.0000I

" LPMD =

0.0000

M CH NO. = 2 ,010 9

CH = 2775oCT = 0 0000

mo 0

o 0

o0

0 Oo 0.25 0. 5 0 0.75 1.0 -ho 02 0.0 075 10

PSI

FMg ue 2 7

C o n t i n u e

d

82

O FLRT DELTR NING

RLPHR = 4.0000LAMOR = 70.0000MACH NO. = 2.0100CR = 2.7475

)o CT = 0.0000

0

Z)

C\J

8 -

o

0

00,00 0.25 0.50 0.75 1.00

Y/ (B/2)00FLRT DELTR WING0-

RLPHR = 4.0000LRMOR = 70.0000MACH NO. = 2.0100CR = 2.7475

0 CT = 0.0000

0

I

Ci

00

C)0

0

t.o 0 0 .25 0.50 0.75 O1.00/ (B/2)

Figure 28. Sectional Normal Force and Pitching Moment for a

Flat Delta Wing; M = 2.01, a = 4.0.

II83

FLAT TAPERED WING o FLAT TAPERED WING

ALPHA = 2.0000 cRLPHA = 2.0000LAMOR = 54.1154 K LADA = 54.1154MACH NO. - 1.6100 MACH NO. = 1.6100CR = 0.9524 CR = 0.9524Cf = 0.1905 CT = 0.1905Y/IB/2) = 0.0500 (m /B/2) 0.3500

o 0

°3 mo

D

o O0 02 0o .5 'u .. Oo FA TD AR D N

C.Ho 0.25 0'.50 0.75 1.00 To 0 0.25 0.50 0.75 1.00PSI PSI

o FLAT TAPERIED WING 0FLAT TAPEREO WING

O LPHA 2.0000 ALPHR = 2.0000LAMDA = 54.1154 LRMDR 54.1154MACH NO. = 1.6100 IMACH NO. = 1.6100CR = 0.9524 CR = 0.9524

o CT = 0.1905 CT = 0.1905e Y/(B/2) = 0.2000 Y/(B/2) = 0.5000

o 0o

D

C3 0

00

ooo1

9jO0 0.25 0'.50 0.75 1.00 T O0 025 0.50 0.75 1.00

SPSI PSI

1Figure 29. Pressure Distribution on a Flat Tapered Wing; M = 1.61, a = 2.0•

I

II

I

84

Io FLAT TAPERED WING

C" ALPHA - 2.0000LAMOR = 1q, 15 4 ,)MACH NO. = 1.8100CH = 0.9524

CD CT = 0.1905Y/8/2) = 0.7000

C

CDD

00

o 0 CD ( D DC

'b.oo 0'.25 0'.50 0.75 1.00PSI

o FLAT TAPERED WING

CD ALPHA = 2.0000LAMDR = 54.1154MACH NO. 1.6100CH = 0.9524

C) CT = 0.1905mY/(8/21= 0.9000C D-

o C

o-C'

0O0

(D0 (

0

'0.00 0'.25 '0.50 0. 75 1.00PSI

FFigure 29. Continued

I1I

85

Co FLAT -IFPERELU WI GCuj

o; RLPH-R - P.OOUOLRMOR = 54.1154~MACH NO. = 1.610(1CR9 = 0.9'1'24CT = 0.1905

0

)

CD D

CDD

0.00AC NO. = .0 10.10

I RLHR = 2.0024

CT = 0.1905

CD-D

CD

0D

0D

'000 025 0.50 0.75 1- .00

Figure 30. Sectional Normal Force and Pitching Momentfor a Flat Tapered Wing; M =1.61, a =2.0.

86

u-

y

2WING PRESS. LOADINGALPHAI = 10.00

1LAMOR = 0.00CR =1 0.61CT = 0.61M= 1.62

Fig. 31. Typical Multiple Isobars on a Rectangular W ng Planform.

87

0

Ln-Isobars

C0

CDUL)

1.

ILTN

WING PHFSS. LORDING

IRLPHA 10.00 YLRM0R z 70.00C R = 1.79

CT=0.00M 1.01

Fig. 32. Typical Multiple Isobars on a Delta Wing Planf arm.

88)

Isobars C

iU L

C- I

y

CT 79F18C3, 1.38

M~.~ TPc1Iultiple

Isobars On a Cropped

Det Winpan form

89

0

C-

Ln~I-

Isobars

X/

WING PRESS. -ORUIJNL

RLPHR = 10.00

LRMDA = 25.79CR = 1.73

CT = 1.25

M = 1.61

Fig. 34. Typical Multiple Isobars on a Trapexoidal Wing Planform.

I

I

I 90

Table 2

Low AR High ARDelta Trapezoidal TrapezoidalWing Wing Wing

C Experiment .029 .033 .048L

a= 0 Theory .029 .032 .053

(CM) Experiment -.029 -.019 -.064

a= 0 Theory -.029 -.019 -.073

leading edge and wing tip is to be expected and is due to wing thickness,

shocks, and wing twist; the latter, of course, is not significant for the

low aspect ratio wings.

Deformed Wings

These same wings under various kinds of deformation were also

1 analyzed by including vorticity paneling as well as the 3-D theoretical

loading terms in the AC distribution. Deformation shapes of three of

1 these wings are shown in Fig. 35. The delta wing (No. 1) in Fig. 35

was cambered and the slope relative to the chord line of the mean

j camber line was determined to be

e = -3 00825 + tan-'[tan 60 (F- n+n)] (100)

where f=y/(b/2) and =(x-xLE )/c. The camber (see Fig. 35) is uniform in

the spanwise direction and produces a maximum deflection angle of three

degrees relative to the chord line. Although this seems like a small

task, this is actually a severe test of the theory and its agreement with

experimental data. In Figs. 36 through 41, results are presented for threeI

91

Reference Plane

Mea Cabe Pla[n1051e -z~p

-3085i a-[tnj(C ~~ y

Reference Plane

MeCamber Plane laigLg

(c) Wamred aeoda TwingdDetWn

. -tan '0.3 454l i(C3 +T)sn ')

IIN, *

92

CAMBERED DELI WING Mb 1

I ME I A WING

o ALPHA = 6. 00i110 C 0. P,' el. 0O0LAMOR = 70.0000 - ,r [j1I"'rMACH NO . = 2 .0 100 , 1 L . . 2 . rC, I cCR = 2.7475 , -.

o CT = 0.0000 , r j 0 , jY/(B/2) = 0.1000 /'1 ,; U.L'0CO

oD CD

o- \

D: C

ic0 07 ) DC

T - -- e--1 " c

cb o0 0.25 0.5 0 . 75 1 .00 93.0 0O . d, f;. -,) C., ,.r

PSI ) IS I

0 CAMBERED DELTA WING C) H]c [{iE J 'IfL h ,N'

CD co

o-_ ALPHA = 6. O01 O c') AL -FH'

C ). crLLPMDA = 70.0000 !L Y:q 70.,,, ,MACH NO. = 2.11100 L MP'Hv 1,. 1 -L;;:

CA = 2 . 74 75 C R .9 d , !-CD CT = 0.0000 CD 0 .Of)Y/IB/2) 0 0.2500 lo

C) C D~1L ~, l

,::I- IST

ii CD

0 cj

c 0

C3 00 0

o 0 IT 0r

T i"

LM0 0.25 0.50 0 .70 1.00 90. C" U7. '.

PSI P S

Cigure 36. Pressure Distribution on a Cambered Dela Wing; M 2.01, a 6.0.

I0

-- -- o 0 .75I I1 l 0 0.... (l , i~r ... ... iT .. .. III lli '

93

o CAMBERED OELI WING Co CAMBEREO DELTA WING

o " ALPHA = 6.0000 0 ALPHR 6.0000LAMOR = 70.0000 LARt1 70.0000MACH NO. = 2.0100 MRCH NO. = 2.0100CR = 2.7475 CR 2.7475

o CT = 0.00001 CT = 0.0000to T/8B/2) = 0.6000 to T/IB/2) 0.8000

0 CD 0 C

0

0.00 0.25 0.50 0.75 1.O00 C. Ou 01.26 .. -c

PSI Psi

o CAMBERED OELTH WING

o" ALPHR = 6.0000

LAMDA = 70.0000MACH NO. = 2.0100CR = 2.7q75

0 CT = 0.0000wY/fB/2) = 0.7000

c'J

* ~PS[I

IIFigure 36. Continued

!( . .

I.. .

A0

I94

C CAMBERED DELTR WING

ALPHA = 6.0000LAMOR = 70.0000MRCH NO. = 2.0100CR = 2.7475

C" CT = 0.0000

Z0

UC-o

0

¥/ (

)N0

CD--

0oT

CD CAMBERE DELTR WING

I ALPHA = 6.0000LAMDR = 70.0000MRCH NO. = 2.0100CR = 2.7475

0CT = 0.0000

it o

C '

0

C

I° '0.00 0. 25 0. 50 0. 75 1. 00iY/ (B12)

Figure 37. Sectional Normal Force and Pitching Moment for a

Cambered Delta Wing; M = 2.01, c = 6.0.I

95

CAMBERED & TWISTED DELTA WING D CAMBERED & TWISTED DELTA WING

o : ALPHA = 6.0000 o" ALPHA z 6.0000

LAMOR = 70.0000 LAMORA 70.0000MACH NO. = 2.0100 MACH NO. = 2.0100CR = 2.7475 CR = 2.7475CT = 0.0000 C) CT = 0.0000T/(B/21 0.1000 w Y/[B/2J = 0.14000

C) c)-

0- -a- =o

C)0

C o

- 0 O0 0 0 50 0

0 I0

o 0

c'b 0 0'.25 0.50 0.75 1.00 -b. DO 0.?25 0. 50 0J.75 1 .00PSI PSI

Co CAMBERED & TWISTED DELTA WING 0 CAMBERLD & IlISTED DELTA W]Y'OgD C0c ALPHA = 6.0000 C - ALPHA = 6.0000

LAMOR = 70.0000 LAMDA = 70.0000MACH NO. = 2.0100 MACH NO. = 2.0100CR = 2.7475 CR = 2.7475

o CT = 0.0000 CT = 0.0000T/ B/2 ) 0.2500 LD "I t/ 2 ) = 0.500 f)

o o;

CL -= 0- :r

0 0

0 0

O

c" C ._)_ - - C) (

o 0

•

"I ---

T---

g - -

-

CC

C bOO 0'.2S 0'.50 0.75 1'.00 Th.oo 0.o o. o f.) 5G oPSI PSI

Figure 38. Pressure Distribution on a Cambered and Twisted Delta Wing;

M = 2.01, x = 6.0.III

3

I

1 96

ICo CAMBEREO & TWISTED DELTA WING C CAMBERED & TWISTED DELTA WING

o ALPHA = 8.0000 C ALPHA 6.0000LAMOR = 70.0000 LAMOR = 70.0000MACH NO. = 2.0100 MACH NO. = 2.0100

CR = 2.7475 CR = 2.7475CT = 0.0000 C CT z 0.0000Y/(6/2) 0.6000 w Y/ (8/2= 0.8000d"

o

Y/ 200 8 0

C) C 0

o

ACV )Ol 0 0

l0

',00 0.25 U0.5 0 O. 7 S 1.00 6 O. J -0 .50 .75 1. 0 6

PSI PSI

oo C MBERE D TWIS ED DELTA WING

O ALPHA = 6.0000LAMDR = 70.0000MACH NO. = 2.0100CR = 2.7175CT = 0.00000

_Y0 TI/B/2) = 0.7000

0 0c CD C

III Cd-

Th.00 0 .25 0.50 0.75 1.00PSI

IFigure 38. Continued

A

97

o CRMBEED & TWISTE) DELTA WING

o;- RLPHR - 6.0000

LRMDR = 70.0000MRCH NO. = 2.0100CR = 2.7475

C) CT = 0.0000

C)-

0

0

Cu-

0

000 0.25 0.50 0.75 1.00/ (B/2)

0CCRMBERED & TWISTED DELTR WINGI

RLPHR = 6. 0000LRMDR = 70.0000MRCH NO. = 2.0100CR = 2.7475

CM CT = 0.000

o(D

U

0 r

0.00 0.25 0.50 0.75 1.00Y/ (B/2)

Figure 39. Sectional Normal Force and Pitching Moment for a Camberedand Twisted Wing; M = 2.01, cx 6.0.i

I

1!

98

WARPED TRAPEZOIDAL WING C) WARPED TRAPEZOIDAL WING

C ALPHA = 6.0000 c- ALPHA = 6,0000LAMOA = 25.7917 LAMOR z 25.7917MACH NO. = 2.0100 MACH NO. 2.0100CR = 1.7324 CR = 1.7324

C CT - 1.288 CT = 1.2486

1 (B/2) 0.1000 T/I(B/2) 0.5500

Ho 0•

000- =

,300 0.,25 0.50 0.75 1.00 T.00 0.25 0.50 0.75 1.00

PSI PSi

oWARPED TRAPEZOIDAL WING o WARPED TRRPE/OIDHL WIN'

IO ALPHA = 6.0D00 0- ALPHA = 6,00DU3

MACH NO. = 2.0100 MR!;2 Nfl. =2.0100)CR = 1.7324 CR =1.7321

o CT = 1.21188 CT = 1.2'48PL. T D/2 = 0.3500 TI (5/21 = 0.6/0O

(o N

oo

CD_

CD C)i

o 0

9b.oo 0.2-5 00 0.5 .00 .00 0.25 0.00 0.5 1 .00PSI PSl

Figure 40. Pressure Distribution on a Warped Trapezoidal Wing; N = 2.01

0= 6.0.

I_____=_2.791______2579i

MACH .. ..... .

CA = .7324CA = .732

I99

C. WRREO TRRF'EOIDRL WING 0 WREPED TRAPEZOGOGfL WING

Co- RLPHR = 6.0000 " LPHR 6.0000LAMOR = 25.7917 LRMOA 25.791I7MACH NO. = 2.0100 MACH 6O. 2.0100CR = 1.7324 CR 1.7324

o CT = 1.2488 0 CT z 1.2489 T/(8/2] 0.7700 Lo T/ (1312? 0.9700

o

Li L 0C:I)

-)c) C)'.00 c0 COO ..N W)o CD PE a,

C) C)

oD

o 0 C

PSI Ko i

C) WARPED THRPE20IORL WI1NG

o RLPHR = 6.00011

LAMOR = 25.7917MRCH NO. = 2.0100CR = 1.7324

o CT = 1.2486,o Y/ (812) = 0.87000

C)

0 'D

c:C)

C) ) -C)

00

0

C9

F --~r - r .. . ...-

'.00 0.25 0.50 0.75 1.00PSI

Figure 40. Continued

100

CD WRPED TRRPEZO]DR NING0

RLPHR = 6.0000LRMDR 25.7917MRCH NO. 2.0100CR 1.7324

w CT = 1.2488r-

'0.

i Y/ (6/2)

O WAlRPED [RRPEZOI0RL NINO

ALPHA 6.0000LRMOR = 25.7917 "

MACH NO. = 2.0100CR =1.73?'4CT =1.21465

CD_

C)

0 '

Io

I °n

0.0

0.00 0.50 .7 1,0

Tl (6/?)Figure 41. Sectional Normal Force and Pitching Moment for a

Warped Trapezoidal Wing; M 2.01; = 2 6.0.

!- C=14

I

J 101

deformed configurations at a Mach number of 2.01 and angle of attack of

6 degrees. A more complete summary of agreement between the theory and

experimental data is presented in Appendix A.

For the cambered delta wing, agreement at the inboard stations is

relatively good while outboard agreement is rather poor. However, note

that in the regions where most of the wing loading occurs, the agreement

is satisfactory. Additionally, the inclusion of the doublet paneling

terms in the mathematical formulation provides the correct trends and an

added degree of accuracy.

For the same planform shape, linear spanwise twist is added to the

camber deformation such that the mean camber surface is now defined as

el, j = tan-[0.l4054l( -Cn+n)q] (101)

Results of this configuration are shown in Figs. 38 and 39 and in

Appendix A. Similar results are obtained for these cases as was observed

for the cambered (alone) case. Agreement is acceptable at the inboard

stations but is questionable at the outboard stations.

In observing the experimental data for both the cambered alone and

the cambered and twisted wings, the outboard stations (r, ' .5) appear to

have a supersonic leading edge. Whereas the high leading edge sweep

angle gives rise to a subsonic leading edge. In addition, at the ? = 0.5

station, a weak shock appears to be located near = .4 for M = 1.61 but

is not present for M = 2.01 (see Appendix A). Since the theory does not

account for the shocks, the disagreement in this region is not surprising.

The next wing subjected to the theoretical analysis was a "warped"

trapezoidal wing (see Fig. 35). For both M 1.61 and 2.01, the leading

I

102

edge is supersonic and multiple "fold over" regions occur because of the

low aspect ratio of the fact that the wing is trapezoidally shaped. The

equation governing the mean warped camber surface is

= f -tan- [-0.0334547 sin(- --)sin(T)] (102)i,j2

Results for this wing are presented in Figs. 40 and 41 and in

Appendix A. Considering the wing deformation, the agreement is remark-

ably good over the entire wing for both Mach numbers. Note also that the

addition of vorticity paneling to the AC distribution predicts the right

trends and correctly accounts for local wing deformations in the upstream

running Mach cones.

', I

II'I'IU,!

UI . ... . -. . .. " | I -"T

71I

103

Elevon Loading

Very little experimental data exists for loading on an elevon

immersed in the wake of a wing in supersonic flow. The only data found

was Nielsen (Ref. 1) which included the blunt wing trailing edge as

previously discussed. Figure 42 is a comparison of the normal force

coefficient, CN, and hinge moment, C versus wing angle of attack for

an elevon deflection angle of -10'. In this figure, the present theo-

retical results are compared to experimental data and to the theoretical

results from Ref. 1. In the present theory, the blanking effect from

the wing is included which seems to properly account for the normal force

coefficient variation. It is noted also that compressibility or the

Mach number effect, is also properly accounted for, thereby providing

excellent agreement (at least for this case) between experimental data

and theory.

The hinge moment for the elevon is also plotted and agreement here

is not good except at very small angles of attack. Reasons for this

discrepancy may be the result of one of the following problems in the

analysis or perhaps combinations thereof: (1) It is known that viscous

flow in this region could cause large changes in the pressure field as

originally hypothesized in Ref. 1. However, because of the excellent

agreement of the normal force coefficient data from a potential flow

analysis and the schlieren photographs of this region, it is certainly

not conclusive that viscous effects dominate the solution. (2) The

downwash model as outlined in the theory section is inadequate in pre-

dicting the correct downwash velocities. As a matter of fact, if the

downwash distribution is assumed to be constant across the elevon span

g 104

6 -100, Gap =0.0

06 = 2.. Ga200.

M = 2.99 u.1

- -M =2.20 (Ref. 1)

M = 1.62 (Ref. 1)

CC

-0 2.50 5.D2 7.5~ S.C l

;i

Fig. 42. Elevon Normal Force and Hinge Moment Coefficients

versus Wing Angle of Attack, 6 -1~0', Gap =0.0.

LII

105

and chord slightly better results are obtained. However, since the total

normal force is predicted quite well, the chordwise distribution is

apparently in error. From the schlieren photographs (Fig. 23) it appears

that much more of the upper surface of the elevon may be blanked out than

that predicted in the theory. If this were true, the normal force coef-

ficients would decrease slightly and the moment coefficients would

increase (leading edge up) which would give rise to an increase in the

associated hinge moment. (3) The assumed location of the trailing

vortices relative to the elevon is perhaps incorrect. (4) In analyzing

the experimental data from Ref. 1, there is possibly a problem in inter-

pretation and conversion to the reference areas and lengths used in the

present theoretical analysis.

In conclusion, it is recognized that the flow field immediately

behind the blunt wing trailing edge and elevon leading edge is not well

understood. Detailed experiments to measure pressure and loads accom-

panied by schlieren photographs are required before an adequate model can

be postulated.

I Finally, Figure 43 is a plot of the theoretically predicted normal

force and hinge moments for the elevon when a gap exists between the

elevon and the wing. Similar results are obtained as for the "no gap"

case but the influence of the gap is clearly identified and the present

theory predicts the correct trends.1III

106

Cap= 0.,

-~~ Gap = 4.011 - -

N 2.99

M = 2.20

M = 1.62

//

N= 2.99

-100

Fig. 43. Theoretical Normal Force and Hinge Moment Coefficientsversus Wing Angle of Attack for Gap 0.0 and 4.0",

'II

VIII. REFERENCES

1. Selden, B. S., Goodwin, F. K., and Nielsen, J. N., "AnalyticalInvestigation of Hinge Moments on Missile Trailing-Edge ControlSurfaces," Final Report, U.S. Army Missile Command, September 1974.

2. Glauert, H., "A Theory of Thin Airfoils," British AeronauticalResearch Committee, Rep. and Memo. 910, 1924.

3. Glauert, H., "Theoretical Relationships for an Airfoil with HingedFlap," British Aeronautical Research Committee, Rep. and Memo.,

1095, 1927.

4. Halsey, N. D., "Potential Flow Analysis of Multielement AirfoilsUsing Conformal Mapping," AIAA Paper No. 79-0271, Jan. 1979.

5. Halsey, N. D., "Potential Flow Analysis of Multiple Bodies UsingConformal Mapping," Master of Science Thesis, California StateUniversity, Long Beach, California, Dec. 1977.

6. Ashley, H., "Some Considerations Relative to the Prediction ofUnsteady Air Loads on Interfering Surfaces," Symposium on UnsteadyAerodynamics for Aeroelastic Analyses of InterferinSurfaces, Part I,NATO AGARD-CP-80, April 1971.

7. Woodward, F. A., "Analysis and Design of Wing--Body Combinations atSubsonic and Supersonic Speeds," J. of Aircraft, Vol. 5, No. 6,Nov.-Dec. 1968, pp. 528-534.

8. Lan, C. E., "A Quasi-Vortex Lattice Method in Thin Wing Theory,"J. of Aircraft, Vol. 11, No. 9, 1974, pp. 516-527.

9. DeJarnette, F. R., "Arrangement of Vortex Lattices on Subsonic Wings,"Vortex-Lattice Utilization Workshop, NASA SP-405, NASA LangleyResearch Center, May 1976, pp. 301-323.

10. Cunningham, A. M., "Oscillatory Supersonic Kernel Function Methodfor Interfering Surfaces," AIAA Journal of Aircraft, November 19 4.Vol. 11, No. 11.

11. Cunningham, A. M., "Unsteady Subsonic Collocation Method for WingsWith and Without Control Surfaces," ATAA Journal of Aircraft,June 1972, Vol. 9, No. 6.

12. White, Richard B., And Landahl, Marten T., "Effect of Caps on theLoading Distribution of Planar Lifting Sair 'aces," AIAA Journal,Vol. 6, No. 4, April 1968, pp. 626-631.

107

I

108

13. Landahl, M., "Pressure-Loading Functions for Oscillating Wings withControl Surfaces," AIAA Journal, Vol. 6, No. 2, February 1968,pp. 345-348.

14. Purvis, J. W., "Simplified Solution of the Compressible SubsonicLifting Surface Problem," Master's Thesis, Aerospace EngineeringDepartment, Auburn University, August 1975.

15. Burkhalter, J. E., and Purvis, J. W., "An Aerodynamic Analysis ofDeformed Wings in Subsonic and Supersonic Flow," Final Report onU.S. Army Contract DAAG29-77-C-0069, Aerospace Engineering Depart-ment, Auburn University, December 1977.

16. Carlson, H. W., and Miller, D. S., "Numerical Methods for theDesign and Analysis of Wings at Supersonic Speeds," NASA TN D-7713,1974.

17. Vaughn, Milton E., Jr., "Pressure Distribution on an Ogee Wing inSupersonic Flow," Master of Science Thesis, Aerospace EngineeringDepartment, Auburn University, Alabama, 1981.

18. Harmon, S. M., and Jeffreys, I., "Theoretical Lift and Damping inRoll of Thin Wings with Arbitrary Sweep and Taper at SupersonicSpeeds - Supersonic Leading and Trailing Edges," NACA TN 2114, 1950.

19. Malvestuto, F. S., Margolis, K., and Ribner, Ii. S., "TheoreticalLift and Damping in Roll of Thin Sweptback Wings of Arbitrary Taperand Sweep at Supersonic Speeds - Subsonic leading Edges and Super-sonic Trailing Edges," NACA TN 1860, 1949.

20. Ashley, H., and Landahl, M., Aerodynamics of Wings and Bodies,Addison-Wesley, Reading, Mass., 1965.

21. Ashley, H., Widnall, S., and Landahl, N., "New Directions in Lifting

Surface Theory," AIAA Journal, Vol. 3, No. 1, January 1965, pp. 3-16.

22. Pitts, W. C., Nielsen, J. H., and Kaattari. (;. E., "Lift and Centerof Pressure of Wing-Body-Tail combinations at Subsonic. Transonic,and Supersonic Speeds," NACA Report 1307, 1957.

23. Purvis, J. W., "Analytical Prediction of Vortex Lift," AIAA Paper79-0363, 17th Aerospace Sciences Meeting, 1979.

24. Abernathy, J. M., "An Analysis of Gap Effects on Wing-Elevon Aero-dynamic Characteristics," Master of Science Thesis, Aerospace Engineer-ing Department, Auburn University, June 1980.

I

109

25. Tinling, Bruce E., and Dickson, Jerald K., "Tests of a Model Hori-zontal Tail of Aspect Ratio 4.5 in the Ames 12-Foot Pressure WindTunnel. I-Elevon Hinge Line Normal to the Plane of Symmetry,"NACA RM A9Hlla, October 1949.

26. Abbott, Ira H., and Von Doenhoff, Albert E., Theory of Wing Sections,Dover Publications, Inc., New York, N. Y., 1959.

27. Jones, J. Lloyd, and Demele, Fred A., "Aerodynamic Study of a Wing-Fuselage Combination Employing a Wing Swept Back 630. - Character-istics Throughout the Subsonic Speed Range with the Wing Camberedand Twisted for a Uniform Load at a Lift Coefficient of 0.25," NACA

RM A9D25, August 1949.

28. Evvard, John C., "Use of Source Distributions for EvaluatingTheoretical Aerodynamics of Thin Finite Wings at Supersonic Speeds,"NACA Report 915, June 1949.

29. Landrum, E. J., "A Tabulation of Wind-Tunnel Pressure Data andSection Aerodynamic Characteristics at Mach Numbers of 1.61 and2.01 for Two Trapezoidal and Three Delta Wings Having DifferentSurface Shapes," NADA TN D-1394, 1962.

30. Landrum, E. J., "A Tabulation of Wing-Tunnel Pressure Data andSection Aerodynamic Characteristics at Mach Numbers of 1.61 and2.01 for a Reflex Cambered Wing and a Cambered and Twisted WingHaving the Same Swept Planform," NADA TN D-1393, 1962.

II

APPENDIX A

'! I

II

!II

TABLE A

FigureNumber Deformation Equation

Al-A28 No deformation

Cambered Delta WingA29-A32

e -3.00825 + tan-'[tan 6.0'(x - xy + y)]

Cambered and Twisted Delta Wing

A33-A6 0& tanI H(tan S. 0 )(xy - xy- + y2 )]

Warped Trapexoidal Wing

A37-A40 ~ ~ e -tan -' [- 0. 033654 [Tsin ( - Y) sin ( x)I

Ill

112

o FLAT TRAPEZOIDAL WING 0 FLAT TRAPEZOIDAL WING

o ALPHA = 2.0000 C1 ALPHA r 2.0000

LAMDA = 25.7917 LAMOR = 25.7917MACH NO. = 1.6100 MACH NO. = 1.6100CR = 1.7324 CR = 1.732U

,n CT = 1.2488 CT = 1.218e,-", T/18/2) = 0. 1000 LJY1(B/2) - 0.5530

C:) CD 0 uc;-

00

Co 0

,10 0' 5 i . '75 1' 00 o' .00 0. 2r -0.50 0. 75 1 .O00

Q FLAT TRAPEZOIDAL WING CD FLAT IRPEZOIDAL WING

C" ALPHA = 2.0000 °nALPHA = 2.0000

MACH NO. = 1.6100 MC N{. = 160CH = 1.73241 CH = 1.7324t

Ln CT = 1.2488 Q T = 1.2L488""Y/(B/2) =0.3500 Y/ T1B12) 0 . 6 ir, O

CD CDCJ•0 Oo 0

0 o

o 0

c: CD

0 0

b.00 0'.25 0.50 0. 75 1.00 cT.oo 0.25 0.50 .75 1.00

PSI PSI

Figure Al. Pressure Distribution on a Flat Trapezoidal Wing; M = 1.61,

o = 2. 0.

- T (/2 0350 j.- ........... /0. . .. .. , , . =; . --,r _T . .L5 . . .

113

oFLAT TRAPEZOIDAL WING cm FLAT TRAPEZOIDAL WING

O; ALPHA = 2.0000 0- LPHAR = 2.0000LAMDA = 25.7917 LAMDA 25.7917MACH NO. = 1.6100 MACH NO. -1.6100CA = 1.7324 CR 1. 7324

U)CT = 1.2488 CT = 1.2488Y/ (6/2) =0.7700 [j (6 (/2) =0.9700

~ o~ooor, LL2o 0-:

ch. o o 0'.25 0'.50 01.75 1.00 '0.00 0.25 0.50 0.75 1'.00PSI PSI

o3 FLAT TRAPEZOIDAL WINGclJ

o ALPHA = 2.0000LAMOA = 25.7917MACH NO. = 1.6100CR = 1.7324CT = 1.2488

- T/(B/2) = 0.8700c.00

00

00

0n

0Q

0D(

U,03

00 0'. 25 0'.0 07 '0

PS

Figure Al. Continued

I114

o FLRT THRPEZOIDRL WING

C\1

o RLPHR = 2.0000

LRMDR = 25.7917MRCH NO. = 1.6100CR 1.7324

L CT = 1.2488C -0

U-0

0

0

O0I 1 1 70.00 0.25 0.50 0.75 1.00

Y/ (B12)

0SFLRT TBRHPEZO]DRL WING

RLPHR = 2.0000LRMOR = ;5.7917MRCH NO. = 1.6100CR = 1.732tlCT = 1.248R

LI)

0_

'D

10

10

0

I Tl (B/2

FLire A2. Sectional;j 0.75 Frc and Pitching Moment for a

I Flat Trapezoidal Wing; M = 1.61, a = 2.0.

Cli UU

I

115

FLAT TRAPEZOIDAL WING 0 FLAT TRAPEZOIDAL WING

.° ALPHA = 2.0000 C" HILPHA = 2.0000LAMOP 25.7917 LAMfi = 25.7917MACH NO. = 2.0100 MACH NO. = 2.0100CR = I.7324 CR 1.1324CT = 1.2488 m CT 1= .2488Y/(B/2) = 0.1000 - Y/(B/2) = 0.5500

o 0

Lfl Llw

o 0•0 0 ( (

00

o0

b.oo 0.25 0.50 0.75 1.00 Th.00 0.25 0.50 0.75 1.00

PSI PSi

o FLAT TRAPEZOIDAL WING C FLR1 TRAPEZOIDAL WING

8-ALPHA = 2.0000 C3 ALPHI 2.0000LAMOA = 25.7917 LAMDA = 25.7917MACH NO. = 2.0100 MACH NO. = 2.0100CR = 1.7324 CR = 1.7324CT = 1.2488 CT = 1.2488Y/ (/21 = 0.3500 - Y/(B/2 = 0.6700

o 0

o 0

ST -

r 1 I - 1

O0 0.25 0'.50 0'.75 O1.00 . 0.25 0.50 0.75 1.00PSI PSf

Figure A3. Pressure Distribution on a Flat Trapezoidal Wing; M = 2.01,= 2.0.

I

116

FLAT TRRPElOIDA[ WING c FLT TRAPEZOIDAL WING

CD ALPHA = 2.000U CD RLFHR = 2.0000

LAMOR 25.794i LAMOR = 25.7917MACH NO. 2.0100 MACH NO. = 2.0100CR = 1.7324 CH 1.7324CI = 1.2488 CT 1.2488

- Y/tB/2) 0.7700 - Y/(B/2) 0.9700

09

o 0

0

C)c--C

C:) C)o 0oo

c~ciM 0.50 0.70 1 ct.00 0.95 0.50 0.75 10PSI PS I

o FLIl 1RPqPEZ0ORLA WING

(: - LPHO = 2.0000LflIMOP = 25.7917MACH NO. =2.01D0

CR = 1.7324LCT = 1.2488

1/(8/21 0.8700

CD0

LIC)

I°00

0...0 0.25 0.o0 0.75 1.00PSI

FI Figure A3. Continued

117

cJ FLHT TRRPEZOIORL WING

RLPHR = 2.0000LRMCR 25.T71MA1FCH NO. 2.0100]CR - 1.732q1

u) CT 1.2488

Z C)

UC)

0

- -'0'.00 0.25 0.50 0.75 1 0C

Y/ (B/2)

FLAT TRRPEZ1]IORL WNG

RLPHR = 2.0000LRMOR = 25.7917MRCH NO. = 2.0100CR = 1.7324

j u) CT = 1.24188

0

(D

~0.00 0.25 0. 50 0. )

I Y! (BI)

Figure A4. Sectional Normal Force and Pitching Moment for a

j Flat Trapezoidal Wing; M = 2.01, = 2.0.

I

II

118

C FLqT TRRPEZOIORL WING m FLAT TRAPEZOIDAL WING

o; ALPHI, 4.0000 A LPHA = 4.0000LAMOH = 25.7917 LAMOA = 25.79171MACH NO. = 1.6100 MACH NO. = 1.6100CA = 1.732L CR = 1.7324

C3 CT = 1.2488 CT = 1.2488T/(B/2) = 0.1000 m YI(B/2) 0.5500

Co 0 Co=; 0

l3 (D ____c

o 0

9b.O0 0.25 0.50 0.75 1.00 T'.oo 0.25 0.50 0.75 1.00

PSI PSIC FLAT TRAPEZOIDAL WING FLAT TRAPEZOIDAL WING

oALPHA = 4.0000 A LPHR = 4.0000LAMOA = 25.7917 I LRMOA = 25.7917MACH NO. = 1.6100 MACH NO. z 1.6100CR = 1.7324 CR = 1.7324

o CT = 1.248 CT = 1.2188!") Y/(B/2) z 0.3500 m' C) Y/(8/2) 0.6700

#1 °~0

S.oo 0'.25 '.50 0.75 1.00 h.oo 0.25 0.50 0'.75 1.00

PS I P,.,

Figure A5. Pressure Distribution on a Flat Trapezoidal Wing; M =1.61,

a= 4. 0.

119

oFLAT THHPEZOIDHIl WJIl4 C) FLHI liflFOZOIDFO. WING

o; ALPHA ~4 .OOU1J RLPHH- 4.0000gLAMOR dLS.7 9 121.7917MACH NO. 61c MA~CH Nil. - euCR 17. Q1 L 13 2

M CT I0 L1t CT I c48brn U. TI(/l 70C0 CD Y/ ffi/2, 0.9*10 K

CC

CD C

o oD

0 C C

Oil--0 C2 C. SO UC L 0 0 .7, 1O

CS C

S I . 73?I

oD LT I RP~fIA . 248

05. P7(

c3iL0

Fiur A5 CotCue

I120

C) FLm 1HRHPEZOIDRL WING

RLPHR = 4.O000LRMOR = 25.7917MRCH NO. = 1.6100CR = 1.7324CT

= 1.2488

CD

CC

C) C

U - (D

CC

CD

'O0.0 0.25 0.50 0.75 1.00

Y/ (B/2)CD

FLRT TRRPEZOIORL WING

RLPHR = 4.0000LRMOR = 25.7917MRCH NO. = 1.6100CR = 1.7324

M CT = 1.248B

CD-

4 C)

C) C

C))

C)-

CD

I) -- --

'0.00 0.25 0.50 0.75 1 . 0Y/7 (B/2)

Figure A6. Sectional Normal Force and Pitching Moment for a Ft-

Trapezoidal Wing; M =1.61, a 4.0.

121

C3 FLAT TRAPEZOIDAL WING oFLAT TRAPEZOIDAL WING

C)ALPHA = 4.0000 ALPHA = 4.0000LRMOR = 25.7917 >-~LAMDA - 25. 7917MACH NO. = 2.0100 MACH NO. -2.0100 ICH 1.7324 CR =1.7324 I

oCT = 1.2488 CD CT - 1.2488Y/IB/2) =0.1000 LJ T (8/21 0.5500 [

0:)

o 0 0 (

0 CD

0 (D0(

o 0

Th.oo 0 .25 -0'.50 0 1.75 1'.00 0h.00 0'. 25 -0.50 - 0.75 1.00PSI PST

oFLAT TRAPEZOIDAL WING C0 FLAT TRAPEZOIDAL WING

CD ALPHA = 4.0000 0ALPHA 4.0000LAMOA = 25.7917 f- LAMOR 2S.7917MACH NO. = 2.0100 -,MACH NO = 2.0100CR = 1.7324 IICR = 1.7324

Co CT = 1.2488 CT = 1.2488Y/(6/2) 0.3500 Y/ (B/2) 0.6700

C3o

C3 C

0 D CD

a) 0

oa 0

Th. oo 0.2 0.0 S 0. 7 1.00 Th.oo 0.25 0.50 0.75 (.00PS I PS I

Figure A7. Pressure Distribution on a Flat Trapezoidal Wing; M 2.01,ax 4. 0.

Ew

I

122

o FLAT TRAPEZOIDAL WING C FLAT TRAPEZOIDAL WING

o ALPHA = 4.0000 - ALPHA = 4.0000LAMOR = 25.7917 LAMOA = 25.7917MACH NO, = 2.0100 MACH NO. = 2.0100CR = 1.7324 CR = 1.7324

o CT = 1.2488 CT = 1.2488YTIB/21 0.7700 Y/(8/2) = 0.9700

o

C)- Dez~ m-m

o 0

TO.00 0.25 0. 50 0.75 1.00 ThO0 0.25 0.50 0.7 .0

PS I PS I

o FLAT TRAPEZOIDAL WING

O" ALPHR = 4.0000LAMDAP 25.7917 {-.MACH NO. : 2.0100CR = 1.7324

DCT = 1.2488

1I18/2) = 0.8700

*1 0.M

Ljc- cbc)00

'0.00 0.25 0.50 0.75 1.00

PSI

Fig. A7. Continued.

IB

I

123

O FLAT TRAPEZOIDAL WING

" RLPHR = 4.0000LAMOR = 25.7917MACH NO. = 2.0100CR = 1.7324

o CT = 1.2488

C)

U

0.25 0.50 0.75 1.00Y(/ (B/2)I

FLAT TRAPEZOIDAL WING

0\0

ALPHA = 4.0000LAMOR = 25.7917MACH NO. =2.0100CR 1.7324

0 \

CT 1.248

XCL

U

C0

0.00 0.25 0.50 0.75 1.00

Y/ (B/2)

Figure A8. Sectional Normal Force and Pitchimg Moment for aFlat Trapezoidal Wing; M 2.01, ot 4.0.

124

CD FLAT TRRPLZII[RL WING C FLAT TRAPEZOIDAL WING

- ALPHA = 10.6000 ALPHA = 10.0000LAMDA = 25. 1917 LAMDA = 25.7917MACH NO. = 1.6100 MACH NO. = 1.6100CR = 1.7324 CR = 1.732qCT = t.2488 CT = 1.2488-YI(B/2) = 0.1000 'rl/(B/2) 0.5500

00

00o In o

L)CD: 0 (Do (D

UC)C C)° °N0 0

o 0T___ .... - -U I . 0r-------1--00o 0T25 0.50 0.75 1.00 Thb.oo 0.25 0.50 0.75 1.00

PSI PSI

o FLAT TRANPEZOIOOL WING D FLAT TRAPEZOIDAL WING- ALPHA = 1.0.0000 -i ALPHA = 10.0000 I

LAMDA = 25.7917 1 LAMOA = 25.7917MACH NO. = 1.6100 jMACH NO. = 1.6100CR = 1-7324 CR 1 .7321..

LACT = 1.21488 0CT = 1.2488r-- T/(B/21 = 0.3500 -- T/(B/21 0.6700o -o-

C) ~C)

0

0.2 0.0 07 0CD .5 10

I °n

o 0T ,

o 0'. 25 0.50 o .75 - '. oo h'. oo o. 25 o.0s 0'. 7 1.00o

SPSI PSI

Figure A9. Pressure Distribution on a Flat Trapezoidal Wing; M = 1.61,A = 10.0.

II

-n CT' = .28 IIC 128I I- .. .. . .. . .Y / 8 2 = 0 ... .1- 0. -... .- = 0.6 70 0

..... 0ll "i I

II

125

o FLAT TRRPEZOIOAL WING FLAT TRAPEZOIDAL WING

RLPHR 1 10.0000 RLPHR = 10.0000LAMOR = 25.1917 LRMDR = 25.7917MRCH NO. = 1.6100 MRCH NO. = 1.6100CR = 1.7324 CR = 1.7324

o C= 1.2488 (D CT = 1.2488o T/(6/2) = 0.7700 Y/IB/2) = 0.9700

0

0o

Q Lfl U,

aC - 07M C

Lo ) C LflDC C ) (

00

o 0

'0.00 0.25 0.50 0.75 1.00 -.00 0.25 0.50 0'.75 1.00PSI PS1

o FLRT TORPEZOIDAL 1NG0

ALPHA = 10.0000LAMOR = 25.7917MACH NO. = 1.6100

D CR = 1.7324w CT = 1.248B

T/(B/2) = 0.8700o yo

0•

.1 L°

U6,

(D C

,00 0.25 0.50 0.75 1.00PSI

Figure A9. Continued

!II

NNN

126

o FLRT TRRPEZOIOAL- WING0C.

- ALPHA = 10.0000LRMDR = 25.7917MRCH NO. = 1.6100CR = 1.7324t

Ln CT = 1.2483

C

C) C DoC;ItnU CD

C)C

001

'0.00 0.25 0.50 0.75 1. 00Y/ (B/2)

C!o FLAT TRRPE/OIOAL WING

1 LPHA 10.0000LAMOR = 25.7917MACH NO. = 1.6100CR = 1.7324nCT = 1.2488

CC

0LI)0

-- o.

I L)

0I,

. I I1

0.00 0.25 0.50 0.75 1.00Y/ (B/2)

Figure AlO. Sectional Normal Force and Pitching Moment for aFlat Trapezoidal Wing; M = 1.61, c = 10.0.

1~127

o FLAT TRAPEZOIDAL o FL/I TRRPEZOIDala)

co- ALPHA 10.000 0 ALPHA 1O.0000

LRMDA 25.7911 LRMH- 25 7917MACH NO. 2.0100 MACH NO. = 2.0100CR = 1.7324 CR = 1.

7324

CT = 1.2488 CT = 1.2488w Y/(B/21 = 0.1000 w Y/(B/2) 0.5500

0 L

C) 0D

U C- (Dci-

0 01

CDD

CD Cj C

o ,Co cCe

0 0

o- T

S o 05 .0. 1 L. 0o U. .50 0.75 1. OPPS I PSI

o FLAT TRAPEZOIDH.. FLAT 1RHFEO1IDAL

Co ALPHR z 10. 000' PI Pi 1 0.001JO.LRMDR z 25. 7917 LOMDOMACH NO. = 2.01O MRCH NC. = 2.0100CR = 1.7324 CR z 1.73291CT = 1.248b CT 1= I P48

wTY/ B/2) 0.3500 i .' -

o 0J i -,C5

CCD

UC3D

FiueAl-rsueDsrbto naFa rpzia ig .0 11.,

ci=10.0.

I

128

oFLAT TRAPEZOIDAL FLAT TRAPEZOIDAL

ALPHA = 10.0000 ;- ALPHA = 10.0000LAMOR = 25.7917 LAMOR = 25.7917MACH NO. = 2.0100 MACH NO. = 2.0100CR - 1.7324 CR = 1.7324CT = 1.2488 CT = 1.2488T/(B/2) 0.7700 Y /(8/2) 0.9700

JO w0-Co

o 0

o 0

93.oo 0.25 0.50 0.75 1.00 V.00 0.20 0.50 0.75 1.00

PSI PSI

o FLAT TRAPEZOIDAL

8" ALPHA = 0.D000LAMOR = 25.7917MACH NO. = 2.0100CA = 1.7324CT = 1.2488Y/18/2) 0.8700

0

I

I

93r00 0.25 0.50 C.n.00PSI

I Figure All. Continued

I

2!. _____

I

129

o FLRT TRAPEZOIDRi.

- ALPHR = 10.0000LAMOA = e5.7917MRCH NO. = .0100CR = 1.7324

U) CT = 1.2488p.-

o

C)ZL

00CC)

h0.o0 0.25 0.50 0.75 1.00

Y/ (B12)0o FLRT THRFEOI0HL

RLPHR = 10.0000LAMOR = 25.7917MRCH NO. = 2.0100CR = 1.73P4

r CT = 1.2688

CL

0

U (D

IU

I DC).

0.00 0.25 (.50 0.75 1.00i Y/ (B12)

Figure A12. Sectional Normal Force and Pitching Ioment for aFlat Trapezoidal Wing; M = 2.01, a = 10.0.

I

.. . ,.. ..I_ : L , --i -r .

130

o FLAT DELTA WING C3 FLAT DELTA WING

o ALPHA = 4.0000 0" ALPHA 4.0000LAMDA = 70.0000 LAMOR 70.0000MACH NO. = 1.6100 MACH NO. m 1.6100CR = 2.7475 CR = 2.7475

o CT = 0.0000 0 CT = 0.0000Y/1(/2) 0.1000 cr D Y/(8/2) = 0.4000

0

o N

C 'D C1D

ID C

0 CD

0'0 '25 0'.50 0'.75 1.0Ol.00 0.25 0.50 0.75 10

PSI PS[

SFLAT DELTA WING C FLAT DELTA WING

5"ALPHA = 4.0000 P' LPHA .0000LAMDA = 70.0000 ,oLRMDA = 70.0000 FMACH NO. = 1.6100' MACH NO. = 1.61O00

CA = 2.7q75 CR 2.7475C3 CT = .0000 CT =0.0000

Y/(8/2) 0.2500 Y11812) =0.5000

ICDAL

u 0

o D

'.00 01.25 0.50 0.75 1.00 _.00 0. 2L, 0.50 .7 1.00

PSI PSI

Figure A13. Pressure Distribution on a Flat Delta Wing; M 1.61, 0x 4.0.

CR!2745C .17! T=000 T=000

T/(/1 020! D T E/) 050o, _ _ -... _ -__----ID.. . .._

I

131

o FLAT DELTA WING C FLAT DELTA WING

o; ALPHA = 4.0000 C)- ALPHA 4.00000 LAMDR = 70.0000 LAMOR = 70.OOO0

MACH NO. = 1.6100 I MACH NO. = 1.6100CR = 2.71475 CR = 2.7475

o CT = 0.0000 o )D = 0.0000Y/(8/2) = 0.6000 T 15/2) 0.8000

D D 0 •D (D (D

o CD

0 N) CJ

-0 c5

oD I D m

oD CD=)

O0 002 '5 5 1O..oo 0.25 0.50 0.75 10.Do 0.25 0.50 0.75 1.00

PSI PSI

oD FLAT DELTA WING

ALPHA = 4.0000LAMOA = 70.0000

co) MACH NO. = 1.6100CR = 2.7475

O I m CT = 0.0000C"5/ ID (B/21 0.7000

C

CC0

.0? 0.25 .50 0.75 i.00

Figure Al> Continued

I132

o FLRT DELTR WING

o- RLPHA R 4.0000LRMDR = 70.0000MRCH NO. = 1.6100CR = 2.7475

o C1 = 0.0000

C)

- CD

U

C2D

C-

0

C,

'0.00 0.25 0.50 0.75 1.00Y/ (B/2)

CoFLRT DELTn WING

I RLPHR = 4.0000LRMDR = 70.0000MRCH NO. = 1.6100CR = 2.7475

0 CT = 0.0000

I ~U _- - -- -- -- -

C

C_

C

0.00 0.25 0o50 0.75 1.00Y/(B12)

Figure A14. Sectional Normal Force and Pitching Moment for a

Flat Delta Wing; 14 = 1.61, jx 4.0.

I

133

FLAT DELTA WING o FLAT DELTA WING

" ALPHA = 4.0o0 C0 ALPHA = 4.0000LANDA = 70.0000 LMAMO = 70.0000MACH NO. = 2.0100 MACH NO. = 2.0100CR = 2.7475 CR = 2.7475

o CT = 0.0000 CT = 0.0000YT/B/2) = 0.1000 , T/(6/2) 0.Y000

o c

C) C)

o o C

Q) C)

0):

C))

o •

-- cl - C) r

5 0 5 .O

C- -C)-

'.00 0.25 0.50 0.75 11.00 0 00 0.5 0.50 .75 1,06PSI PSI

o FLAT DELTA WING C) FLAT DELTA WING

o" ALPHA = 4.0000 CD- PLPHO q. 0oo0,LAMDA = 70.0000 LAMDAR 70.0000MACH NO. = 2.3100 MACH NO. = 2.OICCCR - 2.7475 CH = 2.7q75CT - 0.0000 CT = 0.0000Y/[8/2) = 0.2500 m T/ (B/2) = 0.5000

Q)o CD

C) C

-o CD

o co7 F . . "HI -- 1-1

'.0O0 0. 25 0.507 0. 75 1 .1 0 0-J, ( . p5 C, 5 0 0. ,:5 1 . ;PSI PSI

IFigure A15. Pressure Distribution on a Flat Delta Wing; N 2.01, x. 4.0.

I

I!

I

134

o FLAT DELTA WING FLAT DELI WING

o API-A 1.0CD ALPHA z 4.0000

LAMDR 70.0000 MACH NO. = 2.0100

MACH NO. = 2.0100 ACR = 2.7275

CR = 2.7U475 CT = 0.0000

CT = 0.0000 C .

Y/ /B1'2 0.6000 m Y/(B/2) 0.8000

oI0 C) 0

.)o-

)

CD CLO

0: 0

-C,oZ 0

- CD

0 1 0 2

.0 0.25 0.50 0.7b 1.00 .,PSIPS

C FLAT DELTA WING

AALPHA = 4.0000)LAMDR = 70.0000

MACH NO. = 2.0100, CR = 2.71475

CT = 0.0000o Y(B/2) 0.7000

ca

CD C

-n0

o o

o

C0.oo 0.25 U.5O 0. 1' . LGPSI

I A15. Continued.

I

I135

Q FLAT OELTA WING

C - fLPHA = 4.0000LAMOR = 70.0000MRCH NO. = 2.0100CR = 2.7475

Co CT = 0.0000

0ZC

C,Z

U

0

0

(D.o 0.25 0.50 0.75 1.00

Y/ 1B/2)

CFLAT DELIR WING

RLPHR = .0000LRMDR = 70.0000MACH NO. = 2.0100

CD o CR = 2.7475CT = 0.0000

0

0iCI

C

0

0.00 0.25 0. 5 0 0. 75 1.00Y/ (B/21

Figure A16. Sectional Normal Force and Pitching Moment for a

Flat Delta Wing; M = 2.01, x = 4.0.

Lm.n' mm

I

136

o FLAT DELTA WING o FLAT DELIA WINGCon

ALPHA = 10.0000 AL.PHR = 10.0000LRMOA 70.0000 LAMOR 70.0000MACH NO. = 1.6100 MACH NO. = 1.6100CR = 2.7475 CR = 2.7475

o CT = 0.0000 0 CT = 0.0000T/!B/2) = 0.1000 Y Y/(B/2) : 0.4000

0 C

C) C

CD)

C C)CD C.)

0 C

C)C 0o" 0

c0 O0 0. 25 0.50 0.75 1.00 '0.O0 0.2 0.50 0. 75 1. O0PSI PSI

C FLAT DELTA WING C FLAT DELTR WING

O ALPHA; = 10.0000 C:- LP-i = I10. 0000LAMDR = 70.0000 LAMOR = 70.0000MACH NO. = 1.6100 MACH NO. = 1.6100CR = 2.7q75 CR = 2.7q75

o CT = 0.0000 0 CT = 0.0000w Y/ B/2 = 0.2500 '1(0/21 0.5000

5C) L-CA

0 :C- V) CDoc C)C

C)0 CD

a Co 0

T.0 0'. 25 .50 00.75 .0:. o 0.25 0. 50 0.75 I .OFrPSI PS I

Figure A17. Pressure Distribution on a Flat Delta Wing; M = 1.61, = 10.0.

I

137

FLAT DELTA WING 0FLAT DELTA WING

ALPHA = 10.0000 A LPHA = 10.0000LAMDI = 70.0000 LRMDR = 70,0000

MRCH O. = 1.610 MfCH NO .6100CT = 0.0 000 C 000

0AC NO./~ = .6 00 MAC NO.2 0 .6 00oocL M A 0070 0 00

.600 C 0 - 0? 0

CTo00 0CT=0 00

B 2008 0

0 (D0D C

0~~I 0LR OE)WN

0 (DPH C0.0

0C

c 0.D ' 5 0 o 07 .0'~o 0 .a 07 C

0T 00

2 0 0 0,00T=000

Y/8/) .70

0 D ) (

cf -0 02Figure.7 1. Cotiue

PS

FiueA7Iotne

138

O FLRT CELTR WING

RLPHR = 10.0000LRMDR 70.0000MRCH NO. = 1.6100CR = 2.7475

nCT = 0.0000

C)-

C) C

Lf)

U)

oJC

C)

0.0 0.25 0.50 0.75 1.00

Y/ (B/2)

0 FLRT DELTFI WING

RLPHR = 10.0000LRMOR = 70.0000MRCH NO. 1.6100CR = 2.7475

o CT = 0.0000

C)

U C Q

C))

L)00 0 00

, D Wig;M1. a 0000 0.25 0.50 0.75 i.50

Figure A18. Sectional Normal Force and Pitching Moment for a

I Flat Delta Wing; M = 1.61, ( = 10.0.

I

IlI '''' '' ... " _...,L, FIU. .... ...,, - ' "

9

139

FLAT DELTA WING C FLAT DELTA WINGOD OD

ALPHA = 10.0000 0- ALPHA = 10.0000LAMDA = 70.0000 LRMDR 70.0000MACH NO. = 2.0100 MACH NO. = 2.0100CR = 2.74,75 CR = 2.7475CT = 0.0000 C CT - 0.0000Y/(B/2) = 0.1000 Y Y/(B/2 = 0.14000

o 0L. C- L -- ( D C

0 CD

CAD-c oD CD D

0 0o

0O 0.25 0'.50 0'. 75 1.0o 0. Do 0.25 0.50 (.75 1.00PSI PSI

o FLAT DELTA WING C FLAT DELTA WING

o ALPHA = 10.0000 C- ALPHA = In.O0O0LAMDR = 70.0000 LAMPA 70.0000MACH NO. = 2.0100 MACH NU. = 2.0100CR = 2.7475 CR = 2.775CT = 0.0000 CT = 0.0000Y/(B/2) = 0.2500 1/ 2 0/2) 0. CLiCo o" K_

O 0

o ooC

IDI

C) C)

Q) CCDC

c .00 0'.25 0'.50 0'. 75 1. 00 ) 00 0 . 25 0.5 0. 5 1.00

PSI PSI

Figure A19. Pressure Distribution on a Flat Delta Wing; M = 2.01, x = 10.0.

I

140

o FLAT DELTI WING FLAT DELTA WING

C LPHR = 10.0000 ALPHA 110.0000LAMOR = 70.0000 1LAMOR 70.0000MACH NO. = 2.0100 MACH NO. r 2.0100CR = 2.7475 CR 2.7475CT = 0.0000 cCT = 0.0000

CD

AD M D

o/!B/2= 0.6000

T/[B/21 0.8000

o) 0 0

0- c):). "0 0

ctoo o2s o.so . .oo 0.o2S 0.5 .5 oo

PSI PSI

o FLAT DELTA WING

SALPfI = 10.0000LAMOR = 70.0000MACH NO. = 2.0100CR = 2.7475CT = 0.0000Y/(B/2= 0.7000

1L

o

iI 8 2 =,.7 00'"'-CDC

U 0 0.2 0 .O O S I.r0

0

0.25 0.50 0.75 1 Of,• PSI

Figure A19. Continued

II

141

o FLRT DEI TO WING0

RLPHR = 10.0000LRMDR = 70.0000MRCH NO. = 2.0100CR = 2.7475

un CT = 0.0000f,-

00

00 U) c0)N

C;CD

C

0 00 0.25 0.50 0.75 1 .00Y/ (B/2)

0FLRT DELTR WING

Cu_

RLPHR = 10.0000LAMOR = 70.0000MRCH NO. = 2.0100CR = 2.7475CT = 0.000

-it

I

C)N

(D

0

00.00 0.25 0.50 0.75 1.00

Y/ (B12)

Figure A20. Sectional Normal Force and Pitching Moment for aFlat Delta Wing; M - 2.01, a 10.0.

A

I

142

o FLAT TAPERED WING C FLAT TAPERED WING

ALPHA = 2.0000 ALPHA 2.0000

LAMOA = 54.1154 LAMOR = 54.1154

MACH NO. = 1.6100 MACH NO. = 1.6100

CR = 0.9524 CR = 0.9524

CT = 0.1905 CT = 0.1905

Y/(8/2 = 0.0500 m/ (B/2) 0.3500

0- ru

";- U

0

C3 00

.00 0'.25 0.50 0.75 1.00 Ch.00 0.25 0.S0 0.75 1.00•PSI PSI

FLAT TAPERED WING 0 FLAT TAPERED WING

O ALPHA = 2.0000 ALPHR = 2.0000LAMDA = 54.1154 LAMDR = 54.1154

MACH NO. = 1.6100 MACH NO. = 1.6100

CR 0 0.9524 CR = 0.9524o CT = 0.1905 CT = 0.1905

Y/(9/2) = 0.2000 m Y/(B/2) = 0.5000

CDC

0 -cm 0

c)00

0 050 o C) ~ (DC) C)

0 ~ a (D c)W))CCC

CD C) C C

.0 0'.25 0'.50 0.75 1.00 Th.oo 0.25 0.50 0.75 2.0O

PSI PSI

Figure A21. Pressure Distribution on a Flat Tapered Wing; M = 1.61, = 2.0.

I

143

FLAT TAPERED WING

ALPHA = 2.0000LAMOA 54.1154MACH NO. = 1.6100CR = 0.9524CT = 0.1905Y/l(6/2= 0.7000

0C)

N C-

C3

0a

Ob.0o 0.25 0.50 0.75 1.00PSI

o FLAT TAPERED WING

o" ALPHA = 2.0000LAMOR = 54.1154MACH NO. = 1.6100CR = 0.9524

o CT = 0.1905Y/lB/2) = 0.9000

0

Q)

0LJC)

i-

0.00 0.25 0.50 0.75 1.00

PSI

I

I

1l ,,________ __,_.,____._,_.__.______- FI-

S IF [144

O FLRT TRPEPEO WINGClJ

C; ALPHA = 2.0000

LRMOA = 54.1154IMACH NO. = 1.6100CR = 0.9524

LCT = 0.1905

0

)

¥I (BIDFL)T CPED NN

c_

00 L.

C)D

6.00 0.25 0.50 0.75 1.00Y1/ (B/2)

0

0

MACH NO. = 1.6100Ln CR = 0.9524

CT 0.1905

L)

-)

CDC

C;.D

00.00 0.25 0.50 0.75 1.00j Y/ (B/2)

Figure A22. Sectional Normal Force and Pitching Moment for a

Flat Tapered Wing; M = 1.61, a 2.0.

I

145

FLAT TAPERED WING 0 FLAT TAPERED WING

ALPHR = 2.0000 0" ALPHA 2.0000LAMOR - 54.1154, LAMDR = 54.1154MACH NO. = 2.0100 MACH NO. = 2.0100CR = 0.9524 CR = 0.9524CT = 0.1905 CT = 0.1905Y/B/2) = 0.0500 YI/(B/2) 0.3500

o 0

o 0

T.00 0'....25 0.50 0.75 100uT. O0 0.25 050 075 3.0

A LPHA 2.0000 oRLP'HR 2.0000LAMOR = 514.1154 LARD 54.1154MACH NO. =2.0100 IMPCH NO. -2.0100CR = 0.9524 CR 0.9524

oCT = 0.1905 CT = 0.1905Tl!Bl21 = 0.2000 T/I1BI2] =0.5000

o ,0 0

000

0 i0 0 0 20.5 1 0

--1------- --- --- --

.0 0.25 0.50 0.75 1.00 T0.00 0.25 0.50 0.7S 1.00PSI PSI

A Pressure Distribution on a Flat Tapered Wing, M = 2.01, = 2.0.

IL

I' =

!RC NO 200 AHN.=•00

146

o FLRT TAPFRF[) WING

RLPHR = 2.0000LRMOR = 54.1154MRCH NO. = 2.0100CR 0.952LICT 0.1905TI (612) = U.7000

0

0

0.o

0

* 0

0 @@ 0 @

0

'0.00 0.25 0. D .75 1.00PSI

o FLRT THPL flED W NrCO ALPH( 2. 00 !1

LAMOR = 51.1154MACH NO. = 2.0100CR = 0.95211

C) CT = 0.1905T/(18/2) = 0.90 -

a-

1)5-

(Do

00

.0n o '. 25 0.,50 0.75 L.013PSI

Figure A23. Continued

I

!

I147

o FLRT TRPFRLD VJINL

- RLPHFI z 2.0000LRMDR = 54.1154MRCH NO. = 2.0100CR = 0.9524CT = 0.1905

d -

0

0

L5

)

0 --

O0 0.25 0,50 0,75 1,00Y/ (B/2)

0SFLRT TRPERED WING

RLPHR = 2.0000LRMDR = 514.1154MRCH NO. = 2.0100CR = 0.952q

LCT = 0.1905

0

X 0

LO_

0

I00.00 0 .2 5 0. 5 I 0.75 1.00i Y/ (B12)

Figure A24. Sectional.Normal Force and Pitching Moment for

a Flat Tapered Wing; M =2.01, a = 2.0.

I

148

o FLAT TAPERED WING FLAI TnrEREO WING

o ALPHA = q.0000 C- ALPHA 4 4.0000LRMOR 54.1154 1LARU 'j4.1154MACH NO. = 1.6100 MACH NO. = 1.6100CT = 0.1905 CT = 0.1905

Y/(B/2) 0.0500 I/ (B/2) = 0.3500

o C);

r0 L3,C3]

C C)

C"0CO ,

o C.o-7c) '

'.00 0.25 0.50 0.75 1.00 0.0o 0.25 0.50 0.75 1.00

PSI PSI

o FLAT TAPEREO WING CD FLAT IAPEREU WING

" ALPHA = q.0300 C RLP = q.000LAMOR = 54.1154 , LMA = 54.1154MACH NO. = 1.6100 MACr NO. -1.600CR = 0.9524 CR 0.952L4

o CT = 0.1905 (D CT = 0.1905Y/(B/2) 0.2000 1' (B/,0 0. 5C0 G

c)C

.1!IUCD

o CC)

0- C, CD C) 0 (D

o C-)

.OO 0' 25 0.50 0. 71, 1 .('. 00 0.25 0. 50 C]. 1.f

PSI PSI

Figure A25. Pressure Distribution on a Flat Tapered Wing; M 1.61, a = 4.0.

I _________

149

FLAT TRPEFHED WINGo" ALPHA = 4 .0000

LAMO = 54 .1154MACH NO. = 1.6100CR = 0.9524

co CT = 0.190SY/(B/2) = 0.7000

C)"0

D(

C) (D Ca C )c)no

O~ 0 0 .D 25 0.50 .75 . 00

PSI

c:) FLAT TAPLHlE[ WING030L

" FALPHA = 4.,0D O

A LPHA = 4. G 00LnM0A = 51.11S1MRCH NO. :|60

CR = 0.9524CT = 0. 1905

Y/ (/21 = 0.9000I0 __CT. .1 0

/ (2 ) . O . 5 D

PSIl

C) DUCD 0

O)CDo C 0

00':.00 0.25 0."0 O.7is 15

Psi

Figure A25. Continued

150

FLAT TRPERED WING

RLPHR = 4.0000LAMDA = 54.1154MACH NO. = 1.6100CR = 0.9524

Lf) CT = 0.1905

co 0

- 0

C -

/ (D /2

U.

V)

C)

C)

CD

'0.00 0.25 0.50 0.75 1.00

YT/ (B12)

0SFLHT rPERED WINGc_I RLPHR = 4.0000

LAMD = 54.115I4MRCH NO. = 1.6100CR = 0.9524

U) CT = 0.1905c_

L)

If)

CD_1 'i

Ici°

'.oo 0.25 0.50 0.75 1.00

Y/ (B/2)

Figure A26. Sectional Normal Force and Pitching Moment for

a Flat Tapered Wing; M = 1.61, x = 4.0.

II 'l 2ZI'I 'I: '=

II

151

o FLAT TAPERFO WING o FLAT TAPERED WING

- ALPHA = 10.0000 ALPHA 10.0000LAMOR 54.1154 LRMDR 5141154MACH NO. = 1.6100 MACH NO. = 1.6100CR = 0.9524 CR = 0.9524

C CT = 0.1905 CT = 0.1905T/(B/2) 0.0500 T /(B/2) = 0.3500

C3o

0- o m

000

PSI PS I

oFLAT TAlPERED WING FLRT TAPERED WINO

(0 (0

- ALPHA = 10.900 ALPHA = 10.0000LAMOR = 54.1154 LRMOFT 54.L11514MRCH NO. = 1.6100 MACH NO. 1.6100CR = 0.9524 CR = 0.9524

o CT = 0.1905 CT = 0.1905NT/(B/2) = 0.2000 1/IB/21 = 0.5000

oCDC) 0 0

o0

O 00 0'.25 0.50 0.75 1.on 6.00 0.25 0.50 0. i5 1.00

PSI PSI

Figure A27. Pressure Distribution on a Flat Tapered Wing; M = 1.61, a = 10.0.

Linmmmmmm

152

o FLAT TAPERED WING

RL.PHP = 0.n0000LAMOR = 514.1154JMACH No. =1.6100

5T 0.DO

00

CDC)

PSI

C$o FLAT TAPERED WING

ALPHA = 10.0000LPMOR = 54.1?54MACH NO. =1.6100CR =0.9524CT =0.1905

('.1 l'(6121 0.9000

-C3

(D0

C))

CTh. 00 0'.25 0.50 0.715 1.00PSI

Figure A27. Continued

I153

C FLRT TFPEREO WING

o- RLPHA = 10.0000

LAMOR = 54 1154MRCH NO. = 1.6100CR = 0.9524

o CT = 0.1905

C;-~0

CZ

Z =r0-

0

C

.ooo o25 0.50 0.7 1u .00Y/ (B/2)

0

CFLRT TRPEREO WING

C)I RLPHR 10.0000

LRMDR = 54.1154MRCH NO. = 1.6100CR = 0.9524CT = 0.1905

C-

II

0

XCD

0CCCj

0-

.00 0.25 0.50 0.75 1.00

Y/ (B/2)

Figure A28. Sectional Normal Force and Pitching Moment for a

Flat Tapered Wing; M = 1.61, Ct = 10.0.

Lz~

I

1-54

c CAMBERED DELTA WING 0 CAMBERED DELTA WING

C) ALPHA = 6.0000 ALPHA = 6.0000LAMOR = 70.0000 LAMDA 70.0000MACH NO. = 1.6100 MACH NO. = 1.6100CR = 2.7475 CR = 2.7475CT = 0.0000 CT = 0.00001/(B/2) = 0.1000 Y '/(B/2) 0.4000

000

tO0 00

AMEEDLT IG 0 ABRDDLAWN00

- --- -- --CI 0 '.5 50 .7 .(0c0 .2S 0 .50 0.75 1.00

PSI PSI

oCAMBERED DELIA WING C3 CAMBERED DELTA WING

o ALPHA = 6.0000 ALPHA = 6.0000LAMOR = 70.0000 LAMOA = 70.0000MRACH NO. = 1.6100 MACH NO. = 1.6100CR = 2.7475 CR = 2,7475CT = 0.0000 C CT = 0.00001/(B/2) 0.2500 co Y!(B/21 0,5000

o 0

IDI

0.500 0

Figure A29. Pressure Distribution on a Cambered Delta Wing; N 1.61, c = 6.0.

II

1

155

o CAMBERED DELTA WING o CAMBERED DELTA WING

o ALPHA = 6.0000 0 ALPHA = 6.0000LAMDR = 70.0000 LAMOA = 70.0000MACH NO. = 1.6100 MACH NO. = 1.6100CR = 2.7475 CR = 2.7475

o CT = 0.0000 CT ='0.0000Y/tB/2) 0.6000 L Y/18/2) = 0.8000

oo

0 C

0 o

o.

O0 0.25 01.50 075 1.00 0.25 050 0.75 l.0

PSI PSI

CAMBERED DELTA WING

o ALPHA = 6.0000LAMOA = 70.0000MACH NO. = 1.6100CR = 2.74-75

\a CT = 0.0000Y/(B/2= 0.7000

0 CD C

o

0o 0.25 0.50 '.7 1.00

PSI

Figure A29. Continued

CI

f0

... ... . .. . .. "0 ,.. . . . : . . I

156

C CAMBEREO DELTA WIN[

C" ALPHA = 6.0000LAMOR = 70.0000MRCH NO. = 1.6100CR = 2.7475

C"° CT = 0.0000

01

o

CDD

0

'0.00 0.25 0.50 0.75 1.00

Y/ (B/2)

CAMBERED DELTA WING

7- ALPHA = 6.0000LRMDA 70.0000MRCH NO. = 1.6100CR = 2.7475

oU CT = 0.0000t

0C) C

w C

L_0

I I0.00 0.25 0.50 0.75 1.00

Y/ (B/2)

Figure A30. Sectional Normal Force and Pitching Moment for a

Cambered Delta Wing; M = 1.61, c = 6.0.

.

157

o CAMBERED DELTA WING c CAMBERED DELTA WING

ALPHA = 6.0000 ALPHA = 6.0000LADA = 70.0000 LAMOR 70.0000MACH NO. = 2.0100 MACH NO. = 2.0100CA = 2.7475 CR = 2.7475

o CT = 0.0000 0 CT = 0.0000Y/(B/2) = 0.1000 (, Y/(B/2) = 0.4000

o ci-

o c:i

U 8c - uc C -

C3

0 0oI O0 o

oo 0'. 25 0'. so 0. 75 1.0 '00 02 '.5 O.''10PSI PSI

oCAMBERED DELTA WING c, CAMBERED DELTA WINGS00

o ,LPHA = 6.O0 - ALPHA = 6.0000LANDA 7o.oo LAMOP 70.0000MACH NO. = 2.0100 MACH NO. = 2.0100CA = 2.7475 CA = 2.7475

C CT = 0.0000 CT = 0.0000Y/IB/2) = 0.2500 (D Y/(B/2) = 0.5000

o ci

o 0i

00 0, O

0~C C)) )

.0 O• 25 0 . 50 0 . 75 !.00 h. OO 0.25 0.50 0.75 1. ocPSI PSI

I Figure A31. Pressure Distribution on a Cambered Delta Wing; M = 2.01, a = 6.0.

1I

L 'm m mm

158

oCAMBERED DELTA WING oCAMBERED DELTA WING

o LHD600 ALPHA =6.0000

ALPMDA = 76.0000 LAMOA 70.0000

MAN. = 70.000 MACH No. =2.0100MCH NO =2. .0100 CA 2.71475

CT 2 0.00

0CT = 0.0000 K:B2 080Y/(B/2) =0.6000

Y//2 .000

C

N0 c'0

0 C)

ch.oo 0.25 0.50 0.75 1.00 PSI .2 .0 0.5 10PSI S

o CAMBERED OELIA WING

O ALPHA 6.0000LPMOR 70. 0000MACH NO. = 2.0100CR =2.71475

0 CT = 0.0000w T/fB/2) =0.7000

0 0C

cbo 0.2 05

Figure A31. Continued

I

CAMBEIOE DELITA N1NG

RLPHR = 6.0(,30LRMOR = 70.00MRCH NO. = 2.0100CA = 2.7475

C) CT z0.0000

C -

I U)

I0

Luc3

00

C

0.00 0.25 0. 50 0.75 t.00Y/ (B/2)

C0o CMBERED DELTA WING

RLPHR = 6.0000LRMOR = 70.0000MRCH NO. = 2.0100CR = 2.7475

0 CT = 0.0000

-

C

UU0

CL

0

0.O0 0.25 0.50 0.75 1. 00Y/ (B/2)

Figure A32. Sectional Normal Force and Pitching Moment for a

Cambered Delta Wing; M =2.01, ot = 6.0.

£

I

160

0 CAMBERED & TWISTED DELLA WINC, C CAMBERED A IWISTED DELTA WING

o ALPHA = 6.0000 0- ALPHA = 6.0000LAMOA = 70.0000 LAMOR 70.0000MACH NO. = 1.6100 MACH NO. = 1.6100CR = 2.7475 CR = 2.7475

o CT = 0.0000 = 0.0000Y/(8/2) = 0.1000 0 YI(B/21 = 0.4000

o C) D 0 01

CD)

C.JD

C3 0

.D c

% C)o~ 0

-cTo 0.2 5 0.500 .75 i. 0 9jo. oo 0.25 O.so 0.75 io0PSI PSI

C3 CAMBERED & TWISTEC DELIR W:NG - Ci MHEREU A TWISTEE DE.T T

o ALPHA = 6.0000 D A LPiH = c.00r

LAMOR = 70.0003 LAMc = 73.000,MACH NO. = 1.610n MACH NO. 7 1.610-CR = 2.7475 CH 2.7475

C CT = 0.0000 C CT = 0.C000Y/(B/2) = 0.2500 Ti,(0.2 = 0.5000

C3 C:-L\c CD

C))o C)

• C) 0 C,

T C) ) .

.00 0.25 0.50 0.75 1.0f o. O0 0.25 0.50 0.75 1.00PSI Psi

Figure A33. Pressure Distribution on a Cambered and Twisted Delta Wing;

M = 1.61, a = 6.0.

1 l

161

CAMBERED T "WISTED DELIH NiNC CRMBERf & TWISTEDl DELTP WING

00 c" ALPHA =- b.0000

A ALPHA = 6.0000 LRLAMA z 70.0000

LRMOA = 70.0000 ACH NO. =

1.6100

MACH NO. = 1.6100 C 2.747 5

0 CR = 0.-0005 cT= .0

CRC

CT YI/(2 2) 0.8 00

0 (

ic

0- 1060['0

Y /1 : ((/ ,) 0 .20 0

L CD

00

C" ci

00(

0 N~

0.25 0.50 0.75 1 Th oo 0.25 0.50 0.0 7.5o.2 o5s 1SPS I

o CAMBERED 4 T EISTE [ 3K f HING

A RLPHA = 6.00010

LRMD 70.0000MACH NO. = 1.61009CR = 2.7475

CT = 0.0000T/ 0B/21 = 0.(100

C)C I

00~c 0.2 C .0 07

('a

0

0 --

'bO0 0.25 0.50 0.75 1.O0

PSI

Figure A33. Continued

I.3

i I162

0 CRMBF[U!O & IWIUfV 13 IF TP NIOS

- LPHR = 6.0000LRMDR = 70.0000MRCH NO. = 1.6100CR 2.7475

C) CT = 0.0000(D

o -U

'.O0 0.25 0.50 0.75 l .OuY/ (B/2)

cO CRMBERED & TW]SIED DLTR INU

RLPHA = 6.0000LRMDR = 70.0000MRCH NO. 1.6100CR = 2.74175CT = 0.0100

0i

00

C

C

C

d__

0.00 0.25 0. 50 0.7F I. i0Y/ (6/?)

Figure A34. Sectional Normal Force and Pitching Moment for a

Cambered and Twisted Delta Wing; M - 1.61, a - 6.0.Ii /

I

163

o CAMBERED & TWISTED DELTA WING o CAMBERED 4 TWISTED DELTA WING

m" ALPHA = 6.0000 C0" ALPHA = 6.0000LAMOR = 70.0000 LRMOR = 70.0000MACH NO. = 2.0100 MACH NO. = 2.0100CA = 2.7475 CR = 2.7475CT = 0.0000 D CT = 0.0000

LO(B/2) = 0.1000 T/(B/2) 0.4O000

o= 0

D

cj- - C L

Ui c

0 0W a)

T i00

O0 025 0.7/5 i'. 0 cbO0 0.25 0.50 0.75 .O0PSI P 5

o CAMBERED 4 TWISTED DELTA WING c0 CRMBERED & TYISIED DELTA WING

RLPHR = 6.0000 ALPHA = 6.0000LAMDA = 70.0000 LAMOR = 70.0000MACH NO. = 2.0100 MACH NO. = 2.0100CA = 2.71475 CR = 2.7475

o CT = 0.0000 CT = 0.0000Y/(6/2) 0.2500 T/(B/2) = 0.5000

o 0

G- (-CZ 0

0 0 0~a C) (D C)

0 C

0"0 0.75 1.00 O 0.25 0.50 0.75 1. 00

PSI PS I

Figure A35. Pressure Distribution on a Cambered and Twisted Delta Wing;

M 2.01, ¢. = 6.0.

164

o3 CAMBERED 4 TW15TED DELTA WING C)CAMBEREC & TWISTEn DELTA WING

o; ALPHA = 6.0000 0 ALPHA - 6.0000LAMDA z 70.0000 LAMOR = 70.0000MACH NO. -2.0100 MACH NO. =2.0100CR = 2.7475 CHR 2.74~75

0CT = 0.0000 0CT = 0.0000inY/(B/2) 0.6000 00 Y/(B/2i 0.8000

0 C)

U e; 0m 0D 0 C 0

C0 Q

0 0

'b.oo 0.25 0.50 0. 75 1.00 Th.DOu 0.25 0.50 0.75 1.00p SI PS I

0 CAMBERED & TWISTED DELiA WING

o ILPH'A = 6.0000LAMOP 70.00C

N. -= 2 f) OI L7475

0CT OU0inY/ (8/d, 0. 7000

c;- c0 0D \ \D

c~C)

00

'b .oa Q.25-. -- - 1-0

0

FiueA3.Cntne

FI

j 165

CAMBERED & TWISTED OLLR WING

" RLPHR = 6.0000LRMDR = 70.0000MRCH NO. = 2.0100CR = 2.7475

0 CT = 0.0000

)O0

UC-0

CD

0

C

0.00 0.25 0.50 0.75 1.00Y/ (B/2)

0CD CRMEPO & TWISTED OELIq HING

RLPHR = 6.oqn.LRMOR = 70.-10MACH NO. = 2.0100CR = 2.74750 CT = 0.0000

Cj

0

C

C

~0.00 0 .25 0.50 0. 75 1.00Y(B/2)

Figure A36. Sectional Normal Force and Pitching Moment for aCambered and Twisted Delta Wing; M 2.01, a~ 6.0.

Io

I°

166

O WARPED TRAPEUOIDAL WING D WARPED TRAPEZOIDAL WING

c" ALPHA = 6.0000 0 ALPHA = 6.0000LAMOR = 25.7917 LAMOR = 25.7917MACH NO. = 1.6100 MACH NO. = 1.6100CR = 1.7324 CR = 1.7324

0 CT = 1.2488 0 CT = 1.2488

o 00.O './2 0.1000 O' "5 10/21 = 0.5500

00

00

PSI PSI

0WARPED TRAPEZOIDAL WING C) WARPED TRRqPEZOIDAL WING

o 0o

NALPHA = 6.0000 C0 ALP0 = 6.0000LAMoA = 25.79170 LHOR = 25.79-17

MACH NO. 1.6100 MACH NO. = 1.6100CH = 1.7324 CA = 1.7324

0 CT = 1.24188 CT = 1.2482Y/(B/2) =0.3500 CD Y/(B/2) = 0.6700

0 00 0

0 - C - 0 (

Oh ...0.25 0'.50 0.7-5 1.0 Th cOO 0T. 2. 0.50 0.7 ? 1 .O0

CD WAPE TRPZCAD~10oWRE RAEODLW

O0

PSI PSI

Figure A37. Pressure Distribution an a Warped Trapezoidal Wing; M =1.61,

a = 6. 0.

L7z

MAHN.=160IMC D .10CR =I1.7324CR-=.1.732

167

Q HARPED TRAPEZOIDRL WING 3 WARPED TRAPEZOIDAL WING

ALPHA = 6.0000 C" ALPHA = 6.0000LAMOR 25.7917 LMAMO= 25.7917MACH NO. = 1.6100 MACH NO, = 1.6100CR = 1.7324 CR 1.7324CT 1.2488 CT = 1.2488Y/(6/2) = 0.7700 W T1(8/21 0.9700

C3 0O

o C)L c- uc -

o CD

o o

'b 0.2 '05 0.71'000 0i. 5 0.50 0.75 1.00

PSI PSI

o WARPED TRAPElOIDPL WING

o ALPHA = 6.0000cLAMOA = 25.7917MACH NO. = 1.6100CR = 1.7324

o CT = 1,2.488LY /9/2) 0.8700

0

0

IPS

&)__

o

0o000

Figure A37. Continued

ILIn.i..__ _ _ _ _ __ _ _ _ _ __ _ _ _C

168

o WARPED TRAPEZOIDAL WING

ALPHA = B.0000LAMDR = 25.7917MACH NO. = 1.6100CR = 1.7324CT = 1.2488

I-

U

U-)

0

'0.00 0.25 O.50 0.75 1 .[0

Y/ (B/2)0C WARPED TRAPEZOiRL WINO

7- ALPHA = 6.0000LRMOA = 25.7917MACH NO. " 1.0100CR = 1.7324CT = I.2488

I

0.0 0. 25 0. 50 0. 75 1 .0

Figure A38. Sectional Normal Force and Pitching Moment for aWa;e 7 :rpezoidal Wing; M 1.61, a 6.0.

C:)

169

,- WARPED TRAPEZOIDAL WING 0 WARPED TRAPEZOIDAL WING

L. .0" ALPHA = 6.0000C A L P H A = 6 .0 0 0 0 M O - 2 5 7 1

LAMOR = 25.7917 M'- LANOA = 25.7917MACH NO. - 2.0100 MACH NO. = 2.0100CR = 1.732L4 II CR = 1.732u,

CT = 1.2488 CT = 1.2488CY(8/2) 0.1000 Y/(B/2) = 0.5500

0

U0 CD00 00_)

00

00

0 025 PS. PS I

PSIPSSWARPED TRAPEZOIDAL WING 0C WARPED TRAPEZOIDAL WINGGo. P" £ H A

= 6 .0000

RLPHR = 6.0050 LR 2.0000

LAMOR = 25.7917 LtD 571

MACH NO. = 2.0100 MACH NO. = 2.0100

CR - 1.7324 CR = 1.73248

CT = 1.2488 Cl = 1.2466Y •f2) 0.3500 w Y/(B/2) 0.6700

8l-2 0= D.50

00

0-: 00

°.0 -

U8- 0 ° c:)

00

0o

00

C3 0

0 0 7 1

9i.oo 0.25 0.50 0.75 1.00 0.o 0.25 0.50 0.75 1.0PSI PSI

Figure A39. Pressure Distribution on a Warped Trapezoidal Wing; M = 2.01,

6.0.

Naa

SI

170

o WARPED TRAPEZOIDAL WINGC WARPED TRAPEZOIDAL WING

P 6 LPHA 6.0000o"RLPHR = 6.0000

LMR = 25.7917 LAMOR = 25.7917MRCH NO.= 2.0100 MACH NO. = 2.0100MA H NO 2 0 0 C H 1732q

CR = 1.7324 C= 1.241

CT = 1.2488 c/6 1.2 78 0

Y/(Bl2) = 0.7700

wc-

c'DC

( CD

(0 D (D

0 .25 0.50 o..00 0.25 0.50 0.75 1.00

0 0.50 0.75 1.00PSIPS

3 WARPED TRAPEZOIDAL WING

o ALPHA = 6.0000

LAMDA = 25.7917MACH NO. = 2.0100CR = 1.7324CT = 1.2488

w T/(B/2) = 0.6700

Lc

o C

.CD

c o0 0.25 0.50 0 T75 1.0

PSI

Figure A39. Continued

I171

C WARPED TRAPEZOIDRL WING0

ALPHA = 6.0000LAMOR = 25.7917MACH NO. 2.0100CR = 1.732U

nCT = 1.2488

C)

U

InCd

C

'0.00 0.25 0.50 0.75 1.00Y/ (B/2)

CO WRRPEO TRfAPEZOIORL WING

ALPHR = 6.0000LAMOR = 25.7917MACH NO. = 2.0100

CR = 1.7324

CT = 1.2486

II

C)

U'

L)

0 0

0-

Th.00 0.25 0.50 0.75 1.00Y/ (B/2)

Figure A40. Sectional Normal Force and Pitching Moment for aWarped Trapezoidal Wing; M4 2.01, ot =6.0.

-o~

I4

I

APPENDIX B

:1

SUBSONIC COMPUTER PROGRAM USERS GUIDE

I. General Description of the Main Program and Subroutines forthe Subsonic Program

MAIN PROGRAM - This is the executive segment which primarily controls and

coordinates the various tasks performed by the subprograms. Other

functions performed by the main program include: (a) all input and the

majority of output; (b) non-dimensionalization; (c) calculation of the

boundary conditions (the "B" matrix); (d) computation of all total

characteristics, both exact results and numerical integrations (this

includes wing-elevon, fuselage, suction, and all combinations of these).

BLOCK DATA - This data gives coordinates of a non-planar mean line (XTB,ZTB)

as well as the slope (ZPTB) for use in calculating wing deformations.

The present data is for the deformed Hawk wing.

SUBROUTINE SURFIN - This routine computes the value of the surface integral

(the "A" matrix) by summing the effects of each grid panel at a control

point. It is called once for the wing and once for the elevon for each

control point.

SUBROUTINE SECLM - The sectional loading characteristics are calculated for

the wing or elevon in this subroutine. It is called after the pressure

loading coefficients are determined.

SUBROUTINE PRESUR - This subroutine is used, after the pressure loading

coefficients are determined, to calculate and print ACp at NSPAN by

NCORD points on the lifting surfaces.

173

~z.LL!

)174

]SUBROUTINE GEOM - This subroutine sets up the geometry of both the grid

panels and the control points, and then centers the control points

inside the grid element in which they appear. Basically, a cosine

distribution is used for both the grid elements and control points.

This routine is called once for the wing and once for the elevon.

SUBROUTINE INTV - This routine contains the actual equations of the sur-

face integral and, therefore, is utilized by SUBROUTINE SURFIN.

SUBROUTINE INVERT - Gauss elimination is used in this subroutine to

"invert" the A matrix and solve for the unknown leading coefficients,

B 's.nm

SUBROUTINE HINGE - This contains the equations of the closed form solution

for the elevon hinge moment.

SUBROUTINE TLUP - This is a "table-lookup" routine which uses the data

given by the variables in BLOCK DATA to interpolate for wing mean line

slopes used in determining deformation angles. It is called by

FUNCTION WARP.

FUNCTION WARP - This function uses either the table-lookup or a function

definition to find wing/elevon deformation. It is called by both the

main program to determine deformation of control points (included in

the boundary conditions) and by SUBROUTINE SURFIN to determine deforma-

tion of grid panels.

175

II. INPUT DATA

A. Data Cards:

Card No. Format Variable List

1 18A4 WING

2 7F10.0 SPAN,CR,CT,LAMDA,SAREA,CBAR,MACH

3 F10.0,15 ALPHA,KSTOP

4 1015 NC,NS,NX,NY,NCF,NSF,NXF,NYF,NSPAN,NCORD

5 5F10.0 CRF,CTF,DELTA,ALTWMX,EPS

6 F1O.0,15,2FlO.O HNL,IDEF,AMP,DB

(Additional 3F10.O ALPHA,DELTA,ALTWMXcards)

("Additional cards" allows more than one case of the givengeometry to be run with different boundary conditions.There should be (KSTOP-1) additional cards.)

B. Variable definitions (all linear dimension units must be consistent)

WING - Alphanumeric problem description

SPAN - Total wing span

CR - Wing root chord

CT - Wing tip chord

LAMDA - Wing leading edge sweep angle

SAREA,CBAR - Reference area and reference length. If 0.0 is input,the total area and mean aerodynamic chord will be

calculated and used.

MACH -Mach number

ALPHA - Angles of attack of wing, in degrees. Leading edge upis positive.

KSTOP - Number of cases of different boundary conditions that areto be run.

I1

176

NC,NS - Number of chordwise and spanwise control points (re-spectively) on the wing (typically 4x4).

NX,NY - Number of chordwise and spanwise (respectively) grid panelson the wing (typically 45x45).

NCF,NSF - Chordwise and spanwise control points on the elevon(typically 3x4). NSF must be equal to NS.

NXF,NYF - Chordwise and spanwise grid panels on the elevon(typically 20x45). NYF must be equal to NY.

NSPAN,NCORD - Number of spanwise by chordwise points at which ACdistribution is to be printed out. P

CRF - Elevon root chord

CTF - Elevon tip chord

DELTA - Elevon deflection angle, in degrees. Trailing edge downwardis positive.

ALTWAX - Maximum twist angle of the wing tip, in degrees. Leadingedge down at the tip is positive.

FPS - Gap distance between wing and elevon.

HNI, - Hinge line location from elevon leading edge.

IDEF - Deformation check. If IDEF=l, the Table-Lookup Routine isused. If IDEF=O, a function definition is used.

AMP - Amplitude of a function deformation.

DB - Diameter of fuselage.

III. OUTPUT DATA

Major Headings Are:

A. INPUT DATA - Dimensional imput as it was given on the data cards.

B. ACTUAL VALUES USED IN PROGRAM - Variables non-dimensionaliz~d

by b/2.

C. CONTROL POINTS - Location based on non-dimensionalization oflocal wing chord by b/2. Gap distance is not included in XO.

P AD-AI~i '675 AUBURN UNIV ALA ENGINEERING EXPERIMENT STATION, F/6 20/~4AN AERODYNAMIC ANALYSIS OF DEFORMED WINGS IN SUBSONIC AND SUPER--ETC(U)

APR Al J E BURKHALTER, J M ABERNATHY DAA629-78-0-0036

UNCLASSIFIED ARO-15666.4-A-E NL

~3 EIENDI+

I

177

D. BOUNDARY CONDITIONS

I - Control point number.

B - Boundary condition matrix values.

XO,YO - Control point location as above, except with gap distanceincluded.

ALTW - Local twist angle at the control point (degrees).

TAU -Local deformation angle at the control point (degrees).

E. POLYNOMIAL COEFFICIENTS - B 's of pressure loading function.nm

F. PRESSURE DISTRIBUTION

ELEVON LEADING EDGE - Location in percent local chord (gap included)

Y - Nondimensional spanwise location of the given pressure.

X - Nondimensional chordwise location in percent local chord(gap included).

XDIS - Chordwise location in terms of nondimensionalization by b/2.

DCP - AC (pressure loading)P

C - Local chord length of the wing only.

G. SECTIONAL CHARACTERISTICS

CLW - cc, of wing only.

CMW - c2cm of wing only

CLE - cc, of elevon only

CMH - c2 c of elevon onlym

CCL - cc z of total wing-elevon.

CCM - c2c of total wing-elevon

XCP - Center of pressure in percent local chord behind the leadingedge.

Y - Nondimensional spanwise location of the sectional characteristics.

C - Local wing chord nondimensionalized by b/2.

I1

178

CF - Local elevon chord nondimensionalized by b/2.

CTDT - Total local wing-elevon chord (gap not included)

CDW CCD of wing only

CDE - ccD of elevon onlyDi

CCD - ccD of total wing-elevon

H. TOTAL FORCES AND MOMENTS

TOTAL LIFT, etc. - Characteristics of wing-elevon combination.

CORRECTED LIFT - Lift multiplied by sin u.

LIFT AND MOMENT DUE TO SUCTION - Contribution of suction only.Lift value is correct only when o>O. Both terms are incorrect

for 6#0.

FUSELAGE LIFT AND MOMENT - Contribution of fuselage only.

WING-FUSELAGE... - Wing-elevon fuselage characteristics.

TOTAL CONFIGURATION...- Wing-elevon-fuselage-suction characteristics.

HINGE MOMENT ABOUT ELEVON HINGE LINE - Elevon hinge moment.

IV. NON-PLANAR CONFIGURATION OPTIONS

A. Wing Twist - Statement numbers 2310 and 4540 calculate wing twistwith a linear distribution along the span (YO(J)). Other (spanwise)

variations may be used by substituting the proper equation.

B. Deformations

1. Statement numbers 2200 and 5430 must be deleted if the elevon

is to be deformed in addition to the wing.

2. The FUNCTION WARP routine supplies the deformation angles

a. If the input variable IDEF is 0, the equation for WARPgiven by statement number 9430 is used to compute deforma-tions. The variable (also input) AMP may be used tomodify this equation. The supplied equation must be for

the slope (dz/d ) of the mean line. Different functionscould be used at different spanwise locations if thevariables YO(J) in statement number 2160 and YA in state-ment number 5450 are included in the argument list, andthe appropriate modifications are made to FUNCTION WARP.

b. If IDEF > 1, a table lookup routine is used to compute de-

formations based on a numerical table given by the variablesin BLOCK DATA.

...... . .. .. . .. . , .. . . . .. . .. .t- .' .. = ; .. - N . . " "

-

1I

SUPERSONIC COMPUTER PROGRAM USERS "UIDE

I. General Description of the Main Program and Subroutinesfor the Supersonic Program

MAIN PROGRAM

This is the executive portion of the program and basically controls

the sequencing of the other subroutines. All data to be read into memory

is completed here as well as all nondimensionalization. Decisions are

made on the type of pressure solutions desired and appropriate transfer

of control is made. Geometry for the vorticity paneling for the wing

and elevon is completed. Wing pressure solutions are called and the

trailing legs vorticity strengths are computed. Most of the entire

elevon solution is complete in this program segment and appropriate

computational output is generated.

Subroutine BLOCK DATA

This subroutine establishes the spanwise locations at which the

chordwise pressure distribution si computed. If one wishes to alter these

locations, it must be done in this subroutine.

Subroutine DAUGHN

This subroutine calculates the downwash of any arbitrary location on

the elevon due to the trailing vortices on the wing. This routine is

called after the wing loading and trailing vorticity strength is estab-

lished.

179

| , . ..I. .

180

Subroutine EXACT

This subroutine calculates the pressure distributions on the wing or

elevon using planar three-dimensional supersonic flow theory. This

subroutine calls the load coefficient subroutine where sectional forces

and moments are calculated.

Subroutine LODCOF

This subroutine takes the AC arrays and integrates the chordwisep

distribution using Simpson's Rule. It integrates the spanwise distribu-

tions using trapezoidal integrations.

Subroutine DOWN

This subroutine calculates a numerical value for the downwash on

the wing only at some (x,y) location. It is used primarily to check

boundary condition agreement.

Subroutine SURF

This subroutine computes the surface integral over the forecone

from the field point (xy) by summing the contribution from each subpanel.

In doing so it loads the "A" matrix to be inverted later for the unknown

polynomial coefficients, BL.

Function CW

This function evaluates a cosine function which is used frequently

in the program.

Subroutine INTV

This subroutine evaluates the results of the downwash integral over

a subpanel.

!

.. .. 1

181

j Subroutine INVERT

This subroutine is a matrix "inversion" routine which actually uses

a Gaussian reduction to find the unknown coefficients, BL.

Function PRESS

This function evaluates the appropriate expression to find the ACp

at a point using supersonic three-dimensional theory.

Subroutine ELLIP

This subroutine numerically evaluates an elliptic integral used in

Function Press.

Subroutine THA

This subroutine establishes the deformation equations of the deformed

wings and returns the local wing slope relative to the wing chord line.

Subroutine SUM

This subroutine performs a summation step used often in Subroutine

SURF.

Subroutine ETHA

This subroutine is identical to subroutine THA except the deformation

equations define local slopes on the elevon surface.

II. Input Data

A. Data Cards:

Card No. Format Variable List

1 815 IDOWN, IWGEX,IWGVP,IELEX,IELVP,IPRINT, IWGDF, IELDF

2 615,2FI0.0 NC,NS,NX,NY,NCORD,NSPAN,VAG,DB3 12A4 CONFIG

4 4F10.0 SPAN,CR,CT,LAMDA5 4F10.O SWING,CBRW,MACH,ALPHA

6 415,FI0.0 LVON,NCE,NXE,NCRE,CRE7 5F10.0 CTE,GAP,DELTA,XH,SIV

8 2FI0.0 THIK,THTED

!

I

I182

Additional configurations can be run by adding sets of cards

4 through 8. After the last set to be run, SPAN should be

read in as 0.0.

B. Variable Definition (all linear dimension units must be consistent)

IF (IDOWN = 1) a subroutine is called in which the downwash iscomputed at all pressure computed points on the wing.

IF (IWGEX = 1) uses exact solution for wing.

IF (IWGVP = 1) uses vortex paneling also for wing solution.

IF (IELEX = 1) uses exact solution for elevon.

IF (IELVP = 1) uses vortex paneling also for solution of elevon.

IF (IPRINT = 1) a more extensive printout is produced at inter-mediate computation points.

IF (IWGDF = 1) the wing has deformation as specified in functionTHA. Otherwise wing deformation is zero (planar).

IF (IELDF = 1) the elevon has deformation as specified in functionETHA. Otherwise elevon deformation is zero (planar).

NC is the number of chordwise control points for the wing.

NS is the number of spanwise control points for the wing.

NX is the number of grid points in the chordwise direction.

NY is the number of gridpoints in the spanwise direction.

NCORD is the number of chordwise points at which the pressure iscomputed for the wing.

NSPAN is the number of spanwise points at which the pressure iscomputed for the wing.

VAG is the angle in degrees that the vortices leave the wingreference plane.

DB is the dimensional body diameter.

1 CONFIG is the alphanumeric configuration description.

SPAN is the dimensioi,a total wing span.

1

I183

ICR is the exposed wing root chord.

CT is the wing tip chord.

LAMDA is the wing leading edge sweep angle in degrees.

SWING is the dimensional reference area. If 0.0 is read in, theprogram will compute and use the wing area as the referencearea.

CBRW is the dimensional reference length. If 0.0 is read in,the program will compute and use the mean aerodynamic chord

as the reference length.

MACH is the freestream Mach number.

I ALPHA is the angle of attack of the wing alone in degrees.

IF(LVON = 0) the elevon is omitted from all computations and

the wing alone case is analyzed.

NCE is the number of control point locations on the elevon inthe chordwise direction.

NXE is the number of subelements on the elevon in the chordwisedirection.

NCRE is the number of points in the chordwise direction on theelevon at which pressure is to be computed.

I CRE is the elevon root chord.

CTE is the elevon tip chord.

1 GAP is the gap distance between the wing TE and the elevon LE.

DELTA is the elevon deflection angle in degrees relative to wingtrailing edge down is positive.

XH is the dimensional distance from the Y axis to the elevon

I hinge line.

SIV is the nondimensional chordwise location at which thewing trailing vortices are started.

THIK is the dimensional wing TE thickness.

THTED is the (positive) wing surface slope at the trailing edgein degrees.

I!

I

184

III. Output Data

Major Headings Are:

A. Input Data - Dimensional or nondimensional data as it wasgiven on the data cards.

B. Actual Values used in Program - Variables after non-dimensionalization by b/2.

C. Pressure Distribution -

D. Sectional Properties as Follows:

YA (X LE) CA CCL CCMY XAC XCPA

(n) (xLg) (c) (cc) (c ( ac) (X ac)

yE. Total load coefficients from "exact" theory

CN - Total normal force coefficient for wing only.

CLT - Total load coefficient.

CMTY - Total moment coefficient.

YBAR - Spanwise location for mean aerodynamic chord.

SIAC - Chordwise location for aerodynamic center.

CMC4 - Moment coefficient about the quarter chord.

CD-CDMIN - Induced drag.

SAREA - Reference area.

CBAR - Reference length.

F. [For deformed wings, the following is also output]

1. (a) Control point location; XO, YO, local chord; CA,

The x location of the last row of control points is re-located at trailing edge of wing and new locations areprinted out.

(b) XO and YO are the nondimensional control pointlocations on the wing.

II

185

(c) Deform angle is the local deformation angle.

(d) B is the boundary condition (right-hand side

side of equation 57).

2. (a) x and y are the dimensional location of the controlpoints.

(b) B is the boundary condition (reference equation 59)

(c) WW is a dummy variable representing part ofboundary condition term.

(d) BOD DBLET UPWSH is the body doublet upwash.

3. (a) Polynomial coefficients are the coefficients whichdescribe AC loading distribution on the wing(Reference Pequation 54).

4. (a) Pressure tdistribution for wing using doubletpaneling. The left-hand column, PSI, are thenondimensional chord-wise locations; the top roware the nondimensional spanwise locations at whichthe pressure is computed.

5. (a) The next output is the sectional coefficientsfor the deformed wings (reference section D, above).

(b) Next is the total coefficients for the deformedwings (reference section E, above).

G. 1. XG and YG are the x and y locations of the startingpoint of the trailing vortices on the wing.

2. GMA are the strengths of the trailing vortices.

H. Pressure distribution for elevon using exact functions(reference section 4(a)).

I. This section contains the sectional properties for the elevon(reference section D).

J. Total force coefficients for the elevon (reference section E).

K. Conversion to Neilsen's Data.

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