Solution of the Two-Dimensional Navier-Stokes Equations Sparse Matrix Solvers EDED Bender and PDKD Khos

incinnati, Cincinnati, OH

AlAA 25th Aerospace Sciences Meeting January 12-15, 1987lRen0, Nevada

For permission to copy or republish, contact the American Institub of Aeronautics and Astronautics 1633 Btolldny, Naw York, NY 10019

Solution of the Two-Dimensional Navier-Stokes Equations Using Sparse Matrix Solvers

* ** Erich E. Bender and Prem K. Khosla #87* ~ Y 9 . 9

University of Cincinnati Cincinnati, Ohio, 45221

Abstract

The use of direct sparse matrix solvers in the solution of the Navier-Stokes equations is investigated. The Yale Sparse Matrix Package and its implementation in the solution algorithm is described. The streamfunction-vorticity form of the Navier-Stokes equatiofls are discretized and linearized and the resulting system of equations are solved using this package. Several viscous Plow problems are investigated, including flow in a cavity and flow around a NACA0012 airfoil. Massively separated flow around a sine wave airfoil is investigated and high Reynolds number solutions are obtained. A solution of the unsteady flow around a Joukowski airfoil at high angle of attack is presented.

Introduction

Most relaxation methods slow down when the number of grid points increase. Usually some type oP acceleration is applied in order to achieve a converged solution. Typically, conjugate gradient or variants thereof, multi-grid and Chebyshev techniques have been used in literature. For algebraic qquations arising from large Reynolds number flows on stretched grids, most of the acceleration techniques require special considerations and become problem dependent. There is no single method which can be reliably used to solve flow problems without fine tuning the codes and that will work for a large class of problems. Separated flow regions, for example, require special consideration when computed by an iterative multigrid type strategy.

In view of these problems, the authors have elected to revisit the class of direct solvers. Typically, the direct solvers have been rejected in the past due to problems of operational efficiency, large storage requirements, and stability on fine stretched grids. However, in the past decade a great deal of progress has been made in addressing these problems and many algorithms have been developed that take advantage of matrix sparsity and structure. In spite of these advancements, such techniques still require a considerable amount of memory. However with the ac4vances in computer technology, the price of memory has diminished to the point that large memories (16 megabytes) are even available in

a. micrlocomputers. Storage considerations no longer . play a dominant role in the selection of a

SOlUti~n technique; therefore, other criteria, such a3 speed, robustness, and flexibility, can be given more emphasis. Under these circumstances direct fdethods become more attractive. With this in mind, a sparse matrix package from Yale University (ref. 1 ) has been considered for the

solution of viscous flow problems. This package has been used by Vanka and Leaf (ref. 2) for the fully coupled solutions of the incompressible viscous flow in a driven cavity and the flow in a channel with a sudden expansion. They compared the performance of the package with iterative techniques and showed the direct method to be considerabley more efficient. Vanka (ref. 3-51 has also w e d the package for the numerical solution of combustion problems.

In the present paper, a direct solver is used to solve the linearized equations in two fashions; first, a direct solution of the entire flow domain and second a direct solver procedure that consists of the iterative solution over subdomains of the flowfield. Th@ latter procedure uses the direct solver on overlapping subdomains and is quite efficient; it requires less memory than the fully direct solution. Depending on the number of subdomaim, the'longer wavelengths of the solution error can be treated more effectively. This latter method is also well suited for parallel processing. Vanka (ref. 4.5) has used this subdomain Jtrategy with good results for internal flow and combustion problems. These techniques have strong robustness, speed and flexibility for two dimensional problems; however, they are quite complex. Application to three dimensional problems perhaps is still beyond the computational capabilities of most computers; however, the subdomain strategy applied to 3-D problems is worth investigating.

The choice of the present techniques for application to the Navier-Stokes equations has been largely dictated by the flow problem to be addressed. For example, unsteady flow using the streamfunction-vorticity formulation cannot be computed by AD1 or other consistent methods due to the lack of a time term in the equation for streamfunction. In view of this limitation, Osswald, Ghia and Ghia (ref. 6.71 used a direct solver for the solution of the streamfunction equation and the AD1 procedure for the vorticity transport equation. The solution procedure treats the equations in an uncoupled fashion and thereby is limited by first order accuracy in time and stability restrictions on the time step.

Another application is the problem of massive separation. This problem has been addressed by Smith and Cheng (ref. 8) using matched asymptotic expansions and triple deck theory, by Rothmayer and Davis (ref. 9) using interactive boundary layer theory, and by Ramakrishnan and Rubin (ref. 10) using the reduced Navier-Stokes equations. As predicted by Smith and Cheng, the separated flow sOl~ti0nS computed by these authors breakdown at some critical Reynolds number; this is indicative

* Graduate Research Assistant, Dept. of Aerospace Engineering & Engineering Mechanics, Student Member AIAA. **Professor. Dept. of Aerospace Engineering and Engineering Mechanics, Member AIAA.

Copyright O Amefiaa Imtltute of Aerwautia a d Ascroaauiks. lac.. 1987. All rights resewed.

1

1 of the the onset of breakaway or massive ' separat ion or of a s ingu la r i t y i n the system of

equations. The need t o compute such flows without the addit ional problems associated with an i t e r a t i v e procedure, requir ing under or over

- r e l axa t ion , has been the overr iding fac tor i n the choice of a d i r ec t solver .

This approach has been invest igated fo r a number of model and flow problems. Solutions t o

, - the diffusion equation, the streamfunction- vo r t i c i t y form of the Navier-Stokes equations, and the boundary region equations fo r a curved duct

. have been obtained by t h i s d i r ec t solver ' procedure. Some of these a r e discussed below.

Governing Equations

The equations governing 2-D incompressible viscous flow can be wri t ten i n general orthogonal coordinates a s

where the streamfunction $ and the ~ 0 r t i C i t y o , i n terms of the transformed ve loc i t i e s u, and u;, a r e defined as

and the sca le functions h, and h,

h, - ( ( x ~ ) ' + ( Y ~ ) ~ )

h, - ((x,,)' + (Y,,)')

a r e defined by

Uniform flow is specif ied f a r from the a i r f o i l giving the r e l a t i ons

$,, - y U-cosa - x U-sina n n w = o

uhere a is the angle of at tack and U- is the

freestream velocity.

On the a i r f o i l surface, $ - 0. To enforce the no-slip condition on the surface a second wder two point formula by Wood ( r e f . 11 is sppl led. Ih orthogonal coordinates the formula is

A t the outflow boundary the governing equations a re used with the $ and w terms

E € EE neglected.

The governing equations a r e expressed i n f i n i t e difference form using cent ra l differences and a l l nonl inear i t ies a r e f u l l y quasi l inearized t o second order. The differencing f o r the convective terms i n t he v o r t i c i t y equation allows f o r t he option of using f i r s t order upwind differencing with the cent ra l difference obtained as a deferred correct ion ( t h i s is K-R differencing r e f . 1 2 . The vo r t i c i t y equation a t the outflow, having only f i r s t der iva t ives i n E , is differenced a t the c e l l midpoint i n order t o maintain second order accuracy i n the d i rec t ion ; conventional cent ra l differencing is used in the n direct ion. The equations a r e second order accurate i n the converged s t a t e .

The Yale Sparse Matrix Package

The YSMP is an ef fec t ive s e t of programs fo r solving l a rge , sparse systems of l inear equations. Only nonzero coef f ic ien ts of the matrix a r e stored. The programs keep track of nonzero coef f ic ien ts generated by the LU decomposition and a l l oca t e s s torage f o r them.

The package contains a s e t of four programs, one f o r symmetric matrices and the other three f o r nonsymmetric matrices. The f i r s t nonsymmetric code allows f o r solving of several right' hand s ides ; it performs the LU decomposition only once if t h e same matrix is t o be used fo r each case. The second code combines the LU decomposition and back subs t i t u t i on s t eps and therefore does not allow t h i s f l e x i b l i t y ; however, i t needs l e s s than half as much storage as t h e first code. This code a l s o employs a compressed s torage scheme t o reduce s torage overhead. The th i rd program fea tures the f l e x i b i l i t y of the first code with the compressed s torage scheme of the second. Since the difference equations i n the present problems a re nonlinear the associated matrix changes a s the so lu t ion evolves. This requires tha t the matrix be recomputed and reinverted each i t e r a t i on . Due t o t h i s property and the reduced storage requirements of the second code, the second code is wed f o r the present study.

These codes requi re tha t the matrix is be s tored i n a spec ia l format. The nonzero coef f ic ien ts a r e s tored in a vector A i n r o r w i s e order. An integer vector J A of the same s i z e s to re3 the column number of the corresponding coef f ic ien t . F ina l ly an integer vector I A of length N+l, where N 1s the number of unknowns of the system, contains the S t a r t i ng posi t ion of each row i n A , and IA(N+l) is s e t t o the length of A plus one. A code has been wri t ten by the authors f o r t he s tudies herein, t o allow storage of the difference equations generated on a rectangular gr id i n the specif ied format.

The ordering of equations has a atrong e f f ec t on the amount of f i l l - i n t ha t occurs i n the fac tor iza t ion s t e p and therefore a f f e c t s the amount of s torage and time required f o r solut ion. Several methods e x i s t f o r ordering equations i n order t o reduce f i l l - i n . These include the nested d issec t ion method of George ( r e f . 13) and the a l t e rna t e diagonal.ordering ( r e f ; 14) . A program supplied with the YSMP implements the minimum degree algorithm ( ref . 13) . The method a s implemented with the package, however, is only good f o r symmetric matrices. If the matrix of A

is nonsymmetric, a s is i n the present case, t he : algorithm can be applied t o the symmetric matrix A

m

+ A' with good r e su l t s . The authors have wri t ten

a code t o generate A + i n the YSMP format. . Storage and time reductions of a fac tor of two or

more a re typical using the minimum degree algorithm.

Solution Procedure

I f the e n t i r e flow f i e l d is solved a t once the procedure is merely Newton's method applied t o a large s e t of nonlinear algebraic equations. An

' i n i t i a l so lu t ion is assumed and the l inear ized equations a r e generated using t h i s solution. This system is solved using the YSMP and the l inear ized system is updated using the new solut ion. If the system is quasi l inearized, t he method converges very rapidly (5 - 15 i t e r a t i ons ) . Examples of the amount of computer s torage required by the fac tor iza t ion s t e p is given i n f ig . 1. Five d i f fe ren t problems a r e shown, each with several gr id s izes . The simple poisson equation was solved on a uni t square using the f i r s t nonsymmetric 'code mentioned above (NDRV) , and the second code (TDRV). Grid s i z e s from 11 X 11 t o 101 X 101 were used. The viscous flow i n a ' c av i ty and a sudden expansion involved three coupled coupled equations per gr id point; these so lu t ions gere obtained by Vanka and Leaf ( re f . 2). Three dimensional flow i n a curved duct was invest igated by the authors using a parabolized Wavier-Stokes approach. This problem consisted of d i r e c t l y solving the difference equations a t s ~ C ~ e S s i V e cross planes and marching i n the streamwise direction. 21 X 21 and 41 X 41 grids were used t o resolve the cross planes.. There were four difference equations per gr id point i n t h i s problem. A s can be seen, moderately l a rge aroblems can be solved e a s i l y with four megabytes 2f memory. In addit ion, experience of the authors ?as shown the d i r e c t method t o be subs t an t i a l l y f a s t e r than i t e r a t i v e methods.

For la rger problems the amount of ava i lab le 2omputer memory may prevent the d i r e c t so1ut;ion of ;he e n t i r e flow domain. I n t h i s case, t he flow lomain can be divided in to overlapping subdomains. Zach subdomain is solved sequential ly and the jolution is relaxed i t e r a t i ve ly . Experience has ;horn tha t having the subdomains overlap r e s u l t s :n a s ign i f i can t increase i n the r a t e of :onvergence.

I t should be noted tha t the subdomain . l t rategy lends i t s e l f t o pa ra l l e l processing. By 'ar the most expensive par t of t h i s method is the

'.ime required t o obtain the LU decomposition. !owever, the decompositions fo r the subdomains can )e obtained independently of one another and

. .herefore can be done simultaneously. I n t h i s ianner f a s t e r execution times can be expected.

Driven Cavity

. A s a f i r s t t e s t the subdomain s t ra tegy was ctsted on the driven cavi ty problem. The case $resented here is fo r Re - 1000. I t was solved on

101 x 101 uniform grid. The cavi ty w a s divided nto e ight adjacent horizontal rectangular ubdomains wi th seven subdomains overlapping t h e i r oundaries, for a t o t a l of f i f t e e n subdomains.

The so lu t ion converges i n f o r t y i tb ra t ions . The K - R differencing w a s needed f o r s t a b i l i t y fo r the first 10 i t e r a t i o n s , but no underrelaxation was required. The computations were performed with a b t of i n f in i t y .

The so lu t ion is shown i n f i g s . 2a and 2b. These r e s u l t s agree well with the f i ne grid r e s u l t s of Ghia e t al ( r e f . 15).

NACA0012 Air fo i l

The incompressible flow around a NACA0012 a i r f o i l a t zero angle of a t tack was computed using both f u l l y d i r ec t and subdomain s t ra teg ies . A 105 x 50 H-grid was used t o d i s c re t i ze the flow domain. Due t o poor grid reso lu t ion of the blunt nose region, first order upwind differences were used near the nose t o suppress s p a t i a l o sc i l l a t i ons .

Results f o r Re - 1000 and Re = 10000 a r e shown i n f i g s . 3 and 4. No underrelaxation was necessary. For Re - 1000, uniform flow was spec i f ied 'as an i n i t i a l condition. For Re - 10000 a smoother i n i t i a l condition was heeded fo r s t a b i l i t y : t he Re - 1000 so lu t ion was used.

Convergence h i s t o r i e s fo r the Re - 1000 case a r e shown i n f i g 5. For the subdomain case the flow f i e l d was s p l i t i n t o two horizontal subdomains with a t h i r d overlapping subdomain. A s can be seen, the convergence r a t e using three subdomains is v i r t u a l l y the same a s solving the e n t i r e f lowfield d i r ec t ly . The ac tua l computing time f o r one i t e r a t i o n of t he subdomain case was only 0.67 times the time f o r the f u l l y d i r e c t case. This is because the time required t o do the LU decompositions increases super l inear i ly with the number of unknowns. Therefore the time fo r three so lu t ions of the subdomains, with approximately 5000 unknowns each, was l e s s than doing one so lu t ion of the e n t i r e gr id with 10500 unknowns. The memory required f o r the LU '

decompositions f o r t he subdomain case was 170,000 words f o r the smallest subdomain and 280,000 fo r the la rges t : t he f u l l y d i r e c t so lu t ion required 630,000 words. Solut ion time was about 43 minutes an i t e r a t i o n on a Charles River Data Systems Universe 68 ( a MC68000 based microprocessor with 4 Mbytes of memory): t h i s is approximately 33 times slower than the University of Cincinnat i ' s Amdahl 470.

Sine Wave Air fo i l

This problem has been considered due t o its importance t o the study of massive separation. Smith and Cheng ( r e f . 8) have shown the existence

. of asymptotic s t eady ' s t a t e massive separation so lu t ions a t high Reynolds numbers. Here so lu t ions of t he f u l l Navier-Stokes equations have been calculated using the f u l l y impl ic i t procedure described above. It is f e l t t ha t t ha t the robustness of the procedure with respect t o i t e r a t i v e methods could give ins ight i n t o the existence of such flows.

Ramakrishnan and Rubin ( r e f . 10) solved the reduced Navier-Stokes (RNS) equations using the coupled s trongly impl ic i t method t o inves t iga te t h i s problem. They invest igated thickness r a t i o s between 0.04 and 0.056 and Reynolds numbers from

100,000 t o 600,000. A t a Reynolds number of 600,000 an i n s t a b i l i t y was encountered i n t h e separa t ion bubble. The present f u l l y i m p l i c i t algorithm has been app l ied t o t h i s problem t o - determine i f s o l u t i o n s at higher Reynolds numbers

. can be obtained.

The s i n e wave a i r f o i l is descr ibed by

I n add i t ion a s p l i t t e r p l a t e is a t t ached t o t h e t r a i l i n g edge and extends t o x - 1.5. A th ickness r a t i o of t / c - .04 has been used f o r a l l

. ' t e s t s . A 205 x 58 orthogonal g r i d was generated

using t h e Schwartz-Christoffel mapping of Davis

( r e f . 16) . A mesh spacing of 2x1 oL5 a t t h e a i r f o i l su r face and an exponent ia l s t r e t c h i n g were used t o reso lve t h e boundary l ayer . The s t r e t c h i n g f a c t o r is increased a f t e r t h e first 20 po in t s i n order t o put t h e o u t e r boundary a t a minimum of 20 chord lengths . 150 po in t s were d i s t r i b u t e d uniformly a c r o s s t h e a i r f o i l and an exponential s t r e t c h i n g was used ahead and behind of t h e a i r f o i l .

So lu t ions were generated f o r Reynolds numbers from 100,000 t o 800,000. I n o rder t o make t h e s o l u t i o n procedure a s i m p l i c i t as poss ib le t h e e n t i r e g r i d was solved simultaneously and n o , overlapping g r i d scheme was used.

The s k i n f r i c t i o n c o e f f i c i e n t is shown f o r these cases i n f i g . 6. A t a Reynolds number of 900,000 an i n s t a b i l i t y occurs i n t h e separa t ion bubble ( f i g . 7) and a s o l u t i o n h u l d no -longer be obtained. Although s o l u t i o n s were obtained a t higher Reynolds numbers than those i n ref. 10, i n c i p i e n t separa t ion was a l s o assoc ia ted with higher Reynolds numbers. I n f i g s . 8 and 9 comparison between t h e p resen t r e s u l t s and those of r e f . 10 a r e shown. For t h e RNS results separa t ion s t a r t s a t a Reynolds number of 100,000 while t h e present r e s u l t s do no t s e p a r a t e u n t i l about 300,000. The RNS s o l u t i o n a t 400,000 looks s i m i l a r t o t h e present h igher Reynolds number r e s u l t s i n f i g . 6. One poss ib le cause f o r t h i s discrepency was thought t o be t h e absence of streamwise viscous terms i n t h e RNS approximation. I t was f e l t t h a t these terms may no t be n e g l i g i b l e a t reattachment due t o t h e s t e e p streamwise g rad ien t s there. However, numerical tests were considered neg lec t ing these terms i n t h e present system and no apprec iab le change from the o r i g i n a l s o l u t i o n was d-iscerned. S imi la r t e s t s were c a r r i e d ou t by Ramakrishnan and Rubin ( r e f . 17) by adding streamwise viscous terms t o t h e i r RNS system: once again no s i g n i f i c a n t change i n t h e i r o r i g i n a l s o l u t i o n s was-obtained. A t present t h e . cause of the discrepency i n Reynolds number is unclear ; however, the q u a l i t a t i v e flow breakdown

. n t l a r g e Reynolds numbers has been confirmed and -equires f u r t h e r s tudy. The p resen t Navier-Stokes

. r e s u l t s were computed using s i n g l e p rec i s ion a r i thmet ic . This has not caused problems previously , however a t t h e high Reynolds numbers encountered he re t h i s could be a problem due t o the extremely fine'meshes required. Double p rec i s ion a r i t h m e t i c and f i n e r meshes are now being used t o i n v e s t i g a t e t h e problem f u r t h e r .

Unsteady Problems

he unsteady flow about an a i r f o i l a t l a r g e ang les of a t t a c k has been r e c e n t l y given ex tens ive s tudy (ref. 6,7,181. Osswald and Ghia s o l v e t h e g - w sys tem ' in an uncoupled fash ion , us ing a d i r e c t s o l v e r f o r $ and a time c o n s i s t e n t A D 1 procedure f o r w. A d i s t i n c t advantage of t h i s method is that t h e poisson equat ion f o r $ is l i n e a r and has time i n v a r i a n t c o e f f i c i e n t s . This r e s u l t s i n a time i n v a r i a n t matrix f o r t h e f i n i t e d i f fe rence r e p r e s e n t a t i o n of the $ equation. The time consuming mat r ix f a c t o r i z a t i o n has t o be performed and s t o r e d only once. The time marching procedure then c o n s i s t s of an A D 1 sweep f o r w and a d i r e c t s o l u t i o n f o r I), us ing t h e f a c t o r i z e d matrix and t h e updated values of w. Uncoupling the equat ions however, f o r c e s the boundary cond i t ions f o r w a t t h e no-s l ip s u r f a c e t o be lagged. This results i n first order accuracy i n time and l e a d s t o s t a b i l i t y r e s t r i c t i o n s on t h e time s tep . I n add i t ion , t h i s method cannot be appl ied t o the f u l l y coupled non l inear system of equat ions , a s would be encountered i n compressible flow. A f u l l y coupled s o l u t i o n of t h e $ a w system has been m v e s t i g a t e d here. The f i n i t e d i f f e r e n c e equat ions a r e l i n e a r i z e d i n time and solved f o r a t each time s t e p . While t h i s method needs a new f a c t o r i z a t i o n each time s t e p , it does not s u f f e r s t a b i l i t y l i m i t a t i o n s and is second o rder a c c u r a t e i n time and space.

The d i f f e r e n c i n g scheme is e s s e n t i a l l y t h e same as i n t h e s t eady case. The K-R d i f f e r e n c i n g is abandoned i n o rder t o o b t a i n f u l l y i m p l i c i t c e n t r a l d i f f e r e n c i n g and thereby maintain second o rder accuracy f o r t h e time varying so lu t ions . The time term he lps r e i n f o r c e t h e main diagolial of the w equat ion s o sucn d i i f e r e n c i n g is not necessary. The time term is represented by a three point second o rder backward d i f fe rence .

This procedure has been t e s t e d on a 122 t h i c k Joukowski a i r f o i l a t 15 d e p e e s angle of a t t a c k and a Reynolds number' of 1000. A conformal mapping ( r e f . 19) was used t o generate a 249 X 54 orthogonal CRgrid. The mesh s i z e normal t o t h e

a i r f o i l is approximately 4 X 1 0 ' ~ a t t h e a i r f o i l surface . A hyperbol ic tangent s t r e t c h i n g is used t o r e s o l v e t h e boundary l a y e r and t o g e t a f a i r l y . even d i s t r i b u t i o n i n t h e normal d i r e c t i o n c l o s e t o t h e a i r f o i l . An exponent ia l s t r e t c h i n g is then used t o p lace t h e o u t e r boundary a t l e a s t 20 chord l e n g t h s from t h e a i r f o i l . Fig. 10 shows t h e g r i d near t h e a i r f o i l .

Prel iminary r e s u l t s have been obtained using time s t e p s of 0.05 and 0.1. Fig. l l a shows contours of t h e s t reamfunct ion a t t ' - 10 and f i g . 1 l b d e p i c t t h e v o r t i c i t y contours. The method has proved t o be very robus t with no s t a b i l i t y l i m i t a t i o n f o r t h e time s t e p s t e s t e d . The t e s t s were run on a Microvax I1 with a CSPI 6420 a r r a y processor having 16 megabytes of memory. The s o l u t i o n requ i red 15 minutes of processor time per time s t e p and t h e f a c t o r i z a t i o n requ i red 7.1 Megabytes of memory. Work is i n progress t o o b t a i n q u a n t i t a t i v e r e s u l t s f o r comparison with o t h e r work.

If a subdomain s t r a t e g y is t o be used f o r unsteady problems, it is d e s i r a b l e t o avoid

i t e r a t i o n a t each t i m e s t e p s i p c e t h i s inc reases the computer cos t s . A time c o n s i s t e n t approach r e q u i r i n g only one o r two p a s s e . through t h e subdomains is desi red. I n such a case , t h e

. d i f f e r e n c e equat ions on t h e subdomain boundaries i n t h e i n t e r i o r of t h e flow f i e l d r e q u i r e s p e c i a l treatment. A s an example, the.'heat equat ion is considered and is w r i t t e n i n t h e fol lowing d i f f e r e n c e form:

where 6: and 6' a r e c e n t r a l d i f f e r e n c e opera to r s Y

f o r approximating Txxand T YY'

On a subdomain

boundary, where f o r example y - cons tan t , 6'T Y i J

cannot be represented i m p l i c i t l y a t t h e n th time l e v e l . In order t o keep t h e method second order

accura te , 'T" on t h e subdomain boundary is id

replaced with an i n t e r p o l a t i o n of t h e form

The f u l l y i m p l i c i t d i f f e r e n c e equat ion is . solved i n t h e subdomain i n t e r i o r , and t h e modified e q u a t i o n . 1 ~ solved on t h e subdomain boundary.

Numerical t e s t s have shown t h i s s t r a t e g y t o be unstable . However, i f t h i s s t r a t e g y is app l ied t o generate an i n i t i a l p red ic to r s o l u t i o n , a cor rec ted s o l u t i o n can be computed using overlapping subdomains. I n t h e c o r r e c t o r pass , 6'T on t h e subdomain boundary is now known from

Y i j the predic ted s o l u t i o n , . s o t h a t i n t e r p o l a t i o n i n time is not necessary. Numerical t e s t s have shown t h i s method t o be s t a b l e , f o r a l l time s t e p s , f o r the heat equat ion on a u n i t square.

Attempts t o apply t h e subdomain s t r a t e g y t o the $ w system f o r t h e unsteady a i r f o i l problem have t h i s far have been unsuccessful. The major d i f f i c u l t y seems t o be t h e l ack of a time term i n the $I equation. This f a c t o r a long with high g r i d aspect r a t i o s on subdomain boundaries seem t o be causing an i n s t a b i l i t y . Adding an a r t i f i c i a l time term t o t h e $ equa t ion-on t h e subdomain boundary removes t h e s t a b i l i t y problem. However, i t e r a t i o n is then requ i red at each time s t e p ; an i n e f f i c i e n c y t h a t t h i s method w a s t r y i n g t o avoid. These problems a r e being inves t iga ted . It would '

seem, however, that t h e problems caused by t h e lack of a time term can be avoided if a formulation where time terms a r e present i n a l l squat ions were used.

Conclusions

The Yale Sparse Matrix Package has been used f o r t h e s o l u t i o n of t h e Navier-Stokes equat ions .for both s teady and unsteady problems. The r e s e n t method has proven t o be very robus t with l i t t l e o r no dependence on underre laxat ion f a c t o r s and is competit ive with i t e r a t i v e techniques wi th nespect t o computation time. Steady s o l u t i o n s of flow i n a c a v i t y and about a NACA0012 a i r f o i l have been obtained using both t h e f u l l y d i r e c t method snd t h e subdomain s t r a t e g y . The subdomain s t r a t e g y has been proven t o be e f f i c i e n t f o r these 2roblems. This s t r a t e g y requ i red l e s s computer

s to rage and time than t h e f u l l y d i r e c t method. High Reynolds number s o l u t i o n s have been obtained f o r flow over a s i n e wave a i r f o i l us ing t h e f u l l y coupled approach. Again, no underelaxat ion was necessary t o ob ta in t h e so lu t ions . The problem is being s t u d i e d by t h e au thors t o see i f higher Reynolds numbers SOlutiOnS than those a l ready obtained can be found. Prel iminary work on unsteady flows around a i r f o i l s a t high angle of a t t a c k is promising. The f u l l y coupled s o l u t i o n s do not have r e s t r i c t i o n s on time s t e p . The use of a time c o n s i s t e n t subdomain s t r a t e g y needs i n v e s t i g a t i o n f o r t h e streamfunction-vorticity equations.

Acknowledgements

The au thors would l i k e t o thank S.V. Ramakrishnan and S.G. Rubin f o r many use fu l d iscuss ions . This 'work was supported i n p a r t by t h e A i r Force Of f ice of S c i e n t i f i c Research Contract no. F49620-85-C-0027 and i n p a r t by the NASA t r a i n i n g g r a n t NGT36a004-800.

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ig. 1. Computer Storage Required for ?k ':.. 1.:

2a. Streamfuncri rr C:. '.x~rs for Driven Cavit-. Le - C - ' .

Vertlcdl a x i s exirggwirled t'or clitrtty

3a. Streamf~nctio~ Zontours, NACA0012 Airfoil, Re - 1000.

Fig. 2b. Vorticity Contours for Driven Ca..-' '.- ., Re = 1000.

Fig. 3b. Vorticity Contonrs, NACA0012 Airfoil, Re = 1000.

Vcdrt ical ;I% 13 ex.lt:~:t'r'dtcd f o r c1ar.i t y

Fig. 4a. Streamfunction Contours, NACA0012 Airfoil, Re = 10,000.

"+. , -- . --. . , r -1--.---I 1. 00 2. 00 3.00 4. 00 5. 00 6. 00 7.00 8. 00

ITERWrION NO. X10 a

ig. 5. Convergence Sistory - Airfoil, Re = LOCO.

1,. 7 . Skin Friction Coefficient Vs. X Showing Instability in Separation Bubble (6th Iteration).

Verticdl a x i s rxirgguratud f o r C 1 ; l f ' l C Y '

Fig. 4b. Vorticity Contours, NACA0012 Airfoil, Re = 10,000.

Fig. 6. Skin Frinction Coefficient Vs. X for Sine Wave Airfoil.

Fig. 8. Skin Friction Coefficient Vs. X. Comparison between Full Navier-Stokes and RNS Solutions. Re = 100,000.

3 g . 9 . Skin Friction Coefficient Vs. X. Comparison Between Full Navier-Stokes and RNS Solutions. Re = 400,000.

'ig. Ila. !?treaiiunction Contours for Joukowski ILrfoil, oL = 15 and Re = 1000.

Fig. 10. Computational Grid for Joukowski Airfoil.

Fig. llb. Vorticity Contours for Joukowski Airfoil, e- 1s and Re - 1000.