THE NAYIER-STOKES EGUATIONS(U) FLORIDA UNIY GAINESYILLEP J MCKENNA 19 NAY 95 AFOSR-TR-85-9625 RFOSR-93-633S

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NATIONL BUREA OF STANDAROSMICROCOPY RESOLUTION~ TEST CHART

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PYOSR-TR. 85-0625'

Far-Field Boundary Conditions in Numerical

Solutions of the Navier-Stokes Equations

L°O* (.

P.J. McKenna

LA. DTIC* E.LECTE

Final Report AFOSR Grant 83-0330 918

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17. COSATI CODES I 1S. SUBJECT TERMS (Continue on rerse if neceauiiry and identify by block number)FIELD GROUP I SUB.R. artificail boundary conditions, Navier-Stokes equations

19. ABSTRACT (Continue on truerse if necemary and identify by block number)

The purpose of this project was to investigate the artificial boundary conditionswhich must be imposed at the boundary of a numerical grid. Numerical experimentswere performed which evaluated various boundary conditions of this type. Ofparticular interest was the question of whether linearization around the flowat infinity provides an adequate choice of boundary condition for the fullynonlinear equations. Ifflow-outflow conditions as well as several choices ofperiodic boundary conditions were considered. The inflow-outflow conditions workedreasonably well for two dimensional problems. For the periodic conditions,results were mixed. Concerns about over specification when using pure periodic

*" conditions proved to be groundless. However, the method failed when attempting touse periodicity inthe two sideways traveling periodic waves.

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I INTRODUCTION

With the advent of high speed computers, new approaches to

the solution of engineering problems have arisen. One such field

is Computational Fluid Dynamics, in which numerical techniques

such as finite differences are used to numerically integrate

partial differential equations governing the physical phenomena.

one problem with this approach is that on the boundary of the

numerical grid, artificial boundary conditions must be imposed.

Usually, these conditions are based on a linearization around the

flow at infinity, and are thus non-physical. In this paper, we

perform a series of numerical experiments, in which we evaluate

various boundary conditions of this type, and investigate whether

* the linear model is an acburate representatioll of the fully non-

* linear equations.

The flow of fluid around obstacles in two dimensions is

described by the compressible Navier-Stokes equations

au + aE +F 0a() at a 0

where

U p E= pu

2pu Pu -xx

pv puv -ZxY

L pe pue - uOxx - vZxy ix

* 2-*

.. ,.. . .. ,. ... -;. . .. - *:-=._ . .. ... .. j - . . . . . . - = , . , ' . - w

'00

2

F = pv

pUV - Z

", 2 yypve - a - uz - qy

2 +xx (3)

" Bu Vxy By x

Accession For

= 2 P + 2 IU NTIS GRA&I.yy -P (y DTIC TAB

Unannouned 0Justificatlon

ix y Zy By .

Dis.D_ tribut ien/

Availaility codesp p RT = (T) ail ,,o

D Special

e = CvT + (u 2 + v2)/2.

OII

Here, the four variables pUve represent the physical oo,quantities of density, x- and y- components of velocity, and I'SEC"',

"- internal energy.

This nonlinear system of mixed parabolic-hyperbolic type in

two space dimensions and time, with four independent variables

must be solved in an exterior region in R2 . The geometry will

depend on the particular physical situation that one is

attempting to model.

The situation we shall be interested in occurs in modelling

.';;,%v%-;;-.~-

9. . . . . . . . .

3

• .flight conditions, in which the conditions at infinity are pre-

* scribed with a large u-velocity and v-velocity zero and pre-

scibed p. and e. Fluid flows around an obstacle in x-y space.

Usually, this equation is solved numerically using a finite

difference scheme of the Lax-Wendroff type, such as the

MacCormack ADE method. Since these calculations can only be made

on a finite grid of points in xy space, an artificial far-field

boundary is created. This boundary ought be sufficiently far

". away from the object around which the fluid is flowing so that

*[ local phenomena are not neglected by omitting part of the region

of fluid flow. on the other hand, the farther away the region

is, the more grid-points need to be included aad thus the more

expensive and time-consuming the computations become.

One then has the problem of deciding what effect this new

* boundary has on the solution of the problem. Because of the

viscosity terms in (1) and the additional artificial viscosity

introduced by the finite difference schemes, some boundary condi-

" tions must be imposed.

As we shall show in this report on numerical experiments,

considerable care must be exercised in the choice of the boundary

conditions. If one is interested in steady state flow, then one

-. starts off with an initial approximation, and hopes that the

errors in the numerical solution propagate out of the region as

transitory disturbances in the physical variables. One then

expects to converge to the steady state flow.

300

4

We have previously shown that the incorrect choice of

boundary conditions can give rise to some of the following

* phenomena: i) reflecting boundary conditions, in which the

disturbances in the physical variables represented by the

* difference between the steady state flow and the initial

* conditions are not allowed to exit through the far-field boundary

but instead continue echoing within the grid, and giving rise to

spurious oscillatory solutions; ii) under-specified boundary

conditions, in which large errors are introduced before

convergence takes place.

In this paper, we first discuss the theory for simple linear

* hyperbolic systems in one and two space dimension. We then

analyze several computational experiments in the light of this

* theory in two space dimensions.

II A REVIEW OF THE LINEAR CASE

The method used in the calculations which are the subject of

this paper is the MacCormack alternating-direction explicit

A-

5

scheme. [6) [15). This is a multistep efficient scheme which

reduces in the linear case to the Lax-Wendroff scheme. For a

diagonal N x N matrix A, this scheme approximates the equation

Ut + AU X = 0 by

(2) n+l e At. n n

-X Uj+l-Uj_ 1 1

2 At 2 n nn+ - (U. -2U+U_

AX j+1 I 1 1

As usual, {j j represents the space step and n

*v represents the time step. If we are consideri ng the equation

Ut + AU X = 0

on the region [(x,t), Ox1l, t>01 then the 'analytic solution is

determined by the initial conditions and boundary conditions at

x = 0 and x = 1. If the first k eigenvalues are positive and the

remaining N - k are negative then the quantities ul---uk must be

. prescribed at x = 0 and Uk+l---uN must be prescribed at x = 1.

* Thus if WI - (Ul, u2 ,---uk, 0 0---0) and WII = (0, o,---0,

Uk+l---uN) then for well-posedness, the boundary conditions must

* be

W = f(t) + BoWI at x = 0

W = g(t) + B1WI at x = 1.

F-2";.:'?:-:- _ L-;- -_".-?.-,'<.',m .'.-. -,:-.:-..-....'................-......-.-..-...............-......-.......-....-......- ..-. ... . -

6

-This gives a total of N boundary conditions. If either4.

kx(n-k) matrix B1 and the (n-k)xk matrix B0 are non-zero, then

the boundaries are reflecting, that is a wave in WI travelling

left to right will be reflected as a wave in WII running right to

left.

If a boundary is supposed to be non-physical, then it should

not be reflecting, since the reflections would depend on the

location of the artificial or numerical boundary.

NOW let us consider the difference scheme of (2). If the

grid points are given by {xj) with x = 0 and xi = 1, then it is

clear from (2) that 2N boundary conditions are required. Thus we

-. must prescribe boundary conditions in such a way as to least

-. affect the closeness of the numerical to the analytic solution.

This situation has been the subject of several papers. In

[3], Gustaffson and Kreiss point out the danger of over-

specification. By this is meant that all u. are specified at

both endpoints. This might be tempting because one might argue

*" that if the boundary onditions ui---uk are given constants at

x = 0, then eventually these same values will be assumed by these

variables at x = 1. However, in [4) it is pointed out that

* convergence may or may not occur, depending on whether the number

' of grid points is odd or even.

A method which works well, as pointed out in [3], is to

impose

*i ' .** *

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

" " , -""-. ', .' . ,-''-, - "'°o "° "3 ."". °"" ".' " ' "°' " '"-" '.-" "..","-'-" ""%""" " ""- . ". .' . ."" "",~. ~ * . * .... . . . . . . . .

Ai 7

""U, (1,t) = 0 1 -C i k.aui

-- (Ot) = 0 k+l i 4 n.

or in the numerical scheme

uij =uij- 1 i k.

u.,O = ui n.

This introduces small errors at the outflow but these errors

do not propagate upstream. This is proved analytically by Parter

[101.

In [12], many different numerical boundary conditions are

- given. The conclusion is that upwind differencing at the point

of outflow

u +l ; un + xiA-)(u -uj AX J-1 J

is most pccurate, although it converges with the same speed as

the previously discussed ux = 0.

The most serious error which one could make would be to

prescribe conditions at the wrong end. In other words, since u1

*: is right running, this would involve prescribing u, at x = I and

imposing ul= 1 at x = 0. This would result in convergence to a

. .U° . - . . . . - . . . .

-;' ."""""%" ""-" ." , ' "' "-". -"."- ' .. -, ' ..-. ' " ,,.""' ""2 . , - -, .- "k "."- "," ,.- .". ." "'" .- , .

--~~~.-.-...,L ,....... ...... . .... ... _.... . .. , ,:,.., . .. :.:.... ,...

8

steady state which depends on the initial conditions.

* III THE NAVIER-STOKES EQUATIONS AND CHARACTERISTIC VARIABLES:

We now begin our discussion of the equations of gas

dynamics. We will neglect viscosity for the purposes of this

analysis. We will assume that the flow is one-dimensional and

subsonic and that the deviations from free-stream solutions are

small. This will allow us to neglect second order terms.

There are many forms of this equation, but the one most

suitable for the present discussion is

"" /aU AU/_+ a- =0

at A x

where

A= 0 1 0

u2U-) (3-y) u y

(y-l)4 3 - yeu _j 3 u 2pp z-(y- I - y

* and

U(u)

or in terms of physical variable

" (3) --U + _L- - 0'-at axai i

ax+-=

• "- , .. ::.,'i¢.u,: , --, ' Z..'.<,,:-,.-_,:.' : - ,- ; .. . .' .".*- -_. .-*- _ -'' . .{ . .''. '.'.'

9

where

A =M AM

and M 0 01 U P-U/p 1/p u[P

2/ ) u 2 (l-y)u (Y-1)

Here we make the key assumption that deviations from the

free stream are going to be sufficiently small that we can treat

the entries in the matrix A as being approximately constant (at

least locally). Denote these frozen variables by 0-subscript.

*- We then make the substitution

(4) W (1 0 -1/c 0

S) 1/p1/poco (:)and when this is substituted into ( ) we obtain

awI awl 1*a(5) -- t- +u 0-ax = 0 w = p -

cO

aw2 aw2 1at + (u0 +c 0 ) =0 w2 =u + p00 wp 0c0

aw 3 +W 3 Wat 0 0 ax 3 P-u-

o . . . . * . . . . . ..... . . . . . . . . . . . . . .

0

Notice now how this breaks down into two separate cases. on the

one hand, if flow is supersonic then all wave motion is in the

left to right direction. In this case all analytic boundary

conditions should be prescribed at the left hand side and only

numerical boundary conditions prescribed at the right hand side.

Since the substitution (4) is equivalent to

p = K 1 + (p0/2c0 )(K 24K 3)

u = 1/2(K 2-K3)

p P 0 Co/2(K 2 +K3 )

it follows thtt prescribing all physical variables at the inflow

and prescribing ap/ax = au/ax = ap/;x = 0 at ihe outflow is

legitimate in terms of analytical and numerical requirements in

the supersonic case.

However, we must now consider the case of subsonic flow. In

this case the situation is completely different. Here, two of

the variables W1 and W2 go left to right with velocities u0 and

* u0 + c0 respectively, whereas one of the variables runs right to

left with velocity co-u 0 . The variables W and W3 have no clear

physical significance, yet it is only in considering these variables

that the full wave structure of the equations (5) or (6) can be

understood. Thus, one would be led to predict, for small

deviations from free stream conditions, that the best boundaryI

conditions would be, for an interval (0,L)

,- --. . .. .* "* . -.. *.,b -.'- ',-. . . , - . -- . -

7 T)

11

(6) W (0,t) = K1 dW (Lot) 0

1 dW

W (O,t) = K dW2 (Let)

dW3d (O,t) = 0 W (Lt) =1K3dx33

Note the curious aspect of these boundary conditions. In

order to prescribe the numerical values KI and K2, we need to

.* know accurately all three physical variables at some distance to

..- the left. However, only the two combinations K1 and K2 are pre-

scribed. This can be summarized by saying that while we have

*used all three pieces of information upstream, we have done so in

*such a way that one degree of freedom remains thus allowing the

waves in W3 to exit without reflections.

On the basis of the linearized model, various other combi-

nations would be well-posed. For example, it is possible to

prescribe K3 in terms of either KI or K2 at the outflow x = L.

Thus at the outflow one may prescribe

W 2 (Lt) = 3 (t) + clW1 (Lt) + c2W2 (Lt)

For example if cI = 0, c2 = 1, then this amounts to putting

(7) u (Lt) = 1/2 F3

(i.e.' we prescribe velocity at the outflow).

12

SAlternatively, we might take c, = 0, c2 = -1 and we would get

(8) p (L,t) = ((p0 C0 )/2)F3

(i.e. we prescribe pressure at the outflow).

Many other combinations are possible, but as remarked in section

*II, all these will cause errors in the initial data to be

reflected back into the medium as waves running from right to

left. For example, we would predict that an error in W3 would be

reflected back as an error in W1 if we use boundary condition

(8). As we shall see, this is exactly what happens.

At the inflow end, we may prescribe W1 and W2 in terms of

W3. Thus the following boundary conditions are well posed;

(9)(a) W, (Ot) = F1 + clW 3 (Ot)

(b) W2 (Ot) = F2 + c2 W3 (Ot)

For example, choosing c2 = +1 in (9b) corresponds to

u(O,t) = (1/2)F 2

(i.e. prescribing u at the inflow) and c2 = -1 corresponds to

p(O,t) = (p0C0/2)F 2

(i.e. prescribing p). One can prescribe the combination (u,p) by

first choosing c2 = 1 (thereby prescribing u) and then choosing

Cc1 0/c0 , thereby prescribing p in terms of a given F, and a

". " ,o' . % m" ' " "'"*. . . . . . ..- . . ..". ..""" , o ."" . " °

°"•" •"' ' o""•","' ' " "' ° "

-' ' ',- "

J -J

.- "" "" I

= :" ":- " ' ' "'- '-:i i' i,..il" " " "" € <

"< "':; "' :":"" ':; "f:'': : 'u '"""-" ; i " "'" ; i :,' -

-. VI TWO DIMENSIONAL RESULTS

* Having now understood the phenomena which can occur when

calculations are made with one space dimension, we now consider

" the more complicated situation of two space dimensions.

In this situation, we shall solve the Navier Stokes equation

i (1), again by the standard MacCormack A.D.E. scheme on a 20 x 20

. grid, which is a very simplified model of a wind tunnel. We

shall continue to consider flow close to the free stream flow

(M = 0.5) previously studied in the one dimensional case. The

*j physical values are given earlier. Figure 8 shows the geometry

. of the situation. The fluid is flowing in at the top right

corner of the grid and flowing out at the bottom left.

We have three distinct types of boundaries to consider. We

Shave the inflow and outflow boundaries (as before) and in

addition, two sidewall boundaries, where the fluid is flowing

*. parallel to the boundary.

We shall continue to impose one dimensional boundary

conditions of the type given in the first section on the inflow

and outflow, along with the additional condition v = 0. This

says the fluid flow is one dimensional at the inflow and outflow,

and could be physically reasonable.

When we come to the sidewall conditions we must undertake

another one-dimensional analysis. Thus, we assume all variables

are constant in the x-direction and variation only lakes place in

the y-direction.

* J * - **, * .. * . *.*i, * *~

* r*.~***.** % *****. \ . . .

.1

This leads to the set of equations

(6) Vt + AVy 0

where

V p A 0 0 1 0

u 0 0 u 02

v u12(y-l) -u(Y-I) 0 y- 1

e e0- 0o

Freezing the coefficients of A in (6), we substitute

(7) = p + 2c 0

1T 2 = v + p p

o-I

T = -v +0-p3 POc 0

T u4

Equation (6) is then transformed to

"~~~~~ ~~ ~~~~~~~~~.... ..... .. . .....-.. .... -....... .. . .---- . . ., - .°.- - , -

(8) Tit =0

T2 t + C0 T2y= 0

T3 t - c 0 T3y 0

T =04t

Thus, if we only consider deviations transverse to the free

stream flow, we have on the basis of the linearized model, four

non-physical variable T1 , T2 , T3 , T4 , two of which are u and

entropy. T, move with zero velocity in the y direction, one of

• .which, T2, moves with speed co in the positive x direction, and

one of which moves with speed co in the negative x direction.

" Thus one dimensional theorypredicts that at.the sidewall, we

should impose

y =0 y =L

Tly = const Tly = 0

T 0 T3 const

. Ty=0 T4y = 0T4y =0

*This, together with the inflow and outflow conditions gives the

.* following set of boundary conditions

DI

* . A A. AA.

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

0

* Inflow

V =V

1 (1

P1 = [P2 + +l U

1 2 .O. 2

PO ~k P P2+kP1 = k1 + [ k - 2

. where the free stream values are specified in the characteristic

variables k1 = P - P./c0 2 , k3 = U p + p=/Poco and where the zero-

subscript refers to frozen variables

Outflow

Vn = Vnl

1- = [U + p /(poco ) k

2 n-U n----

"= POCO [k + Un_ + P Poco)]

Pn-l + PO [k +O Pn-l]

l2n c0n-1 u n- o

III

-P0 -..

,o0

Top wall

Popo [v _ 1 + f- k 3 1Pj 2 1 PoCo 3

v. = /2k 3 + 1/2(vj_1 + Pj 1 /PoCo)

pp.

P PO (vj 1 + J- k) + pj - pj 1-pj 2c 0 jl pc2c Polo 3 -1 c -

uj = U

Bottom wall

u 1 = U=

S1 "1/2[v 2 + 2 - p 2/(Poco) ]

P1 - [-v 2 + k 2 + P 2 /(Poco ) ]

Pl= P2 + (P /2c ) [-V2 + k2 - P 2 /P0 co]

where the free stream values are used to specify the

characteristic combinations

W2 v + p 0 c 0 "3 =V - pr/PoCo •

..0

The purpose of the present grant was two-fold: first to

expand the previous work to situations where the flow was at an

angle a to the artificial boundary, but is primarily steady-

state, and second, to see how the artificial boundary conditions

* apply in period situations where convergence to free stream is

'" not expected.

The first objective is clearly desirable, simply to save in

additional computing time. The requirement that the grid not be

rectangular with boundaries either parallel to or perpendicular

to the free stream at infinity clearly can be used to remove a

large number of points from the grid. The entire boundary can be

chosen approximately equidistant from the object whose flow char-

acteristics are being studied.

The second objective was chosen because of its importance in

modelling flow in the compressors of turbines. If one is

attempting to design turbine compressors, one must have knowledge

of how the flow in the area between the stators and the rotors is

*. behaving. This is almost impossible to measure.

On the other hand, to accurately model the entire inside of

the compressor requires a computational capability far exceeding

the largest of today's computers. Thus one models only the flow

past two blades of the rotor, and uses periodicity to extend the

calculations to the entire rotor. The flow coming in past the

- " " ."- .. "-, "".-... .

C)

stator is regarded as a periodic inflow term.

The following pages summarize the results of the research on

these two projects.

- Section I Flow at an angle of incidence a.

The previously outlined boundary conditions were used.

* Computations were done on the basis of the inflow or outflow

* depending on the flow perpendicular to the surface. For computa-

* tional simplicity, we chose a rectangular grid, but choose

* u., = V = 300 This results in a 450 angle of incidence or

exit. We impose an initial 10% disturbance in W2, the right

* running characteristic variable and observe what happens as it

* exits. Recall that this disturbance runs at an angle of 450 to

the flow. The following plots for W2 at various iteration

* numbers show it exiting without any surprising phenomena, similar

* to the one-dimensional case. By the sixty-fifth iteration, the

flow has almost converged to steady state. There appear to be

* some small reflections (as shown in the last plot) in W3, but

nothing to worry about.

Thus we can conclude that the inflow-outflow boundary

* conditions work reasonably well. Subject to a rotation of

coordinates, they can work on any geometry - the only decision is

whether we are at an inflow or outflow point.

2 II. The case of periodic BCs

We now consider some experiments on the case of periodic

boundary conditions. We study flow parallel to the rectangular

* grid, perpendicular to the inflow and parallel to the side-walls.

The most obvious boundary condition is pure periodic.

S1 = Sn-1' Sn = S2for all physical variables S. This, it was

* feared might lead to over specification, but these fear appeared

* groundless. A sight-traveling wave would exit to the right, and

* reappear at the left in a predictable and correct manner.

The next objective was to see if this could be done by only

using periodicity in the two sideways traveling periodic waves.

* The reason for this was to minimize storage problems for the more

* complicated turbine problem already discussed.

Curiously, this method failed. Furthermore, the failure

- seems intrinsic to the whole idea, as we shall demonstrate.

First the W2 wave appears in iteration 5 and proceeds to

exit. Periodicity requires that it reappear, and this can be

* seen in iteration 20. However, already, a sharp spike

*(reminiscent of over specifica tion is appearing in W3 by

iteration 20. As far as iteration 40 W2continues to progress

from right to left, but at iteration 55, sharp spikes begin to

develop in W2 and W3. This seems to suggest overspecification at

the corners, but as far as we know, this is not the problem.

Shortly afterwards the program crashes. The relevant portion of

the program for the boundary condition implementation is

included.

C2

0 CP

CP

C

m -v

0 -d

F-4

Z 00

aP

C2

at4F

41

C3

W2.

Z 0

ca

44M

CQ1

I9w

u4=

Nf

i ,,

C' a

z" 0 O 0 0 I" -Z O

LL

i:0

, , ,,

II0

ifin

IIl

crCP0mCz-0 ''

z a.

o -a

- F.. -

a

0

CYC

m -

E--4

Lna

al

Id 0

0 p

47

ay

zcz cm

0a

0a

II

Z 'V

a"0 1"0 O - -

zA

.,... .-...: ..; -,-. :, -."- : - . ....... ,..,...... ....- ..-.... ,..,-..- ,.,...-.. ,- ,- .- ,- .. .. .. .. .. .. . . ., . .

iin

0

CP

ca

U,

0P

-T..

!0

CD D

.°La

CD

Z 0r

2 °

H- •iC

;R0

' °,

eC

22 . .. . .. - . - . . . , . . - . .- ... - -. . . . . . . . . . . . . . . . . .~ ~

IE 0

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Finally, we attempted to run periodic characteristic B.C.s

on a one-dimensional basis. We felt that since the problem

* manifested itself at the corners, if we removed the corners the

problem should go away. We next show what happened in this

situation.

First we see the one-dimensional wave in W2 moving to the

right, exiting and reappearing by the iteration 20. However,

sharp spikes appear in W3 by this time. W2 appears to behave

* well until iteration 85, but either reflection or some non-linear

instability lies caused W3to grow catastrophically. Clearly

this represents failure of the boundary conditions, even in the

one-dimensional case.

Various minor variations were tried, but all produced the

same basic phenomena.

-. ".,,. .... -

* .

C)

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Conclusion

Non-reflecting characteristic boundary conditions work well

* so long as the flow is basically returning to steady state. This

* was seen in the first series of numerical experiments.

However, at this stage of development they cause strange

- instabilities which prevent efficient modelling of periodic

* phenomena. The reasons for this are not understood at the

present.

BIBLIOGRAPHY

[11 Engquist, B., and Majda, A., "Absorbing BoundaryConditions for the Numerical Simulation of Waves", MathComp. Vol. 31, pp 629-651, (1977).

(21 Gottlieb, D., and Turkel, E., "Boundary Conditions forMulti-Step Finite Difference Methods for Time DependentEquations", J. Comp. Phys. Vol. 26, pp 181-196 (1978).

(31 Gustaffson, B., and Kriess, H.O., "Boundary Conditions forTime Dependent Problems with an Artifical Boundary",Jour. Comp. Phys. Vol. 30, pp 333-351 (1979).

(41 Gustaffson, B., Kreiss, H.O., and Sundstrom, A.,"Stability Theory of Difference Approximations for MixedInitial Boundary Value Problems II", Math. Comp., Vol. 26,pp 649-686 (1972).

[51 Kreiss, H.O., "Stability Theory for DifferenceApproximations of Mixed Initial Boundary Value ProblemsL", Math. Comp., Vol. 22, pp 703-714 (1968).

(61 MacCormack, R.W., "Numerical Solutions of the Interactionsof a Sh7ock Wave with Laminar Boundary Layer", LectureNotes in Physics, Vol. 19, Springer Verlag (1976).

(71 McKenna P.J., Graham, J.E., and Hankey, W.L., The role offar-field boundary conditions in numerical solutions of

the Navier-Stokes equations, AFWAL Tech. Report TR. 81.

[81 Moretti, G., Comment on "Stability 4spects of DivergentSubsonic Flow", AIAA Jour., Vol. 19i".No. 5., p 669 (1981).

[91 Oliger, J., and Sundstrom, A., Theoretical and practicalaspects of some initial boundary value problems in fluiddynamics, SIAM J. of Appld. Math., 35 (1978) pp. 419-446.

[101 Parter, S.V., "Stability, Convergence and Pseudo-Stabilityof Finite Difference Equations for an Over-Determined

b problems", Numer. Math, Vol. 4, pp 277-292 (1962).

[i] Rudy, D.H., and Strikwerda, J.S., "Boundary Conditions forSubsonic Compressible Navier-Stokes Equations", Computersand Fluids, Vol. 9, pp 327-338 (1981).

[121 Rudy, D.H., and Strikwerda, J.C., "A Non-reflectingOutflow Boundary Condition for Subsonic Navier-StokesCalculations", Jour. Compu. Phys. Vol. 36, pp 55-70

•.(1980).

"k L

113) Schlichting, H., "Boundary Layer Theory". Seventh Edition,McGraw Hill N.Y. (1979).

[14) Shang, J.S., oscillatory compressible flow round acylinder, Proc. AIAA 20th Aerospace Meeting, Jan 11-14(1982). Orlando, Florida.

[151 Shang, J.S., "Numerical Simulation of Wing FuselageInterference", AIAA 19th Aerospace sciences Meeting,AIAA-81-0048, Jan 12-15, 1981.

[16) Steger, J.L., Pulliam. T.H., and Chima, R.V., "AnImplicity Finite Difference Code for Inviscid and ViscousCascade Flow", AIAA Preprint-80-1427, AIAAA 13th Fluid andPlama Dynamics Conference, July 14-16, 1980.

[17) Yee, H.C., "Numerical Approximation of Boundary Conditionswith Application to Inviscid Equation of Gas Dynamics",NASA Technical Memorandum 81265, March 1981.

[18] Yee, H.C., R.M., Beam, and Warming, R.F., "Stable BoundaryApproximations for a Class of Implicit Schemes for theOne-Dimensional Inviscid Equation of Gas Dynamics", AIAAPaper 81-1009, presented at Palo Alto, CA, June 1981.

b.

.

".

FILMED

10-85

DTIC

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