Vsaero Theorical Manual

7/15/2019 Vsaero Theorical Manual

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NASA Contractor Repo_ _237_7 _177

Program VSAERO_eory DocumentA Computer Program forC_latingNonlinear Aerodynamic Characteristicsof Arbitrary Configurati__

Brian Maskew .....

CONTRACT NAS2-11945SEPTEMBER 1987

N/LS/X


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NASA Contractor Report 4023

Program VSAERO Theory DocumentA Computer Program for CalculatingNonlinear Aerodynamic Characteristicsof Arbitrary Con_gurations

Brian MaskewAnalytical Methods, Inc.Redmond, Washington

Prepared forAmes Research Centerunder Contract NAS2-11945

National Aeronauticsand Space AdministrationScientific and TechnicalInformation Office1987


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TABLE OF CONTENTS

section

LIST OF FIGURESLIST OF SYMBOLS1.0 INTRODUCTION

i.I General ....................1.2 Objectives ..................1.3 Strategy ...................

2.0 FORMULATION2.1 Flow Equation .................2.2 Boundary Conditions .............2.3 Choice of Singularity Model ..........

3.0 NUMERICAL PROCEDURE ................3.1 Geometrical Description ...........

3.1.1 Patch . _ . . .............3.1.2 Panel Geometry .............3.1.3 Panel Neighbors ............3.1.4 Wakes .................

3.2 Matrix of Influence Coefficients .......3.2.1 Velocity Potential Influence Coefficient

3.2.1.1 Doublet Contribution .....3.2.1.2 Source Contribution .....

3.2.2 Velocity Vector Influence Coefficients3.2.2.1 Doublet Contribution .....3.2.2.2 Source Contribution .....

3.2.3 Matrix Assembly ............

3.3 Matrix Solver .................3.4 On-Body Analysis ...............3.5 Wake-Shape Calculation ............3.6 Boundary Layer Analysis ............3.7 Off-Body Analysis ...............

3.7.1 Velocity Survey ............3.7.2 Off-Body Streamline Procedure .....

aS_ No.

iiivi

111

368

16161621252631

3340

414446

4848575859596O

i


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TABLE OF CONTENTS (CONCLUDED)

_No.4.0 DISCUSSION OF RESULTS ............... 62

4.1 Swept Wing .................. 624.2 Wing with Strake ............... 624.3 Nacelles ................... 644.4 Wing-Flap Cases ................ 684.5 Body Alone .................. 744.6 Wing-Body Cases ............... 74

5.0 CONCLUSIONS .................... 84

6.0 REFERENCES .................... 85

APPENDICES:

A: SUBROUTINE FLOW CHART ................ 88

B: BIQUADRATIC INTERPOLATION .............. 96

ii


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LIST OF FIGURES

F_No.123456789

i0ii1213

14

15

1617

18

192021

Title _No.

Section through the Idealized Flow Model . . 4Three of the Possible Internal Flows .... 9Outline of the Steps in the VSAERO Program . 17General Arrangement of the Configuration . . 18Configuration Hierarchy .......... 19Patch Conventions ............. 20Sections Defining Patch Surface ...... 22Panel Geometry ............... 23Panel Neighbor Information ......... 27Panel Neighbors across Patch Edges ..... 28Wake-Grid-Plane Scheme ........... 29Wake Arrangement .............. 30Model used in the Evaluation of the SurfaceIntegrals ................. 35Evaluation of the Surface Integrals for OneSide of a Panel .............. 36Treatment of Symmetry and Ground-PlaneProblems .................. 39Evaluation of the Line Integral ...... 43Evaluation of Perturbation Velocity on PanelK (All Neighbors Available and on Same Patchas Panel K) ................ 49Evaluation of Perturbation Velocity on PanelK when a Neighbor is not Available ..... 52Force Contribution from Panel K ...... 53Global and Wind Axis Systems ....... 55Streamline Calculation ........... 61

iii


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LIST OF FIGURES (CONTINUED)

F__ No. Title22 Comparison of Calculated Chordwise Pressure

Distribution on a Thin Swept Wing ...... 6323 Comparison of Calculated y-Component of Veloc-

ity at Two Stations on a Wing with Strake;NACA 0002 Section .............. 65

24 Comparison of Internal Pressure Calculatedusing a Range of Methods Showing Leakage ofLow-Order First Generation Model ....... 66

25 Comparison of Calculated Pressure Distribu-tions on a Nacelle with Exit Diameter/Chord.3;NACA 0005 Section .............. 67

26 Comparison of VSAERO Calculation with ExactSolution for Williams (26) Two-Element Case B

(a) Main Element .................(b) Flap .....................(c) Paneling ...................

697071

27 Comparison of Pressure Distributions on anAspect Ratio 6 Wing with Part-Span Fowler Flap= 16

(a) Section at _ = .33 ............. 72(b) Section at _ = .6 .............. 73

28 Comparison of Calculated and Exact PressureDistributions on a 4:1 Spheroid at _ = 5 . . 75

29 Pressure Distributions on a Wing-Body Config-uration at _ = 4 , M_ = .6

(a) Above Wing-Body Junction Pressures ...... 76(b) Spanwise Cut through Present Calculations . . 77

3O Swearingen Metroliner Wind Tunnel Model . . . 7831 Comparison of Calculated and Experimental Pres-

sure Distributions on the Swearingen MetrolinerWind Tunnel Model at e = 0.0, M_ = 2.0

(a) Buttline Cut, Y = 9.0 ............(b) Buttline Cut, Y = 18.0 ...........

7979

iv


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LIST OF FIGURES (CONCLUDED)

F_No.32

33

34

TitleComparison of Calculated and Experimental Pres-sure Distributions on the Swearingen MetrolinerWind Tunnel Model at e = 12 , M = 2.0

(a) Buttline Cut, Y = 9.0 ............(b) Buttline Cut, Y = 18.0 ............

8O8O

Comparison of Calculated and Experimental Pres-sure Distributions on the Swearingen MetrolinerWind Tunnel Model at e = 16 , M_ = 2.0

(a) Buttline Cut, Y = 9.0 ............(b) Buttline Cut, Y = 18.0 ...........

8181

Calculated Streamlines and Separation Zone onthe Swearingen Metroliner Wind Tunnel Model at= 16 ................... 83

APPENDIX A

A-I

A-2

A-3

Flow Chart of Main Program (Deck MVP); theViscous/Potential Iteration Loop .......

Flow Chart of Subroutine VSAERO in PotentialFlow .....................

Flow chart for Major Subroutines in the Bound-ary Layer Analysis (Subroutine BLCTRL) ....

89

9O

95

APPENDIX B

B-I Biquadratic Interpolation i00

v


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LIST OF SYMBOLSAREAB ,CJK JK

DD ,EJK JK

Ei,j,kKKWPU,IWPL

Panel areaPerturbation velocity potential influence coefficientat the control point of panel J for a uniform distribu-tion of unit source and unit doublet, respectively, onpanel KDiagonal vector in a panelInfluence coefficient for the normal component ofvelocity induced at the control point of panel J by auniform distribution of unit source or doublet,respectively, on panel kPosition vector of a panel corner in the panel localcoordinate system with origin at panel centerOrthogonal unit vector system along the x,y,z axes,respectively, of the global coordinate systemFactor for the local contribution to (Eq. (13))Program subscripts of upper and lower surface panelsshedding a wake column (Figure ii)

NNABORNABSIDiRrrSsSMP,SMQV

Orthogonal unit vector system in a local region; n isnormal to the surface and , m tangentialNumber of panelsProgram subscript of neighboring panel on side iProgram subscript of adjacent side of the neighboringpanel in side i (Figure 9)Position vector of a panel corner point in the G.C.S.Position vector of a point relative to an element, ds,of the surfaceModulus of Surface (solid) of the configurationDistance along the surfacePanel half-median lengths (Figure 8)Velocity vector

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VV

Wx,y,z

LIST OF SYMBOLS (CONTINUED)

Modulus of VPerturbation velocityWake surfaceCartesian coordinates in the Global Coordinate System(G.C.S.)--body-fixed frame of reference

YIncidence of G.C.S. x-axis degreesVorticity vector densityBoundary layer displacement thicknessLocal coordinates in the and m directions, respect-ively, of the panel local coordinate systemDoublet singularity densityTotal velocity potentialVelocity potential contribution, or perturbation com-ponentDensitySource singularity density

V

N

. _ _Laplacian operator, I_-_ + j_-_+ k _z

Dot product of vectorsCross product of vectors

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LIST OF SYMBOLS (CONCLUDED)

c

e

iJ,K

LN

P

S

U

WOO

Center

Edge of the boundary layerInner region

Pertaining to panel J,K

Lower surface

Normal component

Pertaining to the point, P; also used as a subscript

Pertaining to the surface

Upper surface

Pertaining to the wake outer edge of the boundary layerReference onset condition

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1.0 INTRODUCTIONi.i GeneralVSAERO is a computer program for calculating the subsonic

AEROdynamic characteristics of arbitrary configurations havingVortex Separation and strong Vortex/Surface interaction. Thisdocument describes the theory and numerical procedures used forthe method. An earlier report (I) describes the input and use ofthe computer program.

1.2The initial objectives of the new program were to analyze

general wing configurations with multiple part-span, high-liftdevices and to give special attention to edge-vortex separationsand to close vortex/surface interactions. Later objectivesbrought in the modeling of fuselage and canard surfaces leadingto the present capability for arbitrary configurations includingcomplete aircraft.

1.3At the beginning of the program development there was only

one clear cut choice for approaching the solution to the abovenonlinear problem. This was an iterative viscous/potential flowcalculation using a potential flow panel method coupled withintegral boundary layer methods, with wake-shape iteration calcu-lations included in the potential flow analysis. No other ap-proach appeared to offer the feasibility of achieving the aboveobjectives. Finite difference field methods, for example,treating more general flow equations, could not be developedwithin a reasonable time scale for application to the complexgeometry of multiple part-span, high-lift devices. In the mean-time, the surface integral approach of the panel method offered apowerful and practical engineering tool which had the capabilityto represent the complex geometries involved in the objectives.Furthermore, the low computing time of the panel method was anattractive feature in view of the complex iterative loops madenecessary by the nonlinear effects of viscosity and wake loca-tion.

In the chosen strategy for the development of the VSAEROprogram the capabilities of two earlier programs were combined:(i) VIP3D (2) offered a viscous/potential flow iterative methodfor transport wings with full-span, high-lift devices; and (ii)QVORT, a quadrilateral-vortex panel method (3) and (4), offeredan iterative wake relaxation procedure. Neither method had acompletely satisfactory singularity model or geometry package forthe new objectives. Accordingly, in the development of VSAEROthese features were given prime consideration.

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Various singularity models were investigated for the newrogram including symmetrical singularities (5), high-orderormulations and numerical integration. Finally, a low-order

formulation based on internal Dirichlet boundary conditions waschosen. This formulation is described in Section 2. The numeri-cal procedure for the method, which employs quadrilateral panelsof constant doublet and source distributions, is described inSection 3. The description includes details of the extendedgeometry package for the VSAERO program. Correlation studies areexamined in Section 4; these include a number of basic test casesinvolving exact, nominally exact and also experimental data.

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2.0 FORMULATION2.1 Flow EquationIn order to obtain a solution to the flow problem the sur-

face of the configuration is assumed to be at rest in a movingairstream. Regions of the flow field that are dominated byviscous and rotational effects are assumed to be confined to thinsurface boundary layers and wakes. The rest of the flow field isassumed to be inviscid, irrotational and also incompressible.

Figure 1 shows a streamwise section taken through a wing andits wake. The section indicates the surface, S, of the wing, thesurface (upper and lower) of the wake, W, and an outer surface,S_, which encloses the complete flow problem at infinity. Thetotal surface, S + W, divides space into two regions: the exter-nal region which contains the flow field of interest and theinner region which contains a fictitious flow. Velocity poten-tial fields, _ and _i, are assumed to exist satisfying Laplace'sequation in the flow field and in the inner region, respectively;i.e.,

V25 = 0 in the flow fieldand

V2_ i = 0 in the inner region.Next, apply Green's Theorem to the inner and outer regions

and combine the resulting expression (see Lamb (6)). This yieldsthe following expression for the velocity potential, Cp, anywherein the two regions, expressed in terms of surface integrals ofthe velocity potential and its normal derivative over the bound-ary surface:

_W_S

n (V# - V#i)dSS_W_S

(i)

where r is the distance from the point, P, to the element, dS, onthe surface and n is the unit normal vector to the surfacepointing into the fluid.

The first integral in Eq. (i) represents the disturbancepotential from a surface distribution of doublets with density (- i ) per unit area and the second integral represents the con-tribution from a surface distribution of sources with density-n.(V - V# i) per unit area. The singularities, therefore, rep-resent the jump in conditions across the boundary; the doublet


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r

s

u

Figure i.Section

thrugh the

Idealized Flow MOdel.


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density represents the local jump in potential and the sourcedensity represents the local jump in the normal component ofvelocity.

Examine Eq. (I) separately over the three surfaces, S, W,S_, as indicated in Figure I.

The surface at infinity, S_, can be regarded as a largesphere centered on P. The local conditions at that boundaryconsist only of the uniform onset velocity, V , the disturbanceat that distance due to the configuration having essentiallydisappeared. The surface integrals in Eq. (i) taken over S_ thenreduce to _p; i.e., the velocity potential at point, P, due tothe onset flow.

The upper and lower wake surfaces, W, are assumed to beinfinitesimally close to each other for the present problem;i.e., thin wakes. (A simplified thick wake representation isdescribed in References 7 and 8.) In this case the correspondingupper and lower elements can be combined and the i term for thispart of the integral disappears. Furthermore, if we neglectentrainment into the wake surface, then the jump in normalvelocity component, n (re U - VCL), is zero and so the sourceterm in Eq. (I) disappears for the wake contribution. (Note thatentrainment modeling _Q/_ be included given a distribution ofthe entrainment normal velocity.)

With the above conditions, Eq. (i) becomes:

S

Sn (V - Vi)dS

W- _L ) n V (1) dW + _

P (2)

The normal, n, on surface, W, points upwards in this case.If the point, P, lies on the surface, S, then the integral

becomes singular on S. To evaluate the local contribution thepoint, P, is excluded from the integral by a local sphericaldeformation of the surface centered on P. For a smooth surfacethe local deformation is a hemisphere and the local contributionobtained in the limit as the sphere radius goes to zero is


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%(-i)p, which is half the local jump in potential across thesurface at P. Note that the first potential in the expression ison the side of the surface on which p lies, or, more con-veniently, the sign can be changed for the point on the otherside. For example, if P is on the iIiH_ide surface of S, then Eq.(2) becomes:

/fP - 4_ (_ - i ) n V( )dS - %( - i) pS-P

i//_ rS

n (re - V i)dS

w

(3)

2.2 Boundary Conditions

A solution to Eq. (3) must satisfy a number of known bound-ary conditions which can be imposed--or in the case of the wake--implied at the outset. On the surface, S, the external Neumannboundary condition must be satisfied.

n- V ffi-V N - n V S on S (4)where V N is the resultant normal component of velocity relativeto the surface. V N is zero for the case of a solid boundary withno transpiration; non-zero values are used to model boundarylayer displacement effect, entrainment and also inflow/outflowfor engine inlet/exhaust modeling. V s is the local velocity ofthe surface; this may be composed of several parts due to bodyrotation (e.g., pitch oscillation (9), or helicopter blade rota-tion (10)), arbitrary motion (Ii), body growth (12), etc. Forthe present problem the surface of the configuration is assumedsteady in a body-fixed coordinate system and so V s is zero.

Hence,

n V = -V on S (5)NThe wake surface, W, cannot support a load and so the doub-

let distribution, CU - eL' on W must satisfy a zero-force condi-tion.


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Consider an element, dW, of the wake surface with local unitnormal, n. The local vorticity vector associated with the doub-let distribution, CU - @L' has the density

y = -n x V (u - L ) (6)The elementary force exerted on the element, dW, in the

presence of the local mean velocity, V, is (from the Kutta-Joukowski Law):

6F = pV x ydWwhere p is the local density.it follows that

And since the force must be zero,

V x 7=0.Substitute for y from Eq. (6):

and expandV x {n x _ (@U- @L )} = 0

n V V( Cu - @L ) - V nV(@ U - @L ) = O. (7)

Equation (7) is satisfied when V n = 0; i.e., the surface,W, it aligned with the local flow direction, and V V(@ U - @L) =0. In the latter case, the gradient of the doublet distributionin the direction of the local mean flow is zero; in other words,the wake doublet distribution is constant along mean streamlinesin the wake surface. The doublet value on each streamline is,therefore, determined by the condition at the point where thestreamline "leaves" the surface, S. Clearly, at the outset ofthe problem with both the doublet distribution and wake locationunknown, an iterative approach is necessary to obtain a solution.When a converged solution is obtained the upstream edge of thewake, W--and hence the trailing edge of the surface, S--carriesno load and so the Kutta trailing-edge condition will be satis-fied. At the outset, therefore, the Kutta condition is im_simply by shedding the trailing-edge potential jump (@u - @.) asescribedconstant down each "streamwise" line on an initially _rwake surface. An explicit Kutta condition in which the doubletgradients on the upper and lower surfaces at the trailing edgeare equated to cancel has been used on occasion, but in theformulations described below, this does not appear to be neces-sary.

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2.3 Choice of Singularity Mode]In principle, a given flow field can be constructed from an

infinite number of combinations of doublet and source distribu-tions on the surface, S, each combination producing a differentflow in the inside region. In practice, however, there may beonly a small number of singularity combinations which are suit-able as a basis for a well behaved numerical model; i.e., onethat is accurate, convenient to use and robust (insensitive touser abuse).

A unique combination of singularities can be obtained in Eq.(3) in a number of ways. One way is to specify one of thesingularity distributions and to solve for the other usingboundary conditions only on one side of the boundary. There areseveral examples of this: in the source-only formulation (13)the doublet distribution is set to zero; in the doublet-onlyformulation, e.g., (3) and (4), the source distribution is set tozero; in program VIP3D (2), the doublet distribution (or vortici-ty) is specified and the method solves for the source distribu-tion; the reverse of the latter has also been used, i.e., speci-fying the source distribution--usually related to the thicknessdistribution--and solving for the doublet or vorticity distribu-tion.

Another way of achieving a unique combination of singulari-ties is to apply certain constraining relationships on the singu-larity distributions. The "symmetrical singularity method" (5)is one example of this and was examined briefly at the start ofthe VSAERO program development.

One characteristic that separates "good" singularity modelsfrom "bad"ones from the numerical point of view is that the flowfield generated in the inner region by a "good" model is ratherbenign and related to the boundary. This is usually not the casefor a "bad" model. In other words, when passing through theboundary, S, the jump from the internal flow to the external flowshould be small--thus requiring a minimum of perturbation fromthe singularities (Eq. (3)). "Bad" singularity methods wereobserved to have very large internal cross flow between the upperand lower surfaces of wings and required high panel densities inorder to achieve acceptable accuracy in the flow solutions. Thisresulted in a push towards high-order formulation which, forsubsonic flows at least, proved quite unnecessary.

One way of achieving "good" characteristics is to treat theinternal flow directly in Eq. (3). This is, in fact, another wayof obtaining a unique singularity distribution--in this case byspecifying boundary conditions on both sides of the surface, S.Earlier methods (14), (15) specified the _L_ on the insidesurface of S. Alternatively, the inner velocity potential, i ,can be specified directly in Eq. (3). This is referred to as aninternal Dirichlet boundary condition. Three possible internalflows were examined for the VSAERO program, Figure 2 . Two of


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8 8II II

E_ZB_r_zOrD oII II.,-I

@-,.-tI-.-I

0-,-t,.q

0I.-1

rd

I-I0-;-I

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N

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8 /r-t _ /IIIII

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_JB

CXl

-,-I


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these are described below, the third one, in which the internalflow was aligned with the wing chord line (e.g., i = -xV_ cos _ )was examined in a two-dimensional pilot program only. Althoughthis worked very well, the formulation was not as convenient asthe other two. Even so, it would be a useful member in a pos-sible general scheme in which the user could select a suitableinternal flow for each component in a complicated configuration.

(i) Zero Flow Inside

Consider first the interior stagnation condition which wasimplied by earlier doublet-only codes (e.g., References 3 and 4)and which used the exterior Neumann boundary condition, V n =0. For the present formulation, set = constant = 0, say inEq. (3), givlng i

= _ i // - V( )ds -p 0 4zS-P

I ll! n . V#dS4_ JJ r

S

W

(8)

The doublet distribution is now the total velocity potentialat the surface and the source distribution is now the totalnormal velocity of the fluid at the surface, S. This mustsatisfy the external Neumann boundary condition (Eq. 5); i.e.,

n V = -V NThis term can be used directly in the second integral of Eq. (8).

For the solid boundary problem, i.e., zero transpiration, VNis zero and the source term disappears leaving

4---_ n V )dS -S-P

p

W(U - eL ) n

I0

V(1)dW + _op = 0 (9)


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This is by far the simplest formulation for the lifting casewith no transpiration. It is, in fact, even simpler than theoriginal zero-lift source method based on the external Neumannboundary condition (13). In a low-order formulation (i.e., con-stant doublets on flat panels) the method has some minor problemsassociated with numerical differencing for velocities and withthe effect of panel arrangement on accuracy (16). These problemscan be alleviated by separating into a known part and an un-known part--the latter being as small as possible.

The most convenient breakdown is to use the onset flowpotential as the known part; i.e., = _ + . The known part,_, and its velocity field, V_, are used directly in the formula-tion. The solution for and the subsequent numerical differen-tiation for surface velocity are then less prone to numericalerror.

For the general case with transpiration, inlet flows, jetsand unsteady conditions, the source term must be present and sothe advantage of the doublet-only formulation diminishes. In themeantime an alternative "zero internal perturbation" formulationhad become popular in other methods and is considered below.

(ii) Internal Flow Equal to the Onset FlowThe internal Dirichlet boundary condition, i = _, was usedearlier by Johnson and Rubbert (17) and also Bristow and Grose

(18) in high-order formulations. With . = _ and %p = _, Eq.(3) becomes: i

S-P W

//r " " (V# - V#_)dS = 0S

(I0)

where _, the perturbation potential on the exterior surface, isnow the doublet density; i.e.,

The source distribution in this formulation is(ii)

4_ = - n (V - V_oo)

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Expanding this expression and substituting for n V from theexternal Neumann boundary condition in Eq. (5):

4zg = V N n V oThe source distribution is therefore established at the outset.

Again, for solid boundary conditions, VN is zero; for themore general case here, V N may have two parts representing (i)boundary layer displacement effect using the transpirationtechnique (2), and (ii) inflow/outflow for engine inlet/exhaustmodeling.

Thus,

4_ =-u (Vs 6*) + VNORM - n Ve (12)

The boundary layer term is set to zero for the first poten-tial flow solution; on subsequent iterations it is updated bycoupled boundary layer calculations. The VN__ term has a posi-. U _Vltive value for outflow and negative for infI_w.

Compared with the zero internal flow formulation (Eq. 9),the present formulation (Eq. (i0) is more forgiving for "bad"panel arrangements (e.g., Reference 16) and is the current formu-lation in the VSAERO program. The reason for its better behaviorprobably arises from the smaller jump in the flow condition fromthe inner to the outer flow; i.e., the singularity strengths aresmaller. However, there are situations where this is not thecase; e.g., a wing at large angle of attack or a powered nacellein zero onset flow. In other words, it may be desirable toprovide a capability that allows the internal flows to be speci-fied independently of the onset flow so that they can comply moreclosely with the shape of each component.

Once the doublet solution is known, then Eq. (3) can beapplied to determine the velocity potential field, the gradientof which gives the velocity field. For the present formulation,Eq. (3) becomes:

ff ffS S

WV(!)dw + _r p (13)

12


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Where the doublet, _, and source, o, are taken from Eqs.(ii) and (12), respectively and _w = Cu - eL is the wake doubletdistribution. The factor, K, has different values depending onthe location of the point, P; if P lies off the surface, then K =0; if P lies on a smooth part of the surface, then K = 2z if P ison the outside, and K = -2_ if P is on the inside; if P is at acrease in the surface, then K takes the value of the appropriatesolid angle contained at the crease. In each case when P is onthe surface, then the surface integral terms in Eq. (13) excludeP.

If the potential field has been computed at a mesh of pointsthen the velocity field can be generated using local numericaldifferentiation; i.e.,

Vp = - VCp (14)Alternatively, the velocity field can be obtained by taking

the gradient of Eq. (13) directly with respect to the coordinatesof point P, thus forming the source and doublet velocity kernels.(Thls is the present approach in VSAERO.)

ffS S

1V (r)dS

f/ i_w V(n V(?))dW + VooW

(15)

(The kernels will be developed further in Section 3.) The off-body velocity field is used in the wake-relaxation cycle, in off-body streamline calculations and for general flow-field informa-tion.

The discussion so far has been concerned with thick con-figurations having a distinct internal volume enclosed by thesurface, S. If pars of the configuration are extremely thin(e.g., thickness/chord ratio < 1%, say) or are wing-like andremote from the area of interest, then these parts may be repre-sented by open surfaces. When the upper and lower parts of thesu@face are brought together, the corresponding upper and lowerelements can be combined and Eq. (2) becomes

13


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S Sn .(re U - V#L)dS

WIf the normal velocity is _ through the sheet thenthe term n (V_U - V_L) = 0 and the source term disappears.(This is _ot a necessary step; the source term could be left inthe equation for simple modeling of thickness effects.) Removingthe source term does not preclude the use of non-zero normal

velocity at the surface; the only restriction is that the normalvelocity be 9_II]IQ_ through the surface. Equation (16) canthen be written:

Y //p = _n V (r)dS + _W n " V (r)dW + _ps w

(17)

where _ - 14_ (u -eL ) is the jump in total potential across thesheet.

In order to satisfy the external Neumann boundary condition(Eq. (5)) the velocity expression is required. Hence, taking thegradient of Eq. (17) with respect to the coordinates of P,

S

//_wV


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S

W(19)

This is the basic equation for the unknown doublet distribu-tion on thin surfaces (referred to as Neumann surfaces in VSAEROin accordance with their primary boundary condition). Again, thevelocity kernels will be developed further in Section 3.

In the VSAERO program, thin and thick components may existsimultaneously in a configuration.

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3.0 NUMERICAL PROCEDURE

The treatment of a given flow problem can be broken downinto a number of distinct steps, each step having a certainfunction. These steps are shown in the VSAERO program outlinedin Figure 3. The first step is to define the surface geometry ofthe configuration and to form the panel model. A m_i_ix ofinfluence coefficients is then formed; this represents a set ofsimultaneous linear algebraic equations resulting from a discre-tization of the flow equations in Section 2.0. The __Q_ion tothese equations provides the surface doublet distribution fromwhich the surface velocities and, hence, surface streamlines,pressures and body forces and moments can be _la/_. With thesurface singularity distributions known, the -W_ calcula-tion can proceed, using computed off-body velocities. The newwake shape must then be repanelled and the matrix of influencecoefficients modified for the next pass through the wake-shapeiteration loop.

The surface flow solution can be passed to the boun_j_ycalculation. The computed boundary layer displacementgrowth rate is passed back to the matrix procedure where the

right-hand side terms of the equation are modified for the nextpass in the viscous/potential flow iteration loop, Figure 3.Finally, when both iteration loops have been completed, 9_

for velocity and streamline paths are processed.

The flow chart showing the relationships of subroutineswhich perform the above steps is given in Appendix A. Details ofthe steps are described in the following Subsections.

3.1 Geometrical Description

The surface geometry of the configuration to be analyzed isdescribed in a global _oordinate _ystem (G.C.S.) which is a body-fixed frame of reference, Figure 4. The configuration is dividedinto a number of parts for convenience, Figure 5. The functionsand use of these parts are described in detail in the ProgramUser's Manual (i). Briefly, the two major parts are the (solid)SURFACE and the (flexible) WAKE. On a complicated configurationthere may be several surfaces and several wakes. The surface andwake are further subdivided as described below.

3.1.1 Patch

Each surface is represented by a set of quadrilateral PANELSwhich are assembled into a regular array of rows and columnswithin PATCHES, Figure 6. Patches are grouped together under twoheadings (Figure 5), COMPONENTS and ASSEMBLIES for user con-venience (I).

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Z_0

_ZH_

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N

X

//

//

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//

z,--, .._10 L_Z0I'-- "'0 Z

0._.1

I---ILlO0Z0ILlZ

00_..j _--0 co....i >--t..o o0

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48/114

//

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HU

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36


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whereRNUM = SM * PN * (B * PA - A * PB)DNOM = PA * PB + PN 2 * A * B * SM 2

PN = PJK " nKPJK = RcjA = lai

- Rc K

B = IblPA = PN z * SL + A1 * AM (= a { x (a x s)})PB = PN 2 * SL + A1 * BM = PA - A1 * SMSM = s m

SL = s AM = a - mBM = b mA1b

= AM * SL - AL * SM (= n (s a))is the position vector of the jth control pointrelative to the end of this side, Figure 14 (i.e., b =a- s)

The subscript, i, has been omitted from the above variables; a, band s are constructed for each side in turn, noting that a =1bi_ 1

There is a limiting condition as PN--the projected height ofcontrol point J from the mean plane of panel K--goes to zero:

I = _ within the strip (DNOM0)

The positive sign occurs for points to the right of the sideand negative to the left. If PN 0-, all the above signs onare reversed.The total velocity potential influence coefficient for the

unit doublet on the panel is the sum for all four sides, from Eq.(27).

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CjK = _ CjK ii=l

(28)

The four arc-tangent functions in this summation can becombined into a single arc tangent; however, extreme care isrequired to ensure the resultant angle is in the correct quadrantin the range, 2_.

For plane of symmetry and/or ground effect problems thecontribution to CjK from image panels (having the same singulari-ty strength as the real panel) is evaluated simply by repeatingthe calculation for the _ panel influencing the images ofcontrol point J. This is possible because the geometrical rela-tionship between the image of panel K and the point J is identi-cal to that of the actual panel K and image of point J (Figure15). Hence, for a symmetrical problem in ground effect therewould be four contributions for each panel, Figure 15. Thistreatment is more convenient than considering the image panelsdirectly. The vertical plane of symmetry is the y = 0 plane inthe G.C.S., while the ground plane is the z = 0 plane. Thus, forground effect problems the configuration height and orientationabove the ground require care during input (i).

The above expression for CjK is used when panel J is closeto panel K. When panel J is remote from panel K, i.e., in thefar field, then a far-field formula is used under the assumption

n rthat the quantity, 3 , in Eq. (25) is essentially constantrover the panel. Then r = PJK and

PNjK * AREA K (29)CjK =PJK

where

PN

n K

JK n= PJK K= the unit normal to panel K

AREA K = the area of panel K (see Subsection 3.1.2)

PJK = IPjKI, is the distance between the control points, Kand J

Equation (29) is the contribution from a point doublet at

MQ"-sRc_)Kith orientation n K and is used when PJK > RFF * (SMP +The quantity, (SMP + SMQ)K, is the "size" of panel K (seeSubsection 3.1.2) and RFF is the user-defined, far-field radiusfactor which has a default value of 5 in the program. This value

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oII

.4-I

u)u_O(9

_J

fl#

o_

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-,-Ir0O_

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Ill,--I

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0L9

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t/lOo (D

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,-tI,._OL_C;

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-,-4

39


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was established following initial test cases on a wing and on awing-body configuration. Use of the expression gives a signifi-cant reduction in computing time; however, a user may use thenear-field equation (Eq. (25)) throughout the configuration if hewishes simply by giving a very large value for RFF in the inputfile (i).

3.2.1.2 Source Contribution

The velocity potential influence coefficient for a unituniform source distribution on the panel, K, is, from Eq. (21),

B JK //ir dSPane i K

The integral is treated for each side of the panel as in the caseof the doublet. Thus, for one side

Yb

BjKi = _ rl r2 1

Y=Ya _=0

where r I, r 2, Ya' Yb are defined under Eq. (26). This becomes:

where

BjK ' = A1 * GL - PN * CjK '1 1

A + B + s 1GL = _ LOG I_ + B - sS

(30)

and s = Isl

AI, PN, C jK i, A and B are defined in Eq. (27). (Note that asecond LOG term arises in the above integral but its total con-tribution from all sides of a closed polygon cancels exactly.)

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Again, the total source velocity potential influence coeffi-cient is obtained by summing the contribution from all four sidesof the panel; i.e.,4

= _ (31)jK BjK ii=l

Symmetry and/or ground-effect treatment is the same as thatdescribed for the doublet contribution.

The expression for B_K for points that are in the far fieldfrom panel K is, using a slmilar approach to that for the doublet(Eq. (29)),

AREA KBjK - PJK (32)

With the above expressions for CjK, BjK substituted in Eq.(20), the zero internal perturbation formulation is similar tothat given by Morino (19), who applied Green's Theorem directlyto the external flow.

3.2.2 Velocity Vector Influence Coefficients3.2.2.1 Doublet Contribution

The velocity induced at control point, J, by a constantdoublet distribution of unit value on panel K is, from Eq. (24),

// i]_ =- Vj (n " V(r))dS (33)JKPane i K

The kernel is one of the terms resulting from the expansionof a triple vector product. Hence, V can be writtenUJK

Panel KBut, Vj V(_)_ = 0.

{(n x Vj) x V(I) + nVj V(1)} dSr

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Hence,

or

Thus,

Pane i K

(n x Vj) x V(1)dS

iUjK .Jr

by Stokes' Theorem.

i-__ r _3drV_jK = _/ r

(34)

(35)

The integral can be performed along the straight line ofeach side of the panel, Figure 16. Hence, the contribution fromside i is

V = a x b * (A + B), (A * B - a b)I_jK " a x b a x b A * B1

This expression is singular when the point, P, is in linewith the side (a x b = 0). However,

a x b a x b = (A * B - a b)(A * B + a b).

And so an alternative expression is

V_JK.

l

= a x b * (A + B) (36)A * B * (A * B + a b)

This becomes singular only when the point, P, lies on theside. The singularity is avoided by applying a finite-core modelon the vortex line. (The core radius can be set by the user(i).)

The total velocity coefficient induced by the unit doubleton the panel is the sum of Eq. (36) over all four sides.

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u

GJ

H

-,-I

O

O4J

,--Ir_

43


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V_j K = _ V _JKi=l 1(37)

For points in the far-field, the integrand in Eq. (33) istreated directly as a point doublet; i.e.,

or

V = - Vj(n K V(1)) AREA_JK r K

V = - AREA * Vj {nK " r_/_JK K r 3

Thus,2 5

V = AREA K * {3 * PNjK * PJK PJK nK}/PJK (38)lajK

The above variables are defined in Eq. (29).

Cases with a vertical plane of symmetry and/or ground effectare treated as before by evaluating the influence of the _9__panel at each im___ point rather than vice versa. With thisapproach the velocity vector contribution must have its componentnor_l_l to the symmetry plane reversed in sign in order to obtainthe actual contribution from the i_ panel acting on the realcontrol point.

The normal velocity influence coefficient, EjK, in Eq. (23)can now be evaluated; i.e.,

EjK = nj V _JK

Where the velocity influence, V_j K, is taken from either Eq. (37)or Eq. (38) depending on the distance of the point, J, from panelK.

3.2.2.2 Source Contribution

The velocity induced at control point, J, by a constantsource distribution on panel K is, from Eq. (24),

// ioj K = - Vj (?)dSPane i K

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and taking the gradient with respect to the coordinates of pointJ, this becomes

v// rrPane i K

(39)

The integral is treated for each side of the panel as in thecase for the velocity potential evaluation. From side i, thecontribution is

VOjK ' = GL * (SM * - SL * m) + CjK n1 1(4O)

Where GL is defined under Eq. (30), and SM, SL and CjK i aredefined under Eq. (27). The total for the panel is the sum overall four sides.

4VOjK = _ V_jK" (41)

i=l 1

For points in the far field, the panel's source distribu-tion is concentrated in a single source at the panel center. Inthis case the source velocity coefficient becomes (from Eq. (39))

AREAK * PJKVjK = 3 (42)

PJKThe above variables are defined in Eq. (29).

Cases with a vertical plane of symmetry and/or ground effectare treated as before by evaluating the influence of the ealpanel at each im_i.q_ point rather than vice versa. With thisapproach the velocity vector contribution must have its component

to the symmetry plane _ in sign in order to obtainthe actual contribution from the ima_g_ panel acting on the realcontrol point.

Finally, the normal velocity influence coefficient for thesource term in Eq. (23) can be evaluated; i.e.,

DjK = nj VajKThe source velocity coefficient, Vo , is taken from either

JKEq. (41) or Eq. (42), depending on the distance of point J frompanel K.

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3.2.3 Matrix AssemblyThe matrix equations, Eq. (20) and (22), can now be formed

by assembling the singularity influence coefficients for all thesurface and wake panels in the configuration acting at the con-trol point of each surface panel in turn. In the VSAERO programthe surface and wake contributions are computed separately: thesurface matrix of influence coefficients, once computed, remainsfixed throughout the treatment of a configuration while the wakecontribution varies with each wake-shape iteration. The wakecontribution is, in fact, recomputed and combined with the wake-shedding panel influence coefficients after each wake-shape cal-culation.

Consider the wake influence in Eq. (20); i.e.,

NW

_W CjLL=I

Now from the boundary condition imposed on the wake (Eq.(7)) the value for _W is constant down each wake column and hasthe value of the 3ump in total potential at the start of thecolumn (from Eq. (I0)); i.e., for the Mth column,

_WM -- CKWPU M - CKWPL M (43)

is constant down the column; KWPUM, KWPL M are the upper andlower surface panels shedding the M th wake column (Subsection3.1.4).

Thus,E E_W M _KWPU M _KWPL M + _ _KWPU M KWPL M

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And the wake influence contribution can be written

NW NWCOL

_ _W CjL = _ IPHCjM * (_KWPUML=I M=I

- UKWPL M)

+ PHCjM * (_KWPU M

(44)

whereNWP M

PHCjM = _ CjLL=I

NWCOL = Total number of wake columnsand

NWP M = Number of wake panels on the M th columnThe result of the summation of the doublet influence coeffi-

cient down a column, i.e., PHCjM, is therefore added to andsubtracted from the associated upper and lower wake sheddingpanel influences (KWPUM, KWP[M) , respectively.

It is tempting to discard the last term of Eq. (44) becausethe quantity (_ - _ ) vanishes in the limit as theKWPU M KWPL M

upper and lower control points on the wake-shedding panels ap-proach the trailing edge in high density paneling. However, withpractical panel densities, this quantity will rarely be zero and,in _act, Youngren et al. (20) noticed that a serious error incirculation results if the term is omitted for thick trailingedges. The term is, in fact, readily evaluated and is combinedwith the source terms to form the known "right-hand side" of theequations.

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3.3 Matrix Solver

With the matrix of influence coefficients formed, the solu-tion routine can proceed. VSAERO employs either a direct or aniterative matrix solver depending on the number of unknowns(i.e., number of panels). The direct solver is Purcell's vectormethod (21) and is used when the number of panels is less than320. When the number of unknowns is greater than 320, then a"blocked controlled successive under- or over-relaxation" (22)technique is used based on the Gauss-Seidel iterative technique.

The iterative technique proceeds by forming inverses ofsmaller submatrices centered around the diagonal and by multi-plying the residual error vector with the stored inverse of eachsubmatrix. The submatrix blocks are formed automatically withinthe code on the basis of complete columns of panels. An optionis provided for the user to specify his own blocks when thearrangement of patches and panels becomes difficult in awkwardgeometrical situations. Placing two strongly interacting panelsin an order which prevents their inclusion in the same block canlead to poor convergence and sometimes divergence in the blockedGauss-Seidel routine.

3.4 On-Body Analysis

The matrix solver provides the doublet (i.e., the perturba-tion velocity potential) value for each surface panel in theconfiguration. These values and the source values are now as-sociated with the panel center points in the evaluation of thevelocity on each panel.

The local velocity is:

V= - V_or

V = V + v (45)

The perturbation velocity, v, is evaluated in the panel'slocal coordinate system; i.e.,

v = VL + VMm + VNn (46)

The normal component is obtained directly from the sourceform, Eq. (12); i.e.,

VN = 4_ (47)

The tangential components, VL, VM, are evaluated from thegradient of _, assuming a quadratic doublet distribution overthree panels in two directions in turn, Figure 17. Thus, whenevaluating the velocity on, say, panel K, the four immediateneighboring panels, NI, N2, N3, N4, are assembled together with

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Z

Z

Z Z

Z

Z

Z Z

,-I(D

O_-,-I

49


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the neighboring _ides of those panels, NSI, NS2, NS3, NS4. Thisinformation is obtained directly from stored neighbor information((NABOR(I,K), NABSID(I,K), I=1,4), K=I,NS), see Subsection 3.1.3.Surface distances are constructed using the SMP, SMQ informationfor each panel (Subsection 3.1.3), noting that the distance fromthe center of panel K to the midpoint of either side 1 or side 3is SMQ K and, similarly, the distance to the midpoint of eitherside 2 or side 4 is SMPK. The additional distance from themidpoint of a side of panel K to the midpoint of the neighboringpanel of that side depends on the NABSID value. For example, theadditional distance to the neighbor, NI, from side 1 of panel Kis SMQNI if NABSID(I,K) is 1 or 3 and is SMPNI if NABSID(I,K) is2 or 4; the former is always the case when the neighbor is in thesame patch as panel K (Figure 17) and the latter will occur whenthe neighbor is on a different patch with a different orienta-tion.

Consider the regular case where the neighboring panels areavailable and are on the same patch as panel K. Then the deriva-tive of the quadratic fit to the doublet distribution in the SMQdirection on panel K can be written

DELQ = (DA * SB - DB * SA)/(SB - SA) (48)

where

SA = - (SMQ K + SMQNI)

SB = SMQ K + SMQN3

DA = (_NI - _K )/SA

DB = (_N3 - _K )/SB

A similar expression based on SMP rather than SMQ and usingneighbors, N4 , N2, instead of NI, N3, gives the gradient DELP inthe SMP direction.

DELQ and DELP provide the total gradient of _ and, hence, byresolving into the and m directions, the perturbation tangen-tial velocity components (Eq. (46)) can be written

VL = -4z * (SMP * DELP - TM * DELQ)/TL

and

VM = -4z * DELQ (49)

The vector, TL, TM, with modulus, SMP, is the center of side2 relative to the panel center expressed in the local coordinatesystem of panel K.

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The complete set of panel neighbors is not always availablefor the above derivative operation. Neighbor relationships arecancelled by the program across each wake shedding line to pre-vent a gradient being evaluated across a potential jump. Theprogram also terminates neighbor relationships between panelswhen one has a normal velocity specified and the other has thedefault zero normal velocity. In addition, the user can preventneighbor relationships being formed across a patch edge simply bysetting different ASSEMBLY numbers for the two patches and,finally, the user can actually terminate neighbor relationshipson a panel-by-panel basis in the input. This is applied whereverthere is a potential jump (e.g., where a wake surface butts up tothe fuselage side) or where there is a shape discontinuity in theflow properties (e.g., a sharp edge) or where there is a largemismatch in panel density from one patch to another.

When a panel neighbor is unavailable on one or more sides ofa panel, then the doublet gradient routine locates the neighborof the neighbor on the opposite side to complete a set of three.For example, if neighbor N2 is not available, Figure 18, theneighbor on the opposite side is N4 with adjacent side NS4.Going to the far side of panel N4, i.e., away from panel K, theneighboring panel is NABOR(MS4,N4) with adjacent sideNABSID(MS4,N4), where

MS4 = NS4 2; i< MS4k< 4If this panel is available then the set of three is com-

plete, the gradient now being taken at the beginning rather thanthe middle of the quadratic. If the panel is not available thenthe program uses simple differencing based on two doublet values.(If no panel neighbors are available, then the program cannotevaluate the tangential perturbation velocity for the panel K.)

With the local velocity known, the pressure coefficient isevaluated on panel K:

2pK= 1 - V K (50)The force coefficient for a configuration or for parts of a

configuration are obtained by summing panel contributions. Thecontribution from panel K is (Figure 19)

AF Kq_ - CPK * AREAK * BK + CfK * AREAk * tk (51)

where

qoo is the reference dynamic pressureCP K is the pressure coefficient evaluated at the panelcenter, Eq. (50)

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Z

.Q,-4.,-Ir_>

0

Z1.40a_-,..4Z

0..Q.a

Z

0

-IJ-;-I0

>0-,-Ir_..Q

0,-4-,-40 >-,-I.l-J

0,-4 _:

_I -,-I

rag,.-i

-,-I

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\\

0

@0_ D_

0

04J

-,-I

0(DO0

,--fID

-,-I

N

53


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AREA K is the panel area projected onto the panel's meanplane

nKtK

Cf K

is the unit vector normal to the panel's mean planeis the unit vector parallel to the panel's mean planeand in the direction of the local (external)velocity; i.e., t = V /IV iK K Kis the local skin friction coefficient

The force vector for a patch is then

K 2Fp . AF K

K=K 1

(52)

where KI, K_ are the first and last panel subscripts in thepatch. The force coefficient vector is

K2 AF KC F = Fp/q_ SRE F = _ _ /SREF

P K=K 1(53)

where S is the reference area specified by the user.REFThe components of the force coefficient vector are printed

out in both wind axes, C L, CD, CS, and body axes, Cx, Cy, C z,(i.e., the global coordinate system), Figure 20.

For symmetrical configurations the total force vector isdoubled prior to printout. This means that the printed sideforce value, i.e., the component normal to the plane of symmetry,is actually twice the side force on one side of the configurationrather than being cancelled to zero for the total configuration.

The skin friction coefficient, CfK, in Eq. (51) is zero forthe first time through the potential flow calculation. Duringthe viscous/potential flow iteration cycle the Cf values arecomputed along the surface streamlines passing tSrough user-specified points. The program then sets the local Cf value oneach panel crossed by a streamline. Panels not crossed bystreamlines have Cf values set by interpolation through localpanels that have been set, provided these are in the same patch.Patches having no streamlines crossing them are left with zero Cfvalues on their panels; consequently, it is up to the user to

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C D

Force Coefficients

C L

--Wind Axes

Wind Axes Side-Wash VectorLies in Global x,y Plane

C s

C

Y

Global CoordinateSystem (Body Axes)

Figure 20. Global and WAnd Axis Systems.

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adequately cover the regions of interest with streamline requests(i). If a patch is folded (i.e., a wing) and/or there is adividing stream surface approaching the patch, then streamlinesshould be requested on both sides of the attachment line.

The contribution to the moment coefficient vector from theK th panel is:

AC tM K q_ SRE F L (54)

where R K is the position vector of the control point on thepanel, K, relative to the moment reference point; L=SSPAN, thereference semispan for the x and z components of moment (rollingand yawing moments); and L=CBAR (the reference chord) for the ycomponent of the moment (pitching moment). CBAR and SSPAN arespecified by the user (i).

The summation is carried out on the basis of patches, com-ponents and assemblies. Summation of the forces and momentsacting on user-specified sets of panels can also be obtained.

Patches with IDENT=I have an additional output of localforce and moment contribution from the summation down each patch.This is useful for "folded" patches representing wing-type sur-faces.

Patches with IDENT=3 include both upper and lower surfacevelocity and pressure output. On these patches the doubletgradient (Eq. (49)) provides the jump in tangential velocitycomponent between the upper and lower surfaces. The programfirst computes the mg_]/l velocity (i.e., excluding the localpanel's contribution) at the panel's center then adds to this onehalf of the tangential velocity jump for the upper surface andsimilarly subtracts for the lower surface.

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3.5 Wake-Shape CalculationWith the panel singularity values known, the velocity vector

can be computed at any point in the flow field using the panelvelocity influence coefficients and summing over all surface andwake panels (Eq. (18)). In the wake-shape calculation procedurethe velocity vector is computed at each wake panel corner pointand each "streamwise" edge; i.e., sides 1 and 3 of a panel,Figure ii, is aligned with the average of its two end velocities.

The wake shape calculation is performed in a marching pro-cedure, starting at an upstream station and progressing throughthe set of vertical wake-grid planes, Figure ii. In each wake-grid plane the full set of (streamwise) wake lines intersectingthat plane is first assembled and the full set of velocitiescomputed at that station. In assembling the set of wake linesfor treatment, certain lines can be left out, e.g., lines onwakes identified as rigid by the user (i); individual lines thathave been identified as rigid, for example, the line where thewing wake butts up to the fuselage side; and lines that arecoincident with other lines that are already in the set.

The step from one wake-grid plane to the next is performedfor all lines before another set of velocities is computed.Thus, for a given set of points, RWi,M; i=l, NL M, in the M thwake-grid plane, the set of points in the next wake-grid planeis:

j JRWi,M+ 1 = RWi, M

J J+ Vi * DX/IVx I; i=l, NL M

l(55)

thwhere NL_ is the _umber of wake lines being moved in the Mwake-@_ia plane; V_ is the mean velocity across the interval forthe i TM line in th_ jL** predictor/corrector cycle.

(vJ Jl 1= i,M + Vi,M+I /2 (56)J is the total number of predictor/corrector cycles in use (de-fault is 2).

o J j-iV is set of Vi,M, and Vj-I l,m+l I,M+I

RW i,M+l

is computed at the point,

In each predictor/corrector cycle the complete set of pointsin each wake-grid plane are moved and the current movement vector

j j-id =RW -RW

1 I,M I,M

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is applied to all the points downstream on each line, before thenext set of velocities is computed. A limiting movement can beapplied as a damper in difficult situations.

When all the wake-grid planes have been processed, the wakepanel parameters are reformed using the new corner points. Themodified set of influence coefficients can then be computed forthe next solution.

3.6 BQundary Layer AnalysisThe solution from the on-body analysis (Subsection 3.4) can

be transferred to the boundary layer analysis. Two integralboundary layer analyses are provided in the VSAERO program: thefirst method, which is described in Reference 23, provides theboundary layer characteristics along computed (external) surfacestreamlines. The location of a point (panel) on each streamlineis provided by the user who is, therefore, responsible for en-suring that a family of streamlines adequately covers the regionsof interest. The method, which is applicable for wings andbodies, includes the effects of surface curvature and streamlineconvergence/divergence under the assumption of local axisymmetricflow. The surface streamline tracking procedure is described in(8) and (24) and is based on a second-order surface streamfunction formulation. The streamline boundary layer calculationcan be expected to break down in regions of large cross flow.However, the procedure has given surprisingly good guidance forthe onset of separation for a wide range of practical problems.

The second method, which is installed in the VSAERO programand which is described in Reference 2, includes a cross-flowmodel. This method is an infinite swept wing code and is appliedin a strip-by-strip basis across wing surfaces, with the userspecifying which strips are to be analyzed.

The boundary layer module is fully coupled with the VSAEROinviscid code in an outer viscous/potential iteration loop,Figure 3. In further iterations, the displacement effect of theboundary layer is modelled in the potential flow calculationusing the source transpiration technique; i.e., the boundarylayer displacement term appears in the external Neumann boundarycondition equation as a local non-zero normal velocity component(Eq. (12)), and is thereby related directly to source singulari-ties in this formulation. The boundary layer calculation alsoprovides a separation point on each streamline, thereby defininga boundary of separation for a family of streamlines. The cri-terion for separation is currently a vanishingly small skinfriction coefficient (in the direction of the external stream-line).

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3.7 Off-BQ_y AnalysesWhen the iterative loops for the effects of wake shape and

boundary layer are complete, the VSAERO program offers optionsfor field velocity surveys and off-body streamline tracking.These options, which are discussed separately below, both use theoff-body velocity calculation, Eq. (18), based on a summation ofthe contributions from all surface and wake panels added to theonset flow.

3.7.1 Velocity SurveyOff-body velocity calculations are performed at user selec-

ted points. Normally, the points are assembled along straightscan lines. These lines in turn may be assembled in planes andthe planes in volumes. The shape of the scan volume is deter-mined by the parameter, MOLD. Two options are currently avail-able (i).

MOLD=I Allows a single point, points along a straightline, straight lines within a parallelogram orwithin a parallelepiped.

MOLD=2 Allows points along radial lines in a cylindricalvolume.A number of scan boxes can be specified, one after the

other. A MOLD=0 value terminates the off-body velocity scan.The location of scan lines within the scan boxes can be

controlled in the input. The default option is equal spacinggenerated along the sides of the box. The location of pointsalong a scan line can be controlled also in the input, but again,the default option is equal spacing.

If a scan line intersects the surface of the configuration,an intersection routine locates the points of entry and exitthrough the surface paneling. The scan line points nearest toeach intersection are then moved to coincide with the surface.Points falling within the configuration volume are identified bythe routine to avoid unnecessary velocity computations. Thesepoints are flagged in the printout. The input parameter, MEET,controls the calls to the intersection routine: MEET=0 (default)makes the routine active, while MEET=I switches it off; clearly,if the selected velocity scan volume is known to be outside theconfiguration volume, then there is no need for the program tocompute where the scan lines intersect the configuration surface.

The velocity routine, VEL, computes the velocity vector ateach point by summing the contributions from all surface panels,wake panels, image panels (if present), onset flow, etc. The

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routine includes near-field procedures for dealing with pointsthat are close to the surface panels or wake panels. The near-field routine involves a lot of extra computation; if, therefore,it is known that a scan box is well clear of the configurationsurface and wake then it is worthwhile turning off the near-fieldroutine to avoid unnecessary computation. This is controlled bythe input parameter, NEAR; NEAR=0 (default) keeps the near-fieldroutine active while NEAR=I switches it off.

3.7.2 Off-Body Streamline Procedure

The numerical procedure for calculating streamline paths isbased on finite intervals, with a mean velocity vector beingcalculated in the middle of each interval, Figure 21. Thevelocity calculation point, RP, is obtained by extrapolation fromthe two previous intervals on the basis of constant rate ofchange in the velocity vector direction; i.e.,

RP = R + .5 s t (57)n nwhere R n is the previous point calculated on the streamline, s_is the present interval length, and t is a projected tangen_vector.

)( >= \{ n Sn-I kn_ - t + (58)+ s 1 n-2 tn-In-2 n-IVn- 1

where tn_ 1 - ]Vn_ll etc. V n-imiddle of the previous interval.

is the velocity calculated in the

RP does not necessarily lie on the streamline path. Thecalculated velocity, V n, at RP, is used to evaluate the nextpoint on the streamline; i.e.,

s n V nRn+ 1 = R n + i Vn} (59)

An automatic procedure is included in the routine whichchanges the interval length, s n, in accordance with the change intangent direction. If a large change in direction is calculated,the value of sn is decreased and the calculation is repeateduntil the change in direction is within a specified amount. Ifthe change in direction is smaller than a certain amount, thenthe interval length for the next step is increased.

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InTErVALENGTHS

tn - 2 S - _" %t t_'_ "*'_'_ '-nRn- 2 ....... Vn- I

Vn.2 n- 1\VELOC ITY CALCULATEDIN MIDDLE OF EACH INTERVALGIVES LOCAL TANGENT VECTOR

STREAMLINE PATH /",,_///

//

NEXT VELOCITYCALCULATIONPOINT

Figure 21. Streamline Calculation.

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4.0 DISCUSSION OF RESULTSCorrelation studies discussed below include exact and nomi-

nally exact solutions of a number of basic test cases, some ofwhich had proved difficult for earlier low-order panel methods.In addition, some comparisons are made between calculated andexperimental data.

4.1 _e_RL_Kia_Figure 22 shows the chordwise pressure distribution at _ =

0.549 on a thin, swept wing at 5 incidence. The wing has a mid-chord sweep of 30 , aspect ratio 6, taper ratio 1/3, and a NACA0002 section. A nominally exact datum solution (25) for the casewas computed by Roberts' third-order method using a 39(chordwise) by 13 (spanwise) set of panels. The present calcula-tions, based on the i = 0 formulation, use i0 spanwise panelswith both 80 and 20 chordwise panels on a "cosine" spacing,giving increased panel density towards leading and trailingedges. Roberts' datum paneling is heavily weighted towards theleading edge: he has, in fact, 4 more panels ahead of the mostforward panels in the present 80 cosine spacing case.

The comparison in Figure 22 indicates that the low- andhigh-order methods give essentially the same solution where thecontrol point density is the same. The present calculationsusing 20 panels chordwise deviate from the datum solution nearthe leading and trailing edges where the control point densitiesare different. Good agreement is restored in those regions bythe 80-panel case which, at the trailing edge at least, has asimilar panel density to the datum case. (Note: not all the 80-panel pressure values are plotted in the central region of thechord in Figure 22.)

4.2 Wing with StrakeThe fact that surface velocities are obtained by numerical

differentiation in the present method might lead to inaccuraciesin awkward situations. So far, no serious trouble has beenexperienced. Any problem that might arise with the derivativescheme (e.g., in a region of mismatched paneling) can usually bealleviated by cutting panel neighbor relationships at that sta-tion , thus forcing a forward or backward differencing (Subsec-tion 3.4). One situation that could be difficult and which hascaused a problem for other methods is the evaluation of thespanwise velocity component, Vy, in the neighborhood of the kinkon a wing with strake. In the case considered in Reference 25,the leading-edge kink occurs at 25% of the semispan and theleading-edge sweep is 75 inboard and 35 outboard. The trail-ing-edge sweep is 9. Again, the wing section is a NACA 0002 andincidence 5 .

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Figure 22.

--1.4Tr _-- ---e-- ROBERTS,DATUR ('THIRD-OEDER')(39 X 1:3)_ REFERENCE25

- I. 2 "11 o PANELS% 20cHo

" I. 0 I'I_ PANELSCp

80 CHORDWISE20 CHORDWISE"SINE" SPACING

PRESENTCALCULATIONS10 SPANWISECOLUMNS

0

0 .2 .4 .6 .8 1.0 X/CComparison of Calculated Chordwise Pressure Distri-butions on a Thin Swept Wing. NACA 0002 Section;

= 5 .

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The chordwise distribution of Vy is plotted in Figure 23 atstations n = .219 and .280; i.e., just inboard and just outboardof the kink, respectively.

Two higher-order datum solutions are plotted (25) fromRoberts' third-order method and from Johnson and Rubbert's (17)second-order method. These datum solutions agree on the outboarddistribution but disagree by a small amount on the inboard cut.The disagreement might be caused by the different ways the inter-polation is set up across the kink.

The present calculations are in close agreement with thedatum solutions and favor the second-order solution near theupstream part of the inboard cut. The case was run as part of a"shakedown" exercise of the code and used 946 panels in a 44(chordwise) by 20 (spanwise) array with 66 panels on the tip-edgesurface. The chordwise paneling was on a "sine" distributionwith density increasing towards the leading edge. Such a highnumber of panels was probably not needed for this case, but thecalculation took approximately 2 minutes on a CDC 7600.

4.3In the past, low-order solutions for internal flows have

suffered badly from "leakage" problems in between control points.The present code was, therefore, applied to two nacelle cases.The first of these is a simple through-flow nacelle formed byrotating a NACA 0010 about the nacelle axis such that the minimumarea is one fourth of the inlet and exit areas, Figure 24. Thisexample, taken from Reference 18, shows how the earlier, firstgeneration, low-order analysis, because of the leakage, cannotmatch the internal flow calculated by the higher-order methods.However, the low-order, VSAERO solution (with the internal poten-tial set to zero) is in close agreement with the high-ordersolutions and evidently does not have this leakage problem.

The second nacelle case considered is also an open nacelle,but with an exit diameter 0.3 of the chord. The nacelle surfaceis generated by a NACA 0005 section tilted leading edge out by 5 from the x-axis.

Two higher-order datum solutions (25) are shown in Figure25. for the chordwise pressure distribution. These are due toHess and also Roberts.

Roberts' solution shows a slightly lower pressure inside theduct compared with Hess's solution. The present calculations,based on the %i = _ formulation, use I0 panel intervals aroundhalf the circumference and with two chordwise distributions, 26cosine and 48 sine, around the complete section. Both resultsare in very close agreement with the datum solutions with atendency for slightly lower pressure inside the duct comparedwith Hess's solution. This could be caused by the slightly

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.6

0 I .2 .6 .7 .8 .9 1.04 .5

-.6T = .219

Vy -----4k-- ROBERTS HIGHER-ORDER DATUM---e---- RUBBERT SOLUTIONSo6 U _) PRESENT CALCULATIONS

PIECEWISE CONSTANT SINGULARITY PANELS

-.6 -._ = .280Figure 23. Comparison of Calculated y Component of Velocity at

Two Stations on a Wing with Strake. NACA 0002 Sec-tion; _ = 5 .

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-2

/--ICp _ I

/.0 0.2II II\

I--EXACT (HIGH-ORDER AXI- \

HESS SYMME_ _IC) JO BRISTOWB HESS (LOW-ORDER SOURCEA VSAERO C )

0.4 0.6 0.8 1.0X/C

FROMREF.18

Figure 24. Comparison of Internal Pressure Calculated usinga Range of Methods Showing Leakage of Low-OrderFirst Generation Model.

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-I

25HESS _ HIGH-ORDER----- ROBERTS )PRESENT CALCULATIONS:CHORDWISE PANELLING0 26 COSINE SPACING | 10 PANELS ONA 48 SINE SPACING W HALF CIRCUMFERENCE

0

0 .2 .4 .6 .8 1.0X/LFigure 25. Comparison of Calculated Pressure Distributions ona Nacelle with Exit Diameter/Chord .3. NACA 0005

Section; _ = 0.

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smaller duct cross-section area generated by the present i0 flatpanel representation. Again, there is no leakage problem withthis formulation.4.4 Wing-Flap CasesCases with slotted flaps have proven troublesome for some

earlier, low-order panel methods, especially those using a speci-fied doublet or vorticity singularity distribution and employing"Kutta points" where tangential flow is specified just downstreamof the trailing edges (e.g., (23)). In closely interactingmulti-element configurations the interference of the downstreamelement acting on the "Kutta point" of the upstream element canresult in a circulation error; in addition, the solution can besensitive to the _ of the prescribed loading distribution onthe upstream element--this is quite different from that on theregular single-element wing section.

Two wing-flap cases are considered below for the VSAEROprogram. The first case is one of the exact two-dimensional,two-element cases provided by Williams (26). Figure 26 shows theclose agreement between the VSAERO calculations and the exactsolution for configuration B on both the main element, Figure26(a) and the flap, Figure 26(b). Part of the high aspect ratioaneling in the VSAERO calculation is shown in Figure 26(c), andas ii0 panels chordwise on the main and 60 on the flap.

The second wing-flap case involves a part-span flap. Theinset in Figure 27 shows a general view of a wing-flap configura-tion at 16 incidence. The wake shape is shown after one itera-tion cycle. The wing has an aspect ratio of 6 and the basicsection is a NACA 0012. The 30% chord Fowler flap is deflected30 and the flap cutout extends to _ = .59 on the semispan. Thesurfaces of the wing tip edge, the flap edge and the end of thecove are covered with panels.

Vortex separations were prescribed along the wing tip edgeand the flap edge. The present calculations of surface pressuresare compared with experimental data (27) at two stations inFigure 27. A section through the flap at _ = .33 is shown inFigure 27(a) and one just outboard of the flap edge at n = .6 inFigure 27(b).

Boundary layer calculations have not been included in thiscalculation yet, and so the agreement between the results at thisincidence (16 ) is probably too close. The lower peak suction inthe flap data, Figure 27(a), may be caused by interference from anearby flap support wake.

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...I

O6. $

Z l_, _ CO t_lo

! !CO nj0F- c_(-_ b.l_:


82/114

to

I

0

I I

0c._ ILl

la.,I

0

I

0

0

'70

0X

o

o

o

0

00

A

0

-,-I


83/114

71


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I0

0

-M

II

r0

0-,-I0u_

rO

04J

0

m

r_0010-,-I

_o

0 0

0._1_0 I

0 rOL3_

(D

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0o

B

m

m

_LI'_

D

D

m

m

--0

0IIc- ,-_oo 0O0-r--tu c_0

.-Q t_

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4.5The case of 4:1 spheroid at 5 angle of attack was run to

test the prediction accuracy of body cases. Figure 28 shows thevery good agreement between the VSAERO prediction and the exact(non-lifting) solution. The calculation has i0 panels around thehalf-body and 20 longitudinally, with the case being run with thevertical plane of symmetry. Small discrepancies in the Cp ~ Zplot are probably partly due to the control points being on theinscribed polygon (i.e., a plot of Cp ~ G would look better, butthe Cp ~ Z plot is more practical for the general case.

4.6 Wing-Body CasesConcave corners have posed a problem to some low-order

methods in the past, making it necessary to use high panel densi-ties. Also, such a region might cause a numerical differentia-tion scheme to misbehave. Two wing-body cases are consideredhere. Figure 29(a) shows a longitudinal cut through a pressuredistribution on a wing-body case from Reference 28 for e = 4 .The cut passes just above the wing-body junction. The presentcalculations compare well with other solutions and are in goodagreement with experimental results.

A spanwise cut through the present calculations at X/L = .48is shown in Figure 29(b) together with the surface geometryshape. The pressure distribution passes smoothly from the bodysurface onto the wing surface even though relatively large panelsare present on the fuselage side (only 6 around the half sec-tion).

In this calculation the wing and body form one completevolume in which _ = _.1

The second wing-body case is the Swearingen Metroliner windtunnel model. Figure 30 shows the VSAERO paneling and it will benoticed that there has been no treatment of the intersectionregion; i.e., the wing and fuselage panel models are from_ wing-alone and body-alone runs. This is not a recom-mended practice; however, it is interesting to observe that--atleast in the present configuration--the results are good andcompare very well with experimental data right into the junctionregion. (Note that the doublet gradient procedure has not usedpanel information that falls inside the fuselage or wing.)

Comparisons between experimental (29) and calculated pres-sure distributions are made in Figure 31, 32 and 33 at two span-wise stations, y = 9.0 (y/b/2 = .22) and y = 18.0 (y/b/2 = .43).The location of the body side is at y = 5.0 (y/b/2 = .12). Thecalculations include one wake relaxation iteration. Superimposedon each plot is the geometry of the local cross section.

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C P

CP

-0.20

-0.16

-0.12

-0.08

-0.04

0.2

0.4

0.6

0.8

1.

x = 0.07822

EXACTQ VSAERO

-.3! I I-.2 -.i 0

x = 0.75806

0.I 0.2

(a) Cp _ z.

0.4

-I .0 I i I I t-0.8 -0.6 -0.4 -0.2 0(b) Cp ~ x.

Figure 28. Comparison of Calculated and Exact PressureDistributions on a 4:1 Spheroid at _ = 5

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!

I (_,o Q !

001v)o

0-,-I0om

0It_

-,-I

o0

0

II

4J

0-,-I4JtO

-,-Iu_0U

0I

-,-I

0

0.,-I

-,-I

-,-I

_ mul II

01.4-,-t

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CROSSSECTION SHAPE ------

Cp VALUES I(PRESENTCALCULA-TIONS)

UPPER SURFACE vLOWER SURFACE

a

-I.5

m

dmm

--5

0

.5

D,m

a

fmm

am,

! .I

\7

il VP

/^ A

Immmm .w & D

vV V

A A A

U

V

0 .25 .5 .75 1.0 1.25Y

. _L__.__

1.5

(b) Spanwise Cut through Present Calculations atX/L = .48Figure 29. Concluded.

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r_r_o

E_O

f__J

o

r_

78


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-2.e

-1 .S

-1.0-- Z

e --

O.S --

1.0

-4

-6

-8

ii i

(_) [XPERZMEHT (2!I I--- USAERO _. LO_,[eueeL'euar_csuer_

4_ 44 46I -O .i .2 .3 .4

48 58 52 $4X' I5 .6 .7 .8 ._ 1.0

X/C

(a) Buttline Cut Y = 9.0.

$6

-I.S

-I.0

Cp -e.S

.O.S

1.0

Figure 3i.

--m a"

!-_.. Q-

Z-a

-4-

-6-

r_ v

42 44 4G 48

XPERZMENT (29)I--- VS_ERO ueeel su_'A

fA LOUt[R SU_F_r_

BSO S8 S4X

0 .I .2 .3 .4 .5 .6 .7 .8 .9 1.0/C

(b) Buttline Cut Y = 18.0.Comparison of Calculated and Experimental PressureDistributions on the Swearingen Metroliner Wind TunnelModel at e = 0.0, M = 2.0.

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-3 -- Z

I --

-4-

-6-

-8

-le4Q

%--rl

___ EPE_InE.*i(29___!___--- VS.E_O WUPI

!I

4_ 44 46i0 .i .2 .3

48 Se 52X

SURFACE

.4 .5 .6 .7 .8

LOWER SURFACE

I.9 1.0X/C

(a) Buttline Cut, Y = 9.0.

- s- F

Cp -4

e

Figure 32.

0 , q EXPERIMEMT (29)J--- VSAERO UPPER SURFACELOWER SURFACE

(b) Buttline Cut, Y = 18.0.

Comparison of Calculated and Experimental PressureDistributions on the Swearingen Metroliner Wind TunnelModel at e = 12 , M = 2.0.

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-le

-8

-6

-4

-2

-- z -e

-4

-6

-6

-re

2

4o

ii f -

!IAI 1_"6 _'-Ia _' -

I r fO EXPERZMENT (29} UPPER

"-" VSAERO _ LO_ERr, I IPREDICTED SEPt ATION., [

0 0 0

4a 44 46 48 5eI x0 .I .'2 ;3 .4 .5 .6 .7

52 54I

.8 .9 1.0

5F

(a) Buttline Cut, Y = 9.0.

-8--

--&

cp --4 --

0-- --

Figure 33.

2 -

-2

-4

-6-

I_i iI

I ""q '_)..,.I _'I

PREDICTED SEPARATION

_D- _g_a- - _ -,',- -'-

42 44 46 41 50x

!0 .I .2 .3 .4 .5 .6 .7

SZ 54

I.6 .9 1.0(b) Buttline Cut, Y = 18.0.

Comparison of Calculated and Experimental PressureDistributions on the Swearingen Metroliner WindTunnel Model at _ = 16 , M = 2.0.

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Figure 31 shows the comparisons at _ = 0. The experimentalresults at the inboard station indicate a small separation zonenear the trailing edge and show slightly lower Cp values near theleading edge compared with the calculated data. The latter doesnot include a viscous/potential iteration for this case. Theoutboard station shows a much closer agreement between the re-sults (Figure 31(b)).Figure 32 shows the comparisons at _ = 12 . The results arein remarkably good agreement except for the separated zone nearthe trailing edge, which has grown to approximately 20% of thechord.Figure 33 shows the comparison at _ = 16 . The calculationsinclude one viscous potential iteration and the figure indicatesthe calculated location of separation which is in reasonableagreement with experimental data. Figure 34 shows the calculatedsurface streamlines and separation zone on the wing for thiscase.

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83

0

0.)

0(D

t_-,-t

4_0_J0

0-;-4

r_C_ ,

-_1 r_,.--I

0c/1 ,--I'CSrd

_1 -,-.-I


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5.0 CONCLUSIONSThe formulation of the VSAERO program is described and a

number of basic test cases examined. The panel method module ofthe program, based on simple quadrilateral panels with constantdoublet and source, has produced results of comparable accuracyto those from higher-order methods given the same density ofcontrol points. The calculated results compare very well withexact, nominally exact and also experimental data cases. Prob-lems associated with earlier low-order panel methods do notappear with the present formulation based on internal Dirichletboundary conditions. Moreover, the low computing cost of thesimple panel method makes it practical for applications to non-linear problems requiring iterative solutions, e.g., for wake-shape and surface boundary layer effects.

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6.0 REFERENCESi. Maskew, B., "Program VSAERO, A Computer Program for Calcula-

ting the Nonlinear Aerodynamic Characteristics of ArbitraryConfigurations, User,s Manual", NASA CR-__, November1982.

2. Dvorak, F.A., Woodward, F.A. and Maskew, B., "A Three-Dimen-sional Viscous/Potential Flow Interaction Analysis Method forMulti-Element Wings", NASA-_, July 1977.

3. Maskew, B., "Calculation of the Three-Dimensional PotentialFlow around Lifting Non-Planar Wings and Wing Bodies using aSurface Distribution of Quadrilateral Vortex Rings", Lough-borough University of Technology, Dept. of Transport Tech.Report TT 7009, September 1970.

4. Maskew, B., "A Quadrilateral Vortex Method Applied to Con-figurations with High Circulation", Paper No. i0, Presentedat the NASA Workshop on Vortex-Lattice Utilization, LangleyResearch Center, NASA SP-405, May 1976.

5. Maskew, B. and Woodward, F.A., "A Symmetrical SingularityModel for Lifting Potential Flow Analysis", AIAA J. _,September 1976.

6. Lamb, H.Y., _[IL_iL[l_mics, 6th Ed., Dover Publications, NewYork, Article No. 58, 1945.

7. Maskew, B., Rao, B.M. and Dvorak, F.A., "Prediction of Aero-dynamic Characteristics for Wings with Extensive Separa-tions", Paper No. 31 in 'Computation of Viscous-InviscidInteractions', AGARD CP-291, February 1981.

8. Maskew, B., Vaidyanathan, T.S., Nathman, J.K. and Dvorak,F.A., "Prediction of Aerodynamic Characteristics of FighterWings at High Angles of Attack", Analytical Methods Report8405a, Prepared for Office of Naval Research, Arlington, VA,under Contract N00014-82-C-0354, March 1984.

9. Maskew, B., "Influence of Rotor Blade Tip Shape on Tip VortexShedding--An Unsteady Inviscid Analysis", Proc. 36th AnnualForum of the AHS, Paper 80-6, May 1980.

i0. Summa, J.M., "Advanced Rotor Analysis Methods for the Aero-dynamics of Vortex/Blade Interactions in Hover", Paper No.2.8, Presented at 8th European Rotorcraft and Powered LiftAircraft Forum, Aix-en-Provence, France, August-September1982.

ii. Maskew, B. and Dvorak, F.A., "Prediction of Dynamic StallCharacteristics using Advanced Nonlinear Panel Methods",Analytical Methods Report 8406, Final Report, Prepared under

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Contract F49620-82-C-0019 for AFOSR/NA, Bolling AFB, D.C.,April 1984.12. Maskew, B., "Calculation of Two-Dimensional Vortex/SurfaceInterference using Panel Methods", NASA _, November

1980.

13. Hess, J.L. and Smith, A.M.O., "Calculation of Non-LiftingPotential Flow about Arbitrary Three-Dimensional Bodies",Douglas Aircraft Company Report No. ES 40622, March 1962.

14. Martinsen, E., "The Calculation of the Pressure Distributionon a Cascade of Thick Airfoils by Means of Fredholm IntegrandEquation of the Second Kind", NASA TT F-702, July 1971.

15. Maskew, B., "A Surface Vorticity Method for Calculating thePressure Distribution over Aerofoils of Arbitrary Thicknessand Camber in Two-Dimensional Potential Flow", Loughbor

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Vsaero Theorical Manual