A Generalized Vortex Lattice Method for Subsonic and Supersonic Flow Applications

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NASA Contractor Report 2865

Luis R. Miranda, Robert D. Elliott, and William M. Baker

,8 J

CONTRACT NAS 1-12972 DECEMBER 1977

N/ /X

NASA Contractor Report 2865

A Generalized Vortex Lattice Method for Subsonic and Supersonic Flow Applications

Luis R. Miranda, Robert D. Elliott, and William M. Baker

Lockheed-California Company Burbank, California

Prepared for Langley Research Center under Contract NAS1-12972

National Aeronautics and Space Administration

Scientific end Technical Information Office

1977

TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS ...... iv

SUMMARY ...... i

INTRODUCTION ...... i

THEOP_TICAL DISCUSSION ...... 3

The Basic Equations ...... 3 Extension to Supersonic Flow ...... 5 The Skewed-Horseshoe Vortex ...... 9 Modeling of Lifting Surfaces with Thickness ...... 14 Modeling of Fusiform Bodies ...... 15 Computation of Sideslip Effects ...... 17

THE GENERALIZED VORTEX LATTICE METHOD ...... 18

Description of Computational Method ...... 18 Numeric 8.1 Considerations ...... 2O

COMPARISON WITH OTHER THEORIES AND EXPERIMENTAL RESULTS ...... 21

CONCLUDING REMARKS ...... 22

APPENDIX A USER'S MANUAL FOR A GENERALIZED VORTEX LATTICE METHOD FOR SUBSONIC AND SUPERSONIC FLOW APPLICATIONS ...... 37

THE VORLAX COM.VJTER PROGRAM ...... 39

PRACTICAL INPUT INSTRUCTIONS ...... 4o

INPUT CARD IMAGE DESCRIPTION ...... 46

PROGRAM OUTPUT ...... 61

RECOMMENDATIONS FOR THE EFFICIENT USE OF THE VORLAX PROGRAM .... 93

ApPENDIX B COMPLETE PROGRAM COMPILATION AND EXECUTION FOR A GENERALIZED VORTEX LATTICE METHOD FOR SUBSONIC AND SUPERSONIC FLOW APPLICATIONS ...... 97

HARDWARE AND SYSTEMS ...... 99

PROGRAM- UNIFIED VORTEX LATTICE METHOD FOR SUBSONIC AND sUPERSONICFLOW ...... i01

iii Page

APPENDIXC WAVEDRAGTOVORLAXINPUTCONVERSIONPROGRAM(WDTVOR).. . 249 SUMMARY...... 251

INTRODUCTION...... 251

NOMENCLATURE...... 253

PROCEDURE...... 253

PROGRAM- WAVEDRAGTOVORLAZINPUTCONVERSIONPROGRAM(WDTVOR).... 279

APPENDIXD VORLAXTOWAVEDRAGINFCTCONVERSIONPROGRAM(VORTWD).. . 327

SUMMARY...... 329

INTRODUCTION...... 329

NOMENCLATURE...... 33O

PROCEDURE...... 331

PROGRAM- VORLAXTOWAVEDRAGINPUTCONVERSIONPROGRAM(VORTWD).... 341

REFERENCES...... 371

_.v LIST OF ILLUSTRATIONS

Figure Page

I Definition of integration regions for the computation of principal part ...... 23

2 Sweptback horseshoe vortex ...... 24

3 Modeling of thick wing with horseshoe vortices ..... 25

4 Vortex lattice collocation ...... 26

5 Modeling of fusiform body with horseshoe vortices ...... 27

6 Conventional approaches for analysis of wing in sideslip . . 28

7 Lattice geometry for analysis of wing in sideslip ...... 29

8 Generalized vortex lattice model of wing-body configuration ...... 3O

9 Theoretical comparison of arrow-wing lift slope and aerodynamic center location ...... 31

10 Theoretical comparison of arrow-wing drsg-due-to-lift factor ...... 32

Ii Theoretical comparison of chordwise loading for delta wing . 33

12 Theoretical comparison of chordwise loading for sweptback rectangular wing ...... 34

13 Comparison with experimental pressure distribution on wing-body model at Math = 0.5 ...... 35

14 Theoretical comparison of pressure distribution on ellipsoids at zero angle of attack in incompressible flow ...... ' .... 36

A-I Control surface nomenclature ...... 4l

A-2 Modeling of thick wing with horseshoe vortices ...... 42

A-3 Vortex lattice collocation ...... 44

A-4 Modeling of fusiform body with horseshoe vortices ...... 45

A-5 VORLAX case data deck setup ...... 47

V Figure Page

A-6 Generalized vortex lattice model of wing body configuration ...... 49

A°7 Vortex lattice panel representation for twin engine fighter ...... 95

C-I Typical WAVE DRAG format dataset ...... 257

C-2 Graphic representation of WAVE DRAG dataset ...... 259

C-3 Listing of output dataset in VORLAX format ...... 26O

C-4 Result of converting circular fuselage VORLAX dataset back to WAVE DRAG format ......

C-5 Plot of figure C-4 dataset ......

C-6 Result of converting flat fuselage dataset back to WAVE DRAG format ...... 16of

C-7 Plot of figure C-6 dataset ...... 26_3

C-8 Converted dataset before editing for VORLAY run ...... 269

C-9 Converted dataset after editing for VORL_[ run .... 27?-

C-IO Plot of dataset edited for submittal to VO_qL_Y ...... 276

C-II Portion of output from submittal to VORLAX ...... 277

D-I VORLAX dataset with smarts cards included ...... 33_

D-2 WAVE DRAG format dataset resulting from conversion ..... 336

D-3 Planform plot from figure D-2 dataset ...... 338

D-4 Isometric plot from figure D-2 dataset ...... 339

vi A GENERALIZED VORTEX LATTICE METHOD FOR SUBSONIC AND SUPERSONIC FLOW APPLICATIONS

Luis R. Miranda, Robert D. Elliott, and William M. Baker Lockheed-California Company

SUMMARY

A vortex lattice method applicable to both subsonic and supersonic flow is described. It is shown that if the discrete vortex lattice is considered as an approximation to the surface-distributed vorticity, then the concept of the generalized principal part of an integral yields a residual term to the vorticity-induced velocity field. The proper incorporation of this term to the velocity field generated by the discrete vortex lines renders the present vortex lattice method valid for supersonic flow. Special techniques for simulating nonzero thickness lifting surfaces and fusiform bodies with vortex lattice elements are included. Thickness effects of wing-like components are simulated by a double (biplanar) vortex lattice layer, and fusiform bodies are represented by a vortex grid arranged on a series of concentrical cylindrical surfaces. The analysis of sideslip effects by the subject method is described. Numerical considerations peculiar to the application of these techniques are also discussed. A summary comparison of the results obtained by the method of this report with other theoretical and experimental results is presented. This method has been implemented in a digital computer code identified as VORLAX. A users manual for the VORLAX program is contained in Appendix A. A complete Fortran compilation and executed case are contained in Appendix B. Appendices C and D describe input conversion programs useful for transforming input between the VORLAX and NASA Wave Drag programs.

INTRODUCTION

The several versions or variations of the vortex lattice method that are presently available have proven to be very practical and versatile theoretical tools for the aerodynamic analysis and design of planar and nonplanar configurations. The success of the method is due in great part to the relative simplicity of the numerical techniques involved, and to the high accuracy, within the limitations of the basic theory, of the results obtained. But most of the work on vortex lattice methods appears to have concentrated on subsonic flow application. The applicability of the basic techniques of vortex lattice theory to supersonic flow has been largely ignored. The method presented herein allows the direct extension of vortex lattice techniques to supersonic Mach numbers. The equations allowing this extended application are derived in the next section starting from the first order vector equations governing inviscid compressible flow. They are then applied to the particular case of a skewed-horseshoe vortex with special attention given to the supersonic horseshoe. In the following theoretical discussion, the basic arguments involved in the simulation of thickness and volume effects by vortex lattice elements are presented. This particular modeling of the above effects represents an alternative, with somewhatreduced computational requirements, to the method of quadrilateral vortex rings (refs. 1 and 2). The simulation of thickness and volume effects makespossible the computation of the surface pressure distribution on wing-body configurations. The fact that this can be donewithout having to resort to additional types of singularities, such as sources, results in a simpler digital computer code.

2 THEORETICALDISCUSSION The Basic Equations

Wardhas shown, (ref. 3), that the small-perturbation, linearized flow of an inviscid compressible fluid is governed by the three first order vector equations: v = ®% V. -.-Q, = v (l)

on the assumption that the vorticity _and the source intensity Q are known functions of the point whose position vector is R. The vector_ is the perturbation velocity with orthogonal Cartesian components u, v, and w, and • is a constant symmetrical tensor that for orthogonal Cartesian coordinates with the x-axis aligned with the freestream direction has the form

i - M O 0 q' = 0 __ (2) [20 0 j

where M_ is the freestreamMach number. If 82 = I-M2, then the vector_ has the components_ = 82 u T + v _+ w-k. This vector was first introduced by Robinson (ref. 4), who called it the "reduced current velocity". If_ denotes the total velocity vector, i.e., _ = (u_ +u) T + v _ + w_, and p the fluid density, then it can be shown that for irrotational and homentropic flow

p _ = p_ u--_+ p_ + higher order terms (3) where the subscript _ indicates the value of the quantity at upstream infinity, e.g., u_ = u_ i. Therefore, to a linear approximation, the vector_ is directly related to the perturbation mass flux as follows: (4)

The second equation of (1), i.e., the continuity condition, shows that for source-free flows (Q = 0), w is a conserved quantity.

Ward has integrated the three first order vector equations directly without having to resort to an auxiliary potential _anctlon. He obtained two different solutions for V (R), depending on whether 82 is positive (subsonic flow), or negative (supersonic flow). These two solutions can be combined formally into a single expression if the following convention is used:

K = 2 for _2 > 0

K = 1 for B2 < 0 1/2

R, = Real part of {(X-Xl)2 + _2_y-yl)2 + (Z-Sl)2]}

3 = Finite part of integral as defined by Hadamard(refs. 5 and 6). The resulting solution for the perturbation velocity _ at the point whose position vector is R1 = x 1 1 + Yl j + Zl k, is given by

V ( ) = - 2--_ " RB dS S

dS i R - R I 2_K R 3 S

< dV (5) + 2--_ _ dV 2wK R3

V V

This formula determines the value of V within the region V bounded by the surface S. The vector _ is the unit outward (from the region V) normal to the surface S. Furthermore, it is understood that for supersonic flow only those parts of V and S lying within the domain of dependence (Mach forecone) of the point R I are to be included i_n the integration.

For source-free (Q_O)_ irrotational (_0) flow, equation (5) reduces to

2_K n w V i dS + 82 ]J R--R-RIf- dS (6) s s B

This is a relation between V inside S and the values of n.w and _ <-v on S, but-these two quantities cannot be specified independently on S.

To determine the source-free, irrotational flow about an arbitrary body B by means of equation (6), assume that the surface S coincides with the wetted surface of the body, with any trailing wake that it may have, and with a sphere of infinite radius enclosing the body and the whole flow field about it, namely, S = SB +S W + S •

This surface S divides the space into two regions, V e external to the body, and Vi internal to it. Applying equation (6) to both V e and Vi, since the integrals over S_ converge to zero, the following expression is obtained:

4 A _ (R) V 1 L {Nx AV(R)} ×_ dS (7) =2wKl • R_ dS - 2wK R3j S B + S W S B + S W

where N = Hi= -he is the unit normal to the body, or wake as the case may be, positive from the interior to the exterior of the body, A _ --We - w--i,and A q = Ve - vi. Here the subscripts designate the values of the quantities on the corresponding face of S. The first surface integral can be considered as representing the contribution of a source distribution of surface density • A _, while the second surface integral gives the contribution of a vorticity distribution of surface density N × A V.

If the boundary condition of zero mass flux through the surface S B + SW is applied to both external and internal flows

• 0 _e =_ • (P_o + P_) = 0

N. :N. (8) then the condition N . A w = 0 exists over S B + SW, and the flow field is uniquely determined by

-- x R-RI (9)

S B + Sw where _ (R) = N × A V is the surface vorticity density.

Extension to Supersonic Flow

In order to extend the application of the vortex lattice method to supersonic flow, it is essential to consider the fundamental element of the method, the vortex filament, as a numerical approximation scheme to the integral expression (9) instead of a real physical entity. The velocity field generated by a vortex filament can be obtained by a straightforward limiting process, the result being

(lo) c where ['= lim V. 8

7-_oo

6-- o

8 is a dimension normal to V, and d_ is the d_stance element along Y. In the classical vortex lattice method, applicable only to subsonic flow, the vorticity distribution over the body and the wake, i.e., over the surface S B + SW, is replaced by a suitable arrangement of vortex filaments whose velocity fields are everywhere determined by equation (lO). This procedure is no longer appropriate for supersonic flow. For this latter case, it is necessary to go back to equation (9) and to derive an approximation to it. This is done in the following.

If the surface S B + SW, which defines the body and its wake, is considered as being composed of a large number of discrete flat area elements T over which the surface vorticity density V can be assumed approximately constant, then equation (9) can be approximated by the following equation:

N f_ 9(RI ) = - 2nK / RR-R13 J=l Tj where N is the total number of discrete area elements T . When the point whose position vector is R1 is not part of Tj, the integral over this discrete area can be approximated by the mean value theorem as follows:

R3_ Cj S where Cj is a line in Tj parallel to the average direction of V in rj, 80" is a distance normal to Cj, and d_ is the arc length element along Cj. Thi@ means that the velocity field induced by a discrete vorticity patch rj can be approximated for points outside of T j by some mean discrete vortex line whose strength per unit length is yj _T. But if the point R1 is part of the discrete area T, the integral in equatlon (_i) has an inherent singularity of the Cauchy hype due to the fact that R = R 1 at some point within _. In order to evaluate the integral expression for this case, consider a point close to R1 but located Just above T by a distance _. As indicated in figure l, the area of integration in r is divided into two regions, A r_¢ and Ae. Obviously, the integral over A __e has no Cauchy-type singularity, Hadamard's finite part concept being sufficient to perform the indicated integration. Thus,

6 _x-- dS= lim + lim R3_ e-..o _--,.O T A e A T-_

d_ (13) = lira I(_) + V _ _< R_RIR3_ _-.._O C

The last integral in equation (13) represents the conventional discrete vortex line contribution whose evaluation presents no difficulty. In order to determine the integration denoted by I(e) assume that, for simplicity, the coordinate system is centered at the point RI, and that the x-y plane is determined by the discrete area T. Then, if y denotes the modulus of y,

(14) , sinA - x cosA -)2 dx dy Y!X2 _ B 2 (2 + e2_ A

where A is the angle between the y-axis and the direction of the vorticity in • , and B 2 = -8_ > 0 (supersonic flow). The components of the vector cross product _ ×(R-RI) = _ x R which are not normal to the plane of _ have been left out of equation (14) because, when the limit operation ¢--o is carried out, they will vanish. The area A¢ is bounded by a line parallel to the vorticity direction going through x -(I+B)e and by the intersection of the Mach forecone from the point (o, o, s) with the z-plane, consequently, if the integration with respect to x is performed first,

(_) f ix2_B2,t F (y2- x + 2)}3/2 dx } dy

ty -(I+B) e where t = tan A , and kl, _2 are the values of y correspondim_ to the intersection of the line x=ty -(l+B)¢ with the hyperbola x -- -B_y 2 + e2" Let = e2(l+RB)_2(l+B)_ty _(B2_t 2) y2, then the finite part of the x- integration yields

l(e) = 'Y cosA+ B2ty (y2(ty-(1+B)_)+ )Jf 47-z i_ dy = 7 cosA _ B2_-(I+B) _y-(B2-t2)y 2 dy (16) B 2 J y 2 + ¢2

Since ¢ is a very sm_ll quantity, the variation of y in the interval (kl, 42) is going to be equally small, and, therefore, the quantity within brackets in the last integrand of equation (16) can be replaced by a mean value and taken outside of the integral sign. The same is not true of the term 1/_-_ since it will vary from co for y = 41, go through finite values in the integration interval, and then again increase to eo for y = _. With this in mind, and if denotes a mean value of y, I(¢) can be written as

I(¢) = Y CosA B2_-(l+B) gt_ - (B2-t 2) _2 ___ dy (17) B 2 ,,,2 2 y + ¢ J _-

But 41, _2 are the roots of ty-c = -B_y 2 + 2 , i.e., they are the roots of the polynomial denoted by _. Thus

_= _ 2(l_2B)_2(l+B)_ty -(B2_t 2) y2 = B__t 2 " _(_l_y)(y__ ) (18)

Introducing this expression for ^/--_ into (17), and taking the limit ¢--o, the following value for I(¢) is obtained:

_2 dy I(o) = lira I(¢) = - Y cos A (19) ¢ -.-o B 2 B2-t 2 _ _(41-Y) (Y- _ )

The integral appearing in equation (19) can be easily evaluated by complex variable methods; its value is found to be

dy = (20) _i 41-Y) (Y- _ ) 41

8 The contribution of the inherent singularity to the velocity field, within T, induced by the vorticity patch T, and dented herein byw*, is therefore given by

. _2 lim I(_) _cosA _B 2 t2 (21) 2w _ o 2

This contribution is perpendicular to the plane of T, and it has only physical meaning when B 2 > t 2, i.e., when the vortex lines are swept in front of the Mach lines. It is expression (21), taken in conjunction with equations (ll) and (12), that makes the vortex lattice method applicable to supersonic flow.

The Skewed-Horseshoe Vortex

Velocity fields due to complex vortex curves can be generated by the linear superposition of fields induced by simple vortex geometries. For instance, the velocity field due to a horseshoe line vortex can be obtained by the addition of the corresponding fields induced by three rectilinear segments: a transverse skewed segment, and two trailing leg_ figure 2. Therefore, the determination of the velocity field due to a line vortex segment of constant, but arbitrary, sweep is the fundamental building block in the formulation of aerodynamic influence coefficients of complex three-dimensional vortex lattices. Choosing a coordinate system such that the vortex line lies in the plane z = O, the conventional discrete vortex contribution to the velocity at a point whose coordinates are (Xo, Yo' Zo) Ls given, in Cartesian components, by the following expressions

u = + 2.-"-'_ (X_Xo)2 + _2( (y_yo)2 + Zo2) 3/2

(22)

,v = - rZ° 22 K I (x-x°)2 + _2( (y_yo)2 + Zo2) } 3/2

(X-Xo) dy - (y-yo) dx w = + 2wKF ,82 _C I (x-x°)2 ÷ _2( (y_yo)2 + Zo2) I 3/2

9 where rrepresents the circulation per unit length of discrete vortex line length, and the integrations are to be carried out along that part of C which satisfies the conditions

(x- Xo )2 + _2 ((Y- Yo )2 + Zo2). >0

and

x-x <0 if M >i. o

For the transverse leg of the horseshoe vortex, the coordinate x appearing in equations (22) can be expressed as a function of y, i.e., x = ty, and the indicated integrations carried out between the limits y = -s and y = +s, figure 2. By defining the following auxiliary variables

XI = XO + ts x 2 = x 0 - ts

Yl = Yo + s Y2 = Yo - s

X* = x ° - ty °

The resulting formulas giving the velocity components induced by the skewed transverse rectilinear vortex filament can be written as follows:

u = + 2_Kr___&z. x + 1 I tXl _2+ 132y( 1 2+z 2 2 2 ] *2 (t2+_) Zo 2 ,_,/X i 2 + Yl o ) _/X2 + _ (Y2 +Zo )

tx 2 + _2y 2 V _ m rz . t "I tXl + _2yl - 2_K -2 + (t%_2)z°2 _/xl2 + _2 (yl%Zo2) _/x22÷ _2 (y2%Zo2) (23)

F r x * I .txl + _2yl W x + (t2+_2) Zo x 2 + tx2 + _2y2"" ] 2_K *2 2 [ _/ 1 _2 (yl2+Zo 2 llfX2 2 + #,,2,2 tY 2 +Z o 2,)

In the above expressions, the coordinates (Xo, Yo, Zo) of the receiving field point are measured with respect to the midpoint of the rectilinear vortex segment, the x-y plane of the coordinate system coinciding with the plane defined by the x-axis and the vortex itself.

F The case of a rectilinear vortex segment parallel to the x-axis (t = _ ) is of special importance since the trailing legs of a horseshoe vortex are generally assumed to be parallel to the x-axis. Since dy = 0, equations (22) become

u = 0

FZo '_2 _ dx

v - A {(X-Xo12+ ((Y-Yo)2 + Zo21Y21 (2 I

- F(Y-Y°2trK) ,82 _ { (X-Xo)2 + fl2@((y-yo)2dx + Zo2 }3 /2

If _he vortex segment extends from x = x i to x = xf, the above integration yields

u = 0

XO - xf - -- 0 1 1 (2_) 2 2 2 V = _ZO2wK LIj o X _2- X, (Xo-Xl)_ + ((yo-y) + zo ) (Xo_Xf)2 + f12((yo_y) + Zo ) Cyo-y) 2 + Zo 2

+ _2 ((yo_y)2 + Z° 2) (Xo-Xf) {,2 +x ° I_2- xf((yo-y) + Zo2) ] (yo-y)Yo 2- ÷y z 2 0

For a conventional horseshoe vortex whose trailing legs stretch to downstream infinity, equations (25) would give the contribution of the port leg with the following substitutions

x. = _ if M_

• = x - 4-_ 2 ((Yo+S) 2 + z 2) i,f M. >I XI O O

xf - ts

y ------S

ii Likewise, the contribution from the starboard trailing leg can be computed by introducing the following values into equations (25)

x. = +ts 1

xf = _ if M_

xf = x° _ __132 ((Yo-S)2 + z°2) if M® >l

y = 4"8

Combining these results with equations (23), the formulas defining the flow field induced by a discrete vortex consisting of a skewed segment and two trailing legs parallel to the x-axis (the skewed-horseshoe vortex) are obtained. Keeping in mind Hadamard's finite part concept, and after introducing the following notation 2 txl + _ Yl F1 =

_/Xl 2 + _2 (yl2+Zo 2)

2 tx2 + _ Y2 F 2 = __ _2 _x2 2 + (y2 2 + zo 2 ) (26)

x 1 G 1 = + C ; (M®<'I: C = i; M. >I: C = O)

_/- Xl 2 + _2 (Yl 2 + Zo 2)

x 2 G 2 = + C ; (M_ i: C = 0) I 2 _2 2 2) x2 + (Y2 + Zo

then, the horseshoe vortex induced field formulas can be expressed as follows:

12 r Zo (Fz - F 2) u (xo, yo, Zo) = + 2_--_x*2 + (t 2 + _2) z 2 o

(FI F 2 ) t G 1 + Q2 I v (xo, Yo' Zo) = + F I 2 2 2 2 _ (27 2w---KZ° [ - *2 _2 2 x + (t2+ ) z° Yl + Zo Y2 + Zo

F (FI - F2) Yl Y2 w (Xo, Yo' Zo) - 2_K *2"-- t 2 2 + 2 2 GI 2 2 G2 Ix + ( + 132) z Yl + z Y2 + z 1 o o o

The finite part concept determines the value of the constant C appearing in the definition of G 1 and G 2.

A notable simplification of equations (27) occurs for supersonic flow when the receiving point (Xo, Yo' zo ) is in the plane of the horseshoe, namely zo = 0. First, the values of the axialwash and sidewash, u and v, vanish identically; secondly, the upwash expression becomes

(28 w (xo, Yo' 0) r IFI-F 2 G 1 G2)

Equation (28) is applicable to both subsonic and supersonic flow in its present format. But for the supersonic flow case, the fact that the constant C of the G functions becomes null due to the finite part concept allows further simplification of the upwash equation. Introducing the corresponding values of the F and G functions, the er)anded version of equation (28) is

Yl tx2 + _2y2 _g i - ,. w (xo, Yo' O) 2wK x tx I ++ 2 Yl 22 + Y22 ] (29

xl/Y 1 x2/Y 2 +

._Xl 2 + _2 Yl 2 ,_x2 2 + _12 y22 l

13 Sincein factorsx = Xoof i (xtYoI _ = _yl)Xl- %/x12tYl = +x2-_2 ylty2 ],-Itheandrea[rrangement(x 2 _ tY2) Jx22of equation+ _2 Y22(2i)-i finally reduces the upwash formula to

w (x , yo O) = F i _Xl Yl _ _x22 + _ Y22 o - 2_----K" --_x Yl Y2 (30 ) l 2+_2 2 2 I

When the field point (Xo, Yo' O) is within the distributed vorticity patch which is approximated by the discrete horseshoe vortex, e.g., the control point associated with the horseshoe, the upwash given by equation (30) has to be complemented by the distribution due to the generalized principal part of the upwash integral, as given by equation (21). If _x is the distance, measured in the x-direction, occupied by the distributed vorticity ¥ which has been lumped into the discrete transverse vortex leg of circulation _ the relationship between the F of equation (21) and the Fof equations (27) and (30) is

F = _cos A 6x (31)

Modeling of Lifting Surfaces with Thickness

The method of quadrilateral vortex rings placed on the actual body surface (ref. l) provides a way of computing the surface pressure distribution of arbitrary bodies using discrete vortex lines only. Numerical difficulties may occur when the above method is applied to the analysis of airfoils with sharp trailing edges due to the close proximity of two vortex surfaces of nearly parallel direction. An alternative approach, requiring somewhat less computer storage and easier to handle numerically, consists in using a double, or biplanar, sheet of swept horseshoe vortices to model a lifting surface with thickness, as shown schematically in figure 3. This constitutes an approximation to the true location of the singularities, similar in nature to the classical lifting surface theory approximation of a cambered sheet.

All the swept horseshoe vortices, and their boundary condition control points, corresponding to a given surface, upper or lower, are located in a same plane. The upper and lower surface lattice planes are separated by a gap which represents the chordwise average of the airfoil thickness distribution. The results are not too sensitive to the magnitude of this gap; any value between one half to the full maximum chordwise thickness of the airfoil has been found to be adequate, the preferred value being two thirds of the maximum thickness. Furthermore, the gap can vary in the direction normal to the x-axis to allow for spanwise thickness taper. On the other hand, the chordwise distribution, or spacing, of the transverse elements of the horseshoe vortices have a significant influence on the accuracy of the computed surface pressure distribution. For greater accuracy, for a given chordwise numberof horseshoe vortices, the transverse legs have to be longitudinally spaced according to the cosine distribution law

(32) xjv - x ° =c 2 [1 - cos (n-_2Jl)l

where x_ - x o represents the distance from the leading edge to the midpoint of the Swept leg of the Jth horseshoe vortex, c is the length of the local chord running through the midpoints of a given chordwise strip, and N is the number of horseshoe vortices per strip. The chordwise control point location corresponding to this distribution of vortex elements is given by

xjC - x ° = c 1 - cos _

The control points are located along the centerline, or midpoint line, of the chordwise strip (fig. 4). Lan has shown (ref. 7) that the chordwise 'cosine' collocation of the lattice elements, defined by equations (32) and (33), greatly improve the accuracy of the computation of the effects due to lift. His results are directly extendable to the computation of surface pressure distributions of wings with thickness by the biplanar lattice scheme presented herein.

The small perturbation boundary condition

(34)

is applied at the control points. In equation (34), n : _ +m_ +n_, and _' =m_ +nk, where _, m, and n are the direction cosines of the normal to the actual airfoil surface. Equation (34) implies that J_ul <

Modeling of Fusiform Bodies

The modeling of fusiform bodies with horseshoe vortices requires a special concentrical vortex lattice if the simulation of the volume displace- ment effects, and the computation of the surface pressure distribution, are to be carried out. To define this lattice, it is necessary to consider first an auxiliary body, identical in cross-sectional shape and longitudinal area distribution to the actual body, with a straight baricentric line, i.e., without camber. The cross-sectional shape of this auxiliary body is then approximated by a polygon whose sides determine the transverse legs of the horseshoe vortices. The vertices of the polygon and the axis of the auxiliary body (which by definition is rectilinear (zero camber) and internal to all possible cross sections of the body) define a set of radial planes in which

15 the bound trailing legs of the horseshoe vortices lie parallel to the axis (fig. 5). As the body cross section changes shape along its length, the corresponding polygon is allowed to change accordingly, but with the constraint that the polygonal vertices must always lie in the sa_rle set of radial planes. The axial spacing of the cross-sectional planes that determine the transverse vortex elements, or polygonal rings, follows the cosine law of equation (B2). The boundary condition control points are located on the auxiliary body surface, and in the bisector radial planes, with their longitudinal spacing given by equation (33).

The boundar_ _ condition to be satisfied at these control points is the zero mass flux equation.

where all the components of the scalar product _ . K = _2_ u + my + nw are to be retained. Thus, equatio_ (35) is a higher order condition than equation (34). The use of this higher order boundary condition, within the framework of a linearized theory, is not mathematically consistent. There- fore, it can only be justified by its results rather than by a strict mathematical derivation. !n the present treatment of fusiform bodies, it has been found that the use of higher order, or exact boundary conditions is a re- quisite for the accurate determination of the surface pressure distribution.

The fact that the vector _: instead of V, appears in the left hand member of equation (_5)j. requires some elaboration. First, it should be pointed out that for small perturbations w . n_ V n'. Furthermore, for incompressible flow (8 = I), the vector _ is _dentical to the perturbation velocity _. Con- sequently, the boundary condition eqL_tion (3_) is consistent with the continuity equation, _. w = O_ to a first order for compressible flow, and to any higher order for incompressible flow. But when a higher order boundary condition is applied in compressible flow to a iinearized solution, it should be remembered that this solution satisfies the conservation of _, not of V, i.e., . _ = O. Thus, the higher order boundary condition should involve the reduced current velocity, or perturbation mass flux, vector w, as in equation (35), rather than the perturbation velocity vector _.

The body camber, which was eliminated in the definition of the auxiliary body, is taken into acco_mt in the computation of the direction cosines _,m, and n, which are implicit in equation (35). Therefore, the effect of camber is represented in the boundary condition but ignored in the spatial placement of the horseshoe elements. This scheme will give a fair approximation to cambered fusiform bodies provided that the amount of body camber is not too large.

16 Computation of Sideslip Effects

The aerodynamics of an isolated wing in sideslip can be analyzed by two different approaches depending on the coordinate system chosen. In one approach, the coordinate system consists of wind axes, the longitudinal axis being aligned with the free-stream velocity vector, figure 6. This formulation of the problem is known as the skewed-wing approach, and a first order solution obtained within this framework will give the dominant effects of sideslip, even for the case of zero dihedral. The other approach, also shown schematically in figure 6 and known as the skewed free-stream approach, is based on a body-axis formulation of the problem and the corresponding first order solution, though it may be adequate for large dihedral, will fail to produce the significant effects of sideslip for low or zero dihedral. To compute the sideslip effects correctly within the framework of a skewed free-stream formulation, it is necessary to solve partial differential equations containing second order terms of the perturbation velocities. This implies a much more involved computational procedure than that required for the solution of the first order perturbation equations (i) On the other hand, the application of the skewed-wing approach to anything more complex than an isolated wing in sideslip, such as might be the case with a configuration with wing, fuselage, and nacelles, becomes geometrically very complicated.

The approach adopted herein is a combination of the two approaches men- tioned above, formulated with the objective of obtaining reasonably accurate sideslip effects using only a first order perturbation solution but without all of the geometrical complications inherent in the skewed-wing approach. Basically it is assumed that the vortex lattice representing the configuration and its vortex wake consists of both bound and free elements or legs; the vortex filaments that model rigid surfaces are considered bound, and those that constitute the wake are the so-called free elements, figure 7. The bound portion of the lattice, containing both transverse and trailing, or chordwise, segments, is invariant in a body axis system, the chordwise legs being parallel to the x-axis. The free legs of the lattice are not actually force free, rather they are assumed to extend to downstream infinity parallel to a predetermined direction which is proportional to the angles of attack and sideslip. If the proportionality factors are unity, then the free portion of the lattice would be invariant in wind axes.

After the circulation strengths of the above lattice geometry are solved for under the appropriate boundary conditions, the pressure coefficient distribution is computed in accordance with the higher order expression

2 c - IU_ u + v_ v) (36) p 2 q_

where U_ and V_ are the components along the x and y body axes of the freestream velocity vector of modulus q_ ; the corresponding perturbation velocity components are denoted by u and v, as usual. The use of equation (36) instead of the linear approximation

17 2 Cp 2 U_ u (37) q_

is required for the correct computation, within the present framework, of the rolling moment due to sideslip of a planar wing. This is due to the fact that the bound trailing legs, being defined in body axes, are not lined up with the free-stream flow and therefore, according to the theorem of Kutta, they con- tribute to the normal force. This contribution is represented by the second order term in equation (36), namely, V_ v. Even though this contribution is of second order, it must be included in the computation of the differential rolling moment due to sideslip, since this quantity itself is of the second order for a planar, or nearly planar, wing.

THE GENERALIZED VORTEX LATTICE METHOD

Description of Computational Method

The four items discussed in the preceding section, i.e., the inclusion of the vorticity-induced residual term w* for supersonic flow, the biplanar scheme for representing thickness effects, the use of a vortex grid of concentrical polygonal cylinders for the simulation of fusiform bodies, and the special lattice geometry consisting of both bound and free elements for the analysis of sideslip effects, have been implemented in a computational procedure herein known as the generalized vortex lattice (GVL) method. This method, outlined in what follows, has been codified in a Fortran IV computer program (VORLAX).

The basic element of the method is the swept horseshoe vortex whose trailing legs has both bound and free segments. The latter segments may trail to downstream infinity in any arbitrary, but predetermined, direction whereas the bound trailing legs are laid out on the proper cylindrical control surfaces in a direction which is parallel to the x body axis. Figure 8 illustrates schematically the representation of a wing-body configuration within the context of the present method. In this illustration, the streamwise arrangement of the lattice follows the cosine distribution law, equation (32), but both chordwise and spanwise distributions of vortex lines can be independently specified to be either of the cosine or of the equal spacing. To each horseshoe vortex there corresponds a control point which is placed midway between the bound trailing legs of the horseshoe; the longitudinal location of the control point is determined by equation (33) if the cosine chordwise distribution has been chosen, otherwise it is located halfway between the transverse legs, as required by quarter-chordthree-quarter-chord rule.

The direction of floatation of the wake vortex filaments is defined by the two angles _v and 6_ shown in figure 8, the former being proportional to the angle of attack, and the latter being proportional to the sideslip angle. The proportionality constants are part of the program input, the recommended values being 0 for the sideslip constant, and 0 or 0.5 for the angle of attack constant.

18 The velocity field induced by the elementary horseshoe vortex is given by equations (27) when above constants are both zero, and by somewhat more complicated expressions which take into account the kinks in the trailing legs when either one or both of the wake floatation parameters are nonzero. Though not presented here, these expressions can be easily derived through the application of equations (23).

When the influence induced by a horseshoe vortex upon its own control point is being evaluated, the contribution from the generalized principal part, as given by equation (21), is included in the normalwash if the free-stream is supersonic• Furthermore, for the supersonic case, the simplified downwash formula, equation (30), is used whenever the receiving point is in the same plane of the inducing horseshoe.

The horseshoe vortex velocity field is used to generate the coefficients of a system of linear equations relating the unknown vortex strengths to the appropriate boundary condition at the control points. This linear system is solved by either a Gauss-Seidel iterative procedure, known as controlled successive over-relaxation (ref. 8), or by a vector orthogonalization technique, i.e., Purcell's Vector method (ref. 9). If the inverse process is desired, i.e., synthesis or design instead of analysis, the linear system of equations is used to compute the slope distribution (surface warp) required to achieve a specified surface loading; this involves a straightforward matrix multiplication process. Mixed cases, i.e., design and analysis, are easily handled by proper grouping of the boundary condition equations.

The pressure coefficients are computed in terms of the perturbation velocity components, the computation being carried out according to either one of three possible ways, as follows:

la If the surface under consideration is assumed wetted by the flow on both sides (zero thickness panel) and the configuration sideslip angle is zero, then a net loading coefficient is computed based on the local value of the spanwise vorticity, namely, ACp = 2y cosA;

1 When the configuration sideslip angle is not zero, the net loading coefficient of a zero-thickness surface is calculated through the use of the higher order expression (36); and

o When surface pressure coefficients are computed, i.e., the panel under consideration is assumed wetted by the flow on one side only, the isentropic flow relationship giving the pressure coefficient in terms of free-stream Mach number and local-to-freestream velocity ratio is resorted to.

The force and moment coefficients are calculated by numerical integration of the pressure coefficient distribution with due account being given to the edge forces. If cosine chordwise lattice spacing is specified by the VORLAX program user, the computation of the leading edge suction of zero thickness

19 panels is carried out according to Lan's procedure (ref. 7), whose application to supersonic flow is made possible by the generalized vortex-induced velocity field formulas presented in this report. If equal chordwise lattice spacing is specified, the contribution of the leading edge suction singularity to the forces and moments is calculated by the technique indicated by Hancock in reference i0; this approach is not nearly as accurate as Lan's, the magnitude of the leading edge suction being significantly underestimated.

The VORLAX computer program has the capability of analyzing symmetrical and asymmetrical cases as well as configurations in steady state angular rotation about any or all of three axes, parallel to the coordinate axes, going through the input moment reference center. Steady state angular rotation cases are treated by the subterfuge of assuming a nonuniform onset flow, this onset flow being defined by the values of the angular rates and distance of the field point to the rotation center.

Ground proximity effects are analyzed by the method of images, i.e., the configuration is mirrored about the ground plane; the flow around the airplane and its image then contains a stream surface which coincides with the ground plane due to the symmetry of the arrangement. In this modeling of a configuration in ground proximity, it is assumed that the trailing vorticity wake floats to downstream infinity parallel to the plane of the ground.

Numerical Considerations

At supersonic Mach numbers, the velocity induced by a discrete horseshoe vortex becomes very large in the very close proximity of the envelope of Mach cones generated by the transverse leg of the horseshoe. At the characteristic envelope surface itself, the induced velocity correctly vanishes, due to the finite part concept. This singular behavior of the velocity field occurs only for field points off the plane of the horseshoe. For the planar case, the velocity field is well behaved in the vicinity of the characteristic surface.. A simple procedure to treat this numerical singularity consists of defining the characteristic surfaces by the equation

(X-Xl)2 C B _ (Y-Yl + (Z-Zl (38) where C is a numerical constant whose value is greater than, but close to, i. It has been found that this procedure yields satisfactory results, and that these results are quite insensitive to reasonable variations of the parameter C.

Another numerical problem, peculiar to the supersonic horseshoe vortex, exists in the planar case (field point in the plane of the horseshoe) when the field point is close to a transverse vortex leg swept exactly parallel to the Mach lines (sonic vortex), while the vortex lines immediately in front of and

2O behind this sonic vortex are subsonic and supersonic, respectively. Thi_: problem can be handled by replacing the boundary condition ecumti_:_nfor such sonic vortex with the averaging equation

-YI*-I + 2 Yi* - YI*+I : 0 (39) where YI* is the circulation strength of the critical horseshoe vortex, _ud ¥I*-i and YI*+I are the respective circulation values for the fore-mud-aft adjacent subsonic and supersonic vortices. The axialwash induced velocity component(u) is meededfor the com!_utation of the surface pressure distribution, and for the formulation of the bcundary condition for fusiform bodies. Whenthe field point is not too close to the generating vorticity element, the axialwash is adequately described by the conventional discrete horseshoe vortex representation. But if this point is in the close vicinity of the generating element, as may occur in the biplanar and in the concentrical cylindrical lattices of the present method, the error in the computation of the axialwash due to the discretization of the vorticity becomes unacceptable. This problem is solved by resorting to a vortex-splitting technique, similar to the one presented in reference ]I. Briefly, this technique consists of computing the axialwash induced by the transverse leg of a horseshoe as the summation of several transverse legs longitudinally redis- tributed, according to an interdigitation scheme, over the region that contains the vorticity represented by the sing3e discrete vortex. '?his is _]one only if the point at which the axialwash value is required lies withim a given near field region surrounding the original discrete vortex.

COMPARISON WITH OTHER THEORIES AND EXPERImeNTAL RESULTS

Conical flow theory provides a body of exact results, within the context of linearized supersonic flow, for some simple three-dimensional configurations. These exact results can be used as bench mark cases to evaluate the accuracy of numerical techniques. This has been done rather extensively for the CVI, method, and very good agreement between it and conical flow theory has been observed in the computed aerodynamic load distribution and all force and moment coefficients. Only some typical comparisons are presented in this report, figures 9 through 12.

Finally, the capability of computing surface pressure distributions by the method of this paper is illustrated in figures 13 and 14. CONCLUDINGREMARKS

The present vortex lattice method, in the form of a computer program, has the capability to calculate the aerodynamic load distribution at subsonic and supersonic Machnumbersfor arbitrary nonplanar configurations. It has been found to be a very useful preliminary design tool, particularly when aircraft configurations whosemission requirements involve both subsonic and supersonic flight are considered. It is also capable of the inverse process, namely, the computation of the surface warp required to achieve a given load distribution. Correlation with experimental data and with results from other theories showsa good agreement not only in the overall force and moment coefficients due to lift, but also in the distribution of the load coefficients. The schemesshownfor the simulation of thickness and volume effects, which allow the computation of surface pressure distribution by using only vortex lattice singularities, appear adequate for most practical purposes, though experience in this respect is somewhat limited.

The treatment of sideslip cases by the present method does not require higher order solutions, as is necessary for the skewed free-stream approach, and it is not as geometrically complicated as the skewed-wing formulation. Yet the analysis of complex configurations in sideslip still requires care and caution due to the numerical anomalies that may result from the interaction among aircraft components, such as a horizontal tail or a body, and the "free" trailing legs of the horseshoe vortices.

Additional capabilities that can be added to the present computer code, and that would enhance the value of the method as a preliminary design tool include the following:

Incorporation of an optimization algorithm based either on Lagrange multipliers or on a gradient method, to design the surface warp for minimum drag under specified constraints.

Application of the technique of reference ii, or of some other adequate technique, for the simulation of jet exhaust effects, with particular attention to its extension to supersonic flow.

Introduction of a design procedure for the calculation of the geometry required to achieve a given surface pressure distribution, i.e., synthesis of both camber and thickness. The biplanar vortex lattice simulation of a thick lifting surface is well suited for the development of such a design procedure when combined with an iterative scheme.

22 Uoo

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T/ =0.55&1.0 T/C=9.0% 048 15 ---b/2--_

-0.8 A_ -0.6 -_ 7 = 0.15 -0.4

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Figure 13.- Comparison with experimental pressure distribution on wing-body model at Mach = 0.5.

35 -0.6 LID =

-0.4

L/D = 8 Cp o.2

•_-- EXACT POTENTIAL FI..OW o [] GVL METHOD

0.2 0.4 0.6 0.8 1.0 X/L

Figure 14.- Theoretical comparison of pressure distribution on ellipsoids at zero angle of attack in incompressible flow. 36 APPENDIX A

USER 'S MANUAL FOR A GENERALIZED VORTEX LATTICE METHOD FOR SUBSONIC AND SUPERSONIC FLOW APPLICATIONS

THEVORLAXCOMPUTERPROGRAM

A computer program has been developed for the aerodynamic analysis and design of arbitrary aircraft configurations in subsonic and supersonic flow. This computer program, herein identified as VORLAX,has been codified in FORTRANIV for use on the CDC6600, the IBM 360, and the IBM 370 digital computer systems. A complete compilation and executed case in CDCFORTRANIV is contained in Appendix B. Two auxiliary interface programs, treated in Appendices C and D, provide for input data transformation from NASAWaveDrag format (Reference 12) to VORLAXformat (Appendix C) and for transformation from VORLAXto NASAWaveDrag format (Appendix D). The VORLAXprogram is based on a generalized vortex lattice (GVL) method which extends the applicability of vortex lattice _echniques to a broader range of problems than has heretofore been considered. In this program, the configuration is represented by a three-dimensional, generally nonplanar, vortex lattice; the basic element of the lattice is the skewedhorseshoe vortex whose induced velocity formulas have been generalized for subsonic and supersonic flow. Thickness effects can be simulated by a double (biplanar or sandwich) lattice arrangement. Fusiform bodies can be modelled by a concentric cylindrical lattice of polygonal cross sections. The computational capabilities of the program include the following: • Surface pressure or net load coefficient distribution. • Aerodynamic force and momentcoefficients. @ Surface warp (camber and twist) design in order to achieve input pressure distribution. • Longitudinal/lateral stability derivatives. • Ground and wall (wind tunnel interference) effects. • Flow field survey. • Symmetrical/as_etrical configurations and/or flight conditions. The limitations of the VORLAXprogram are characteristic of methods based on inviscid linearized potential flow theory, as follows: • Attached flow. • Small perturbation flow. • Flow entirely subsonic or supersonic (no mixed transonic flow). • Straight Machlines. • Rigid vortex wake.

39 PRACTICAL INPUT INSTRUCTIONS

In defining a configuration for the program input, a master frame of reference X- Y- Z- is assumed. The X-Z plane is the centerline plane, the Z axis directed upward, and the X-axis pointing in the downstream direction; the Y axis points to starboard. The origin of the system can be any con- venient point in the X-Z plane. In general, the configuration can be made up of symmetrical and asymmetrical components, and in defining the symmetrical components only the starboard elements need be specified.

The configuration to be input is divided into a set of major panels; up to 20 of these panels can be input, symmetrical components being counted only once. For instance, a wing with straight leading and trailing edges, and with linear lofting between the root and tip, constitutes a major panel. Complex planforms, and nonlinear changes in twist and airfoil sections are described by defining more than one panel for a given wing. The computer program will then subdivide each major panel into a number of smaller elementary panels spaced chordwise and spanwise, i.e., a finer mesh lattice is generated internally. The chordwise and spanwise spacing is specified by the user, two options being available: (i) the semicircle or cosine distribution so well known in airfoil and wing theory, and (2) the equally spaced distribution.

Up to 2000 elementary panels can be used in the definition of a given configuration. Any consistent system of length and area dimensions can be used in the specification of the configuration length, but it is recommended that the system of units used be such that the largest length dimension does not require more than three digits to the left of the decimal point. Other- wise, significant digits may be lost in the output printout.

Wing thickness effects can be taken into account within the context of control surface theory. By control surface theory one means that the exact linearized theory is used to evaluate induced velocities along a given mean surface, known as the control surface, and these values enter into the computation of the bounda_ conditions which are satisfied at this control surfac, rather than at the actual boundary surface. The control surface equivalent of a typical two-dimensional airfoil is illustrated in figure A-I. The assumption inherent in control surface theory is that the induced velocities vary very little in the vicinity of the surface. Experience has shown that for the majority of practical configurations the loss in accuracy is negligible, and is more than compensated for by the increase in computational efficiency. Any wing-like component with thickness is then represented by a double set of panels, one for the upper surface and one for the lower surface as sho_ _" schematically in figure A-2.

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42 distribution. The results are not too sensitive to the magnitude of this gap; any value between one half to the full maximum chordwise thickness of the airfoil has been found to be adequate, the preferred value being two thirds of the maximum thickness. Furthermore, the gap can vary in the direction normal to the x-axis to allow for spanwise thickness taper. On the other hand, the chordwise distribution, or spacing, of the transverse elements of the horseshoe vortices have a significant influence on the accuracy of the computed surface pressure distribution. For greater accuracy, for a given chordwise number of horseshoe vortices, the transver_e legs have to be longitudinally spaced according to the cosine distributicn law

Xj - x 0 =-_" - COS

v where xj - x o represents the distance from the leading edge to the midpoint of the swept leg of the Jth horseshoe vortex, c is the length of the local chord running through the h/dpoints of a given chordwise strip, and N is the number of horseshoe vortices per strip. The chordwise control point location corresponding to this distribution of vortex elements is given by

xj- x° =_ - cos _

The control points are located along the centerline, or midpoint line, of the chordwise strip (figure A-3).

The modeling of fusiform bodies with horseshoe vortices requires a special concentric vortex lattice if the simulation of the volume displace- ment effects, and the computation of the surface pressure distribution, are to be carried out. To define this lattice, it is necessary to consider first an auxiliary body, identical in cross-sectional shape and longitudinal area distribution to the actual body, with a straight baricentric line, i.e., without camber. The cross-sectional shape of this auxiliary bod_: is then approximated by a polygon whose sides determine the transverse legs of the horseshoe vortices. The vertices of the polygon and the axis of the auxiliary body, which by definition is rectilinear (zero camber) and internal to all possible cross sections of the body, define a set of radial planes in which the bound trailing legs of the horseshoe vortices lie parallel to the axis (figure A-h). A_ the body cross section changes shape along its length, the corresponding polygon is allowed to change accordingly, but with the constraint that the polygonal vertices must always lie in the same set of radial planes. As in the case of the biplanar representation of thickness effects, cosine axial spacing should be used for the analysis of fusiform bodies. The effect of body camber is taken into account by independently specifying the camber of the baricentric axis of the body.

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45 INPUT CARD IMAGE DESCRIPTION

The cards that constitute the input data deck are described in the following paragraphs. All format specifications are given in FORTRAN IV.

While some 19 cards are described, this does not mean that input for a case consists of 19 cards. Rather, these should be thought of as card types. Furthermore, not all card types will be included in a given case, those to be included or deleted being a function of some of the input values as shown in figure A-5.

CARD (I): TITLE In columns i through 80 write any alpha- numeric identification heading.

CARD (2): IS_LV Method to be used for solving the system of linear equations relating the boundary conditions to the vorticity strength, in integer format, in column 2. IS@LV = O: Gauss-Seidel relaxation with accelerated convergence (under- or over-relaxation); IS@LV = i: Purcell's vector orthogonalization method.

LAX Chordwise, or streamwise, spacing of vortices, integer quantity punched in column 12, LAX = O: vortices are collocated at the percent chord (X/C) values determined by the cosine law X/C - 0.5 (l-cos((2K-l) z/2N)), where K varies between i and N, N being the number of chordwise vortices; LAX = i: vortices are collocated according to the equally-spaced quarter-chord law X/C = (4K-3)/(4N). The cosine law is reco_r_aended for greater accuracy for a given number of vortices.

LAY Spanwise, or lateral, spacing of vortices, integer quantity punched in column 22. LAY = O: vortices are spaced at intervals (elementary vortex span) given by the cosine distribution law _b = bp(COS((J-l)_/M)-cos (JTr/M))/2, where Ab is the vortex element span, bp is the panel span, and J varies between i and M, M being the spanwise number of vortices in a given panel; LAY = i: vortices are equally spaced along the panel span, i.e., Ab = bp/M. The cosine spacing is recommended for enhanced accuracy, but

46 ®

Figure A-5. - VOBLAX case data deck setup for most cases the difference in the results between cosine spanwise spacing and even spanwise spacing is negligible.

REXPAR Over-relaxation parameter, in FI0.0 format starting in column 31. This parameter is intended to accelerate the Gauss-Seidel relaxation process, and/or make it convergen when it might otherwise diverge. Blank, or zero, input, implies that the program will compute internally the optlmum over- relaxation value. If a positive quantity between 0.01 and 0.99 is input, this becomes the value of the over-relaxation parameter that the program will use, the optimum value being overridden. If IS_LV = I, this parameter is not used, and therefore, not a required input quantity.

HAG Height above ground of the moment reference center, in FI0.0 format starting in column 41. If it is punched equal to zero, or left blank, the height above the ground is infinity, i.e., no ground effect. If a quantity different than zero is input, then the ground effect will be computed by the method of images, the height being given by the input value, in consistent units.

FLCATX Longitudinal vortex wake flotation factor, in FI0.0 format, starting in column 51. If zero, or blank, then the trailing vortex legs being shed from the corresponding trailing edges, extend to infinity parallel to the X-Y plane. If a value different fro_ zero is input, then the trailing vortex legs shed from the trailing edges form an angle [email protected] with the X-Y plane, where ALPHA is the freestream angle of attack. (See figure A-6)

FL_ATY Lateral vortex wake flotation factor, in FI0.0 format, starting in column 61. If zero or blank, then the trailing vortex legs being shed from the corresponding trailing edges extend to infinity parallel to the X-Z plane. If a value different from zero is input, then the vortex legs shed from the trailing edges form an angle

48 0 .H

.r4

0 0 # 0 ,0

t_2

% 0.-I %

0 %

I1)

t_9 _v = FLCATY'BETA with the X-Z plan, where BETA is the freestream angle of slideslip (see figure A-6).

ITRNAX Maximum number of iterations allowed for the Gauss-Seidel relaxation method, in 13 format right-adjusted to column 80. If no value is input, the code will make ITRMAX = 99 by default. If IS_LV = i, i.e., the vector orthogonalization solution is resorted to, then ITRMAX is not a required input.

CARD (3): NNACH Number of Mach numbers to be analyzed, in 12 format; i.e., integer value of NMACH in column 2. NMACH_ 7.

NACH Mach numbers in FI0.0 format starting in column II.

CARD (4): NALPHA Number of angles of attack in I2 format; i.e., integer value of NALPHA in column 2. NALPHA s 7.

ALPHA Angles of attack in degrees in FI0.0 format, starting in column ii.

CARD(5) : LATRAL Asymmetric flight or configuration flag. 0 in column 2 = symmetric flight and symmetric configuration about the X-Z plane. I in column 2 = asymmetric flight and/or asymmetric configuration.

PSI Sideslip angle in degrees in FI0.O format starting in column II. + = wind coming from left side of nose. Input O. or blank when LATRAL is O. Used to obtain static stability derivatives such as Cn_ , Cy_, etc.

PITCHQ Pitch rate in degrees/second in FIO.0 format starting in column 21. + = nose up pitch. Used to obtain dynamic stability derivatives such as Cm_, CLq , etc. LATRAL may be 0 when PITCH_ is nonzero.

RCLLQ Roll rate in degrees/second in FI0.0 format starting in column 31. + = left roll. Input 0. or blank when LATRAL is 0. Used to obtain dynamic stability derivatives such as C_p, Cnp , etc.

5O YAWQ Yaw rate in degrees/second in FI0.0 format starting in column 41. + = left yaw or airstream component washing from left to right across nose of airplane. Input O. or blank when LATRAL is O. Used to obtain dynamic

stability derivatives such as Cnr' CY r' etc.

VINF Reference free stream velocity in FI0.0 format starting in column 51. If no value is input, VINF is automatically set equal to i.0 by the program. This parameter is only used when any of the angular rates is different from zero. It enters in the computation of the equivalent flow angle. For instance, if VINF = WSPAN/2 (wing semispan) and ROLLQ = 5.73, then pb/2V = 0.i, and the rolling moment coefficient printed out by the program will be exactly one-tenth the value of the stability derivative C_ • Likewise, if VINF = CBAR/2 (half the mea_ aerodynamic chord) and PITCHQ = 5.73, then the difference between the output pitching moment coefficient and the pitching moment coefficient for the case PITCHQ = 0. will

be equal to one-tenth of the Cm derivative. q CARD (6)- NPAN Number of major panels that will define the configuration, in 12 format; i.e., integer value of NPAN in columns i-2 right adjusted to 2. NPAN $20.

SREF Reference area for force and moment coefficients, in FI0.0 format starting in column ii.

CBAR Pitching moment coefficient reference length, in FI0.0 format starting in column 21. Usually mean aerodynamic chord length.

XBAR Abscissa of moment reference point, in F10.0 format starting in column 31. X-coordinate in master frame of reference.

ZBAR Ordinate of moment reference point, in FI0.0 format starting in column hl. Z-coordinate in master frame of reference.

51 WSPAN Total wing span in units consistent with SREF and CBAR, in FI0.0 format starting in column 51. If left blank, a value of 2.0 will be assuaged by the program.

CARD (7): XI X or longitudinal coordinate of the leading edge of one side of a major panel. Usually taken as the most inboard side in the case of wings. Input in FI0.0 format starting in column i.

YI Y or lateral coordinate of leading edge of first side of a major panel. Input in FI0.0 format starting in column Ii.

ZI Z or vertical coordinate of leading edge of first side of a major panel. Input in FI0.0 format starting in column 21.

CCRD1 Chord length of first side of major panel measured from XI, YI, ZI above in the positive direction of, and parallel to, the X axis. c_D (8): X2 X or longitudinal coordinate of the leading edge of the second side of the major panel described on card (7). Usually taken as the most outboard side in the case of wings. In the case of a closed curved panel, e.g._ a cylindrical segment representative of a nacelle, X2 would be identical to XI. Input in FI0.0 format starting in column I.

Y2 Y or lateral coordinate of leading edge of second side of the major panel described on Card (7). Input in FI0.0 format starting ir column ii.

Z2 Z or vertical coordinate of leading edge of second side of the major panel described on Card (7). Input in FI0.0 format starting in column 21.

CCRD2 Chord length of second side of major panel measured from X2, Y2, Z2 above in the positive direction of, and parallel to, the X axis.

Note: Columns 41-80 of cards (7) and (8) are not read. Thus, any informatior useful for identification purposes may be written there.

52 Note that the side edges i and 2 of a major panel, which have just been described by the input in cards (7) and (8), define the direction of the positive normal for that panel. This is determined by the feet-to-head direction for an observer standing on the panel looking upstream (dc_m the negative X-axis) and with panel edge i to his left, and panel edge 2 to his right. In this analogy it is assumedthat gravity is not a factor in that am_bserver could be standing on the bottom of a wing, for example, equally as easily as he might stand on top. In the case of curved panels, such as a nacelle component, the direction of the positive normal loses its meaning for the panel as a whole, but it is still applicable to each one of the longitudinal, or chordwise, strips that make up the major panel. CARD(9) NV_R Numberof spanwise elements or vortices that will be used to represent the p_nel, in FI0.0 format starting in column i. NV_R_i00.0. RNCV Numberof chordwise vortices that will be used to represent the panel, in FI0.0 format starting in column ]I. R_CV_50. The program, using NV_Rand RI_CV, will subdivide the panel under consideration into a gr_d of NV@R x RNCV swept horseshoe vortices collocated in accordance with the values of the LAX and LAY parameters. Note that chordwise and lateral distributions are ingependent, e.g., a cosine chordwise spacing (LA_<=O) is compatible with equal spanwise distribution (LAY = I), and vice versa. The corresponding control points at which the boundary conditions are satisfied are collocated according to the law (X/C)contro I = 0.5 (J-cos (K_/N)), K varying between I and N, where N is the number of chordwise vortices, if LAX = 0. If LAX = I, then the control points are placed at (X/C)contro I = (LK-I)/(hN), n_mely, according to the equally spaced three-quarter chord distribution. The spanwise location of the control points is always at the centerline of the elementary swept horseshoe vortices.

To determine the surface slope at each element control point, the program uses straighz line element lofting between the two longitudinal, or butt line, edges of the major panel.

53 In addition to the above limitations to the values of NV@R and RNCV, if more than one major panel is used in the description of the configuration, the following should be observed:

NPAN If LATRAL=O, NV_R x RNCV _<2000 1

NPM_ If LATRAL=I, NV#R x RNCV x IQUANT _<2000 I

IQUANT being defined further down.

SPC Leading edge suction multiplier, in FI0.0 format starting in column 21. 0. = no suction, i. = i00 percent leading edge suction. Nonzero values are recommended for all panels whose leading edges are wetted by the airstream. The program has the capability of computing the effects of free leading edge vorticity (leading edge vortex flows) by a localized application of the Polhamus analogy. This computation is triggered by inputting the SPC parameter as a negative quantity. When this is done, the sectional leading edge suction vector will be rotated normal to the camber surface at the leading edge, instead of the corresponding attached-flow tangential orientation, and the forces and moments will be computed using the rotated suction vector.

PDL Planar/curved panel flag, in FI0.0 format starting in column 31. 0. = planar major panel is to be described (including warped planar.) PDL = 999. (or >360.) = a curved major panel is to be described.

CARD (i0): PHI Polar coordinate angle of radius vector when defining the subpaneling of a curved major panel. Omit this card when PDL = 0. PHI is the angle measured from the horizontal in a plane parallel to the Y-Z plane. PHI = 0 coincides with a line parallel to and in the positive direction of the Y-axis. Positive values of PHI are measured counterclockwise when viewed from the rear of the aircraft.

54 PHI is input in FIO.0 format starting in column i. This and the subsequent input RO constitute a polar coordinate pair. The number of pairs to be input = NV_R + i. Four pairs per card may be input in FI0.O format starting in co]_nn i. As many cards as necessary are used. While the location of the origin from which the polar angle, PHI, is arbitrary the first PHI, R_ polar coordinate pair must coincide with the YI, ZI rectangular coordinates input on Card (7). Likewise, the last input polar pair must coincide with the Y2, Z2 of Card (8).

Re Radius vector from arbitrary origin when defining the subpaneling of a curved major panel. Input in FIO.0 format. Each R_ is part of a PHI, R_ polar coordinate pair.

CARD (il): AINCI Tangent of the angle subtended by major panel root chord]_ne, or first edge (described in Card (7)), and the positive X-axis, in FIO.0 format starting in column i. Sign convention is determined by observing the edge i chordline and the X-axis from edge 2. The edge i chord is then rotated counterclockwise until it is parallel to the X-axis. If the angle rotated through is less than 90 degrees then the angle, and consequently its tangent, are considered positive. If it is greater than 90 degrees, then AINCI is negative.

AINC2 Tangent of the angle subtended by major panel tip chordline, or second edge (described in Card (8)), in FI0.0 format starting in column ii. Sign convention is determined by observing edge 2 and the X-axis, looking in the direction from edge 2 towsrd edge I. The edge 2 chord is then rotated counterclockwise until it is parallel to the )[-axis. If the angle rotated through is less than 90 degrees, then the sign is positive; otherwise it is negative.

ITS Surface flag input as a two place integer in columns 21 and 22, right-adjusted to column 22. ITS = 0 or blank indicates that the panel is considered as a lifting surface of zero thickness, i.e., both its upper and

55 lower surface are wetted by the external flow. ITS = 01 means that only the panel upper surface is wetted by the real external flow. ITS = -i indicates that only the panel lower surface is wetted by the real external flow. A double panel setup can then be used to represent a wing-like component with non-zero thickness, as previously illustrated in Figure 2. Notice that the XI, YI, ZI, X2, Y2, and Z2 values to be input correspond to the control surface plane, and not to the actual chordal plane. The results are not critically sensitive to the separation between the upper surface panel (ITS = 01) and the lower surface panel (ITS = -i), a separation of two thirds the thickness ratio of the airfoil being a good average value to use.

NAP Number of percent chords or stations along the chord (CCRDI and C_RD2) at which the camber, or surface, ordinates are to be input. Input as a two-place integer in columns 31 and 32, right-adjusted to column 32. Maximum value of NAP is 50. A NAP = 0, i, or 2 will be interpreted as a flat wing and no subsequent camber cards will be expected. If ISYNT, on this same card, is to be input as i, i.e., a design case, then NAP should be 0 or blank.

IQUANT Symmetry flag with respect to X-Z plane input as an integer in column L2. IQUANT = 0 or 2 indicates there is a mirror image of the panel on the opposite side of the X-Z plane. IQUANT = i indicates the panel is unique to the side for which it is being input.

ISYNT Design/analysis flag input as an integer in column 52. ISYNT = 0 or blank indicates that the panel has been defined geometrically and only analysis is to take place. ISYNT = I indicates that the panel camber is to be designed by the program to support a specified pressure distribution. If NAP on this same card was input >2, then ISYNT should be zero or blank.

56 NPP Nonplanar parameter, input as on integer in column 62. NPP = 0 indicates that all the vortex filaments representing a given surface lie in the cylindrical surface whose directrix is the leading edge of the panel, and whose generatrices are all parallel to the X-axis. NPP = i denotes that the transverse vortex. filaments are located on the actual body surface, but the bound trailing legs are parallel to the x-axis. This parameter affects the definition of ZC I and ZC 2 on cards 16 and 18.

CARD (12) CI Pressure coefficients along the first, or root, edge of major panel defined on Card (7). If ISYNT = 0, then this card is omitted. Units are dimensionless, _p/q. Format is 8FI0.O starting in column i with as many cards as necessary to input RNCV values of CI.

The desired values of the aerodynamic loading are defined at the chordwise location of the vortex lines. Thus, if LAX = 0, the corresponding X/C points follow the cosine distribution (1-cos ((2K-I)w/2N))/2; if LAX = I, then the definition points are located by the law (4K-3)/hN. In the above expressions K ranges between 1 and N, N being equal to RNCV, the chordwise number of vortices.

CARD (13): C2 Pressure coefficients along the second, or tip edge of major panel defined on card (8). If ISYNT = 0_ then this card is omitted. Format is 8 FI0.O starting in column I with as many cards as required to input RNCV values of CI. Linear interpolation between corresponding values of CI and C2 is used to obtain _C at intermediate spanwise values for the su_panels.

XAF Chord percent values at which camber, or surface ordinates will be supplied for the major panel, in 8FI0.0 format starting in column i using as many cards as necessary to define NAP values of XAF. If NAP is 0_ I_ or 2_ then a flat uncambered surface is implied and this card is omitted. These chord percents

5Y need not be equally spaced, but the same set applies to both the root and tip chords, or edges of the panel. Second order Lagrange or sliding parabola interpolation is used to interpolate between input XAF points to obtain the surface slope value at the control point of each subpanel.

For a panel representing the surface of a non-zero thickness airfoil or body, ITS W 0, the Lagrange interpolation is modified to a fractional power (1/2) Lagrange method in the neighborhood of the leading edge. This allows the precise definition of the surface slopes at the control points for a blunt leading edge.

CARD (15 ) : RLEI Leading edge radius, in percent chord, of airfoil section at the first, or root, edge of panel, in F10.0 format between columns 1 and 10. This card exists only if ITS @ 0, PDL <360.0 and NAP >2, i.e., an airfoil with non-zero thickness is being simulated. Other- wise, it must be omitted.

CARD (16): ZCl Camber ordinates or surface ordinates of root chord of the major panel described on Cara (7) in units of percent chord, in 8F10.0 format, using as many cards as necessary to input NAP values, each corresponding to an XAF of the Card (14) series. Omit this card if NAP is 0_ 11 or 2. If a lifting surface, such as a wing, is being simulated by a zero thickness panel then the ordinates of the wing camber line should be input here. If the wing thickness is being simulated by the sandwich or biplanar, method, i.e., a separate panel for upper and lower surfaces, then the surface ordinates for upper or lower should be input here. If a curved panel is being simulated (i.e., PDL >360.) and NPP = 0 then ordinates of the panel streamwise edge should be input. These might represent the mean line of a shaped cowl for a flow through nacelle, for example. In this simulation all shed vortices will lie in the same cylindrical surface determined by the leading edge of the curved panel and the X direction. If NPP = 1

58 and PDL > 360.0, the ZCI array represents the camber of the bod_ axis.

CARD (17) : RLE2 Leading edge radius, in percent chord, of airfoil section at the second, or tip, edge of panel, in FIO.O format between columns i and i0. This card exists only if ITS _ O, PDL < 360.0 and NAP > 2, otherwise it must be omitted.

CARD (18): ZC2 Camber ordinates, or surface ordinates of the second, or tip chord of the major panel, or area ratios of the major panel in 8FIO.O format, using as many cards as necessary to input NAP values, each corresponding to ar XAF of the Card (14) series. Omit this card if NAP is O, i, or 2. Linear spanwise interpolation is used to obtain intermediate values.

If a curved panel is being simulated (PDL > 360.) and NPP = i then area ratios in percent are expected here. These should represent the ratio of the cross sectional area of the closed polygonal surface being simulated at the XAF station under consideration divided by the area of the reference polygon times I00. Note that ZC2 is entered in percent where a value of i00 represents a section exactly the size of the reference polygon. Values greater or less than i00 are permitted down to and including O. The reference polygon is that input via the PHI-RO pairs on Card I0. In this simulation it is presumed that all stations along the panel have the same shape as the reference polygon, and the transverse vortices are located on the actual body surface of the curved panel.

This concludes the input for the first major panel of the configuration. If there is more than one panel, then start over with Card (7) and work down to this point. Panels may be input in any sequence.

After the last panel is described_ continue with Card (19).

CARD (19): NXS Number of X-stations that will define the spatial flow field survey grid. NXS = 00 means no survey desired. Maximum value of NXS is 20. Input as a two-digit integer in columns 1 and 2, right-adjusted to column 2.

59 NYS Number of Y-stations that will define the butt line values of the survey grid. NYS = 00 for no survey. NYS and NZS (following) may be any positive integer subject to NXS x NYS x NZS _2000.

Input as a two digit integer in columns Ii and 12.

NZS Number of Z-stations that will define the water line values of the survey grid. NZS = O0 for no survey. Input as a two digit integer in columns 21 and 22.

CARD (20): XS X station values for the spatial flow field grid. Omit this card if NXS=0. Input i_n 8FI0.0 format starting in column i, using as many cards as necessary to define NXS values.

CARn (21): YNCT Beginning of grid in the butt line direction. Input in FI0.0 format starting in column i. Omit this card if NXS = 0.

DELTAY Y-spacing of the grid. There will be NYS butt line planes equally spaced a distance DELTAY apart. Input in FI0.0 format starting in column i!.

ZNCT Beginning of grid in the water line direction. Input in FI0.0 format starting in column 21.

DELTAZ Z-spacing of the grid. There will be NZS water line planes equally spaced a distance DELTAZ apart. Input in FI0.0 format starting in column 31.

This ends the input description for a single case.

Consecutive data sets or cases can be submitted at the same time. The program will always identify the presence of a new set by th_ corresponding title card (Card (i)).

6O PROGRAMOUTPUT

The program output is processed by a standard 132 characters-per-line printer. The output from each configuration is preceded by a printout of the input data cards. This printout is not an exact image of the input deck; rather, it is the version of the deck as the code sees it, namely, the default value of an input parameter is printed if there is a corresponding blank in the input card. Also, data within a format field are lined up for clarity in identification, even though in the input deck such data maybe arbitrarily located within its field. The input deck data is followed by a list of the major geometric parameters used by the program and generated from the input data deck. Next, the componentand total force and momentcoefficients are printed out for a given flow condition (Mach number, angle of attack, angle of sideslip, and rotational velocities). These are followed by a tabulation of the location of all the vortex elements, the pressure coefficients, the circulation strengths, and other ancillary information. If a flow field survey about the configuration has been requested, then the flow parameters (velocity components, flow angles, Math number, and pressure ratios) at a series of field grid points will be listed. If other flow conditions have been analyzed, the sametype of output will follow for each one of them, starting with the listing of the componentand total force and momentcoefficients. If other configurations have been input, then the output will continue with the listing of the corresponding input data deck, and so on. Rather than describing the output format in detail, a glossary of the output terms, arranged in sequential order of appearance, and a sample computer output, Table A-I, are presented. Numberingindex for major panel identification. For cases where LATRAL= i, the 1-number preceded by a double asterisk in the PANELGEOMETRYlist denotes that the panel is the mirror image (about the X-Z axis) of the panel with the same1-number but without asterisks.

XAPEX =XI (see input terminology). YAPEX =YI (see input terminology). ZAPEX =ZI (see input terminology). PDC =PDC(see input terminology).

LESWP Sweepof the panel leading edge, in degrees. Positive for sweepback,negative for sweepforward. CSTART =C@RDI(see input terminology). TAPER Panel taper ratio, C_RD2/C@RDI.

61 PSPAN Panel span.

NVCR =NVCR (see input terminology).

RNCV =RNCV (see input terminology).

SPC =SPC (see input terminology).

SURF Panel surface area.

CN Panel normal force coefficient, referenced to its own surface area.

CL Panel lift coefficient, wind axes, referenced to its own surface area.

CY Panel lateral force coefficient, wind axes, referenced to its own surface area.

CD Panel drag coefficient, wind axes, referenced to its own surface area.

CT Panel leading edge thrust coefficient, referenced to its own surface area.

CS Panel leading edge suction coefficient referenced to its own surface area.

CM Panel pitching moment about moment reference center divided by (freestream dynamic pressure X SURF), wind axes.

CRM Panel rolling moment about moment reference center divided by (freestream dynamic pressure X SURF), wind axes.

CYM Panel yawing moment about moment reference center divided by (freestream dynamic pressure X SURF), wind axes.

SREF =SREF (see input terminology).

WSPAN =WSPAN (see input terminology).

CBAR =CBAR (see input terminology).

CLTCT Total (summation over all panels) lift coefficient referenced to SREF.

CDT_T Total pressure drag coefficient, referenced to SREF.

CYTCT Total lateral force coefficient, wind axes, referenced to SREF.

62 CMTCT Total pitching moment coefficient about moment reference center, wind axes, referenced to SREF and CBAR.

CRTCT Total rolling moment coefficient about moment reference center, wind axes, referenced to SREF and WSPAN.

CNTCT Total yawing moment coefficient about moment reference center, wind axes, referenced to SREF and WSPAN.

E Oswald's efficiency factor.

S Perimetral, or spanwise, index of vortex element.

C Chordwise, or streamwise, index of vortex element, 1 denotes leading edge element, and C value equal to RNCV corresponds to the last, or trailing edge element. x/c Percent chord location of bound, vortex line.

X, Y, Z Coordinates of horseshoe vortex element centroid (center point of bound vortex line).

CH@RD Local chord length.

SLCPE Surface slope at boundary control point.

ITS Flag which indicates type of panel surface (see input terminology), A zero value means that the panel is considered as a zero thickness lifting surface. A positive unit value (1) denotes that the panel is the upper surface of an airfoil-like element. A negative unit value (-i) corresponds to the lower surface. In the case of body-like components, 1 denotes the external, or wetted, surface. For a flow-through nacelle arrangement, 1 stands for the external surface, and -I for the internal surface.

DCP Local loading coefficient (_C = C - C , ) if panel ITS = O. P P_ p_ If ITS # 0, then DCP is the local pressure coefficient (Cp).

CNC Sectional normal force coefficient times local chord.

CN Sectional normal force coefficient.

DL Local dihedral, in degrees.

CMT Sectional pitching moment coefficient about local quarter chord.

GAMMA Vortex element circulation strength, divided by freestream velocity. zc/c If the design option is being invoked, the resulting surface warp is printed out instead of GAMMA. This surface warp is expressed in fraction of the local chord, and it includes both camber and twist.

CTC Sectional thrust coefficient times local chord.

63 CDC Sectional pressure drag coefficient times local chord.

ITRMAX Maximum allowable number of relaxation cycles if Gauss-Seidel solution is used (see input terminology).

EPS Tolerance, or minimum, iteration change for relaxation process.

ITER Actual number of relaxation cycles.

BIG Actual iteration change for relaxation process. Relaxation will stop when BIG 5 EPS, or when ITER = ITRMAX.

If a flow field survey about the configuration has been requested (NXS > 0 in the input deck, see input description), then the following parameters will also be output as part of this survey:

X, Y, Z Coordinates of the survey grid points (not to be confused with the X, Y, Z coordinates of the vortex centroids previously described), referenced to the configuration master coordinate frame. The X, Y, Z coordinates are determined by the XS, YNOT, DELTAY, ZNOT, DELTAZ input values (see input terminology).

U Dimensionless velocity (freestream velocity at infinity assumed to be unity) along the X-direction (body axes).

V Dimensionless velocity along the Y-direction (body axes).

W Dimensionless velocity along the Z-direction (body axes).

EPSL@N Upwash angle in degrees, this angle is measured with respect to the X-axis in a plane parallel to the X-Z plane.

SIGMA Sidewash angle in degrees. This angle is measured with respect to the X-axis in a plane parallel to the X-Y plane.

CP Local field pressure coefficient (CP = (Pstatic-Pinf)/qinf).

ML0C Local Mach number.

P/PT_T Local static-to-total pressure ratio for isentropic flow.

P/PINF Local to-freestream static pressure ratio for isentropic flow.

64 TABLE A-I. SAMPLE OUTPUT

VEHICLE (6L1607-32/-14 WING) N|TH O/U NACELLES. (1/10 SCJLE;. J_mSmlC CRUISEo 1 0.10 0.0 0.0 0.0 11

O.SO0 2.550

S.O00

0.0 0.0 0.0 0.0 1.000

11 9676.80 107.2280 191.0400 -5.4500 129.0000

Z ********** 0.0 0.0 4.'_762 352.3OqO FUSELJGF (FUSIFORN P,NELl 0.0 0.0 -q. 7868 _52.39Q9 FUSELAGE (FUS|FORM PANFL) 3.0000 48.0000 0.0 999.b000 NVOR,RNCV,SPC,POL 7.1815 90.00 7.1815 35.00 7.1815 -23.74 7.1815 -90.00 0.0 0.0 1 29 2 0 1 0.0 3.4052 6.8104 10.2157 13.6209 17.02hl 20.&313 73.P365 54.4825 _rY.2411 30.64?0 34.0522 37.45?4 40.862? 44.7670 47.6731 60.1989 63.5641 66. 9693 70.3?46 75.6044 77.18_0 80.5007 83.9955 87.4007 90.8059 94.2111 97.6163 100. 0000 -1.0785 -0.8726 -0.6739 -0.5?50 -0.4597 -0.4512 --0._881 -0.5931 -0.76|6 -0,9293 -1.0423 --1.1501 -1.278_ -I._015 -I._7_3 -1.7472 -1.9552 -2.0687 -2 • 1283 -2.1737 -2.1793 -7.17_7 -?.1254 -2.C733 -1.8927 -1.7281 -I. 5125 -I.2571 -I.0409 0.0 16.0000 30.2500 45.0000 59.2900 72.2500 S%.O000 I00.0000 100.0000 100,0000 tO0.O000 IO0.O000 I00.0000 I00.0000 IO0.O00C lO0.O00C 100.0000 100.0000 I00.0000 100.0000 I00.0000 I00.0000 _.O000 72.2_00 $9.2900 45.0000 30.2500 16.0000 o.0

_.4100 6.5740 -5.4060 144.8o40 wINC, PANFL NP. 1 L_.2SO0 22.3940 -5.44?0 92._70 w|NG PANEL Nn. 1 4.0_0 10.0000 I.oooo o.0 NVOR,RNCV,SPCtPnL 0o0 0.0 0 13 7 0 0 0.0 1.0000 5.0000 lO.O00C 20.000C 30.0000 4n.r_00 50.r000 70.0000 80.0000 o0.0000 lO0.O000 0.0 -0.0129 -0.1367 -0.4899 -1.258o -1.o_7_ -_._770 -?.88h_ -3.1954 -3.4439 -3.666_ -3.9008 -4.1644 0.0 0.0529 0.2700 0.3078 0. I14_ -0.?_30 -0.6_2_ -I.I_OO -1.6545 -2.1469 -2.6232 -3.0692 -3.4807

146 • 2500 22.39&0 -5.4470 _2.5¢_7C wING PJNFL N_. Z 126.8800 31.2410 -6.9000 65.0[q0 Wing PiPEt Nn. ? 3.0000 10.00oo 1.0000 O.C NVDR,RNCV,SPCePDL 0.0 0.0 o 13 ? O 0 0.0 1.0000 5.0000 IO.OCAO 20.0000 30.0000 _O.CCCO 60.0000 70.0000 eO.O000 o0.0000 100.OG00 0.0 0.052V 0.2700 0._078 0.1145 -0.?_30 o._7_ -1.6545 -2 • 1469 -2.6232 -3.0692 -3.4807 0.0 0.0138 0.1708 U.349_ 0.4138 0._277 0.I_8 -C.C78_ -0.3554 -0.6585 -0.9846 -1.3231 -I.6o77

4 ********** I'M.SeO0 31.7410 -_.900_, 65.0010 WING P&NFL NO. 3 190.S400 36.8040 -?.?lbQ '_2.eBTG WING PANFL Nn. 2.0000 IO.O000 1.0000 0.0 NVOR,RNCVtSPC,PPL 0,0 0.0 0 13 ? 0 0 0.0 1.0000 5.00_0 10.0(_0 20.0000 30.0000 _0.0000 50.0000 &O.O000 ?O.O00C 80.0000 qO.OCO0 lO0.OOOO 0,0 0.0138 0.1708 0.34q2 0.4138 0.3?77 0.IS38 -O.O78E -0.35S4 -0.6585 -0.9846 -1.32_I -!.6677 0.0 0.0151 0.0224 0.2623 0.3095 0.2567 0.137_ -0.022_ -.O.2227 -0._511 -0.7002 -0.9_06 -1.2_62

65 TABLE A-I. - Continued

190.5400 36.8040 -7.7160 52._E70 WING PANEL N_. 4 222.5800 48.6830 -q.9740 31.1720 WING PANEL NO. 4 4.0000 lO.OOCO 1.0000 O.0 NVOR,RNCV,SPC+POL 0.0 0.0 0 13 2 0 0 0.0 l.O00C 5._00(, I0.0000 20.0000 30.0000 40.0000 50.0000 bO.0000 70.00_0 RO.O000 _0.00_0 100.0000 0.0 0.01_1 0.0774 0.2623 0.30_5 0.2567 0.1378 --0.0226 -O.2227 -0.4511 -0.7002 -0._6C6 -1.2362 0.0 -0.0257 -0.1123 -0.11_5 0.0513 0.1283 0.192_ 0.2310 0.263I 0._919 0.3117 0.327? 0.3368

222 .$800 48.6_30 -8.9740 31.1720 WING PANEL NO. 5 248.0000 64.5000 -lO.O00O I_.4%00 WING PANEL NO. 5 5.0000 lO.O00O 0.O 0.0 NVORtRNCVtSPC,PDL 0.0 0.0 0 13 0 0 0.0 1.0000 5.0000 IO.C(,OO ?0.0000 30.0000 40.0000 50.0000 60. OOOC 70.0000 80.0000 90.0000 100.0000 0.0 -0.02_7 -0.1123 -0.11_ 0.0513 0.1283 O. IP_3 0.2310 0.2631 0.29_I 0.3112 C.3272 0.3368 0.0 -0.0411 -0.1800 -C.3137 -0.4371 -0.4731 -0.4628 -'0.3651 -0.2057 -0.0051 0.2674 0.5811 0.9103

e4e*ee**ee "1 e_et_ee 20Q.4000 22.3040 -3.4000 3_.30n0 UPPER NACELLE PYLON 209.4000 22.3040 -5.4470 3_ .3C00 UPPER NACELLE PYLON I .O000 I0.0000 0.0 C.O 0.O 0.0 3 0 2 0 0

20_ .4000 22.39,,0 3.38_0 39.3000 UPPER NJCELLE 209.4000 22.30_,f_ 3.3840 3_.3000 UPPE_ NJCELLE 6.0000 10.OOC(: O.0 999.0_0 NVOR,RNCV,SPC_PDL 90.00 3.39 2(, 30.00 _.3920 -30.00 3.3920 -90.00 3.3920 -150.00 3.39'_0 -_I0.00 3._9P0 -_TO.(_) 3.3920 0.0 0.0 1 3 2 0 I 0.0 12.1522 I00. 0000 0.0 0.0 n.o I00.0000 100.000C 100. 0000

209. 1000 2_'. 3940 -_;. 4470 39.4000 LOWER NACELLE PYLON 209.1009 22.3940 -7.2000 3_._000 LOWER NACELLE PYLON I • 0000 10.0000 O.0 0.0 0.O 0.0 0 0 2 O 0

IC _w_ _ _ ?09. 1000 ?t2.3940 -7. 2000 39.4000 LOWER NACELLE 209. I000 22.3040 -7. 2000 39.4000 LOWER NACELLE o.0000 10.0000 O.0 q99.0000 NVOR,RNCV,SPC,PDL 90.00 3.2000 30.00 3.2C00 -30.00 3.2000 -eO.O0 3.2000 -15G.O0 3.2000 -210.00 3.2000 -2¥0°00 3.2000 0.0 0.0 1 3 2 0 I 0.0 1i.9148 tO0. 0000 0.0 0.0 0.0 l _0. 0000 I00.0000 I00.0000

248.0000 64.5000 -10.0000 19.4S000 FIN PANEL NO. • 259.5398 64.5000 -2. 8000 14.1 100 FIN PANEL NO. 1 4.0000 10.0000 0,0 0.0 NVORtRNKV_SPCtPDL 0.0 0.0 0 O 2 0 0

0 0 0

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92 RECOMMENDATIONS FOR THE EFFICIENT USE OF THE VORLAX PROGRAM

The definitions of the input and output data presented above should suffice for the running of any arbitrary configuration through the VORLAX program. But in order to achieve an efficient and accurate use of the capabilities provided by the generalized nonplanar vortex lattice method, the following recommendations should be adhered to:

When running a new configuration for the first time, run only one flow condition point, i.e., NMACH=I and NALPHA=I, to check out the input deck. This should be repeated for both subsonic and supersonic Mach numbers if the analysis of the given configuration will extend through both flow regimes.

Use this relaxation solution (IS@LV=O) whenever possible. This is due to the fact that the relaxation procedure computational time varies somewhat proportionately to the second power of the total number of vortices making up the configuration, whereas the vector solution time varies as the third power. In addition, the vector solution involves an order of magnitude increase in the data trans- fer operations between the computer storage and core regions.

• When major panels are in tandem, and lying approximately in the same plane, the spanwise distribution of vortex elements should be identical to prevent the trailing vortex legs shed by the fore panel from running through the control points of the aft panel, in which case, pressure distributions may show spurious oscillations with possible consequent effects on aerodynamic coefficients.

In specifying the vortex lattice density of a given configuration as a whole, or of a particular major panel, by the quantities NV_R and RNCV, three different degrees may be considered: sparse, medium, and dense. To illustrate the case, the starboard panel of an aspect ratio 3 straight-tapered wing is sparsely latticed if NV@R=!0 and RNCV=5, say; if NV_R=I6 and RNCV=IO, then the panel may be said to have a medium density grid. A finely latticed, or dense, grid would be obtained for the values NV_R=30 and RNCV=20. In the above examples, the grid density was considered to be uniform in both the chordwise and spanwise directions. It is obvious that mixed-type lattices like dense-spanwise-sparse-chordwise are also possible. The type of lattice, when correctly specified, is a powerful tool for the program user. It allows him to achieve the lowest computational cost for a given type of required data, as follows:

(i) Stability and control type data (accuracy is essential only in the force and moment coefficients, good definition of the aerodynamic load distribution is not required, drag coefficient accuracy not too critical): sparse lattice.

93 (2) Load distribution and drag type data (good definition of the aerodynamic load distribution, both spanwise and chordwise_ is required, accurate drag coefficient values are needed): medium density lattice.

(3) Surface pressure distribution type data (good definition of !,h_ surface pressure distribution -- control surface theory -- i_ essential, such as for wing design): lattice with maximmn chordwise density and with medium spanwise density.

In addition to the lattice, or vortex grid, density, the type of vortex distribution (cosine and equal spacing) is highly sig_ifie_ in determining the accuracy of the results. It has been found thst chordwise cosine spacing (LAX=0) is superior in accuracy to th,_ chordwise equal spacing (LAX=I). In terms of spanwise spaein6_ no significant differences have been observed between cosine (LAY=0) and equal spacing (LAY=]), the cosine spaein_ appearing to l_e slightly more accurate. For the same number of vortices, there: is no difference in the computational cost between cosine and equ_ spacing lattices.

When the fusiform body representation is being used (PDL=360 and NPP=I), the number of sides of the polygon defining the body cros_ section should be kept as low as possible, e.g., the cross seetio_ of a body of resolution can be adequately represented by a hexago_. Also, when a very slender body with pointed nose is being eonsidere_i_ the nose (and afterbody if it is also pointed) should be arbitraril;v blunted in the input definition in order to minimize the numericaJ difficulties caused by the crowding of the vortical singularities in the body nose region.

Figure A-7 illustrates a typical vortex lattice modelof an advanced t_di_- engine tactical fighter. This particular lattice model, with even chordw:i_r and spanwise spacing (LAX=I, LAY=I), is considered sparse, and quite adequate for stability and control work, both longitudinal and lateral. In this model, both the fuselage and nacelles are represented, or simulated, by flat plate elements, i.e., NPP=0. Obviously, this body simulation does not allow the computation of surface pressures, but it is adequate, and the most computa- tionally efficient, for stability and control work as well as for load di_- tribution and drag data.

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APPENDIX B

COMPLETE PROGRAM COMPILATION AND EXECUTION FOR A GENERALIZED VORTEX LATTICE METHOD FOR SUBSONIC AND SUPERSONIC FLOW APPLICATIONS

HARDWAREANDSYSTEMS

The VORLAXprogram has been run on several different IBM computer systems at Lockheed. All classified work conducted by the AdvancedDevelop- ment Projects Division, "Skunk Works", has been run on a 360/65 and no information can be presented about operation on this system. All non- "Skunk Works" areas such as commercial and military engineering perform aerodynamic and loads analysis using VORLAXon the IBM 360/91. Furthermore, a 370/168 has been used to assure that VORLAX can be efficiently run on this system if scientific applications are removed from the 360/91 system.

The program has been used for a wide range of aircraft configurations and a good idea of the best operational procedures has been established. The program has been run at sizes ranging from 65K words (260K bytes) to 85K words (340K bytes). An initial attempt was made to use a central memory region of 65K. This proved extremely inefficient because the number of accesses for I/O to peripheral storage soared with the imposed limitation of small buffers for each logical unit used. The program became I/O bound and larger buffers were called for. By going to large buffers, approximately 3000 words are accessed with a single read or write. With the smaller buffers, the number of words per access might be 600 - a factor of five smaller. When large cases are run on VORLAX, 5 to lO million words may have to be accessed; it is readily apparent that the number of reads and writes can become overwhelming if the buffering is not adequate.

Almost all cases run on VORLAX use less than 20 minutes of central processor time on the 360/91. Lockheed has demonstrated that a 25 minute case will run in about 15 minutes on a Control Data Cyber 175. It is, however, incumbent on the user to ensure that the system control language is properly adjusted to optimize operation of the program and the computer on which it is running.

Since small central core size is usually an objective, it is worth noting that due consideration has been given to various methods of reducing VORLAX central memory requirements. All the obvious methods of core reduction have been found to have disadvantages. In particular, by studying the tree struc- ture of the program, it was determined that overlay would not significantly reduce size and it would introduce inefficiences. Consequently, in order to reduce core size requirements, the actual array sizes in the program have been minimized to allow reasonable configuration complexity without un- necessary waste. But more importantly, mathematical techniques were used in the problem solution that reduce the need to have large matrices in central memory. One, two, or three elements are brought into central memory, processed, and shipped back to peripheral storage to minimize memory requirements. This results in enormous savings in central memory.

Since the Spring of 1975, the source coding has almost doubled in size

99 with a comparable increase in program flexibility and capability and yet central memory requirements have only increased about 20 percent. This was made possible only through continued attention to the optimization of the operation of the program on Lockheed's computer complex. It is of on-going concern to continue to study possible means of improving program operational efficiency, and to this end several novel ideas are being actively examined for their possible future incorporation to the computer code.

lO0 PROGRAM

UNIFIEDVORTEXLATTICEMETHOD FORSUBSONICANDSUPERSONICFLOW (VOP_AX)

COMPLETE COMPILE AND EXECUTION

IN CDC FORTRAN

i01

C{)NTROL._VRL X o AAMA [ NC_) g/16/7 VORLAX VORLAX C VORLAX C VORLAX C L {_00 CCCC _, _, H H EEEFE EEEEE l)Lh)i) VORLAX C L 0 0 C K K H H E E 0 O VORLAX C L 0 0 C K K H H E E D D VORLAX C L 0 0 C KK HHHHH EEErE EEEEE I) 0 VOHLAX C L f) _ C K K H H E E I) l) VORLA_ C L D 0 C K K H H E E b 0 VORLAX C LLLLL 000 CCCC K K H H EEEFE EEEEE Ot)bF)O VORLAX C VORLAX C VORLAX C VORLAX C PPpPP RR_RR 000 PPPPP RRRRR IT)'(( _EEEE ;TTTI AAA RRRRR Y Y VORLAX C P P R R D I) p P R R 1 E T A A R R Y Y VORLAX C PPPPP RRRR_' 0 0 PPPPP RRRRR I EEEEE T AAAAA RRRRR Y VORLAX C P R R O 0 P R R I F. 1 A A R R Y VORLAX C P R R 0 0 v R R I E T A A R R Y VORLAX C P R R 000 P R R fill( 'E'_EEEE T A A R R Y VORLAX C VORLAX VORLAX

LOCKHEED PROPRIETARY DATA

THIS DATA CONTAINS INFORMATION AND/OR DESIGNS WHICH ARE THE PROPERTY OF LOCKHEED AIRCRAFT CORPORATION. RECIPIENT, BY ACCEPTING THE SAME, AGREES THAT THE DATA WILL NOT BE REPRO- DUCED, TRANSFERRED TO OTHER DOCUMENTS, DISCLOSED OUTSIDE OF THE RECIPIENT ORGANIZATION, OR USED FOR MANUFACTURING OR PROCUREMENT WITHOUT THE EXI'_ESS WRITTEN PERMISSION OF LOCKHEED AIRCRAFT CORPORATION. THIS I.IMITATION MAY BE REMOVED OCTOBER 1977.

LOCKHEED PROPRIETARY DATA

C VORLAX C GENERALIZED VORIEX LATTICE PROGRAM (S(IRSOI_IC/SUPERSONIC NoNPIANAR) VORLAX C LUIS R. MIRANDA (DEUT. 75-_I) IBLOG. f-,]G /(;_]3) 847-68I_/ VORLAX C WILLIAM M. I]AKER (F)EPI. 80-34) /BLOG. 67 /(_13I _.7-_537/ VORLAX C _,_,r_ LOCKHEED-CALIFORNIA COMPANY., HU_BANK• cALIFORNIA _'_'*_ VORLAX C COMPUTER SERVICES .;{)F_NUMBER _-565 VORLAX C VORLAX

C VORLAX C VORLAX

C VORLAX C V V 000 RRRRR L ,_ X X VORLAX C V V n 0 R R L A A X X VORLAX C V V 0 0 RRRRR L AAAAA X VORLAX C V V 0 0 R R L A A X _ VORLAX C V 000 R k LLLLL A A X X VORLAX C VORLAX

C VORLA_ C VORL AX PROGRAM vORLA_. ( INPUT, OUIPU [, TAPES=INPUI • T APEB=OUIPUT. TAPE I , T APE2 ' VORLAX _'TAPE3, I APE4, TAPET, TAPEg,_ IAPE l | •TAPPI2) VORLAX Co • • VORLAX C.o.OFFINITION OF VARIABLES STOREO IN C,OMMON t-'fLOCKS. VOI;'LAX C... VORLA_

zo3

C..,B2 COMPRESSIBILITY FACTOR (M eL2 - I,) VORLAX C,,.CO, CL, CM, CN PANEL DRAG, LIFT, PITCHING MOMENT, AND _OPMAL VORLAX

Clod FORCE COEFFICIENTS, VORLAX C...CX, CY PANEL X- AND Y-FoRCE COEFFICIENTS. VORLAX C.,.DL DIHEDRAL OF CHORDWISE ROW OF HORSESHOE vORT! VORLAX C,,, cES. VORLAX C.,.FN, FY PANEL NORMAL AND Y-FOHCF PER UNIT Q. VORLAX C.,.IH, IQ ANGLE OF ATTACK AND NACH NUMBER INDICES. VORLAX C...NI TOTAL NUMBER OF CHORDWISE ROWS OF VORTICES. VORLAX C.,.RM PANEL ROLLING MOMENT PEP UNIT Q, VORLAX C,.,SX SPANWISE LOCATION INCEX, VORLAX C,,.XS ABSCISSAE OF FLOW FIELD SURVEY CROSS-PLANES, VORLAX C.,,¥M PANEL YAWING MOMENT PER UNIT Q. VORLAX C,,,YY V-COORDINATE OF HORSESHOE VORTEX CENTRUID_, VORLAX C.,,ZC NORMAL CAMBER COORDINATr IN FRACTION of CHORD. VORLAX C..,ZZ Z-CoORDINATE OF HoRSEsHoE VORTEX CENIRUII)_. VORLAX C..,BIG MAXIMUM CHANGE PER RELAXATION CYCLE. VORLAX C.,.CDCt CNC CHORDWISE CD AND CN lIMES LOCAL CHORI), VORLAX C,.oCRMt CYN PANEL ROLLING ANO YAWING MOMENT COEFFICIENTS, VORLAX C,.,DCP PRESSURE COEFFICIENTS (FIIHER LOAD OR VORLAX

Ceee SURFACE). VORLAX C.,.EPS ACCEPTABLE FINAL RELAXATION CYCLE CHANGF. VORLAX C,o,HAG HEIGHT ABOVE GROUND OF MOMENT REFERENCE VORLAX

CeDe CENIER. VORLAX C.o. ITS_ JTS PANEL AND STRIP FLOW EXPOSURE FLA_S. VORLAX C,,,LAX,LAY LATTICE CHOROWISE ANC SPANWISE OISTRIBUTION VORLAX Co.. FLAGS, VORLAX C.,.NPP PANEL CHORDWISE NONPLANARIIY FLAG. VORLAX C...NXS, NYS, NZS NUMBER OF FLOW FIELD SURVEY PLANES (ORTHOGD VORLAX C,o, NAL TO THE COORDINATE AXES). VORLAX C,,.PDL PANEL 5PANWISE NONPLANARITY FLAG. ALSO PANEL VORLAX

Coee PANEL DIHEDRAL IF PANEL IS FLAT IN THE SPAN VORLAX

CeDe wiSE DIRECTION. VORLAX C.,,PSl YAw ANGLE (POSITIVE IHEN FLOW COMES FROM VORLAX C.,. BORTSIDE). VORLAX C,.,RCS CROSS-SECTION RAoIUS VECTOR. VORLAX C,,.RLM APPROXIMATE VALUE OF EIGENVALUE RADIUS _F VORLAX

CoRD NORMALmASH MATRIX, VORLAX C. •. SLE SLOPE AT LEADING EDGE. VORLAX C,.,SPC PANEL LEADING EDgE SbCTION FACTOR. VORLAX C.,,TNL TANGENT OF LEADING EDGE SWEEP, VORLAX C,,.TNT TANGENT OF 1RAILING EDGE SWEEP. VORLAX C...VSP TANGENT OF SKEWED vORTEJ LINE, VORLAX C,..VSS SEMISPAN OF cHORDWISE VORTEX ROW. VORLAX C.,,VST ACTUAL HORSESHOE vORIEX SEMISPAN, VORLAX C...XAF CHORDWISE PERCENT COOrdINATES _T WHICH rAMBEH VORLAX

C... IS INPUT, VORLAX C,o.XTE TRAILING EDGF ABSCISSA OF VORTEX ROW. VORLAX C.,.ALFA ANGLE OF ATTACK (IN RAOIANS), VORLAX C,o,ALOC COMPONENT OF FREE-STREAM AND ONSET FLOW% NOR VORLAX

CeI. HAL TO THE SHRFACE AT THE CONIROL POINI_. ALSO VORLAX

CeoG AUXILIARY ARRAY, VORLAX C.,.BETA PRANDTL-BLAUERT FACTOR, VORLAX C.,°CBAR REFERENCE CHORD, VORLAX C,,.CMTC CHORDWISE TORSIONAL NONFNI AHOUT QUARTEP CHORD VORLAX

Ce_. TIMES LOCAL CHORq, VORLAX C.,.CSUC PANEL LEAOING EDGE TPHUSI PER UNIT Q. VORLAX

i05 C,..ORAG PANEL INDUCED DRAG PER UNIT Q. V0RLAX C..,NEAD PANEL DESCRIPTION INFORMATION. VORLAX C...IDES DESIGN (SYNTHESIS) FLAG. VORLAX C...IPAN HORSESHOE STRIP PANEL INDEX. VORLAX C.,.ITER ACTUAL NUMBER OF RELAXATION CYCLES. VORLAX C...LIFT PANEL LIFT PER UNIT Q, VORLAX C.,.MACH FREE-STREAM MACH NUMBER. VORLAX C,,.NPAN NUMBER OF PANELS OEFINEN IN THE DATA INPUT. VORLAA C...NVOR NUMBER OF PANEL cMoRDwISE STRIPS OF VORTICES. VORLAX C.,,RNCV CHORDWISE NUMBER OF hORSESHOE VORTICES FOR A VORLAX Ceee GIVEN PANEL. VORLAX C...SLEI, SLE2 SURFACE SLOPES AT LEADING EDGE OF PANEL SIDE VORLAX COlD EDGES. VORLAX C,..SMAX NI. VORLAX C...SREF CONFIGURATION REFERENCE AREA. VORLAX C,..SURF PANEL SURFACE AREA (PLANFORM AREA). VORLAX C,,,VINF REFERENCE FREE STREAM VFLOCITY. VORLAX C,..XBAR ABSCISSA OF MOMENT REFERENCE CENTER. VORLAX C,..XSUC PANEL LEADING EDGE THRUST PER UNIT Q. VORLAX C,,.YAWQ YAW RATE (OEGS/SEC). VORLAX C..,YNOI BUTTLINE ORIGIN OF FLOw FIELD SURVEY GRID . VORLAX C...ZBAR ORDINATE OF MOMENT REFERENCE CENTER. VORLAX C...ZETA INCIDENCE OF STREANWISE VORTEX ROW (STRIP) VORLAX C.°. CHORDLINE. VORLAX Co..ZLEI_ ZLE_ LEADING EDGE OFFSET OF CAMBERLINE AT THE VORLAX Ce.. PANEL SIDE EDGES. VORLAX C.,.ZNOT wATERLINE ORIGIN OF FLOw FIELD SURVEY GRID. VORLAX C...AINC1, AINC2 CHORDLINE INCIDENCE AT sIDE EDGES OF PANEL. VORLAX C...ALPHA ANGLE OF ATTACK (DEGS), VORLAX C...CDIOT TOTAL INDUCED DRAG COEFFICIENT. VORLAX C..°CHORD CHORD LENGTH MEASUREC ALONG CENTRLINE Or VORLAX C... STREAMWISE ROW OF HORSESHOE VORTICES. VORLAX C...CLIOT TOTAL LIFT COEFFICIENT. VORLAX C..,CMTGT TOTAL PITCHING MOMENT COEFFICIENT. VORLAX C...CNTO_ TOTAL YAWING MOMENT COEFFICIENT. VORLAX C..°CRTOT TOTAL ROLLING MOMENT COEFFICIENT. VORLAX C...CYTOT TOTAL SIDE FORCE cOEFFIcIENT. VORLAX C...DNDXI, DNDX2 SURFACE SLOPES AT THE CONTROL POINTS MEASURED VORLAX Coil ALONG THE PANEL SIDE EDGES. IF DESIGN I_ IN VORLAX COB. VOKED, THEY ARE THE LOAD COEFFICIENTS AT THE VORLAX Cee. LOAD POINTS ALONG THE PANEL SIDE EDGES. VORLAX C.,.GANMA HORSESHOE VORTEX CIRCULATION , ALSO USED VORLAX Ceee AS TEMPORARY DATA STORAGE. VORLAX C...ISOLV FLAG DETERMINING SYSIEN Of SOLUTION FOR THE VORLAA CeI, BOUNDARY CONDITION EQUATIONS. VORLAX C..'LESWP PANEL LEADING EDGE SWEEp (OEGS). VORLAX C°..NMACH NUMBER OF MACH NUMBERS PER CASE. VORLAX C...ONSET ONSET FLOW VELOCITIES GENERATED BY THE VORLAX CaB. ROTATION OF THE CONFIGURATION ABOUT THE VORLAX Ceeo MOMENT REFERENCE CENTER. VORLAX C...PSPAN PANEL SPAN. VORLAX C...RFLAG COEFFICIENT MULTIPLIER IN SYSTEM OF BOUNDARY VORLAX Ceo, CONDITION EOUATIONS. VORLAX C..;RNMAX NUMBER OF VORTICES FOR A GIVEN CHORDWISF ROW. VORLAX C...ROLLO ROLL RATE (DEGS/SEC). VORLAX C...SLOPE SURFACE SLOPE AT THE LATTICE CONTROL POINTS. VORLAX C...SYNTH PANEL SYNIHESIS (DESIGN) FLAG. VORLAX

1o6 £...TAPER _ANEL TAPER RATIO. VORLAX C...TITLE CASE TITLE. VORLAX C...WSPAN CONFIGURATION REFERENCE WING SPAN. VORLAX C...XAPEX, YAPEX, ZAPFK COORDINATES OF PANEL FIPST SIDE EDGE LE_DING VORLAX

CJl, EDGE (PANEL APEX). VORLAX C...CSTART CHORD LENGTH OF PANEL FIRST SIDE EDGF. VORLAX C...OELTAY, DELTAZ 8U[TLINE AND WATERLINE SPACING OF FLOW FIELD VORLAX

Coco SURVEY GRID. VORLAX C...FLOATX, FLOATY VORTEX WAKE FLOAIATION PARAMETERS. VORLAX C...INTRAC PANEL LATERAL NONPLAnARITY PARANETEN. VORLAX C...INVERS i)ESIGN (SYNTHESIS) FLAG. VORLAX C...IQUANT PANEL SYMMETRY FLAG. VORLAX C...ITRMAX MAXIMUM ALLOWABLE NUMBER OF RELAXATION cYCLES. VORLAX C,..LATRAI. CONFIGURATION OR FLICHT CONDITION SYMMETRY VORLAX

Co.. FLAG. VORLAX C...MOMENT PANEL PITCHING MhMENT PER UNIT Q. VORLAX C...NALPHA NUMBER OF ANGLES OF AITACK PER CASE. VORLAX C...NPANAs TO_AL NUMBER OF PANELS THAT HAVE TO HE TAKEN VORLAX

C..o INTO ACCOUNT IN THE ACTUAL COMPUTATION VORLAX

C.., pROCESS. VORLAX C...PHIMED ANGULAR COORDINATE OF CROSS-SECTION RADIU_ VORLAX

Ceoo VECTOR. VORLAX C...PITCHU pITcH RATE (OEGS/SEC). VORLAX

Ce., VORLAX C...... • ...... IPER UNIT Q[ MEANS (PER UNIT FREE-STREA_ VORLAX C...... _.....DYNAMIC PRESSUREf, VORLAX C... VORLAX C VORLAX C VORLA_

107 76/76 OPl=] FIN 6.5+610n I0/04/76 iO•lS.13

CONTROL*VHLX,AERO VORLAX 307 Cee. VORLAX 30H SUBROUTINE AERO IEW, ITOIAL) VORLAX 309 C... VORLAX 31_

C...PURPOSE TO COMP*JTF FORCE AND _OMENI OAIA BY INIEBRAIION f_F VORLAX 311 CIQ. PRESSURF OISIRIBUIION "DATA ANC 8Y IAKING INTO ACcOtlNI VORLAX 3t_ Ceee EDGE SUCTION FORCFS, VORLAX 313 Cue. VO_LAX 31_ C...INPUI CALLING SEQUENCES VORLAX 31_ CJee EW = NO_MALWASB AT LEADING EI)GE INFLUENCE COEFFIcIENI VORLAX .}16 Coil MATRIX (REIRIEVEI) ROw BY _OW FROM UNIT 9}, VORLAX 317 Cea. COMMONS VORLAg 318 CeDe X, Y, l• H2, DL, SX, yY, 7Z, CCP, JTS* LAX, PSI, VORLAX 319 CeDe SPC, VSS, VST, XTE, AIFA, MACF, NPAN, NVOR, SREF, VORLAX 320 C•o'o VINF, X_AR, YAW_, iBAR, ZETA, CHORD, _AWMA, LES_, VORLAX 37I Ce•a PSPAN,, RNMAX, NOLL_, SLOPE, TAPFR, CSIART, IhUANT, VORLAX 3_?

CeDe LATRAL, NPANAS, PITCH,S, VORLAX 3_3 Cele VORLAX 3_ C. • •OUTPIII CALLING SEOUENCE_ VGRLAX 3_5 C•a• NONE. VORLAX 3_& Co•• COMMON< VORLAX 3Z7 Co•• CD_ CL, CM, CN, CX, CY, F.J, FY, QM, YM, CDC, CNC, VORLAX 3?8 C•oo CRM, CYM, CMTC, CSUC, DRA_, LIFT, SURF, XSUCw VORLAX 329 Co•• CDTOI, CLTOI, CMThI, cNIOT, C_IOI, CYTOT, MOMENT• VORLAX 3_0 Col• VORLAX 3_I

C•. .SUBROUIINES VORLAX 3}2 C.• • CALLED NONE. VORLAX 333 C•• VORLAX 334 Co• .DISCUSSION SUbrOUTINE AERO C()MPUIES THE AEROBYNAMIC FORCE A_D VORLAX ]35 C•. MOMENT cOEFFICIENTS BY INTEGRAIING IHE PRESSURE VORLAX 336 C•• DISTRIBIJTION AND COMPLITING THE I.FADING EDGE SUCTION VORLAX 337 Co• FORCES IN ACCORDAr_CE wITH LAN_ PROCEI_URE. IHFRE ARE VORLAX 3_R C•• IHNEE CI ASSES OF COEF_ICIFNTS, A_ FULLqWS _ (i) TOTAL VORLAX 33_ C,•. CONFIGURATION COEFFICIENTS, (2) PANEL COEFFICIENTS, VORLAX 3_0 Ce•• ANO (3) STRIPWISE, OR CHORDWISE, COEFFICIENTS. ALL VQRLAX 34I Clll TOTAL C()EFFICIENTS ARE RFcEREBCEn TO WIND AXES• cLASS VORLAX 3_Z Ce•• (P) AND (_) COEFFICIE_JIS ARE HEFFRENCEO EITHER To BODY VORLAX lw3 Co•• OR To WIND AKES AS RF.)UIRF0 BY THE CORRESPONDING ChEFF. VORLAX 3_ Cool DEFINITION. AERO SUHR(gUTINE IS CALLE0 BY MAIN FOp EVERY VORLAX 3_5 C••• 4NGLE OF ATTACK AIcL) MACH ,qUHBE_ COMHINATIhN. V0RLAX ]46 C VORLAX 3g_ C VORLAX 348

_o8 74/74 0PT=I F'TN _.S*_10J 10/04/76 10.IS.l_

CONTROLOVHLX.BOUNUY g/IV/7B VORLAX 747 CQoe VORLAX 74H SUBROUTINE BOUNDY (IIOIAL) VORLAX 7_9 Cloe VORLAX 750 C...PURPOSE 10 CALCULATE THE ONSET FLOw COMPONENT NORMAL TO VORLAW 751 C.,. THE BOUNDARY SURFACE AT THE VOwTFX LATTICE CON VORLAX 7_ C... TROL POINTS. VORLAX 7_3 Cos. VORLAX 754 C...INPUT CALLING SEQUENCE VORLAX 755 C.., ITOTAL = TOTAL NUMHER OF HORSE_HnE VORTICES. VORLAX 756 C... COMMON VORLAX 757 C,,. ALFA9 PSI. VINFt PITCHQ+ ROLLG, YAWQ, LAX, RNMAX, VORLAX 75_ C... SX_ CX_ X. YY_ ZZt DL, CHORD, XH_R, ZHAR, SLOPE, VORLAX 7_9 C... RFLAG. VORLAX 76_ CI.+ VORLAX 7_1 C...OUTPUT COMMON VORLAX 7h2 C... ALOC, ONSET. VORLAX 763 Coee VORLAX 7_4 C,..SUBROUTINES VORLAX 76_ C...CALLED NONE. VORLAX 766 C,ss VORLAX 7_7 C.,,DISCUSSION THE ONSET FLOW COMPONENT NORMAL TO THE BOUNDARY VORLAX 7hH Ctoe AT THE VORTEX LAITICE CONTROL _OINTS IS CALCULATED VORLAX 7_9 Ctee BY PROJECTING THE FREE-STqEAM VELOCITY VECIOR VORLAX 77U C,lo ALONG THE SURFACE NORMAL AND I_KING LNTO ACCOtJNT VORLAX 77I C+eo A RIGID BODY ROTATION A_OIIT TP_ POINT (_BAR, 0, VORLAX 77_ C+os ZBAR). THE ONSET FLOW NORMAL CUMPMONENT IS ALt)C. VORLAX 773 Ci.e ONSET DENOTES THE RIGID BODY _tJTATI()N INOtJCED VOHLAX 77_ Ct,o VELOCITY COMPONENT ALONG THE X-AXIS. BOTH ARRAY VORLAX 77% C,eo ALOC AND ONSET+ ARE DIMENSIONLESS, I.Eot THEY ARF VORLAX 776 C_e. REFERENCED TO THE FREF-STQEAM VELOCITY. VORLAX 777 Cs.o VORLAX 778 Co.. VORLAX 77g

lo9 74/74 DPT=I FTN _._*_IOA ]0/0_/76 I0•15•13

CONTROL*VRLX.GAUSS '_ ; ) 3/16 VORLAX B56 CI•. VORLAX 857 SUBROUTINE GAUSS (ITOTAL, REXPAR, FW, X_T) VORLAX 8_

Coo VORLAX 859 tee ,PURPOSE VORLAX 860

Ce. T 0 oVEN-RELAXATION{• VORLAX 86I VORLAX B62

Co- •INPUT CALLING SEQUENCES VORLAX 86] C.• ITOTAL = TOIAL NUMBER OF HoPSEHH()E V_}_I)CES_ VORLAX 864 Ce• REXPAR = RELAXAIION PARAMEIER* VORLAX 865 Co. EW = ROw OF NORMALWASH MATRIX. VORLAX 866 C.a COMMON< VORLAX 867 C.. CX, SX, LAX, ALOCt IDES, CHORD, PNMAX, INVERS_ VORLAX 8_8 CoG ITRMAX. VORLAX 869 C,o VORLAX 870 Co. ,OUTPUT CALLING SEQUENCE% VORLAX 87I C•• XRI = AI_XILIARY VECTOR USEo Ik C.S.O.I_. S_LUIION. VORLAX 87_ C.• COMMON< VORLAX 873 Co.. BIG_ DCP, EPS9 HLM, SI_E_ ]TER, GAMMA_ SLOPE. VORLAX 874 Cei• GAMMA IS IRE SOLUTION VECTOR OF _()UDARY CONDITIOn, VORLAX 875 C••. EQUAl,ORS. I. E., HORSESHOE VO_VIFX CIHCULATION VORLAX 876 C•o• SIRENGTMS. VORLAX 677 C.•• NOTE_ IF INVERS = i (I_ESIGN P_OCFSS} 1HEN GAMM$_ IS VORLAX 878

C.•• PART INPUT AND PART OUTPUT. VORLAX 87q C. VORLAX 8RO C• • . SURNOHT INFS VORLAX BRI C. • .CALLED NONE• VORLAX 8R_ C• VORLAX 893 C• •.DISCUSSION 1HIS SUBROUTINE SOLVES FOR THE CIRCULATION STRENC, IH VORLAX 8R4 C• == oF THE HORSESHOE VORTICES THAI SATISFY THE 8.C. OF VORLAX 885 C• i. NO MASS-FLUX ALON_':, THF NORMAL I0 THE SURFACE AT [HF VORLAX _6 C• CONrROL POINTS. THIS SOLUTION LS PERFORMED ITERA VORLAX 897 C• TIVELY NOW BY ROW BY _JSING THE C.$.0._. METHOD. IF VORLAX _ C• A L,IVEN NOW IS PART OF A PANEL. IO HE I)ESIGNED,, I.E., VORLAX 8R9

C• IDES = 1, I_IEN INSTEAn OF >OLVINc, FOR GAMMA(= XP_)), VORLAX BgO

C• .o THE COMPUTAIION OF THE '-,Lr)PE _'ISI(;'TH_!IIO('J AL[)N[_ VORLAX 8qI C• THAI ROw IS F'ERFOh'MEr) BY ,-_',TPI', *.'UL TI_LIC;_ITON, VORLAX 892 C• SLOPE = EW _'GAMMA. VORLAX 8q3 C. °• VORLAX Rq_ C. VORLAR _g_

Ii0 7¢174 OPT=l FIN 4.5+_IOA I0/04176 I0o15,13

CONTROLtVRLX.GEOM 9/16/76 VORLAX I077

Coo. VORLAX 1078 SUBROUTINE GEOM (ITOTAL) VORLA_ I07_ Cme* VORLAX I0_0 C.,.PURPOSE TO COMPUTE THE VORTEX LATTICE bEnMETNY AND SURFACE VORLAX I0_I

Ceo, SLOPES AT THE CONTROL POINTS. AL_O TO COMPUTE THE VORtAX lOR2

Coee LOAD DISTRIBUTION AT THE CORRESPONDING LOAD POINTS VORLAX I0_3

CeIe IF DESIGN PROCESS IS INVOKED (IDFS = |). VORLAX lOa_

Ceo, VORLAX I0_5 C,o,INPUT CALLING SEQUENCE% VORLAX I08_

Coo, NONE. VORLAX I0_7

Cee. COMMON( VORLAX |OR_

Cme. DL_ ITS, LAX, LAY, NPP, PI)L, RC5, VSS, NPAN, NVOu, VORLAX 10_9

Cee. RNCV, SLEI, SLE2, ZLEI, It E2, AI_ICI, AINC2, ONDXI, VORLAX lOqO

C_.e DND_2, GAMMAt LESwP, PSPA_, SY_T_, TAPER, XAPEX, VORLAX lOq]

Cie. YAPEX, ZAPEX, CSTART, INTRAC, IQIIANT, LATRAL, VORLA_ IOq2

Ceo. PHIMED. VORLAX lOq3

Ceo. VORLAX 109_ C...OUTPUT CALLING SEQUENCE( VORLAX lOq5

Cii= [TOTAL = TOTAL NUHBER OF HORSESHnE VORTICES. VORLAX 1096

Ceo, COMMON_ VORLAX lOq7

Ceoe X, Y, Z, CX, DL, NI, SX, YY, 2Z, OCP, JT5. 5LE, VORLAX IO_R

Coo, INL, TNI, VSP, VS5, VST, XIE, ALnC_ ]PAN, SMAXt VORLAX IOq_

C,oo ZEIA9 CHORD, RNMAX, SLOPE, NPANAg. VORLAX i100

Coee NOTE( DC AND VS5 MAY _E EITHE_ INPUT OR OUTPUT VORLAX Ilnl

CQee DEPENDING ON CONFIGURATION COkDITION5. VORLAX I1o_

Coco VORLAX II,)3 C...SUBROUTINE5 VORLAX II0_ C...CALLED NONE. VORLAX 1105

Cel. VORLAX II_6 C...OISCUSSION THE VORTEX LATTICE GEOMETRY IS LA{D OUT PANEL HY VORLAX 1107

C.,e PANEL BASED ON THE GEOMETRIC A_b VORIEX OISIRIBU VORLAX IIOR

Ce.. TION CHARACTERISTICS SPECIFIEC F_R THE GIVEN PANEL VORLAX llOq

C*., IN THE INPUT DATA IINPUT SUBROUTINE). EACH PANEL VO_LAX 1110

C.., 15 SUBDIVIDED INTO A NUMB_ OF X-AXIALWISE 5IRIP_ VORLAX 11_i

C... (NVOR), EACH STRIP CONTAINING A &IVEN NUMHER (RN_AX = VORLAX

Co,o RNCVI OF HORSESHOE VORTICES WFUSF BOUND TRAILING LFG5 VORLAX 1113

Cmee COINCIDE WITH THE X-AXIALWISE EDGES OF THE STRIP IF VORLAX

C,*. THE PARAMEIER NPP = Oo WHEN NPP = I THERE IS NO VORLAX 1115

C..e LONGER A CONIINUOU_ STRIP OF VORTICES SINCE {HEY ARE VORLAX 1116

Coo. NOT LOCATED IN THE SAME PLANE+ IN IHIS CASE {NPP = VORLAX 11|7

Coe. I) IHE STRIP BECOMES AN ARRAY OR ROW OF VORIICE5 VORLAX

C... LOCATED IN TANDEM BUT WHOSE SPANS ARE NOT NECE5 VORLAX Illq

Coo. SARILY EQUAL. EACH STRIP _R VOMTFX HOW IS IOENTI VORLAX 11_0

Col, FIED BY AN INDEX (SX). EACH HORSESHOE VORTEX IN ^ VORLAX IIZI C.e= GIVEN STRIP OR ROw IS II)E_TIFIEO BY A SECOND INDEX VORLA_ II_?

C.o, (CX}_ THE VALUE CX = ] D_NOTIkO THE LEAOTN_ EI)GE VORLAX

Co.. ELEMENT, AND CX = RNMAX = RNCV OFNOTING IHE LAST, VORLAX I_

C... OR TRAILING EDGE, HORSESH)E OF THE ROW. THEREFnR_ VORLAX

C,,. EACH AND EVERY HORSESHOE VORTex _ UNI(_UELY I()FN VOmLAX

C,.. rIFIED _Y EITHER AN OVERALL I_)Ex (WHICH RUNS FR_ VORLAX IP7

Cmll I TO ITOTAL) OR BY THE PAIR OF VALUES (CX, SX). VORLAX IP_

Coe. THE SPATIAL LAY-OUT OF IHE VORTEX LAITICE C,)RRE5 VORLAX I_

Ce,* PONDING TO A GIVEN PANEL r)_PENUS _N THE VALUES Or VONLAX

C..e IWO PARAMETER5_ POL A_D NPP. ]_ PIlL .LE. 3_0.0 I_EN VORLAX

Co.e THE IRA_SVERSE VORTEX SEG_Er_I5 OF THE SAME VALUE OF VORLAX I_?

Ce.e CX FORM A CONTINUOUS STRAIGHT LI_IE_ BUr IF PDL ._;]. VORLAX

ill 7_It4 ()PT=] FTN 4o5+_1U_ 10104176 I0.I_.13

Cioo 360.0 THEN THE TRANSVFRSE VORIEx SEGMENTS OF SAMF CXI VORLAX 113_ C*** THOUGH S/ILL CONTINU011S* FURM A POLYGONAL LINE VORLAX 113_ C.=, WHEN P_uJECTEO VN A PLANE NORMAL TO [HE X-A_]So IF VORLAA !136 C,,, NPP = 0 IHEN ALL THE TRANSVERSE vONTEx SEGMENTS VORLAX 1137 C=i. OF A GIVEN ROW (SAME SX) LIE IN THE SAME PLANE+ VORLAX 113_ C.I, BuT |F _PP = I IHEN THE TRANSVERSE SEGMENTS OF A VORLAX 113q C,,J ROW ARE LAID ON THE ACI(IAI. BUDf _|)RFACE. THE BOUKID VORLAX lifo C.I. 1RAILING LLGS OR SLGMENTS ARE aLwAYS PARALLEL TO VORLAX 11_1 C,I, THE X-AXIS (_JP IO IHE IRAILING EnGE OF THE GIVE_ VORLAX 11_2 Co** STRIP OP ROW). VORLAX 1143 CI,I VORLAX 1144 C*** VORLAX

112 76/74 OPT=| FTN 4,5+qlOA 10/0_/76 10,15,13

CONTROL*VRLX.NAP q/14/76 VORLAX 1971 CQe, VORLAX 197_ SUBROUTINE MAP (EW, EWX, EWY, ITOT_L) VORLAX I_73 Cell VORLAX 1976 C...PURPOSE 10 COMPUTE THE FLOW FIFL() AfObI THE CONFIGURATIOq. VORLAX 19tS Co.. VORLAX I976 C...INPUT CALLING SEQUENCE_ VORLAX 1977 CQoo EW • UPWAS_ INFLUENCE COEFFICIENT MATRIX (REIRIEvED VORLAX 19/8 Coe= ROW BY ROW FROM UNIT 3). VORLAX 197q Cee. EWX = AXIALWASH INFLUENCE COEFFICIENT MATRIX (RETRIEVED VORLAX lgRO Ceee ROW BY ROW FROM UNIT W). VORLAX lqal Cee= EWY = SIDEWASH INFLUENCE COEFFICIENT MATRIX (RETRIEVED VORLAX lq_? C_e. ROW BY ROW FROM UNIT ?). VORLAX 1983 CQi= ITOTAL = TOTAL NUMBER OF HORSESHOE VORTICES. VORLAg 1984 Cel. COMMON_ VORLAX 19q% CeDe IH_ IQ. NT_ XS, NXS, NYS. NZS, PRI. ALFA, MACH, YNOT° VCRLAX 1986 Ces, ZNOI9 ALPHA, GAMMA_ RNMAX. DELIAY, DELTAZ. VORLAX 19_7 Coco VORLAX 19RR C...OUTPUI CALLING SEOUENCE_ VORLAX 19R9 CiJl NONE. VORLAX Igq_ Cell COMMON_ VORLAX Igql Coo. NONE. VORLAX 19q_ Coa, DIRECT PRINT( VORLAX 1OR3 Cei. XS_ YKI. ZKI = FIELD GRID POlhl COORDINATES (HODY AXIS VORLAX Igq_ Ceoo SYSTEM). VORLAX l_q% Coil VX, VF, WF = TOTAL DIMENSIONLESS (REFERENCEU rO vEIO VORLAX 199_ Ceet CETY AT _)PSTOEAM iNFINITY> VELOCITY COMPU- VDHLAX log7 Coo. NENTS ALONG THE X-Y-Z AXES RESPECTIVELY VORLAX IgqR Coel (BODY AXIS SYSTEM). VORLAX 19qq C... EPSLON = UPWASH FLOW ANGLE IN DEGREES. VORLAX _0_}0 SIGMA = SIDEWASH FLOW ANGLE lh DEGREES. VORLAX _n[ C°l. CP • LOCAL PRESSURE CqEFFICIENI. VORLAX ?On_ Ce.. RN = LOCAL MACH NUMBER. VORLAX _h;)3 Ce.. PPIOl = (LOCAL SIATIC PRESSuRE}/(TOIAL PRESSURE}. VORLAX _0_)4 CeD. PIE = (LOCAL STATIC PRESSbARE)/(FPEE STREAM STATIc PRtS VORLAX _OnS Co.. SIJRE). VORLAX _006 CDel VORLAX _007 C...SURROUTINES VORLAX 7_)_ C...CALLEO NONE. VORLAX 2()_9 C°.. VORLAX ?Oiu C...D_$CUS$1ON FLOW FIELD OUANTIT]ES ARE COMPUIFD AT THE NODAL DOINIS VORLAX _0}} Ceo. OF A ]-()'GRID DEFINED AROUND IRE CONFIGURATION HY A SET VORLAX _)l_ Ceo, OF ORTHOGONAL PLANES ( X = CONbT._ Y = CONSI.. AqD V_RLAX _01_ C°.I Z = CONSI. PLANES). THE VFLOCIIIFS ARE CALCULATEn qY VORLAX 2_14 CJoo THE USE Of INFLUENCE COEFFICIEJ_T MATRICES RASED _}r_ F_E VORLAX _(}I5 Cel. vORTEX LATIICE REPRESFNIATION uF THE CONFIGURATInN. VORLA_ £Qlh Cee. THESE MATRICES ARE COMPUTED IN SUBROUTINE SURVEY AND VORLAX _017 Co,o SIOREO IN UNITS 3° 4. AND 7 (ONE MATRIX AND ONE =_NIT VORLAX _OIH Cel. PER VELOCIIY COMPONEI}. THE PfiES_URE RATIOS AN() _EIATED VORLAX ROIQ Ce.e FLOW QUANTITIES _RE ChMPUTED THROUGH THE USE OF ISFN VORLAX _0_0 Ceo. TROPIC FLOW RELATIONSHIPS. VORLAg _0?| C VORLAX _0_ C VORLAX 20?]

i13 76176 OPT=I FIN 6.5,410a 10/0_/76 I0.15.13

CONT_OL_VRLX.MATRX 9t16/76 VORLAX _167 Coco VORLAX _16_ SUBROUTINE MAT_X (EW_ EU_ ITOTAL) VORLAX R1_9 C.me. VORLAX PLSO C.o.PURPOSE TO GENERATE THREE AERODYNAMIC iNFLUENCE COEFF. MaIPl VORLAX _151 CeDe CE_ (I) THE NORMALWASH AT THE CnNIRUL POINTS VORLAX _IS_ CeDe (EW / uNIT I)+ (2) THE AXIALWASH AT THE CONTROL POINTS VORLAX _lR3 Ceao (EU / UNIT 2)_ AND (3) THF NORMAI WASH AT THE LEA_}. EDGE VDRLAX ZIS_ C,o, (EW / UNIT 9). THESE MATRICES _EPRESENT THE INDUCED VORLAX 2155 C.e° VELOCITY FIELD DUE TO IHE HORStSHOE VORTICES OF THE VORLAX _ISO Cee. LATTICE. VORLAX 21SI C.o° VORLAX _15_ C...INPUT CALLING SFQUENCE_ VORLAX _IS_ CeIe ITOTAL = TOTAL NUMH_R OF HORSESHOE VORTICES. VORLAX 2160 Ceea COMMON_ VORLAX _l_! Cee. Xt _2_ CX_ DL. NT. SX. YT. ZZ. HAG. LAX. NPP_ PSI. INT. VORLAX _16@ CeDe VSP* VST0 XTE. ALFA. {PAN. XHA_. lhAR_ CHORD. RFI AG_ VORLAX _163 Ceo. RNMAX_ _LOPE. FLOATX. FLO_TY. [NvERSt LATRAL. VORLAX _16_ Cee. VORLAX _I65 C...OUTPUT CALLING SEQUENCE_ VORLAX _I6_ Cem, EW = CONTROL POINT NORMALWAS_ MATRIX (STORED ROW BY VORLAX _16T CeDe HOW IN UNIT }). VONLAX _16B Cee. EW = LEADING EDGE NORMALWASH MATRIX (STORED ROW _Y VCRLAX _169 C... ROW IN UNIT _). VORLAX _]70 C... EU "= AXIAl.WASH MATRIX (STqHED _OW HY ROW IN UNIT 2). VORLAX _171 C... COMMON< VORI.AX 2172 C..° NONE. VORLAX 2173 C,oo VORLAX 2174 C...SUBROUIINES VORLAX 211S C...CALLED WASH. UxVEL, V_RLAX 2176 Ceee VORLAX 2177 C...DISCUSSION THE ELEMENTS OF THE I_FLU_NCE ¢OFFFICIENT MATRICFS VORLAX 2178 C.o. ARE GENERATED HY COMPUTING THE CnRRESPONDING vFLqCITY VGRLAX 217q C..e INDUCED AT THE (KI.JI) CONTROL PnINT HY THE (K.J) VORLAX 21_0 C..e HORSESHOE VORTEX OF H_IT STREhbTH. IF K = KI AND VORLAX 2IHI Cee. j = J( (SELF-INFLUENCE( THEN THE PRINCIPAL PAOT nF VORLAX _IB_ Ceeo THE DOWNWASH INTEGRAL IS ADDE_ TO THE CONPUTATIO_I hF VORLAX PIN3 CeDe THE CORRESPONDING EW COEFFICIENT. ALSO IF THE CU_IROL VORLAX _I_6 C.ot POINT IS WITHIN A GIVEN NEAR FJF_D RADIUS OF THE VORLAX 2]HS Coco INOUCING HORSESHOF VO, IEX. THE AxTALWASH CUNIRIO+_TION VORLAX 2IQ_ Coo° IS COMP(_TED RY INTERDIGITAIE{) VUoTEX SPLIFTING. VORLAX 21_1 Coco VORLAX 21_ Ce,. VORLAX _la_ 74174 OPT=| FIN 4,S÷4lOA |0/04/7_ !0.15,13

CONTROLOVRLX.INPUT _/16/76 VORLAX 1_1_ Cool VORLAX 1417

SUBROUTINE INPUT VORLAX 141B Coo. VORLAX I419 C..,PURPOSE TO READ IN INPUT DATA AND PREPARE SUCH DATA FhR VORLAX |_0 Ceeo USE IN THE GENERATION OF vORTEx LATTICE GEOMETRY VORLAX 1471 CeeQ TO BE OONE IN SUBROUTINE GEON. VORLAX I4_Z

C.o. VORLAX I_ C...INPUT CALLING SEOUENCE_ VORLAX I_¢

Coo. NONE. VORLAX I_25

Glee COMMON( VORLAX 1_26 Cool LAX. VORLAX |_T Col. VORLAX 14?_

C...OMTPUT CALLING SEQUENCE_ VORLAX 14_q Coo. NONE. VORLAX 1430 Coo, COMMON_ VORLA_ 14_I Coo. DL, XS. ITS, NPP, NXS. NYS, NZS. PDL, PSI. RCS, VORLAX 1432 Coee SPC) VSS, CBARp HEAD. NACH. NPAN. NVOR, RNCVt SL_I, VOHLAX 1433 Coo. SLE2. SREF, VINF, XHAR. YAWQ, YNATt ZBAR, ZLEI. VORLAX 143_ Ceml ZLE2, ZNOT, AINCl, AINC2. ALPha. DNOXI. DNDX2. VORLAX I_}5 Cole GAMMA. I ESWP, NMACH, PSPAN. ROLL,. SYNIH. TAPER. VORLAX 143_ Coo. WSPAN, XAPEX, YAPEX, ZAPEX, CSIAPT. UELTAY, DELT_Z. VORLAX 14_7 Coot INTRAC. INVERS. IQUANT, LATRAL, NALPHA, PHIMEI_. VORLAX I_38 Caet PITCHQ. VORLAX 1439 Cool VORLAX 14_n

C...SURROUT INES VORLAX I_l C...CALLED NONE. VORLAX 14_ CJ.. VORLAX I4.3 C...OISCUSSION A MASTER FRAME OF REFERENCE IS A_SUMEO IN DEFINING VORLAX 14_ C... A CONFIGURATION. THIS FRAME OF REFERENCE IS AN VORLAX I_5 Celt ORIHOGONAL CARTESIAN CDORqINATE _YSTEM. T_E X-Z VORLA_ 14WO C... BEING THE CENTERLINE PLANE WIT, THE X-AXIS POINIIN& VORLA_ 14w7 C... DOWNSTRrAM. AN() THE Z-AXI_ DI_CTEU UP_ARD_ IHE VORLAX IWW_ C... Y-AXIS _OINTS To SIARHOAmq. TPE oRIGIN OF T_E SYs VORLAX 1440 Co.. TEM CAN BE ANY CONVEVIENT DO,hi IN THE X-_ PLANE. Ve_LAX 14_0

Co., THE CONFIGURATION CAN BE ._ADE UP hF SYMMEIRICaL VORLAX ]451 C..I (ABOUT THE X-Z PLANE) AND/OR A_YMMETRICAL COMPO- VORLAX 145P C..Q NENTS. aND IN DEFINING IHE SYMMETRICAL CO_PONFNI_ V_RLAX )4%3 C..o ONLY THE STARHOARD ELEMENTS ARt _PECIFIED. THE VORLAX 145W Co.. CONEIbUPATIDN TO RE INPUT IS DIVIDED INTO a SET VORI_AX 1_55 Cte. OF MAJOR PANELS+ UP TO 20 OF IHE_E PANELS CAN BE VORLAX I_ C... INPUT. _YMMETRICAL COMPONENTS (LFFT + RIGHT) HEIHG VORLAX 14%1 Co.. COUNTED ONLY ONCE. FOR INSTANCE. a WING OF ZERO VOPLAX I_SH Co.. THICKNESS AND WITH STRAIGHT LEADING ANI) TRAILING VORLAX I_ Coo. EDGES, AND WITH LINEAR LOFTINE METWEEN RO_)I AND VORLAX I4_l) C..Q TIP. CONSTITUTES a MAJOR PANEL. COMPLEX PLANFORMq. VORL&X l_l C.eo AND NON-LINEAR CHANGES IN TWIST AND AIRFOIL_ S_CIIOHS VORLAX 1W_P C..o ARE DESCRIBED BY DEFINING MORE THAN ONF PANEL FUh A VORLAX 14_ Co.. GIVEN WING. SUBROUTINE INPUT PWEPARE_ THE DATA SuE VORLAX 14_ Co.. CIFIED FOR EACH MAJOR PANEL 50 THAT THEY CAN LATER VO_l.Ax Iw_ C..t @E USED IN SURROUIINF GEO_ TO bENER_TE [HF PWOPE_ VO_LAX 1_ C.eo VOWTEX I ATTICE FOP EACH PANEL. VCRLAX l_l Co.. AN AIRFOIL WITH THICKNESS CAN HE REPRESENTEO HY A VO_LAX l_ C=.. DOUBLE vORTEX SHEET, I. E.. BY a)FFINING g_{} MAJO_ PA VORLAX I_q C... NELS ARPANbED IN A #BIPLANE_, OR _SANO#ICH_. FASwlhN_ VORLAX IW_(_ C... ONE PANEL REPRESENTING THE UPPIR SURFACE .)F THE _I _ VONLAX 1#71 FOIL. aND THE OTHER PANEL REP_EI_NTING THE LOwFR VOWLAX l_l_

i15 76/74 OPT=! FTN _. S+_, 1C,_ L ()/0,W/76 10.15o13

CoJ= SURFACE OF THE SA_4E AIRFOIL. VORLAX I_73 Ctog FUSIFORM BODIES A_E M;IDEI_I ED HY F_EFIN[NG AN A_IXI_ IARY VORLAX ]_7_ Celo _OL)YIOE'_TICAL IN CROSS-_EcrIUF, AL SHAPE ANf) LL).'qGIFUI_I VORLAX I_7% Cooo NAL AREA L)LSTRIBI_TION TO THE ACT_AL EIOt)_' HUT wII_()(ll VORLAX I_7_

Coel CAM_ER. IHE AUXILIARY BODY CRO'-,S-SECTIUN IS APP_(I_I VORLAX Iw77 Cet. MATED BY A POLYbO_4 _/HI)SE SIDES [}FTERMINE THE IRA_,_S VORLAX lwTR Coo= VERSE LFGS OF THE HURSESHt)E VOw(rICES. THE VERTICFS VORLAX ]_79 Cleo OF IHE POLYGON ANT) THF AHKILiA_Y BUOY AXI_ t)EFINF VORLAX I_O Coet A SET OF RADIAL PI ANES IN WHICPl THF HOUND IWAILI_G VORLAX I_HI Coot LEGS OF THE HORSESHOE VORTICES LIE PARALLFL r_b IHE VORLAX Iw_2 Coco AXIS,_ AS [HE CROSS-SECTIO,,_ CH_OFS SHAPE ALUN(, [_E VORLAX I_H3

Cleo AXIS.p THE POLYGON CHA_,_GES ACCO_I)TN(_LY F_UT WITH IHE V_RLAX 1_ Ciol CONSTRAINT THAT THE P()LYGC_NAL VF-_TICLS MUST ALWAYS VORLAX I_8_ Cooo LIE IN THE SAME SET OF _AI_IAL F'LANES. THE EII)DY C?,MRER VORLAX IWH_ Cooo IS SPECTFII-O INUEPEr_F)FNTL. Y. VORLAX I_H7 CJea VORIAX I_ Cooe VORLAX IwH_ Ciol VORLAX I_O

i16 7_,_/74 OPT=I FTN ;,_,*410a 10/0_/7_ 10,15,13

_/|5/76 VORIAX 2524 CONTROLeVRLX,PRE5$ VOPLAX 2525 Ceoo VORLA_ 2526 SUBROUTINE PRESS (ITOTAL, EU) VORLAX 2527 C VORLAX 2528 C VORLAX 2529 C..oPURPOSE TO COMPUTE PRESSURE LOAD COEFFICIENTS (CPLOWER VORLAX 2530 Ctel CPUPPER), OR SURFACE PRESSURE COFFFICIENTS* FROM THE VALUES nF THE INDUCED VELOCITIES AND CIRCULATION VORLAX 2531 CJao VORLAX 2532 Ce+. STRENGTHS. VORLAX 2533 Cle+ VORLAX 2534 C...INPUT CALLING SEQUENCE_ ITOIAL = TOTAL NUMBER OF HORSESHOE VORTICES. VORLAX 2535 Ce.. VORLAX 2536 C..e _U = AXIALWASH INFLUENCE COEFFICIENT MATRIX (RETQI_VED VORLAX 2537 Ceoe ROw BY ROW FROM uNIT 2), VORLAX 2538 Ce.* COMMON< VORLAX 2539 C*ee B2* CX, DL+ SXt JTS. PSI, TNL. [NIt ALFA, YAWQ, cH_RD, VORLAX 2540 Ce*° GAMMA, ONSET, RNNAX, SLOPE. WSPAN. VORLAX 254! Coee VORLAX 2542 C°°.OUTPUT CALLING SEQUENCE_ VORLAX 25_3 C-,, NONE. VORLAX 2544 Co°. COMMON5 VORLAX 2545 C... OCP. VORLAX 2546 Cee+ VORLAX 2547 C,..SUBROUTINES VORLAX 25W8 C,,,CALLED NONE. VORLAX 2549 Ce°= VORLAX 2550 C,.,DISCUSSION THIS SUHROUIINE COMPUTES THE PHE_SURE COEFFICIENT ARRAY VORLAX 2551 Ceoe DCP, EACH ELEMENT OF DCP CORRE_PhNDS TO A GIVEN VORLA_ 2552 Coee HORSESHOE VORTEX OF IHE LAITICE+ AND IT IS ASSUNFD TO VORLAX 2553 Cee= ACT AT THE HORSESHOE CENTROID, I.E=+ IHE MI= VORLAX 2554 Ceo, THE TRANSVERSE+ OR SKEWED, LEG* AN ELEMENT OF THF DCP ARRAY I_ EITHER A LOAD COEFFICIENT (IF THE SURFACE VORLAX 2555 Ce°o IS ASSUMED WETTED ON BOTH FACE_, I.E., JTS = ITS = I), VORLAX 2556 C,e= VORLAX 2557 Ceel OR A SURFACE PRESSURE COEFFICIENT (IF THE SURFACE IS ASSUMMED WETTED ON ONE SInE ONLY, I.E,+ JTS = If_ = 0). VORLAX 255R Co..

Cmee IN THE cOMPUTATION OF LOAD COEFFICIENTS, THE cIRcULA VORLAX 2559 VORLAX 2560 Ceo= IION STRENGTHS_ THE FREE sTREAM vELOCITY COMPONENTS, VORLAX 2561 Ceee AND THE ONSET FLOW DUE TO ANY ANGULAR ROTAIION OF THE VORLAX 2562 Ceoe CONFIGURATION ARE TAKEN INTO JCCNUNT* THE EFFECT OF THE INDUCED AXIALWASH IS IGNOREO (SECONO ORDER ERROR)° VORLAX 2563 Ceo° VORLAX 256_ Ce*e IN THE COMPUTATION OF SURFACE PRESSURE COEFFICIENTS, VORLAX 2565 C.9. THE AXIALWASH IS ALSO TAKEN INTO ACCOUNT, IF NPP = I+ VORLAX 2566 Coee THE WAXIALWASH# IS NO LONGER A TRUE AXIALWASH (X-AXIS VORLAX 2567 Cleo VELOCITY COMPONENT}, RATHER IT REPRESENTS THE VELOCITY VORLAX 256R Ceeo COMPONENT TANGENTIAL TO THE SbRFACE BUT WITH THE SIDE

Ce*o WASH LEFT OUT. SURFACE PRESSURE cOEFFICIENTS ARE LIMI VNRLAX 256g IED BY THE ISENIROPIC VAL_IES CORRESPONDING TO SIAGNA VORLAX 2570 Cloo VORLAX 2571 Cee* TION AND 70 PERCENT OF VACUUM FOQ THE GIVEN FREE-STREAM VORLAX 2572 Ceo° MACH NUMBER. VORLAX _573 Coo° VORLAX 2574 Ceo*

ll? 7417_ OPT=| FTN 4,5._IOA I0/04176 I0.IS.13

CONTROL°VRLX.PRINT 9/17/76 VORLAX 2768 Ceee VORLAX 2769 SUBROUTINE PRINT (ITOTAL. EW. EWX. EWY] VORLAX 2770 Coee VORLAX 2771 C...OURPOSE TO PRINT PROGRAM DATA. VORLAX 2772 C VORLAX 2773 C....INPUT CALLING SEQUENCE_ VORLAX _774 Ce•l ITOTAL = TOTAL NUMBER OF'HORSE_HNE VORTICES. VORLAX 2775 Co•• EW_ EWX. EWY = STORAGE ARRAYS bET ASIDE FOR CALLING VORLAX 2776 Ces. SUBROUTINE MAP. VORLAX 2777 CeDe COMRON5 VORLAX 277B teed CD, CL. CM. CM, CX, CY. DL. IM, ICJ. YY, ZC, ZZ, VORLAX 2779 Coee BIG. CDCt CNC, CRM. CYM, nCP, EP_. HAG. JTS, I ak. VORLAX 27R0 Csu. NXS, PDL• PSI. RLH, SPC, CRAB* CMTC, CSUC, HEAD, VORLAX 2781 Ce.• IPANt ITER. LIFT. MACH, NPAN, NV_R, RNCV. SREF, VORLAX 27RZ Ceo. SURF. VINF, XBAR. XSUc, YAWQ, ZRAR, AINCI, AINC?. VORLAX 2783 Cee. ALPHA, CDTOI. CHORD, CLIOT. CNTOT. CHEf)I, CYTOT. VORLAX 278_ C••. DNOXI, DNOX2, GAMMA, IS(}Lv, LE_wP. PSPAN, RNNAX. VORLAX 2785 Co•o ROLLQ, sLOPE. SYNTH. TAPER. TIILE_ WSPAN. XAPEX. VORLAX 2786 Ce•. YAPEX, ZAPEX. CSTART, FLOAIX_ FLnATY, INVERS, VORLAX 2787 Ceso IQUANT. IIRMAX. LATRAL. MNMENI. PITCHQ. VORLAX 2788 Cemo VORLAX 2789 C.••OUTPUT CALLING SEDUENCE_ VORLAX 2790 Cmoo NONE. VORLAX 279| CeDe COMMON< VORLAX 279_ _oee NONE. VORLAA _793 Ce,e VORLAX _794 Co• ,SUBROUTINES VORLAX Z795 •CALLED ZNORM, MAP• VORLAX _796 C_. VORLAX 2797 C•. .OISCUSSION THE DATA PRINTED OUT HY THIS SUBROUTINE ARE ARRA_JGFD VORLAX 2798 Coo IN THREE GROUPS. AS FOLLOwS_ VORLAX 2799 C.• (I} INPut OATA_ THESE AWE DATA WHICH ARE TAKEN FPOM VORLAX 2800 C•Q THE INPUT DECK ANn HAVE BEEN CONVERTED IN A FORMAT VORLAX _801 Ce• SUITAHLE FOR USE @Y THE PR(JGPAM. VORLAX 7807 C.•. (2) PANEL AND TOTAL CONFIGURAIION FORCE AND MOMENT VORLaX C•.. COEFFICIENT DATA. VORLAX ?BO_ C••. (3) DATA WHICH ARE RELATED EIIHER TO CHORDWISE STRIPS VORLaX _805 C•.e OR TO INDIVIDUAL HORSESHOE VORTICES. VORLAX 2B06 CoDe THESE THREE DATA GROUPS ARE P_LNTED SEQUENTIALLY• VORLAX _BO7 CQ•• IF A FLOW FIELD SURVEY HAs BEEN pEQUESIED. THE CqRRES VORLAX 2808 Ce•. PONDING DATA ARE PRINTED OUT THROUGH SUBROUTINE _AP. VORLAX 2809 Ceo• VORLAX _810

L'L8 74/14 Ol-'F=l _ TN 4.5_410a LO/O4/Tb 10°15,13

CONTROL*VRLX.SURVEY 9/I 7/76 VORLAX 3055 C,.. VORLAX 3056 SOkROL_[INE SURVEY ([ _ t_;,. LL-IY_ IlOIAL) VORLAX 3OR/ VORLAX 3058 C... PIJRPOSE TO GENERATE IHREE AERODYNAMIC INFLUENCE COEFFICIFNT VORLAX 30£9 Ci.o MATRICES <_ (I) TrIE UPWASH AT T_E FLOW FIELD SURVEY VORLAX 3060

Co,o POINTS ([W I (]t_IIT 3}v, (2} [HE a_TALWASH AT THE Fi OW VORLAX 30_I Ci.. FIELD S_IRVEY POINTS (EWX / UNII 4)'_ AND ('_} THE VORLAX 30fi2 Co.. SIDEWASH AT THE FLOW FlEin SURvEy POINIS (EWY / _INI[ 1). VORL. AX 3063 C.o. THESE MATRICFS REPRESENT THE Iq'_I)q_CED VELOCITY FIFLr] VORLAX 3064 C.°° DIIE TU THE HORSESHOE VORTICES ()F THE LATTICE. THIS FLOW VORLAX 306q C .... FIELD Tq _IEASI)RED Ar THE NODAL PnINTS OF a SPECIFIFD VORLAX _066 C.4. ]-D GRID. VORLA× 3067 C... VORLAX 306_ C. • • INPUT CALLING SE L)UENCt <_ VORLAX 306g C.,J ITOTAL : TOTAl NU_IBEI_ OF ._oRSE_HhE VOF(T[CES. VORLAX 3070 Ce.. COMMON< VORLAX 3011 Cooo X, B2, CX. I>i.. NT, S×, XS. YY, Z7, HAG, LAX, NXS, VORLAX 301P Ci.o NYSt NZ_, PSI, I_I. VSP. VS[_ XTF* ALFA, YN()Tq X_AR* VORLAX 3073 Ct.. ZBAR_ ZNOI. CHE)RI)° HN_4AX, _)EI. TAY, DELIAZ_ FLOATX. VORLAX 3014 C... FLUATY, t AIRAL. VORLAX 3075 C... IORLAX 3076 C. • .OiJ IPZ_[ CALLING %F(_UFNCL <_ IORLAX 3077 C... EW = UPwASH _ATRI.( (STORED ROW BY _OW IN HNIT 3) VORLAX 3078 EWX -- AXIALWASH MAId, IX (ST(_REC ROW flY ROW IN HN[T 4) VORLAX 3079 C... EWY = SIDgWASH NA[_IX IST;_RE-D _)W HY ROW IN UNIT 7) VORLAX 3080 COMMON< V_RLAX 30RI N()NE, VORLAX 30R2 Co.. VORLAX 30_3 C • •. _I H_ROUT I NES VORLAX 30R4 C.._CALLED WASH_ UKVEL. VORLAX 30R5 VORLAX 30R6

C. . ,I) ISCUSS ION {HE EI.E_ENTS nF THE [_IFIHKNCE COEFFICIENT MATRICFS ARE VORLAX 30fl7 Co.. GENERATED BY COMPIJTING IHF VELOCITY INDUCEO AT A FlOW VORLAX 30_fl C.,o FIELD S_RVEY PDINF HY THE (K_ .J) HORSESHOE VORTEx OF VORLAX 308Q UNIt STPENGFH. IF IHE FlOW FIELr) SURVEY POINT I= VORLAX 3090 wITHIN A GIVEN NEAR FIELD RADIUS OF THE INDHCING VORLAX 3091

C,o_ HORSESHOE VORTEX 1HEN THE AXIALWASH CONTRIBUTION IS VORLAX 3097 C.., COMPUTEr) HY INTEROIbI_AFFn VORfEx SPI_IrTING. THE F_OW VORLAX 3093 FIELD St_RVEY POINTS ARE [.IL NOI_AI POINTS ¢)F A 3-n GRIO VORLAX 30Q4 C._e DEFINED HY A SET OF THREE ORT_OGONAL PLANFS. THESE VORLAX 3095 Clll PLANES ARE SPECIFIEO HY T_E I_F'UT VALUES OF XS, yNoT_ VORLAX 3096 Ce_, DELTAY, ZNOT_ AND DEt IAZ. VORLAX 30q_ Co., VORLAX 30q_ C¢ol VORLAX 3099 Co., VORLAX 3100

m_9 74174 OPT=! FTN 4.5+_|0_ 10/04/76 LO.IS.I3

CONTROLeVRLX.UXVEL 9/[4176 VORLAX 1355 Cool' VORLAX 3%56 SURROUTINE UXVEL (Xt Y, Z_ St T_ R_o TOLZ_ U) VORLAX 1357 Co.. VORLAX 3}5_ C...PHRPOSE I0 COMPL}TE THE AXIALWASH INDUCE() BY A SKEWEO VORLAX 3359 C..i RECTILINEAR VORTEX SEGMENT oF dNIT CIRCULATION. VORLAX 1_60 C,o. VORLAX 33_I C...INPUT CALLING SEQUENCE_ VORLAX 336_ C... X, Y, Z = OWTHOGO_AL CARTESIAh CnORDINATES OF VORLAX _3_3 ClJ. RECEIVING POINT MEASURED IN A REFERENCF FRAME VORLAX %36_ Co.. CENTERED KI THE MIDPOINT OF VORTEX SEGMENT+ VORLAX 3365 C... IHE X-AXIS IS P_RALLEL TO THE X-AXIS OF THE VORLAX 3366 Co.. MASTER (CONFIGURATION) COORDINATE SYSIFM. THE VORLAX 33_7 Co.. Y-AXIS IS NORMAl TO THF X-AXIS HUT LIES IN VORLAX 33_ CI.. IHE PLANE DETER4INEC BY THE X-AXIS AND THE VORLAX 33_9 C.io VORTEX SEGMENFt AND THE Z-AXIS IS NNRMAL TO VORLAX }37O C.la SUCH PLANE. VORLAX 337] C.°. S = SEMISPAN OF VORTEX SEGMENT. VDRLAX 3_72 C... I = TAN(_ENT OF SWEEP ANGLE OF vORTEX SEGMENT. VORLAX 3373 C.o. B2 = COMPRESSIBILIIY FACTqR (= MaCH _12 - |.0). VORLAX 3374 C.e. TOLZ = rlUMLRICAL TOLERANCE CON3TANT. VORLAX 3375 Ci.. COMMON_ VORLAX 317_ CJo. NONE. VORLAX 33/7 CoB. VORLAX 137_ C...OUTPUT CALLING SEQUENCE_ VOHLAX }379 C... U = X-AxiS VELOCITY CnMPONENT (AXIALWASH) INDUCEn RY VORLAX _3_0 C,e. SKEWED VORTEX SEHMENT OF bNIT INTENSITY. VORLAX 3t81 C.o. COMMON_ VORLAX 3382 C.e. NONE. VORLAX 3383 C... VORLAX 3384 C...SU_ROUTINES VORLAX 338S C. ..CALLED NONE. VORLAX 33R6 C. VORLAX t3RT C. ..DISCUSSION THIS SUBROUTINE IS CALLED WHEN THE AXIALWASH IS VORLAX 3388 C. COMPUTE_} IN ACCORDANCE WITH VORTFx SPLITTING SCHrNF. VORLAX 3389 C. .l ONLY THE AXIALWASH IS COMPUIEC. AND ONLY THE TRAx.S VORLAX 3390 C. i. VERSE SEGMENT OF IHE HORSESHOE IS TAKEN INTO ACCmJNT. VORLAX %3qI C. THE SAME COMMENTS PRESENIEU IN SuHROUTINE WASH VORLAX 3392 C. RE_ARDING THE NUMERICAL SINGULARITY IN THE VICINITY VORLAX 3393 C. OF THE CHARACTERISTIC SURFACES (MACH CONES) ARE VORLAX 3394 C. .l ALSO APPLICABLE HERE. SUHROUTINE UXVEL IS A OIRErI VORLAX 3395 C. .o COPY OF PARTS OF SUBROUTINE WASH. VORLAX 139_ C. i. VORLAX 3397 C. .o VORLAX 339_

120 ?4/76 OPT=I FTN 6.5+610a 10/06/76 10,15.13

CONTROLOVRLX.VECTOR 9/16/76 VORLAX 34_ VORLAX 3485 Co.+ SUHROLFTINE VECTOR (ITOTAL, N_I, AV, EWt VoRI_ VORK, VO_L) VORLAX 3486 C VORLAX 3687 VORLAX 3488 C C. • .PURPOSE TO SOLVE THE LINEAR SYSIEW OF HOHNDARY CONDITION VORLAX 34_g VORLAX 3690 C.+. EQUATIONS HY PURCELLIS VECTOR URTHOGONALIZATION C... METHOD. VORLAX 36gl C..e VORLAX 3492 C..,I_IPUT CALLING SEOUENCE_ VORLAX ]4q3 C.+. ITOTAL = TOTAL NUMBER OF HORSESHOE VORTICES. VORLAX 3496 VORLAX 14q5 Coe. NXI = ITOTAL + I VORLAX 3496 Coco EW = ROW OF NORMALWASH INFLUEhCE COEFFICIENT MAIPlX. Ceee COMMON_ VORLAX 3_g7 Cee. ALOC, VORLAX 3_98 VORLAX ]4qq Ceee C...OuTPUT CALLING SEQUENCE_ VORLAX 3500 VORLAX 3501 Ce_. VOR| : 4UglLIARY COMPUTATIONAL ROW VECTOR. VORLAX 3502 Cool VORK : FXTENDED SOLUTION ROW VECTOR, Cell VORL : AUXILIARY CUMPLiTATTONAL RnW VECTOR, VORLAX 35N3 VORLAX 35N4 Coco COMMQN_ VORLAX 3505 CIel bAMMA. C... VORLAX 350_ C,.,SUBROUTINES VORLAX 35(17 C...CALLEO NONE. VORLAX 3508 C,,. VORLAX 3509 C,..OT$CUSSION THE 80UHOA_Y CONDITION EQUATIOP_S ARE SOLVEO DIRECTLY VORLAX 3510 Cee. BY A VECTOR OR[HO(iONALIZATION PROCEDURE (PURCELL#S VORLAX 3511 VORLAX 351_ Ceel VECTOR uETHOD), S_TS OF LINEARLY INOEPENDENT VECTORS VORLAX 3513 C.e. ARE CONSTRUCTED WHICH ARE SUCCESIVELY ORTHOGQNAL TO C.e. EACH ROW. WHEN ALL ROWS H4VE BEEN CQNSIDERED THERE IS VORLAX 3516 VORLAX 3515 Cee. ONE VECTOR WHICH IS NORMAL (OfiIHoAONAL) TO ALL ROWS VORLAX 3516 C... AND CONTAINS THE SOLUTION VECTOR, NO MATRIX INVERSION C..- IS INVNI VED AND ONLY ONE ROW OF THE COEFFICIENT _ATRIX VORLAX 3517 Coco IS REQUIRED AT A TIME, AN_ ONCE oPERATEO ON CAN RE VORLAX 3518 VORLAX 3519 C..e OVERWRITTEN. IN ADDITION, TWO AIJxILIARY ROm VECToRs Ceee ARE NEEDED FOR TEMPORARY sTORAbE OF INTERMEDIATE VFCTOR VORLAX 35_0 Cee. VECTOR DATA. VORLAX 35_I C... VORLAX 35_2 VORLAX 35_3 Cee.

121 CONTROLeVRLX,WASH _/|_/76 VORLAX 36|T VORLAX 3612 SUBROUTINE wASH (Xt Y, Z, S_ T, _, Ue vt Wt AA, A4, TE, el, MM) VORLAX 3_13 Co,. VORLAX 3_14 C,,,PURPOSE TO COMPOTE IHE IHREE VELOCITY COMPONENTS INDUCEO AT A VORLAX 3615 CiJ, GIVEN PnINI BY A GE_EL_ALIIED F(}RSESHOE VORTEX OF UNIT VORLAX 361& Cole STRENGTH. VORLAX 3617 C VORLAX 3_lR

C,.,INPUT CALLING SEUUENCES VORLAX 361g Cell X, Y_ Z = ORTHOGONAL CA_fEsIAk COORUINATES OF VORt.AX 3620 C,.t RECEIVING (FIELO OR CO_TROL) POINT MEASURED VORLAX 362l Ceee IN A REFERENCE FRAME CFNIEREO AT THF MIDPOINT VORLAX 362_ Coo, OF THE IRANSVEI_St V(_TFX SLGJaENT (NORSFSHOE VORLAX 3_3 Col, VORTEX CENT_OID)* TFt X-AXIS IS PARALLFL TO VORLAX 3_?_ Cee, THE X-AXIS ,IF THE HA_TFR (CUNFIGURAT[O =) VORLAX _O25 COOPl)INaTE sYSTF_, TrIE Y-AXIS IS NOf_MAl T(} V0_LAX }62_ Cal. THE X-AX]_ _UI LIES IN THE PLANE DETERMINED VORLAX _O_7

C,t. HY THE X-ARTS IISELF ANn THE TRaNSvERS_ IEG VORLAX _R_R Call OF THE HORSF_HOE, ANO THE L-aXIS IS NO¢_M_L VORLAX 36_g Cal. TO SUCH PLANE, VORLAX 3630

CaJ, S = HORSESHOE _URIEX SEMIS_N. VO_LAX 3b_l Coo. T = TANGENT I)F SWEEP _NI_I_F oF I_^NSVERSE I_EG, POSITIVE VORLAX 3_ 17 Ce,, FOR SWEEPBACK, VORLAX )6_J CeJ, B2 = COMPRESSIBILITY FACT,W (= _ACtl _2 - 1.0). VORLAX 3_A

C,e. AA_ AM = {)]RECTION ANGLES (ANEL_g OF FLOAIAIION) OF VORLAX 36 _5 CIe, FREE (WAKED TRAILING LEGS. VORLAX 36}6 C,,J TE = 1Ap=GENI oF TRAILING C_)(3E SWFFP ANGLE. VORLAX _h31 CT = AVERAGE LENGTH OF UOUND TnAILING LEGS OF HORSESHOE VORLAX _h)R (DISTANCE BETWEE;_ VORTEX CE_JTROID AND TRAILING VORLAX 3h_g Ci,= EOAE MEASURED ALnNG X-AXIS). VORLAX _hqD C,,m MM = FL(_ATING WAKE COHPLIT_TIOk FI a(,. VORLAX 36gi Ce,, COMMON_ VORLA_ 36_2 Cel, NONE. VORLAX _6_'_ C,o, VO_LAX 36q_ C,.,Ot)TPUT CALLING SEQUENCE5 VORL_AX 3645

C..i U_ V, W : ORTHOOONAL VELOCI[Y COwPONENTS INOUCED RY V_NLAX 3h4_ C.,- GENERALIZED HORSESHOE vORIEX OF UNIT VORLAX 3h41 C,,e CIRCULATION INTFNSIIf, VORLAX 36w_ Co.. VONLAX 3h_9 C,,,SUBROUTINES VORIAX 365(I

Co,,CALLED NONE. VORLAA 36_I C_ee VU_LAX 36%_ _,,,DISCUSSION THE [GE_ERALIZEOI HORSESHOE V(]_TFX ELEMENT CONSISTS t)F VORLAX 3653 FIVE LF(;S tJR SEGMENTS, OF WHICH THREE ARE IR(IUNU( ANt) VORLA_ 36%4 IWU ARE [F_EE( OR IF(OATI;_(,I. IHE liOUN{) LF(_S ARt. THE VQ_LAX 365S C=== SKEWED, OR SWEPT, TRA'_SVERSE SE(;_ENT AND TH_ TWO VORLAX _g_ _e.e TRAILI_I_, UR CHORF)WISE, F|LAME_Tg EXTENDINb F;_()M _4E V(}_LA_ 3hg/ C,e_ ENDS OF THE TRANSVERSE LEG TU IHn TRAILING EDGE. THE VO_LAX 3h%h

C_ee FLOATIN(, TRAILING LEGS ARE THE SFMI-INFINITE LINFS VORLAX 3659 WHICH START AT 1HE TRAILING ED(_E AND CONTINUE TO VORLAX 3600 DOWNSTNFAM INFINIIY ACCORDING IO A PRESCRIHED O[PLC_IION VORIAR $661

Ce=, DETERMI_IEO BY THE FLO&TAIION A_GIES AA AN() AM. I,EgE VO_LAX 3062 FIOATIN(; TRAILING LEGS CONSTITUTF THE CONTINUATION VONLAX 36h] C,== IN THE WAKE OF THE BOUND TRAILING SEGMENTS. VORLAX 3_h_ AT SUPFPSONIC NAC_ NHWBERS THE V_LOCITY INDUCE() ,Y A VORLAX 36_ DISCRET_ _ORSESHOF VORTEX IIECOME_ VE,_Y LARGE IN t_F \:_RLAX 3_6h VICINITY OF THE NIACH AFICnNES o_NERATEO HY THF S_EWEt) VI_LAX ]_7

122 7417_ OPT*I FTN _,_*410A LO/O_I7b |O.I_,L:_

C*** LEG OF THE HORSESHOE, AT THE CHARACTERISTIC ENVEI._)HE V0_LAX 36_H ITSELF, THE INDUCED VELOCITY VANISHES DUE TO THE FINITE VORLAX 3_q

C,e* PANT CONCEPT. THI_ 51NGUL&R AEHAv|ON OCCURS ONLY FOR VORLAX 3670

C*** FIELD P(IINTS OFF THE PLANE OF rHF HONSESH_}E, TO ^V_I_ VCNLAX 367L TH|$ NUMERICAL SINGtJLARITY THE CHARACTERISTIC SUNFICES VOHLAX 367_

Ceoe (NACH CONES) ARE TAKEN TO HE GIVEN HY VORLAX 3_73 IX - Xl) "e2 • B2 "((Y - Yl) ,o_ , (Z -ZI) ''P) /CLITOFF VORLAX 3_7_

Co,, MH_RE CIITOEF IS A NUNERIC&L CONSTANT WHOSE VALUE I_ VORLAX 367_

C,., LESS THANe BUT CLOSE TOo 1,0 • VORLAX 3h7_

C,e* VORLAX (677 VORLAX _67R

C*** VORLAX _b7_

123 7_474 OPT=I FTN 4.5*410A 10/04/16 10,15.13

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247

APPENDIXC WAVEDRAGTOVORLAXINPUT CONVERSIONPROGRAM

249

SUMMARY

The purpose of a program called WDTVOR, developed to convert the Wave Drag input geometry into the VORLAX input geometry description, is to save time, improve accuracy, and reduce human drudgery when configurations for which the geometry was first digitized in the Wave Drag format are also to be analyzed on the VORLAX program.

The present version of WDTVOR contains the option to convert fuselages to flat plates having the correct planform area or to a simulation having hexagonal cross sections. All wings and planer surfaces are converted to zero thickness panels although the wing camber effects are preserved. Engine pods are converted as curved panels approximated by hexagons.

INTRODUCTION

In the analysis of an airplane design, what is basically the same geometric body is described by several different geometric models, each of which is unique to the discipline for which it is designed; yet at the same time there exist certain elements common to all models. For example, the NASA Wave Drag format (Reference 12) emphasizes the enclosed volume of the aircraft and correct spatial relationship of components while the VORLAX method of aerodynamic analysis (unified subsonic and supersonic) uses a paneling procedure similar to but different in detail from the NASA Wave Drag format in order to reduce computation time.

The program described herein, called WDTVOR, is for direct transformation from WAVE DRAG to VORLAX input format. A comparison program, called VORTWD, transforms data from VORLAX to WAVE DRAG formats and is documented in Appendix D.

Presently the simulation form into which Wave Drag data is converted is limited to that most commonly used in VORLAX analysis as follows:

GENERAL

Cosine law spacing of streamwise vortices and equal interval spacing of lateral vortices is assumed.

251 A single Machnumberfrom the first wave drag case card is picked up and output in the VORLAXdataset. Three angles-of-attack, 0, 5, and i0 are also included.

Symmetric Flight is assumed.

The value of CBARis arbitrarily set to 1.0. The momentcenter coordinates, XBARand ZBAR,are set to zero.

All major planar panels are set up for three spanwise vortices (NV@R= 3) and i0 chordwise vortices (RNCV= i0). Full leading edge suction is assumed on each panel ($PC = 1.0) except for the fuselage panel. WINGS

• Wing camber is preserved in the transformation.

• Wing thickness is not preserved. Converted wing panels are zero thickness.

FUSELAGE

• Both circular and noncircular cross section fuselages may be converted to a single zero thickness panel of trapezoidal shape. The planform area of the zero thickness fuselage is made equivalent to the planform area of the Wave Drag simulation and the vertical location of the fuselage is set equal to that of the leading edge of the wing root chord so as to effectively "seal" the wing-fuselage juncture in the VORLAX simulation.

Optionally fuselages which are circular in the Wave Drag simulation may be transformed to a hexagonal cross-section simulation in VORLAX. With this option transverse vortices are located on the actual body surface. Also a wing-fuselage seal is assured by holding the fuselage diameter constant in the region from leading to trailing edge of the first wing chord and by assuming that one vortex of the reference polygon lies in the plane of the first wing chord.

• If the Wave Drag simulation was a cambered circular one, the fuselage camber is preserved in the VORLAX simulation.

If the fuselage was of abritrary cross section in Wave Drag no camber information is transferred to VORLAX. At present it is possible to convert arbitrary fuselages into only the zero thickness planform simulation in VORLAX.

NACELLES

• Circular nacelles are transformed as closed hexagonal panels using the option that transverse vortices are located on the actual body surface.

252 FINS

• Each fin is treated as a single panel of zero thickness.

CANARDS

• Each canard is treated as a single panel of zero thickness.

After a Wave Drag data set has been converted to VORLAX format it may be validated by converting it back into Wave Drag format and a configuration plot made. This plot will, of course, be different from a plot of the original Wave Drag dataset by virtue of the zero thickness panel representations.

NOMENCLATURE

CPU Central Processor Unit.

FUSTYP Flag for fuselage type. Use 0 for single trapezoidal panel fuselage simulation. Use i for hexagonal (curved) panel fuselage simulation.

VORLAX A Unified Non-Planar Vortex Lattice Method for Subsonic and Supersonic Flow Described in this report.

WDTVOR Wave Drag to VORLAX data conversion Program described in this appendix.

VORTWD VORLAX to Wave Drag data conversion program described in Appendix D.

PROCEDURE

. Prepare a dataset in WAVE DRAG format if it does not already exist. See Reference 12 for the WAVE DRAG input description. Be sure at least one case card is included with the dataset. It is necessary to instruct the conversion program as to whether the VORLAX simulation of the fuselage is to be a single trapezoidal panel or a curved (hexagonal) panel. This is accomplished by appending to the front of the WAVE DRAG dataset a card in namelist format specifying either FUSTYP = 0 for a single trapezoidal or FUSTYP = i for curved (hexagonal) simulation. Figure C-I is a listing of an example WAVE DRAG dataset showing how the namelist input is placed at its beginning. Figure C-2 is a graphic representation of the Figure C-I dataset obtained from the Configuration Plot Program of Reference 12.

253 . Submit the dataset with the WDTVOR program. A compilation listing of the program is included at the end of this appendix.

m Examine the output from the above submittal. Output consists of a listing of the input dataset in card image form (see Figure C-!) followed by a listing of the output dataset in the VORLAX format. Figures C-3 and C-8 are sample datasets as output from WDTVOR program. Figure C-3 was based on the curved (hexagonal) panel VORLAX fuselage simulation while Figure C-8 was derived from the single trapdzoidal panel fuselage.

Because of the simplification that takes place in WDTVOR, e.g., zero thickness panels, it may be desirable to generate a graphic representation of the VORLAX dataset. This is accomplished by using the VORLAX to WAVE DRAG conversion program described in Appendix B followed by the Configura- tion Plot Program described in Reference 12. Figure C-4 is a sample of a simplified dataset after transformation back into WAVE DRAG format. Figure C-5 is a plot of the Figure C-4 dataset. Figure C-6 is a listing of the WAVE DRAG format for the single trapezoidal panel fuselage simulation and Figure C-7 is the corresponding plot.

. Generally, a certain amount of editing is required to prepare a VORLAX dataset generated by transformation via the WDTVOR program for an actual run on VORLAX. Figure C-8 is a listing of a converted dataset as output from the conversion program and Figure C-9 is the same dataset after it has been edited preparatory to a VORLAX run.

An explanation of the notes of Figure C-9 follows:

NOTE i - The value of the over relaxation parameter, REXPAR was changed from -0.02 to 0.0 implying that the program is to compute internally the optimum over relaxation value.

NOTE 2 - The number of angles of attack, NALPHA, was reduced from 3 to i and the single ALPHA selected was 5.0 degrees.

NOTE 3 - The number of major panels, NPAN, was changed from 12 to 13. The pitching moment reference length, CBAR, was changed from 1.0 to 1022.28. The XBAR and ZBAR of the moment reference point were changed from zero to 1910.4 and -800.4, respectively.

NOTE 4 - The number of spanwise vortices, NVOR, and the number of chordwise vortices, RNCV, were adjusted to provide a better subpaneling representation.

NOTE 5 - Wing Panel Number 5 was completely eliminated. It was actually a null panel having little or no span which was included in the original WAVE DRAG simulation for another purpose but which would have caused errors in VORLAX if it has been retained.

254 NOTE 6 - A vertical panel was added to simulate a nacelle pylon which was not included in the original WAVE DRAG simulation.

NOTE 7 - The spanwise location of the nacelle was moved outboard from 214. to 233.94 to coincide with a major panel break in the wing. Alternatives to this procedure would be to create an extra major panel having an edge at the 214. span location or making sure that one of the wing vortices resulting from subpaneling coincides precisely with the engine pylon spanwise location. The above precautions must be taken whenever major panels intersect.

NOTE 8 - The variable area nacelles were changed to fixed area ones by setting the ZCI values to i00. This makes each nacelle cross- section equal to that of the reference polygon, in this case the inlet area. This was a compromise made to simplify the nacelle pylon simulation. With constant area nacelles the pylon could be simulated by a single rectangular panel. Retention of the variable area nacelles would have required several panels to seal the pylon area between the wing and engine.

NOTE 9 - The location of the nacelle was moved outboard from 204. to 233.94, otherwise, Note 7 applies.

NOTE i0 - Fin Panel #2 which was on the aircraft centerline was completely eliminated. Since the desired VORLAX run was to be symmetric about the pitch plane the centerline fin was not required. Furthermore, its inclusion would have un- necessarily increased computing cost because its presence alone triggers off certain asymmetric calculations.

NOTE Ii - The horizontal tail (labeled canard) root chord dimensions were changed to coincide with the side of the fuselage. Actually, omly that portion of the horizontal tail outboard of the VORLAX fuselage simulation is included.

NOTE 12 - A vertical panel was added to seal the gap between the plane of the horizontal tail (Z = -40.) and that of the fuselage (Z = 54.96).

. It is a good idea to submit the edited dataset to the VORTWD program (Appendix D) to recheck the editing just done. This step is optional. It probably wouldn't be done if only minor editing was involved in Step 5. If major editing was done, such as the addition of pylons and sealer panels as in Figure C-9, then it is recommended. The results of converting and plotting the dataset of Figure C-9 is shown in Figure C-10. Note the absence of centerline fin, the addition of engine pylons and horizontal tail sealer panels and the constant diameter nacelle simulation.

255 o The edited dataset may then be submitted to the VORL_P_ progr_r,. A portion of the VORLAX output resulting from this submittal is presented in Figure C-II.

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APPENDIX D

VORLAX TO WAVE DRAG INPUT CONVERSION PROGRAM (VORTWD)

327

SUMMARY

A program developed to convert the VORLAX input geometry description into the Wave Drag input geometry description has two purposes: I) to permit plotting of the configuration geometry in wire frame form as a check on input errors; 2) to save time and reduce human drudgery when configurations for which the geometry was first digitized in the VORLAX format is also to be analyzed for wave drag.

While the present version of VORTWD does not convert all VORLAX input options, it does handle the most common ones. It is recommended that all newly created VORLAX data sets be converted and plotted to validate the input geometry.

INTRODUCTION

In the analysis of an airplane design, what is basically the same geometric body is described by several different geometric models, each of which is unique to the discipline for which it is designed; yet at the same time there exist certain elements common to all models. For example, the NASA Wave Drag format (Reference 12) emphasizes the enclosed volume of the aircraft and correct spatial relationship of components while the VORLAX method of aerodynamic analysis uses a paneling procedure similar to but different in detail from the Wave Drag format.

The program described herein, called VORTWD is for direct transformation from VGRLAX to Wave Drag input format. To accomplish this, certain data must be added to the VORLAX dataset to give the program sufficient "smarts" as to which VORLAX panels are associated with which _ave Drag components.

A companion program called WDTVOR converts data from the Wave Drag to VORLAX formats and is described in Appendix C.

Presently the VORTWD program is limited as to the types of VORLAX data sets it can handle as follows:

WINGS

• Wings may have camber but not thickness.

All wing panels must have the same number (NAP) and values (XAP) of percent chord stations for the cm_ber definition. Flat wings without camber can also be handled.

329 All wing panels must span the distance from leading to trailing edge of the wing. That is, separate panels representing leading or trailing edge flaps are not properly treated at present.

Twist applied to a panel via AINCI and AINC2 in VORLAX is ignored in the transformation.

FUSELAGE

VORLAX fuselage simulations consisting of either a single trapezoidal panel of zero thickness or a curved panel having polyagonal cross sections may be converted.

NACELLES

Circular nacelles represented by curved panels in VORLAX can be converted. Nine is the maximum number of nacelles. Nacelles treated as vertical flat plates are better treated as fins.

FINS

Zero thickness panels without camber are treated; one panel per fin maximum. Six is the maximum number of fins and fin panels.

_ANARDS (Horizontal Tail)

Zero thickness panels without camber are treated; one panel per canard maximum. Two is the minimum number of canards and canard panels.

Two plot control cards compatible with the Configuration Plot Program (Reference i) are automatically added to the converted dataset.

NOMENCLATURE

Symb o i Description

NWN GP Total number of VORLAX panels making up the wing description. If wing is cambered, then all panels must have the same number of percent chords (NAP) and the same values of percent chords (XAF) in the camber description. This is a Wave Drag requirement but not a VORLAX requirement. Input as an integer, no decimal point. Maximum value is 20.

33O Symbol Description

NWING VORLAX panel numbers making up the wing. VORLAX panels are numbered according to their input sequence, not by any alpha- numeric description appearing to the r__ht of th_ VORLAX data cards. Wing panel numbers need not be consecutive but they must be listed in a sequence proceeding from root to tip. Input as integers separated by commlas. The maximum number of values is 20 and the number listed must be equal to the !'_NGP value.

NFUSP The total number of panels making up the fuselage. The fuselage is ass_ned to consist of only one major fuselage segment in terms of Wave Drag. Input one as an integer.

NFUSE The VORLAX pane], number describing the fuselage.

NP Number of curved panels describing pods or engine nacelles. Maximum value is 9.

NPPAN List of VORLAX panel numbers making up the pod or nacelle descriptions. One panel per pod. Presently, such panels should be curved panels, i.e., PDL>360. If a nacelle is being simulated as a vertical flat plate or plates, then it would be more appro- priately treated as a fin or fins for Wave Drag purposes.

NF Number of fins to be described. _ymmetric fins need be described only once. Maximum value is 6.

NFPAN List of VORLAX panel numbers making up the fin descriptions. Maximum number is 6 and number listed must be equal to the NF value. That is, one panel per fin.

NCAN Number of canards (horizontal tails) to be described. MaximmTl value is 2.

NCANP List of VORLAX panel numbers making up the canard (horizontal tail) descriptions. Maximum number is 2 and number listed must be equal to NCAN. That is one panel per canard.

PLOTSZ The maximum dimension in inches of the plot to be made from the plot cards appended to the converted dataset. Ssm_e definition as PLOTSZ in the Configuration Plot program (Reference I).

PROCEDURE

I. Create a VORLAX dataset if it does not exist already.

331 2, Add smarts cards to the front of the VORLAX dataset. A typical dataset with smarts cards included is shown in Figure D-I. Note that all such cards are input in namelist format in which nothing may be entered in column one; only columns 2 through 80 may be used. The first namelist card must contain &INPUT and the last card &END. Input quantities are input as VARIABLE NAME = VALUE, as shown in Figure D-I. Variables may be named in any order. Definitions for the various smarts variables are found in the nomenclature section.

. Submit smartened dataset to the VORTWD conversion program. Figure D-2 shows the output resulting from conversion to Wave Drag format.

. Validate the converted dataset by submitting it to the Configuration Plot program (Reference i). Figure D-3 and D-4 shows the plotted results of the Figure D-2 dataset.

. If the configuration is to be analyzed for Wave Drag; the converted dataset must be edited to supply thickness distributions for the wing and tail surfaces and if the VORLAX fuselage simulation was the single trapezoidal kind, then it must be given volume.

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370 REFERENCES

1. Maskew, B.: Calculation of the Three-Dimensional Potential Flow Around Lifting Non-Planar Wings and Wing-Bodies Using a Surface Distribution of Quadrilateral Vortex-Rings. Loughborough University of Technology TT 7009, 1970.

. Tulinius, J.; Clever, W.; Niemann, A.; Dunn, K.; and Gaither, B.: Theoretical Prediction of Airplane Stability Derivatives at Subcritical Speeds. North American Rockwell NA-72-803, 1973. (Available as NASA CR-132681.)

. Ward, G. N.: Linearized Theory of Steady High-Speed Flow. Cambridge Univeristy Press, 1955.

4. Robinson, A.: On Source and Vortex Distributions in the Linearized Theory of Steady Supersonic Flow. Quart. J. Mech. Appl. Math. I, 1948.

, Lomax, H.; Heaslet, M. A.; and Fuller, F. B.: Integrals and Integral Equations in Linearized Wing Theory. NACA Report 1054, 1951.

. Hadamard, J.: Lectures on Cauchy's Problem in Linear Partial Differ- ential Equations. Yale University Press, 1923.

. Lan, E. C.: A Quasi-Vortex-Lattice Method in Thin Wing Theory. Journal of Aircraft, Sept. 1974.

. Bratkovich, A.; and Marshall, F. J.: Ite_ative Techniques for the Solution of Large Linear Systems in Computational Aerodynamics. Journal of Aircraft, Feb. 1975.

, Purcell, E. W.: The Vector Method of Solving Simultaneous Linear Equations. Journal of Mathematical Physics, Vol. 23, 1953. i0. Hancock, G. J.: Comment on "Spanwise Distribution of Induced Drag in Subsonic Flow by the Vortex Lattice Method". Journal of Aircraft, Aug. 1971. ii. Lan, E. C.; and Campbell, J. F.: Theoretical Aerodynamics of Upper- Surface-Blowing Jet-Wing Interaction. NASA TN D-7936, Nov. 1975.

12. Craidon, Charlotte B.: Description of a Digital Computer Program for Airplane Configuration Plots. NASA TM X-2074, 1970.

371

1. REPOR_NO. -.... -Z 50_;ERNMENTACCE_S%---_N---O_...... _i-n Ei:,Pi_'T:S' i_._TALOGNO. NASA CR-2865 _. TITLE AND SiJBTITLE ..... "...... 5.'REPOR'T'DATE December 1977 A GEh%RALIZED VORTEX LATTICE _,r _(fi) FOP, SUBSONIC '61PERFORMING ORGCODE AND SUPERSONIC FLOW APPLICATIONS m 7. AUTHOR(S) r"_.PERFORMING ORG-REPORT NO. Luis R. Miranla_ Robert D. F lliott_ and LR 28112 Wil!i_n M. Baker [10. WORK UNIT NO. 9. PERFORMING ORGANIZATION NAME AND ADDRESS LOCKHEED-CALl FORNIA COMPANY P.O. BOX 551 111. CONTRACT OR GRANT NO. BURBANK, CALIFORNIA 91520 NAS 1-12972 13. TYPE OF REPORT AND PERIOD 12. 'SPONSORING"_'GENCY NAME AND ADDRESS .... COVERED

14 SPONSORING AGENCY C'ODE NASA Langley Research Center Hampton, Virginia 23665 | , , ,45. SUPPLEMENTARY NOTES Langley Technical Monitor: Harry H. Heyson Topical Report

16. ABSTRACT A vortex lattice method applicable to both subsonic and supersonic flow is described. It is shown that if the discrete vortex lattice is considered as an approximation to the surface-distributed vortJcity, then the concept of the generalized principal part of an integral yie3ds a residual term to the vorticity-induced velocity field. The proper incorporation of this term to the velocity field generated by the discrete vortex lines renders the present vortex lattice method valid for supersomic flow. Special techniques for simulating nomzero thickness lifting surfaces and fusiform bodies with vortex lattice elements are included. Thickness effects of wing-like components are simulated by a double (biplanar) vortex lattice layer, and fusiform bodies are represented by a vortex gri4 a_'r'ahged o_J a series of" concentrical cylindrical surfaces. The analysis of sideslJ_ effo,:is by the subject method is described. N_merical consideratiot_s peculiar to the. a[q/Lication of these techniques are also discussed. A s_:_ary compa]'ivon of the results obtained by the method of th:is _-'eport witL ©ther theo_-et_c_3 and experimental results is presented. The method h_s been im_i_:'.,_nt_._d_m '_ digital computer code identified as VORLAD(. A users m_nu,t_! i_ inc]uded along with a comp]ete Fortran compilation and executed ca_. Also _.c]_ded are conve_"_ion proL_'a_s useful for transforming input betwe-m VORLAX and the NASA Wave Drag program.

17. KEY WORDS (SUGGESTED BY AUTHOR(S) ) '!18. DISTRIBU'TION STATEMENT Aerodynamics Vortex-lattice method Unlimited - Unclassified Subsonic flow

Supersonic flow Subject Category 02

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