NASA Technical Paper 3637

Guide to AERO2S and WINGDES Computer

Codes for Prediction and Minimization of Drag

Due to Lift

Harry W. Carlson

Lockheed Martin Engineering & Sciences .. Hampton, Virginia

Julio Chu and Lori P. Ozoroski

Langley Research Center • Hampton, Virginia

L. Arnold McCullers

ViGYAN, Inc. • Hampton, Virginia

National Aeronautics and Space Administration

Langley Research Center • Hampton, Virginia 23681-2199

November 1997
Available electronically at the following URL address: http://techreports.larc.nasa.gov/ltrs/ltrs.html

Printed copies available from the following:

NASA Center for AeroSpace Information

800 Elkridge Landing Road

Linthicum Heights, MD 21090-2934

(301) 621-0390

National Technical Information Service (NTIS)

5285 Port Royal Road

Springfield, VA 22161-2171

(703) 487-4650
Contents

Symbols ......................................................................... vii

Abstract ........................................................................... 1

1. Introduction ..................................................................... 1

2. Basic Lifting Surface Solution ....................................................... 2

2.1. Grid System and Lifting Surface Definition ......................................... 2

2.2. Subsonic Linearized Theory Solution .............................................. 3

2.3. Supersonic Linearized Theory Solution ............................................. 6

3. Theoretical Leading-Edge Thrust ..................................................... 8

4. Attainable Leading-Edge Thrust and Vortex Forces ...................................... 9

5. Aerodynamic Forces and Moments .................................................. 13

6. Wing Analysis Method for AERO2S Code ............................................ 16

6.1. Wing With Flaps ............................................................. 16

6.2. Wing in Combination With Second Surface ........................................ 20

6.3. Two-Dimensional Airfoil ....................................................... 20

7. Wing Design Method for WINGDES Code ............................................ 20

7.1. Candidate Surfaces ............................................................ 21

7.2. Influence of Leading-Edge Conditions on Wing Design ............................... 23

7.3. Selection of Leading-Edge Surfaces .............................................. 23

7.4. Selection of General Camber Surfaces ............................................. 24

7.5. Summary of Design Process .................................................... 25

7.6. Evaluation of Design .......................................................... 26

7.7. User Control of the Design ...................................................... 26

7.8. Flap Design Feature ........................................................... 27

7.9. Design in Presence of Interference Flow Fields ...................................... 29

7.10. Supersonic Empirical Corrections ............................................... 29

8. AERO2S Computer Code and Its Use ................................................ 32

8.1. Wing Surface Numerical Representation ........................................... 33

8.2. Vortex Force Options .......................................................... 34

8.3. Optimization Capabilities ....................................................... 34

9. WINGDES Computer Code and Its Use .............................................. 35

9.1. Wing Surface Numerical Representation ........................................... 36

9.2. Design Surface Options ........................................................ 36

9.3. Mission Adaptive Design ....................................................... 37

9.4. Flap System Design ........................................................... 38

9.5. Design by Iteration ............................................................ 39

9.6. Leading-Edge Surface Control ................................................... 39

9.7. Design With Moment Restraints ................................................. 39

9.8. Interference Flow-Field Design .................................................. 40

ooo !11
9.9. Empirical Correction .......................................................... 40

10. Drag Synthesis ................................................................. 40

11. Examples of Prediction of Aerodynamic Performance .................................. 42

11.1 Aspect Ratio 2 Wing-Body ..................................................... 43

11.2. Cranked-Wing Supersonic Fighter ............................................... 43

11.3. Subsonic Transport ........................................................... 46

11.4. Supersonic Transport Wing .................................................... 46

11.5. Supersonic Transport Wing-Body-Horizontal Tail ................................. 46

11.6. Two-Dimensional Airfoil ...................................................... 46

12. Examples of Design for Aerodynamic Performance Optimization ......................... 47

12.1. Supersonic Transport Cruise Surface Design ....................................... 47

12.2. Supersonic Transport Low-Speed Flap Design ..................................... 49

12.3. Nonsymmetrical Wing Cruise Surface Design ..................................... 51

13. Concluding Remarks ............................................................ 51

Appendix A--AERO2S Code Input and Output .......................................... 52

A1. Wing Planform--Required Input ................................................. 52

A2. Wing Planform--Optional Input ................................................. 52

A3. Wing Camber Surface--Required Input ........................................... 52

A4. Section Parameters for Attainable Thrust Calculation--Required Input ................... 53

A5. Flight Conditions--Required Input ............................................... 53

A6. Leading- and Trailing-Edge Flaps--Optional Input .................................. 53

A7. Solution Convergence Criteria---Optional Input ..................................... 54

A8. Canard or Horizontal Tail--Optional Input ......................................... 55

A9. Optional Two-Dimensional Airfoil Solution ........................................ 55

A10. Code Output Data ............................................................ 56

Appendix B--W-INGDES Code Input and Output ........................................ 57

B 1. Wing Planform--Required Input ................................................. 57

B2. Wing Planform--Optional Input ................................................. 58

B3. Wing Camber Surface--Optional Input ............................................ 58

B4. Section Parameters for Attainable Thrust Calculation--Required Input ................... 58

B5. Flight Conditions--Required Input ............................................... 59

B6. Solution Convergence Crtefia---Optional Input ..................................... 59

B7. Design Specifications--Required Input for Design Mode .............................. 59

B8. Design Specifications--Optional Input for Design Mode .............................. 60

B9. Additional Optional Input ....................................................... 61

B10. Reflex Surface Design--Optional Input for Design Mode ............................ 61

Bll.

B12.

B13.

Flap Design--Optional Input for Design Mode ..................................... 61

Design Surface Smoothness Control---Optional Input ................................ 61

Interference Flow Field---Optional Input .......................................... 62

iv
B 14. Empirical Corrections--Optional Input ........................................... 62

B15. Code Output Data ............................................................ 62

Appendix C--Estimation of Idealized Minimum Drag and Maximum Suction Parameter ......... 64

References ....................................................................... 73

Tables ........................................................................... 74

Figures .......................................................................... 88

Symbols

The computer symbols used in the text, tables, and figures are defined in appendixes A and B.

A candidate surface weighting factor

A e equivalent area due to lift

b 2

AR aspect ratio, -ff

b span

Axial force

CA axial- or chord-force coefficient,

qS

CO drag coefficient, Drag

qS

ACD drag coefficient due to lift, C o - CD, 0

CD,0 drag coefficient at ot = 0" for configuration with symmetry about wing reference plane

Lift

CL lift coefficient, --

qS

CL,de s design lift coefficient

CL,op t optimum lift coefficient, lift coefficient corresponding to maximum value of suction

parameter

CLc' theoretical lift-curve slope at tx = 0 °, per deg

Ct rolling-moment coefficient about lateral center of gravity for nonsymmetrical wings,

Rolling moment

qSb/2

Cm pitching-moment coefficient, Pitching moment

qS_

fro,de s design pitching-moment coefficient

Normal force

CN normal-force coefficient,

qS

Pt- P

Cp pressure coefficient, --

q

ACp lifting pressure coefficient

Cp, c pressure coefficient on cambered wing at _ = 0 °

(Cp, c) c component of Cp, c due to pure camber loading (contribution with no leading-edge

singularity)

component of Cp, c due to flat wing loading (contribution with leading-edge singularity)

pressure coefficient on flat wing at ct = 1°

limiting pressure coefficient used in definition of attainable thrust

pressure coefficient at specified initial point

Cp,/

Cp, lim

Cp, o

Cp,vac

c

CA

CA,c

vacuum pressure coefficient, - --

local chord

mean aerodynamic chord

section axial-force coefficient

component of cA due to basic pressure loading of camber surface at ct = 0 ° acting on camber

surface

vii
CA,tic

C e

Cle

Cm

CN

CN, c

(C,v,c)c

(CN,c)f

CNdr

c R

Cr

c t

¢t

ct,f

cte

e x

ey

e ! , e 2, e 3, e4

Fc(x')

F/ (x')

f

h

J

K

gdes

rs

r_

k

components of cA due to basic pressure loading of flat wing at ot = 1° acting on camber

surface

element chord at element midspan

streamwise chord for leading-edge design area

section pitching-moment coefficient

section normal-force coefficient

component of cN due to basic pressure loading of camber surface at _ = 0° acting on camber surface

component of CN,c due to pure camber loading (contribution with no leading-edge

singularity)

component of CN,c due to flat wing loading (contribution with leading-edge singularity)

component of cN due to basic pressure loadingof fiat wing at 0t = 1° acting on camber surface

section resultant force coefficient, _ + c_

wing root chord, chord at y = 0

theoretical section thrust coefficient

attainable section thrust coefficient

theoretical section leading-edge thrust coefficient for flat wing at _ = 1°

streamwise chord for trailing-edge design area

exponent of x used in definition of candidate camber surfaces

exponent of y used in definition of candidate camber surfaces

exponents used in calculating attainable thrust

normal-force integration factor for pure camber contribution to basic cambered wing loading at a = 0 ° acting on camber surface

normal-force integration factor for basic pressure loading of flat wing at _ = 1° acting on

flat surface; normal-force integration factor for flat wing contribution to basic cambered

wing loading at a = 0° acting on camber surface; axial-force integration factor for basic

pressure loading of flat wing at a = 1° acting on camber surface

rate of growth of lifting force per unit distance along equivalent body axis

location correction factor for code perturbation velocity

altitude

index of wing element longitudinal position within code grid system and index used in

identification of candidate surfaces

index of wing element lateral position within code grid system and index used in identification of candidate surfaces

constant used in curve-fit equation

(CL,des) opt,exp

design lift-coefficient factor,

( C L,des ) opt,th

suction parameter factor, (Ss'max)exp

(Ss,max)th

C t

attainable thrust factor, fraction of theoretical thrust actually attainable, --

c t

arbitrary constant used in definition of pressure distribution

ooo

viii
kQ

k o kf

kh,1, kh,2

kn

k,

kvor

kb k2

LE

I

le

le, v

M

Me

mle

mte

P

Pl

q

R

r

ri

S

SS

SST

s

TE

t

U

U, I;, W

Au

(A.d2)o

(Au ')o,c

constant used in attainable thrust curve-fit equation

constants used in curve fitting of code perturbation velocities and pressure coefficients for

integration purposes

constants used in hinge-line singularity correction

arrow wing notch ratio, see figure 40

constant used in candidate design surface definition

constant used in definition of vortex force distribution

constants used in definition of camber surface slope

leading edge

overall wing length

effective length of body of revolution representing distribution of equivalent area due to lift

as defined by area rule cutting planes

effective length of body of revolution representing distribution of volume as defined by

area rule cutting planes

Mach number

equivalent Mach number used in place of M n to account for values of Cp.li m differing from

Cp,vac

normal Mach number, see figure 12

multiplying factor for tangent of leading-edge flap deflection angle

multiplying factor for tangent of trailing-edge flap deflection angle

free-stream static pressure

local static pressure

free-stream dynamic pressure

Reynolds number based on mean aerodynamic chord

linearized theory perturbation velocity influence function

leading-edge radius

leading-edge radius index, (r/c)rl

(xlc) 2

wing reference area

C L tan (CL/CL)- AC o

suction parameter,

C L tan (CL/CL_) -C2/(rcAR)

supersonic transport

distance along section camber line

trailing edge

theoretical section leading-edge thrust

free-stream velocity

perturbation velocity in x, y, and z direction, respectively

longitudinal perturbation velocity difference across wing lifting surface as fraction of freestream velocity

limiting value of leading-edge thrust parameter Au,_ -_ at wing leading edge

limiting value of leading-edge thrust parameter Au4r'_ at wing leading edge for cambered

wing at ¢x = 0 °

ix
( Au,41-XP)o,f

Au c

Auf

V

x, y, z

Ax

Ax o Ax R, Ax L

Xcg

x h

Xmc

x

v

x e

X , ,

Ycg

go

Z0, Zl, Z2

O_

O_des

a_t

c_t

a0

7

_c

_C,H

_e,n

_te,s

£

eo

n

limiting value of leading-edge thrust parameter Au 4t_ at wing leading edge for flat wing at

o_=1 °

value of Au for cambered wing at _ = 0 °

value of Au for fiat wing at c_ = 1°

configuration total volume

Cartesian coordinates, positive aft, right, and up, respectively

longitudinal spacing of grid lines used in establishment of code wing grid system

longitudinal distances employed in influence function

longitudinal center of gravity

distance from wing leading edge to flap hinge line

longitudinal moment center

distance in x direction measured from wing leading edge

distance in x direction measured from wing element leading edge

values of x' at which camber surface z ordinates are specified

lateral center of gravity

limiting value of singularity parameter ACp,,/_' at x' = 0

camber surface ordinate at x0, Xl, and x 2, respectively

angle of attack, deg

angle of attack corresponding to design lift coefficient, deg

range of angle of attack for full leading-edge thrust, deg

angle of attack for zero thrust, deg

angle of attack corresponding to zero lift, deg

ratio of specific heats, 1.4

incidence of canard reference plane with respect to wing reference plane, positive with

leading edge up, deg

incidence of horizontal-tail reference plane with respect to wing reference plane, positive

with leading edge up, deg

leading-edge flap deflection angle measured normal to hinge line, positive with leading

edge down (segmented flap deflection specified as inboard/outboard), deg

leading-edge flap streamwise deflection angle, positive with leading edge down (segmented

flap deflection specified as inboard/outboard), deg

trailing-edge flap deflection angle measured normal to hinge line, positive with leading

edge down (segmented flap deflection specified as inboard/outboard), deg

trailing-edge flap streamwise deflection angle, positive with leading edge down (segmented

flap deflection specified as inboard/outboard), deg

angle between line tangent to wing section camber surface and camber surface reference

plane, deg

value of e at wing leading edge

location of section maximum thickness, fraction of chord
A

Ah,le

Ah,te

Ale

k

Z'N, _'M, _'R

_t

Subscripts:

adj

av

b

c

cot

emp

eval

exp

f

le

max

min

opt

pre

rep

te

th

tot

vis

vor

14,

sweep angle of wing constant percent chord line, deg

leading-edge flap hinge-line sweep angle, deg

trailing-edge flap hinge-line sweep angle, deg

wing leading-edge sweep angle, deg

angle between intersection of distant fore Mach cone with wing plane and wing longitudinal axis, deg (see fig. 39)

Lagrange multipliers

Mach angle, sin -1 (I/M), deg

section maximum thickness

azimuth angle in frontal projection between line connecting field point with wing longitudinal axis and wing plane, deg (see fig. 39)

adjusted

average

aerodynamic characteristics due to basic pressure distributions alone (no thrust or vortex

forces)

cambered wing

corrected

empirical

evaluated

experiment

flat wing

leading edge

maximum

minimum

wing section normal to leading edge

optimum

previous

replacement

trailing edge

theoretical

total

viscous

vortex

wave

xi

Abstract

The computer codes, AER02S and WINGDES, are now widely used for the analysis and design of airplane lifting surfaces under conditions that tend to induce flow

separation. These codes have undergone continued development to provide additional

capabilities since the introduction of the original versions over a decade ago. This

code development has been reported in a variety of publications (NASA technical

papers, NASA contractor reports, and society journals). Some modifications have not

been publicized at all. Users of these codes have suggested the desirability of combining in a single document the descriptions of the code development, an outline of the

features of each code, and suggestions for effective code usage. This report is

intended to supply that need.

1. Introduction

The computer codes, AERO2S and WlNGDES, are

now widely used for the analysis and design of airplane

lifting surfaces under conditions that tend to induce flow

separation and degrade performance. These codes have

undergone continued development to provide additional

capabilities since the introduction of the original versions

over a decade ago. This code development has been

reported in a variety of publications (NASA technical

papers, NASA contractor reports, and society journals).

Some modifications have not been publicized at all.

Users of these codes have suggested the desirability of

combining in a single document the descriptions of the

code development, an outline of the features of each

code, and suggestions for effective code usage. This

report is intended to supply that need.

A method for estimation of attainable leading-edge

thrust introduced in reference 1 provides the fundamental

basis of a system applicable to partially attachedpartially separated flow. The original computing code

employing the attainable thrust numerical method which

was applicable to analysis of a single lifting surface with

twist and camber at subsonic speeds is described in

reference 2. A modification of the analysis method to

provide for the handling of simple hinged leading- and

trailing-edge flaps is described in reference 3. A further

modification to permit the analysis of a wing surface in

combination with a second lifting surface such as a

canard or a horizontal tail is described in reference 4. The

present version of this code is designated "AERO2S."

The wing-design computer code described in reference 5 provides for the design of a wing mean camber

surface (twist and camber in combination) to minimize

drag for given design lift and moment conditions. The

design method defines an optimum combination of candidate surfaces rather than the more usual optimum combination of loadings. In the design process, attainable

leading-edge thrust is taken into account to provide the

mildest possible camber surface which meets the design

requirements. The use of candidate surfaces provides an

additional capability to design mission adaptive camber

surfaces which restrict changes to designated areas of the

planform. The design code was later modified to provide

for the design of leading- and trailing-edge flap deflection schedules as described in appendix A of reference 3.

The present code also has provision for the design of

reflexed wing surfaces in the vicinity of engine nacelles

and for the handling of asymmetrical planforms. These

two modifications, as well as several others, were not

previously documented. The design code is applicable to

both subsonic and supersonic speeds and provides analysis as well as design capabilities. The present version of

this code is designated "WINGDES."

A survey of research on wing design for reduction of

drag due to lift at supersonic speeds reported in reference 6 led to the development of an empirical correction

to account for real flow effects not covered by linearized

theory methods. This correction provides an adjustment

to the design process so that the wing design may be

optimized with nonlinear penalties associated with

excessive camber surface severity taken into account.

The empirical correction, now incorporated into the

WINGDES code, results in a milder camber surface than

would otherwise be found. A second empirical correction

provides a more realistic estimate of the achievable performance of the design. A further modification to the

application of the correction, introduced in the present

paper in section 7.10, now permits the additional benefit

of reduced camber surface severity associated with an

attainable thrust design to be included in the performance

estimate.

Reference 7 describes a recent revision of the attainable thrust prediction method used in both codes. The

newer method that provides for a greater range of airfoil

shapes from very sharp to very blunt leading edges is

based on experimental data for a wider range of

Reynolds numbers than the previous method. References 8 and 9, in addition to references 2 to 7, give examples of correlation of code results (both AERO2S and

WlNGDES) with experimental measurements and offer
adviceon applicationof the codesto problemsof

interest.

Thisreportis intendedto provideanunderstanding

ofcodecapabilitiesandguidanceineffectiveapplication

toproblemsof practicalinterestbutnotademonstration

of codevalidity.Themanypreviouslyreportedcorrelationsof codedatawith experimental resultsusedto

validatethemethodsarereexaminedonlyif theyprovide

informationpertinento futurecodeusage.Readers

interestedprimarilyin applicationof thecodeswill find

informationdescribingcodeinputandoutputdatain

appendixA for AERO2Sand in appendixB for

WINGDES.Examplesofcodeusegiveninsections11

and12offerguidanceinuseof thecodesforproblemsof

interest.Data used in these examples are given in table I

for AERO2S and in table II for WINGDES and serve as

models for the preparation of code input.

The use of area rule concepts to provide a further

understanding of design for drag minimization at supersonic speeds is the final topic of this report. Mathematical relationships presented in appendix C describe the

dependence of vortex drag, wave drag due to lift, and

wave drag due to volume on configuration geometric

characteristics and establish minimum values for each of

these drag components. The strategies for the estimation

of drag minima outlined in appendix C have been incorporated into a computer code, CDMIN.

2. Basic Lifting Surface Solution

The primary component of both the WINGDES and

the AERO2S codes is a modified linearized solution for

the forces and moments acting on twisted and cambered

lifting surfaces of arbitrary planform. Forces obtained by

integration of pressure distributions on the zero thickness

lifting surfaces used in these codes do not include a

leading-edge thrust contribution arising from the high

velocities and low pressures generated by a flow around

the leading edge from a stagnation point on the wing

lower surface, l However, methods are available that provide estimates of not only the theoretical leading-edge

thrust but also the amount of this force that can actually

be realized. A means of estimating attainable leadingedge thrust and also the vortex force generated as a result

of leading-edge flow separation is included in the modified linearized theory solution used in these codes.

Among the unique features of the linearized theory

methods used herein are (1) solutions obtained by pure

iteration and (2) the use of leading-edge singularity

tUnder some circumstances (wings with symmetrical sections at

negative angles of attack, for example), the stagnation point may

occur on the upper surface. In either case, leading-edge thrust may

be developed.

parameters to identify and separate velocity distribution

components with and without singularities. The first feature permits an easy code expansion to accommodate

more wing elements for greater accuracy as computer

capabilities improve. The second feature permits more

accurate determination of leading-edge thrust distribution for wings with twist and camber and provides

improved pressure distribution integration techniques.

2.1. Grid System and Lifting Surface Definition

The linearized theory solutions are obtained by an

iterative solution of influence equations for an array of

wing elements representing the wing planform as

depicted in figure 1. Only a right-hand wing panel is necessary because of the lateral symmetry of the wing and

all the flow properties. Only recently has the WINGDES

code been modified to permit solutions for asymmetric

planforms. When that option is selected, the entire planform

is represented as a right-hand panel and the imposition of

symmetry is revoked. In figure 1 only a small number of

elements are shown for the purpose of illustration; in

practice several hundred to as many as 2000 dements

would be employed. The elements are superimposed on a

rectangular grid so that the inboard and outboard element

boundaries lie along unit values of the spanwise parameter _y and the midspan leading and trailing edges lie on

unit values of the chordwise parameter xlAx. The scaling

of the wing from model or airplane dimensions to code

dimensions is chosen to provide the desired number of

elements in the spanwise direction. The distance Ax controis the chordwise spacing of the elements; it is selected

by specification of an element aspect ratio ELAR, which

is constant for all but the leading- and trailing-edge elements. Each element is assigned a number and a record is

kept of the number assigned to the leading- and trailingedge elements in each chordwise row. The i andj indices

(i(x/Ax) and j(_y)) are used in determining the order of

solution; elements are selected f'n'st according to advancing values of the index i then according to advancing values of the index j. The order of solution thus marches

inboard to outboard and front to rear.

As shown in figure 1(b), the wing representation for

supersonic speeds employs elements with unswept leading and trailing edges, a limitation made necessary by the

peculiar nature of supersonic flow. For subsonic speeds,

swept leading and trailing edges are acceptable and provide a more accurate solution. For subsonic solutions,

element corner points at the wing leading and trailing

edges are found by interpolation of the scaled input planform definition. These points determine the leading-edge

sweep of the first element and the trailing-edge sweep of

the last element in each chordwise row identified by the

index j(_y). Sweep angles for element leading and trailing edges at a specified span location are found from
' ' /J" / '1 / '

- -_- - 47_ 7/-_- -/- - +/- -I- -'- -

I I I J I

f i i i J

Sketch 2-1

simple geometry for a superimposed arrow wing planform as indicated in sketch 2-1.

Camber surface ordinates at the lateral midpoint of a

given chordwise row of elements are found by linear

interpolation of input ordinates just inboard and outboard

of that location. The wing surface slopes are obtained by

a curve fit of the interpolated code input camber surface

coordinates. The curve-fit equation has the form

z = z o + kl(x'-x' o) + kz(X'- x_))2

As shown in sketch 2-2, the interpolated input camber

surface ordinates are chosen so that one ordinate x_) is at

or ahead of the element leading edge, one ordinate x' 1 is

within the element, and one ordinate x_ is at or behind

the element trailing edge. With the constants k 1 and k 2

chosen to pass the curve through these three points, they

can then be used in definition of the element surface

slope expressed as

dz

__ = k I +k2x e

dx'

r

where x e is the distance from element leading edge

and k 1 and k2 are redefined to correspond to the new

origin. Stored values of k 1 and k 2 allow subsequent

I

_______._

xi

Element ----

z2

Zo

Sketch 2-2

recalculation of surface slopes anywhere within the

element. The slope at the element three-quarter point

(the control point) is used in satisfying boundary

conditions. As discussed subsequently in sections 6

and 7, the codes repeat the basic linearized theory solution for two or more wing surfaces. One or more of these

wing surfaces has the slopes just described; the remaining surface has a constant slope equal to the tangent of

(x -- 1° (dz/dx = --0.01745).

2.2. Subsonic Linearized Theory Solution

Each trapezoidal element used to represent the wing

for the subsonic solution is assumed to have an associated horseshoe vortex with a bound leg along the quarterchord and trailing legs extending to infinity along the

extensions of the inboard and outboard chords as shown

in sketch 2-3. At any point in the plane of the wing, the

downwash velocity created by the vortex is given by

_rt _ Au c e

W

U

where

Au

C e

k

longitudinal perturbation velocity difference

across wing surface

element average chord

subsonic influence function

In terms of the geometry system used, the subsonic influence function is given as

_,

_x_ L J(AXR)2+(A_y+0.5) 2

_ (A y- 0.s___]) : Ax____ L!an 1

J(AXL)2*(A y-0.5) 2 ]

+

Ax R

ff(AXR) 2 + (A_y - 0.5) 2

(1)

The three terms in equation (1) represent, respectively,

the bound leg, the left trailing leg, and the right trailing

leg. The geometric quantities represented in the equation

are illustrated in sketch 2-4. The sign convention is such

that the Ax quantities in the sketch are positive. The sign

convention for the Ax quantities in equation (1) and
Vortex point -_

._-.--...-- Ce -........--._

_ Control point G

Sketch 2-3

Influencing

element __ AXL

/= AxC

_- Field

I' point

Sketch 2-4

sketch 2-4 is the reverse of that given in reference 2 so

that distances behind an influencing element may be represented as positive quantities. In addition, the sign of the

influence function k has been changed so that the function has the same sign as the induced velocities.

A graphical representation of the subsonic influence

function R for an unswept element is given in figure 2.

The vertical velocities generated by a given element are

directly related to this function. Behind the bound leg

and between the two trailing legs, only downwash is generated. Ahead of the bound leg and outside the trailing

legs, only upwash is generated. The singularities which

arise at the bound and trailing legs are shown only in part

because values of R are truncated for locations closer to

the lines than one tenth the element span. These singularities dictate the need for a precisely defined grid and element system to represent a lifting surface. Clearly an

unstructured random array of elements could lead to

coincidence of singularities with control points and result

in an unacceptable infinite local velocity. Even a near

coincidence could result in unstable solutions. The grid

system chosen places element inboard and outboard

boundaries on streamwise lines with equal lateral spacing. The element control point location at the midspan of

the element and at three quarters of the element chord

provides a structured array of the vortex system so that

4

the influence of each element is dependent on the

strength of its vortex and is not subject to distortions

caused by geometric coincidences. The depiction of the

influence factor in figure 2 shows clearly the relative

advantage of high wing aspect ratios. A second element

just to the left or right of the influencing element can,

because of its upwash field, generate a given amount of

lift at a smaller angle of attack than would an isolated

element. The smaller angle of attack of course results in a

smaller drag for a given lift. On the other hand, a second

element directly behind the first would, because of the

downwash field, require a large angle of attack to generate the same lift, and a higher drag would result. The relatively small effect of the lifting element at locations

ahead of the element explains the applicability of the iterative solution.

The downwash at any point in the plane of the wing

induced by the complete wing may be found by a summation of the contributions of all the individual elements.

At the control point of a field element at which the

boundary condition of no flow through the element is to

be met, the downwash velocity is given as

=_'-_t( au Ce+ _--_

where the quantities with stars (-A-) refer to the field element and the summation includes all elements but the

field element itself.

or

The boundary condition is met when

l

au* : .c. dx j Ce R c e

(2)

The perturbation velocities are converted into pressure

coefficients in accordance with the linearized theory

assumption

AC_ = 2 Au*

Before the solution by iteration is begun, perturbation velocities for all elements are set to zero. Then, each

element in turn is considered as a field element, and a

perturbation velocity for that element satisfying the

boundary condition is found. This new velocity replaces

the old one in the velocity table, and the calculation proceeds to the next element. As described in section 2.1,

the order of solution proceeds from inboard to outboard

at the front of the wing and repeats this inboard to outboard sweep at successive rearward positions. As a
meansof verifyingthesolutionconvergence, theabsolutevalueof thevelocitydifferencesbetweensuccessive

iterationsiscalculatedforeachelement,andanaverage

valueofthisdifferenceforthewholewingisfound.The

iterationprocessidiscontinuedwhentheaveragedifferencefortwosuccessiveiterationsislessthanonehalfof

1percent(0.005)of theaveragepressureloadingof the

fiatsurfaceat ct = 1°.

As a means of reducing computational time, only

elements relatively close to the field point are considered

in the first iteration. As the iteration process proceeds

and the convergence criteria are approached, the region

of influence considered is expanded. The influence

region is related to the convergence criteria in such a way

to ensure that, at least for the last two iterations, the

whole wing is included. An element is excluded from the

summation if A_y is greater than

4 + 2_/0.005/(CNVGP ) [(JBYMAX) - 2]

or Ax L or Ax R (whichever is less) is greater than

2 Ax + _/0.005/(CNVGP ) [(XMAX)(SCALE) - 2 Ax]

where CNVGP is the value of the average difference

ratio for the previous iteration.

Because of the element system used to represent the

wing, it was known at the outset that there would be

errors in the numerical solution in the region of the wing

leading edge. These errors were anticipated to be systematic in nature and thus predictable so that corrections

could be made. To study the numerical solution errors,

the coded solution was modified slightly to permit a

solution for a two-dimensional wing. Typical code

results for a flat (uncambered) two-dimensional wing at

ct = I ° are shown in figure 3. Velocity distributions are

shown for uniform chord elements at the left of the figure

and for a smaller chord first element at the right. The

code results are compared with the exact linearized theory solution which is

or

Au = 20t -

C

with 0_ in radians. The plot of the singularity parameter

Au,4_r]_ allows a more critical comparison of numerical results with the exact linearized theory. The numerical result velocities are assumed to act at the element

quarter-chord because the center of pressure for the critical leading-edge elements tends to occur at this point. As

shown at the left of the figure, with uniform chord

spacing only the first element result is in error. For a

smaller first chord as shown at the right, code results for

the first two elements (but only the first two elements)

behind the leading edge were found to be in error.

When other first element chords were employed,

results such as those shown in the upper plot of figure 4

were obtained. The well-behaved nature of the errors

suggested that a correction could easily be made. The

correction is made to location, rather than to the velocity

itself, because a shift in location will correct fiat wing

data but will not introduce appreciable errors in data for

surfaces with pure camber Ioadings. As shown in

sketch 2-5, if the location x'/Ax is multiplied by a correction factor f, the singularity parameter now expressed

as Au,,!f(x'/Ax) will follow the dashed-line curve.

Thus, it is a simple matter to find a new x' location,

defined by the factor f, which will produce agreement

with the exact linearized theory. The required factor for

each of the code data points is shown in the middle plot

of figure 4. Also shown is a curve fit to the correction

factor data defined by

f = 1 + 0.36

1.25 - x'/Ax

x'/Ax

+ 0.18 sin _ixit x, < 0.5 )

and

1.25 - x'/Ax

f = 1 + 0.36 x'/Ax

+O'18sinl'25-x'/AXl.5 rt (X'_x > 0.5)

The singularity parameter obtained when the location of

the velocity is defined by the factorfis shown in the bottom plot of figure 4.

The simple correction derived from the twodimensional results was found to be equally valid in three

/

/

7

f

ii i I i Original

I

X'

Sketch 2-5

5
dimensions.Typicalcoderesultsfor constant-chord

wings(right-handpanelonly)ofvarioussweepanglesat

M = 0.01 are shown in figure 5. The singularity parameter is shown as a function of chordwise position for a

midspan section. No erratic behavior of the first two elements is seen. Results for other sweep angles between 0°

and 80 ° at other Mach numbers up to 0.8 were similar.

The iterative solution converged quite rapidly to a

reasonable approximation of fully converged results as

estimated by extrapolation and as given by vortex lattice

methods using matrix inversion. However, when stringent convergence criteria are applied, as required to

obtain accurate leading-edge singularity information,

many iterations may be necessary. An example for a

constant-chord wing with Ale = 40 ° (right-hand panel

only) is given in figure 6. The first and second iterations

give the general character of the solution. More than

four iterations are required before sufficiently accurate

leading-edge perturbation velocities are provided. The

code convergence criteria previously discussed are met

after the tenth iteration in this example. For more complex planforms and for severely cambered wings, more

iterations are required. For some of the examples shown

later, up to 70 iterations were required.

2.3. Supersonic Linearized Theory Solution

Each rectangular element used to represent the wing

for the supersonic solution is assumed to have an associated horseshoe vortex system with the same geometry as

the subsonic solution except that the bound leg is

unswept. The vortex strength, however, is different. At

any point in the plane of the wing, the downwash velocity created by the vortex is given by

Rw = _R a. ce U 2n

where

Au

C e

k

longitudinal perturbation velocity difference

across wing surface

element average chord

supersonic influence factor

In terms of the geometry system used, R is given as

= J Ax2 -(A_y-0.5) 2 JA 2

_ x C + (A_y + 0.5) 2 (3)

Axc( A_y - 0.5) Axc( A_y + 0.5)

The quantity Ax c is again defined as shown in

sketch 2-4. For an unswept bound leg, Ax c = Ax R = Ax L.

A graphical representation of the supersonic influence factor R is given in figure 7. As with the subsonic

distribution, only downwash is generated behind the

bound leg and between the two trailing legs. The upwash

generated outside the trailing legs, however, is confined

to a region behind Mach lines from the bound leg corner

points. The aerodynamic advantage of high aspect ratio,

clearly shown for subsonic speeds, applies to supersonic

speeds only if additional outboard elements are arranged

to lie behind the Mach line.

The downwash at any point in the plane of the wing

induced by the complete wing may be found by a summation of the contributions of all the individual elements.

At the control point of a field element where the boundary condition of no flow through the element is to be met,

the downwash velocity is given as

= \dx) (4)

or

2 (dz_* 1

Au*= _..

R c e

(5)

and in accordance with linearized theory assumptions

ACp = 2 Au*.

The strong influence along Mach lines dictates a

more rigid element geometry than does the subsonic

solution. Trapezoidal elements used in the subsonic solution could result in coincidence of the Mach line with an

element control point; this coincidence gives erratic

results. Even a near coincidence could result in large

local velocities and unstable solutions. The rectangular

elements employed in the supersonic solution ensure that

such a coincidence cannot happen. The Mach line follows a path that places it no closer to the control point of

any element than a longitudinal distance equal to one

half the nominal element chord Ax; this is true even for

leading- and trailing-edge elements. The longitudinal

separation distances are based on the grid space occupied

by the element and are unaffected by changes in the element chord Ce.

For a supersonic solution, the iteration procedure

employed for subsonic flow is not required. Because of

the limited regions of influence, a single pass is sufficient. The solution proceeds from inboard to outboard of

successive rearward positions as does the subsonic solution but is terminated when the last element on the wing

is reached.

The enforced simplicity of the supersonic solution,

however, brings with it a disadvantage not encountered

in the subsonic solution. There is a strong tendency

toward longitudinal oscillations in perturbation velocities. The oscillations are subdued in part by an aftdement sensing technique described in reference 10. The

6
aft-element sensingtechniqueinvolvesthedetermination • . "1_ •

of prehmmary Au results for a given field element

and for the element immediately following combined

with a subsequent fairing or smoothing of these preliminary results. The fairing is applied to the velocity potential (i.e., the integral of the velocity) rather than to the

velocity itself because of the noticeable better behavior

of the velocity potential in regard to the absence of

discontinuities. The procedure which is described in

detail in reference 10 is not repeated here.

Generally, the numerical method employing aftelement sensing gives rather smooth distributions of the

lifting pressure coefficient as evidenced by the numerous

comparisons of numerical method results with exact linearized theory given in reference 10. But for very highly

swept leading edges, a tendency is found for the formation of oscillations in fiat wing pressure distributions.

These oscillations center on the correct solution and thus

create no large problems in the determination of overall

wing forces and moments. Nevertheless, to find a means

of suppressing or smoothing these oscillations which for

very highly swept wings can become large is desirable.

An exploration of the causes of the oscillations and the

development of supplementary smoothing strategy is

described in the following paragraphs.

An example of extreme oscillations in lifting pressure coefficient given by the basic supersonic analysis

system is shown in sketch 2-6. The data shown are for a

delta wing with Ale = 75 ° at a Mach number of 1.41

(_ cot A = 0.27). The parameter Au x,4x,4x_y,derived from

theoretical distributions of pressure loadings on flat delta

wings, permits inclusion of data for several adjacent

spanwise stations near the mid-semispan, and compensates for the 1/,fx-' decline in pressure aft of the leadingedge singularity. This and similar plots for other sweep

angles and Mach numbers show a wavelength of the

oscillations which correlates well with the parameter

tanAlet_ + 1. As shown in sketch 2-7, the code array of

the rectangular elements for two adjacent spanwise stations dictates such a pattern. Because of the rectangular

element structure and the nature of supersonic flow, any

influence of the outboard span station on the inboard station is delayed to the chordwise position shown.

These considerations suggest a fairing which covers

a number of elements equal to the absolute value of the

local parameter tanAle/[_ + 1 and takes into account the

nature of supersonic pressure distributions• This fairing

can be accomplished by a least-squares curve fit of one

of the pressure variation forms illustrated in sketch 2-8.

Each curve passes through an initial (xo,Cp,o), and the

least-squares solution is used to determine the factor k

giving the best fit. In the code, a solution is found for

each form, and the form giving the smallest value for the

I

tan Ale

X'

Sketch 2-6

tan Ale

/ tan Ale /

_Y _

Sketch 2-7

(_,Cp,o)

[_ _'°fl+k(_-a)l

Cv _ %. + ,(x'- x_)

_CP'°II + k! _' - II:

X'

Sketch 2-8

sum of the squares of the errors is selected. The process

begins with Au of the first element behind the leading

edge. After application of the curve fit, the point immediately behind the initial point is given a new value defined

by the factor k of the selected form. Then the process is

repeated as often as necessary by advancing one element

rearward and by using the just replaced value as a new

initial point. In the region of the trailing edge, when the

remaining points are less than the defined number, new

values are found for all points.

7
Sketch2-9 showsthe Au distributionwith and

withoutsmoothingat themid-semispanof theexample

75° deltawingatM = 1.41. Some irregularities remain,

but they are minor compared with the original large

oscillations.

The two-step smoothing process for the supersonic

solution just described was not required for the subsonic

solution. However, errors in the subsonic solution in the

region of the wing leading edge were corrected as previously described by relocation of calculated velocities. A

comparable correction is also required in the region of

the wing leading edge for the supersonic solution. The

correction function covers only the first element behind

the leading edge instead of the first two. This correction

provides the original initial point for the previously

described supplementary smoothing process which is

applied to all elements. For aft elements, the perturbation

velocity rather than its location is adjusted.

Figure 8 shows the ratio of AUcode to Auth for

leading-edge elements of a series of flat delta wings with

different values of the leading-edge sweep parameter

13cot A. The data cover all code span positions up to the

maximum permissible for a given value of 13cot A. The

code velocities are assumed to act at the element quarterchord and the ratios are plotted as a function of the

element chord. Most observed scatter in the data is due to

inclusion of inboard span stations where a stable numerical solution has not yet developed. For these leadingedge elements, a curve expressed by the equation

AUcode l-(l-_e] TM

r(x') -

AUth ff213 cot Ale

was found to adequately represent the data of figure 8.

The corrected perturbation velocity location for leadingedge elements is

• Ce

Xco r --

4[F(x')] 2

As in the subsonic solution, the location rather than the

velocity itself is changed because a shift in location corrects flat wing data but does not introduce appreciable

errors in data for surfaces with pure camber loadings.

3. Theoretical Leading-Edge Thrust

Both the subsonic and the supersonic linearized theory solutions apply to lifting surfaces of arbitrary planform with arbitrary distributions of twist and camber.

The character of the calculated perturbation velocity and

pressure coefficient distributions is strongly influenced

by leading-edge flow conditions. For flat lifting surfaces

An

/

• Without smoothing

With smoothing

Sketch 2-9

(except for wings with supersonic leading edges),

leading-edge singularities are present at all angles of

attack other than zero. At positive angles of attack, the

upwash just ahead of the leading edge and the perturbation velocity just behind the leading edge become infinite. The presence of twist and camber alters the flow

conditions so that the singularities usually disappear not

at zero angle of attack but at some other angle of attack.

Because the theoretical leading-edge thrust is directly

related to the strength of the leading-edge singularity, the

angle of attack corresponding to the disappearance of the

singularity has been designated as the angle of attack for

zero thrust O_zt. At other angles of attack, singularities

form at the leading edge just as they do for the flat wing

at angles other than zero.

A means of identifying the angle of attack for zero

thrust offers considerable advantages in calculation of

theoretical leading-edge thrust and in integration of perturbation velocity and pressure distributions to obtain

forces and moments for wings with arbitrary twist and

camber. The following discussion describes the calculation of tXzt and its use in the calculation of theoretical

leading-edge thrust. The use of tXzt in the separation of

perturbation velocities and pressure distributions into

components with and without singularities for employment in integration techniques is discussed later in

section 5.

In the WINGDES and AERO2S codes, results coveting a range of angle of attack are obtained by combining the solution for the input cambered wing (considered

to be at tx = 0°) with a solution for a flat wing of the same

planform at tz -- 1°. An example of these basic solutions

for a 40 ° swept leading-edge, constant-chord wing (one

panel only) is shown in figure 9. The mean camber surface is an arc of a circle with a radius selected to give a

leading-edge slope of dz/dx = 0.0875 (e o = 5°). Results

for the cambered wing are given at the top of the figure,

and results for the flat wing are given at the bottom. Note

that the cambered wing as well as the flat wing displays a

leading-edge singularity.
Figure10showsresultsfor otheranglesof attack

obtainedby combiningsolutionsforthecamberedand

theflatwingsbythefollowingexpression:

sin

Au = Au c + Auf sin 1°

The angle of attack of 1.8 ° was chosen for this illustration because at or near that angle the leading-edge singulaxity vanishes. The velocity distribution for this case

may be considered to be a pure camber loading. For this

constant-curvature surface, the velocity distribution

closely follows a curve defined by

Au = k cjx( ) c 1 -

or

 .jx--

c

=k c x--

c

j 1 xc

A distribution of this form is used in the subsequent analysis of leading-edge thrust characteristics.

Figure 11 illustrates how the angle of attack for a

vanishing singularity at a given spanwise station may be

found directly. Singularity parameters in the form

Au4'x v- are shown for the first three elements of both a

cambered wing at ct = 0 ° and a flat wing at ct = 1o. From

previous observations of the nature of cambered and flat

wing velocity distributions, assuming a leading-edge singularity parameter of the following form is reasonable:

X t •

Au '= kI dc-x' +kc 7,/c-x

where the first term on the right-hand side represents a

flat wing contribution and the second term, a pure

camber contribution. Curve fits of the data for the first

two elements using this equation are shown as the dashed

lines. The singularity parameter values at the wing

leading edge are designated as "(Au,,/%-')o f" for the flat

wing and "(Au,4_')o,c" for the cambered wing. It now

becomes clear that the angle for a vanishing singularity

or, in other terms, the angle for zero leading-edge thrust

is simply

-( Au4_')o,c

_zt --

(Au47')o,i

Using relationships developed in reference 2 shows

that the section leading-edge thrust coefficient is

related to the singularity parameter by the following

expression:

c t = -_ -bJtan2Ale+ _ 2 [(Au,4%-')o] 2 (6) 2S

where the positive sign of _]2 applies to subsonic flow

and the negative to supersonic flow.

For the fiat wing,

sin c_

(Au JXT)o (Au4x)°'f si---ff1_

and

7_ b 2 _2 2

(sin ot )2 _Jtan + [(Au_x')o,f] Ct'f = \sin 1°) 2 Ale

For the cambered wing, the theoretical leading-edge

thrust is assumed to act in the same fashion as it does for

the flat wing except that the center point, at which c t is

zero, occurs at tXzt not ct = 0 °. With the definition of Otzt

and Ct, f the section theoretical leading-edge thrust coefficient at any angle of attack that may employ camber may

be found by use of the expression

(sin ot - sin Otzt-_2

ct = ct,f_," sin ! ° )

This method provides a more accurate result than a direct

application of equation (5), which does not recognize the

existence of an angle of attack for zero thrust.

4. Attainable Leading-Edge Thrust and Vortex

Forces

The original method for the estimation of attainable

leading-edge thrust was introduced in reference 1. An

improved method described in reference 7 has recently

been developed that provides for a greater range of airfoil

shapes from very sharp to very blunt leading edges. This

new method is based on a wider range of Reynolds numbers than available for the previous method. An additional aesthetic appeal of the new approach is a

relationship between flow over the wing surface and the

corresponding two-dimensional airfoil analysis more

consistent with actual flow patterns. Because development and use of the new method are covered in great

detail in reference 7, only an outline of the method with

emphasis on the calculation steps is given herein.

The attainable thrust method is based on (1) the use

of simple sweep theory to permit a two-dimensional

analysis, (2) employment of theoretical airfoil codes to

define thrust dependence on section geometric characteristics with pressures limited to a vacuum, (3) generalization of the thrust dependence on limiting pressures to
includemorerealisticallyachievablepressures,and

(4) theexaminationof experimental two-dimensional

airfoildatato definethemorerealisticlimitingpressure

dependenceonlocalMachandReynoldsnumbers.The

applicabilityof themethodwasdemonstratedbycomparisonsof theoreticalandexperimental aerodynamic

characteristics for a series of wing-body configurations.

The relationship between streamwise airfoil sections

and the normal sections used in method development is

illustrated in figure 12. Steps in the solution process may

be outlined as follows.

For each of a large number of wingspan stations, the

following terms define the normal section geometric

characteristics:

Cn = C cOS Ale

't)n _ _ 1 c c cosA:e

1In = 1]

r)n r !

c = c cos2"Ale

(rlc)nrl

ri, n --

('_/C)2n

In addition, for each span station the normal Mach

number, Reynolds number, and normal section thrust

coefficient are defined as

M n = M cos AIe

c

=R n

Rn 6 cos Ale

c 1

Ct, n = Ct

Cn cos2Ale

or

Cav 1

ct, n = c t Cn cos2Ale

with c t nondimensionalized as in AERO2S and

WINGDES. The theoretical thrust coefficient c t is supplied by the lineaxized theory lifting surface analysis.

This coefficient accounts for variations of the theoretical

leading-edge thrust coefficient with such factors as freestream Mach number, wing planform, and wing twist and

camber. Then for each span station, the limiting pressure

coefficient is calculated by use of the following equation:

l ].4 -2 Rn × 10-6

Cp,li m = ._ ---

yM n R n × 10 -6 + K

(7)

where

K = 108(l-M")

e 4 = 0.028Mn 0'75

The equivalent Mach number

Me = _JJ 1 + ( yCp, limi__'-_n2)2-I (8)

TCp, li

which accounts for differences between the limiting pressure coefficient and the vacuum pressure coefficient, is

then used in the calculation of the attainable thrust factor.

The attainable thrust factor K t is found by use of the

following equation:

g t

1 + (x/c)_ 2

- k a Cl,tl_tl

e 3

(T,/C)n(lq/O.5)elqe2 l

with K t limited to values no greater than 1.0 where

(9)

(0.,4i,.o_1.o_

_(1-Me] 0.48(l+ri°3n)

+ °'11 jk )

e I = O_ tu.,+ri, ,,_ . 0.16n -0.7)

e 2 = 1.6riO'l 0-3.0

010

e 3 = 0.32ri," n -0.3

The expression defining the exponent eI differs slightly

from that given in reference 7. The change was made to

remove irregularities in the variation of K t with leadingedge radius that sometimes occur for airfoils with a very

forward location of the maximum thickness and very

small leading-edge radii. For standard and near-standard

airfoils (n = 0.3 to 0.5 and ri = 0.25 to 0.35), this change

has a negligible effect.

10
Theratioof attainabletheoretical thrustK t for each

normal airfoil section is then applied to the theoretical

thrust for the wing at the wing spanwise station from

which the normal section was derived to arrive at a spanwise distribution of attainable thrust. The attainable

thrust section coefficient then is

C t = Ktc t

where c t is the full theoretical leading-edge thrust coefficient defined by the attached flow linearized theory lifting surface solution.

For a given wing and a given set of test or flight conditions, the theoretical leading-edge thrust coefficient c t,

given by a linearized theory code, varies with span position and wing angle of attack. The attainable thrust factor

K t determined by the attainable thrust method varies with

those quantities and with other factors including local

leading-edge sweep angle and wing section characteristics. The effect of wing twist and camber (or flap deflection) is accounted for in the calculated value of c t.

Because the attainable thrust factor is dependent on the

theoretical thrust, it too is influenced by twist or camber

and flap deflection. The same spanwise integration techniques used in the lifting surface code can be employed

to calculate wing attainable thrust coefficients.

As discussed in section 7, it may be desirable for

design purposes to know how much a local leading-edge

deflection angle may be changed from the local flow

alignment condition (presumed to be defined by tXzt) and

still retain attached flow and full theoretical thrust. As

described in reference 7, this angle-of-attack range Actft

may be found by a two-step process. First, Kt is set to a

value of 1.0 (attainable thrust equals full theoretical

thrust) and the corresponding value of the theoretical

thrust coefficient is found. Then the equation relating the

full theoretical thrust coefficient to angle of attack is

applied to a flat wing (CZzt = 0) to define the limiting

angle to give

Alxft = sin-l(sin 1° Ict-_f /

For a flat wing, full theoretical thrust at a particular wing

spanwise location is developed over a range from -Afffft

to Aft.ft. For a wing with twist and camber, full theoretical

leading-edge thrust is presumed to develop over a range

from Otzt - Aotft to Otzt + Ac_ft.

For very thin flat wings with sharp leading edges,

little or none of the theoretical leading-edge thrust can

actually be realized. The real flow about these wings

cannot remain attached to the surface as it negotiates the

turn around the leading edge from a stagnation point on

the wing lower surface. A detached vortex flow pattern,

with embedded circulation, forms and induces an additional lifting force. If the flow reattaches to the wing

upper surface ahead of the trailing edge, the attached

flow potential lift associated with the change in vertical

momentum from just ahead of the leading edge to just

behind the trailing edge is not lost. Thus the wing lift is

increased by the amount of the vortex induced lift. As

discussed in section 7, for conditions which tend to create a vortex flow which does not reattach ahead of the

trailing edge, wing camber or leading-edge flaps can usually be employed to reduce vortex strength and bring

about an earlier reattachment. Polhamus (ref. 11) established a relationship between the normal force induced

by the separated vortex flow and the theoretical leadingedge thrust. According to the Polhamus suction analogy,

the suction vector ct/cos Ale is assumed to rotate to a

position normal to the wing surface, where it affects the

normal force rather than the axial force. Because the

present method treats a partially developed leading-edge

thrust, considering a partial development of the vortex

force seems logical. The simplest approach is to equate

the vortex force with the undeveloped thrust as follows:

Cf -- CI, a

CVO r --

cos Ale

This flow separation always causes an increase in drag.

The increase in lift cannot compensate for the loss of

leading-edge thrust. An ability to predict these changes,

however, is very useful in airplane design trade-off studies. Also, an understanding of the mechanism of flow

separation can be helpful in the search for strategies to

reduce separation and improve performance.

The suction analogy provides no information on the

point of application of the vortex force vector. An

implied assumption is that it acts just behind the leading

edge. Because the vortex flow field can act at locations

which under some conditions may be far removed from

the leading edge, accurate estimates of the vortexinduced normal force, axial force, and pitching moment

can be made only with some knowledge of the location

of the vortex flow field. Both the WlNGDES and the

AERO2S codes offer the following options for the location and distribution of vortex forces.

Option 0: The vortex force is assumed to act perpendicular to the wing reference plane at the wing

leading edge. This option, which provides no contribution of vortex force to the wing axial force, was at

one time the only choice available in the AERO2S

predecessor code SUBAER.

Option 1: For delta wings and delta wing derivatives,

the vortex force center may be located through use of

11
12

anempiricalrelationshipderivedwiththeaidof figtire 13. In this figure, data for uncambered delta

wings with sharp leading edges are used to define, in

an approximate fashion, the lateral location of the

center of the vortex flow pressure field as a function

of the wing local semispan (x cot Ale ) and the angle

of attack. The fairing of the data represented by the

equation

Yvor 1

x cot Ale 1 + t,_ 0t

has been found to be applicable to a range of sweep

angles from about 30 ° to about 80 ° . An approximate

location of the center of the vortex pressure field

may be provided for wings that depart from the delta

planform and for wings which may employ twist and

camber or deflected flaps. The vortex center Xvo r

shown in sketch 4-l may be calculated by use of the

following equations:

Xvo r = 0.0 (O_zt - Ao_ft _< (X < O_zt + Ao_ft )

Xvor fScot l Y °zt

(0t > 0tzt + A_ft )

' - Y [1+ Jtan (IXzt- Aotft- _)1 Xv°r cot Ale

(_X< Otzt - Aotft )

in which Ale is the local leading-edge sweep angle,

Otzt is the wing angle of attack for a local leadingedge thrust of zero, and Ao_ft is the range of angle of

attack for full thrust. This formulation locates the

vortex center aft of the leading edge when full thrust

is not realized. However, it does not account for the

initiation of leading-edge separation at points along

the leading edge other than the apex of the superimposed delta wing. This option, which is the default

for both the WlNGDES and the AERO2S codes, has

been used for the vast majority of the correlations

with experimental data reported in the references.

Option 2: An alternate and very simple means of

locating the vortex force center is given by Lan and

Chang in reference 12. When applied to the present

numerical method, the location of the vortex force

center is

t

Xvo r = c t Car

A1e

--_ x',.r_

Sketch 4-1

For options 1 and 2, the distribution of the force is

assumed to take the form shown in sketch 4-2. Mathematically this form can be expressed as

:<l X--lX or,

Over an element chord x_- x'1, the incremental normal

force due to the vortex is given by

ACN,vor

Car _1

kvorF , X'vor ( . ' sin Xl 1]

: G[(x2-x'l)----_-isln_ x2 -

--It

Xvor X vor ]]

and over the entire interval from 0 to 2Xvo r by

1

ACN,vo r = -- 2kvorXvo r

Cav

so that

ACN,vor Cav

kvo r -

2Xvor

The factor kvo r establishes the magnitude of the

ACp vor distribution action on the wing surface. Because

the surface may be cambered and may include deflected

ACp,vor

i

XVO r

X I

Sketch 4-2
flaps,contributionstoaxialforceaswellasnormalforce

arepresent. If thevortexcenterliesaftofthelocalchord

midpoint,partof thevortexforcedoesnotaffectthe

wingandislost.

5. Aerodynamic Forces and Moments

Section aerodynamic coefficients are found by

integration of the section pressure distributions, for

which the pressure coefficient is assumed to be given by

Cp = 2 Au. Because perturbation velocities are obtained

by superposition of cambered and flat wing solutions, the

pressure coefficient may be expressed as

sin ct

Cp : Cp, c +Cp, s sinl o

or

sin a

Cp = 2 Au c + 2 Au f sin 1°

As shown in sketch 5-1 the pressure acting on the airfoil

camber surface produces an incremental section normal

force given by

where

dc N : Cp ds cos e = Cp dx'

dx'

cos E : --

ds

and an incremental section axial force given by

dc A = Cp ds sin E = -Cp d_.__zdx'

dx'

The section coefficients may be expressed as

C N : -- Cp dx'

Cav

1 _c dz dx' C A = -- Cp dx""-;'

Cav 0

1

C --

m

Cav

c cp (x' + xle- Xmc) ax'

In order to account for leading-edge singularities where

appropriate and to avoid them where not appropriate, the

integrations are performed by parts. The total section

normal-force coefficient (exclusive of thrust or vortex

forces) is given by

sin a

c N : CN, c -t- CN, f sin 1°

dc A

Sketch 5-1

x'

The section normal-force coefficient generated by

the flat wing pressure distribution for a = 1o, CN,f ' is

obtained by the integration depicted in sketch-5-2.

Within the limits of a given element (x I to x_), the pressure distribution is assumed to have the form

Cp,f = 2 Au = 2kf _,-1

with the constant kf defined so that the curve passes

through the Cp value at the element quarter-chord (or the

corrected location for the first two elements). The incremental section normal-force coefficient for this element

is given by the integral

= Cp,f dx' = 2kf -1 dr'

The integration may be performed through use of the following substitutions:

fix_ - = cot

0

1

2

and

dx" = c sin0d0

X'

Cp,f

\

xi

Sketch 5-2

13
with

Theresultis

cos0= 1-2 x--

c

dCN, f = f f(x')f p,f

where

c 0

Ff(x') = _ tan _ (02 - 01 + sin 02 - sin 01)

with

0=

0 2 = COS 1-

cos-' Ii /

The section normal-force coefficient generated by flat

wing pressure distributions for _ = 1° is simply the integral of the incremental coefficients as follows:

= dCN, f dx' CN'f 0

which, as carried out in the code, is merely a summation.

The section normal-force coefficient generated by

the cambered wing pressure distribution c N c is the result of two integrations. First, as indicated in' sketch 5-3,

the cambered wing pressure distribution is separated into

two parts. This separation is accomplished through use of

the angle of attack for a section thrust coefficient of zero

Otzt as follows:

%,c = (%,<)s+(%,c)c

(%,c)f = --O_zt%,f

The integration of the flat plate component (Cp c)e is

handled in the same fashion as was the basic Jflat

wing pressure distribution at o_= 1° previously discussed

to obtain a cambered wing normal-force contribution

(CN,c) e. The integration of the pure camber loading

(Cp,c)c may be explained through the use of sketch 5-4.

Within the limits of a given element, the pressure distribution is assumed to have the form

Cp = 2 Au = 2kc,,/x'(c - x')

Cp,c

(cp,c),,

(Cp,c)l

+

Sketch 5-3

with the constant kc defined so as to pass the curve

through the Cp value at the element quarter-chord (or the

corrected location for the first two elements). The incremental section normal-force coefficient for the element is

given by the following integral:

(dCN'c) c _l

When integrated,

where

Fc(X' ) =

(dCN,c) c = Fc(X" ) (Cp,c) c

2J(l_X_' X-'cjc-2 1-cjc

-c.,x, x: c

+_1.-1(2x2 )l. -1(2x2 )l sm _-1 -,_ sin _--_-1

Then

dCN, c = (dCN,c) c + (dCN,c) f

and

CN, c = _ 1 _ c dCN, c dx"

Cav 0

which is obtained as a simple summation.

14
Ccp,c)c /

/

xl ,_

\

\

\

\

Sketch 5-4

The total section pitching-moment coefficient

(exclusive of thrust or vortex forces) is found in a similar

fashion as follows:

sin

Cm = Cm,c + ¢ra,f sin 1°

1 1 fc dCN,f (x'+ -Xmc ) dx' Cm'f - c 0 Xcg Cav

Cm'c C

Cav

dCN, c (x' + Xcg - Xmc ) dx'

The total section axial-force coefficient (exclusive of

thrust and vortex forces) is given by

sin

CA = CA,c + CA,flc sin 1°

The section axial-force coefficient generated by the flat

wing pressure distribution for a = 1° acting on the

cambered wing surface c A,f/c is obtained by the integration depicted in sketch 5-5. As before within a given element, the pressure distribution is assumed to have the

following form:

Cp = 2Au = 2kf_,-1

The camber surface slope within the element is assumed

to be

d__._zz= kl +k2(x,_x,l)

dx"

with an average or midchord slope of

= k_ +ka(X'2L2x'_) ,z

dx'

Cp,f

\

Sketch 5-5

The incremental section interference coefficient for this

element is given by the integral

tz ,,

dCA,f/c = -f_2 Cp -_-xax

(x'2-x'l)] 4 ----2kf Ikl + k 2 _ _x,

With the exception of the negative sign and the average

slope term, the integral is identical to that for dCN, f.

Therefore

I (x,_x;!l dca,fl c = - k 1 +k2------_-- Ff(x') Cp,f

with Ff(x') as previously defined. Then

CA,f/c = -- If c dCA,fl c dx'

Cav 0

is obtained as a summation. The approximation resulting

from the use of the average slope is acceptable because

element chords can be reduced to make the change in

slope small compared with the average value. Studies

discussed in sections 8 and 9 help to establish element

chords (defined by element aspect ratio) necessary to

provide acceptable accuracy of the numerical solution.

As indicated in sketch 5-6, the section axial force

due to the cambered wing pressures acting on the camber

15
dC_c

dCN,c 1

Sketch 5-6

surfaces dCA,c is calculated from the previously discussed

cambered wing section normal-force coefficient dCN, c as

dz

dCA,c = dCN,c "dx

with dz/dx evaluated at the element midchord to represent an average slope. Then

CAw = -- dCA, c dx'

Cav

is obtained as a summation.

An overall view of the way components of the section force coefficient are combined with thrust and vortex forces is given in figure 14. The cambered wing at its

reference condition (assumed to be ot = 0°) produces a

normal-force coefficient CN, c. At other angles, the normal force is increased by the flat wing loading increment

given by cN,f. sin o/sin 1°. In addition, there may be a

small change in normal force due to a component of the

attainable thrust acting in the normal-force direction

c t sin e o. Another, often larger, increment can result from

the vortex force Cvor that arises when the attainable thrust

is less than the full theoretical thrust.

The cambered wing at zero angle of attack produces

an axial-force coefficient CA, c. An important interference term contributing to axial force at all other angles of

attack is produced by the flat wing loading on the camber

surface. That increment is given as CA,fl c sin o/sin I o.

This term is primarily responsible for the performance

benefits of twisted and cambered wings. Finally, at

angles of attack other than that for zero section thrust,

there is an attainable thrust contribution, c t cos e o.

A sample of code-generated section force coefficients is shown in figure 15. The wing planform used in

the code included the body. The most noticeable breaks

in the general shape of the distributions are explained by

the wing-body juncture at a semispan fraction of 0.127

and a cambered wing spanwise load distribution that was

designed to be constant to the 0.625-semispan station and

linear from there to the tip. The quantities Ct, f and Ctzt

16

are used as described to give theoretical section thrust

coefficients as a function of angle of attack.

As has been discussed in section 5, with the exception of c t and Cvor, the section force coefficients shown

in figure 15 may be obtained by fairly simple operations

involving the angle of attack and the basic section parameters illustrated in figure 14. The attainable thrust and the

vortex force coefficients, although predictable, have no

simply defined dependence on angle of attack. In

employing the attainable thrust method to calculate these

coefficients, separate calculations for each span station at

each angle of attack are required; the results are stored

for subsequent use in the section force coefficient

buildup.

The code techniques for integration of section forces

to obtain overall wing characteristics are very simple.

The section force coefficients c N and c A are assumed to

be constant over the span of a given section. Because the

section force coefficients are nondimensionalized by the

wing average chord, the wing coefficients CN, CA, and

C m for a given angle of attack are determined by a simple summation covering the wing semispan which is then

divided by the semispan to produce coefficients based on

the reference area. Wing lift and drag coefficients are

defined as

CL = C N cos o[- C A sin o_

C D = C N sin ot + C A cos o_

6. Wing Analysis Method for AERO2S Code

The AERO2S computer code uses lifting surface

analysis methods developed in sections 2 through 5 for a

wing which may employ leading- and trailing-edge flaps

in combination with a canard or a horizontal tail surface.

6.1. Wing With Flaps

The analysis of a wing with flaps is constructed from

contributions of four basic lifting surface solutions, a

twisted and cambered surface, a leading-edge flap surface, a trailing-edge flap surface, and a flat surface at

ot = 1°. These components are illustrated in sketch 6-1.

For the analysis of a wing with flaps in combination with

a canard or horizontal tail, a second lifting surface solution with, in general, a different planform is combined

with the first four component surfaces. This second lifting surface does not, however, include leading- or

trailing-edge flaps. Lifting surface solutions for the wing

in combination with a second surface are performed in an

iterative fashion so that the mutual influence of one on

the other is taken into account.
In representationof leading- and trailing-edge flaps

as component surfaces in a solution by superposition, it

is important to conform to the same rules as for the wing

representation. The flap surfaces must be represented by

the same grid system, and the flap surface elements must

also be placed so that inboard and outboard element

chords lie along unit values of the spanwise parameter

13y.The code provides an automated adjustment to input

flap planform definitions to meet this requirement. As

shown in sketch 6-2, flap chords are defined by tabular

inputs which extend from the wing centerline to the

wingtip. New entries in the table are required only when

the chord changes. A discontinuity in chord requires

entries at the closely spaced but not identical span stations. The automated code adjustment alters the flap

planform to provide trapezoidal elements with inboard

and outboard boundaries coincident with unit values of

the spanwise parameter _y as shown in sketch 6-3. The

adjustment preserves the original flap area.

For input flap surfaces, described as a spanwise distribution of flap chord and streamwise deflection angle,

values of surface slope for each of the wing elements are

determined within the code. Solutions for the longitudinal perturbation velocities corresponding to these new

surfaces are performed simultaneously with the solutions

for the fiat and cambered wings. For the surfaces composed of flat panels, an assumption of lifting pressures

proportional to the sine of the deflection angle rather than

the tangent (the surface slope) gave better agreement

with experimental data. For curved camber surfaces.

however, the sine substitution resulted in a tendency to

underestimate the camber drag and overestimate the

wing performance. Thus, for the wing analysis method,

the lifting pressures are defined as

sin ct

Cp = 2 Auf tan l ° (Flat wing)

sin file

Cp = 2 Aule tan _51e - 2 AUle cos 81e (LE flap)

sin _te

Cp = 2 Aute tan _te

- 2 Aute cos _te

Cp = 2 Au c

(TE flap)

(Cambered surface)

Section force coefficients are found through the integration techniques described in section 5. Now, however,

the cambered wing coefficients include deflected flaps as

well as camber surface contributions. Both solutions are

handled as described for the cambered surface except

that, for the flap contribution, additional cos _le and

cos _te terms are introduced. Sketches 6-4 and 6-5 illustrate the necessity for this correction. As shown in

Sketch 6-1

Yle Cle Yte Cte

0 0 0 0

8.0 0 8.0 0

8.1 5.0 8.1 5.2

38.5 2.2 28.0 4.0

28.1 0

38.5 0

Sketch 6-2

I I I I I I I I I

Y

Sketch 6-3

sketch 6-4, an incremental force for the cambered wing

section may be defined as

dc N = dc R cos 8

= _Cpds cos 8

= fCpdX

because dr = ds cos 8.

17
dc A

__ dCN_ /_dc R

Sketch 6-4

But for the flap surface shown in sketch 6-5,

dc N = dc R cos 8

= fCpds cos 8

= fCpdxcos 8

because dx = ds.

These refinements, which introduce a cos 8 term in

both the lifting pressure and the normal-force coefficients for deflected flap surfaces, increase the accuracy

of the code application to flap systems.

Code results for additional flap deflections may be

obtained by the use of input flap deflection multipliers.

Solutions may be obtained for wing surfaces composed

of all possible combinations at leading-edge and trailingedge flap settings provided by the original deflections

and by the flap deflection multipliers. Up to 25 pairs of

leading-edge and trailing-edge flap deflection schedules

may be treated simultaneously. Solutions obtained by

using the multipliers are determined by a perturbation

process and are not as accurate as solutions for the

original or nominal input deflections. When increased

accuracy is required or when the change in performance

with the change in deflection must be evaluated--as in

the construction of suction-parameter contour maps--

individual solutions without recourse to multipliers may

be required.

dc A

Sketch 6-5

18

For additional flap deflection angles, the user may

specify factors role and mte for the leading and trailing

edges, respectively, to produce results for new deflection

angles:

81e = tan -1 (role tall 81e,o )

8te = tan - 1 (rote tan 8te,o )

where the subscript o indicates the original input flap

deflections. These factors act on the complete input spanwise schedule of flap deflections. Thus the magnitude

but not the form of the deflection distribution is changed.

When the variations of element chord and pressure coefficient with cos 8 (for both influencing and field element) are taken into account, the flap-induced pressure

coefficients for the new deflection are

tall 81e,o cos 81e,o

X role cos281e _ tan 81e COS 81e

2

COS 81e,o

ACp,te = 2[(Aute,o + ,_ tan 8te,o COS 8te,o )

x rote c°s2 8re 7t tan 8re cos 8te 1

25 15 COS te,o J

These relationships are found by consideration of

equation (2) which shows the effect of the summation of

the influence of all elements except the field element in

producing the field element pressure coefficient for the

input nominal deflection. For a different flap deflection,

this influence is multiplied by the ratio of deflection

angles role and by the ratio of the cosine of the deflection angles squared to form an estimate of the influence

of all the other elements for the new deflection. The

adjustment is based on the assumption that the new

deflection is constant for a sufficiently large region in the

vicinity of the field point so that the multiplication factor

represents the change in influence of all the other elements to a sufficient degree of accuracy. Then with this

new estimate of the influence of all other elements,

equation (2) is applied again to find the field element

pressure coefficient for the new deflection angle. These

new pressure coefficient distributions are integrated in
thesamewayasthosefor theoriginal deflections to

obtain force and moment coefficients.

The present code provides an improved accounting

of hinge-line singularities in determination of wing

forces and moments. The technique used is illustrated in

figure 16. In the original code, only wings with smooth

camber surfaces were treated, and there was no provision

for flaps. When the code was expanded to cover leadingand trailing-edge flaps, two new surfaces were added, but

the fairing and integration techniques were not changed.

The flap surface loadings were simply added to the existing camber surface component. Thus, the fairing for a

case with leading- and trailing-edge flaps and a camber

surface with z = 0 would appear as shown at the top of

figure. As can be seen, the code fairing is not well suited

to the character of the flap loadings. In spite of this handicap, acceptable results were obtained when a sufficient

number of chordwise elements were used. However,

there is a tendency for part of the singularity loading to

be lost in the integration process. To provide a partial

remedy, the adjustment procedure illustrated in the middle portion of figure 16 is now used. The adjustment to

the camber surface pressure distribution is made only for

contributions of the leading- and trailing-edge flap surfaces. A curve of the form

kh,2

ACp = kh, 1 -1+ _h--X, I

is fitted to the ACp data for a given element and for the

preceding and following elements. The integrated area

under this curve is compared with the integrated area

under a linear fairing, and the difference is represented as

an adjustment to ACp extending over the given element.

When this adjustment is made for all the chordwise elements, a revised distribution, as shown at the bottom of

figure 16, is obtained. The original code integration procedures, when applied to the revised distribution, account

for the lost singularity areas and provide an improved

integration in the calculation forces and moments.

The attention given to the proper handling of theoretical singularities may be questioned because it is wellknown that singularities do not develop in the real flow.

The following discussion, which enlists the aid of

sketches 6-6 and 6-7, can be used to justify this special

treatment.

The actual flow about an airfoil with a deflected

leading-edge flap can be quite different from that

depicted in figure 16. A flow field similar to that shown

in sketch 6-6 would be expected to form. The boundary

layer separates from the surface at the leading edge and

encloses a region of circulation whose strength is related

X v

Sketch 6-6

Acp

Separated

I, _ _ Attached

/ -_ t I

X'

Sketch 6-7

to the theoretical leading-edge thrust. Because the

theoretical thrust for the deflected leading-edge flap is

relatively small (compared with the theoretical thrust for

the flat wing), the vortex force associated with the circulation within the separated region is also relatively small.

The relationship between separated and theoretical

attached flow for efficient flap systems may be further

explored with the aid of sketch 6-7. The dashed line represents the mean camber surface of a flap system airfoil

section and the corresponding theoretical attached flow

lifting-pressure distribution. For the same section in separated flow, most of the force distribution (excluding the

vortex force) is caused by the flow external to the separated region, a flow described by a streamline which

emanates from the wing leading edge and reattaches on

the wing upper surface. The short-dash-long-dash line

represents an effective mean camber surface for the

separated flow and the corresponding lifting-pressure

distribution.

If, as shown at the bottom of sketch 6-7, the separated flow returns to the surface in the vicinity of the

19
leading-edgeflaphingeline,theefficiencyof thesystem

isclosetothatof thefullyattachedflowdepictedatthe

topof thesketch.Thiscanhappenbecausethechangeof

downwardmomentumof theflowovertheflapitselfis

approximatelythesamefor separatedandattachedflow;

therefore,thelifting forceis approximatelythesame.

Thedistributionof thisforceovertheflapis different

fromtheattachedflow,perhapsasshowninthepressure

distributionof thesketch;however,becauseof theuniformslopeoftheflapsurface,theintegratedforceinseparatedandattachedflow is approximatelythe same.

Thus,forreasonablyefficientflapsystems,theattached

flow solutioncanmodelthe forcesgeneratedin the

actualseparatedflow,eventhoughtheattachedandseparatedflowfieldshaveentirelydifferentstructures.This

equivalency, however,iscompleteonlyif allthetheoreticalforce(singularitiesincluded)is takenintoaccount.

Theadditionalcontributionof thevortexforcemaybe

accountedfor byemployment of thePolhamusanalogy.

Severeflow separationi whichtheseparatedflow

doesnotreattachto anyportionof thewingcanbring

aboutextremeperformancepenalties.ThentheAERO2S

codeis no longerapplicable.Whenflow separation

occursonthetrailing-edgeflapafterremainingessentiallyattachedovertherestof thewing,coderesultsare

alsoquestionable. However,asshownlater,trailing-edge

flap flow separationis not nearlyas disastrousas

leading-edgeflapflowseparationthatdoesnotreattach

atoraheadoftheleading-edgeflaphingeline.

6.2. Wing in Combination With Second Surface

Figure 17 depicts the way the present modified code

represents a typical wing-canard configuration. The

inboard and outboard boundaries of the second surface

are made to lie along unit values of the spanwise parameter 13y. The code routine that provides for this adjustment

also changes the second-surface chords to preserve the

surface area (but not the aspect ratio). There may be

some overlap of the canard and the wing in the region of

the wing-body juncture. This overlap and the change in

aspect ratio introduce a small error in the numerical

solution. This error and other errors associated with

discretization are reduced as the number of spanwise

subdivisions is increased. Second surface elements are

numbered in the same manner as those of the wing, and a

code index distinguishes between the two surfaces. The

order of solution still marches from front to rear and

inboard to outboard, and either a wing or a second surface element is taken as its turn comes up.

The computer code has no provision for vertical displacement of the two lifting surfaces, but comparison

of theory and experiment for two canard-wing

configurations with considerable vertical displacement of

the canard above and below the wing (ref. 4) indicated

that this vertical displacement might not impose a significant problem. Further work is required to establish the

applicability and limitations of the AERO2S code to this

type of problem.

6.3. Two-Dimensional Airfoil

As explained in section 4, the effects of Reynolds

number and Mach number on attainable thrust are

derived empirically by use of available two-dimensional

airfoil experimental data. The accuracy of the system

thus depends on the availability and accuracy of these

experimental data. A new set of two-dimensional airfoil

experiments tailored specifically for this problem would

make a significant contribution to the technology. In

particular, a better definition of limiting pressure coefficients at high Reynolds numbers for low Mach numbers

is needed.

The problem in calibrating the attainable thrust

method is to find by iteration values of Cp lim that give

axial-force coefficients fitting the experir'n'ental data as

they break away from the full theoretical thrust curve. To

facilitate any future recalibration of the system, the

AERO2S code has been modified as described in appendix A to provide a two-dimensional airfoil solution. In

obtaining this solution, the values of Cp lim used in the

current calibration may be replaced by _bitrary selected

values. When a chosen XMCPLT value, acting as a multiplier of current values, gives a curve for AC A matching

the experimental data for a range of angle of attack near

the breakaway point, the corresponding limiting pressure

coefficient included in the output data is taken to be the

value for the input Mach number and Reynolds number.

A collection of such data for a large number of airfoils

and test conditions then can be used to generate a revised

curve-fit equation in a manner similar to that used in the

original development.

7. Wing Design Method for WINGDES Code

The design method employed in the WINGDES

computer code is directly dependent on the lifting surface

analysis methods, discussed in sections 2 through 5,

which are applied to a series of candidate wing surfaces.

Considerations of leading-edge flow condition in combination with drag minimization techniques are used to

find an optimum combination of these surfaces. Inclusion of the influence of attainable thrust on the design

dictates that the solution be found by an iterative process.

The process begins with the evaluation of the aerodynamic characteristics of a code input surface. Except

20
forspecialdesignpurposesdiscussedlater,thatsurfaceis

fiat.Theimportantdesigninformationsuppliedbythis

evaluationincludesthe angleof attackat whichthe

designlift is achieved,thespanwisedistributionof the

angleof attackfor zerothrust,andtherangeof angleof

attackforfullthrust.Thisinformationisusedtotailorthe

wingsurfaceintheleading-edgergionto providedistributedleading-edgethrustforceswhichcompensatefor

failuretoachievethefulltheoretical leading-edgethrust.

Becausethischangein thewingsurfacechangesthe

overallwinglift coefficientatthedesigncondition,introducingadditionalincremental wingsurfacesithennecessarytorestorethedesignlift coefficient.TheLagrange

methodof undeterminedmultipliersis usedto find a

combinationof additionalsurfaceswhichproducesthe

necessarylift incrementwith a minimumaxial-force

coefficient.Thetailoringof theleadingedgeplusthe

combinationof additionalsurfacesdefinesa newwing

surfacewhoseaerodynamiccharacteristics maybedeterminedbyreapplicationof evaluationmethods.Thenew

surfacegenerallyhasa differentangleof attackforthe

designlift coefficientandadifferentdistributionof the

angleof attackforzerothrust;thesedifferencesnecessitatearevisedtailoringof theleading-edgesurfaceanda

reviseddefinitionof theadditionalsurfaces.Thusa

solutionby iterationisrequired;thediscussionsin sections7.1through7.4elaborateonthestepstakenin this

process.

7.1. Candidate Surfaces

To provide data for use in the optimization process, the evaluation methods are applied to a series of candidate surfaces to evaluate normal-force, axial-force, and pitching-moment coefficients and interference axial-force coefficients as

well. The candidate surfaces are given in the following table:

Type Surface Defining equation

Input I z (defined by table of coordinates)

Flat 2 z = -(tanl°)x '

General camber

surfaces

Trailing-edge

camber surfaces

Leading-edge

camber surfaces

3

4

5

6

7

8

9

10

11

12

13

14

15

64

Z = ksyey'3(x') ex'l

z = ksyey'3(x') ex'l

. e_, 3. ..ex,l

z = Ksy (x)

z = ksyeV4(x') exl

ev.l (x,)ex,2 z = ksY"

z = ksyey'2(x') ex'2

i ev 3 , ex,2

z = Xsy " (x)

Z = ksyey'a(x') ex'2

z = ksyey'l(x" -c + Cte) ex'te

e

Z = ksY Y'2(x - c + Cte) ex'te

. ey.3 , ex,te

Z = rsy (x -c+cte )

e,

z = ksY Y'4(x' - c + Cte )ex'te

z = (tanl°)x '(1- 5

"_ ICz = _ (tan 1°)% (clc < x')

The first surface is defined by an input TZORD table described in appendix A. For special design problems, the user

may choose to use such a surface. However, for conventional design purposes, a surface with z = 0 everywhere is preferable and this surface is provided by a code default. The second surface is a flat surface at an angle of attack of 1°.

21
Surfaces3to 10affecttheentirewingandarecalledgeneralcambersurfaces. Thecodeusermayselectadesired

numberof thesesurfacesfrom0to8tobetakenintheorderlisted(forinstance,if foursurfacesarecalledfor,surfaces

3to6 areused).Theordercannotbechanged, butotherexponentscanbesubstitutedfor ey,1, ey,2, ey,3, and ey,4 and

for ex,! and ex, 2. The code uses all eight surfaces unless the user chooses otherwise; the code default for the number of

general camber surfaces is 8. Typical general camber surfaces for a delta wing with default exponents are illustrated in

figure 18.

Surfaces 11 to 14 are intended to cover a wing trailing-edge region for special purposes such as design of mission

adaptive surfaces or selection of trailing-edge flap geometry. If desired, these surfaces can be used as additional general

camber surfaces by setting the trailing-edge surface chords equal to the wing chords and selecting a value of ex,te

different from ex, ! and ex, 2 but greater than 1.0. The user may select a desired number of these surfaces up to 4 to be

taken in the order listed. The code default for the number of trailing-edge modifying surfaces is 0. Typical trailing-edge

surfaces are also illustrated in figure 18.

The remaining surfaces serve the purpose of modifying the wing leading-edge region. They are designed to have a

much larger effect on leading-edge surface slope dzldx than any other surfaces (except the fiat surface at ct = 1°) and,

thus, to exert a strong influence on the important design factor, the wing angle of attack for a local leading-edge thrust of

zero. One leading-edge modifying surface is used for each wing spanwise station from root to tip. Each surface has the

specified surface ordinates only for a strip one unit wide centered on that particular station. Everywhere else, the surface

has an ordinate of zero. A typical leading-edge modification surface for the third of seven semispan stations for a

subsonic design Mach number is shown in sketch 7-1. For supersonic speeds, the shape would be similar but the leading

edge would be unswept.

Sketch 7-1

Since the optimization process is critically dependent on these leading-edge surfaces, the user has no option for

reducing the number. The user may, however, select the area to be affected by the leading-edge modification by entering

a tabular schedule of Cle versus span station to replace the code default table which sets Cle at all span stations equal to

the wing root chord. Reduced areas for the leading-edge modification could very well give an optimized wing design

with better performance than that given by the conservative code default. However, very small leading-edge modification areas could lead to erroneous results. Section aerodynamic and geometric data at span stations where fewer than

2 or 3 elements cover the chord of a leading-edge modification surface could be suspect. The number of elements N in a

given chord may be approximated as

Cle (JBYMAX)(ELAR) N=_-_ (Subsonic speeds)

Cle JBYMAX

N - b/2 _ (Supersonic speeds)

Because computational costs tend to increase as the fourth power of JBYMAX and the second power of ELAR, an

increase in the element aspect ratio is the more efficient means of providing for increased definition. At supersonic

speeds the only recourse is to increase JBYMAX.

22
7.2. Influence of Leading-Edge Conditions on

Wing Design

Wing aerodynamic performance is critically dependent on leading-edge flow conditions. If, as depicted in

sketch 7-2, the wing section thickness and leading-edge

radius are large enough to retain attached flow and full

leading-edge thrust for a given set of flight conditions,

there is little need to depart from a flat lifting surface. If,

on the other hand, the wing section is very thin with little

or no possibility for the development of leading-edge

thrust, comparable aerodynamic performance can be

obtained only by shaping the wing camber surface as

shown in sketch 7-3 to distribute the pressures so that as

much of the section lifting force as possible is generated

on the forward portion of the section where a thrust force

can be generated. For wing sections with thickness and

radius which are appreciable but not large enough to generate full thrust at the design condition, a compromise

may be made by introducing just enough camber to

reduce the angle between the upwash vector and the

mean camber surface to a value which permits attached

flow. Such an intermediate solution is depicted in

sketch 7-4. As discussed subsequently, the attainable

thrust prediction method provides the basic information

required in a design process which takes advantage of the

possibilities for thrust generation to reduce the severity

of the design camber surface.

7.3. Selection of Leading-Edge Surfaces

The design process begins with the evaluation of the

aerodynamic characteristics of the code input surface.

Except for special purpose designs, that surface is flat

(the code default surface), and such a surface is used for

illustrative purposes. The input surface is not allowed to

change in the design process, and thus the greatest potential for drag minimization is permitted with an input surface which places no restraints on the design. Among the

information provided by the code evaluation of the input

surface is a spanwise distribution of the range of angle of

attack for full thrust, which might appear as shown in

sketch 7-5. For angles outside this range, the attainable

thrust levels are less than the full theoretical values. The

evaluation of the input surface also provides an estimate

J

Sketch 7-2

Sketch 7-3

Sketch 7-4

Al::tft

--------t

y

b/2

Sketch 7-5

of the angle of attack required to generate the design lift

coefficient which is designated the "design angle of

attack."

The object of the design process is to alter the wing

angle of attack for zero thrust distribution to create a relationship similar to that shown in sketch 7-6 wherein the

upper limits of the range of full thrust for the cambered

wing are coincident with the design angle of attack. This

process gives a design with the mildest camber surface

capable of a theoretical aerodynamic efficiency comparable with the full theoretical leading-edge thrust efficiency. A milder camber surface offers an obvious

advantage in wing structural simplicity. An additional

advantage for supersonic speeds is the reduced real flow

drag penalties associated with camber surface severity

discussed in reference 6. The supersonic empirical correction first devised in reference 6 has now been modified to provide a means of incorporating an estimate of

these penalties in performance predictions.

The code design is carried out by iteration. For any

design iteration, the leading-edge surface weighting

factors are set equal to Otde s - (O_zt + A_ft ). For a flat

23
._ S ctzt + A°q't

/./"

I

y

b12

Sketch 7-6

I_les

input surface with _zt = 0°, leading-edge surface

weighting factors for the first iteration might be as shown

in sketch 7-7.

The resultant surface defined by the addition of the

spanwise distribution of leading-edge surface weighting

factors alters the wing lift and pitching-moment

coefficients. The optimization procedure, described in

section 7.4, is then used to find additional surfaces

Y___

b/2

Sketch 7-7

(general camber surfaces) which restore the wing lift

coefficient to the design value and introduce a moment

increment to approach the design pitching-moment coefficient with the least possible chord force. Because these

general camber surfaces have an influence on the spanwise distribution of the angle of attack for zero thrust and

the design angle of attack, it is necessary to evaluate

these quantities and then find a distribution of incremental leading-edge surface factors to rematch the upper

limit of the range of full thrust with the design angle of

attack.

7.4. Selection of General Camber Surfaces

The Lagrange method of undetermined multipliers is used to define general camber surface weighting factors which

minimize the wing axial-force coefficient while producing a specified increment in normal-force coefficient and, if

desired, a specified increment in pitching-moment coefficient. For the design of an asymmetrical wing, an additional

restraint which eliminates a rolling moment about a specified center of gravity may also be applied. Application of this

method to the problem of selecting an optimum combination of loadings was covered in some detail in reference 10. For

the present application, the following set of equations is used to establish the strength of each of the candidate surface

factors:

i=n

ECA,2iA i + _.NCN,2

i=2

+ _.MCm,2 + _.RAY2 CN, 2 = 0

i=t,1

___CA,niA i +

i=2

i=n

_ CN,iAi +

i=2

i=n

Y Cm,iAi +

i=2

i=n

.dAyiCl,iAi +

i=2

_.NCN,n + _.MCm,n + _.RAYn CN, n = 0

0 + 0 + 0

0 + 0 + 0

0 + 0 + 0

i=N

=- _ CN,iAi

i=n+l

= Cm,des - Cm,pre + Cm,cor -

i=N

= _ Cl,pr e + Cl,co r -

i=N

E Cm,iAi

i=n+l

AYiCN,iA i

i=n+l

24
where

n

N

Cm,pre

Cm,cor

Ayi

Cl, pre

Cl,cor

number of general camber surfaces and trailing-edge camber surfaces

total number of camber surfaces, n + JBYMAX

pitching-moment coefficient evaluated in previous iteration

pitching-moment coefficient correction based on differences between design and realized pitching-moment

coefficients in previous iterations

distance between center of pressure of candidate surface i and wing center of gravity

rolling-moment coefficient evaluated in previous iteration

rolling-moment coefficient correction based on differences between design and realized rolling-moment coefficients in previous iterations

If a rolling-moment restraint for an asymmetrical wing is not applied, the terms in the bottom row and the column

just left of the equal sign are eliminated. If a pitching-moment restraint is not applied, the next to bottom row and the

second column from the equal sign are eliminated. With the surface factors evaluated by standard numerical procedures

for solutions of simultaneous equations (up to 15), surface slopes and pressure distributions of the optimized surface are

found by linear combination as follows:

i=N

(d_)o = Zai(_x)i

i=l

i=N

ACp,o = EAiCp,i

i=1

7.5. Summary of Design Process

As described, the design process is carried out by

cycling through the following steps:

1. Definition of aerodynamic characteristics of wing

surface including spanwise distribution of angle of

attack for zero thrust and design angle of attack

2. Definition of incremental leading-edge surface factors to match upper limit of range of full thrust with

design angle of attack

3. Definition of general camber surface factors to

minimize wing axial-force coefficient while mainraining design lift coefficient and approaching

design pitching-moment coefficient

The iteration is stopped when, from one iteration to the

next, the design angle of attack changes by less than

0.01 ° and the design pitching-moment coefficient

changes by less than 0.001.

Although the code design procedures were developed specifically to take advantage of attainable thrust in

an attempt to define mild camber surfaces which yield

aerodynamic performance comparable with that attainable with full theoretical thrust, the code design features

can be used in other ways. For example, a design which

provides a margin of error in meeting leading-edge alignment conditions for thrust recovery can be achieved by

designing for wing sections with zero thickness and

using a subsequent evaluation with actual wing section

thickness and leading-edge radius to give performance

estimates. Such a design, however, can be expected to

suffer more severe real flow drag penalties at supersonic

speeds than the milder surface design; these penalties

may now be estimated through use of the revised empirical correction.

Two principal goals of the design approach of this

paper, as applied to a sharp leading-edge wing, are the

alignment of the wing leading edge with the local

upwash and the generation of a significant amount of

normal force in the vicinity of the leading edge to create

a distributed thrust to replace the lost concentrated

leading-edge thrust. At first glance, these two goals may

appear to be contradictory. The alignment of the leading

edge with the local flow gives a loading of zero at the

leading edge which, of course, cannot produce the

desired thrust. The saving feature of the design concept is

the rapid curvature of the surface behind the leading edge

due to the optimized combination of candidate surfaces.

This permits the rapid development of thrust producing

loadings immediately behind the leading edge. The handling of leading edges in linearized theory has always

created problems such as theoretically infinite pressures

for flat surfaces and theoretically infinite slopes for wing

design surfaces. However, this is a very localized condition and, except in the immediate vicinity of the singularities, the solutions are reasonable. The failure of

25
numericalmethodsto producethesesingularitiesposes

norealhandicap.

7.6. Evaluation of Design

Although evaluation methods are used to determine

the aerodynamic characteristics of each of the candidate

surfaces, the results for the optimized combination of

surfaces may not provide a true representation of the

aerodynamic efficiency of the wing design. Because the

evaluation data provided in the code design mode are

based on the direct addition of surface ordinates and

aerodynamic coefficients for up to 64 different candidate

surfaces, each of which may introduce numerical calculation errors, an accumulation of errors is a possibility.

To provide a better assessment of the aerodynamic

characteristics of the just completed design which is consistent with evaluation of other wing surfaces, a special

code feature has been provided. Unless the code user

chooses not to use this feature, the many contributing

surfaces are consolidated to provide a single camber surface which in combination with a flat surface at o_= 1o is

used in a standard evaluation. When this code default

option is exercised, the code creates, through interpolation and extrapolation, a table of camber surface ordinates to replace the original input (or code default)

surface and then performs the normal evaluation procedures beginning with the determination of code geometry

information for this new surface.

7.7. User Control of the Design

The code user can find a number of ways in which to

exert an influence on the design beyond the normal selection of design Mach number, Reynolds number, lift coefficient, and if desired, pitching-moment coefficient. As

mentioned previously, the user may exercise some control over the candidate surfaces to be used in the design.

The use of this capability for special purpose designs is

illustrated in one of the code application examples given

in section 12.

The user may also affect the design by overriding the

distribution of leading-edge surface factors provided by

the code. Because these factors are determined by a

numerical iteration process, the design may result in a

wing surface with irregularities in the spanwise variation

of camber surface ordinates. As described later, the user

may substitute a smoothed set of leading-edge surface

factors and redesign the wing to produce a camber surface without irregularities.

The provision for alteration of the distribution of

leading-edge surface factors may also be used for another

purpose. When aerodynamic data for a camber surface

design from the code evaluation mode differ significantly

from the data developed in the design mode, there is a

possibility that the wing performance may be improved

by user control of the leading-edge surface factors. There

are two primary ways in which the design as evaluated

may differ from the design goals. First, as shown in

sketch 7-8, differences may occur between the design

goal and the evaluation in the spanwise distribution of

the angle of attack for zero thrust. This difference is a

measure of the failure to provide the proper relationship

between leading-edge surface slope and the local

upwash. For reasons discussed previously, the evaluation

data must be considered as the more accurate. Any tendency for the design data to underestimate or to overestimate this angle can be compensated for by an adjustment

to the leading-edge surface factors used in the design.

Second, as shown in sketch 7-9, the evaluation data may

give an optimum lift coefficient that does not correspond

to the design lift coefficient. Again the evaluation data

must be regarded as the more accurate and a correction

may be made by an adjustment to the leading-edge surface factors. The following equation has been found to

provide a revised leading-edge surface factor distribution

that offers improved performance most of the time when

either or both of the preceding discrepancies are

significant:

CL,des

Ale,adj - CL,opt (Ale + tXzt,goa1- t_zt,eval)

0t, zt

m

"__ Evaluation

t

y

b/2

Sketch 7-8

Ss

CL,des

', CL,opt

,, ,,

,

I t

I I

I I

I I

I I

cL

Sketch 7-9

26
For the user's convenience, a listing of the leading-edge

factors used in the design and a listing of suggested

values which may lead to improved performance are

provided.

7.8. Flap Design Feature

The design method employed in the WINGDES

code provides for the definition of a wing lifting surface

employing twist and camber which is generally smooth

without discontinuities in surface slope. The method can,

however, be extended to the design of simple hinged

leading- and trailing-edge flap systems. Use of an

attached flow design method for optimization of specialized lifting systems which obviously lead to appreciable

flow separation is justified by the following logic.

If it were possible for thin wings to achieve attached

flow and develop the full theoretical leading-edge thrust,

high levels of aerodynamic efficiency could be achieved

without the use of twist and camber or of flaps of any

type. As depicted in sketch 7-10, a thin flat wing could,

if the flow remained attached to the surface, develop a

lifting-pressure distribution with very high suction pressures in the vicinity of the leading edge (a singularity in

the case of a fiat section with no thickness). The high

pressures acting on the nose of the wing section produce

a leading-edge thrust which counteracts a large portion of

the drag distributed over the remainder of the section.

For two-dimensional flow, the counteraction is complete,

and theoretically the drag disappears. Even for a flat section with no thickness, thrust is theoretically developed

because in the limit, as the thickness approaches zero, the

integral of the pressure acting on the projected frontal

area tends to remain constant.

However, in the real flow there are severe limitations

on the levels of suction pressures that can be achieved.

When the high suction pressures associated with attached

flow cannot be achieved, the flow tends to separate from

the surface, and flow patterns and pressure distributions

such as those shown in sketch 7-11 may result. The

x'

Sketch 7-10

X _

Sketch 7-11

/\

Sketch 7-12

shading represents a separated flow region with an

embedded circulation. Although the actual leading-edge

thrust may disappear, the force associated with the singularity is not lost, but according to the Polhamus analogy

(ref. 11) is redistributed to appear as a normal force

instead of a thrust force. Because the gain in normal

force cannot compensate for the loss in thrust, the aerodynamic efficiency becomes much poorer.

As shown in sketch 7-12, use of a leading-edge flap

can make the problem less severe. The theoretical

attached flow lifting pressures in the vicinity of the leading edge are much reduced. The one singularity at the

leading edge is replaced by two singularities, one of

lesser strength than the original at the leading edge and a

second at the flap hinge line. Thus, a distributed thrust

force replaces the concentrated leading-edge thrust of the

flat wing. For the proper deflection angle, pressures in

the vicinity of the two singularities can be made comparable. This distributed lifting pressure acting on the frontal projected area of the flap produces a theoretical thrust

force approaching that of the concentrated leading-edge

singularity of the flat wing. Because of the generally

reduced pressures required to produce nearly the same

level of theoretical thrust, that thrust is much more likely

to be achieved or approached in the real flow. Within

limits, the required pressure levels for the achievement of

27
a distributed thrust nearly equivalent to that of the flat

wing singularity can be controlled by selection of the flap

chords and deflections. Even with flow separation, the

thrust force is preserved if reattachment takes place at or

ahead of the hinge line and if no hinge-line separation

Occurs.

A trailing-edge flap can also be used to improve real

flow wing performance. As shown in sketch 7-13, a

deflected trailing-edge flap can increase the wing loading

so that the required lift can be generated at a lower angle

of attack. This in turn reduces the required loading and

singularity strength at the leading edge and thus

decreases the chance of separation in the real flow. In a

sense, the trailing-edge flap causes the remainder of the

wing section to act as a large-chord leading-edge flap.

As shown in sketch 7-14, the effects of leading-edge

and trailing-edge flaps can be combined to further reduce

the theoretical pressure peaks and decrease the chances

for real-flow separation. Now there are three singularities

associated with the turning of the flow, and for properly

selected deflection angles, pressures in the vicinity of

singularities can be made comparable in strength. Possibilities for attached flow are further enhanced by a small

leading-edge radius and an effective radius at the hinge

lines (created in part by the boundary layer). As discussed previously, even with flow separation at the

leading edge, good performance of the leading-edge flap

J

x'

Sketch 7-13

X t

Sketch 7-14

may be retained if the separation is sufficiently localized.

Such flow patterns are termed "predominantly attached."

As discussed in section 11.2, flow separation over the

trailing-edge flap area only is not likely to be nearly as

detrimental as separation at the wing leading edge.

An extension of the preceding arguments for the use

of leading- and trailing-edge flaps would lead to multisegmented flaps fore and aft and an approach to the

continuously curved camber surface illustrated in

sketch 7-15. Such a surface, which may be derived from

the WlNGDES code, provides for the complete elimination of singularities and an approach to a uniform pressure distribution which should maximize the possibilities

for a fully attached flow. Therefore, it is reasonable to

use such a wing design solution as a guide in the selection of flap chords and deflections to approximate that

surface, its loading, and its aerodynamic performance.

Use of the flap design feature may be illustrated with

the aid of figure 19. For flap selection, a mission adaptive or restricted area design is performed. Leading-edge

and trailing-edge modification surfaces must be

employed, but no general surfaces are used. The chords

of these surfaces should be input as the chords of the

flaps themselves. The code then designs a restricted area

camber surface for leading- and trailing-edge areas

whose chords are set to 1.5 times the flap chords. The

original camber surface (a flat surface or a milder camber

design such as for supersonic cruise) is then superimposed by rotation and translation on the new design. Differences in leading- and trailing-edge ordinates are then

used to calculate flap deflections which approximate the

designed camber surface. The flap deflections thus

obtained are not necessarily optimum deflections but

only approximations. Iterative use of the AERO2S evaluation code can help provide a better estimate of true optimum deflections. Generally, the code user finds it

necessary to modify the code-generated spanwise deflection schedules to correspond to practical flap segmentation schemes.

jr _

X _

Sketch 7-15

28
7.9. Design in Presence of Interference Flow

Fields

The WlNGDES computer code now permits the

design of a wing lifting surface with flow fields of other

airplane components such as fuselage, nacelles, or

canards taken into account. This design may be accomplished by the addition of a table describing the interference pressure distribution on the wing surface generated

by the other airplane components. This pressure field

enters into the optimization process but, unlike the other

ioadings, remains unchanged throughout the design

process.

The interference pressure field must be supplied by

the user. Normally, these pressures are found by the use

of some other aerodynamic analysis code capable of handling the designated airplane components. Most times,

two computer runs of this other code are required; one

has all the airplane components represented, and the

other has only a mean camber surface that matches as

closely as possible the fixed input camber surface

(surface 1) of the wing design code. The interference

pressure field for the wing design code is then defined as

the difference between these two loadings. By using the

appropriate WlNGDES options, the design surface may

include only the wing outboard of the wing-body juncture or it may include the complete lifting surface, in

which case a new fuselage camber surface is generated.

When an interference pressure field is employed in

the lifting surface design, normal forces generated by

that field influence the wing design and are included in

the listed wing forces and moments for the code design

mode. For the code evaluation mode, these forces are

excluded. Proper values of lift, drag, and suction parameter for the more complete configuration can be found

only if all forces generated by the additional component

are taken into account. In the absence of accurate axialforces as well as normal-force increments, exclusion of

both increments can be expected to give a more realistic

performance estimate than inclusion of one or the other.

A special code option provides for the design of a

reflexed surface in the vicinity of engine nacelles to take

advantage of the nacelle pressure field. This option is

Sketch 7-16

intended for use at supersonic speeds where the influence

of the nacelles is confined to a relatively small area of the

wing near the trailing edge as shown in sketch 7-16. The

reflex design is performed by use of trailing-edge surfaces corresponding to the affected area which must be

specified by the user. When the reflex surface design

option is used, the usual surface ordinates are replaced by

a set more tailored to this particular problem. For this

design, trailing-edge camber surface ordinates are

defined by

ev , ex

Z = ks(Cte) "(x - c + Cte )

The code exponent defaults are as given in the following table:

Surface ey ex

11 1.0 1.5

12 1.0 2.0

13 2.0 1.5

14 2.0 2.0

This definition provides a design reflex surface reasonably well matched to the pressure fields generated by

nacelles with a continual growth in cross-sectional area

from the lip to the last station affecting the wing surface.

With the choice of only four candidate surfaces, a

detailed design cannot be expected. Actually, a detailed

design with more undulations in the surface may be self

defeating if it leads to boundary-layer separation.

7.10. Supersonic Empirical Corrections

A study of the application of the WINGDES code to

the design and analysis of wings for supersonic speeds

reported in reference 6 led to the development of empirical corrections which have now been incorporated in the

code. Extensive comparisons of theory and experiment

for twisted and cambered wings reveal a consistent

pattern in which maximum suction parameter is overestimated by the linearized theory, and the required surface

for given flight conditions is less severe than that given

by the linearized theory. Analysis of data including pressure distributions indicates that an unrealistic theoretical

prediction of the magnitude of the wing upwash field is

the primary cause of the discrepancy and that the use of a

theoretical design lift coefficient less than the desired

operational lift coefficient offers an appropriate correction for linearized theory design methods. A further

analysis of the data led to the development of an empirical method for the selection of the proper design lift

coefficient and for the estimation of achievable aerodynamic performance. The empirical factors derived in

reference 6 are shown in figure 20.

29
Useoftheplotsoffigure20inselectingadesignlift

coefficientandestimatingwingperformancemaybe

summarizedasfollows.First,for theselecteddesign

Machnumber,usethefactorKde s from the top plot to

define a replacement design lift coefficient:

(CL,des)re p = KdesCL,de s

for use in the computer code definition of the lifting surface ordinates and theoretical performance including

(Ss,max)th. Then use K S from the bottom plot to estimate

the maximum suction parameter

Ss 0

-1

/-- Envelope

I I

.1 .2

CL

(Ss,max)em p = Ks(Ss,max)th, b

+ [Ss,th - (Ss,th) b] K_osCL_

that can actually be achieved. The code value of

(Ss,max)th,b used in this expression is taken as the value

at (CL,des)re p even though (CL,des)re p differs from

CL,de s. As illustrated in sketch 7-17 for a wing design

family of various values of CL,de s, there is very little

variation of Ss,ma x with C L. This expression is somewhat different from that used in reference 6. The changes

were made to make the empirical correction more appropriate for wings designed to take advantage of attainable

thrust. Such designs have less severe camber surfaces

and can be expected to have less severe drag penalties

than those given by the original method.

The first change is the application of K S to a suction

parameter that represents only forces due to the basic

pressure loading, not the total forces with attainable

thrust and vortex forces included. The wing designs

employed in wind tunnel tests that supplied data for

establishment of K S did not account for the presence of

thrust or vortex forces and, in fact, prevented the attainment of any significant contribution of these forces at

design conditions. The wing twist and camber were

designed to produce leading-edge surfaces aligned with

the local upwash. For the less severe camber surfaces

that result from inclusion of attainable thrust benefits in

the design process, the drag penalties should be less

severe. The empirical estimate takes this into account by

applying the factor to a suction parameter representing

only the basic pressure forces and by handling thrust and

vortex contributions separately.

The second term in the previous expression represents the contribution of thrust and vortex forces. As

shown in sketch 7-18, which depicts typical code results,

suction parameter increases (drag decreases) due to the

presence of attainable thrust at the code replacement

design lift coefficient. For an idealized design, attainable

thrust is equal to theoretical thrust for a limited range

of lift coefficients up to (CL,des)re p. Beyond that point,

Sketch 7-17

(CL,des)rep

SS

CL,des

I D

I I

CL

Sketch 7-18

Ss, th

(SS, th)b

AS S

theoretical leading-edge thrust begins to be lost (K t less

than 1.0) and vortex forces begin to develop. The net

result is that a maximum theoretical suction parameter is

achieved at or near (CL,des)re p. At the design condition

CL,de s, where the wing with reduced twist and camber

resulting from attainable thrust considerations actually

operates, the drag reduction due to attainable thrust and

vortex force contributions is even greater than at

(CL,des)re p. At the higher CL, the stronger upwash field

results in a larger theoretical leading-edge thrust. The

decreasing attainable thrust factor K t and the increasing

theoretical thrust compensate in such a way that keeps

the suction parameter increment nearly constant. Thus as

shown by the typical code results of sketch 7-18, the drag

reduction of the design lift coefficient can the approximated by assuming the suction parameter increment

Ss,th - (Ss, th)b at (CL,des)re p is applicable at CL,de s as well.

30
Thisrevisedempiricalmethodforestimatingachievable

suctionparametersshouldprovidea betterestimatefor

wingsdesignedtotakeadvantageof attainableleadingedgethrust.Forwingsdesignedwithouthrustconsiderations(leading-edgesurfacesalignedwithlocalupwash),

therevisedmethodgivesresultsthatarelittledifferent

fromthoseobtainedbyuseof theoriginalmethod.

Theempiricallycorrectedragcoefficientat the

designlift coefficientcanbeestimatedby useof the

expression

C a

(CD,des)em p = CD,O + L,des + [l - (Ss,max)emp] nAR

2

X (CL,de s tan CL'des CL'des/ CLc t _AR J

which is derived from a rewriting of the suction parameter definition. The lift-drag polar near the cruise lift coefficient may be approximated as

2

C L

CD'emp = CD'O + rtAR

+ ( 1 - Ss,emp)(C L

t

where for any CL the suction parameter Ss,em p is found

from the curve for the uncorrected code (Ss,th)b for a

value of CL equal to Kde s times the actual C L. The parameter is corrected for the overestimation tendency by subtracting an increment between the theoretical and

empirical suction parameter (assumed to be constant over

a range of CL) and by adding an incremental suction

parameter representing the attainable thrust and vortex

force contribution. In equation form,

Ss,em p = (Ss,th)b rdesCL -- ( 1 -- Ks)(S S max)th b

+ [Ss,th - (Ss,th) b ] Kdo,CL•

As indicated by this expression, for a given CL, both the

suction parameter associated with basic pressure loadings and the increment associated with attainable thrust

and vortex force contributions are found from code

results for a different and lower lift coefficient KdesC L.

This correction as now implemented in the WINGDES

code, for both the design and evaluation mode, incorporates a procedure to identify the optimum lift coefficient

(CL,th)b,op t and the maximum suction parameter

(Ss,max)th,b USed in defining the empirical correction.

This procedure is used to represent as accurately as possible the aerodynamics of the surface being analyzed,

which in the design mode may not represent a fully optimized surface for the specified design conditions.

Sketch 7-19 illustrates a typical application of the

procedure.

When applied to wings designed without consideration of attainable thrust or vortex forces (including the

wings used to supply data for establishment of the empirical corrections), the revised method gives only slightly

higher estimates of performance than the original method

but still has good agreement with experimental data. For

S S

SS'th

i-=-- (Ss'th)blKsCL

.... -__ ......... (Ss max'thb

// / / ", _ Empirical estimate, Ss emp

// ,

i/ / //, :

// ," //

t I

o

c L

Sketch 7- ! 9

31
wingsdesignedtotakeadvantageofattainablethrust,the

revisedmethodpredictsperformancenoticeablygreater

thanthatgivenbytheoriginalmethod.Thisexpectation

is reasonablebecausethemildersurfaceof thatdesign

wouldbeexpectedto sufferlessfromnonlinearpenalties

associatedwithtwistandcamber.However,additional

windtunneldataareneededtovalidatetheapplicationof

thepresentcorrectiontothattypeof design.

Thenatureof thedifferencesbetweentherevised

andoriginalempiricalmethodfor typicalcoderesults

maybeseeninsketch7-20.Thedatashownarefora70°

sweptleading-edgearrowwingdesignedforaseriesof

designlift coefficientsat aMachnumberof 2.05.The

wingsectionshadamaximumthicknessratioof 0.03at

the50-percent-chord station,andastandardleading-edge

radiusindexof 0.3. Dashed-linecurvesshowcoderevisedempiricalestimatesfordesignlift coefficientsof

0.04,0.08,0.12,0.16,and0.24.Thesolid-linecurve

labeled"Revisedmethodenvelope"showsthedependenceof theestimatedsuctionparameter ontheseverity

of thetwistandcamberasdictatedby thedesignlift

coefficient.Fortheoriginalmethod,theenvelopegivesa

constantvalueof about0.4foralldesignlift coefficients

including0.0,whichcorrespondsto a flat wing.But,

codeevaluationresultsfor a flat wing showsuction

parametersat lowlift coefficientsthatareconsiderably

greater.Withtheoriginalmethod,theuserwasadvised

tousediscretionasshownbythefollowingquotefrom

reference6:

"Forsmallvaluesof CL,de s, the factor will

properly define the mild surface required, but

application of the factor K s may underestimate

the performance. The suction parameter for a

properly designed twisted and cambered wing

will not be less than SS for a fiat wing at the

same CL. For a better performance estimate, use

whichever value is greater."

The revised method eliminates the need for user intervention by providing a transition from the results for a

flat wing at and near CL = 0 to the corrected results at

large design lift coefficients. A proper empirical correction should provide such a transition; only the shape of

the curve in the intermediate range of lift coefficients is

in question. The revised method seems to give a reasonable shape for this curve.

If the need to impose moment constraints on an optimum combination of loadings arises, the following

expression can be used to define a replacement design

moment coefficient for use in the design code:

OC m

(Cm'des)rep = fro'des - _)C-_ [CL'des - (CL'des)rep ]

Ss

.6

.5

.4

.3

.2

.1

0

_= 2.05; R = 2.0 x

106

S Revised methodtheond_lOS pe

Flat wing

m

I I I

.2 .3

Sketch 7-20

This strategy provides a design moment at (CL,des)rep,

which in combination with a moment increment due to

the difference between CL,de s and (CL,des)re p yields

the desired Cm,de s at CL,de s.

Although wing design codes provide for the design

of surfaces meeting specified moment constraints, this

capability should be used cautiously for supersonic

cruise vehicles. The imposition of seemingly mild constraints for Cm,de s Can have large detrimental effects on

lifting efficiency.

The empirical corrections outlined in this section

have now been incorporated in the WINGDES code as a

user option. In the design mode, Kde s is used in the selection of a replacement design lift coefficient and K S is

used as a correction in the estimation of realistically

achievable aerodynamic performance. This option may

also be used in the evaluation mode to provide corrected

performance estimates for wings designed by other methods. The correction is applicable to any wing design

based on wing shaping to utilize the upwash field in the

recovery of leading-edge thrust that otherwise would be

lost. When using this correction to aerodynamic performance, remember that the correction is primarily applicable for a narrow range of lift coefficients on either side

of the one where maximum suction parameter is

achieved.

8. AERO2S Computer Code and Its Use

The theoretical concepts and the numerical methods

outlined in sections 2 through 7 have been incorporated

32
Actual planfo_

I

JBYMAX

_ __,_ 8

Sketch 8-1

Actual planform_

..._......-.-_

_/_ JBYMAX

Sketch 8-2

into two computer codes: one primarily for analysis and

the other primarily for design. This section discusses the

analysis code and its use.

The wing analysis computer code AERO2S provides

a subsonic analysis of wing-canard and wing-horizontal

tail combinations that may employ wing leading- and

trailing-edge flaps. The numerical method is based on

linearized theory potential flow solutions for a lifting surface with zero thickness represented by an array of horseshoe vortices. A solution by iteration rather than by

matrix inversion is used. The code also provides for an

estimate of attainable leading-edge thrust and the forces

caused by separated leading-edge vortices. A description

of the code input and output data is given in appendix A.

8.1. Wing Surface Numerical Representation

Effective use of the AERO2S code requires that the

user make an intelligent choice of the numerical model

used to represent the wing lifting surface. The number

and distribution of code elements are controlled by the

entries JBYMAX and ELAR. Figure 21 shows the results

of a study of the sensitivity of the AERO2S code

numerical solution to the wing planform representation

for an aspect ratio 2 wing-body used for theoreticalexperimental correlations to be presented later. In this

figure, drag coefficient at a Mach number of 0.60 and a

represented lift coefficient of 0.30 are shown as a function of the element spacing in the spanwise direction,

JBYMAX, and the element spacing in the chordwise

direction, ELAR. The scale chosen for this presentation

is linear in the inverse of these quantities to enable an

extrapolation of results to those presumably attainable

with a very large number of elements approaching

infinity. For some selected data points, the number of

elements representing the complete wing (right- and lefthand panels) is shown in parentheses. As shown at the

left of the figure, a reasonably stabilized solution is

obtained with JBYMAX of 12 or greater with the element aspect ratio set at 2.0. As shown in sketch 8-1, a

JBYMAX value of 12 gives a much closer approximation to the actual planform of the wing-body than does

either a value of 4 or 8. The planform representation

should be a primary concern in selection of an appropriate JBYMAX value. At the right of the figure, an ELAR

value of 2.0 to 4.0 is seen to be sufficient to give a reasonably converged solution for both the flat and twisted

and cambered configurations.

Additional information on choice of wing numerical

representation was obtained for a cranked-wing supersonic fighter used in a later example of theoreticalexperimental correlation. In a manner similar to that for

the previous configuration, a study of the sensitivity of

the numerical solution to the lifting surface representation was conducted. As shown in figure 22, a JBYMAX

value of 8 and an ELAR value of 4.0 were found to be

sufficient to give a reasonable approach to converged

drag coefficient values for the representative lift coefficient of 0.8 at a Mach number of 0.50. Sketch 8-2 confirms the conclusion that a JBYMAX of at least 8 is

required for planform representation. The selected ELAR

of 4.0 in conjunction with a JBYMAX of 8 places an

average of 4.3 elements between the wing leading edge

and the flap hinge line. Generally two or more elements

should be placed within the smaller flap chords of a

given configuration. A formula given in appendix A

provides a means of estimating the average number of

elements in a given chord.

For wing-body configurations, the code generally

gives better results, particularly in pitching moment,

when the body (or fuselage) is modeled as part of the lifting surface planform. The body camber is believed to be

modeled best by mean camber surface ordinates--the

33
samerepresentationusedforthewing.Thisrepresentationgenerallyrequiresomegraphicalworkto define

stationsandordinatesfor anairfoilsectionto represent

thelifting surfaceinboardof thewing-bodyjuncture.

Themeancambersurfaceordinateissimplythehalfway

pointbetweenthebodyupperandlowersurfaces.This

procedurewasusedin generatingnumerical modelsfor

thetheoretical-experimental correlationsshownlater.

8.2. Vortex Force Options

The vortex force created when the flow about the

wing can no longer remain attached beyond the leading

edge can have a significant impact on aerodynamic performance. The code user has some options for the calculation of this force, which are outlined in appendix A.

The default option (IVOROP = 1) has been found to generally give a vortex force prediction in better agreement

with experimental data than the other two options for

wings with more than moderate sweep. However for

unswept wings and for two-dimensional wing sections,

option 0, in which the force acts at the leading edge, is

the appropriate choice.

The code user can also exercise a degree of control

over the spanwise location of the vortex flow-field origin

which can provide a better modeling of the vortex field

for special planforms such as sweptforward wings. For a

typical sweptback wing with a curved leading edge, the

default option (YAPEX = 0.0) gives a vortex location as

shown in sketch 8-3. As shown, the vortex center (indicated by the dot) moves away from the leading edge as it

progresses from inboard to outboard locations. As can

happen in real flow, for the most outboard station shown,

the vortex center is aft of the wing trailing edge. A vortex

center track, the dashed line, can be formed by connecting the dots. For sweptforward and M-wings, a better

representation of the vortex is given by using the proper

value of YAPEX as shown in sketch 8-4. Because the

vortex center track extends on both sides of the YAPEX

location, this option cannot be used indiscriminatingly.

The AERO2S code has recently been modified to

provide a solution for two-dimensional airfoil sections.

The primary purpose of the modification is to provide

a simplified means of recalibrating the method for

predicting attainable thrust should the need and opportunity arise. The procedures used to conduct such a

recalibration are thoroughly discussed in reference 7 and

are treated briefly in appendix A.

8.3. Optimization Capabilities

The AERO2S code does not provide any direct

design capability; however, information supplied by the

code may be used in an optimization process for the

selection of leading- and trailing-edge flap deflections.

34

YO

YO --

YO --

Sketch 8-3

YO-- _ _

The optimization process involves the construction of

performance contour maps with data from multiple runs

of the code covering a matrix of deflection-angle schedules. For wings which have only a single leading-edge

flap angle and a single trailing-edge flap angle, the

matrix may represent the angles themselves. If the flap

deflections are described by a schedule of flap angles that

vary with spanwise position, the matrix should be composed of multipliers of the tangents of the angles which

are used to create flap-angle schedules. Generally, a new

computer code run should be made each time the angle

(or the angle schedule) is changed. The code provides for

generation of a matrix of data in a single run as outlined

in appendix A. This capability, however, should be used

only for small flap deflections--angles small enough so

that little difference exists between the tangent of the

angle and the sine of the angle. Where there is doubt, the

safe approach is to use multiple runs.
Thefirst stepin theconstructionof performance

contourmapsis thepreparationof plotssuchasthose

shownin sketches8-5and8-6.Suctionparameter S s is

plotted as a function of both leading- and trailing-edge

factors. Data points from the code represented by the

dots are faired to provide a family of curves. These two

plots, in combination rather than a single plot, provide

more data for construction of contour maps and also aid

in the detection of errors which might mar the results.

The dashed-line curve in the sketches connects maximum suction parameter points from each of the faired

curves. The maximum value of the dashed-line curve

indicates the peak suction parameter and the deflection

angle at which it is reached. Sometimes this may be the

only optimization information needed. If however, as is

generally true, there are constraints on the design (for

example, c,n or 0t), contour maps are also required.

Sketch 8-7 illustrates the construction of one contour

line that would appear in a complete map. This sketch

shows, as an example, a contour line for SS = 0.72. The

dots represent points for SS = 0.72, which were taken

from sketches 8-5 and 8-6 at the intersection of each

curve with the line for S s = 0.72. These points alone sufficiently define a closed circuit along which the suction

parameter remains constant at S S = 0.72. In other situations where the dot array is meager, the dashed-line

curves of sketches 8-5 and 8-6 provide additional information to help define the contour. These dashed-line

curves transposed to sketch 8-7 show locations where the

contour line when crossing the location must be parallel

to the plot axes. Other contour lines are constructed in a

similar manner to produce complete maps such as those

shown in section 11.2. These maps permit identification

of the estimated maximum suction parameter and the

required deflection angles. As shown in one of the examples of code application, other contour lines such as

pitching moment or angle of attack can be added to the

plots to provide for the selection of maximum performance subject to appropriate design constraints.

S S

.8

.6

8te factor

1.0

-- .5

S S 0 72

.4

I I

0 l 2

_lefactor

Sketch 8-5

file factor

1.0

.8 .5

- S s = 0.72

S S .6

.4

I I I

0 1 2

_te factor

Sketch 8-6

_lefactor 1

t I

I

x

1 " I

1 2

8te factor

Sketch 8-7

9. WINGDES Computer Code and Its Use

The theoretical concepts and the numerical methods

outlined in sections 2 through 7 have been incorporated

into two computer codes: one primarily for analysis and

the other primarily for design. This section discusses the

design code WINGDES and its use.

The wing-design code WINGDES generates an optimized twisted and cambered lifting surface for a given

wing planform operating at specified flight conditions,

provides the corresponding lifting pressure distribution,

and gives wing force and moment data. The code provides an analysis of the designed surface and may be

operated in an analysis-only mode. Supersonic and subsonic speeds can be handled, but it is not a code for transonic speeds. Because the solution is based on the use of

candidate surfaces, it can provide a twisted and cambered

surface restricted to specified wing regions (a mission

adaptive design) as well as whole-wing design. The code

35
alsoprovidesforthedesignof leading-andtrailing-edge

flapstoapproximateheshapeof missionadaptivecambersurfacedesignsandto approachtheirperformance.

Theactualperformanceof wingswithflapsshouldbe

evaluatedbyuseoftheAERO2Scode,whichisnotlimitedby themildcambersurfaceassumptionsnecessary

forthedesigncode.

Thenumerical methodisbasedonlinearizedtheory

potentialflowsolutionsfor a lifting surfacewithzero

thicknessrepresentedbyanarrayof horseshoevortices.

A solutionbyiterationratherthanbymatrixinversionis

used.Thecodealsoprovidesforanestimateofattainable

leading-edgethrustandof theforcescausedbyseparated

leading-edgevortices.Attainableleading-edgethrust

considerationsplaya directpartin thedesignprocess,

butvortexforceestimatesdonotexceptforareduction

of designlift coefficient(andcambersurfaceseverity)

causedby thevortexlift contribution.A descriptionof

thecodeinputandoutputdataisgiveninappendixB.

9.1. Wing Surface Numerical Representation

As with the AERO2S code, the user must make an

informed choice of the numerical model used to represent the wing lifting surface. For subsonic speeds, the

controlling code entries JBYMAX and ELAR may be

selected according to the guidelines for the AERO2S

code offered in section 8. For supersonic speeds the user

has no control over ELAR which is set to a default value

of 1.0/[3. The remaining control, JBYMAX, must be set

to a value large enough to ensure convergence, which

typically is much larger than that required for subsonic

speeds. Sketch 9-1 shows results of a sensitivity study

for a supersonic transport wing used for theoreticalexperimental correlations in section 11.4. In this sketch,

Co

1.0

.010- u

o

.008 -

[]

.006 -

.004--

.OO2-

0- I

10

10

JBYMAX

.8 .6 .4 .2 0

I I I I I

0 0 0 0 0(30 .........

O [] [] [] or'r1 .........

o Fiat

ta Twisted and cambered

o

O_ O_

I I

20 40 oo

JBYMAX

Sketch 9-1

36

drag coefficients at a Mach number of 2.40 for a representative lift coefficient of 0.12 are shown as a function

of element spacing in the spanwise direction. For both

the flat and the twisted and cambered wing, a reasonable

convergence is obtained for JBYMAX of 25 or more. For

this example, JBYMAX of 40 was used. A safe procedure is to use the code maximum value of 50. The supersonic solution runs quite fast and there usually is little

time penalty for the additional spanwise elements.

The design procedure employed in the code is

intended to provide the mildest possible camber surface

that yields an aerodynamic lifting efficiency comparable

with that of a fiat wing with full theoretical leading-edge

thrust by using to the fullest extent any thrust that may

actually be developed. For such a design, the upper limit

of the range of full thrust is made to coincide with the

design angle of attack. For a more conservative design,

one more comparable with previous design methods such

as reference 10, an alternate approach may be taken. In

this alternate approach, the wing design is performed

with TBTOC and TBROC set to zero for the entire semispan. After the design run, a second run with actual values of TBTOC and TBROC is performed to estimate the

wing performance. For this design, the range of attainable thrust provides a factor of safety on either side of the

design to minimize the effect of failures of the design

procedures to properly match the surface to the upwash

field in the vicinity of the leading edge.

For wing spanwise stations at which the leading edge

is supersonic (13 cot Ale > 1), no theoretical leading-edge

thrust is developed and the range of full thrust is zero. In

the design process, however, the code still makes use of

the calculated O_zt, which for a twisted and cambered

wing is generally not zero. For this situation the value of

O_ztmay be considered to represent the zero leading-edge

loading condition resulting from an alignment of the

leading-edge surface with the local flow ahead of it.

Maintenance of the same flow alignment condition, as

for subsonic leading edges (O_zt+ o_ft= OCdes),was found

to yield a more efficient surface than other strategies and

was adapted for supersonic leading edges also.

9.2. Design Surface Options

For the design code, the body (or fuselage) may be

modeled as part of the lifting surface, a procedure

described in section 11. This modeling generally provides a more accurate pitching-moment prediction for

analysis purposes. However, for design problems, the

highly swept leading edge at the wing apex and the large

root chord can introduce exaggerated wing surfaces that

are not practical for realistic airplanes. Sketch 9-2 illustrates the nature of the design surface obtained by use of

code default options for a wing-body design problem.
Theleading-edgesurfacescoverall theplanformwitha

chordequalto thewingrootchordatall spanstations

(thecodedefault).Generalcambersurfacesalsocover

theentireplanform.Notrailing-edgesurfacesareused.

As shownin thesketch,for a normaldesign,thewing

rootchordhasalargerincidencethanthatof aflatwing

developingthesamelift. Thislargerincidenceoccurs

becausetheleading-edgesurfacechangesrequiredfor

thrustrecoveryalsoresultin a lossof lift thatmustbe

compensated forbyincreasingliftingforceselsewhere.

Thewingcodeprovidesa numberof optionswhich

canbeusedtocontrolthecharacterof thedesignedsurface.Forspecialdesignproblems,theusermaywantto

explorevariousalternatedesignapproaches. Forexample,to reducetheincidenceof thewingrootchordand

thecabinfloorangle,thegeneral designsurfacesmaybe

restrictedto thewingoutboardof thewing-bodyjuncturebyuseof theentryYFUS.Asshownin sketch9-3,

thisresultsin a muchreducedbodyincidence.If the

leading-edgesurfacedefaultchordsareretained,asin

thisexample,thebodycenterlineactuallyhasa slightly

negativelocalangleof attackrelativetothewingreferenceplane.Howeverthelocalangleofattackof thewing

surfacejustoutboardof thewing-bodyjuncturemaybe

largerthanthatof theoriginaldesigntocompensatefor

thelossof lift generatedby thebody.Thisalternate

designprocedurecanbeexpectedtoproduceadragpenalty,andaproblemmayoccurin thematingof thewing

andbody.

Anotherdesignoptionthatmaybeexploredisillustratedinsketch9-4.Forthisdesign,thegeneralsurfaces

arecompletelyeliminatedby settingNGCS= 0, and

trailing-edgesurfacesaresubstitutedto performtheir

functions.Trailing-edgesurfacechordscanbeselected

aszeroinboardatthewing-bodyjunctureandtochange

fromzeroatthatlocationtovaluesapproachingorequalingthelocalwingchordatmoreoutboardstations.Likewise,leading-edgesurfacechordscanbeselectedtobe

zeroinboardof thewing-bodyjunctureandto change

fromzeroatthatpointto largervaluesatmoreoutboard

stations.Asshowninsketch9-4,theleading-edgechords

maybelargerthanthelocalwingchords.In fact,large

leading-edgeareachordsaregenerallydesirable.The

codedefaultgivesvaluesequalto thewingrootchord.

Althoughthesevaluesareprobablylargerthanthosefor

a trueoptimum,theyproducegoodperformanceand

avoidproblemsthatmightoccurwithchordsthataretoo

small.Asshowninthesketch,thisprocedurealtersthe

characterof thedesignsurfaceandoffersanotherdesign

choicewithitsowndragpenalty.

Anotherstrategythecodeusermayemploytocontroltheshapeof thewingsurfaceisuseatableof incrementsforwingtrailing-edgesurfaceordinatestoalterthe

Sketch 9-2

Ybody f

Sketch 9-3

Ccte

Sketch 9-4

lifting surface after the design of the surface and before

evaluation. As described in appendix B, a table of

changes for wing trailing-edge ordinates, TDELZTE,

alters wing airfoil sections to produce a desired trailingedge shape without appreciably affecting the leadingedge thrust recovery design. An example of such a

change in trailing-edge shape is shown in sketch 9-5.

This procedure offers a powerful means of controlling

the wing shape but again with possibly large drag

penalties.

9.3. Mission Adaptive Design

The WINGDES code provides for an automated

design of mission adaptive wing surfaces. In this design

process, only portions of the wing defined by leadingedge and trailing-edge design areas enter into the design.

The rest of the wing remains as defined by the camber

surface input tables. This original camber surface may be

input by the user, may have been retained from a previous design run, or may be the program default (a flat surface). Sketch 9-6 shows a wing-body with designated

leading- and trailing-edge design areas. The numerical

representation of the wing surface (JBYMAX and

ELAR) should be chosen so that a sufficient number of

elements are within the leading-edge surface chords and

the trailing-edge surface chords to adequately define the

sometimes highly curved designed camber surfaces. The

smallest nonzero chords should accommodate two or

more elements. Appendix B gives a formula which may

37
Sketch9-5

cto- 

.,....I

Sketch 9-6

be used to estimate the number of elements in a given

chord. For the mission adaptive design, the number of

general surfaces must be set to zero to prevent camber

surface changes outside the designated design areas.

9.4. Flap System Design

The WINGDES code may also be used in the design

of low-speed flap systems. This option (activated by

setting IFLPDES = 1) uses an automated flap-fitting

technique, which is described in reference 3 and in section 7.8, applied to an appropriate mission adaptive

design. When the flap design feature is used, leadingedge and/or trailing-edge design surfaces must be used,

and NGCS must be set to zero. The chords of these surfaces are input as the chords of the flaps. The code then

designs a restricted area camber surface for leading- and

trailing-edge areas whose chords are set at 1.5 times the

flap chords. The original camber surface (a flat surface or

a milder camber design such as for supersonic cruise) is

then superimposed by rotation and translation on the new

design. Differences in leading- and trailing-edge ordinates are used to calculate flap deflections which approximate the designed camber surface. Note that the flap

deflections thus obtained are not necessarily optimum

deflections but only approximations. The code aerodynamic characteristics do not apply to the configuration

with flaps but to the smooth designed surface from which

the flap geometry was derived. For subsonic cases, use of

the AERO2S code can help provide a better estimate of

true optimum deflections and the flap system aerodynamic characteristics. See table II(e) for an example of

flap design code input.

For the design of mission adaptive surfaces and the

selection of flap geometry, a straightforward application

of the code tends to underemphasize the contribution of

loadings provided by the camber surface or flaps in the

region of the wing trailing edge. This underemphasis

results in only a small penalty in theoretical performance

because the wing leading-edge shape is still proper for

the design conditions. As a practical matter, however,

additional loading of the trailing-edge surfaces reduces

the need for leading-edge camber which introduces drag

penalties not fully covered by the linearized theory. A

recommended procedure that uses trailing-edge camber

and/or flap deflections to increase the theoretical performance and produce a more practical design is as follows:

Perform a whole-wing design for the entire wing

planform at the design lift coefficient, Mach

number, and Reynolds number. Use no moment

restraint if performance alone is the concern but

specify a design moment if trim conditions must

also be considered. This design provides an indication of performance potential and aids in the selection of local design area chords within limitations

imposed by wing structural restraints. Generally,

chords should be as large as structural considerations allow; however, the whole-wing solution

sometimes shows areas where leading-edge design

area or flap chords may be reduced or eliminated.

. Perform a restricted area wing design for the same

flight conditions and for a moment coefficient

Cm,de s at the design lift coefficient as given by the

whole-wing solution. Imposition of the design

moment ensures that adequate consideration is

given to trailing-edge contributions to lifting efficiency. For a mission adaptive design, input the

desired area chords subject to the considerations

discussed in the first step. To perform a flap design,

input a design area chord equal to the actual flap

chord and activate the flap design (FLPDES) feature of the code to provide a spanwise flap deflection schedule.

. For flap designs, examine the code output flap

deflection schedule and modify it as necessary to

meet design restraints such as those imposed by

spanwise segmentation. Also, because the theoretically recommended deflections are only approximations and not true optimums, experience may be

applied in modifying results, particularly in reducing large indicated angles. For subsonic speed,

application of the wing evaluation code AERO2S

to the selected flap system helps in defining more

accurately the optimum deflections and the flap

system performance.

An alternate process for making the design moment

selection is given for the supersonic transport low-speed

flap design in section 12.2.

38
9.5. Design by Iteration

In addition to the preceding strategies, the code user

may make wholesale changes in the wing surface by creating a new set of wing surface ordinates with extensive

modifications evaluated by using the analysis mode of

the WINGDES code. Design by iteration using the analysis mode of the WINGDES code is highly recommended.

The results of the analysis mode, not the design mode,

must be accepted as the better estimate of actual aerodynamic performance. The design method provides candidate designs but sometimes candidate design with serious

deficiencies for practical incorporation into practical airplane configurations. The two modes of the code used in

combination can help the designer evaluate designs in

which aerodynamic performance is compromised as little

as possible to satisfy other restraints.

9.6. Leading-Edge Surface Control

Code-determined weighting factors for the leadingedge modification surfaces are subject to numerical inaccuracies which may produce z ordinates that do not have

a smooth variation with respect to the y dimension. In

addition to the leading-edge surface weighting factors

used in the design, the code also provides a listing of

suggested replacement values that, as described in

reference 5, may lead to improved performance. By

using this option (IAFIX = 1), the user may substitute a

smoothed set of leading-edge surface factors (TAFIX)

for the code-tabulated values. With the present code, two

runs are required: the first finds the nonsmoothed values

and the second operates with the smoothed values.

The table of suggested TAFIX values may sometimes show negative values. Negative values are provided to apply to design cases in which the initial surface

is already twisted and cambered. If this initial surface has

too severe a leading-edge camber for the new design

conditions, it may need to be reduced, and a negative factor is appropriate. For the usual designs (the code default,

for example), occasional negative values should be

disregarded.

When using the suggested replacement leading-edge

surface weighting factors TAFIX, one particular situation requires an additional explanation. For the standard

automated design process which attempts to achieve a

surface with minimum distortion from a flat surface, suggested redesign TAFIX values may take on the appearance shown in sketch 9-7. Strict adherence to the

suggested distribution would produce a surface slope discontinuity that is neither necessary nor desirable. A better distribution is indicated by the dashed line. This

distribution not only provides a smoother surface but also

represents a design with a greater margin of safety. The

suggested TAFIX values shown in sketch 9-7 result from

a wing design with values of O_zt and Ao.ft shown in

sketch 9-8. Inboard of the 30-percent-semispan station,

the range of full thrust Actft is greater than the design

angle of attack, and no camber or twist is required. However, a moderate amount of camber and twist dictated by

the TAFIX distribution shown by the dashed line gives

an equally good design phase aerodynamic efficiency

and provides a design surface that does not suffer as

severe performance penalties if evaluation phase results

do not completely meet the design goals.

9.7. Design With Moment Restraints

For the design of a camber surface covering the

whole-wing planform in which moment restraints are

imposed, a straightforward application of the code yields

a reasonable solution. However, for more than mild

moment restraints a somewhat better performance is

obtained by using the following steps:

1. Perform whole-wing design at design lift coefficient, Mach number, and Reynolds number but

impose no moment restraint.

2. Perform second whole-wing design at same

conditions but with desired moment restraint; for

this case, also impose set of leading-edge surface

I

0 1

y

b12

Sketch 9-7

/

1

y

b/2

Sketch 9-8

39
weightingfactorsTAFIXasdefinedintableofsuggestedvaluesgivenbyfirstdesign(withappropriatesmoothing).

Thisprocesswillproduceamorenearlyoptimumsurface

inthecriticaleading-edgeregion.Therequiredmoment

incrementis suppliedby thegeneralsurfacescovering

theentirewing.Theleading-edgesurfaceis thuscompromisedtoalesserdegreebythemomentrequirement.

9.8. Interference Flow-Field Design

The computer code permits the design of a wing lifting surface with flow fields of other airplane components, such as fuselage, nacelles, or canards, taken into

account. This design may be accomplished by the

addition of a table TCP and associated input data

(appendix B) describing the interference lifting pressure

distribution on the wing surface generated by the other

airplane components. This pressure field and the surface

on which it acts, described by an input table, enter into

the optimization process but, unlike the other surfaces

and loadings, remain unchanged throughout the design.

The interference pressure field must be supplied by

the user; normally, it is found by the use of some other

aerodynamic analysis code capable of handling the

desired airplane components. Most times, two computer

runs of this other code are required: one has all the airplane components represented and the other has only a

mean camber surface that matches as closely as possible

the fixed input camber surface (surface 1) of the wing

design code. The wing design code interference lifting

pressure field is then defined as the difference between

these two loadings. By using the appropriate wing design

code options, the design surface may include only the

wing outboard of the wing-body juncture or may include

the complete lifting surface, in which case a new fuselage camber surface is generated.

The code permits a design of the wing camber surface in the vicinity of nacelles which further optimizes

the lifting efficiency of the wing by taking advantage of

the pressure field created by the presence of the nacelle.

The pressure field must be known and input as an interference flow field TCP and associated input. (See appendix B.) The design is accomplished by employment of

trailing-edge wing surfaces TBTEC and associated input

(appendix B) which represent in general the boundaries

of the interference flow field. Trailing-edge surface

chords which adequately define effective boundaries of

the pressure field must be supplied by the user. This

option is used primarily for the design of supersonic

cruise wing surfaces. In this case, the intersection of an

envelope of Mach cones emanating from the nacelle lip

with a plane representing an average wing camber

4O

surface provides an adequate definition. This option is

activated by input of the index IREFL = 1.

9.9. Empirical Correction

An empirical method for selection of design lift

coefficient and estimation of achievable aerodynamic

performance for supersonic speeds described in

reference 6 and in section 7.10 may be included in the

code calculations by input of the following index:

IEMPCR empirical correction index (this index set to 1

implements empirical corrections; index set to

0, code default, bypasses this feature)

This correction adjusts the design lift coefficient to

account for the tendency of linearized theory to overestimate the magnitude of the upwash field for supersonic

speeds. It also corrects the estimated aerodynamic performance to compensate for the tendency of the theory to

be overly optimistic.

10. Drag Synthesis

Use of the WINGDES and AERO2S codes to predict

aerodynamic performance for complete airplane configurations is based on the assumption that estimates of other

drag contributions may be obtained separately and

combined with estimates of drag due to lift in a simple

additive fashion. The most accurate drag estimate of the

complete configuration would seem to require that all

major drag contributions and possible interactions be

taken into account. However, reasonable estimates of the

drag may often be found without consideration of interactions because as discussed later the interactions are

often negligible or tend to compensate for one another.

Sketch 10-1 depicts a typical lift-drag polar curve

and shows the main contributions to the drag. For efficient flight at a lift coefficient which maximizes the liftdrag ratio, the drag due to lift is about one half the total.

This contribution is estimated by the WINGDES and

AERO2S codes. The codes evaluate the drag by an integration of pressures acting on the lifting surface and a

computational process which accounts for leading-edge

thrust forces. Linearized theory concepts also allow the

drag due to lift to be separated into two fundamental

components: (1) vortex drag associated with the spanwise distribution of the lifting force and the resultant

downwash behind the wing and (2) wave drag due to lift

associated with the longitudinal distribution of lift and

the resultant disturbance waves propagating into the surrounding air. This separation of drag-due-to-lift contributions can be of value in analysis of wing performance and

in the search for optimum designs; however, it is not a

part of the drag breakdown employed in the system discussed herein. A discussion of the use of far field or area
ruleconceptsin designfor dragminimizationandfor

estimatingminimumachievablevaluesof dragisgiven

inappendixC.

TheWINGDESandAERO2Scodesactuallywork

withforcesperpendicular andtangenttoawingreference

planeratherthanlift anddragdirectly.Therelationship

betweentheseforcesisdepictedinsketch10-2.Theuse

of CN and CA rather than CL and CO permits a more elementary separation of the contributions to wing forces.

As shown in subsequent analysis, curves of CA versus

are particularly useful in comparisons of the performance

of a twisted and cambered wing with that of the corresponding flat wing. The drag-due-to-lift contribution is

calculated by the WlNGDES and AERO2S codes for lifting surfaces with zero thickness which may employ twist

and camber and deflected flaps. Wing section thickness

enters into the determination of attainable leading-edge

thrust but is considered to generate no drag attributable to

the generation of lift.

The thickness- or volume-induced drag is considered

separately (not in the WINGDES or AERO2S codes).

This drag arises from thickness-generated pressure fields

acting on the configuration surfaces. It is generally calculated for a configuration without twist and camber at zero

angle of attack. Thickness pressure drag generally arises

only when the speed of sound is exceeded in some part of

the flow field.

For low subsonic speeds, the thickness pressure distributions on airfoils tend to have the form depicted in

sketch 10-3. Pressure distributions on bodies of revolution have a similar form. Because the airfoil generates

pressure disturbances that travel at the speed of sound, its

presence is felt well upstream of the airfoil itself. A stagnation point (zero velocity) at the leading edge is followed immediately by an acceleration of the flow around

the leading edge. The low speeds in the vicinity of the

leading edge create the pressure peak shown in the

sketch. A second stagnation point at the trailing edge

which makes its presence felt well upstream decelerates

the flow to create maximum velocities and minimum

pressures near the airfoil midchord. The distribution of

pressures acting on the airfoil surface creates a thrust

force on the aft portion of the airfoil which counteracts

the drag force created by the forward portion. Thus in

idealized inviscid flow, no thickness pressure drag

occurs at subsonic speeds.

For supersonic speeds, the pressure distributions on

airfoils and bodies of revolution have a completely different character as depicted in sketch 10-4. The airfoil

shape shown here is similar to that shown in sketch 10-3

except for the elimination of any bluntness so that stagnation points are avoided to give a flow that is supersonic

everywhere. Here the pressure change associated with

Co

cp

cp

Drag due to lift

Thickness wave drag 1

Skin friction drag _ CD'O

I

0

CL

Sketch 10-1

Sketch 10-2

CD

jk

Sketch 10-3

Sketch 10-4

41
the airfoil leadingedgecannotpropagateupstream

againstthesupersonicfreestreamto influencetheflow

aheadof theairfoil.Beyondtheairfoilleadingedge,the

flowacceleratestohighervelocitiesandlowerpressures.

In contrastto thatfor subsonicflow,thisacceleration

continuesall thewayto therearof thebodyor airfoil

becausepressuresgeneratedbythetrailing-edgedecelerationcannotpropagateupstream; nowarningof thecoming changeis given.For anairfoilwith foreandaft

symmetry,thepressurejumporshockattherearisequal

to thatatthenose.Forsupersonicflow,noaft surface

thrustisavailabletocounteract theforwardsurfacedrag,

anda substantialthicknesspressuredragis created.

Becausethediscontinuousriseinpressurewhichpropagatesintotheflowfieldsurroundingtheairfoilorbody

resemblesthebowwavecreatedbyaboat,thisdragcontributionisoftencalledwavedrag.

Thethicknessorwavedragcontributionmaybecalculatedbymethodsdescribedin reference13.Thecomplexofcomputingcodesdescribedinreferences14to 17

containsa codeimplementingthe methodof reference13.A newandefficientimplementationof this

methodis availablefromthesamesourceasAERO2S

andWINGDES.

Theskinfrictiondragwhichmustbetakeninto

accountforbothsupersonicandsubsonicspeedsmaybe

calculatedby methodsdescribedinreference18,which

alsoareimplementedinthecomplexofcomputingcodes

ofreferences14to 17.

actingonthebodysurfacewouldleadtoacounteracting

increaseinbodypressuredrag.Thecompensationmight

notbeexact,butonlyacarefulestimateof bothof the

opposingeffectswouldyield a betterestimatethan

neglectingtheinteractions.

Skinfrictiondragandtheassociatedboundarylayer

doesnotnormallyhaveadirectinteractionwithdragdue

to lift. Theprimaryinfluenceof theboundarylayeris

generallyto alterthethickness-induced pressurefields,

which,asjustshown,tendtohaveonlyasmalleffect.

However,undersomecircumstances, thenormally

attachedboundarylayermayseparatefromtheconfigurationsurface.Sucha majorchangein flowconditions

couldhavestrongimplicationsconcerningtheapplicability of predictionmethodsderivedwithanattachedflow

assumption. Themostsevereflow separationtendsto

occurat wing leadingedgesfor wingswithoutflaps

atlargelift coefficientsandfor wingswithdeflected

leading-edgeflapsatlowlift coefficients.Correlationsof

experiment andtheory(refs.3to9)sometimesshowsignificantdiscrepanciesfor theseconditions.But for an

efficientflapsystemwithdeflectionsselectedto minimizeseparation, the predictedperformancegenerally

agreesreasonablywell with the measuredresults.

Subsequent discussionsof somesamplecorrelations

showwhyanattachedflowmethodcanbeapplicableto

detachedflowprovidedthattheseparationis mildand

localized.

Thesimplifiedanalysisandsynthesisof dragrepresentedinsketch10-1excludesinteractionswhichgenerally arenot largeenoughto be thecauseof serious

concern.For example,all theexperimental-theoretical

correlationsofreferences2 to9weremadewithoutconsiderationof possibleinteractionsbetweendragdueto

lift andeitherthicknesspressuredragor skinfriction

drag.Thefollowingdiscussionhelpsto showwhy in

manycasestheseeffectsarenegligible.

Theinteractionbetweendragdueto lift andthicknesspressuredragisgenerallynegligiblefor configurationswith nearverticalsymmetry.Thickness-induced

pressurefieldsonthewingtendtobeofequalmagnitude

fortheupperandlowersurface.Anylossof lift onthe

uppersurfaceiscounteractedbyanalmostequalgainon

thelowersurface.Evenfor arrangements withoutverticalsymmetry,opposingforcesareatworkthatresultin

onlysmallinteractionforces.Considerabodyofrevolutionlocatedbelowaliftingwingsurface.Forabodywith

a continuallyincreasingradius,thethickness-induced

pressurefieldactingonthewingcouldcreateanappreciablelifting forceleadingto asignificantreductionin

dragdueto lift. However,thepositivepressurefield

belowthewing associatedwith thegenerationof lift

11.Examples of Prediction of Aerodynamic

Performance

Extensive correlations of code theoretical results

with wind tunnel experimental results given in references 2 to 9 have provided a validation of the WINGDES

and AERO2S codes. A few correlations are repeated

herein to provide the reader with examples that serve to

illustrate the application of the codes to practical problems of interest. Where necessary, theoretical results

have been updated to reflect the present status of codes

which have undergone continuous improvement.

In the correlation figures, an attached-flow

computer-code solution that includes no leading-edge

thrust forces and no separated leading-edge vortex forces

is shown by the short-dashed line. Code-estimated

forces, which include attainable thrust and the effects of

a separated vortex whose strength is determined by the

Polhamus leading-edge suction analogy and whose location is given by delta wing empirical data (vortex

option 1), are shown by the solid line. For reference, drag

upper and lower bounds are also shown. The theoretical

lower bound CD, 0 + (C2/TtAR) is the drag for a wing

with an elliptical span load distribution and full

42
theoreticaleading-edgethrust.A theoreticalupper

boundCD,O + C L tan(CLICL_ ) is the drag for a flat

wing with no leading-edge thrust and no vortex forces.

The drag at tx = 0 ° for a flat wing CD, o was obtained

from experimental data wherever possible.

11.1 Aspect Ratio 2 Wing-Body

A comparison of code results with experimental data

from reference 19 for both a flat and a twisted and cambered delta wing of aspect ratio 2 in combination with a

simple body of revolution is shown in figure 23. The

wing incorporated a symmetrical 5-percent-thick NACA

0005-63 airfoil section. The twisted and cambered wing

was designed for a trapezoidal spanwise load distribution

at a Mach number of 1.53 and a design lift coefficient of

0.25. The data presented are for a Mach number of 0.61

and a Reynolds number of 3.0 x 106. Input data used in

the AERO2S code analysis are given in table I(a). The

WINGDES code could also have been used for this

example. Theoretical data shown in figure 23 was

obtained for JBYMAX = 12 and ELAR = 4.0.

The drag of a flat wing configuration at oc = 0 °,

CD, o, used in the theoretical analysis was obtained from

the axial-force data presented in figure 23(a). A proper

value of CO, o will reflect the type of flow (laminarturbulent balance) present at lifting conditions. The

experimental C D (and CA) at Ct = 0° of about 0.005

does not provide such a value. Likely a substantially

greater extent of laminar flow is present than at larger lift

coefficients. However, as shown in the axial-force plot,

an appropriate CD, o can be found by fitting the theoretical curve given by the code to the experimental data over

a range of angle of attack near ct = 0 °. The value CD, o =

0.0073 given by this process was used in the theoretical

predictions. See reference 7 for a more complete discussion of the process used to find CD, o from experimental

data and examples of its application in theoreticalexperimental correlations.

In examining the axial force for the flat wing, nearly

full thrust is seen to develop over only a small range of

angle of attack, and only a small portion of the theoretical thrust develops at the largest angles shown. The

present method gives a reasonable estimate of the thrust

actually produced. Because of the failure to produce

thrust, a separated leading-edge vortex would be

expected which in turn would produce a nonlinear

increase in normal force. The normal-force data indicate

that such a vortex force actually is present and is predicted by the present method. The present method is seen

to provide a good estimate of the lift-drag performance of

this wing-body combination. The limits for no thrust and

full thrust provide a broad range of aerodynamic performance possibilities, and thus a reasonably accurate

determination of attainable thrust is a critical part of the

estimation process.

For the twisted and cambered wing (fig. 23(b)), the

axial-force curve is quite different. It is no longer symmetrical and has more negative values of the coefficient

at moderate and large angles of attack. The theory indicates that even without thrust, negative values of axial

force could be achieved. As might be expected, for equal

values of theoretical thrust, the fraction attainable for the

cambered wing is not much different than that for the flat

wing. The experimental increment in axial force at

et = 0° is seen to be larger than the increment predicted

by the code. As for the flat wing, there is a vortex contribution to the normal force. Again the theoretical and

experimental data have good correlation.

11.2. Cranked-Wing Supersonic Fighter

Reference 20 provides subsonic maneuver performance data for a cranked-wing supersonic fighter configuration that employs deflected flaps rather than wing

twist and camber to enhance performance at high lift

coefficients. Data were obtained for a matrix of leadingand trailing-edge flap deflection angles so that maximum

suction parameters and optimum flap settings can be

ascertained and compared with theoretical predictions.

The wind tunnel tests were conducted in the Langley

7- by 10-Foot High-Speed Tunnel at Mach numbers of

0.30, 0.50, and 0.70. The experimental-theoretical correlations presented in the present report are for data gathered at a Mach number of 0.50 and a Reynolds number of

2.9 x 106. The presence of flaps dictates the use of the

AERO2S code, and the AERO2S input data are given in

table I(b).

Figure 24 provides representative data for the longitudinal aerodynamic characteristics for the flap system

depicted in the sketch at the top of the figure with

leading-edge flap deflections of 0°, 15 °, and 30 ° and

trailing-edge flap deflections of 0°, 10°, and 20 °. The

flap deflection pairings shown in figures 24(a), (b),

and (c) were selected to give near optimum performance

(subject to limitations imposed by the constant leadingedge deflection angle measured normal to the flap hinge

line) at lift coefficients of about 0, 0.4, and greater

than 0.8, respectively.

The theory generally predicts the measured performance reasonably well for the undeflected and moderately deflected flaps (figs. 24(a) and (b)). However, for

the severely deflected flaps (fig. 24(c)), noticeable discrepancies appear between experiment and theory for

both high and low lift coefficients. The following discussion helps explain the sources of discrepancies between

theory and experiment that can sometimes arise.

43
In theactualflowaboutflapsystems,theremaybe

seriousdepartures from the idealized flap behavior discussed in section 7 which outlined wing camber surface

and flap system design. These departures may be greater

than those caused by mild leading-edge flow separation,

which permits a predominantly attached flow. An example of the use of axial- and normal-force data in assessing

the nature of departures from attached flow with vortex

separation as calculated by the AERO2S computer code

may be described with the aid of sketches 11-1 and 11-2.

These data represent a wing-flap system that is operating

in a predominantly attached-flow manner for the middle

portion of the range of angle of attack shown. However,

more severe flow separation occurs at higher and lower

angles of attack. At an angle of attack large enough to

cause the separated flow on the upper surface to reattach

aft of the leading-edge flap hinge line but ahead of the

trailing edge, a decrease in distributed thrust on the

leading-edge flap surface and an increase in axial force

CA

\

Sketch 11-1

CN

I

./

/

Sketch 11-2

occur; however, there is little or no loss of normal force.

At angles of attack large enough to prevent reattachment

ahead of the trailing edge for much of the wing, a loss of

normal force as well as an increase in axial force occurs.

These changes bring about drastic losses in performance.

At an angle of attack sufficiently low to cause a separation on the lower surface that originates at the wing leading edge and reattaches ahead of the wailing edge, axial

force is reduced but normal force has little or no change.

For this situation, performance with separated flow may

be better than theoretical performance with attached

flow. At even lower angles of attack, the separated flow

may not reattach ahead of the trailing edge and the normal force is more positive than for attached flow. In the

numerous examples of theoretical and experimental data

correlation given in references 3, 4, and 9, there is only

limited evidence (primarily the present case) of separation at high angle of attack for reasonably efficient flap

systems at lift coefficients below about 0.8. An analysis

conducted in reference 3 indicated that _le n -- 20° and

_te, n = 15° would be more nearly opfi'mum for the

C L = 0.8 design condition. This example with

_le, n = 30° and 8te,n = 20 ° was chosen specifically to

demonstrate the nature and causes of discrepancies that

can occur. The collection of theoretical-experimental

correlations in the references showed many examples of

separation at low angles of attack for highly deflected

flaps.

The data in reference 20 provide sufficient information for the construction of an experimental performance

map for comparison with an analysis code performance

map as shown in figure 25. The contours represent suction parameters for a lift coefficient of 0.45. The code

predicts reasonably well the peak suction parameter and

the deflection angle at which it occurs. Both leading- and

trailing-edge deflections are required for peak performance. At the larger leading-edge flap deflection angles,

the experiment shows poorer performance, perhaps

because of hinge-line separation. Also, a poorer measured performance occurs for the wing with undeflected

flaps than is predicted by the code. This discrepancy may

be caused by, at least in part, a failure of the leading-edge

separated flow to reattach ahead of the wing trailing

edge, particularly on the wing outer panel. The important

point, however, is that at C L = 0.45 the flap system

performance almost matches theoretical expectations;

this indicates that the flow there is predominantly

attached.

The behavior of the contour map data for this wing

points out a difference in the effects on performance of

separated flow in the leading-edge flap region and

separated flow in the trailing-edge flap region. Flow

separation from the trailing-edge flap surface is much

less likely to cause performance penalties than flow

44
separationfromtheleading-edgeflapsurface;thiscanbe

illustrated through the use of sketches. Sketch t 1-3

shows the relationship between lift and drag changes on

the wing as a whole to be the result of a change in the

trailing-edge flap loading. If changes in flow conditions

such as those induced by changes in Reynolds number

cause loading changes restricted to the flap itself, incremental changes in lift and drag are related according to

the expression

dC D

dC L

- tan (a + _te, s)

These considerations cause a loss in flap loading due to

separation to bring about a loss in lift, which is accompanied by a decrease in drag. The net result is that the liftdrag ratio for an optimally deflected flap at a given lift

coefficient is changed very little. An illustration of the

effect of trailing-edge flap separation for the present configuration is given in sketch 11-4. The arrows show the

direction of the relative change in lift and drag caused by

a loss in loading on the trailing-edge flap itself. The

hatched area indicates the magnitude of the change if

50 percent of the theoretical loading is lost. With no

leading-edge flap deflection, the trailing-edge flap

deflection of 20 ° is optimum for a C L- 0.8. At this

,/Co

Sketch 11-3

.20 -

81e,n = 0° _te, n = 20°

CD

.16 -

.12 -

.08 -

.04 B

0 B

Sketch 11-4

Sketch 11-5

CD

.20 -

.16 -

.12 -

.08 -

.04

0-

_ile,n = 30° _te, n = 0°

:ii!ii_

Sketch 11-6

condition, changes in lift and drag tend to occur along a

tangent to the polar curve; thus, little or no performance

penalty occurs. Actually, penalties (drag increases) occur

only for the lift coefficients in excess of 0.8. For lift coefficients less than that at which the deflection is optimum,

the separation brings about a decrease in drag, an effect

noted in the experimental data. This drag reduction is relative to the excessive drag of a surface deflected beyond

the optimum for that lift coefficient. Separation would

not be expected to lead to a drag lower than that of an

optimally deflected surface.

The relationship between lift and drag changes on

the wing as a whole because of a change in leading-edge

flap loading alone, illustrated in sketch 11-5, may be

expressed as

dC D

- tan (et - 81e,s )

dC L

Because of the negative sign, a loss in lift coefficient

caused by a reduction of leading-edge flap loading is

generally accompanied by an increase in drag. When

applied to the present example, changes such as those

shown in sketch 11-6 result. The arrows show the

45
relativechangein lift anddragcausedbyalossinloadingontheleading-edgeflapitself,andthehatchedarea

indicatesthemagnitudeofthechangeif 50percentofthe

theoreticalloadingis lost.Obviously,severepenalties

areassociatedwiththefailureoftheleading-edgeflapto

producetheanticipatedloading.

11.3. Subsonic Transport

The data of reference 21 provide an opportunity to

compare AERO2S code results with measured data for a

highly efficient subsonic transport configuration. Generally little need exists for application of the attainable

thrust prediction method to such a design as this which is

expected to develop almost all the theoretical leadingedge thrust. The example, however, serves to demonstrate the applicability of the method to a continuum of

designs from ones certain to generate extensive separated

flow as in the previous example to ones almost certain to

maintain fully attached flow.

Data for the subsonic transport wing-body tested

at a Mach number of 0.55 and a Reynolds number of

2.25 x 106 is shown in figure 26(a). AERO2S code input

data are listed in table I(c). The WINGDES code also is

capable of handling this example. Generally theory and

experiment have good agreement for all data shown

except pitching moment. The axial-force data follow a

curve corresponding to full theoretical leading-edge

thrust over the whole range of angle of attack shown.

Some of the pitching-moment discrepancy may be attributable to a rather large strut used to mount the model.

Data for a more complete configuration with a horizontal

and vertical tail are shown in figure 26(b). The only difference in the theory shown in the two parts of the figure

is because of an estimated increase in CO, o of 0.0042

caused by addition of the tail surfaces.

11.4. Supersonic Transport Wing

Data from reference 22 for a supersonic transport

wing tested at a Mach number of 2.40 and a Reynolds

number of 3.4 x 106 are shown in figure 27. Results for a

fiat wing in figure 27(a) may be compared with results

for a twisted and cambered wing designed for a Mach

number of 2.4 and a lift coefficient of 0.08 without constraints on leading-edge pressures. The selected design

lift coefficient was chosen through use of a rule of thumb

in an effort to maximize performance at a cruise C L of

0.12. The supersonic test Mach number requires that the

theory be obtained through the use of the WINGDES

code (in its evaluation mode) rather than the AERO2S

code which was used for the previous examples. Code

input data are given in table II(a).

As shown in figure 27(a), for the flat wing there is a

good prediction of all the aerodynamic characteristics,

46

even pitching moment. For the twisted and cambered

wing (fig. 27(b)) the prediction is still good. But as

shown in the suction parameter plot, employment of the

empirical estimate feature of the code (which is

described in detail in section 9.9) brings about a slight

improvement in prediction of the lifting efficiency. Note

that the suction parameter reaches a maximum near the

intended cruise lift coefficient of 0.12.

11.5. Supersonic Transport Wing-BodyHorizontal Tail

The application of the computer code to a highly

swept arrow wing supersonic transport with a horizontal

tail may be examined with the aid of figure 28. Experimental results for this configuration were obtained from

reference 23. The wing is twisted and cambered for

supersonic cruise at a Mach number of 2.20 (wing W 2 of

ref. 24) and is equipped with leading- and trailing-edge

flaps. The leading-edge flap is full span and is broken

into six segments. The trailing-edge flap system is composed of inboard and outboard segments of single-slotted

flaps (ref. 24). AERO2S code input data are given in

table I(d). The tests were run in the Langley 30- by

60-Foot Tunnel at M = 0.09 and R = 4.19 × 106 (based

on C).

The data shown in figure 28(a) are for undeflected

leading- and trailing-edge flaps. The horizontal tail

deflection is -5 ° which was found to be a near optimum

setting for drag minimization. The theory gives a good

prediction of aerodynamic characteristics with the exception of pitching moment for lift coefficients greater than

0.04. The axial-force calculations show the effect of the

attainable leading-edge thrust, and the normal-force calculations show the effect of the vortex force. The mildness of the wing twist and camber is shown by the slight

slope in the axial force without attainable leading-edge

thrust and vortex force.

The data for deflected leading- and trailing-edge

flaps in figure 28(b) show a considerable increase in lifting efficiency for lift coefficients of about 0.5 to 0.7 (suction parameters of 0.65 or more). The drag and suction

parameter are well predicted in spite of some discrepancies in axial- and normal-force correlation. Again, pitching moment is not well predicted. Reference 4 provides a

more detailed study of the prediction of the aerodynamic

performance of this and other configurations with two

lifting surfaces.

11.6. Two-Dimensional Airfoil

The AERO2S code has recently been modified to

provide a solution for two-dimensional airfoil sections.

The primary purpose of the modification is to provide

a simplified means of recalibrating the method for
CA

-.2

-.4

-.6

-.8

Sketch 11-7

predicting attainable thrust should the need and opportunity arise. The procedures used to conduct such a recalibration are thoroughly discussed in reference 7 and are

treated briefly in appendix A.

To illustrate the use of this code feature, an NACA

4409 airfoil has been subjected to a code analysis. The

airfoil description and test data are taken from reference 25. In the development of the attainable thrust

method only symmetrical airfoils were employed, and

that restriction is still recommended. However for this

example, an airfoil with camber was selected to show

that, if necessary, such airfoil data are also applicable.

The code input data are presented in table I(e), and code

results are compared with experimental data in

sketch 11-7. As shown in the sketch, code results were

obtained with values of XMCPLT of 1.0, 0.8, and 0.6.

The object of this exercise is to find the limiting pressure

curve which best matches the experimental data in the

vicinity of the breakaway from the full thrust curve. For

this example Cp,li m = --8.5 becomes a candidate to

replace Cp,li m = --11.7 now used in the attainable thrust

method for a Mach number of 0.06 and a Reynolds number of 8.0 x 106. The section camber causes the slope in

the no-thrust axial-force curve. This slope results in some

additional uncertainty regarding the curve fit and the

selection of the appropriate limiting pressure to match

the experimental data. A collection of data such as these

covering various airfoils and a wide range of Mach

numbers and Reynolds numbers could provide a new and

hopefully more accurate calibration of the attainable

thrust system.

12. Examples of Design for Aerodynamic

Performance Optimization

In section 11, which dealt with prediction of aerodynamic performance, the AERO2S code was employed

for all examples except those for supersonic Mach numbers. The design examples treated in this section require

use of the WINGDES code. Only for one example, which

requires evaluation of a previously designed low-speed

flap system, is the AERO2S code employed.

12.1. Supersonic Transport Cruise Surface

Design

As an example of the use of the WINGDES code in

the design mode, the supersonic transport configuration

shown in figure 29 has been subjected to a multistep

design process. This configuration was derived from the

SST wing treated in section 11.4. The process begins

with a standard wing design for a supersonic cruise Mach

number of 2.40 and a lift coefficient of 0.12. This step is

followed by a design of a wing reflex surface to take

advantage of the nacelle pressure field at the supersonic

cruise Mach number. Next, in section 12.2 the reflexed

wing surface designed for supersonic cruise is subjected

to a mission adaptive (or restricted area) design to define

a candidate flap system for a subsonic design point,

M= 0.30 and C L = 0.6. Finally the AERO2S code is

47
employedtocreateaperformancecontourmapshowing

the effectof variationof flap settingson subsonic

performance.

Asafirststep,thewing-bodyplanformof figure29

wassubjectedto a designfor acruiseMachnumberof

2.40 to optimizethe aerodynamicperformancefor

CZ,de s = 0.12 with no moment restraint. The WINGDES

code input is shown in table II(b). This design is characterized as a "whole-wing design." The design starts with

a flat wing (z = 0 everywhere), and all regions of the

planform are free to be reshaped as necessary. Much of

the required code input data such as the initial camber

surface and the selection of candidate design surfaces is

covered by code defaults. The supersonic empirical

design corrections described in reference 6 are also

implemented as part of the default.

The critical parameters of the design are shown in

figure 30. At the top of the figure is shown a plot of angle

of attack as a function of spanwise position. (The plot

shows 37 spanwise locations rather than 40, as might be

expected, because the input JBYMAX of 40 was reduced

by a code automatic feature to keep the total number of

elements within code limits.) As explained in section 7,

the purpose of the design is to match the upper limit of

the range of full thrust (_zt + At/ft) with the angle of

attack corresponding to the design lift coefficient. In this

way, theoretical leading-edge thrust that is not realized as

attainable leading-edge thrust is recovered as a distributed thrust on the wing camber surface, primarily in the

region just behind the leading edge. For this supersonic

design, which employs an empirical correction discussed

in section 9.9, (CL,des)re p is given by KdesCL,des with

Kde s = 0.57, and the angle of attack for the replacement

design lift coefficient is about 2.7 °. The plot shows that

the design matchup has essentially been achieved. The

large spread of the range of full thrust for the inboard stations simply shows that little or no modification of the

input surface is required there. The suction parameter

plot (middle of fig. 30), however, indicates that the performance at the cruise lift coefficient could be improved

to a slight degree by a redesign which would place the

maximum S s closer to CL,de s. Actually for this design

problem, retaining the first design would be advisable

because of the advantage of its relatively mild camber

surface and the very slight gain a more severe surface

might produce in practice. However, to illustrate the use

of the adjustment procedure for cases where it may be

required, a revised design has been undertaken. The plot

of leading-edge surface weighting factors (bottom of

fig. 30), given as a part of the code output, shows the factors used in the design and also shows revised values that

should lead to improved performance. Because there are

inherent inaccuracies in the numerical solution, and

because the code-recommended values would lead to

48

surface irregularities, a fairing of these data is used

instead. The fairing as shown by the dashed line was

used in a redesign with the data of table II(c) being substituted for the data of table II(b). For this rerun, codecalculated leading-edge surface factors are replaced by

those given in the TAFIX table; otherwise the design

process is the same. The results of the revised design are

shown in figure 31. Figure 31(a) shows the new design

parameters and figure 31 (b) shows the wing camber surface generated by the code. The peak suction parameter

shown in figure 31(a) has not been increased, but it does

occur closer to the cruise lift coefficient. The computergenerated plots of the wing surface show a relatively

small amount of twist and camber.

The wing design may also incorporate a reflexed

trailing-edge surface to take into account the presence of

an interference pressure field induced by nacelles or

other components which are not considered as variables

in the design process. To implement this option, the user

must supply the pressure field information and specify an

index (ICP = 1) to indicate that an interference pressure

field is to be taken into account. The user must also

specify wing trailing-edge design surfaces corresponding

in general to the areas of the wing influenced by the pressure field. An example of the code input for such a

design is given in table II(d). The only differences

between this table and table II(b) are the entries required

to substitute a design incorporating reflexing for a standard design. Results for this design, shown in figure 32,

are similar to results for the initial whole-wing design

shown in figure 30. When the Ale fairing of figure 32 is

employed in a redesign, the peak suction parameter given

by the empirical estimate occurs close to the cruise lift

coefficient as shown in sketch 12-1. At the cruise lift

coefficient of 0.12, the corrected suction parameter given

by the code is about 0.34, which is less than the value of

.6

.5

.4

Ss .3

.2

.1

0

m

m

m

0

(CL,des)rep

I

,' CL,de s

, Theory

I

irical

/ ; i

I I

I I

I

.04 .08 .12 .16

cL

Sketch 12-1
0.37forthepreviousdesignwhichdidnotincludethe

interferencepressurefield.Thecodeevaluationof the

suctionparameterdoesnotincludethebeneficialeffect

oftheincreaseinlift causedbythepressurefield.If the

increaseinlift hadbeentakenintoconsideration, thesuctionparameterwouldhavebeengreaterthan0.40,an

obviouslyoptimisticvalue.Propervaluesof lift, drag,

andsuctionparameterforthemorecompleteconfigurationcanbefoundonlyif all forcesgeneratedbytheadditionalcomponent aretakenintoaccount.In theabsence

of accurateaxial-forceaswell as normal-forceincrements,exclusionof bothincrementscanbeexpectedto

giveamorerealisticperformanceestimatethaninclusion

of oneor theother.A pictorialrepresentationf the

reflexedwingsurfacedesignisgiveninfigure33.

12.2. Supersonic Transport Low-Speed Flap

Design

An attractive feature of the WINGDES code is the

capability for performing runs in succession. For example, a supersonic design run could be followed by an

evaluation of that surface at other Mach numbers----either

supersonic or subsonic--or a second design run. An

example of a supersonic surface design with nacelle

reflexing followed by a subsonic design to determine a

mission adaptive surface and flap deflections to approximate that surface is given. Table II(e) shows the code

input for the supersonic reflex surface design treated earlier but now with provision for a subsequent run for the

design of a flap system for a subsonic Math number of

0.30 and a lift coefficient of 0.6. Note the quantities that

must be reset when consecutive design runs are performed as specified in appendix B. The supersonic

design input of table II(e) is the same as that of table II(d)

with the exception of the TAFIX table addition, which is

very similar to the TAFIX table used in the whole-wing

design.

For the subsonic mission adaptive surface design,

Cm,de s = -0.24 is imposed. For mission adaptive designs,

it is always advisable to employ a Cm,de s that has been

selected carefully. Applications of the mission adaptive

design option have shown that the numerical solutions

tend to call for greater use of leading-edge surfaces and

less use of trailing-edge surfaces than would a true optimum design. The problem does not arise to any appreciable extent for the whole-wing design but does affect

results for the mission adaptive design, where relatively

large surface slopes are needed to generate the required

ioadings on restricted areas. An improved design can be

found by running the mission adaptive design code for a

selected series of design pitching-moment coefficients

and using a plot such as that shown in sketch 12-2. The

unrestrained design provides a suction parameter of

].0 --

SS .5

Cm from

whole-wing design

Cm = -0.24

I

No restraint

I I

I i

_1 I , I ,, I

0 -.1 -.2 -.3

Cm,des

Sketch 12-2

about 0.80 and fro,de s = -0.15, whereas an optimum suction parameter of about 0.87 occurs for Cm,de s = -0.24.

This large negative moment might present a trim drag

problem in an airplane design project. When horizontal

tail contributions or canard trim contributions are considered in the definition of a desired wing moment coefficient and when that moment is specified as a wing design

code input, a better overall design should result. Then the

previously described search for optimum performance of

the wing alone is avoided. An alternate selection of a

design pitching moment can be obtained by using C m

generated by a whole-wing design (with either a flat surface or the supersonic cruise surface as an input) at

CL,de s. This procedure gives Cm,de s = -0.26.

For this design, Cm,de s = -0.24 was used. Results of

the subsonic mission adaptive surface design are shown

in figure 34. These data show the leading-edge tx

matchup, an indicated maximum suction parameter

of 0.88, and the leading-edge factors as used and as suggested for redesign. This pair of back-to-back runs was

repeated with a TAFIX table for the subsonic design as

represented by the dashed line. For this example, the

selected fairing of the leading-edge surface weighting

factors is only slightly greater than the as-used values

because the leading-edge matchup conditions were

almost met and because the suction parameter at CL,de s is

only slightly less than the maximum. Wherever possible,

design surfaces more severe than necessary should be

avoided. Results of this repeat run are shown in

figure 35. As shown in figure 35(a), the leading-edge et

matchup and the suction parameters are changed only

slightly from figure 34. Note the design fails to completely match the design angle of attack at inboard span

positions near the wing-body juncture. This problem is

associated with the small number of spanwise elements.

49
As shown in sketch 12-3, a new design doubling the

spanwise elements (JBYMAX = 20) gives a much better

matchup in the region of the wing-body juncture. However, with present code limitations, this increase in spanwise representation can be accommodated only with a

decrease in element aspect ratio and a corresponding

decrease in chordwise shape representation. When technology advances permit, an increase in the number of

elements the code can handle would be highly recommended. The computer-generated plot of figure 35(b)

shows a mission adaptive design surface much more

severe than the mild supersonic cruise surface. At the

bottom of figure 35(a) is shown the leading- and trailingedge flap deflections provided by the code to approximate the designed mission adaptive surface. The solid

line represents selected flap segmentation and deflection

schedules for use at the subsonic design conditions.

The final step in this exercise involving the SST

design is an evaluation of the performance of the flap

system at the subsonic design conditions. For this purpose, the AERO2S code must be used. An example of

the required input is shown in table I(f). Results of the

flap system evaluation for the nominal flap deflection

schedules of figure 35 are shown in figure 36. As shown

at the top of the figure, the upper limit of the range of full

thrust falls short of the angle of attack for CL,des = 0.6 by

5° or more; this is in contrast to the matchup for the mission adaptive surface. This difference is to be expected

because the flaps are only a poor approximation of the

camber surface. The idea is to balance flow separation

tendencies at the leading edge with those at the hinge line

so that separation is not excessive at either location. The

Body side

25 -

- .... 1---_-_--- ¢¢for

deg

I0

5

o -,/ , , J , ,

0 .2 .4 .6 .8 1.0

Y

b12

Sketch 12-3

mismatch of 5° at the leading edge is expected to result

in the formation of separated leading-edge vortices. But

the separation is hoped to be mild enough so that reattachment occurs at or ahead of the hinge line and performance losses relative to fully attached flow are small.

The suction parameter plot shows a suction parameter of

about 0.77 at CL.de s = 0.6. This value is significantly less

than the value of 0.88 for the continuous mission adaptive surface but nevertheless is a reasonable level for flap

system performance.

Multiple additional runs using input data such as

those of table I(f) with other values of flap deflection

angles can be used in the construction of performance

contour maps as illustrated in figure 37. The deflection

angles are defined as those whose tangents are obtained

by multiplying the tangents of the nominal angles by a

set of factors, 0, 0.5, 1.0, 1.5, and 2.0. The map of

figure 37(a) was obtained by applying these factors to

the nominal leading- and trailing-edge flap deflection

schedules to create input data for a series of code runs.

Construction of this map required 16 individual runs.

The 5 by 5 matrix does not require 25 individual runs

because, as described in appendix A, a single run which

utilizes TXMLEFD and TXMTEFD values of 0.0 gives

valid data for 4 data points (the nominal leading-edge

deflections in combination with the nominal trailingedge deflections, the nominal leading-edge deflections in

combination with zero trailing-edge deflections, the

nominal trailing-edge deflections in combination with

zero leading-edge deflections, and zero deflection of

both leading and trailing edges).

If desired, a full set of data for the construction of

contour maps can be obtained in a single run. For this

purpose, a full set of TXMLEFD and TXMTEFD values

(5 values of each) may be employed in that one run. The

results, however, are not as accurate as those of the multiple run process. A map constructed in this way for the

same configuration is shown in figure 37(b). Some obvious discrepancies are seen, particularly for leading-edge

deflection factors greater than 1.0. This discrepancy is

clearly an example of a case where the simple single run

approach is not applicable and should have been anticipated since leading-edge flap deflections are as high as

56 ° . The single run approach has been used successfully

for smaller deflection angles--angles small enough so

that there is little difference between the tangent of the

angle and the sine of the angle. When in doubt, the safe

approach is to use multiple runs.

The contour map obtained from multiple runs in figure 37(a) shows, for CL,de s = 0.6, a maximum suction

parameter of about 0.77 for a leading-edge flap deflection factor of about 0.8 and a trailing-edge flap deflection

50
factor of about 1.2. Thus, for this example, the optimum

flap settings for CL = 0.6 are estimated to be

_le,s = 21 °/26°/30°

_te,s = 16°/16°/19°

This map also has contour lines for pitching-moment

coefficients to enable selection of optimum flap settings

subject to constraints.

12.3. Nonsymmetrical Wing Cruise Surface

Design

A relatively new addition to the WlNGDES code is

the capability for the design of optimized surfaces for

nonsymmetrical wings such as that shown in sketch 12-4.

As an example of nonsymmetrical wing design, an optimum surface for this elliptical planform was designed for

a lift coefficient of 0.16 at a cruise Mach number of 1.40.

Code input data are shown in table II(f). The rolling

moment was constrained to be zero about an assumed

center of gravity at the midspan. Figure 38 shows the

design results. Figure 38(b) presents a computer drawing

of the wing camber surface. This surface displays considerable camber and a pronounced twist distribution to take

advantage of the increasing upwash along the leading

edge. The plot of angle of attack at the top of figure 38(a)

shows a close matchup of the upper limit of the range of

full thrust with ot for (CL,des)re p. The small part of the

span that has a zero range of full thrust results from the

local supersonic leading edge. As shown in the middle

plot, a maximum suction parameter for the curve with the

empirical correction occurs at a lift coefficient just below

CL,des. For this design there is no need for a redesign

Actual planform

Code input

Sketch 12-4

with modified leading-edge surface weighting factors.

As shown at the bottom of the figure, the code solution

gives a nearly elliptical span load distribution. The

AERO2S code has recently been modified to provide the

capability for analysis of a nonsymmetrical wing.

13. Concluding Remarks

A pair of computer codes, AERO2S and WINGDES,

is now widely used for the analysis and design of airplane lifting surfaces under conditions that tend to induce

flow separation. These codes have undergone continued

development to provide additional capabilities since the

introduction of the original versions over a decade ago.

This code development has been reported in a variety of

publications (NASA technical papers, NASA contractor

reports, and journal articles). Some modifications have

not been publicized at all. Users of these codes have suggested the desirability of combining in a single document, descriptions of the code development, an outline of

the particular features of each code, and suggestions for

effective code usage. This report is intended to supply

that need.

This report describes the development of numerical

solutions of linearized theory for both subsonic and

supersonic speeds that are performed by iteration (rather

than by matrix inversion) and thus are easily adaptable to

provide increased accuracy as more advanced computers

become available. Another important feature of the theoretical solutions is a separation of loading components

with and without the presence of singularities, so that

each may be handled in an appropriate manner to

increase the accuracy of integration techniques used to

evaluate forces and moments. The numerical solutions

also provide for the estimation of attainable leading-edge

thrust which has a powerful effect on the aerodynamic

lifting efficiency that can actually be achieved.

Discussions of the application of the computer codes

to problems of practical interest show how attached flow

theoretical predictions are actually applicable to flows

with detached regions provided that the separation is

mild and localized. Code results are shown to be more

accurate for configuration arrangements which, through

the use of wing twist and camber or flap systems, promote predominantly attached flows and high levels of

aerodynamic efficiency. Design features of the codes

permit the definition of efficient wing camber surfaces

and flap system deflection schedules to minimize separation and maximize performance.

NASA Langley Research Center

Hampton, VA 23681-2199

April 8, 1997

51
Appendix A

AERO2S Code Input and Output

The AERO2S computer code is written in generic

FORTRAN 77 for use on virtually any computer system

with a Fortran compiler having namelist capability. The

first record in the input is a code run identification (title)

that accepts up to 80 characters. The remainder of the

input is placed in namelist format under the name INPT1.

The code is constructed so that successive runs may be

made with a given code entry. To make additional runs,

it is necessary only to add an identification record and

namelist data that are to be changed from the previous

run.

A1. Wing Planform--Required Input

The wing planform information is specified by a

series of leading-edge and trailing-edge breakpoints for a

right-hand wing panel. Up to 30 pairs of coordinates may

be used to describe the leading edge and up to 30 pairs to

describe the trailing edge. The planform input data in

code terminology are as follows:

NLEY number of leading-edge breakpoints (limit

of 30)

TBLEY table of leading-edge y values; beginning at

y = 0; increasing order ofy from root to tip

TBLEX table of leading-edge x values that corresponds to TBLEY table

NTEY number of trailing-edge breakpoints (limit

of 30)

TBTEY table of trailing-edge y values; beginning at

y = 0; increasing order of y from root to tip

TBTEX table of trailing-edge x values that corresponds to TBTEY table

XMAX largest x ordinate anywhere on planform;

includes second surface if present

SREF wing reference area for use in aerodynamic

force and moment coefficients

CBAR wing reference chord for use in aerodynamic

moment coefficients

XMC x location of moment reference center

JBYMAX integer designating number of elements in

spanwise direction (limit of 41)

ELAR element aspect ratio

The size of the wing in code dimensions is controlled

by the entry JBYMAX. The necessary scaling is done

within the code by use of a scale factor 2(JBYMAX)/[3y).

52

The number of complete wing elements N corresponding

to a given JBYMAX may be approximated as

N= 4(JBYMAX)2/_R R)

The code has been written to accommodate 4000 righthand panel elements. Generally, the JBYMAX integer

is much less than the limit of 41. The normal range is

8 to 20. Computational costs tend to increase as the

square of the number of elements.

For flat and mildly cambered wings, an element

aspect ratio approximately equal to the full-wing aspect

ratio is recommended. For small chord leading- or

trailing-edge flaps, the use of a large element aspect ratio

may be necessary to place two or more elements within

the chord. The number of elements in a given chord, Cle

or Cte, may be approximated as

Cle

N = _-_(JBYMAX)(ELAR)

or

Cte

N = b_(JBYMAX)(ELAR)

Because computational costs tend to increase as the

fourth power of JBYMAX and the second power of

ELAR, an increase in the element aspect ratio is the more

efficient means of providing improved definition.

A2. Wing Planform--Optional Input

The code permits the design or analysis of an asymmetrical lifting surface. For this option, the spanwise

geometric data (i.e., TBLEY, TBTEY, TBYC, TBYR)

are input from wingtip to wingtip as positive values

beginning at a span station of zero. The entire wing is

confined to what otherwise would be only a right-hand

panel. To implement this feature, input the index

NSYM nonsymmetrical planform index (set this index

to 1 for asymmetrical planform, code defaults

to 0 to provide a symmetrical planform by

construction of additional mirror image

left-hand panel)

A3. Wing Camber Surface--Required Input

The wing mean camber surface must be specified by

exactly 26 chordwise ordinates at up to 52 span stations.

When fewer than 26 camber coordinates are used to

define the sections, the ordinate tables must be filled with

enough zeros to complete the list of 26. The necessary

section information is as follows:
NYC numberof spanwisestationsatwhich

chordwisesectionsareusedtodefinemeancambersurface(limitof 52)

TBYC tableof y values for chordwise cambersurface sections; beginning at y = 0;

increasing order of y from root to tip

NPCTC number of chordwise stations used in definition of mean camber surface (limit of 26)

TBPCTC table of chordwise stations, in percent

chord, at which mean camber surface ordinates are defined; increasing order from

leading edge to trailing edge

TZORDC table of mean camber surface z ordinates

that corresponds to TBPCTC table; full 26

values for root chord (including zeros for

values in excess of NPCTC) are given first,

followed by similar information for all

spanwise stations in increasing order of y

TZSCALE multiplying factor applied to TZORDC

table to change camber surface ordinates;

default 1.0

The TZORDC table may be multiplied by a scale

factor TZSCALE. This factor may be useful if the original tabulated ordinates are nondimensionalized with

respect to a single measurement (e.g., the wing root

chord) or if it is necessary to evaluate the effect of a

change in camber surface severity.

A4. Section Parameters for Attainable Thrust

CalculationuRequired Input

The following wing section information is required

for the calculation of attainable leading-edge thrust and

leading-edge separation forces:

NYR number of spanwise stations at which information on airfoil sections is supplied (limit

of 30)

TBYR table ofy values for airfoil section information; beginning at y = 0; increasing order ofy

from root to tip

TBTOC table of airfoil maximum thickness as fraction

of chord, (x/C)max

table of section locations of maximum thickness as fraction of chord, r I

table of leading-edge radii as fraction of

chord, r/c

vortex location option as follows:

= 0 full vortex force acts normal to wing

reference plane at wing leading edge;

does not contribute to axial force

TBETA

TBROC

IVOROP

YAPEX

= 1 vortex center given by empirical relationships derived from delta wing

experimental data (default)

= 2 vortex center given by method of Lan

and Chang (ref. 12)

spanwise location of vortex flow-field origin

For special planforms such as forward-swept

wings or other wings with apex away from

centerline, this input can help provide better

estimate of vortex-induced flow fields and

forces (default YAPEX = 0.0)

A5. Flight Conditions---Required Input

The flight or test conditions are specified as follows:

XM free-stream Mach number

RN free-stream Reynolds number (based on c)

x 10--6

NALPHA number of angles of attack to be calculated

(limit of 40)

TALPHA table of angles of attack to be calculated,

deg

NADRd_T number of additional Reynolds numbers

(default 0)

TADRN table of additional Reynolds numbers (limit

of 3)

The commonly accepted practice of performing subsonic calculations for a Mach number of 0 is not appropriate for this code. Realistic estimates of attainable

thrust can be made only if both the Mach number and the

Reynolds number correspond to actual conditions. In

fact, the code stops and writes an error message when

XM = 0 is input.

A6. Leading- and Trailing-Edge Flaps---Optional

Input

The following information makes possible the calculation of loadings and forces on deflected leading-edge

and trailing-edge flaps. If flap data are not desired, simply omit these entries.

NLEFY number of breakpoints in leading-edge

flap chord distribution (limit of 30)

table of y values at breakpoints in leadingedge flap chord distribution; beginning at

y = 0; increasing order of y from root to tip

table of streamwise leading-edge flap

chords that corresponds to TBLEFY table

table of flap deflections in degrees (positive for leading edge down) that corresponds to TBLEFY table

TBLEFY

TBLEFC

TBLEFD

53
NADLEFD

TXMLEFD

number of leading-edge flap deflection

multipliers other than 1.0 (limit of 4)

(default 0)

table of leading-edge flap deflection multipliers (applied as multiplier of tangents of

input flap deflections)

LEFTYPE type of leading-edge deflection as shown

in sketch A-1

number of breakpoints in trailing-edge flap

chord distribution (limit of 30)

table of y values at breakpoints in trailingedge flap chord distribution; beginning at

y = 0; increasing order of y from root to tip

table of streamwise trailing-edge flap

chords that corresponds to TBTEFY table

TBTEFD table of flap deflections in degrees (positive for trailing edge down) that corresponds to TBTEFY table

NADTEFD number of trailing-edge flap deflection

multipliers other than 1.0 (limit of 4)

(default 0)

TXMTEFD table of trailing-edge flap deflection multipliers (applied as multiplier to tangents of

input flap deflections)

CLDES additional lift coefficient for which flap

system aerodynamic performance is to be

specified; code aerodynamic characteristics are given only for angles of attack in

input TALPHA table unless CLDES is

specified

Spanwise tables must begin with y = 0 and extend to

y = b12 (with chords of 0 where there are no flaps). At

spanwise positions where there are discontinuities in

NTEFY

TBTEFY

TBTEFC

1linear (default)

¢

2 parabolic

l

_le,s J

Sketch A- 1

54

either flap chord or deflection, it is necessary to make

closely spaced tabular entries inboard and outboard of

the discontinuity.

The code requires flap deflection angles measured in

the X-Z plane. Flap deflection angles measured normal to

the flap hinge line may be converted to code input angles

by

_le,s = tan-1 (cos Ah,le tan _le,n)

_te,s = tan-I (c°s Ah,te tan 8te, n)

The code provides solutions for wing surfaces composed of all possible combinations of leading-edge and

trailing-edge flap settings provided by the original

deflections (TBLEFD and TBTEFD) and by the flap

deflection multipliers (TXMLEFD and TXMTEFD). Up

to 25 pairs of leading-edge and trailing-edge flap deflection schedules may thus be treated simultaneously. Solutions obtained by using the multipliers (values other than

0 or 1) are determined by a perturbation process; thus,

they are not as accurate as solutions for the original or

nominal input deflections. When increased accuracy is

required, or when the change in performance with the

change in deflection must be evaluated--as in the construction of suction parameter contour maps--individual

solutions without recourse to multipliers are generally

required.

The code also provides an improved accounting of

hinge-line singularities in determination of wing forces

and moments. The technique used is described in appendix B of reference 3. See table I(b) for an example of

leading- and trailing-edge flap input data.

A7. Solution Convergence Criteria---Optional

Input

To determine lifting-surface perturbation velocity

distributions, the code provides for a maximum of

70 iterations. If this number is reached without the convergence criteria being met, the results for the 70th iteration are printed with a warning of the failure to meet the

criteria. The maximum number of iterations may be

increased or decreased by the entry

ITRMAX maximum number of perturbation velocity

iterations (default 70)

The code convergence criteria are met when, for all

four wing surfaces and for two successive iterations, the

average difference in perturbation velocity between iterations is less than half of 1 percent of the average velocity

over the wing. If the average velocity for the camber surface or either of the flap surfaces is less than the average

velocity for the flat surface at ct = 1°, the flat wing

surface value is used instead. In many instances, these
criteriamaybemorestringentthannecessary. If desired,

theconvergencecriteriamaybechangedbytheentry

CNVGTSTperturbationvelocityconvergenceriteria

(default0.005)

A8. Canard or Horizontal Tail---Optional Input

The following set of input data is required for the

handling of wing-canard or wing-horizontal tail configurations. If wing-along solutions are desired, simply omit

these entries.

ILS2 second surface identifier

= 0 no second surface (default)

= 1 canard

= 2 horizontal tail

If a second surface is employed (ILS2 = 1 or 2), the

following planform information must be supplied:

NLEY2 number of leading-edge breakpoints (limit

of 21)

TBLEY2 table of leading-edge y values; need not

begin at y = 0; increasing order of y

TBLEX2 table of leading-edge x values that corresponds to TBLEY2 table

NTEY2 number of second surface trailing-edge

breakpoints (limit of 21 )

TBTEY2 table of trailing-edge y values; need not

begin at y = 0 but initial and final values

must be same as TBLEX2 initial and final

values; increasing order of y

TBTEX2 table of trailing-edge x values that corresponds to TBTEY2 table

For the second surface, the mean camber surface

must be specified by exactly 26 chordwise ordinates at

up to 21 span stations. When fewer than 26 camber coordinates are used to define the sections, the ordinate tables

must be filled with enough zeros to complete the list

of 26. The necessary section information is as follows:

NYC2 number of spanwise stations at which

chordwise sections are used to define mean

camber surface (limit of 21)

TBYC2 table ofy values for chordwise camber surface sections; increasing order of y

NPCTC2 number of chordwise stations used in definition of mean camber surface (limit of 26)

TBPCTC2 table of chordwise stations, in percent

chord, at which mean camber surface ordinates are defined; increasing order from

leading edge to trailing edge

TZORDC2 table of mean camber surface z ordinates

that corresponds to TBPCTC2 table; the

full 26 values for the root chord (including

zeros for values in excess of NPCTC2) are

given first, followed by similar information

for all spanwise stations in increasing order

ofy

TZSCAL2 multiplying factor applied to TZORDC2

table to change camber surface ordinates

(default 1.0)

The following canard or horizontal tail section information is required for the calculation of attainable

leading-edge thrust and leading-edge separation forces:

NYR2 number of spanwise stations at which information on airfoil sections is supplied (limit

of 21)

TBYR2 table ofy values for airfoil section information; increasing order of y

TBTOC2 table of airfoil maximum thickness as fraction of chord, (x/C)max

TBETA2 table of section locations of maximum thickness as fraction of chord, r1

TBROC2 table of leading-edge radii as fraction of

chord, rlc

YAPEX2 spanwise location of second surface vortex

flow-field origin

For special planforms such as forward-swept

surfaces or other surfaces with apex away

from centerline, this input can help provide a

better estimate of vortex-induced flow fields

and forces; default is YAPEX2 = 0.0.

incidence of second surface with respect to

wing reference plane, deg (default DELTA2

= 0.0), positive for leading edge up

for an example of horizontal tail input data.

DELTA2

See table I(d)

A9. Optional Two-Dimensional Airfoil Solution

The code has been modified to provide solutions for

two-dimensional airfoil sections. As described in reference 7, this capability may be useful in recalibration of

the attainable thrust prediction method when additional

experimental data covering a wider range of Mach numbers and Reynolds numbers become available. An input

of JBYMAX = 1 signals a two-dimensional airfoil solution. For this purpose, tables of airfoil section characteristics defining the camber surface and the thickness

distribution with identical listings for a root- and tipchord station must be provided.

A recalibration of the limiting pressure coefficients

used in the attainable thrust prediction method requires

55
aniterativeapproach.Repetitivesolutionswithvarious

valuesof thefollowinglimitingpressuremultiplierare

required:

XMCPLT limitingpressuremultiplier(default1.0)

Thisquantityactsasamultiplierof thelimitingpressure

formulapresentlyusedin thecode.Foreachcoderun,

theoutputlistingincludesthelimitingpressurecoefficientusedin thatrun.As describedin reference7,the

iterationis continueduntil thebestmatchof codeand

experimental axial-forcedataisobtained.Thelistedlimiting pressurecoefficienthenbecomesa revisedvalue

forthespecifiedMachnumberandReynoldsnumberto

beusedinformulationofnewlimitingpressurerelationships.SeetableI(e)for anexampleof two-dimensional

airfoilcodeinput.

AIO. Code Output Data

The printed code results include

1.

.

.

Iteration-by-iteration history of convergence

parameters

Listing of theoretical pressure distributions for

combined camber surface at o_ = 0 ° and for combined fiat surface at ct = 1°; for each code spanwise

station (controlled by JBYMAX), interpolated or

extrapolated pressure coefficients are given for a set

of chordwise stations

Listing of spanwise distribution of section normalforce, axial-force, and pitching-moment coefficients for combined cambered surface at _ = 0 ° and

for combined fiat surface at ct = 1°; interference

axial-force coefficient caused by fiat surface loading acting on camber surface and theoretical thrust

parameters are also printed

. Listing of overall theoretical aerodynamic coefficients CN, CA, Cm, and C D with no thrust and with

theoretical thrust as function of angle of attack

. Listing of spanwise distribution of flat wing angleof-attack range for full theoretical leading-edge

thrust (for wing-alone solution only)

. Listing of overall estimated aerodynamic coefficients including CN, CA, and Cm for basic pressure

loading, AC N and AC A for attainable thrust and

vortex-force increments, and C N, CA, C m, CL, CD,

and SS for total loading

Additional tabulated output data may be selected by

using the following print options:

IPRCPD = 1 theoretical pressure distributions for

each selected angle of attack

IPRSLDT = 1

IPRSLDA = 1

IPRALL = 1

theoretical span load distribution of CN,

CA, C m, and C D with no thrust and with

full theoretical thrust for each selected

angle of attack

estimated span load distribution of CN,

CA, C m, and C D with attainable thrust

and vortex-force effects for each

selected angle of attack

preceding print control options apply

only to first set of flap deflections; select

this option if three preceding options are

to apply to all flap deflection combinations; selection of this option could

result in very large volume of printed

output

56
Appendix B

WINGDES Code Input and Output

The WINGDES code is wriuen in generic

FORTRAN 77 for use on virtually any computer system

with a Fortran compiler having namelist capability. The

first record in the input is a code run identification (title)

that accepts up to 80 characters. The remainder of the

input is placed in namelist format under the name INPT1.

The code is constructed so that successive runs may be

made with a given computer entry. To make additional

runs, it is only necessary to add an identification record

and namelist data that change from the previous run. An

additional capability is provided by the entry NEWDES.

When the code is run in the design mode and NEWDES

is set to 1, a design camber surface is found, the input set

of camber surface ordinates is replaced by camber surface ordinates for the new design, and this new design is

treated as an evaluation case. In the original code, the

default for the entry NEWDES was 0, which provided

for a design of the wing surface but not for a subsequent

evaluation. Now, however, because this feature

was found so useful, the default was changed to

NEWDES = I. When the NEWDES option is used,

successive runs may be employed to evaluate the new

surface at off-design conditions.

B1. Wing Planform--Required Input

The wing planform information is specified by a

series of leading-edge and trailing-edge breakpoints for a

right-hand wing panel. Up to 30 pairs of coordinates may

be used to describe the leading edge and up to 30 pairs to

describe the trailing edge. The planform input data in

code terminology are as follows:

NLEY

TBLEY (NLEY)

TBLEX (NLEY)

NTEY

TBTEY (NTEY)

TBTEX (NTEY)

XMAX

number of leading-edge breakpoints

(limit of 30)

table of leading-edge y values;

beginning at y = 0; increasing order

of y from root to tip

table of leading-edge x values that

corresponds to TBLEY table

number of trailing-edge breakpoints

(limit of 30)

table of trailing-edge y values; beginning at y = 0; increasing order ofy

from root to tip

table of trailing-edge x values that

corresponds to TBTEY table

largest x ordinate anywhere on

planform

SREF wing reference area for use in

aerodynamic force and moment

coefficients

CBAR wing reference chord for use in

aerodynamic moment coefficients

XMC x location of moment reference

center

ELAR element aspect ratio (default 1.0 for

subsonic cases, 1.0/13 for supersonic

cases)

JBYMAX integer designating number of

elements in spanwise direction

(limit of 50)

For subsonic speeds, the element aspect ratio ELAR

is chosen by the user (default 1.0). At supersonic speeds,

a fixed value of ELAR = 1.0/_ is imposed to avoid

computational difficulties, and the user has no option.

For flat and mildly cambered wings at subsonic speeds,

an element aspect ratio one half the full wing aspect ratio

or greater is recommended. For wings with small chord

leading-edge or trailing-edge design areas, using a large

element aspect ratio may be necessary to place at least

two elements within the chord. The number of elements

in a given chord, Cle or Cte , may be approximated as

Cle

N = _ (JBYMAX) (ELAR)

or

c te

N = _-_ (JBYMAX) (ELAR)

Because computational costs tend to increase as the

fourth power of JBYMAX and the second power of

ELAR, an increase in the element aspect ratio is the more

efficient means of providing improved definition. At

supersonic speeds, where ELAR is fixed, the only

recourse is to increase JBYMAX. This parameter controis the size of the wing in code dimensions.

The necessary scaling is done within the code by use

of a scale factor, 2(JBYMAX)/[SPAN (13)]. The number

of complete wing elements N corresponding to a given

JBYMAX may be approximated as

N = 4(JBYMAX) 2 (ELAR'_

\AR J

The code has been written to accommodate 1000 righthand panel elements (2000 complete wing elements).

Generally, the JBYMAX integer is less than the limit

of 50. The normal range is 8 to 20 for subsonic speeds

and 20 to 50 for supersonic speeds. Computational costs

tend to increase as the square of the number of elements.

57
B2.WingPlanform--OptionalInput

The code permits the design or analysis of an asymmetrical lifting surface. For this option, the spanwise

geometric data (i.e., TBLEY, TBTEY, TBYC, TBYR)

are input from wingtip to wingtip as positive values

beginning at a span station of zero. The entire wing is

confined to what otherwise would be only a right-hand

panel. To implement this feature, input the index

NSYM nonsymmetrical planform index (set this index

to 1 for asymmetrical planform, code defaults

to 0 to provide symmetrical planform by construction of additional mirror image left-hand

panel)

Also see section B8 for YCG, CRTST, and CRNSYM

entries. In addition, often specifying NLEC, TBLECY,

and TBLEC input data is necessary rather than allow the

default values to be used. For example, the nonsymmetrical wing example of this report (table II(f)) has a chord of

nearly zero at y = 0, which if used as the default for the

leading-edge surface chord would not produce an acceptable design.

B3. Wing Camber Surface---Optional Input

A wing mean camber surface may be specified by a

set of tabular entries. However, if a flat wing analysis is

performed or if a fiat wing is used as the initial surface in

a design process, these entries are not required. If a wing

surface is input, the section mean camber surface must be

specified by exactly 26 chordwise ordinates at up to

52 span stations. When fewer than 26 camber coordinates are used to define the sections, the ordinate tables

must be filled with enough zeros to complete the list

of 26. The necessary section information is as follows:

NYC

TBYC (NYC)

NPCTC

TBPCTC (NPCTC)

number of spanwise stations

at which chordwise sections

are used to define mean camber surface (limit of 52)

table of y values for chordwise camber surface sections; beginning at y = 0;

increasing order of y from

root to tip

number of chordwise stations used in definition of

mean camber surface (limit

of 26)

table of chordwise stations,

in percent chord, at which

mean camber surface ordinates are defined; increasing

58

order from leading edge to

trailing edge

TZORDC (NPCTC,NTC) table of mean camber surface

z ordinates that corresponds

to TBPCTC table; the full 26

values for root chord (including zeros for values in excess

of NI_TC) are given first,

followed by similar information for NYC spanwise stations in increasing order of y

TZSCALE multiplying factor applied to

TZORDC table to change

camber surface ordinates

The TZORDC table may be multiplied by a scale factor

TZSCALE. This factor may be useful if the original

tabulated ordinates are nondimensionalized with respect

to a single measurement (e.g., the wing root chord) or if

it is necessary to evaluate the effect of change in camber

surface severity. See table II(a) for an example of wing

camber surface code input.

B4. Section Parameters for Attainable Thrust

CalculationmRequired Input

The following wing section information is required

for the calculation of attainable leading-edge thrust and

leading-edge separation forces:

TBYR (NYR)

TBTOC (NYR)

TBETA (NYR)

TBROC (NYR)

IVOROP

number of spanwise stations at which

information on airfoil sections is supplied (limit of 30)

table of y values for airfoil section

information; beginning at y = O;

increasing order of y from root to tip

table of airfoil maximum thickness as

fraction of chord, tic

table of section locations of maximum

thickness as fraction of chord, r1

table of leading-edge radii as fraction

of chord, r/c

vortex location option as follows:

= 0 full vortex force acts normal to

wing reference plane at wing

leading edge; does not contribute to axial force

= 1 vortex center given by empirical relationships derived from

delta wing experimental data

(default)

NYR
YAPEX

= 2 vortexcentergivenbymethod

ofLanandChang(ref.12)

spanwiselocationof vortexflow-field

origin(default0.0)

Forspecialplanformssuchasforwardsweptwingsorotherwingswithapex

awayfromcenterline, thisinputcan

helpprovidebetterestimateof vortexinducedflowfieldsandforces.

B5. Flight Conditions---Required Input

The flight or test conditions are specified as follows:

XM free-stream Mach number

RN free-stream Reynolds number (based on _)

x 10-6

NALPHA number of angles of attack to be calculated

(limit of 19)

TALPHA table of angles of attack to be calculated, deg

The commonly accepted practice of performing subsonic calculations for a Mach number of 0 is not appropriate for this code. Realistic estimates of attainable

thrust can be made only if both the Mach number and the

Reynolds number correspond to actual conditions. In

fact, the code stops and writes an error message when

XM = 0 is input. The TALPHA table must begin with the

smallest angle (most negative) and continue with increasingly larger angles up to the largest angle. A wide range

of angle of attack is required in order to use the code in

the design mode. This range must cover the angle of

attack for CLxle s of the original and all subsequent surfaces. An error message is written when the angle-ofattack range is too small.

B6. Solution

Input

To determine

the input camber

angle of attack of

Convergence Criteria--Optional

perturbation velocity distributions for

surface, the flat wing surface at an

1o, and the candidate camber surfaces

used in the design mode, a maximum of 70 iterations are

provided. If this number is reached without the convergence criteria being met, the results for the 70th iteration

are printed with an appropriate message. The maximum

number of iterations may be changed by the entry

ITRMAX maximum number of perturbation velocity

iterations (default 70)

The code convergence criteria are met when, for all wing

surfaces, the average difference in perturbation velocity

between successive iterations is less than one half of

1 percent of the average velocity over the wing. If the

average velocity for any wing surface is less than the

average velocity for the flat surface at ot = 1o, the value

for the flat wing surface is used instead. In many

instances, these criteria may be more stringent than necessary. If desired, the convergence criteria may be

changed by the entry

CNVGTST perturbation velocity convergence criteria

(default 0.005)

The code wing surface design is also the result of an

iterative process. The iteration is terminated when, from

one iteration to the next, the design angle of attack

changes by less than 0.01 ° and the design pitchingmoment coefficient changes by less than 0.001. If

desired, these criteria may be made more or less stringent

by use of the following entries:

ALPTST angle-of-attack convergence test (default

0.01)

CMTST pitching-moment convergence test used when

a design pitching-moment constraint is specified (default 0.001 )

B7. Design Specifications--Required Input for

Design Mode

The following entries control the solution for the

optimized surface in the code design mode. For the analysis of a specified wing surface, omit the following

entries:

CLDES design lift coefficient (if CLDES is not

specified, the code defaults to 0.0, which

triggers an analysis-only solution)

CMDES design pitching-moment coefficient (if

CMDES is not specified, the code defaults

to 1000.0, which triggers an optimization

solution without moment restraint)

ITRDESM maximum number of design iterations

(default 20)

See section 7.10 for comments on performance penalties that can result from imposition of CMDES

restraints. This capability should not generally be used

for supersonic cruise designs.

In attempting to meet the convergence criteria for

wing design, the code provides for a maximum of 20 iterations. If this number is reached without the convergence

criteria being met, the results for the 20th iteration are

printed with a warning of the failure to meet the criteria.

If desired, the maximum number of design iterations may

be increased or decreased by the ITRDESM entry.

59
B8.DesignSpecifications---Optional Input for

Design Mode

The rest of the design mode entries are optional.

These can be valuable for code user control of the design

process but are covered by code defaults if the user

chooses not to exercise the options. Code defaults provide candidate surfaces which generally provide a camber surface design with good aerodynamic efficiency.

The code user, however, may want to tailor a camber surface solution more appropriate to the problem at hand

and may want to search for solutions offering greater

efficiency.

The user may select the number of general camber

surfaces to be used in the optimization process as

follows:

NGCS number of general camber surfaces covering

entire wing (limit of 8, default 8)

In addition, the user may select exponents that control

the shape of the various general camber surfaces by use

of the following entries:

EXPY 1, EXPY2, exponents of y used in definition of

EXPY3, EXPY4 general camber surfaces (defaults:

EXPYI = 0.0, EXPY2 = 1.0,

EXPY3 = 2.0, and EXPY4 = 3.0)

EXPX 1, EXPX2 exponents of x' used in definition of

general camber surfaces (defaults:

EXPX 1 = 1.5, EXPX2 = 2.0)

To preserve the original surface between the leadingedge modification surfaces and the trailing-edge modification surfaces for a mission adaptive design, NGCS

may be set to zero. In this case, user options for both

leading-edge and trailing-edge modifications must be

employed.

The following entries control the region of the wing

affected by the leading-edge modification surfaces.

Because wing aerodynamic performance is critically

dependent on the surface shape and pressure loading in

the leading-edge region, these surfaces are essential to

the optimization process.

NLEC number of breakpoints used in definition of area of wing affected by

leading-edge modification surfaces

(limit of 30, default 2)

TBLECY (NLEC) table of y values at breakpoints used

in definition of area of wing affected

by leading-edge modification surfaces; increasing order ofy from root

to tip (default 0.0, TBLEY (NLEY))

TBLEC (NLEC) table of Cle values that corresponds

to TBLECY table (default

6O

TBTEX(1) - TBLEX(1 ) for both

entries); see discussion for ELAR

(p. 58) regarding definition of

leading-edge areas; may be necessary to change ELAR or to place

limits on nonzero Cle values

The code employs linear interpolation of tabular chord

inputs to define chords at the required code span

locations.

The following entries control the region of the wing

affected by the trailing-edge modification surfaces and

the streamwise section shape of these surfaces. The code

defaults exclude these surfaces.

NTES number of trailing-edge modification

surfaces (limit of 4) (see fig. 18 for

examples); does not represent number of trailing-edge design areas or

flap surface areas which are controlled by TBTECY and TBTEC

inputs

NTEC number of breakpoints used in definition of area of wing affected by

trailing-edge modification surfaces

(limit of 30)

TBTECY (NTEC) table of y values at breakpoints used

in definition of area of wing affected

by trailing-edge modification surfaces; increasing order ofy from root

to tip

TBTEC (NTEC) table of Cte values that corresponds to

TBTECY table; see discussion for

ELAR (p. 58) regarding definition of

trailing-edge areas; may be necessary to change ELAR or to place

limits on nonzero Cte values

EXPXTE exponent ofx' - (c - Cte) used in definition of trailing-edge modification

surfaces (exponents of y are same as

used in definition of general camber

surfaces) (default 1.5)

Leading- and trailing-edge modification surface chords

must be specified for the entire wing semispan even if the

chords are zero. Where an abrupt change in chord occurs,

values of the chord should be specified for semispan stations just inboard and just outboard of the break. See

table II(e) for an example of leading- and trailing-edge

design surface code inputs.
B9. Additional Optional Input

If desired, the design twist and camber may be confined to the wing outboard of the fuselage. The limit of

the general surfaces is controlled by the following input:

YFUS spanwise station of wing-body juncture; this

entry limits wing general camber surfaces to

values ofy greater than specified (default 0.0)

Leading- and trailing-edge camber surfaces may be limited by existing options; use of these limitations yields a

design lifting surface confined to the wing outboard of

the fuselage.

For an asymmetrical wing design, a design restraint

to eliminate rolling moment may also be applied as

follows:

YCG lateral center-of-gravity and rolling-moment

center as fraction of wingspan (default

1000.0 imposes no restraint)

CRTST absolute value of maximum allowable rolling

moment (default 0.01)

An asymmetric wing reference root chord may be input if

desired as follows:

CRNSYM reference root chord of asymmetric wing

(default computed by code as chord at

midspan)

The code allows for modifications to the designed

lifting surface after generation of the design and before

evaluation of that surface. This option permits an assessment of the effect of practical considerations on wing

aerodynamic performance.

Wing trailing-edge ordinates may be altered to provide less severe lateral surface slopes and to reduce the

incidence of the lifting surface representing the fuselage.

The modification could also be used to provide a straight

line portion of the camber surface at the wing trailing

edge or at trailing-edge flap hinge-line location. The

airfoil section camber lines are altered by use of a formulation that does not change surface slopes at the wing

leading edge so that the design leading-edge thrust condition is preserved as much as possible.

TDELZTE table of changes in wing trailing-edge ordinates expressed as fraction of wing root

chord

This table is used in generation of trailingedge ordinates to replace those listed in

code-generated table of surface ordinates;

entry required for each span station listed in

code output in same order of increasing

span stations.

B10. Reflex Surface Design---Optional Input for

Design Mode

The code permits a design of the wing camber surface in the vicinity of nacelles which further optimizes

the lifting efficiency of the wing by taking advantage of

the pressure field created by the presence of the nacelle.

This option is activated by the input

IREFL reflex surface design index (set this index to 1

if code is used to define reflexed camber surface in vicinity of nacelles; use of this option

requires proper definition of input interference

pressure field and corresponding trailing-edge

design surface; code defaults to index of 0,

which bypasses reflex design feature)

See table II(d) for an example of reflex surface design

code input.

Bll. Flap Design---Optional Input for Design

Mode

The code provides an automated graphical flapfitting technique, described in reference 3, which is activated by the following input:

IFLPDES flap design index (set this index to 1 if code

is used to define spanwise distribution of

leading- and trailing-edge flap deflections

which approximate designed camber surface;

use this option only for design in which

leading- and trailing-edge modification

surfaces are specified; code defaults to index

of 0, which bypasses flap design feature)

When the flap design feature is used, leading-edge and/or

trailing-edge design surfaces must be used, and NGCS

must be set to zero. The chords of these surfaces are

input as the chords of the flaps. The code then designs a

restricted area camber surface for leading- and trailingedge areas whose chords are set at 1.5 times the flap

chords.

B12. Design Surface Smoothness Control--

Optional Input

The following user option provides a degree of control over the smoothness of the camber surface solution:

IAFIX smoothing operation indicator; set

IAFIX = 1 if smoothed values are

supplied (default 0)

TAFIX (JBYMAX) table of smoothed surface weighting factors replacing codegenerated table in same order

of increasing span stations

See table II(c) for an example of surface smoothness control code input.

61
B13.InterferenceFlowField---OptionalInput

The following additional input data provide for a

wing design with pressure fields induced by other airplane components taken into account. For normal code

operation, simply omit these entries. If an interference

pressure field is input, the distribution must be specified

by exactly 26 chordwise positions at up to 52 span stations. When fewer than 26 chordwise positions are used

to define the interference pressure field, the table for ACp

must be filled with enough zeros to complete the list

of 26.

ICP

NYCP

TBYCP (NYCP)

NPCTCP

TBPCTCP (NPCTCP)

TCP (NI_TCP, NYCP)

PFMULT

other airplane componentinduced pressure field indicators; set ICP = 1 if this option

is used (default 0)

number of spanwise stations

at which chordwise sections

are used to define interference pressure field (limit

of 52)

table ofy values for interference pressure field chordwise sections; beginning at

y = 0; increasing order of y

from root to tip

number of chordwise stations

used in interference pressure

field definition (limit of 26)

table of chordwise stations, in

percent of chord, at which

interference pressure field

distributions are defined;

increasing order from leading edge to trailing edge

table of interference pressure

field coefficients that corresponds to TBPCTCP table;

full 26 values for root chord

(including zeros for values in

excess of NPCTCP) are given

first, followed by similar

information for spanwise stations in increasing order of y

multiplier of input pressure

field coefficients (default 1.0)

See table II(d) for an example of a reflex surface design

with an interference pressure field code input.

B14. Empirical Corrections---Optional Input

An empirical method for selection of design lift

coefficient and estimation of achievable aerodynamic

62

performance for supersonic speeds described in reference 6 and in section 7.10 may be included in the code

calculations by input of the index

IEMPCR empirical correction index (this index set to 1

implements empirical corrections; index set to

0, code default, bypasses this feature)

This correction adjusts the design lift coefficient to

account for the tendency of linearized theory to overestimate the magnitude of the upwash field for supersonic

speeds. It also corrects the estimated aerodynamic performance to compensate for the tendency of the theory to

be overly optimistic.

B15. Code Output Data

The code is constructed so that successive runs may

be made with a given computer entry. To make additional runs, it is only necessary to add an identification

record and namelist data that change from the previous

run. An additional capability is provided by the entry

NEWDES. When the code is run in the design mode and

NEWDES is set to 1, a design camber surface is found,

the input set of camber surface ordinates is replaced by

camber surface ordinates for the new design, and this

new design is treated as an evaluation case. In the original code, the default for the entry NEWDES was 0,

which provided for a design of the wing surface but not

for a subsequent evaluation. Now, however, because this

feature was found so useful, the default has been changed

to NEWDES = 1. When the NEWDES option is used,

successive runs may be employed to evaluate the new

surface at off-design conditions.

The wing-design camber surface ordinates are

printed for a reference angle of attack defined by an entry

of ALPZPR (reference angle of attack) or CLZPR (reference lift coefficient). The default is ALPZPR = 0.0.

When CLZPR is specified, the code calculates the corresponding ALPZPR and uses it in the determination of

ordinates.

If the code user desires, span load distribution data

may be printed. If the index IPRSLD is set to 1, section

aerodynamic characteristics, including the separate contributions of basic pressure loadings, attainable thrust,

and vortex forces for each entry in the angle-of-attack

table, are printed. These data are printed only for the

evaluation mode or when the NEWDES option is used in

the design mode.

The printed code results include the following items:

1. Iteration-by-iteration history of convergence

parameters for longitudinal perturbation velocity

solution; in design mode, data given only for most

critical surface of up to 64 surfaces used and for
flatsurfaceatct= 1o;for supersonicsolutionin

whichiterationnotused,printoutisomitted

2. Listing of spanwise distribution of leading-edge

surface factor, angle-of-attack range for full

thrust, and angle of attack for zero thrust; these

data given for evaluation mode and for all iterations in design mode from first (input surface) to

last (optimized surface); for evaluation mode,

leading-edge surface factors all zero

3. Listing of overall theoretical aerodynamic coefficients CN, CA, Cm, C L, and CD with no thrust and

with full theoretical thrust as function of angle of

attack

4. Listing of overall estimated aerodynamic coefficients including CN, CA, and Cm for basic pressure loading, AC N and AC A for attainable thrust

and vortex-force increments, and CN, CA, Cm, C L,

CD, and Ss for total loading as function of angle

of attack; these data given for evaluation mode

and for all iterations in design mode from first

(input surface) to last (optimized surface)

5. Additional listing of overall wing aerodynamic

characteristics including empirical corrections

for optimized surface when IEMPCOR empirical

correction option employed for supersonic design

and evaluation

6. Listing of spanwise distribution of wing-section

aerodynamic characteristics, including separate

contributions of basic pressure loadings, attainable thrust, and vortex forces; these data given

only for evaluation mode (or when NEWDES

option used in design mode) and given only if

print option IPRSLD set to 1

7. Listing of wing-surface ordinates as function of

chord position for each of span stations used in

code solution

8. Listings of lifting pressure distributions for camber surface at t_ = 0° and for flat surface at t_ = 1°

9. Listing of leading-edge surface factors used in

design and listing of suggested replacement values which may lead to improved performance

when NEWDES option used; generally, need for

this replacement arises only when not possible to

provide sufficiently detailed numerical representation of wing to give closely matched aerodynamic characteristics in design and evaluation

modes

10. Listing of suggested spanwise distribution of flap

deflection angles to approximate designed camber

surface and to approach its aerodynamic performance when IFLPDES option used and flap chord

information supplied

As discussed at the beginning of this section, provision has been made for successive runs of the code with a

single computer entry. Under some circumstances, the

following input data quantities may be changed during a

computer run. The new values are used for subsequent

runs unless reset by the code user.

JBYMAX may be reduced to keep number of elements

within code limits; new value which depends

on Mach number retained unless respecified

CLDES reset to 0.0 after design run performed; subsequent runs for evaluation only unless

CLDES respecified

NEWDES reset to 0 after evaluation of newly designed

surface is performed; original input camber

surface description (z = 0 everywhere for

default) is replaced with new values of NYC,

TBYC, NPCTC, TBPCTC, and TZORDC

for designed surface which are retained

unless respecified

TZSCALE reset to 1.0 after rescaling operation performed; any subsequent rescaling activated

by setting TZSCALE to value other than 1.0

applied to newly rescaled surface

TBLEC values multiplied within code by 1.5 when

flap design option (IFLPDES = 1) is

employed; this new table replaces original

input; TBLEC must be respecified if subsequent design performed

CLZPR reset to 1000 at completion of each computer

run; for following run, camber surface ordinates printed for angle of attack of 0° unless

CLZPR respecified

63
Appendix C

Estimation of Idealized Minimum Drag and

Maximum Suction Parameter

For both the AERO2S and WlNGDES codes, the

suction parameter is employed as a means of rating the

relative aerodynamic efficiency of wing lifting surfaces.

In this parameter, the drag of a given wing surface is

compared with that of a flat wing without leading-edge

thrust or vortex forces C L tan (CLICLcL) and with that

of a wing with an elliptical span load distribution and a

corresponding uniform downwash C21nAR. This means

of rating performance is quite appropriate for subsonic

speeds where the vortex drag due to lift with its theoretical minimum of C2/rcAR is the only appreciable contribution. However, at supersonic speeds and whenever the

local flow becomes anywhere supersonic, a new source

of drag--wave drag due to lift--is introduced. As a matter of convenience, the suction parameter definition used

for supersonic speeds is the same as that used for subsonic speeds because, for wave drag due to lift, there is

no simple means of estimating minimum values and the

limits used in the subsonic form are easily reproducible

and not subject to problems of interpretation. The terms

are dependent only on geometric properties and on a flat

surface lift-curve slope, which can be evaluated by any

valid numerical solution of linearized theory. Use of the

same form for both speed regimes, however, means that

supersonic suction parameters cannot approach a limiting

value of 1.0 as they do for low subsonic speeds. However, a process although somewhat complicated may be

used to estimate minimum wave drag due to lift and an

upper limit of suction parameter for supersonic speeds.

In reference 26, Hayes developed a method of calculating wave drag due to lift by consideration of the flow

field created at large distances from the object generating

the lift. The method is closely related to the wellpublicized area rule concept for the calculation of wave

drag generated by the volume or thickness of bodies and

wing-body combinations. Calculations begin with the

creation of a series of equivalent bodies of revolution,

each corresponding to a particular portion of the flow

field propagating away from the body along azimuth

angles ¢ as shown in figure C1. Each of these bodies has

an equivalent cross-sectional area development related to

the summation of the component of lift-generated forces

acting along that azimuth angle. The drag of each of

these bodies may be calculated by use of formulas, 2

2A presentation of these integral formulas is not necessary for the

purpose of this analysis because the drag need be found only for

very specialized cases for which analytic solutions have already

been published.

64

._----- l ------_-

le v

Sketch C- 1

which take into account the second derivative of the

cross-sectional area with respect to the longitudinal distance from the body nose. Hayes shows that the drag due

to lift of the system as a whole may be represented as the

average drag of the series of bodies covering azimuth

angles from 0 ° to 360 °.

At a given point along the axis of an equivalent body

for a given azimuth angle, the equivalent cross-sectional

area is the result of an accumulation of the component of

lifting forces directed along the azimuth line. The summation includes all lifting forces generated by the object

forward of the intersection with the object of a fore Mach

cone from the specified field point as shown in figure C 1.

For simplicity, the lifting object shown in the figure

is a thin delta wing without twist and camber at a very

small angle of attack. As shown in the plan view, the fore

Mach cone intersection with the wing plane has a parabolic shape. For large distances, this intersection is

essentially a straight line with an intersection angle _,.

Through use of some trigonometry, the intersection angle

may be given as

For delta wings with subsonic leading edges

cOt Ale < 1, length of each body up to the point at

which all the forces are accumulated as shown in

sketch C-1 is

I

e

1( 1 cot Ale" _

=l(1 + _ cOt Ale COS¢)

Beyond this point the equivalent cross-sectional area is

constant, and no more drag is created. For delta wings
"_-"_Ale_

I cot Ale _.

.....

,_-- l --1_

_ Ie _-_

Sketch C-2

with supersonic leading edges _ cOtAle > 1 shown in

sketch C-2, the equivalent body length is

Ie = 1(2_] cot Ale cos 0)

Hayes (ref. 26) shows that the equivalent crosssectional area of any point on a body for a given azimuth

angle is given by the relationship

Ae = _-_qfX FL dXo

where FI, is the rate of growth of the lifting force with

increasing distance (for example, lift in pounds per foot

of distance). At the distance le, the point at which all the

lifting force has been taken into account, the crosssectional area will be

Ae,ma x = _2 CL S sin

The negative sign accounts for the occurrence of the

maximum force directly below the lifting object at

0 = -90°. The actual distribution of the equivalent area

with longitudinal distance is dependent on the wing

geometry and the loading distribution. As shown in

sketch C-3, the development may have a quite arbitrary

shape and the drag of a given body may in some

instances be very large. One way of estimating minimum

achievable wave drag due to lift is to assume that the

wing may be shaped to provide an optimum equivalent

area distribution and a corresponding minimum drag for

each equivalent body required for the drag calculation.

Such a limit may not be realistically attainable, but it

does represent an idealized nonzero goal.

An optimum body of revolution and its minimum

drag suitable for this study is given in reference 27. The

A e

I

Sketch C-3

Ae,max

A e

Ae, max

x

Sketch C-4

Sears-Haack body for minimum drag for a given length

and maximum cross-sectional arc has an area distribution

for its front half (up to the maximum area) as shown in

sketch C-4. The drag coefficient of this half-body is

given by

mmax

CD'min- 2 12

This formula may be used to define a minimum wave

drag due to lift for each equivalent body associated with

a given azimuth angle. First the minimum drag equation

is changed to provide a coefficient based on the wing

reference area rather than the body maximum area as

follows:

2

Amax

CD,mi n -

2S 12

Then with the substitution of the previously defined

effective area Ae.ma x and the effective length I e for the

body area and length,

Tt_2C2S fsin 0_

Co,min = 8 L _ Ie )

65
or

-2 2 S (sine') 2

CD'min 8

To find the average drag coefficient for the series of

bodies corresponding to values of 0 from 0° to 360 °,

integrating the term containing sin 0 and Ie is necessary.

As noted previously, le is dependent on the leading-edge

sweep parameter as well as the angle _. Because of symmetry, the average drag can be found by an integration

from 0 = 0° to 90 °. The minimum wave drag due to lift is

thus

x-2 2S 90(sin0?d0 (ACo'min)w= CL- fO \ le/' J

Sketch C-5 shows a plot of the integrand term as a function of the azimuth angle 0 for delta wings with various

values of the leading-edge sweep parameter. A numerical

integration of these curves permits the calculation of an

average drag for each of the selected [3 cot Ale values.

Results of the integration are shown in sketch C-6. This

1.0

.8

.6

/sin _2

lell ]

.4

.2

0 I

0 30 60

_, deg

Sketch C-5

cot Ale

0

.2

N_ .5

1.0

1.5

2.0

I

90

.5

(ACD,rtfin)w .4

_--8_2 C2L(SI12) .3

or

1 f 90 [sin ¢_2 de .2

9°J 0 _ le/l I .l

cot Ale

1 + _ cot Ale

0 .2 .4 .6 .8 1.0

I I I I I

.2 .4 .6 1.0 2.0 4.0

[3cot Ale

Sketch C-6

66
plotusesa scalefortheleading-edgesweepparameter

[_cOtAle that permitsrepresentationof the full

rangeof valuesfrom zeroto infinity.The quantity

13cotAle/(1+ [3cotAle) isplottedonalinearscale.

Althoughthistechniqueforcalculationofminimum

wavedragwasderivedfor adeltawing,it canalsobe

appliedto arrowwings.In fact,thecalculationprocess

maybeemployedin estimationof theminimumwave

dragduetolift forwing-bodyconfigurationswithwings

of arbitraryplanformbysubstitutionof anappropriate

arrowwingfortheactualplanform.SketchC-7helpsto

illustratehowsucha replacement arrowwingmaybe

foundthrougha superpositionof planforms.When,as

shownin thesketch,theactualplanformincludesthe

fuselage,little weightshouldbegivento thefuselage

portionwhichtypicallygenerateslittle of thetotallift.

Thelengthof thearrowwingshouldrepresent alongitudinaldistanceoverwhichthelift maybedevelopedinan

efficientmanner.Likewise,thespanshouldrepresent a

lateraldistanceoverwhichanear-elliptical spanloaddistributionmaybeachieved.

Foranarrowwingasshownbythedashedlinein

sketchC-7,thewingreferenceareamaybeexpressedas

12

S = _(1-kn)(13cotAle )

The minimum wave drag due to lift for any arrow wing

may thus be expressed as

(ACD,n_n)w n -- r; 'tsinO_

2 - _(1-kn)(13c°t Ale) _J0 kie'---'-[-_d_p

CL

x 4

x 3

X _I I

2 __ _"_AI e

Sketch C-7

D

A B

C

D

Sketch C-9

If the actual wing reference area differs from the arrow

wing reference area, the right-hand side of the equation

should be multiplied by the ratio of the reference area to

the arrow wing area.

The effective equivalent body lengths for use in the

drag equation take on values defined in the following formulation which accounts for subsonic and supersonic

leading edges and positive or negative notch ratios:

Ie = x 2 -x 1

I e = (x 3 -Xl)(l + 13cos Ale cos_)

I e = (x 3-x 1)+(x 3-x 1)(13 cos Ale cOs _ - 1)

I e = (x 3 - x 1)(213 cos Ale cos t_)

Use whichever length is greatest.

For some situations, for example, those shown in

sketches C-8 and C-9, the belief that an optimum wing

loading could be maintained over the lengths given by

the formula is not reasonable. Optimum load distributions do not depart to any great degree from that given by

uniform lifting pressures. A uniform load would have a

67
centerof pressureof thecentroidof areawhichfor an

arrowwingis

1 1

x c = 5(x2-xl) + _(x 3 -xl)

For the wing shown in sketch C-8, one half the total lift

would be accumulated when the cutting plane intersection labeled B is reached. No matter how the load is distributed over the rest of the wing, the drag is not likely to

be any lower than that of an equivalent body terminating

at the cutting plane intersection C whose length is

2

I e = 2X c = _(X 2+x 3)

If the arrow wing leading edge is supersonic, for some

cutting plane intersections the equivalent body length

may be greater than 2xc as shown in sketch C-9 where

2

Ie = _(x 2+x 3-2xl) +2(x 3-xl)([_cotAlecos_-l)

Whichever of these two body lengths is longer is taken to

be a maximum not to be exceeded by lengths defined

solely by cutting planes. By use of numerical evaluation

of the integral in the equation for wave drag due to lift,

the minimum wave drag due to lift for arrow wings may

be represented by curves shown in figure C2. A value of

zero for _ cos Ale represents a wing of infinite length

and no span which has no wave drag due to lift. On the

other extreme, 13cos Ale = oo represents a wing of infinite span and no length which has an infinite wave drag

due to lift.

The minimum vortex drag due to lift for arrow wings

may also be represented by figure C2. The minimum vortex drag due to lift is simply

2

C L

( ACo'min)vor - r_AR

For the arrow wing shown in the sketch at the top of figure C2, the span is

2l

b = -if(13 cot Ale )

IJ

and the aspect ratio is

ARb 2 4 [3cot Ale

S _ 1-k n

Thus the minimum vortex drag due to the lift is

(ACD,min)vo r _ I] 1 -k n

2 4_ _ cot Ale C£

68

In contrast to the wave drag, the vortex drag is zero for

cot Ale = 0% which represents a wing of infinite span,

and is infinite for _ cot Ale = 0.

The minimum total drag due to lift obtained by an

addition of these two curves occurs at _ cot Ale = 0.8 for

the delta wings and at lower values of 15cot Ale for more

highly swept leading edges. An important drag contribution so far not discussed in this analysis that has a strong

impact on the choice of leading-edge sweep angle is the

wave drag due to the configuration thickness or volume.

This contribution tends to increase with increasing values

of the leading-edge sweep parameter [3 cos Ale in much

the same way as does the wave drag due to lift. When

this additional drag term is taken into account, the wing

sweep angle for minimization of the complete configuration drag falls well within the subsonic leading-edge

range. The optimum leading-edge sweep parameter

_i cos Ale must be evaluated on an individual basis for

any given design exercise.

The maximum attainable suction parameter may be

expressed as

CL tan( CL/CLc)- [( CD,min)w+ ( C2/_AR) l

Ss,ma x =

C L tan ( CL/CLc *) - (C2/rcAR)

Through use of lift-curve slopes CL_ evaluated by the

WINGDES code, the maximum suction parameters

shown in the plot at the bottom of figure C2 were

obtained. This plot shows that high values of suction

parameter comparable with those attainable at subsonic

speed could be realized only if it were possible to use

wings with high notch ratios. For more practical arrow

wings approaching the delta wing planform, the potential

for high values of suction parameter is much reduced. In

addition, it must be recognized that the high values of

maximum suction parameter for highly swept wings

(13 cos Ale approaching zero) are countered by high

values of vortex drag which tend to negate the wave drag

improvements. It should also be reiterated that the

arrow wing performance maximums given here are,

as mentioned previously, based on highly idealized

assumptions.

Before using the plot for Ss,ma x in figure C2 to estimate achievable values of the suction parameter, an additional factor must be taken into account. Reference 6 has

shown that as Mach number increases, linearized theory

becomes less reliable as an indicator of aerodynamic performance. Achievable suction parameters were shown to

decline with increasing Mach number. A plot of the ratio

of achieved experimental suction parameters to theoretically indicated suction parameters from reference 6 are

repeated in sketch C-10. A final realistic estimate of
maximumsuctionparametermaythusbe foundby

applyingtheK S factor for a given Mach number from

sketch C-10 to the theoretical maximum suction parameter defined by the equation for Ss,ma x or by the curves

of figure C2.

The minimum wave drag due to volume may be estimated through use of another Sears-Haack body of revolution that gives minimum wave drag for a given length

and volume. The drag coefficient of this body is

(Co,o,min) w - 128_ 12S /-_V/2

or

'2Iv/2 128 e,v

( C D, O,min ) w - -_ -'_ l e.-_'-

with le, v representing an effective length for each equivalent body corresponding to a given azimuth angle 0.

Then the minimum wave drag due to volume for a wingbody configuration may be given as

1 ,80l e¢

(CD'O'min)w _ S _ JO (le,v/l)4

If the actual wing reference area differs from the arrow

wing area, the right-hand side of the equation should be

multiplied by the ratio of the arrow wing area to the reference area.

The effective equivalent body lengths take on values

of

I e : X 4

l e = X 1 + (x 3 -Xl)(1 + [3 COt Ale cos (_)

l e = x 4 + (x 3 - Xl)(_] cot Ale cos _ - l) -x 1

I e = (x 3-x 1)(21 ] cot Ale cosO)

Use whichever length is the greatest.

This collection of expressions for drag component

idealized minimums in combination with estimates of the

remaining skin friction drag may be used in studies

aimed at setting performance limits. As an example, consider an arrow wing with all the volume contained within

the planform. For such a configuration, the minimum

wave drag due to volume may be expressed as

128 "V 2/3"3 1 90 1

(Co,0.min)w = _-_[(l-kn)l]cOtAle]2/--_--) _-0 _0 _d*

1.0

.8

SS,exp .6

K S = --

SS, th 4

.2

1 2 3 4

M

Sketch C- 10

where the quantity V213/S is a measure of the configuration volume. Although the same parameters as seen in

the expressions for (ACD,0,min) w and (ACo,0,min)vo r

reappear here, it is no longer possible to construct a plot

covering the range of variables given in figure C2. However, for a given notch ratio and a selected Mach number

and design lift coefficient, as well as a specified volume

and skin friction drag coefficients, informative plots such

as that shown in figure C3 may be prepared.

Figure C3 is intended only to serve as an example of

the application of the minimum drag expressions. These

results are for a simplified and not very practical case.

Nevertheless, they do illustrate the character of the various drag components and the trade-offs involved in the

search for optimized configurations. The design conditions are assumed to be

k n = 0.0 (Delta wing)

M=2.0

h = 50000 ft

S = 8000 ft 2

C L = 0.12

V 2/3

- 0.10

S

Again, the results are shown as a function of the leadingedge sweep angle parameter. Each minimum drag

expression previously discussed is represented as is an

estimate of skin friction drag. The total drag with an

empirical drag correction term is given as follows:

ACD,co r = (Ss,th-Ss,emp) [C L tan CLa _ARJ 1

where Ss,em p is an estimated achievable suction parameter obtained by application of K S for a given Mach number to the theoretical maximum suction parameter. Some

of the wing planforms represented are shown by

69
sketches. As shown by the curves for total CD, an optimum planform will have a leading-edge swept well

behind the Mach line. The optimum 13cot Ale is strongly

influenced by the contribution of wave drag due to

volume.

These idealized minimum drag components could be

valuable for parametric studies of aerodynamic performance dependence on fundamental configuration geometric parameters. They should also be useful in placing

limits on achievable performance for specific airplane

designs to separate reasonable projections from outlandish claims. The strategies for the estimation of drag

minima outlined in this appendix have been incorporated

into a computer code, CDMIN. Although the code derivation as shown here is quite complex, the code itself is

easy to use. The user is required to input only the Mach

number, the wing reference area, the wing span, the configuration volume, and the dimensions x I to x 4 shown in

sketch C-7.

Fore Mach

Intersection

of Mach cone

with win

Field point

Tangent

Top view

ieldpoint

Wing plane

R.=vi.w °s°

Projection / /

Rear view and projection

Ay COS

_xtan tl

_ = tan-I AY = tan-1 IAy cosAy_)/tan _

Side view

Figure C1. Illustration of area rule concepts.

70
cot Ale

1 + _ cot Ale .8 1.0

.4 .4_ Vortex J kn

_c_ i_

-__

0

S s,nlax

krt

0.8

1.0.8 ._

.6

ta,_--e'"ze an a minimum drag due to lift for arrow wings.

C2. Figure

71
.016

.012

%

0 [JcotA

72
References

1. Carlson, Harry W.; Mack, Robert J.; and Barger, Raymond L.:

Estimation of Attainable Leading-Edge Thrust for Wings at

Subsonic and Supersonic Speeds. NASA TP-1500, 1979.

2. Carlson, Harry W.; and Walkley, Kenneth B.: A Computer Program for Wing Subsonic Aerodynamic Performance Estimates

Including Attainable Thrust and Vortex Lift Effects. NASA

CR-3515, 1982.

3. Carlson, Harry W.; and Darden, Christine M.: Validation of a

Pair of Computer Codes for Estimation and Optimization of

Subsonic Aerodynamic Performance of Simple Hinged-Flap

Systems for Thin Swept Wings. NASA TP-2828, 1988.

4. Carlson, Harry W.; Darden, Christine M.; and Mann,

Michael J.: Validation of a Computer Code for Analysis of

Subsonic Aerodynamic Performance of Wings With Flaps in

Combination With a Canard or Horizontal Tail and an Application to Optimization. NASA TP-2961, 1990.

5. Carlson, Harry W.; and Walkley, Kenneth B.: Numerical Methods and a Computer Program for Subsonic and Supersonic

Aerodynamic Design and Analysis of Wings With Attainable

Thrust Considerations. NASA CR-3808, 1984.

6. Carlson, Harry W.; and Mann, Michael J.: Survey and Analysis

of Research on Supersonic Drag-Due-to-Lift Minimization

With Recommendations for Wing Design. NASA TP-3202,

1992.

7. Carlson, Harry W.; McElroy, Marcus O.; Lessard, Wendy B.;

and McCullers, L. Arnold: Improved Method for Prediction of

Attainable Wing Leading-Edge Thrust. NASA TP-3557, 1996.

8. Carlson, Harry W.; and Walkley, Kenneth B.: An Aerodynamic

Analysis Computer Program and Design Notes for Low Speed

Wing Flap Systems. NASA CR-3675, 1983.

9. Carlson, Harry W.; and Darden, Christine M.: Applicability of

Linearized-Theory Attached-Flow Methods to Design and

Analysis of Flap Systems at Low Speeds for Thin Swept Wings

With Sharp Leading Edges. NASA TP-2653, 1987.

10. Carlson, Harry W.; and Miller, David S.: Numerical Methods

for the Design and Analysis of Wings at Supersonic Speeds.

NASA TN D-7713, 1974.

11. Polhamus, Edward C.: Predictions of Vortex-Lift Characteristics by a Leading-Edge Suction Analogy. J. Aircr., vol. 8,

no. 4, Apr. 1971, pp. 193-199.

12. Lan, C. Edward; and Chang, Jen-Fu: Calculation of Vortex Lift

Effect for Cambered Wings by the Suction Analogy. NASA

CR-3449, 1981.

13. Harris, Roy V., Jr.: An Analysis and Correlation of Aircraft

Wave Drag. NASA TM X-947, 1964.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

Middleton, W. D.; and Lundry, J. L.: A System for Aerodynamic Design and Analysis of Supersonic Aircraft. Part 1-

General Description and Theoretical Development. NASA

CR-3351, 1980.

Middleton, W. D.; Lundry, J. L.; and Coleman, R. G.: A System for Aerodynamic Design and Analysis of Supersonic Aircraft. Part 2--User's Manual. NASA CR-3352, 1980.

Middleton, W. D.; Lundry, J. L.; and Coleman, R. G.: A System for Aerodynamic Design and Analysis of Supersonic

Aircraft. Part 3--Computer Program Description. NASA

CR-3353, 1980.

Middleton, W. D.; and Lundry, J. L.: A System for Aerodynamic Design and Analysis of Supersonic Aircraft. Part 4_

Test Cases. NASA CR-3354, 1980.

Sommer, Simon C.; and Short, Barbara J.: Free-Flight Measurements of Turbulent-Boundary-Layer Skin Friction in the

Presence of Severe Aerodynamic Heating at Mach Numbers

From 2.8 to 7.0. NACA TN 3391, 1955.

Hall, Charles E: Lift, Drag, and Pitching Moment of LowAspect-Ratio Wings at Subsonic and Supersonic Speeds.

NACA RM A53A30, 1953.

Campbell, Bryan A.; Hom, Kam W.; and Huffman, Jarrett K.:

Investigation of Subsonic Maneuver Performance of a Supersonic Fighter Cranked Wing. NASA TP-2687, 1987.

Capone, Francis J.: Longitudinal Aerodynamic Characteristics

of a Twin-Turbofan Subsonic Transport With Nacelles

Mounted Under the Wings. NASA TN D-5971, 1970.

Leavitt, Laurence D.: Effect of Empennage Location on TwinEngine Afterbody Slant Nozzle Aerodynamic Characteristics

at Mach Numbers From 0.6 to 1.2--Wind Tunnel Tests.

NASA TP-2116, 1983.

Yip, Long E; and Parlett, Lysle E: Low-Speed Wind-Tunnel

Tests of a 1/lO-Scale Model of an Advanced Arrow-Wing

Supersonic Cruise Configuration Designed for Cruise at Mach

2.2. NASA TM-80152, 1979.

Radkey, R. L.; Welge, H. R.; and Felix, J. E.: Aerodynamic

Characteristics of a Mach 2.2 Advanced Supersonic Cruise

Aircraft Configuration at Mach Numbers From 0.5 to 2.4.

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Sections--Including a Summary of Airfoil Data. Dover Publ.,

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Appl. Math., vol. IV, no. 4, Jan. 1947, pp. 361-366.

73
Table I. Sample Input Data for AERO2S Code

(a) Aspect ratio 2 wing-body

AR 2 DELTA - HALL, RM A53A30 5 PCT - TWISTED AND CAMBERED

SINPTI XM=.61,RN=3.0,JBYMAX=I2,ELAR=4.0,

SREF=578 .0, CBAR=22 . 67, XMC=38 . 12 ,XMAX=60 . 44,

CLDES=. 3,

NLEY=4, TBLEY=0 . 000,1 . 650,2 . 620,17 . 00,

TBLEX=0 . 000,16 . 06,26 .28,55 . 12 ,

NTEY=4 ,TBTEY=0 .000,2 .000,2 .210,17 .00,

TBTEX=60 . 44,60 .44,55 . 12,55 .12 ,

NYC=I2 ,TBYC=0.0,2 . 6,2 . 62,3.4,5 .i, 6.8,8 .5, i0.2, ii . 9,13 .6,15.3,17 . 0,

NPCTC= 9,

TBPCTC=0.000,2.500,5.000,10.00,20.00,40.00,60-00,80"00

TZORDC=0.000,0.000,0.000,0.000,0 000,0.000,0.000,0 000

0.000,0.000,0.000,0.000,0 000,0.000,0.000,0 000

0.409,0.510 0.546 0.446,0 211,0.097,0.057,0 023

0.557 0.665 0.708 0.726,0 406,0.200 0.128,0 082

0.836 0.950 1.014 1 087,1 055,0.524 0.330,0 255

1.114 1.226 1.293 1 390,i 480,1.162 0.725,0 548 0

1.393 1.494 1.564 1 655,1 780,1.815 1.453,1.016,0

1.672 1.761 1.824 1 922,2 060,2.193 2.178,1.972,1

1.950 2.023 2.076 2 163,2.296,2.467 2.553,2.569,2

2.229 2.282,2.324,2 392,2.502,2.665 2.781,2.862,2

2.508 2.538,2.562,2 604,2.675,2.790,2.883,2.962,3

2.787 2.787,2.787,2 787,2.787,2.787,2.787,2.787,2

NYR=4 ,TBYR= 0 . 000,2 . 600,2 . 620,17 . 00,

TBTOC=0.000,0.000, .0500, .0500,

TBROC:0.000,0.000, .0028, .0028,

TBETA=.3000, .3000, .3000, .3000,

NALPHA=I3 , TALPHA=-4 . 0, -2 .0,0 . 0,2 . 0,4 . 0, 6 . 0 , 8 . 0, i0 . 0,12 . 0,14.0,

16.0,18.0,20.0, $

AR 2 DELTA - HALL, RiM A53A30 5 PCT FLAT

$1NPTI TZSCALE=0.0, $

i00.0,

0.000,17"0.0,

0.000 17"0.0,

0.005 17"0.0,

0.053 17"0.0,

0.193 17"0.0,

437 17"0.0

819 17"0.0

524 17"0.0

513 17"0.0

917 17"0.0

031 17"0.0

787 17"0.0

74
Table I. Continued

(b) Cranked wing supersonic fighter

SUPERSONIC CRUISE FIGHTER - CAMPBELL, TP 2687 - DLN:I5,DTN=I0

$INPTI XM=0.50,RN=2.9,JBYMAX=8,ELAR=4.0,IVOROP=I,IPRSLDA=I,

SREF=I63.5,CBAR=I2.4,XMC=20.64,XMAX=33.50,

NLEY=8,TBLEY=0.000,0.470,0.870,1.160,1.190,1.200,5.510,9.550,

TBLEX=0.000,2.000,4.000,6.000,6.540,12.88,26.37,27.84,

NTEY=4,TBTEY=0.000,1.200,1.210,9.550,

TBTEX=33.50,33.50,31.70,29.79,

NYC=2,TBYC=0.00,9.55,NPCTC=2,TBPCTC=0.00,100.0,TZORDC=52*0.0,

NYR:5,TBYR:0.000,1.200,5.510 5.520

TBTOC:0.000,.0160,.0400 .0400

TBROC=0.000,.0001,.0004 .0010

TBETA=.5000,.5000,.5000 .5000

NLEFY=5,TBLEFY=0.000,1 200,4 630,5

TBLEFC=0.000,0 000,3.310,1

9.550,

.0400,

.0010,

.5000,

510 9.550,

080 .4870,

TBLEFD=0.000,4 252,4.252,14.67 14.67,

NTEFY=4,TBTEFY=0.000,1 540,1.550,9.550

TBTEFC=0.000,0 000,i.658, .4870

TBTEFD=0.000,0 000,9.972,9.972

NADLEFD=I,TXMLEFD=0.00 ,

NADTEFD=I,TXMTEFD=0.000,

NALPHA=I4,TALPHA=-6.,-4.,-2.,0.0,2.,4.,6.,8.,10.,12.,14.,16.,

18.,20., S

SUPERSONIC CRUISE FIGHTER - CAMPBELL,TP 2687 - DLN=30,DTN=20

$INPTI TBLEFD=0.000,9.100,9.100,29.42,29.42,

TBTEFD=0.000,0.000,19.94,19.94,NADLEFD=0,NADTEFD=0 S

75
Table I. Continued

(c) Subsonic transport

SUBSONIC TRANSPORT WING BODY - CAPONE, TN D-5971

$INPTI XM:.55,RN:2.25,JBYMAX:I0,ELAR:8.0,

SREF=3674.3,CBAR=21.17,XMC=83.34,XMAX=I71.2,

NLEY=6,TBLEY=0.000,4.000,8.000,11.22,32.19,87.80,

TBLEX=

NTEY=6,TBTEY=

TBTEX=

NYC=I0,TBYC=0

NPCTC=9,

TBPCTC=0.000

TZORDC=0.000

0.610

1.900

3.800

0.659

0.294,

0.000,

0.000,

0.000,

0.000,

NYR:6,TBYR=0.

TBTOC=0.

TBROC=0.

0.000,6.000,20.00,

0.000,4.000,8.000,

171.2,160.0,142.0,

.000,4.0,8.0,11.2,

500,5.000,i0.00,

488,0.943,1.752

010,1.400,2.100

150,2 400,2.800

820,3 840 3.880

700,0 720 0.685

460,0 495 0.512

170,0 240 0.340

128,0 206 0.295

063,0 i01 0.145

0 056,0 090 0.129

000,ii.22,16.44,25

145,0.154,0.125,0.

013,0.038,0.018,0.

TBETA=0.320,0.200,0.300,0.

NALPHA=I3,TALPHA=-2.0,-I.0,0.0,

i0.0, $

60.88,

11.22,

98.11,

11.22,

76.09,105.1,

32.19,87.80,

98.11,114.8,

16.44,25.48,32.19,82.20,87.80,

20.00,

2.970,

3 i00,

3 400,

3 960,

0 505,

0 472,

0 390

0 347

0 170

40.00,60.00

3.965,4.645

4.000,4.600

4.000,4 400

4.120,4 280

0.380,0 270

0.375,0 270

,0.340,0 255

,0.304,0 247

,0.149,0 ii0

0 152,0.133,0 108

.48,32.19,87.80,

i00,0.i00,0.i00,

004,0. 005,0. 005,

400,0.400,0.400,

1.0,2.0,3.0

80.00,100.0,

7 172,12.56,17"0

6 300,i0.30,17"0

5 400,7.650,17"0

4 440,4.600,17"0

0 160 0.000,17"0

0 160 0.000,17"0

0 125 0.000,17"0

0 131

0.064

0.057

0

0

0

0

0

0

0

0.000,17"0.0

0.000,17"0.0

0.000,17"0.0

,4.0,5.0,6.0,7.0,8.0,9.0,

76
Table I. Continued

(d) Supersonic transport wing-body-horizontal tail

SUPERSONIC TRANSPORT WITH DEFLECTED FLAPS AND HORIZONTAL

SINPTI XM=0.09,RN=4.19,JBYMAX=I0,ELAR:6.0,

SREF=I4400.0,CBAR=77.76,XMC=220.91,XMAX=372.0,

NLEY=4,TBLEY=0.0000,7.0000,51.710,81.330,

TBLEX=0.0000,104.70,234.55,280.20,

NTEY=4,TBTEY=0.0000,7.0000,25.800,81.330,

TBTEX=372.00,277.70,277.70,294.97,

NYC=I5,TBYC=0.0000,4.4090,13.482,20.899,23.218,27.202,29.498,

38.294,47.061,49.819,53.701,56.458,67.509,76.399,

81.330,

NPCTC=I0,

TBPCTC=0.000,5.000,10.00

TZORDC=I4.30,15.20,17.50

13.20,13.20,12.79

6.250,6

4.280,4

4.030,4

3.760,4

3.530,4

1.950,2

0.660,0

0.470,0

0.330,0

0.260,0

780,6 990

850,5 250

600,4 940

270,4 600

000,4 370

430 2 800

960 1.200

7OO 0.880

450 0.580

370 0.430

,20.00,30.00,40.00

17.40,15.40

11.24,9.250

6 890,6 200

5 430,5 130

5 200,4 980

4 890,4 660

4 650,4 430

3 120,3 070

1.480,i 600

1.120,i 250

0.760,0 900

0.590,0.720

12.55

7.380

5.320

4.600

4.450

4.220

4 I00

,2 880

1 630

1 330

1 040

0 870,

50.00,60.00

10.00,8 000

5.700,4 250

4.500,3 770

4.030,3 500

3.930,3 430

3.780,3 300 2

3.660,3 200 2

2.680,2 450,2

1.660,i 630 1

1.400,1 420 1

1.130,i 210 1

0.980,i 070 1

80.00

6 000

2 000

2 330

2 470

2 470

38O

35O

000

57O

430

30O

2OO

TAIL - YIP, TM 80152

i00.0,

10.00,16"0.0

.6000,16"0.0

1.000,16"0 0

1.420,16"0 0

1.500,16"0 0

1.560,16"0 0

1.580,16"0 0

1.580,16"0 0

1.480,16"0 0

1.430,16"0 0

1.340,16"0 0,

1.280,16"0 0,

0.300

0.270

0.280

NYR=7,TBYR=0

TBTOC=0

TBROC=0

TBETA=0

0.310

0.270

0.280

00000

04000

00000

50000

NLEFY=I0,TBLEFY=0

TBLEFC=0

TBLEFD=0

NTEFY=I0,TBTEFY=0

TBTEFC=0

TBTEFD:0

000

000

000

000

000

000

0.320 0.350,0.410 0 470,

0.270 0.280,0.290 0 310,

0.280 0.280,0.280 0 280,

4.40900,20.8990,38.2940,

0.02380,0.02870,0.03000,

0.00024,0.00050,0.00048,

0.66000,0.55000,0.58000,

7.000 7.010,17.59,17.

0.000

0.000

17.00

0.000

0.000

0.550,0 620 0

0.330,0.360 0

0.290,0.300 0

51.7000,51.71

0.03000,0.030

0.00040,.0000

790,0.930

430,0.520

330,0.400

0,81.3300

0,0.03000

3,.000013

16"0 0,

16"0 0,

16"0 0,

0.60000,0.60000,0.60000

60,44.42,51.71,64.00,64.01,81.33,

18.84,18.80,18.80,18.78,8.370,6.770,6.770,4.510,

4.300,4.300,12.80,12.80,ii.20,ii.20,17.60,17.60,

17.01,25.80,25.81,36.59,36.60,61.00,61.01,81.33,

15.93,15.93,0.000,0.000,14.50,9.650,0.000,0.000,

30.00,30.00,0.000,0.000,30.00,30.00,0.000,0.000,

NADLEFD=I,TXMLEFD=0.0

NADTEFD=I,TXMTEFD=0.0

ILS2=2,DELTA2=-5.0,

NLEY2:2,TBLEY2=2.0000,23.700,

TBLEX2:323.20,348.99,

NTEY2=2,TBTEY2=2.0000,23.700,

TBTEX2=361.00,355.39,

NYC2=2,TBYC2=2.0000,23.700,NPCTC2=2,TBPCTC2=0.0000,100.0,

TZORDC2=52*0.0,

NYR2:2,TBYR2=2.000,23.70

TBTOC2:0.035,0.035

TBROC2=0.000,0.000

TBETA2=0.500,0.500

NALPHA=I6,TALPHA=-6.0,-4

16.0,18.0,20.0,22.0,24.0

0,-2.0,0.0,2.0,4.0,6.0,8.0,10.0,12.0,14.0,

S

77
Table I. Continued

(e) Two-dimensional airfoil

NACA 4409 AIRFOIL, REPORT 586 - ATTAINABLE THRUST

SINPTI XM=.06,RN=8.0,JBYMAX=I,ELAR=25.0,IVOROP=0,

XMCPLT=I.0,

SREF=2.0,CBAR=I.0,XMC=0.25,XMAX=I.0,

NLEY=2,TBLEY=0.000

TBLEX=0.000

NTEY=2,TBTEY=0.000

TBTEX=I.000

NYC=2,TBYC=0.0,1.0

TBPCTC=0.000,2.500

TZORDC=0.000, .0062

16"0.0,

1.000,

0.000,

1.000,

1.000,

NPCTC=I0,

5.000

.0104

0.000,.0062,.0104

16"0.0,

NYR=2, TBYR=0.000,1.000,

TBTOC=0.090,0.090,

TBROC=.0089,.0089,

TBETA=.3000,.3000,

NALPHA=I3,TALPHA=-8.0,-6

14.0,16

XMCPLT=0.80

$INPTI XMCPLT=0.80, $

XMCPLT=0.60

$INPTI XMCPLT=0.60, $

CALIBRATION

,10.00,20.00,30.00,40.00,60.00,80.00,100.0,

,.0182, .0302, .0374,.0400,.0357,.0235,0.000,

,.0182,.0302,.0374,.0400,.0357,.0235,0.000,

.0,-4.0,-2.0,0.0,2.0,4.0,6.0,8.0,10.0,12.0,

.0, $

78
Table I. Continued

(f) Supersonic transport flap sytem evaluation

SUPERSONIC TRANSPORT EXAMPLE - FLAP EVALUATION

$INPTI XM=.3,RN=2.0,JBYMAX=I0,ELAR=4.0,

SREF=2.65,CBAR=I.85,XMC=2.77,XMAX=5.47,

CLDES=.6,

NLEY=I3,

TBLEY=0.00,0.04,0.10,0.112,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90,1.00,

TBLEX=0.00,0.20,1.00,1.425,1.95,2.55,3.02,3.32,3.56,3.78,4.00,4.22,4.43,

NTEY=I3,

TBTEY=0.00,0.04,0.10,0.112,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90,1.00,

TBTEX=5.47,5.15,4.28,4.075,4.11,4.18,4.26,4.34,4.43,4.51,4.60,4.68,4.76,

NYR=4,TBYR=0.0000,0.5000,0.8000,1.0000,

TBTOC=0.0300,0.0300,0.0300,0.0300,

TBROC:.00030,.00030,.O0050, .00050,

TBETA=0.5000,0.5000,0.5000,0.5000,

NYC:20,

TBYC=0.0000,0.0405,0.0946,0.1486,0.2027,

0.2568,0.3108,0.3649,0.4189,0.4730,

0.5270,0.5811,0.6351,0.6892,0.7432,

0.7973,

NPCTC=I3,

TBPCTC=0.00,

TZORDC=

0.000000,

-0.143945,

-0.241472,

0.000000,

-0.098756,

-0.220214,

0.000000,

-0.034445,

-0.110650,

0.000000,

-0.006233,

-0.039191,

0.000000,

0.003218,

-0.009765,

0.000000,

0.010703,

0.012749,

0.000000,

0.016938,

0.028782,

0.000000,

0.019930,

0.038234,

0.000000,

0.021225,

0.040849,

0.8514,0.9054,0.9595,1.0000,

5.00,10.0,15.0,20.0,30.0,40.0,50.0,60.0,70.0,80.0,90.0,100.0,

-0.004681,

-0.180760

-0.240753

-0.000794

-0.132249

-0.242517

0 002250

-0 051759

-0 131416

0 002784

-0 013038

-0 049239

0 002826

0 000706

-0 006247

0 003150

0.011297

0 014052

0 003687

0 019622

0 032042

0 003996

0 023506

0 044912

0 004165

0 025039

0 046328

-0.019204,

-0.210323,

-0.227979,

-0.008815,

-0.164330,

-0.260285,

0.001000,

-0.070511,

-0.152325,

0 003749,

-0 020937,

-0 047659,

0 004374,

-0 002395,

-0 006488,

0 005279,

0 011690,

0 011896,

0 006449,

0 022285,

0 032985,

0 007093,

0 027108,

0 047363,

0 007439,

0 028834,

0 048703,

-0.037864, -0

-0.230986,

13"0.000000,

-0.020353, -0

-0.193929,

13"0.000000,

-0.002214, -0

-0.090256,

13"0.000000,

0.003652, 0

-0.029711,

13"0.000000,

0.005197, 0

-0.005927,

13"0.000000,

0.006864, 0

0.012018,

13"0.000000,

0.008736, 0

0.025056,

13"0.000000,

0.009734, 0

0.030866,

13"0.000000,

0.010255, 0

0.033458,

13"0.000000,

058624,

034084

006841,

002763,

005492,

008072,

010709,

012077,

012769,

-0 102124,

-0 065313

-0 019173

-0 000819

0 004920

0 009720

0 014054

0 016207

0 017219

79
Table I. Concluded

(f) Concluded

0.000000 0

0.019821 0

0.032310 0

0.000000 0

0 016472 0

0 027540 0

0 000000 0

0 014042 0

0 026038 0

0 000000 0

0 012492 0

0.025285 0

0.000000 0

0.012064 0

0.022008 0

0.000000, 0

0.012104, 0

0.016794, 0

0.000000, 0

0.013056, 0.

0.014202, 0.

0.000000 0.

0.014391 0.

0.018069 0.

0.000000 0

0.015754 0

0 022938 0

0 000000 0

0 016828 0

0 027724 0

0 000000 0

0 017221 0

0 030686 0

NLEFY=I4,

TBLEFY=0.00,.III

TBLEFC=0.00,0.00

TBLEFD=0.00,0.00

NTEFY=I6,

TBTEFY=0.00, .iii

.510,.511

TBTEFC=0.00,0.00

0.00,.235

TBTEFD=0.00,0.00

0.00,16.0

NADLEFD=I,TXMLEFD=0.0,

NADTEFD=I,TXMTEFD=0.0,

004015,

023154,

034719,

003587,

018834,

029189

003272

015617

.027793

.003022

.013523

028579

002919

012933

027143

002687

012968

022316

002574

014303

017284

002450,

016282,

017478,

002307,

018420,

023247,

002143,

020254,

029200,

001989,

021127,

.033050,

0.007141

0.026402

0.036816

0.006314

0.021203

0.029484

0 005721,

0 017326,

0 026923

0 005289

0 014141

0 028468

0 005129

0.013328

0.029196

0.004845

0.013209

0 026000

0 004769

0 014880

0 020543

0 004689

0 017520

0 018368

0 004567

0 020515

0.022945

0.004383

0.023230

0.030140

0.004165,

0.024697,

0.034967,

0.009803,

0.029545,

13"0.000000,

0.008581,

0.024716,

13"0.000000,

0.007716,

0.022147,

13"0.000000,

0.007120,

0.019504,

13"0.000000,

0.006919,

0.017458,

13"0.000000,

0.006657,

0.012882,

13"0.000000,

0.006690,

0.014816,

13"0.000000

0.006741,

0.018114,

13"0.000000

0.006734,

0.022024,

13"0.000000

0.006618,

0.025728,

13"0.000000

0.006396,

0.027894,

13"0.000000

0.012153 0.016236,

0.010531 0.013782,

0.009387 0.012037,

0.008629 0.010923,

0.008392 0.010606

0.008188, 0.010546

0.008372, 0.011099

0.008613 0.011840

0.008790, 0.012535

0.008811, 0.012994,

0.008634, 0.013030,

,.i12,.200,.300,.399,.400,.500,.600,.699,.700,.800,.900,i.00,

,0.00,.248,.248,.248,.248,.156,.123,.iii, .iii, .098,.086,.083,

,26.0,26.0,26.0,26.0,32.0,32.0,32.0,32.0,36.0,36.0,36-0,36-0,

.i12,.189,.190

.600, .700,.800

.203,.234,0.00

.213,.177,.153

14.0,14.0,0.00

16.0,16.0,16.0

.310,.311,.389, .390,

.900,1.00,

0.00,.304,.276,0.00,

.118,.083,

0.00,14.0,14.0,0.00,

16.0,16.0,16.0,16.0,

NALPHA:I6,TALPHA:0 .0,2 .0 ,4 .0, 6 .0,8 .0, I0 .0,12 .0,14 .0,16 .0,18 .0,

20.0,22.0,24.0,26.0,28.0,30.0, $
Table II. Sample Input Data for WINGDES Code

(a) Supersonic transport wing evaluation

CURVED L E WING - DARDEN, TP-2446 T & C NO CONSTRAINT

$INPTI XM=2.4,RN:2.0,JBYMAX=40,IEMPCR=I,

SREF=2.5375,CBAR=I.686,XMC=I.43,XMAX=3.29,

NLEY=II,

TBLEY=0.00,0.10,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90,1.00,

TBLEX=0.00,0.12,0.48,1.08,1.55,1.85,2.09,2.31 2.53,2.75,2.96,

NTEY=II,

TBTEY=0.00,0.10,0.20,0.30,0.40,0.50,0.60,0.70 0.80,0.90,i.00,

TBTEX=2.58,2.60,2.64,2.71,2.79,2.87,2.96,3.04 3.13,3.21,3.29,

NYR=4,TBYR=0.0000,0.5000,0.8000,1.0000,

TBTOC=0.0300,0.0300,0.0300,0.0300,

TBROC=.00010,.00030, .00050,.00050,

TBETA=0.5000,0.5000,0.5000,0.5000,

NYC=II,TBYC=0.00,0.10,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90,1.00,

NPCTC=9,

TBPCTC=0.000,5.000,10.00,20.00,30.00,40.00,60.00 80.00,100.0

TZORDC=.II82, 1083,.0965 .0721

.1345,

.1099,

0357,

0059,

0026,

0018,

0007,

0000,

0000,

1290

1156

0487

0123

0053

0029

0013

0003

0002

0000,.0001

CLZPR:.08,

1195 0968

1127 0972

0557 0560

0157 0184

0073 0090

0039 0048

0018 0023

0007 0013

0005 0010

0002 0004

0494,.0307

0727, 0501

0764, 0544

0485, 0374

0176, 0148

0088, 0075

0046, 0040

0023, 0019

0015, 0016

.0013, 0016

.0015, 0025

0063

0139

0151

0122

0054

0029

0015

0008

0014

0019

0050

-.004,- 013

012,- 035

- 012,- 027

- 011,- 028

- 006 - 017

- 003 - 010

- 002 - 005

- 001 - OO2

.0010 .0O06

.0022 .0023

.0075 .0102

17"0 0

17"0 0

17"0 0

17"0 0

17"0 0

17"0 0

17"0 0

17"0 0

17"0 0

17"0 0

17"0 0

NALPHA=I7,TALPHA=-6.0,-5.0,-4.0,-3.0,-2.0,-I.0,0.0,1.0,2.0,3.0,

4.0,5.0, 6.0,7.0,8.0,9.0,i0.0, $

CURVED L E WING - DARDEN TP-2446 FLAT WING

$INPTI TZSCALE=0.0,CLZPR=.08,IEMPCR=0, $

81
Table II. Continued

(b) Supersonic transport cruise surface design

SUPERSONIC TRANSPORT EXAMPLE - WHOLE WING

$INPTI XM:2.4,RN=2.0,JBYMAX=40,IEMPCR:I,

SREF=2.65,CBAR=I.85,XMC=2.77,XMAX=5.47,

NLEY:I3,

TBLEY=0.00,0.04,0.10,0.112

TBLEX=0.00,0.20,1.00,1.425

NT EY = 13,

TBTEY=0 . 00,0 . 04,0

TBTEX=5.47,5.15,4

NYR=4,TBYR=0.0000

TBTOC=0.0300

TBROC=.00030

TBETA=0.5000

CLDES=.I2,

NALPHA=I9,TALPHA=-2.0,-I.5,-I.0,-0.5,0.5,1.0,1.5,2.0,2.5,

3.0,3.5,4.0,4.5,5.0,5.5,6.0,6.5,7.0,7.5,

DESIGN-INC FUSE M=2.4

,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90,1.00,

,1.95,2.55,3.02,3.32,3.56,3.78,4.00,4.22,4.43,

10,0.112,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90,1.00,

28,4.075,4.11,4.18,4.26,4.34,4.43,4.51,4.60,4.68,4.76,

0.5000,0.8000,1.0000,

0.0300,0.0300,0.0300,

.00030,.00050,.00050,

0.5000,0.5000,0.5000,

(c) Supersonic transport cruise surface design with imposed leading-edge surface factors

SUPERSONIC TRANSPORT EXAMPLE - WHOLE WING DESIGN-INC FUSE - M=2.4

$INPTI XM=2.4,RN=2.0,JBYMAX=40,IEMPCR=I,

SREF:2 . 65, CBAR=I . 85, XMC=2 . 77 , XMAX=5 .47 ,

NLEY= 13,

TBLEY=0.00,0.04,0.10,0.112,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90,1.00,

TBLEX=0.00,0.20,1.00,1.425,1.95,2.55,3 . 02,3.32,3.56,3.78,4.00,4.22,4.43,

NTEY = 13,

TBTEY=0.00,0.04,0

TBTEX=5 . 47 , 5 . 15 , 4

NYR=4 , TBYR= 0 . 0000

TBTOC= 0 . 0300

TBROC= . 00030

TBETA= 0 . 5000

IAFIX=I,

TAFIX=0.00,0.00,0.00,0.00,0.00,0.00,0.00,0.51,1.04,1.56,

2.08,2.60,3.12,3.64,3.64,3.64,3.64,3.64,3.64,3.64,

3.64,3.64,3.64,3.64,3.64,3.64,3.64,3.64,3.64,3.64,

3.64,3.64,3.64,3.64,3.64,3.64,3.64,

CLDES=. 12,

NALPHA=I 9 ,TALPHA=-2 . 0, -i . 5, -i . 0, -0 . 5,0 . 5,1 . 0,1 .5,2 . 0,2 .5,

3.0,3.5,4.0,4.5,5.0,5.5,6.0,6.5,7.0,7.5, $

10,0.112,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90,1.00,

28,4.075,4.11,4.18,4.26,4.34,4.43,4.51,4.60,4.68,4.76,

0.5000,0.8000,1.0000,

0.0300,0.0300,0.0300,

.00030,.00050,.00050,

0.5000,0.5000,0.5000,

82
Table II. Continued

(d) Supersonic transport cruise surface design with reflexing

SUPERSONIC TRANSPORT EXAMPLE - REFLEX DESIGN WITH NACELLE PRESSURES AT M:2.4

$INPTI XM=2.4,RN=2.0,JBYMAX=40,IEMPCR=I,

SREF=2.65,CBAR=I.85,XMC=2.77,XMAX=5.47,

NLEY=I3,

TBLEY=0.00,0.04,0.10,0.112,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90,1.00,

TBLEX=0.00, 0.20,1.00,1.425, i. 95,2.55,3.02,3.32,3.56,3.78,4.00,4.22,4.43,

NTEY:I3,

TBTEY=0.00,0.04,0.10,0.112,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90,1.00,

TBTEX=5.47, 5.15,4.28,4. 075,4 .ii, 4.18,4.26,4.34,4.43,4.51,4.60,4.68,4.76,

NYR=4,TBYR=0.0000,0.5000,0.8000,1.0000,

TBTOC=0.0300,0.0300,0.0300,0.0300,

TBROC=.00030,.00030,.00050, .00050,

TBETA=0.5000,0.5000,0.5000,0.5000,

NTES=4

NTEC=I3,

TBTECY = 0.0,0.i12,0.2,0.25,0.3,0.4,0.45,0.5,0.6,0.7,0.8,0.9,1.0,

TBTEC=0.0,0.0,0.346,0.417,0.424,0.372,0.429,0.428,0.365,0.248,0.120,0.,0.,

IREFL=I, ICP=I,

NYCP=I3,

TBYCP= 0.0,0.i12,0.2,0.25,0.3,0.4,0.45,0.5,0.6,0.7,0.8,0.9,1.0,

NPCTCP = 20,

TBPCTCP = 0.,50.,54.,58.,62.,66.,70.,74.,78.,80.,

82.,84.,86.,88.,90.,92.,94.,96.,98.,I00.,

TCP= 0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,

0.00000 0.00000 0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,

0.00000 0.00000 0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,

0.00000 0.00000

0.00000 0.00000 0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,

0.00000 0.00000 0.00000,0.00000,0.00000,0.00000,0.00000 0.00000,

0 00000 0.03775 0.06469,0.03930,0.00000,0.00000,0.00000 0.00000,

0 00000 0 00000

0 00000 0 00000 0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,

0 00000 0 00000,0.00000,0.00000,0.05550,0.04469,0.07734,0.04332,

0 01874 0 01756 0.01616,0.00607,0.00000,0.00000,0.00000,0.00000,

0 00000 0 00000

0 00000 0 00000 0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,

0 00000 0 05218 0.05195,0.04697,0.07535,0.04200,0.02145,0.01901,

0 01793 0 00941 -.01186,-.04329,0.00000,0.00000,0.00000,0.00000,

0.00000 0.00000

0.00000,0.00000 0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,

0.05156,0.04479 0.08583,0.05949,0.03513,0.02044,0.01760,0.01729,

0 01273,0 04051,0.01629,0.01619,0.00000,0.00000,0.00000,0.00000,

0 00000,0 00000,

0 00000,0 00000,0.00000,0.00000,0.00000,0.00000,0.00000,0.04091,

0 08278,0 i0878,0.14220,0.i0866,0.07703,0.04975,0.03432,0.03134,

0 03108,0 02847,0.02141,0.00706,0.00000,0.00000,0.00000,0.00000,

0 00000,0 00000,

0 00000,0 00000,0.00000,0.00000,0.00000,0.06732,0.05093,0.09089,

0 04847,0 03030,0.02231,0.01977,0.01964,0.01912,0.01517,0.00951,

- 00241,- 02088,-.04112,-.06222,0.00000,0.00000,0.00000,0.00000,

0 00000,0 00000,

83
Table II. Continued

(d) Concluded

0 00000

0 02014

- 01284

0 00000

0 00000

0 03186

0 01154

0 00000

0.00000

0.05134

0.01386

0.00000

0.00000

0.00000

0.03809

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

CLDES=.I2,

0.00000

0 01816

- 02787

0 00000

0 00000

0 02429

0 00860

0 00000

0 00000

0 04636

0 01122

0 00000

0 00000

0 00000

0 03503

0 00000

0 00000

0 00000

0 00000

0 00000

0 00000

0 00000

0 00000

0 00000

0.00000,0.00000

0.01809,0.01793

-.04448,-.06110

,0.05773,0.04634,0

,0.01747,0.01410,0

,0.00000,0.00000,0

0.00000,0.00000

0.01719,0.01452

0.00523,0.00022

.08295,0

.00862,0

.00000,0

.04924,

.00019,

.00000,

,0.04349,0.03549,0

,0.01338,0.01333,0

,0.00000,0.00000,0

0.00000,0.00000

0.04137,0.03638

0.01076,0.01073

.06035,0

.01331,0

.00000,0

.04847,

.01284,

.00000,

,0.00000,0.00000,0

,0.03140,0.02656,0

,0.00000,0.00000,0

0.00000,0.00000

0.02296,0.02581

0.03196,0.02890

.03280,0

.02229,0

.00000,0

.02813,

.01808,

.00000,

,0.00000,0.00000,0.00000,0

,0.02451,0.02361,0.04242,0

,0.00000,0.00000,0.00000,0

0.00000,0.00000

0.00000,0.00000

0.00000,0.00000

.00000,

.04115,

.00000,

,0.00000,0.00000,0.00000,0.00000,

,0.00000,0.00000,0.00000,0.00000,

,0.00000,0.00000,0.00000,0.00000,

0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,

0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,

0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,

NALPHA=I9,TALPHA=-2.0,-I.5,-I.0,-0.5,0.5,1.0,1.5,2.0,2.5,

3.0,3.5,4.0,4.5,5.0,5.5,6.0,6.5,7.0,7.5,

84
Table II. Continued

(e) Supersonic transport reflexed cruise surface design with imposed leading-edge surface factors

followed by subsonic flap system design

SUPERSONIC TRANSPORT EXAMPLE - REFLEX DESIGN WITH NACELLE PRESSURES AT M:2.4

$INPTI XM=2.4,RN=2.0,JBYMAX=40,IEMPCR=I,

SREF=2.65,CBAR=I.85,XMC=2.77,XMAX=5.47,

CLDES=.I2,

NLEY=I3,

TBLEY=0.00,0.04,0.10,0.112,0.20,0.30,0.40,0.50,0.60,0.70,0.80,0.90,1.00,

TBLEX=0.00,0.20,1.00,1.425,1.95,2.55,3.02,3.32,3.56,3.78,4.00,4.22,4.43,

NTEY=I3,

TBTEY=0.00,0.04,0.10,0.112,0.20

TBTEX=5.47,5.15,4.28,4.075,4.11

NYR=4,TBYR=0.0000,0.5000,0.8000

TBTOC=0.0300,0.0300,0.0300

TBROC=.00030,.00030,.00050

TBETA=0.5000,0.5000,0.5000

NTES=4,

NTEC=I3,

0.30,0.40,0.50,0.60,0.70,0.80,0.90,1.00,

4.18,4.26,4.34,4.43,4.51,4.60,4.68,4.76,

1.0000,

0.0300,

.00050,

0.5000,

TBTECY = 0.0,0.112,0.2,0.25,0.3,0.4,0.45,0.5,0.6,0.7,0.8,0.9,1.0,

TBTEC=0.0,0.0,0.346,0.417,0.424,0.372,0.429,0.428,0.365,0.248,0.120,0.,0.,

IREFL=I, ICP=I,

NYCP=I3,

TBYCP= 0.0,0.i12,0.2,0.25,0.3,0.4,0.45,0.5,0.6,0.7,0.8,0.9,1.0,

NPCTCP = 20,

TBPCTCP = 0.,50.,54.,58.,62.,66.,70.,74.,78.,80.,

TCP: 0

0

0

0.

0.

0.

0.

0.

0.

0.

0.

0

0

0

0

0

0

0

0

0

0

0

0

0

82.,84.,86.,88.,90.,92.,94.,96.,98.,i00.,

.00000,0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,

.00000

.00000

00000

00000

00000

00000

00000

00000

00000

01874

00000

00000

00000

01793

00000

00000

05156 0

01273 0

00000 0

00000 0

08278 0

03108,0

00000,0

0.00000

0.00000

0.00000

0.00000

0.00000

0.03775

0.00000

0.00000

0.00000

0.01756

0.00000

0 00000

0 05218

0 00941

0 00000

0 00000

04479

04051

00000

0.00000,0.00000 0.00000,0

0.00000,0.00000 0.00000,0

0.00000

0.00000

0.06469

0.00000

0.00000

0.01616

0.00000

0.05195

-.01186

0.00000,

0.08583,

0.01629,

00000,0.00000,

10878,0.14220,

02847,0.02141,

00000,

0.00000,0.00000,0.00000,

0.04847,0.03030,0.02231,

.00000,0.00000,0.00000,

.00000,0.00000,0.00000,

,0.00000 0.00000,0

,0.00000,0.00000,0

0.03930,0.00000,0

.00000,0.00000,0.00000,

.00000,0.00000,0.00000,

.00000,0.00000,0.00000,

0.00000,0.00000,0.00000,0.00000,0.00000,

0.00000,0.05550,0.04469 0.07734,0.04332,

0.00607,0.00000,0.00000 0.00000,0.00000,

,0.00000,0.00000,0

,0.04697,0.07535,0

,-.04329,0.00000,0

.00000 0.00000,0.00000,

.04200,0.02145,0.01901,

.00000,0.00000,0.00000,

0.00000,0.00000,0

0.05949,0.03513,0

0.01619,0.00000,0

.00000,0.00000,0.00000,

.02044,0.01760,0.01729,

.00000,0.00000,0.00000,

0.00000,0.00000,0

0.10866,0.07703,0

0.00706,0.00000,0

.00000,0.00000,0.04091,

.04975,0.03432,0.03134,

.00000,0.00000,0.00000,

0.00000,0.00000,0.06732,0.05093,0.09089,

0.01977,0.01964,0.01912,0.01517,0.00951,

85
Table II. Continued

(e) Concluded

-.00241

0.00000

0.00000

0.02014

-.01284

0.00000

0.00000

0.03186

0.01154

0.00000

0.00000

0.05134

0.01386

0.00000

0.00000

0.00000

0.03809

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

0.00000

IAFIX=I,

-.02088

0.00000

0.00000

0.01816

- 02787

0 00000

0 00000

0 02429

0 00860

0 00000

0 00000

0 04636

0 01122

0 00000

0 00000

0 00000

0 03503

0 00000

0 00000

0.00000

0 00000

0 00000

0 00000

0 00000

0 00000

0 00000

-.04112,-.06222,0.00000,0.00000,0.00000,0.00000,

0.00000,0.00000,0.05773,0.04634,0

0.01809,0.01793,0.01747,0.01410,0

-.04448,-.06110,0.00000,0.00000,0

.08295,0.04924,

.00862,0.00019,

.00000,0.00000,

0.00000,0.00000,0.04349,0.03549,0

0.01719,0.01452,0.01338,0.01333,0

0.00523,0.00022,0.00000,0.00000,0

.06035,0.04847,

.01331,0.01284,

.00000,0.00000,

0.00000,0.00000,0.00000,0.00000,0

0.04137,0.03638,0.03140,0.02656,0

0.01076,0.01073,0.00000,0.00000,0

.03280,0.02813,

.02229,0.01808,

.00000,0.00000,

0.00000,0.00000,0.00000,0.00000,0

0.02296,0.02581,0.02451,0.02361,0

0.03196,0.02890,0.00000,0.00000,0

.00000,0.00000,

.04242,0.04115,

.00000,0.00000,

0.00000,0.00000,0.00000,0.00000,0

0.00000,0.00000,0.00000,0.00000,0

0.00000,0.00000,0.00000,0.00000,0

.00000,0.00000,

.00000,0.00000,

.00000,0.00000,

0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,

0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,

0.00000,0.00000,0.00000,0.00000,0.00000,0.00000,

TAFIX=0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.4,0.8,

1.2,1.6,2.0,2.4,2.8,3.2,3.2,3.2,3.2,3.2,3.2,3.2,3.2,

3.2,3.2,3.2,3.2,3.2,3.2,3.2,3.2,3.2,3.2,3.2,3.2,3.2,

NALPHA=I9 ,TALPHA=- 2 .0, -i .5, -i .0, -0 .5,0 .5,1 .0,1 .5,2 .0,2 .5,

3.0,3.5,4.0,4.5,5.0,5.5,6.0,6.5,7.0,7.5, S

SUPERSONIC TRANSPORT EXAMPLE - SUBSONIC FLAP DESIGN AT M=.3

SINPTI XM=.3,RN=2.0,JBYMAX=I0,ELAR=4.0,IEMPCR=0,

ICP:0, IAFIX=0,

CLDES=. 6,

CMDES= -. 24,

I FLPDES= 1 ,NEWDES= i,

NGCS=0,

ITRMAX=I00,

NLEC=I2,

TBLECY=0.00, .iii, .112, .200, .300, .400, .500, .600, .700, .800, .900,1.00,

TBLEC =0.00,0.00,.248, .248, .248, .248,.156, .123, .iii, .098, .086, .083,

NTES = 4,

NTEC=I 6,

TBTECY=0.00, .iii, .112, .189, .190, .310, .311, .389, .390,

.510, .511, .600, .700, .800, .900,1.00,

TBTEC =0.00,0.00, .203, .234,0.00,0.00, .304, .276,0.00,

0.00, .235, .213, .177, .153, .118, .083,

NALPHA=I6,TALPHA=0.0,2.0,4.0,6.0,8.0,10.0,12.0,14.0,16.0,

18.0,20.0,22.0,24.0,26.0,28.0,30.0, $

86
Table II. Concluded

(_ Nonsymme_cal wing cruise surface design

NONSYMMETRICAL ELLIPTICAL WING DESIGN

$INPTI XM=I.4,RN=2.0,JBYMAX=40,IPRSLD=I,IEMPCR=I,

CLDES=.I6,YCG=.5,

NSYM=I,

SREF=3.142,CBAR=2.0,XMC=2.0,XMAX=4.0,

NLEY=II,

TBLEY=0.000,.2000,.4000, .6000,.8000,i.000,i.200,i.400,i.600,i.800,2.000,

TBLEX=0.268,0.014,0.161,0.392,0.674,1.000,1.366 1.777,2.239,2.785,3.732,

NTEY=II,

TBTEY=0.000, .2000,.4000, .6000,.8000,i.000,i.200 1.400,i.600,i.800,2.000,

TBTEX=0.269,1.215,1.761,2.223,2.634,3.000,3.326 3.609,3.839,3.986,3.733,

NLEC=2,

TBLECY=0.0,2.0,

TBLEC =2.0,2.0,

NYR=4,

TBYR =0.000,.2000,i.800,2.000,

TBTOC=.0400, .0400, .0400, .0400,

TBROC=.0010, .0010,.0010, .0100,

TBETA=.4000, .4000,.4000, .4000,

NALPHA=I4,

TALPHA=0.00,1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0,9.0,10.0,11.0,12.0,5.25, $

87
[_y

1 2

I

I

I

I

I

..... -I .....

i(x/ax)

3 4 5 6

..... I .....

I

I

7 8 9 10 11

I I I

..... II

I

I I I I I I I I I I

22

0 1 2 3 4 5 6 7 8 9

xlAx

(a) Subsonic.

I

I i

10 11

3 j([_y)

_y

i(x/ax)

1 2 3 4 5 6 7 8 9 I0 11

-- I'- .... "t .......... I .....

i ,17 ,/

I I I I I

' 12

I I I I _I

I I I I

I I I I

k .... _I_ I I _ I

I I I "i

, , __./,

i ' I/' i

' '_"1 1 ' 2 ' 3 4 5

I /I I I

iJ I I I

I I I

I

264 27

/,

I

..... _.... _'-- ' 'I /

I /I

' 22 ' 23 24

':/ "l" ,

.....,'_-/_, ' ,

I

18 19 20

13 14 15

I i I

6 7 8

I I I I I I I I I

I I

 IT-- I

I

,' 28 ', :29

I

I .....

I

0

I

I

I

I

! ....

I

I

,21

I ....

--,,/

9 ',_o

I

I 1 f

0 1 2 3 4 5 6 7 8 9 10 11

x/Ax

(b) Supersonic.

Figure 1. Grid system used in numerical solution of linearized theory.

J(fJy)

88
u

Figure 2. Subsonic influence function.

89
.4

I I_1 I I I_1 I

Uniform chord elements

i i_l I i I_1 I I

Exact linearized theory

Code results (unadjusted)

I1%1

Smaller chord first element

I I I_1 I

Au

.3

.2

.1

0 I I

.03

Au -jxr]-c .02

.01

.04-

0 .2 .4 .6 .8 1.0

x' /c

.2 .4 .6 .8 1.0

x'/c

Figure 3. Typical code results for flat two-dimensional wing. M = 0.01; ot = 1o.

90
Aux,/77_

.12

.08

.04

_Exact linearized theory

• o "---"r-'--_

ee _"_- Code results

"l

x' at element quarter-chord

I I I I

1.8

1.6

f 1.4

1.2

Q

I

I

I

I

I

l

6

l

1.0 _- n

"o.o. • _Curve fit

8.

I I _1_ . :

Aux4_-_

.12

.08

.O4

F Exact linearized theory

x' defined by f factor

I I I I

.4 ,8 1.2 1.6

x'/Ax

Figure 4. Correction of code perturbation velocity location.

91
.02 -

Au x-_-_ .01

o

Q

Ale = 20 °

I I I I I

J

Ale = 40 °

i i I 1 i

.02 -

Au 7xTxTxTxTxTxTxTxTx_ .01

S

Ale = 60 °

_o

i

Ale = 80 °

I i i I i i I I I " i

0 .2 .4 .6 .8 1.0 0 .2 .4 .6 .8 1.0

x'/c x'/c

Figure 5. Code velocity distributions for various sweep angles. Flat wing; M = 0.01; o_= 1°.

92
J

Ale = 40 °

.03 -

.02 _e

.01

0

ml. Au4x4x4x4x4x4x4x4x4x4£% - I

• Code results

........ Fairing of data for 16th iteration

0

D

I i "°---. I

I I

\

I "J

I 1

"1

'1

-•1

.p

"i

I I "'1

0 .2 .4 .6 .8 1.0

X'[C

Figure 6. Illustration of solution convergence. M = 0.01; ct = 1°.

Iteration

16

93
y

Figure 7. Supersonic influence function.

94
12 _ o

AUcode

A

.i

I I I I I

0 .2 .4 .6 .8 1.0

(a) Velocity ratio.

O

O

[]

cot A

O 0.834

[] .574

.362

A .175

Curve-fit equation

AUc°de .6 I_x

AUth

O

cot A

O 0.834

[] .574

.362

A .175

Curve-fit equation

I I I I I

0 .2 .4 .6 .8 1.0

(b) Velocity ratio parameter.

Figure 8. Supersonic numerical solution for leading-edge elements.

95
.2-

Au 0

--.2

.2

Au .1

Ale = 40 °

Cambered wing; _ = 0 °

• • • • .08

au./_ .04

.12 -

-.04

X

Cambered wing; ct = 0 °

Flat wing; ct = 1°

.2

I I I I • I

0 .2 .4 .6 .8 1.0 0

X'/C

Flat wing; (_ = 1°

I I I I • I

.2 .4 .6 .8 1.0

X'/C

Figure 9. Code velocity distribution for cambered wing and flat wing of same planform. M = 0.01.

96
mu

.2

.1-

0

--.I --

O.= 0 °

.12 -

.08 -

aux4_ .04 -

0

-.2 -.04

0_=0 °

An

.2 .12

.1

o_ = 1.8 °

.08

.04

-.1- 0

= 1.8 °

An

Figure 10.

.3-

.2

ot = 3.6 ° (t = 3.6 °

.12 -

.08 -

_x7_

.04

I

1.0 0

I I I I I I I I I

0 .2 .4 .6 .8 .2 .4 .6 .8 1.0

x'/c x'/c

Code velocity distribution for cambered wing at several angles of attack obtained by superposition.

97
Au_

Figure 1 1.

.08

.04

-.04

-.08

-.12

s

t

i

CY"

- -[:]- ........... U]............ ._q:3- ........

/

/

i

i

i

/

J

i

/

/

/

-" 0 Cambered wing

O" [] Flat wing

...... Curve fit

I I I I I I

0 .4 .8 ! .2 1.6 2.0 2.4

.j¢'

Curve fit used to determine section theoretical thrust coefficients.

y M n

Ale

M Streamline

rn

Normal

section

C n C

Figure 12. Relationship between streamwise and normal wing sections used for attainable thrust analysis.

98
Yvor

x cot Ale

Figure 13.

1.0

.6

.4

.2

_'Ale Yvor

_ _ _

Experiment

Ale, deg

O 70 n

[] 6o

_ 74

Empirical equation

I I I

0 10 20 30

_,deg

Location of pressure field at center of vortex as given by experimental data.

/ sin et

_' / Cvo r CN, c + CN,f sTffP

tc__ sinc_

i _--.._ CAc+CA,f/c s_ 1°

cN

CA

0 J.._

/

-_L

CA,c

0

Cvor

CNrc

I

i

I

' _ sin

_c I cosE°

Figure 14. Buildup of section axial- and normal-force coefficients.

99
.10

cN,f .o5

b

ct,f

.001

0

.02 F

CN, c I• • • • • • • •

I

0

.01

CA'_ Io

0 •

CA'f/c L "•'•

-.01

iI.............

I I I I I

0 .2 .4 .6 .8

Y

b12

I

1.0

Figure 15. Spanwise variation of wing section coefficients. Aspect ratio 4 wing-body with twist and camber; M = 0.61.

100
"`%

"`%

,,%

s Mean camber surface

I

/ it

II 1I

Ii /

Fairing of data for integration

_ Fairing with account of singularities

Original code

_ Singularity fairing

Neglected integration

_' _ Linear fairing

_1_,'- ACp adjustment

1

I Adjustment

"-- Fairing of adjusted ata for integration /

Revised code

Figure 16. Flap hinge-line singularity adjustment.

101
[3y

/ I

I l I i I i I i I i I t I i I t I l I i I

0 4 8 12 16 20 24 28 32 36

Figure 17. Typical representation of lifting surfaces in AERO2S computer code. Dashed line indicates original

plan form.

102
General camber surfaces

Figure 18.

Trailing-edge camber surfaces

Typical candidate camber surfaces for delta wing example.

103
Design limit

I_ 1.5Cle

_ Cle---- _

--_ Y _ _i_tsedctflea_;UearfadCeesignsurface

tan- 1('AZle'_

Design limit

1.5Cte _i

ct e

_te

_i_

= tan-I (AZte_

8te,s \ Cte )

Figure 19. Fitting of flap surfaces to wing design surface. Design areas are shaded.

104
gdes

1.0

.6 --

.4 --

.2

0 1 I I I

Ks

1.0 --

.8 --

.6 D

,4 --

.2

0 I I I I

1 2 3 4

M

Figure 20. Empirical method factors used to select optimum design lift coefficients and to predict achievable suction

parameters.

105
co

1.0 .8

I I

.040 -

ELAR = 2.0

Flat

.032

.024

.016

.0O8

0 I

4

JBYMAX

.6 .4 .2

I I I

I

I I I

0 1.0

1

ELAR

.8 .6 .4 .2 0

I I I I I

JBYMAX = 12

Flat

• • • • • •0 ......

I I I I I

.040

.032

.024

C D

.016

.0O8

ELAR = 2.0

Twisted and cambered

JBYMAX = 12

Twisted and cambered

_ I I I I I I I I I

4 8 24 oo 1 2 4 8 ,,0

JBYMAX ELAR

Figure 21. AERO2S code convergence data for aspect ratio 2 wing-body. M = 0.60; R = 3.0 x 106; C L = 0.3.

106
.20

.16

.12

C D I

"0

.20

.16

.12

C D

.08

.20

.16

.12

C D

.08

.O4

4

JBYMAX

1.0 .8 .6 .4

Figure 22.

I I I I

ELAR = 2

No flap deflections

<31e,n= 0% _te,n = O°

I I

.2 0

I I

I I

1.0

I

.8

[

1

ELAR

.6 .4

I I

JBYMAX = 8

No flap deflections

<51e,n= 0°; _te,n = O°

I I

.2

I

• •41•...

I I

ELAR = 2

• • I@ ......

Moderate flap deflections

,_ _ i_o. ,_ _ 10 o

_le,n - _ ' _te,n -

I J I I

JBYMAX = 8

Moderate flap deflections

_le,n = 15°; 8te,n = 10°

I I I 1 I

ELAR = 2 JBYMAX = 8

Large flap deflections

_51e,n = 30°; 6te,n = 20 °

t¢3

I I I 0 I _ I I I

4 8 24 _ 4 16oo

JBYMAX

Large flap deflections

_le,n = 30°; fite,n = 20°

I I

1 2

ELAR

AERO2S code convergence data for cranked-wing fighter. M = 0.50; R = 2.9 x 106; C L = 0.8.

107
With attainable thrust

andvortex force

..... Without thrust or vortex

SS

1.0"- f

.Ol - - - .............. 0

0 Cm

-.12

• .0_}

N . .(_

2 CL

--4 0 4 8

_, deg

(a) Fiat wing.

Figure 23. Theoretical and experimental data for aspect ratio 2 wing-body. M = 0.61; R = 3.0 × 106.

10S
0 Experiment

Theory

With attainable thrust

and vortex force

..... Without thrust or vortex

SS

l.O

.5 --

0

-.5 -

.02 - ,,

.01 -

f

0

CA -.01 -

-.02 -

-.03 -

-.04 m

.8

.6

.4

C N

.2 --

o f

-.2 - I

-4 0

Full thrust

I I I I

4 8 12 16

_t, deg

C

m

co

(b) Twisted and cambered wing.

Figure 23. Concluded.

109
.08 -

.06 -

.04

CA .02 -

0

-.02 -

-.04 D

1.0-

.8 --

.6 --

.4 --

CN

0

-.2 -

-.4

C) Experiment

Theory

With attainable thrust

and vortex force

..... Without thrust or vortex S S

'°f

.5

0

o oo o OOo

C m

CD

i

/ I I I I t

-5 0 5 10 15

ct, deg

.2

.1

0

--.1

--.2

--.3

--.4

.24

.20

.16

.12

.08

.04

I I 0 _ I

-10 20 -.4

C L

-.2 0 .2 .4 .6 .8

C L

(a) Flat wing.

Figure 24. Theoretical and experimental data for cranked-wing fighter. M = 0.50; R = 2.9 x 106.

11o
.08

,06

.04

CA .02

0

C N

© Experiment

Theory

-- With attainable thrust

and vortex force

- - - Without thrust or vortex

SS

1.0

0

1.0 m

.8 m

,6

.4 --

.2

0

--.2

--.4 _ I I

-I0 -5

/

I I I I

0 5 10 15 20

et, deg

2.1 I

0

Cm -.1 -

-.2 -

-.3 -

-.4 -

.24 -

.20 -

.16 -

CD .12 -

.08 -

,04 --

0 - I

--.4

I

/

C o o + CL tan --_ _ /

' Lo _ _t

I i I I I

-.2 0 .2 .4 .6 .8

CL

(b) Deflected flaps, _le,n --" 15 ° and _te,n = 10%

Figure 24. Continued.

111
C A

0 Experiment

Theory

.__----- With attainable thrust

and vortex force

..... Without thrust or vortex 1°f

S S .5

0

o_

.06_ 00i _

.0

2

1.0 - /,/

/o°

/o°°

- 4 _ _ 15 20

-10 -5 0 5 10

t_, deg

2.1 f

0

C m -.1

-.2 -

_.3

-.4 t

.24 --

.20 -

.16 -

CD .12 -

.08 -

.04 --

I cL /

C D,O + C L tan _-'_/ CLo _ O

oL _--_--

-.4 -.2

(c) Deflected flaps, _e,n = 30° and _te,n = 20°"

Figure 24. Concluded.

0 .2 .4 .6 .8

C L

112
30

20

61e,n, deg

10

Theory

Cm

-.12 -.14 -.16 -.18 -.20 -.22 -.24

v i i _ i i

I I I t I I

72 / ! / ', ', ; "_ ; \ ,\ \1

/i/ ;80 i/---_.. : "_ 76',\1

II I I I I

S i i i i

' 80 , ', ',

;X\X._6 \ ',_', I, /;ll

.'X,x.\,x.\ ,__ ', ',/; II

10 20

6re,n" deg

-.14

0

-.16

72

_.6(

Experiment

Cm

-.18 -.20 -.22

.8 If----,--.._ ,.8o X

10

6te,n' deg

Figure 25. Flap system performance map for cranked-wing fighter. M = 0.50; R = 2.9 x 106; C L = 0.45.

20

113
114

%

Figure 26.

Experiment

..... Without thrust or vortex SS

_.

.8-

.6

.4

.2

-.2

°lJ

--4

4 8 12 16

it, deg

.O6

.05

.04

.03

,02 '

.01

0

I

%,o+CL_ c-_z_V/ cL_

/

-.2 0 .2 .4 .6 .8

cL

(a) Wing-bo@.

Theoretica/and experimental data for subsonic transport configuration. M = 0.55; R = 2.25 X I06.
© Experiment

Theory 1.0 --

-- With attainable thrust

and vortex force

SS

C

m

..... Without thrust or vortex

.5

0

--.5

.3

.2

.1

0

--.|

.06

co

.05

.04

.03

.02

.01

0

©

I cL /

CD'O + CL tan CL-'-o_--_/ /

I I I I _ I

4 8 12 16 -.2 0

Or,deg

(b) Wing-body with horizontal and vertical tails.

I I I I

.2 .4 .6 .8

cL

Figure 26. Concluded.

115
CA

@

.010 -

.005 -

0

-.005 -

-.010 -

.30 -

(_ Experiment

Theory

C m

.08 -

.04-

0

--.04

-.08

.25 -

.20 -

,15 -

.10 -

.05 -

0 -"-

-.05 -

-.10- I

-2 0

I I I I I

2 4 6 8 10

ct, deg

SS

%

1.0

.5

0

--.5

-1.0

.06 -

.05 -

.04 --

.03 -

.02 -

.01 -

0- I

-.10

I

-.05

I I I I

0 .05 .10 .15 .20

cL

(a) Flat wing.

Figure 27. Theoretical and experimental data for SST wing-body combination. M = 2.40; R = 3.4 × 106.

t

.25

116
.010]-

.005-

CA 0

I

-.005 -

-.010 -

.30 -

CN

Q) Experiment

Theory

C m

.08 -

.04 --

0

-.04 -

-.08 -

1.0-

.25 -

.20 -

.15 -

.10 -

.05 -

0 ------------_

Y

-.05 -

-.10- I

-2

I I I I

0 2 4 6 8

tx - So, deg

S S

.5-

0

-,5

-1.0 -

.06 -

co

.04

.03

.02

.01

0

.05 -

I _ I I

I0 -.10 -.05

I I I I I

0 .05 .10 .15 .20 .25

cL

(b) Twisted and cambered wing; CL,de s = 0.08.

Figure 27. Concluded.

117
---'-=-

0 Experiment

Theory

-- With attainable thrust

and vortex force

..... Without thrust or vortex

CA

.04-

--.04 --

SS

Cm

1.0

-.5

-1.0

.2-

,1 --

0

--°1 --

0 o

0

0

0

%

.8-

.4-

0

--,4

©

%

.24 -

CD,0

.16 -

.08 -

I I 0 I

10 20 -.4

Ct,deg

-10 0 0 .4 .8

cL

(a) _e,n = 0°; _te,n = 0% _H = --50"

Figure 28. Theoretical and experimental data for arrow wing SST configuration with leading- and trailing-edge flaps

and horizontal tail. M = 0.03; R = 4.19 x 106.

118
.08-

0 Experiment

Theory

Withattainable thrust

andvortexforce

..... Without thrustorvortex

|.0 --

.5 --

S s 0

-.5

-1.0

ca

C N

.04

--.04 --

8.4 I

°l

0_

Oo"_. "

©

C

m

00""

CD

-.1

.lr-

.24 -

.16 -

.08

I 1 I 0

-10 o 10 20

_, deg

(b) _le,n = 13°/34°/35°/35°/19°/29°; _te,n

Figure 28. Concluded.

O0

_ I

--.4

CD,oI+CLtan_LL -_/

I I

0 .4 .8

Q

= 30°; _iH = -5 o.

119
SST planform for whole-wing design

Research "

SST planform for reflex surface design

SST planform for flap system

Figure 29. Wing planform used in SST design study.

120
5 _ ................................. ot for CL,de s

_zt + Actft

3 ........ __--- ct for(CL,des)re p

_,deg 2- ._- _, /

1L// 0

-1- I I _ I I J

0 .2 .4 .6 .8 1.0

y

b/2

SS

.6-

.5

.4 --

.3

.2

.1 --

0

(CL,des)rep

I

' CL,de s

Theory

Empirical estimate

f I

I I

I I

I I

t I

I I I

.04 .08 .12 .16

C L

Ale

5

4

3-

2-

1-

0 I

0

dt'D Suggested

__ /_¢_ As used

/

,' Fairing

.2 .4 .6 .8 1.0

Y

b/2

Figure 30. Critical code design parameters from SST whole-wing design. M = 2.40; CL.des = 0.12.

121
¢t, deg 2

1

0

-1

5 -

m

I _,e I t'1 I I

0 .2 .4 .6 .8

y

b/2

...... ct for CL, des

- - - ot for (CL,des)re p

I

1.0

S S

.6-

,5 --

.4 --

.3

,2 --

.| --

O- I

0

(CL,des)rep

i

CL,de s

_ Theory

I I

//I i

I I

i i

I I

I I I

.04 .08 .12 .16

cL

(a) Critical design parameters.

Figure 31. Code data from SST whole-wing design with imposition of faired leading-edge surface weighting factors.

M = 2.40; CL,de s = 0.12.

122
0

,-1

123
0_, deg

_zt 4- A_f, L

2

o

I • I I I J

-1- 0 .2 .4 .8 1,0 .6

Y

b12

......... _t for CL,de s

(Xzt - Actft

ct for (CL,des)re p

.6

.5

.4

SS .3

0

t..----

0

(CL,des)rep

I

, CL,des

l

_ Theory

y ! _ Empirical estimate

I I

I I

I 1

I I

I I

I I

I I

.04 .08 .12 .16

CL

5 -

3

Ale

oL- L_

0

__ Suggested

"_ _ As used

.2 .4 .6 .8 1.0

Y_L_

b12

Figure 32. Critical code design parameters from SST reflexed surface wing design. M = 2.40; CL,des = 0.12.

124
II

"q.

¢'4

N

cU)

cO

t_

[..

E

0

E

t_

125
20

15

ct, deg 10

5

1.0

.8

.6

SS

.4

.2

.................................. _t for CL, des

Ctzt + Actft

_zt

_zt - A_ft

I I I I

0 .2 .8 1.0

I I

.4 .6

Y

b12

CL, des

t

I

I

_ I I I I I

0 .2 .4 .6 .8 1.0 1.2

cL

120

100

Suggested

Selected

80 "''" "'" As used

Ale 60

40

20

0 I I

0 .2 .4 .6 .8 1.0

Y

b12

Figure 34. Critical code design parameters from SST mission adaptive surface and flap system design. M = 0.30;

CL,de s = 0.6; Cm,de s = -0.24.

126
_, deg

SS

20

15

10

5

].0 --

.8-

.6

.4

.2

Body side

I

i

...... ' ...... _ - - - _ for CL,de s

/ _zt + Actft

//

_ _Ctzt - Accft

I I I I I I

0 .2 .4 .6 .8 1.0

Y

b/2

CL,des

I

I

i

I I I I I I

0 .2 .4 .6 .8 1.0 1.2

cL

40-

30-

81e,s 20

10

°]"

.I °

°

40-

30-

8te,s 20-

10

0 l,_ I 0

0 .5 1.0

Y

b/2

0

J

.5

Y

b/2

1.0

(a) Critical design parameters.

Figure 35. Code data from SST mission adaptive surface and flap system design with imposition of faired leading-edge

surface factors. M = 0.30; CL,de s = 0.6; Cm,de s = -0.24.

127
0

128
Figure 36.

at, deg

Body side

20- I

I

15 ................................. t_ for CL,de s

10 ,/_'_ e _''''r- _ "- atzt + Aaft

atzt

5 atzt - Aatft

/

0 t :_ I I I I I

0 .2 .4 .6 .8 1.0

Y

b12

S S

1,0

.8

.6

.4 ;-

.2_

0 t I

0 1.2

CL,des

I

I

I I

.2 .4 .6 .8 ! .0

%

Code data from SST flap system evaluation (AERO2S), _le,s = 26°/32°/36°; _Ste.s = 14°/14°/16°; M = 0.30.

129
27o127°/30 °

14o/14o/16 o 21°/21°/2 30 4.4Ol51Ol56 °

7o/7°/8 °

36o/43°/48 °

_ie factor 1.C

26o/32°/36 °

1401170/20 °

1,0

fire factor

(a) Nine individual runs.

2,0

2.0

7o/7ol8°

14o114o116 o 21°/21°/2 30 27°/27°/34_°/51 O° °/56 °

1.5

36ol43°148 °

8|e factor !.0

26o/32°/36 °

14°/17°/20 °

1.0

5te factor

(b) Single run with multiplier factors.

Figure 37. SST flap system performance map derived from AERO2S code results. M = 0.30; R = 2.0 × 106; Ct, = 0.6.

130
10

c_,deg

S S

1.0 --

8

2

0 I

0

.8-

.6

.4

i

.2

0- I

0

....... ----_-_--7' ....... °t f°r CL, des

_z + A_

I I I _ ( , CZzt-Acxft

.2 .4 .6 .8 1.0

Y

b/2

(CL,des)rep

I

j CL,des

I I

I I

'i

_E 2e°c_l estimate

.04 .08 .12 .16 .20 .24

cL

cL

.20

.16

.12

.08

.04

0

0 .2 .4 .6 .8 1.0

y

b/2

- .020

- .016

.012

.008

.004

0

C D

(a) Critical design parameters.

Figure 38. Code data from nonsymmetrical elliptical planform wing design. M = 1.40; CL.de s = 0.16.

131
x y

(b) Camber surface.

Figure 38. Concluded.

132

I Form Approved REPORT DOCUMENTATION PAGE OM8No.0z04-0188

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

November 1997 Technical Paper

4. TITLE AND SUBTITLE :5. FUNDING NUMBERS

Guide to AERO2S and WlNGDES Computer Codes for Prediction and

Minimization of Drag Due to Lift WU 537-09-20-02

6. AUTHOR(S)

Harry W. Carlson, Julio Chu, Lori P. Ozoroski, and L. Arnold McCullers

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

8. PERFORMING ORGANIZATION

REPORT NUMBER

L-17546

10. SPONSORING,tMONITORING

AGENCY REPORT NUMBER

NASA TP-3637

11. SUPPLEMENTARY NOTES

Carlson, Lockheed Martin Engineering & Sciences, Hampton, VA; Chu and Ozoroski, Langley Research Center

Hampton, VA; McCullers, ViGYAN, Inc., Hampton, VA.

12,,. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified-Unlimited

Subject Category 05

Availability: NASA CASI (301) 621-0390

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

The computer codes, AERO2S and WlNGDES, are now widely used for the analysis and design of airplane lifting

surfaces under conditions that tend to induce flow separation. These codes have undergone continued development

to provide additional capabilities since the introduction of the original versions over a decade ago. This code development has been reported in a variety of publications (NASA technical papers, NASA contractor reports, and society journals). Some modifications have not been publicized at all. Users of these codes have suggested the

desirability of combining in a single document the descriptions of the code development, an outline of the features

of each code, and suggestions for effective code usage. This report is intended to supply that need.

!14. SUBJECT TERMS

Aerodynamics; Linearized theory; Numerical methods; Attainable thrust

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A07

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