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L

NASA

Contractor

Repo_

_237_7 _177

ProgramA Computer Nonlinear of Arbitrary

VSAERO_eoryProgram Aerodynamic Configurati__ forC_lating Characteristics

Document

Brian

Maskew

.....

CONTRACT SEPTEMBER

NAS2-11945 1987

N/LS/X

_i -_

- Z 2_:-:_

....

-j_

- _L

r- ....

--IZ

__

.Jk

,?_j.. ,/,,_---i-

L. r

Z _ ./___---i..

Z .....

Z -_._ZZ ---

..... ---}

_=---_._..'--_.iii: __.-zz:-

......

__--Z177

_

Z---=__

IIZI_ZZZ[

....

- Zi_-Z

----_

Z

NASA

Contractor

Report

4023

ProgramA Computer Nonlinear of Arbitrary

VSAEROProgram Aerodynamic Con_gurations for

TheoryCalculating Characteristics

Document

Brian Analytical Redmond,

Maskew Methods, Washington Inc.

Prepared for Ames Research under Contract

Center NAS2-11945

National Aeronautics and Space Administration Scientific and Technical Information Office 1987

TABLE

OF

CONTENTS

section

aS_

No.

LIST LIST 1.0

OF OF

FIGURES SYMBOLS

iii vi

INTRODUCTION i.I 1.2 1.3 General Objectives Strategy .................... .................. ................... 1 1 1

2.0

FORMULATION 2.1 2.2 2.3 Flow Equation ................. Boundary Conditions ............. Choice of Singularity Model .......... PROCEDURE ................ Description ........... 3 6 8 16 16 16 21 25 26 ....... Coefficient ..... ..... Coefficients ..... ..... 41 44 46 48 48 57 58 59 59 6O 33 40 31

3.0

NUMERICAL 3.1

Geometrical 3.1.1 3.1.2 3.1.3 3.1.4 Patch Panel Panel Wakes of

. _ . . ............. Geometry ............. Neighbors ............ ................. Coefficients Influence Contribution Contribution Influence Contribution Contribution ............

3.2

Matrix 3.2.1

Influence

Velocity 3.2.1.1 3.2.1.2

Potential Doublet Source Vector Doublet Source Assembly

3.2.2

Velocity 3.2.2.1 3.2.2.2

3.2.3 3.3 3.4 3.5 3.6 3.7 Matrix

Matrix Solver

.................

On-Body Analysis ............... Wake-Shape Calculation ............ Boundary Layer Analysis ............ Off-Body Analysis ............... 3.7.1 3.7.2 Velocity Off-Body Survey ............ Streamline Procedure

.....

i

TABLE

OF

CONTENTS

(CONCLUDED)

_No.4.0 DISCUSSION 4.1 4.2 4.3 4.4 4.5 4.6 5.0 6.0 OF RESULTS ............... 62 62 62 64 68 74 74 84 85

Swept Wing Wing with Nacelles Wing-Flap Body Alone Wing-Body

.................. Strake ............... ................... Cases ................ .................. Cases ...............

CONCLUSIONS REFERENCES

.................... ....................

APPENDICES: A: B: SUBROUTINE BIQUADRATIC FLOW CHART ................ .............. 88 96

INTERPOLATION

ii

LIST

OF

FIGURES

F_No.1 2 3 4 5 6 7 8 9 i0 ii 12 13 Section Three Outline General of through the of

Titlethe Idealized Internal in of the the Flow Flows Model .... Program . . . . .

_No.4 9 17 18 19 20 ...... 22 23 ......... Edges ..... 27 28 29 30 of the Surface 35 for One 36

Possible Steps

the

VSAERO

Arrangement

Configuration

Configuration Patch

Hierarchy

..........

Conventions Defining

............. Patch Surface

Sections Panel Panel Panel

Geometry Neighbor Neighbors

............... Information across Scheme Patch

Wake-Grid-Plane Wake Model Arrangement used in

...........

.............. the Evaluation

Integrals

................. of the Surface Integrals Panel .............. of Symmetry and .................. of the Line Ground-Plane

14

Evaluation Side of a Treatment Problems Evaluation

15

39 ...... Panel Patch 49 on Panel ..... ...... ....... 43

16 17

Integral

Evaluation of Perturbation K (All Neighbors Available as Panel K) ................ Evaluation of Perturbation K when a Neighbor is not Force Global Contribution and Wind Axis from

Velocity on and on Same

18

Velocity Available Panel K

52 53 55 61

19 20 21

Systems

Streamline

Calculation

...........

iii

LIST

OF

FIGURES

(CONTINUED)

F__

No.22 Comparison Distribution of

TitleCalculated on a Thin Chordwise Swept Wing Pressure ...... of VelocStrake; 65

63

23

Comparison of Calculated y-Component ity at Two Stations on a Wing with NACA 0002 Section .............. Comparison of Internal Pressure using a Range of Methods Showing Low-Order First Generation Model Comparison tions on NACA 0005 Comparison Solution (a) (b) (c) Main

24

Calculated Leakage .......

of

66

25

a

of Calculated Pressure DistribuNacelle with Exit Diameter/Chord.3; Section .............. Calculation with (26) Two-Element Exact Case

67

26

of VSAERO for Williams

B 69 70 71

Element

.................

Flap ..................... Paneling ................... Comparison of Pressure Distributions Aspect Ratio 6 Wing with Part-Span = 16 on an Fowler Flap

27

(a) (b)

Section Section

at at

_ _ of

= =

.33 .6

............. .............. and Exact Spheroid at on a Wing-Body .6 Pressure _ = 5 Config-

72 73

28

Comparison Distributions Pressure uration (a) (b)

Calculated on a 4:1

.

.

75

29

Distributions at _ = 4 ,

M_

=

Above Wing-Body Junction Pressures ...... Spanwise Cut through Present Calculations Swearingen Comparison sure Wind Metroliner of Calculated Wind and Tunnel Model .

. .

. .

76 77 78

3O 31

Experimental

Pres-

Distributions Tunnel Model Cut, Cut, Y Y = =

on the Swearingen at e = 0.0, M_ = 9.0 18.0 ............ ...........

Metroliner 2.0 79 79

(a) (b)

Buttline Buttline

iv

LIST

OF

FIGURES

(CONCLUDED)

F_No. 32

TitleComparison of Calculated and Experimental Pressure Distributions on the Swearingen Metroliner Wind Tunnel Model at e = 12 , M = 2.0 (a) (b) Buttline Buttline Cut, Cut, Y Y = = 9.0 18.0 ............ ............

8O 8O

33

Comparison of Calculated and Experimental Pressure Distributions on the Swearingen Metroliner Wind Tunnel Model at e = 16 , M_ = 2.0 (a) (b) Buttline Buttline Cut, Cut, Y Y = = 9.0 18.0 ............ ...........

81 81Zone Model on at

34

Calculated Streamlines the Swearingen Metroliner = 16 ...................

and

Separation Wind Tunnel

83

APPENDIX A-I

A Flow Chart of Viscous/Potential Flow Flow Flow ary Main Program Iteration (Deck Loop VSAERO MVP); ....... in the

89

A-2

Chart of Subroutine ..................... chart Layer

Potential

9Oin the BLCTRL) Bound....

A-3

for Major Subroutines Analysis (Subroutine

95

APPENDIX

B

B-I

Biquadratic

Interpolation

i00

v

LIST OF SYMBOLS AREA B ,C JK JK Panel area velocity potential influence coefficient point of panel J for a uniform distribusource and unit doublet, respectively, on in a panel normal component of point of panel J by a source or doublet, the panel center the x,y,z system (Eq. (13)) panels local axes,

Perturbation at the control tion of unit panel K Diagonal vector

D D ,E JK JK

Influence coefficient for the velocity induced at the control uniform distribution of unit respectively, on panel k

E i,j,k K KWPU,I KWPL

Position vector of a panel corner in coordinate system with origin at panel Orthogonal respectively, Factor Program shedding for unit vector system along of the global coordinate the local contribution to

subscripts of upper and lower a wake column (Figure ii)

surface

Orthogonal unit vector system in a local normal to the surface and , m tangential N NABOR i NABSID i R r r S s SMP,SMQ V Number of panels Program subscript of neighboring panel side of

region;

n is

on side the

i

Program subscript of adjacent panel in side i (Figure 9) Position vector of a panel of a point

neighboring

corner relative

point

in the G.C.S. ds,

Position vector of the surface Modulus of Surface Distance (solid) along

to an element,

of the the

configuration

surface lengths (Figure 8)

Panel half-median Velocity vector

vi

LIST

OF

SYMBOLS

(CONTINUED)

VV

Modulus

of

V velocity

Perturbation Wake surface

W x,y,z

Cartesian coordinates (G.C.S.)--body-fixed

in the frame of

Global Coordinate reference

System

Incidence Y Vorticity Boundary Local ively, Doublet Total

of

G.C.S.

x-axis

degrees

vector layer

density displacement in the local density thickness and m directions, coordinate system respect-

coordinates of the panel singularity velocity

potential contribution, or perturbation com-

Velocity ponent Density Source

potential

singularity

density

V

Laplacian

operator,

. _ I_-_ +

_ j_-_+

k _z

Dot N Cross

product product

of

vectors of vectors

vii

LIST

OF

SYMBOLS

(CONCLUDED)

c e

Center Edge Inner of the boundary layer

iJ,K L N P S U WOO

region to panel J,K

Pertaining Lower Normal Pertaining Pertaining Upper surface surface

component to to the the point, surface P; also used as a subscript

Pertaining Reference

to onset

the

wake

outer

edge

of

the

boundary

layer

condition

viii

1.0

INTRODUCTION

i.i

General

VSAERO is a computer program for calculating the subsonic AEROdynamic characteristics of arbitrary configurations having Vortex Separation and strong Vortex/Surface interaction. This document describes the theory and numerical procedures used for the method. An earlier report (I) describes the input and use of the computer program.

1.2The initial objectives of the new program were to analyze general wing configurations with multiple part-span, high-lift devices and to give special attention to edge-vortex separations and to close vortex/surface interactions. Later objectives brought in the modeling of fuselage and canard surfaces leading to the present capability for arbitrary configurations including complete aircraft.

1.3At the beginning of the program development there was only one clear cut choice for approaching the solution to the above nonlinear problem. This was an iterative viscous/potential flow calculation using a potential flow panel method coupled with integral boundary layer methods, with wake-shape iteration calculations included in the potential flow analysis. No other approach appeared to offer the feasibility of achieving the above objectives. Finite difference field methods, for example, treating more general flow equations, could not be developed within a reasonable time scale for application to the complex geometry of multiple part-span, high-lift devices. In the meantime, the surface integral approach of the panel method offered a powerful and practical engineering tool which had the capability to represent the complex geometries involved in the objectives. Furthermore, the low computing time of the panel method was an attractive feature in view of the complex iterative loops made necessary by the nonlinear effects of viscosity and wake location. In the chosen strategy for the development of the VSAERO program the capabilities of two earlier programs were combined: (i) VIP3D (2) offered a viscous/potential flow iterative method for transport wings with full-span, high-lift devices; and (ii) QVORT, a quadrilateral-vortex panel method (3) and (4), offered an iterative wake relaxation procedure. Neither method had a completely satisfactory singularity model or geometry package for the new objectives. Accordingly, in the development of VSAERO these features were given prime consideration.

1

Various singularity models were investigated for the new rogram including symmetrical singularities (5), high-order ormulations and numerical integration. Finally, a low-order formulation based on internal Dirichlet boundary conditions was chosen. This formulation is described in Section 2. The numerical procedure for the method, which employs quadrilateral panels of constant doublet and source distributions, is described in Section 3. The description includes details of the extended geometry package for the VSAERO program. Correlation studies are examined in Section 4; these include a number of basic test cases involving exact, nominally exact and also experimental data.

2

2.0

FORMULATION 2.1Flow Equation

In order to obtain a solution to the flow problem the surface of the configuration is assumed to be at rest in a moving airstream. Regions of the flow field that are dominated by viscous and rotational effects are assumed to be confined to thin surface boundary layers and wakes. The rest of the flow field is assumed to be inviscid, irrotational and also incompressible. Figure 1 shows a streamwise section taken through a wing and its wake. The section indicates the surface, S, of the wing, the surface (upper and lower) of the wake, W, and an outer surface, S_, which encloses the complete flow problem at infinity. The total surface, S + W, divides space into two regions: the external region which contains the flow field of interest and the inner region which contains a fictitious flow. Velocity potential fields, _ and _i, are assumed to exist satisfying Laplace's equation in the flow field and in the inner region, respectively; i.e., V25 and V2_ i 0 in the inner region. = 0 in the flow field

=

Next, apply Green's Theorem to the inner and outer regions and combine the resulting expression (see Lamb (6)). This yields the following expression for the velocity potential, Cp, anywhere in the two regions, expressed in terms of surface integrals of the velocity potential and its normal derivative over the boundary surface:

_W_S

n S_W_S where r is the distance from the the surface and n is the unit pointing into the fluid.

(V#

- V#i)dS

(i)

point, normal

P, to the element, vector to the

dS, on surface

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

s

r

u

Figure

i. Section thrugh the Idealized Flow MOdel.

density density velocity.

represents represents

the the

local local

jump jump

in in

potential the normal

and the source component of

S_,

Examine Eq. (I) separately as indicated in Figure I.

over

the

three

surfaces,

S,

W,

The surface at infinity, S_, can be regarded as a large sphere centered on P. The local conditions at that boundary consist only of the uniform onset velocity, V , the disturbance at that distance due to the configuration having essentially disappeared. The surface integrals in Eq. (i) taken over S_ then reduce to _p; i.e., the velocity potential at point, P, due to the onset flow. The upper and lower wake surfaces, W, are assumed to be

infinitesimally close i.e., thin wakes. (A described in References

to each other for the present problem; simplified thick wake representation is 7 and 8.) In this case the corresponding

upper and lower elements can be combined and the i term for this part of the integral disappears. Furthermore, if we neglect entrainment into the wake surface, then the jump in normal velocity component, n (re U - VCL), is zero and so the source term in Eq. (I) disappears for the wake contribution. (Note that entrainment modeling _Q/_ be included given a distribution of the entrainment normal velocity.) With the above conditions, Eq. (i) becomes:

S

n (V S

-

Vi)dS

- _L ) n W The normal, n, on surface, W,

V (1) dW +

_ P (2)

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 the point, P, is excluded from the integral by a local spherical deformation of the surface centered on P. For a smooth surface the local deformation is a hemisphere and the local contribution obtained in the limit as the sphere radius goes to zero is

%(-i)p, surface on the

which is half the at P. Note that the side of the surface the sign example, can if P be is

local first on changed on the

jump in potential which p

potential across in the expression lies, or, more

the is con-

veniently, side. For (2) becomes:

for the point iIiH_ide surface

on the other of S, then Eq.

_P

-

4_

/fS-P 4_ S

(_

-

i )

n

V(

)dS

-

%(

-

i) p

i//r n w

(re

-

V i)dS

(3)

2.2

Boundary

Conditions a number of known boundin the case of the wake-S, the external Neumann

A solution to Eq. (3) must satisfy ary conditions which can be imposed--or implied at the outset. On the surface, boundary condition must be satisfied. nwhere V N is the to the surface. no transpiration; layer displacement V ffi-V N n VS

on

S

(4)

resultant V N is zero non-zero effect,

normal component of velocity relative for the case of a solid boundary with values are used to model boundary entrainment and also inflow/outflow

for engine inlet/exhaust modeling. V s is the local velocity of the surface; this may be composed of several parts due to body rotation (e.g., pitch oscillation (9), or helicopter blade rotation (10)), arbitrary motion (Ii), body growth (12), etc. For the steady present in Hence, n The wake surface, CU W, - eL' V cannot on W = -V on S a load a and so the (5) doubcondia problem body-fixed the surface coordinate of the configuration and so Vs is is zero. assumed system

N

support must

let distribution, tion.

satisfy

zero-force

Consider an element, dW, of the wake surface normal, n. The local vorticity vector associated let distribution, CU y The presence Joukowski @L' has the density - L ) the V,

with with

local unit the doub-

= -n

x V (u

(6) element, is (from dW, the in the Kutta-

elementary force exerted on of the local mean velocity, Law): 6F = pV x ydW since

where p is the it follows that

local

density.

And

the

force

must

be

zero,

V Substitute for y from V and expand n V V( Cu x {n Eq. x _

x 7=0. (6): (@[email protected] ) } = 0

@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 distribution in the direction of the local mean flow is zero; in other words, the wake doublet distribution is constant along mean streamlines in the wake surface. The doublet value on each streamline is, therefore, determined by the condition at the point where the streamline "leaves" the surface, S. Clearly, at the outset of the problem with both the doublet distribution and wake location unknown, an iterative approach is necessary to obtain a solution. When a converged solution is obtained the upstream edge of the wake, W--and hence the trailing edge of the surface, S--carries no load and so the Kutta trailing-edge condition will be satisfied. At the outset, therefore, the Kutta condition is im_ simply by shedding the trailing-edge potential jump (@u - @.) as a constant down each "streamwise" line on an initially _r escribed wake surface. An explicit Kutta condition in which the doublet gradients on the upper and lower surfaces at the trailing edge are equated to cancel has been used on occasion, but in the formulations described below, this does not appear to be necessary.

7

2.3In infinite

Choice

of

Singularity

Mode] field can be constructed of doublet and source from an distribu-

principle, a given flow number of combinations

tions on the surface, S, each combination producing a different flow in the inside region. In practice, however, there may be only a small number of singularity combinations which are suitable as a basis for a well behaved numerical model; i.e., one that is accurate, convenient to use and robust (insensitive to user abuse). A unique combination of singularities can be obtained in Eq. (3) in a number of ways. One way is to specify one of the singularity distributions and to solve for the other using boundary conditions only on one side of the boundary. There are several examples of this: in the source-only formulation (13) the doublet distribution is set to zero; in the doublet-only formulation, e.g., (3) and (4), the source distribution is set to zero; in program VIP3D (2), the doublet distribution (or vorticity) is specified and the method solves for the source distribution; the reverse of the latter has also been used, i.e., specifying the source distribution--usually related to the thickness distribution--and solving for the doublet or vorticity distribution. Another way of achieving a unique combination of singularities is to apply certain constraining relationships on the singularity distributions. The "symmetrical singularity method" (5) is one example of this and was examined briefly at the start of the VSAERO program development. One characteristic that separates "good" singularity models from "bad"ones from the numerical point of view is that the flow field generated in the inner region by a "good" model is rather benign and related to the boundary. This is usually not the case for a "bad" model. In other words, when passing through the boundary, S, the jump from the internal flow to the external flow should be small--thus requiring a minimum of perturbation from the singularities (Eq. (3)). "Bad" singularity methods were observed to have very large internal cross flow between the upper and lower surfaces of wings and required high panel densities in order to achieve acceptable accuracy in the flow solutions. This resulted in a push towards high-order formulation which, for subsonic flows at least, proved quite unnecessary. One way of achieving "good" characteristics is to treat the internal flow directly in Eq. (3). This is, in fact, another way of obtaining a unique singularity distribution--in this case by specifying boundary conditions on both sides of the surface, S. Earlier methods (14), (15) specified the _L_ on the inside surface of S. Alternatively, the inner velocity potential, i , can be specified directly in Eq. (3). This is referred to as an internal Dirichlet boundary condition. Three possible internal flows were examined for the VSAERO program, Figure 2 . Two of

@ 8 II II 8 -,.-t I-.-I 0 -,-t ,.q 0 d d 0 u 0 I.-1 X ! II -,-I 0 U O P_ r-t II _/ GJ _J ;..-I ,q -,'-I g 0 -;-II'--I

X

8 /

0 r-I

1-1 I.-I

I

I

-;-..t -,-I -;-I

.P O _J B

I

\E_ Z B _r_

z

CXl N

O rD II .,-I

o II rd

-,-I

I-I 0 -;-I

these flow was this

are

described

below,

the

third

one,

in

which i

the

internal

was aligned examined in worked very

with the wing chord line (e.g., a two-dimensional pilot program well, the formulation was not so, it would in which the component in be a user a useful could

= -xV_ cos _ ) only. Although as convenient as a in a possuitable

the other two. sible general internal (i) flow Zero

Even scheme for Flow each

member select

complicated

configuration.

Inside the interior stagnation doublet-only codes (e.g., exterior Neumann boundary formulation, set i = condition References condition, constant = which was 3 and 4) V n = 0, say in

Consider first implied by earlier and which used the 0. Eq. For the present (3), givlng

_p

=

0

_ 4z // iS-P

- V( )ds -

I ll!4_ JJ S r

n .

V#dS

(8) W

at

The doublet the surface

distribution is and the source

now the total distribution surface, condition

velocity is now S. (Eq.

potential the total This must 5); i.e.,

normal satisfy

velocity of the external

the fluid at the Neumann boundary n V in = -V N

This

term

can

be

used

directly

the

second

integral

of

Eq.

(8).

is

For the solid boundary zero and the source term

problem, disappears

i.e., zero leaving

transpiration,

VN

4---_ S-P

n

V

)dS

-

p

(U W

-

eL )

n

V(1)dW

+

_op

=

0

(9)

I0

This is by far the simplest formulation for the lifting case with no transpiration. It is, in fact, even simpler than the original zero-lift source method based on the external Neumann boundary condition (13). In a low-order formulation (i.e., constant doublets on flat panels) the method has some minor problems associated with numerical differencing for velocities and with the effect of panel arrangement on accuracy (16). These problems can be alleviated by separating into a known part and an unknown part--the latter being as small as possible. The most convenient breakdown is to use the onset flow potential as the known part; i.e., = _ + . The known part, _, and its velocity field, V_, are used directly in the formulation. The solution for and the subsequent numerical differentiation for surface velocity are then less prone to numerical error. For the general case with transpiration, inlet flows, jets and unsteady conditions, the source term must be present and so the advantage of the doublet-only formulation diminishes. In the meantime an alternative "zero internal perturbation" formulation had become popular in other methods and is considered below.

(ii) The earlier

Internal internal by Johnson

Flow

Equal

to

the

Onset

Flow i = _, Bristow _ and %p was used and Grose = _, Eq.

Dirichlet boundary and Rubbert (17) formulations.

condition, and also . = i

(18) in high-order (3) becomes:

With

S-P

W

-

//r S

" " (V#

- V#_)dS

= 0

(I0)

where _, the perturbation now the doublet density;

potential i.e.,

on

the

exterior

surface,

is

(ii) The source distribution 4_ = in n this (V formulation V_oo) is

ii

Expanding external

this Neumann

expression boundary

and substituting condition in Eq.

for (5):

n

V

from

the

4zg The source distribution is

=

VN

n

V

o at the outset.

therefore

established

Again, for solid boundary conditions, VN is zero; for the more general case here, VN may have two parts representing (i) boundary layer displacement effect using the transpiration technique (2), and (ii) inflow/outflow for engine inlet/exhaust modeling. Thus, 4_ (12) the first potenit is updated by term has a posi-

=-u as

(V e 6*)

+

VNORM

-

n

V for

tial

The boundary flow solution;

layer on

term is set subsequent calculations. and negative

to zero iterations

coupled boundary tive value for

layer outflow

The . VN__ U for infI_w.

_Vl

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 formulation in the VSAERO program. The reason for its better behavior probably arises from the smaller jump in the flow condition from the inner to the outer flow; i.e., the singularity strengths are smaller. However, there are situations where this is not the case; e.g., a wing at large angle of attack or a powered nacelle in zero onset flow. In other words, it may be desirable to provide a capability that allows the internal flows to be specified independently of the onset flow so that they can comply more closely with the shape of each component. Once the doublet solution velocity field. is known, potential For the then field, present Eq. (3) can be

applied to determine of which gives the Eq. (3) becomes:

the velocity

the gradient formulation,

ffS

ffS

V(!)dwr W

+ _p

(13)

12

Where

the

doublet,

_,

and

source,

o,

are

taken

from

Eqs.

(ii) and (12), respectively and _w = Cu - eL is the wake doublet distribution. The factor, K, has different values depending on the 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 is on the outside, and K = -2_ if P is on the inside; if P is at a crease in the surface, then K takes the value of the appropriate solid angle contained at the crease. In each case when P is on the surface, then the surface integral terms in Eq. (13) exclude P. If the potential field has then the velocity field can be differentiation; i.e., Vp been computed at a mesh generated using local of points numerical

= - VCp

(14)

Alternatively, the velocity field can be obtained by taking the gradient of Eq. (13) directly with respect to the coordinates of point P, thus forming the source and doublet velocity kernels. (Thls is the present approach in VSAERO.)

ffS S

1 V (r)dS

-

f/W

_w V(n

V(?))dW

i

+ Voo

(15)

(The kernels will be developed further in Section 3.) The offbody velocity field is used in the wake-relaxation cycle, in offbody streamline calculations and for general flow-field information. The discussion so far has been concerned with thick configurations having a distinct internal volume enclosed by the surface, S. If pars of the configuration are extremely thin (e.g., thickness/chord ratio < 1%, say) or are wing-like and remote from the area of interest, then these parts may be represented by open surfaces. When the upper and lower parts of the [email protected] are brought together, the corresponding upper and lower elements can be combined and Eq. (2) becomes

13

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

W If the normal velocity is _ through the sheet then the term n (V_U - V_L) = 0 and the source term disappears. (This is _ot a necessary step; the source term could be left in the equation for simple modeling of thickness effects.) Removing the source term does not preclude the use of non-zero normalvelocity velocity then be at the surface; be 9_II]IQ_ written: the only restriction through the surface. is that the normal Equation (16) can

_p

=

Ys (u

_n

V (r)dS

//+ w jump in

(17) _W n " V (r)dW + _p

where sheet.

_

-

1 4_

-eL

) is

the

total

potential

across

the

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

S

//_wV0 X H E-t

,

X_-11 _0

_U O_ Zl

8I

.1-1 0 I o 0 ,--t

Zl

-l---"

d m

0

-,-I

_H u'l

0 m

17

0 ._.1 I--ILl O0 Z 0 ILl Z

N

0 0 _..j 0 ....i t..o _-co >-o0 t_ u_ 0 C) 0 -,..t 4J

>.0 4J E t_

, C) ..I-} -,-I C)

,.-I Q_ ,-_ 0

,.-I Q_

-,-t

35

,-1 G) r_

O _J

@ O O

,-t r_ tl t_

H U

cr_ (_ 4-1 u_ O O -,-I 4-1 r_ ,-t rd :> F_

,-4

/

t_

/36

where RNUM DNOM PN PJK A B PA PB SM SL AM BM A1 b = SM * PN * PB " nK - Rc K * (B * PA - A * B * PB) * SM 2

= PA = PJK = Rcj = = = = = lai Ibl

+ PN 2 * A

PN z * SL PN 2 * SL s m - m m * SL the

+ A1 + A1

* AM * BM

(= a = PA

{ - A1

x

(a

x s)})

* SM

= s = a = b = AM is

- AL

* SM

(= n vector

(s a)) of the side, jth Figure control 14 point (i.e., b =

position to the

relative as)

end

of

this

The subscript, i, has and s are constructed bi_ 1 There is control point

been omitted from the above variables; for each side in turn, noting that

a, a1

b =

a limiting J from the

condition mean plane

as PN--the projected of panel K--goes to

height zero:

of

Lim

(CjK .) l I

PN+0 +

_ on the edge of the strip = _ within the strip (DNOM0) points 0-, to all the the

(DNOM=0)

and are

The positive negative to reversed.

sign occurs for the left. If PN

right above

of the signs

side on

The total unit doublet on (27).

velocity the panel

potential influence is the sum for all

coefficient four sides,

for from

the Eq.

37

CjK

=

_ i=l

CjK i

(28)

The combined required in the

four arc-tangent functions into a single arc tangent; to ensure the resultant angle range, 2_.

in this summation however, extreme is in the correct

can be care is quadrant

For plane contribution to ty strength as the calculation control tionship cal to

of symmetry and/or ground effect problems the CjK from image panels (having the same singularithe real panel) is evaluated simply by repeating for the _ panel influencing the images of possible of panel panel K because the geometrical K and the point J is and image of point J relaidenti(Figure

point J. This is between the image that of the actual

15). Hence, for a symmetrical would be four contributions treatment is more convenient

problem in ground effect there for each panel, Figure 15. This than considering the image panels is the y z = 0 plane. height and (i). = 0 plane in Thus, for orientation

directly. The vertical plane of symmetry the G.C.S., while the ground plane is the ground effect problems the configuration above the ground require care during is input

The above expression to panel K. When panel J far field, then a far-field n that over the the quantity, panel. Then r

for CjK is remote formula in Eq. and PNjK * PJK

used when panel J is close from panel K, i.e., in the is used under the assumption is essentially constant

r 3 , r = PJK

(25)

AREA

K (29)

CjK

=

where PN nK unit area is normal of panel to K panel (see K Subsection the 3.1.2) control points, K

JK = = K = =

PJK the the

n K AREA PJK

IPjKI, and J

the

distance

between

Equation

(29)

is

the

contribution

from

a

point

doublet

at

MQ"-sRc_)K The orientation with quantity, Subsection 3.1.2) and factor which has a

nK and SMQ)K, used is when "size" PJK > RFF (SMP + is the of panel * RFF is the user-defined, far-field value of 5 in the program. This

(SMP(see + K radius value

default

38

r_ o II 0 L9 U .4-I I o II N t/l O o (D

_,-_ c_ -,-I O O c} ,.-I O

u) u_ O (9

\\J

_ J

J U X

O fl# ,-t H I ,._ O L_ C;

f

\\

_J \

\

\ \\ \

\ \

\ii

Q_

\\I

\

O

\!fl# i

\ \\

\ \ \,--I G) I

! (Mtr..)J

o_ O

\

\Ill ,--I

\\ \

-,-I r0 O_ 4-4,--I O (D -,-I E_(u HI> _o O {.)

-,-4 U

,

/

ig r_

\

I

\\ _ _m O

\

U

39

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

and on a signifiuse the if he the input

3.2.1.2 The uniform

Source

Contribution influence the panel, coefficient K, is, from for Eq. a unit (21),

velocity potential source distribution

on

B JK

=

//ir dS Pane i K of the panel as in the case

The integral is of the doublet.

treated Thus,

for each side for one side

Yb

BjKi

=

Y=Ya _ =0

_ rl

r2

1

where

r I,

r 2,

Ya'

Yb

are

defined

under

Eq.

(26).

This

becomes:

BjK

' =1

A1

*

GL

-

PN

*

CjK

'1

(30)

where GL = _S

LOG

A I_

+ +

B B

+ -

s 1 s

and

s

=

Isl

AI, PN, C jK i, second LOG term tribution from

A

and arises

B

are defined in the above of a closed

in Eq. integral polygon

(27). but cancels

(Note that a its total conexactly.)

all

sides

4O

Again, the total source velocity potential influence coefficient is obtained by summing the contribution from all four sides of the panel; i.e., 4 BjK = _i=l BjK i

(31)

Symmetry described for

and/or ground-effect treatment the doublet contribution.

is

the

same

as

that

from (Eq.

The expression for B_K for panel K is, using a slmilar (29)),

points approach

that to

are in the far field that for the doublet

AREA BjK PJK

K (32)

With the (20), the zero that given by to the external

above expressions for CjK, BjK substituted in Eq. internal perturbation formulation is similar to Morino (19), who applied Green's Theorem directly flow.

3.2.2 3.2.2.1 The doublet

Velocity Doublet

Vector

Influence

Coefficients

Contribution induced at control of unit value on point, panel K J, is, by from a constant Eq. (24),

velocity distribution

V]_ JK

=-

//Pane i K the Hence,

Vj

(n

"

V(r))dS

ifrom be the written

(33)

The of a

kernel vector

is

one

of

terms V

resulting can UJK

expansion

triple

product.

{(n Panel But, Vj V(_)_ = 0. K

x

Vj)

x V(I)

+

nVj

V(1)} r

dS

41

Hence,

(n

x

Vj)

x

V(1)dS

(34)

Paneor

i

K

UjK .Jr

iby Stokes' Theorem.

Thus,

iV_jK = __ _/ r _3dr r (35)

each side

The integral side of the i is

can panel,

be

performed Figure 16.

along Hence,

the the

straight contribution

line

of from

V I_jK "1

= a x

a b

x

b a

* x b

(A

+

B),

(A

*

B A

*

a B

b)

with

This the

expression side (a x

b

=

is singular 0). However,

when

the

point,

P,

is

in

line

a

x

b

a

x

b

=

(A

*

B

-

a

b)(A

*

B

+

a

b).

And

so

an

alternative

expression

is

V _JK.l

= A *

a B

x *

b

* (A

(A + B) * B + a

(36) b)

This side. on the (i).) The on the

becomes

singular

only

when

the

point, a be

P,

lies the

on

the

The singularity vortex line.

is avoided (The core

by applying radius can

finite-core set by

model user

total is

velocity the sum

coefficient of Eq. (36)

induced over all

by four

the

unit

doublet

panel

sides.

42

u

GJ H

-,-I

O O 4J ,--I r_

43

(37) V_j K = _ i=l V _JK 1

For treated

points directly

in the far-field, as a point doublet; V _JK = Vj(n K

the

integrand i.e., AREA K

in

Eq.

(33)

is

V(1)) r

or

V _JK

=

-

AREA K

*

Vj

_ {nK

" r_ / r3

Thus,2 5

V

lajK

=

AREA

K

*

{3

*

PNjK

*

PJK

PJK

nK}/PJK

(38)

The

above

variables

are

defined

in

Eq.

(29). ground effect of the _9__ With this its component

Cases with a vertical plane of symmetry and/or are treated as before by evaluating the influence panel at each im___ point rather than vice versa. approach the velocity vector contribution must have nor_l_l to the symmetry the actual contribution control point. plane from reversed the i_ in sign panel

in order to obtain acting on the real

can

The now

normal velocity influence be evaluated; i.e.,

coefficient,

EjK,

in

Eq.

(23)

EjK

=

nj

V

_JK

Where or K. Eq.

the (38)

velocity depending

influence, on the

V_j

K,

is of

taken the

from point,

either J,

Eq. from

(37) panel

distance

3.2.2.2 The source

Source

Contribution induced on panel at K control is, from point, J, Eq. (24), by a constant

velocity distribution

Voj K

=

-

// iVj (?)dS Pane i K

44

and taking the J, this becomes

gradient

with

respect

to

the

coordinates

of

point

v// rrPane i K The integral is case for the velocity contribution is treated for potential each side evaluation. of the panel as From side in i, VOjK ' = 1 GL * (SM * SL * m) + CjK n 1

(39)

the the

(4O)

Where GL is defined under defined under Eq. (27). The all four sides.

Eq. (30), total for

and the

SM, panel

SL and is the

CjK i are sum over

4 VOjK = _ i=l V_jK" 1 (41)

tion this

For points in is concentrated case the source

the far field, the in a single source velocity coefficient

panel's source distribuat the panel center. In becomes (from Eq. (39))

AREAK VjK = 3 PJK The above variables are defined in

*

PJK (42)

Eq.

(29). ground effect of the eal With this

Cases with are treated as panel at each approach

a vertical before by im_i.q_ point

plane of evaluating rather

symmetry and/or the influence than vice versa.

the velocity to the symmetry the actual contribution control point. Finally, term

vector plane from

contribution must _ in sign the ima_g_ panel

have its component in order to obtain acting on the real

source

in

the normal Eq. (23)

can

velocity influence coefficient be evaluated; i.e.,

for

the

DjK The Eq. (41) panel K. source or Eq. velocity (42),

=

nj

VajK Vo JK the , is taken of from point J either from

coefficient, on

depending

distance

45

3.2.3

Matrix

Assembly

The matrix equations, Eq. (20) and (22), can now be formed by assembling the singularity influence coefficients for all the surface and wake panels in the configuration acting at the control point of each surface panel in turn. In the VSAERO program the surface and wake contributions are computed separately: the surface matrix of influence coefficients, once computed, remains fixed throughout the treatment of a configuration while the wake contribution varies with each wake-shape iteration. The wake contribution is, in fact, recomputed and combined with the wakeshedding panel influence coefficients after each wake-shape calculation. Consider the wake influence in Eq. (20); i.e.,

NW _W L=I CjL

Now

from

the

boundary

condition

imposed

on

the

wake and of

(Eq. has the

(7)) the value for _W is constant the value of the 3ump in total column (from Eq. (I0)); i.e., for

down each wake column potential at the start the Mth column,

(43) _WM -- CKWPU M CKWPL M

is

constant

down

the

column; shedding

KWPUM, the M th

KWPL M wake

are column

the

upper

and

lower surface 3.1.4). Thus,

panels

(Subsection

E

E

_W M

_KWPU M

_KWPL M

+

_

KWPU M

_

KWPL M

46

And

the

wake

influence

contribution

can

be

written

NW

NWCOL

_ L=I

_W

CjL

=

_ M=I

IPHCjM

*

(_KWPUM

-

UKWPL

M)

+

PHCjM

*

(_ KWPU M

(44)

where NWP M PHCjM = _ L=I CjL

NWCOL and NWP M The result down a

=

Total

number

of

wake

columns

=

Number

of

wake

panels

on

the

M th

column coeffito and shedding

cient

of the column,

summation i.e.,

of PHCjM, ,

the doublet influence is therefore added lower wake

subtracted from panel influences It is quantity and the tempting (_

the associated (KWPUM, KWP[M) to discard - _

upper and respectively. last term vanishes

the )

the upper proach

of Eq. in the

(44) limit

because as the apwith

KWPU M lower control trailing edge

KWPL M points on the in high density

wake-shedding paneling.

panels However,

practical panel densities, this quantity will rarely be zero and, in _act, Youngren et al. (20) noticed that a serious error in circulation results if the term is omitted for thick trailing edges. The term is, in fact, readily evaluated and is combined with the source terms to form the known "right-hand side" of the equations.

47

3.3

Matrix

Solver influence VSAERO coefficients formed, the employs either a direct on the solver number of is Purcell's soluor an

tion

With the routine

matrix of can proceed. solver panels).

iterative matrix (i.e., number of

depending The direct

unknowns vector

method (21) and is used when the number of panels is less than 320. When the number of unknowns is greater than 320, then a "blocked controlled successive underor over-relaxation" (22) technique The smaller plying is used based on the Gauss-Seidel iterative technique.

iterative submatrices the residual

technique proceeds centered around error vector with

the the

by forming inverses of diagonal and by multistored inverse of each automatically of panels. own blocks within An option when the

submatrix. the code on is provided arrangement geometrical in an order lead to poor Gauss-Seidel

The submatrix blocks are formed the basis of complete columns for the user to specify his

of patches and panels becomes difficult in awkward situations. Placing two strongly interacting panels which prevents their inclusion in the same block can convergence and sometimes divergence in the blocked routine.

3.4

On-Body

Analysis

The matrix solver provides the doublet (i.e., the perturbation velocity potential) value for each surface panel in the configuration. These values and the source values are now associated with the panel center points in the evaluation of the velocity on each panel. The local velocity is: V= or V The perturbation coordinate system; velocity, i.e., v The normal Eq. (12); component i.e., = VL is + = V v, + v is evaluated in the (45) panel's V_

local

VMm

+

VNn directly from the

(46) source

obtained

form,

VN The gradient

=

4_ VM, are doublet evaluated distribution 17. four from

(47) the over

tangential components, VL, of _, assuming a quadratic in the two directions velocity on, NI, N2, in say, N3, N4,

three panels evaluating neighboring

turn, panel are

Figure K, the assembled

Thus, when immediate with

panels,

together

48

Z Z Z Z

Z Z

Z Z

,-I (D O_ -,-I

49

the neighboring _ides of those panels, NSI, information is obtained directly from stored ((NABOR(I,K), NABSID(I,K), I=1,4), K=I,NS), Surface distances are constructed using the for each panel (Subsection to the the SMPK. 3.1.3), noting the center of panel K is SMQ K and, similarly, side 2 or side 4 is midpoint panel of additional of a side that side distance

NS2, NS3, NS4. This neighbor information see Subsection 3.1.3. SMP, SMQ information the distance 1 from or side 3 of either from the

that

midpoint of either side distance to the midpoint The additional distance of

of panel K to the midpoint depends on the NABSID value. to the neighbor, NI, from

the neighboring For example, the side 1 of panel K

is SMQNI if NABSID(I,K) is 1 or 3 and 2 or 4; the former is always the case same patch as panel K (Figure 17) and the neighbor is on a different patch tion.

is SMPNI if NABSID(I,K) is when the neighbor is in the the latter will occur when with a different orienta-

Consider the regular case where the neighboring available and are on the same patch as panel K. Then tive of the quadratic fit to the doublet distribution direction on panel DELQ where = K can (DA * be SB written DB * SA)/(SB SA)

panels are the derivain the SMQ

(48)

SA SB DA DB

= = = =

-

(SMQ K + -

+

SMQNI)

SMQ K (_NI (_N3

SMQN3 _K )/SA _K )/SB

A neighbors, the SMP

similar

expression instead

based of

on NI,

SMP N3,

rather gives

than the

SMQ

and

using in

N4 , N2, direction.

gradient

DELP

DELQ and DELP provide resolving into the and tial velocity components VL and VM The 2 relative of system vector, to panel the K. = TL, -4z TM, * = -4z *

the total gradient of _ and, m directions, the perturbation (Eq. (46)) can be written (SMP * DELP TM * DELQ)/TL

hence, by tangen-

DELQ with center modulus, expressed SMP, in is the the center of

(49) side

panel

local

coordinate

5O

The complete set of panel neighbors is not always available for the above derivative operation. Neighbor relationships are cancelled by the program across each wake shedding line to prevent a gradient being evaluated across a potential jump. The program also terminates neighbor relationships between panels when one has a normal velocity specified and the other has the default zero normal velocity. In addition, the user can prevent neighbor relationships being formed across a patch edge simply by setting different ASSEMBLY numbers for the two patches and, finally, the user can actually terminate neighbor relationships on a panel-by-panel basis in the input. This is applied wherever there is a potential jump (e.g., where a wake surface butts up to the fuselage side) or where there is a shape discontinuity in the flow properties (e.g., a sharp edge) or where there is a large mismatch in panel density from one patch to another. When a panel neighbor is unavailable on one or more sides of a panel, then the doublet gradient routine locates the neighbor of the neighbor on the opposite side to complete a set of three. For example, if neighbor N2 is not available, Figure 18, the neighbor on the opposite side is N4 with adjacent side NS4. Going to the far side of panel N4, i.e., away from panel K, the neighboring panel is NABOR(MS4,N4) with adjacent side NABSID(MS4,N4), where MS4 = NS4 2; i< MS4k< 4

If this panel is available then the set of three is complete, the gradient now being taken at the beginning rather than the middle of the quadratic. If the panel is not available then the program uses simple differencing based on two doublet values. (If no panel neighbors are available, then the program cannot evaluate the tangential perturbation velocity for the panel K.) With evaluated the local velocity on panel K: CpK= known, the pressure coefficient is

1 - V 2 K

(50)

The force coefficient configuration are obtained contribution from panel K AF K

for a configuration by summing panel is (Figure 19)

or for parts contributions.

of a The

(51) q_ where is the reference pressure Eq. (50) dynamic pressure evaluated at the panel CPK * AREAK * BK + CfK * AREAk * tk

qoo

CP K

is the center,

coefficient

51

0 ..Q .a

Z

0

-IJ -;-I 0

>Z 0 -,-I r_ ..Q

0,-4 -,-4 0 > -,-I .l-J 0 ,-4 _:

.Q ,-4 .,-I r_

>_I -,-I 0 rag ,.-i

Z1.4 0

-,-I

a_-,..4 Z

52

@ 0_ D_ 0

0 4J -,-I

0 \ \ (D O 0

,--f ID

0

-,-I

N

53

AREA K

is the plane is the

panel

area

projected

onto

the

panel's

mean

n

K

unit

vector

normal

to

the

panel's

mean

plane

tK

is the unit vector parallel to the panel's and in the direction of the local velocity; i.e., t = V /IV i K K K is the local skin friction coefficient

mean plane (external)

Cf K The force vector for a patch is then

K2 Fp . K=K 1 AF K (52)

where patch.

KI, K_ are the first The force coefficient

and last vector is

panel

subscripts

in

the

K2 CF P = Fp/q_ SRE F = _ K=K 1

AF K _ /SREF (53)

where

S

REF

is

the

reference of the

area force

specified coefficient

by

the

user. are C x, printed Cy, C z,

The

components

vector

out in both wind (i.e., the global For symmetrical doubled prior to force value, i.e., is actually twice rather than being

axes, C L, coordinate

C D, CS, system),

and body axes, Figure 20.

configurations the total force vector is printout. This means that the printed side the component normal to the plane of symmetry, the side force on one side of the configuration cancelled to zero for the total configuration.

The

skin

friction

coefficient,

CfK,

in

Eq.

(51)

is

zero

for

the first time through the potential flow calculation. During the viscous/potential flow iteration cycle the Cf values are computed along the surface streamlines passing tSrough userspecified points. The program then sets the local Cf value on each panel crossed by a streamline. Panels not crossed by streamlines panels that Patches values have have Cf values been set, set by provided interpolation these are in them it through the same with the local patch. zero user Cf to

having no streamlines crossing on their panels; consequently,

are left is up to

54

CD

Force

Coefficients

CL

--Wind

Axes

Wind Lies

Axes in

Side-Wash Global x,y

Vector Plane

Cs

C

Y Global System Coordinate (Body Axes)

Figure

20.

Global

and

WAnd

Axis

Systems.

55

adequately cover the regions of interest with streamline requests (i). If a patch is folded (i.e., a wing) and/or there is dividing stream surface approaching the patch, then streamlines should be requested on both sides of the attachment line. The panel contribution is: to the moment coefficient vector from

a

the

K th

AC MK

t

q_

SRE F

L

(54)

where R K panel, K, reference and yawing component specified The ponents acting

is the position vector relative to the moment semispan for the x and moments); and L=CBAR of the moment (pitching by the user (i).

of the control point on the reference point; L=SSPAN, the z components of moment (rolling (the reference chord) for the y moment). CBAR and SSPAN are

summation is and assemblies. on user-specified

carried out on the basis of patches, comSummation of the forces and moments sets of panels can also be obtained. have an additional from the summation patches representing output of down each wing-type local patch. sur-

force This faces.

Patches with IDENT=I and moment contribution is useful for "folded"

Patches with IDENT=3 include both upper and lower surface velocity and pressure output. On these patches the doublet gradient (Eq. (49)) provides the jump in tangential velocity component between the upper and lower surfaces. The program first computes the mg_]/l velocity (i.e., excluding the local panel's contribution) at the panel's center then adds to this one half of the tangential velocity jump for the upper surface and similarly subtracts for the lower surface.

56

3.5With

Wake-Shape the panel

Calculation singularity values known, the velocity vector

can be computed at any point in the flow field using the panel velocity influence coefficients and summing over all surface and wake panels (Eq. (18)). In the wake-shape calculation procedure the velocity vector is computed at each wake panel corner point and 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 cedure, starting at the set of vertical grid that plane plane the is full first calculation an upstream wake-grid is performed station and planes, Figure wake full left (i); in a marching proprogressing through ii. In each wakelines intersecting set of velocities

set of (streamwise) assembled and the In assembling lines can be by the user rigid, for fuselage that are

computed at that station. for treatment, certain wakes identified as rigid have been identified butts with up other as wing wake coincident

the set of wake lines out, e.g., lines on individual lines that the line where that the are and lines in the set.

example, side; already

to the lines

The step from one wake-grid plane to the next is performed for 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 th wake-grid plane, the set of points in the next wake-grid plane is: j RWi,M+ 1 = J RWi, M +

JVi * DX/IVx

JI;l

i=l,

NL M

(55)

where NL_ is the _umber of wake lines [email protected]_ia plane; V_ is the mean velocity the i line in th_ jL** predictor/correctorTM

being moved across the cycle.

in the interval

th M for

i

=

(v Ji,M

+

Vi,M+I

Jl 1

/2

(56)

J is fault

the total is 2). o

number

of

predictor/corrector

cycles

in

use

(de-

J is set of Vi,M, and V

j-i is I,M+I computed at the point,

j-I RW

V

l,m+l

i,M+l In each

in

each predictor/corrector wake-grid plane are

moved j

cycle and

the complete the current j-i

set of movement

points vector

d1

=RWI,M

-RWI,M

57

is applied to all the points downstream on each next set of velocities is computed. A limiting applied as a damper in difficult situations.

line, before the movement can be

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

wake The for

3.6

BQundary

Layer

Analysis

The solution from the on-body analysis (Subsection 3.4) can be transferred to the boundary layer analysis. Two integral boundary layer analyses are provided in the VSAERO program: the first method, which is described in Reference 23, provides the boundary layer characteristics along computed (external) surface streamlines. The location of a point (panel) on each streamline is provided by the user who is, therefore, responsible for ensuring that a family of streamlines adequately covers the regions of interest. The method, which is applicable for wings and bodies, includes the effects of surface curvature and streamline convergence/divergence flow. The surface (8) and (24) and under streamline is based the assumption of local axisymmetric tracking procedure is described in on a second-order surface stream

function formulation. The streamline boundary layer calculation can be expected to break down in regions of large cross flow. However, the procedure has given surprisingly good guidance for the onset of separation for a wide range of practical problems. The second method, which is installed in the and which is described in Reference 2, includes model. This method is an infinite swept wing code in a strip-by-strip basis across wing surfaces, specifying which strips are to be analyzed. VSAERO program a cross-flow and is applied with the user

The boundary layer module is fully coupled with the VSAERO inviscid code in an outer viscous/potential iteration loop, Figure 3. In further iterations, the displacement effect of the boundary layer is modelled in the potential flow calculation using the source transpiration technique; i.e., the boundary layer displacement term appears in the external Neumann boundary condition equation as a local non-zero normal velocity component (Eq. (12)), and is thereby related directly to source singularities in this formulation. The boundary layer calculation also provides a separation point on each streamline, thereby defining a boundary of separation for a family of streamlines. The criterion for separation is currently a vanishingly small skin friction coefficient (in the direction of the external streamline).

58

3.7

Off-BQ_y

Analyses

When the iterative loops for the effects of wake shape and boundary layer are complete, the VSAERO program offers options for field velocity surveys and off-body streamline tracking. These options, which are discussed separately below, both use the off-body velocity calculation, Eq. (18), based on a summation of the contributions from all surface and wake panels added to the onset flow.

3.7.1

Velocity

Survey

Off-body velocity calculations are performed at user selected points. Normally, the points are assembled along straight scan lines. These lines in turn may be assembled in planes and the planes in volumes. The shape of the scan volume is determined by the parameter, MOLD. Two options are currently available (i). MOLD=I Allows line, within a single point, points straight lines within a a parallelepiped. along a straight parallelogram or

MOLD=2

Allows volume.

points

along

radial

lines

in

a

cylindrical

A other.

number of scan A MOLD=0 value location of

boxes can be terminates the scan lines

specified, one off-body velocity the scan

after scan. can

the

The

within

boxes

be

controlled in the input. The default generated along the sides of the box. along a scan line can be controlled also the default option is equal spacing. If a scan intersection

option is equal spacing The location of points in the input, but again,

an

line intersects the surface of the configuration, routine locates the points of entry and exit

through the surface paneling. The scan line points nearest to each intersection are then moved to coincide with the surface. Points falling within the configuration volume are identified by the routine to avoid unnecessary velocity computations. These points 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 if the selected velocity scan volume configuration volume, then there is compute where the scan lines intersect The velocity routine, point by summing the panels, image panels switches it off; clearly, is known to be outside the no need for the program to the configuration surface. the velocity vector at from all surface panels, onset flow, etc. The

each wake

VEL, computes contributions (if present),

59

routine includes near-field that are close to the surface field routine involves a lot it is known that a scan box surface and wake then it is routine to avoid unnecessary the input parameter, NEAR; routine active while NEAR=I

procedures panels of extra is well

for dealing with points or wake panels. The nearcomputation; if, therefore, clear of the configuration by

worthwhile turning off the near-field computation. This is controlled NEAR=0 (default) keeps the near-field switches it off.

3.7.2

Off-Body

Streamline

Procedure

The numerical procedure for calculating streamline paths is based on finite intervals, with a mean velocity vector being calculated in the middle of each interval, Figure 21. The velocity calculation point, RP, is obtained by extrapolation from the two previous intervals on the basis of constant rate of change in the velocity vector direction; i.e., RP where R n is the is the present vector. previous interval = R + .5 s t on a (57) the streamline, s_ projected tangen_

n

n

point length,

calculated and t

is

t

=

\{ s

n n-2

+ Sn-I s n-I

)(is

kn_ 1

-

t n-2

>

+

tn-I

(58)

Vnwhere middle RP calculated point on tn_ of 1 the does

1 ' etc. V n-i the velocity calculated in the

]Vn_ll previous not

interval. lie at RP, i.e., on is the used streamline to evaluate path. the The next

necessarily V n,

velocity, the streamline;

sn Rn+ 1 = Rn +

Vn (59)

i Vn}

An

automatic

procedure

is

included

in

the

routine

which

changes the interval length, s n, in accordance with tangent direction. If a large change in direction the value of sn is decreased and the calculation until the change in direction is within a specified the change in direction is smaller than a certain the interval length for the next step is increased.

the change in is calculated, is repeated amount. If amount, then

6O

STREAMLINE

PATH

/

",,_// InTErVAL LENGTHS/ //

tn - 2 Rn2 .......

S

-

_"

%t

t_'_ Vn-

"*'_'_ I

'-n NEXT VELOCITY CALCULATION POINT

Vn.2

n-

1

\VELOC ITY CALCULATED IN MIDDLE OF EACH INTERVAL GIVES LOCAL TANGENT VECTOR

Figure

21.

Streamline

Calculation.

61

4.0

DISCUSSION

OF

RESULTS exact and nomicases, some of panel methods. calculated and

Correlation studies discussed below include nally exact solutions of a number of basic test which had proved difficult for earlier low-order In addition, some comparisons are made between experimental 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 midchord sweep of 30 , aspect ratio 6, taper ratio 1/3, and a NACA 0002 section. A nominally exact datum solution (25) for the case was computed by Roberts' third-order method using a 39 (chordwise) by 13 (spanwise) set of panels. The present calculations, based on the i = 0 formulation, use i0 spanwise panels with both 80 and 20 chordwise panels on a "cosine" spacing, giving increased panel density towards leading and trailing edges. Roberts' datum paneling is heavily weighted towards the leading edge: he has, in fact, 4 more panels ahead of the most forward panels in the present 80 cosine spacing case. The comparison in Figure 22 indicates that the lowand high-order methods give essentially the same solution where the control point density is the same. The present calculations using 20 panels chordwise deviate from the datum solution near the leading and trailing edges where the control point densities are different. Good agreement is restored in those regions by the 80-panel case which, at the trailing edge at least, has a similar panel density to the datum case. (Note: not all the 80panel pressure values are plotted in the central region of the chord in Figure 22.)

4.2

Wing

with

Strake

The fact that surface velocities are obtained by numerical differentiation in the present method might lead to inaccuracies in awkward situations. So far, no serious trouble has been experienced. Any problem that might arise with the derivative scheme (e.g., in a region of mismatched paneling) can usually be alleviated by cutting panel neighbor relationships at that station , thus forcing a forward or backward differencing (Subsection 3.4). One situation that could be difficult and which has caused a problem for other methods is the evaluation of the spanwise velocity component, Vy, in the neighborhood of the kink on a wing with strake. In the case considered in Reference 25, the leading-edge kink occurs at 25% of the semispan and the leading-edge sweep is 75 inboard and 35 outboard. The trailing-edge sweep is 9 . Again, the wing section is a NACA 0002 and incidence 5 .

62

--1.4Tr _--

---e--

ROBERTS, DATUR ('THIRD-OEDER') (39 X 1:3)_ REFERENCE 25 80 CHORDWISE PANELS PRESENTCALCULATIONS 10 SPANWISECOLUMNS

- I. 2 "11

o

% 20 cHo 20 CHORDWISE" I. 0 I'I_ "SINE" PANELS SPACING

Cp

0

0

.2

.4

.6

.8

1.0

X/C

Figure

22.

Comparison butions on = 5 .

of Calculated a Thin Swept

Chordwise Pressure DistriWing. NACA 0002 Section;

63

The stations of the Two Roberts'

chordwise n = kink,

distribution i.e.,

of

Vy just

is

plotted and

in

Figure just

23

at

.219 and .280; respectively. datum method

inboard

outboard

higher-order third-order

solutions and from

are Johnson

plotted (25) and Rubbert's

from (17)

second-order method. These distribution but disagree The disagreement might be polation is set up across The present solutions calculations and favor

datum solutions agree by a small amount on caused by the different the kink. are in close second-order

on the outboard the inboard cut. ways the inter-

datum

the

agreement solution

with near

the the

upstream 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-edge surface. The chordwise paneling was on a "sine" distribution with density increasing towards the leading edge. Such a high number of panels was probably not needed for this case, but the calculation 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 by rotating a NACA 0010 about the nacelle axis such that the minimum area is example, one fourth of the inlet taken from Reference and 18, exit shows areas, Figure 24. how the earlier, This first

generation, low-order analysis, because of the leakage, cannot match the internal flow calculated by the higher-order methods. However, the low-order, VSAERO solution (with the internal potential set to zero) is in close agreement with the high-order solutions and evidently does not have this leakage problem. The with second an exit nacelle diameter a NACA case considered 0.3 of the 0005 section is chord. also The leading an open nacelle edge nacelle, surface out by 5

but

is generated by from the x-axis. Two 25. Hess

tilted

higher-order

datum pressure

solutions

(25)

are

shown These

in are

Figure due to

for the chordwise and also Roberts. Roberts' compared

distribution.

duct

solution shows a slightly with Hess's solution.

lower pressure inside The present calculations,

the

based on the %i = _ formulation, use I0 panel intervals around half the circumference and with two chordwise distributions, 26 cosine and 48 sine, around the complete section. Both results are in very close agreement with the datum solutions with a tendency for with Hess's slightly solution. lower This pressure could be inside caused the by duct the compared slightly

64

.6

0

I

.2 4 .5

.6

.7

.8

.9

1.0

-.6T = I .219

Vyo6

-----4k-- ROBERTS ---e---- RUBBERT

HIGHER-ORDER SOLUTIONS

DATUM

U

_)

PRESENT CALCULATIONS PIECEWISE CONSTANT

SINGULARITY

PANELS

-.6

-._= .280

Figure

23.

Comparison of Calculated Two Stations on a Wing tion; _ = 5 .

y with

Component Strake.

of Velocity NACA 0002

at Sec-

65

-2O

I

--EXACT HESS BRISTOW HESS VSAERO

(HIGH-ORDER SYMME_ (LOW-ORDER

AXI_IC)

\ J

--I

/II II _ I

\

B A

SOURCE C )

FROM REF. 18

Cp

/.

0

0.2

0.4 X/C

0.6

0.8

1.0

Figure

24.

Comparison of Internal Pressure Calculated using a Range of Methods Showing Leakage of Low-Order First Generation Model.

66

25----HESS _ ROBERTS ) HIGH-ORDER

PRESENT CALCULATIONS: CHORDWISE PANELLING 0 26 COSINE SPACING | 10 PANELS ON 48 SINE SPACING W HALF CIRCUMFERENCE

-I

A

0

0

.2

.4

.6

.8

1.0

X/LFigure 25. Comparison a Nacelle Section; of Calculated Pressure with Exit Diameter/Chord _ = 0 . Distributions on .3. NACA 0005

67

smaller duct cross-section area generated by the present i0 flat panel representation. Again, there is no leakage problem with this formulation. 4.4Wing-Flap Cases

Cases with slotted flaps have proven troublesome for some earlier, low-order panel methods, especially those using a specified doublet or vorticity singularity distribution and employing "Kutta points" where tangential flow is specified just downstream of the trailing edges (e.g., (23)). In closely interacting multi-element configurations the interference of the downstream element acting on the "Kutta point" of the upstream element can result in a circulation error; in addition, the solution can be sensitive to the _ of the prescribed loading distribution on the upstream element--this is quite different from that on the regular single-element wing section. Two wing-flap cases are considered below for the VSAERO program. The first case is one of the exact two-dimensional, two-element cases provided by Williams (26). Figure 26 shows the close agreement between the VSAERO calculations and the exact solution for configuration B on both the main element, Figure 26(a) and the flap, Figure 26(b). Part of the high aspect ratio aneling in the VSAERO calculation is shown in Figure 26(c), and as ii0 panels chordwise on the main and 60 on the flap. The second wing-flap case involves a part-span flap. The inset in Figure 27 shows a general view of a wing-flap configuration at 16 incidence. The wake shape is shown after one iteration cycle. The wing has an aspect ratio of 6 and the basic section is a NACA 0012. The 30% chord Fowler flap is deflected 30 and the flap cutout extends to _ = .59 on the semispan. The surfaces of the wing tip edge, the flap edge and the end of the cove are covered with panels. Vortex separations were prescribed along the wing tip edge and the flap edge. The present calculations of surface pressures are compared with experimental data (27) at two stations in Figure 27. A section through the flap at _ = .33 is shown in Figure 27(a) and one just outboard of the flap edge at n = .6 in Figure 27(b). Boundary layer calculations have not been included in this calculation yet, and so the agreement between the results at this incidence (16 ) is probably too close. The lower peak suction in the flap data, Figure 27(a), may be caused by interference from a nearby flap support wake.

68

L)t.D

CD

O?

C)Oh O

-,-I ,--I ,--I -,-I

)GO O

0 4-_ 0

0

...I

O6. $

!0

C)

U r_

k.O(D -,-I

u_ o

_ _ -,-I

0 -_ r_ L_

Z

l_,

_

COo

t_l 0 O_ U] (U r_ 4--_.) L 0

! CO F(-_ _: X

! nj 0 c_ b.l