-11253 FINL REPORT ON AFOSR (RIPt FORCE OFFICE OF SCd NYIFICRESECH) CONTRACT. (U) HOSSACHUSETTS INST OF ECHCOMSIDOE DEPT OF MATHEMATICS S A OASZAO NAY 67

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FINAL REPORT ON

AFOSR CONTRACT F49620-83-C-0064

Steven A. Orszag, Principal InvestigatorDepartment of Mathematics

MITCambridge, MA 02139

Volume 1

DTIC

ELECTEMA 1988

H

VUTREUfIo-N STATKET A 88 1 200Approved for public ree.me;

Dinutbution Unlimited

uNcLASSIFIED

SECURITY CLASSIFICATION OF rI... PAGE AD ,REPORT DOCUMENTATION PAGE

i. REPORT SECURITY CLASSIFICATION IT). RESTRICTIVE MARKINGS

unclassified______________________2&. SECURI1TY CLASSIFICATION AuTHOlrTY 3. OISTRIeuTION/AVAILaiI.Y OF REPORT

2b. oEcL.AssIFicAToNiaowvmGPAoiNG SCHEDULE A f- 4;L 'I PD&4 b~ Ir.4 t'

A. PERFORMING ORGANIZATION REPORT NUMBIERMS 5. MONITORING ORGANIZATION REPORT NUMSERIS)

AFQSR-TR- 8 7 -1 3906a. NAME OF PERFORMING ORGANIZATiON So OFFICE SYMBOL 7a. NAME OF MONITORING ORGANIZATION

ac. AOORESS (city. Stage and ZIP CodaI 7b. AO R? (ct.seat* And ZIP coda)

Departrent of Matheaticsq1I )#lA tFa O-

Carnbridae, % 02139 _____ .~Z-£ qSAL NAME OP PUNOINGISPOINSORING Sb. OFFICE SYMBOL 9. PROCUREMENT INSTRUMAENT rDEN'T1FICATION NUMBERN

ORG ANIZATION (it 0PUiCable,

AFOSR INAJ N A F49620-83--C-0064Sc. AOORIESS (City. State and ZIP Coda) I~ t .bll V' 10. SOURCE OF FUNOING NOS.

Boln i oc aePROGRAM PROJECT TASK WORK UNIT

WsigoD203-48ELEMENT NO. NO. NO. NO.

1 1. TI TL " Ineduaa Securt ty Claaatficationo Final Recort on 6 P 7 A 2-AOR Cotrac F49620-83-C-0064_________________________

12. PERSONAL AUTHOR(S)

*Steven A. OrszaaM3 TYPE OF REPORT 13b. TIME COVERED 14. OATE OF REPORT (Yr.. Wo.. 4.)a ~ 1S. PAGE COUNTli

16. SUPPLEMENTARY NOTATION

17 cCOsATiCODES 18. SUBJECT TERMS tContin&.a onat. .w. .ewy and.dandfy by block 'itu',bFD .

19. ABSTRACT lContinuo oR reverse if necesary~ and identify by blocm number)

We sumrmarize work done under AFOSR Contract F49620-83-C-0064. The major results include:development of renorinalization group techniques for lairge-eddv simulations of turbulentflows, the first direct numnerical. simulation of turbulent spots in channel and boundary-layer f lows, the further development of spectral method~s for turbulence simulations,the identification of secondary instability modies in free shear layers, the developmentof an efficient multi-grid marching methcd for solution of the parabolized Navier-Stokesequations, a mathematical analysis of boundary conditions for the parabolizedccxrressible Navier-Stokes equations, the further develorent of a method to irprovenuerical. solution of singular perturbation problems by use of asymptotic approximations.-

20. DISTRIeuTiONIAVAILABILITY OF ABSTRACT 21. ABSTRACT SECURITY CL-ASSIFICATION

UNCLA3SIPIED/UNLIMITEO tAM ASRT TCUES(~caS$ el

22a. NAM4 OF RESPONSIBLE INOIVIOUA&. 22b. TELEPHONIE NUMBER 22r. OFFI1CE SYMBOL

0D FORM 1473, 83 APR EOITICPP OF I AN4 73 IS OBSOLETE.

-% ',

FINAL REPORT ON AFOSR CONTRACT F49620-83-C-0064

Steven A. Orszag, Principal Investigator

Department of Mathematics

MIT

Cambridge, MA 02139, . - . . ,

-In the-attachtd-paperT-we summarize work done on this researchproject. The major results include:

1. Development of renormalization group techniques for large-eddy

simulations of turbulent flows.

2, The first direct numerical simulation of turbulent spots in channel

and boundary layer flows.

3. The further development of spectral methods for turbulence

simulations.

4, The identification of secondary instability modes in free shear

layers.

5. The development of an efficient multi-grid marching method for

solution of the parabolized Navier-Stokes equations.

6. A mathematical analysis of boundary conditions for the parabol-

ized compressible Navier-Stokes equations.

7. The further development of a method to improve numerical solu-

tion of singular perturbation problems by use of asymptotic ap-

proximations.

Further details are given in the attached papers.

-I-

..............................-..... '. , % % % *%% .5 ," "' ,"#%" " -°.'

List of Papers a

M. Israeli and P. Bar-Yoseph, Numerical Solution of Multi-Dimensional

Diffusion-Convection Problems by Asymptotic Corrections, in Proc.

Fifth GAMM-Conference on Num. Meth. in Fl. Mech., Friedr. Vieweg

& Sohn, 1984.

S.A. Orszag, Instabilities and Turbulence, in Turbulence and Chaotic

Phenomena in Fluids, (ed. by T. Tatsumi), North-Holland (1984).

V. Yakhot and S.A. Orszag, Renormalization Group Formulation of

Large Eddy Simulation, in Non-Linear Dynamics of Transcritical Flows,

(ed. by H. L. Jordan), Springer (1985).

S.A. Orszag, Lectures on Spectral Methods for Turbulence Computa-

tions, in Proc. of International School of Physics 'Enrico Fermi', (ed. by

M. Ghil), North-Holland (1985).

M. Israeli, Marching Iterative Methods for the Parabolized and Thin S

Layer Navier-Stokes Equations, NASA Contractor Report

178028/ICASE Report No. 85-60 (1985)..e

E.T. Bullister and S.A. Orszag, Numerical Simulation of Turbulent Spots

in Channel and Boundary Layer Flows. J. Sci. Comp., in press. S.

(NSPEC '

M. Rosenfeld and M. Israeli, Numerical Solution of Incompressible ,___

Flows by a Marching Multigrid Nonlinear Method, AIAA Journal 25, 5 A-cessin For

(1987).NTIS GA&I(1987). DTIC TAB

U'11nnouncodJu';t i f'-Cnt loll--,

D1 st rl b t i ot,/lvpll

Dist %S

-2- 5

Maurizio Pandolfi/Renzo Piva (Eds.)

Proceedings of theFifth GAMM-Conferenceon Numerical Methodsin Fluid MechanicsRome, October 5 to 7,1983

With 263 Figures

I.1,

Reprint

Friedr. Vieweg & Sohn Braunschweig/Wiesbaden

NV LW'N

131

NUMERICAL SOLUTION OF MULTI-DIMENSIONAL DIFFUSION-CONVECTION PROBLEMS

BY ASYMPTOTIC CORRECTIONS

M. Israeli* and P. Bar-Yoseph**

Technion - Israel Institute of TechnologyHaifa, Israel

SUMMARY

The Booster Method for improvement of the numerical solution of par-tial differential equations by the addition of asymptotic corrections tothe right hand side is presented. It is applied here to the diffusion-convection equation for the case of 'small' diffusion. The correctionterms were used in finite difference and finite element schemes. Thefinite element results were used as reference for checking the performanceof the finite difference schemes. Excellent results were obtained withoutthe use of upstreaming or artificial diffusion. Theoretical expectationswere confirmed.

1. INTRODUCTION

Singularly perturbed initial and boundary value problems for partialdifferential equations appear in various fields of application such asfluid dynamics, heat transfer, transport of atmospheric pollution, etc. Inparticular, such equations appear in diffusion-convection processes. Oftenthe (normalized) diffusion coefficient c becomes small, and thin boundaryor interior layers appear within the region of interest. Consequently,these problems become increasingly difficult to solve numerically by dis-cretization methods.

we would like to avoid the use of a prohibitively large number of gridpoints, as required for resolution by straightforward numerical methodswhen C decreases. To this end, several approaches are possible, such asthe use of nonuniform meshes, adaptive techniques, positive type schemes,etc. The question of applicability of such schemes to multi-dimensionalproblems is presently open.

A different approach is motivated by classical singular perturbationmethods where 'inner' and 'outer' solutions are combined to give approxi-mate solutions. These solutions become more accurate as the equationsbecome stiffer, however, the error is fixed for a given E and cannot beimproved or estimated reliably in most cases of interest.

The Booster Method attempts to combine the asymptotic approach, withknown discretization methods, in order to obtain a numerical method which

improves when £ becomes smaller. At the same time, it keeps the proper-ty that the error can be made arbitrarily small for any fixed e by refin-ing the computational mesh (Israeli and Ungarish [1],[21). For the one-dimensional case, we were able to prove that an improvement by a factor ofcn+l can be obtained where £ is the 'small' parameter and n is theorder of the asymptotic approximation used (Israeli and Ungarish [2). Weexpect similar behaviour in the multidimensional case [1].

In the present paper, we investigate a multi-dimensional applicationto diffusion-convection problems.

nj* Department of Computer Science, Technion.

** Department of Mechanical Engineering, Technion.

.

132

2. FORIULATION

We consider the transport of a quantity q in a rectangular region.The normalized partial differential equation is

2 -L(q) : -EVq +V • Vq = 0 (1)

In the present application the velocity field V is assumed to beknown and q is specified on the boundaries. This problem was often usedas a test case for various finite difference and finite element methods ofsolution and it is well known that most methods fail as the cell Reynoldsnumber (Vjh)/C becomes larger than 0(l) (here h is a representative

mesh size). For example centered schemes develop unphysical oscillationsin space, while uncentered schemes have unacceptable artificial diffusionand are of lower order over the same computational stencil.

The Booster Method uses an asymptotic approximation j(x,y) to the

solution q(x,y) in order to reduce the truncation error in the numerical

scheme.The 'usual' numerical solution Q(x,y) (defined only at grid points)

is obtained fromL N(Q) = f , 2)

where LN is the discrete approximation to the differential operator L.

The improved numerical solution Q is obtained from

L (Q) - f+L (j) -L(j) (3)N N

Here L(j) is the differential operator applied to the approximate solu-

tion. Thus the Booster Method applies an asymptotic correction to the

right hand side of the equation and therefore requires a negligible amount

of extra work. It can be used with any numerical scheme without modifica-tion in the method of solution.

The same basic hpproach of using asymptotic corrections to the righthand side can be used to improve the Standard Finite Element (SFE) method.The resulting Asymptotic Finite Element (AFE) method is described brieflyin the following; for details see Bar-Yoseph and Israeli (41,[5]. .

Suppose that the unit square is divided into elements and that the

variation of q within the given region is approximated by

Q( X) -N.(x)Q. ,(4)

where Qi is the value of the approximate solution at the i-th nodal point

and Ni is the corresponding global trial function (we use the summationconvention, with summation over the nodes within the given region). TheBubnov-Galerkin finite element scheme of eq. (1) is given by

T( N + (N QVV ) h=(N., f)h()C(7TN.,VNiQ.) *v ,.VNi )£ i. h + j i h = N,) h ' l ,2.....m , (5) .

where m is the number of inner nodal points and (*,*) denotes the usual

inner product in L2(M). The subscript h in (',')h denotes an approxi-mation to (*,.) obtained by a quadrature rule.

Our corresponding asymptotic finite element (AFE) scheme for eq. (1)is the following

'a %.~~w arav. ~ ~%~ ~ ~ *,.d. ~ . . - .- ,.

133

E(V NjVNiQi)h + (N ,V'VNiQi) h = (N ,f) hj i (h j ,Vii ) h • ...h''

+ {C(V NjVNiq ) +(N V - (N ,L4) j = ,2. m (6)j iih j iihj

where 4i is the value of the asymptotic solution at the i-th nodal point.Here the terms included in the first line coincide with the SFE scheme,eq. (5) , while the terms in the second line (in curled brackets) representthe correction term which is the essence of the present AFE scheme. ThisAFE scheme can also improve the pointwise error estimate of the SFE schemeby a factor of the O(Cn+l) [4].

Usually the finite element solution supplies values everywhere inside

the elements via the interpolation (4). Applying the' same interpolationusing Qi instead of Qi will not give good results within the elements

especially when there are no nodal points inside the boundary layers. Oneshould use instead the interpolation

q(' x) ()+N.(W)(Qii ) "

which recovers the proper boundary layer behaviour.

3. ASYMPTOTIC SOLUTION

The approximation q(x,y) can be obtained by the method of matchedasymptotic expansions (Cole [3]). Such approximations usually satisfy theboundary conditions and become increasingly accurate as C decreases.Often the error decreases like some power of E depending on the number ofterms used in the construction of the solution (see Table I).

We first construct the zeroth order asymptotic approximation from theouter solution q' where

V Vqo- 0 ,(7)

and from the boundary layer qb satisfying a one-dimensional boundarylayer equation in the direction normal to the boundaries (where the flowexits the computational region). A complete zeroth order solution shouldalso include corner regions, tangential regions, and boundary layersdeveloping from discontinuities in boundary conditions. Restricting our-selves for now to continuous boundary conditions and constant velocityfield not tangent to any boundary, we find that only the exit corner layerhas to be included.

Equation (7) implies that the solution remains constant along stream-

lines, consequently it carries with it the q-values entering the computa-tional region. The difference between these values and the values encount-ered at the exit forces the boundary layers.

we consider the flow in the unit gquare with vertices (0,0), (0,1)

(1,0), (1,1). Let the components of V be u and v (both positive) and

the differences of the exit boundary x - 1 and y - 1 be f(y) and g(x)respectively. Then the structure of the boundary layers will be

- f(y)eu(X1)/c ()v(Y-l)/C (x-)/ V(y-)/)

where p - f(1) - g(l) and f(0) - g(0) - 0 by assumption.The third term in (8) is the corner boundary layer and it satisfies %

the differential equation exactly (in this particular case). The approxi-mation " is obtained by adding q from (7) to qb from (8). It is a

134

uniformly valid approximation in the square, and is easily adapted to par-ticular cases. In our example u - v- 1-12 and the outer solution is

q = siniT(x-y) (9)

corresponding to the boundary conditions

q(x,0) - sin'Tx; q(0,y) -sinry (10)

On the other hand we take

q(x,l) - sinT(x-l) + x, q(l,y) sinlT(l-y) +y , (1)

giving rise to f(y) - y and g(x) = x, which satisfy the requirement of

continuity on the boundary.

4. RESULTS

We solved the differential equation (1) in the unit square with theboundary conditions (10) and (11) on a net with 6, 12 and 24 equal inter-vals in the x and y directions. We used centered three-point differencesfor the first and second derivatives. As we do not have the exact solu-tion for this problem, we used as reference a solution obtained with theAFE scheme employing a uniform mesh of 24 x 24 biquadratic Lagrangian ele-ments (more details appear in [5). The range of C reproduced in thetables was C - 0.01, e - 0.02, c - 0.05.

As a seminorm we used a weighted average of the absolute value ofthe errors. A proper mix of interior and boundary layer points was ob-tained by taking only the 25 mesh points contained in a square near thecorner (1.1).

We observe that (Table I) the error in the asymptotic approximationA(E,h) decreases roughly like C. The error in the regular (Table III)numerical solution N(c,h) decreases very slowly with h and increasesas c decreases.

We note first that the errors in Table II are in all cases smallerthan the corresponding errors in Tables I and III. Moreover the analysisof [2] indicates that the error in the improved solution B(c,h) shoulddecrease with C and h (Table I). In fact it should be proportionalto the product of the previous errors. Table IV presents the ratioK(E.h) - B(E,h)/((A(C,h)N(£,h))). We expect it to approach a constant,the

fact that the values in Table IV are not far from unity makes the boostermethod quite attractive.

We note that the centered method is quite useless by itself. Ourexperience shows that the Booster Method works equally well with otherschemes and other discretization methods.

ACKNOWLEDGEMENT

The first author was partially supported by AFOSR undercontract number P49620-83-C-0064.

. . . . . - " " " - "C ' ". . ", "",""',,,' ",.." ."' ". ," . . '..'.... .,' .. '....,

135

REFERENCES

[1] Israeli, M. and Ungarish, M.: Proceeding of the ICNMFD7, Stanford,1980, Lecture Notes in Physics, 141, Springer-Verlag.

[2] Israeli, M. and Ungarish, M.: Numer. Math. 39, 309-324 (1982).

[3] Cole, J.D.: Perturbation Methods in Applied Mathematics, Waltham,

Mass., Blaisdell Publ. Co., 1968.

(4] Bar-Yoseph, P. and Israeli, M.: An asymptotic finite element methodfor improvement of finite element solutions of boundary layerproblems (to appear).

[5] Bar-Yoseph, P. and Israeli, M.: An asymptotic finite element methodfor 2-D convection-diffusion problems (in preparation).

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4.,S

Lectures on Spectral Methods for Turbulence Computations.

.S. A. UnSZ',(. (*)

3lIotv/rhasetls ostitute of Tch nology - Caitmbridge, . A 02139

1. - Introduction to spectral methods.

Spectral methods are based on representing the solution to a problem asa truncated series of smooth functions of the independent variables. Whereasfinite-element methods are based on expansions in local basis functions, spectralmethods are based on expansions in global functions. Spectral methods are %the extension of the standard technique of separation of variables to the solution %of arbitrarily complicated problems.

Let us begin by illustrating spectral methods for the simple one-dimensionalheat equation. Consider the mixed initial-boundary-value problem '

aU(.r, t) -~et~,1(1.1) -K (0< x<=, t > 0),

(1.2) U(O, ) = u(n, 1) = 0 (t>0)

(1.3) U (X, 0) = A(x) (O< X <n) ."

The solution to this problem is del

(1.4) u(x, t) - a.(t) sin nx.,

(1.5) a.(t) = 1. exp [-/,a t],

where .

2I(1.6) -- f (x) sin 1..

.3.

are the coeflivients of the Fourier sign series cxpan,.ion of I(x).tt

{)Presenlt addre,'-4: IPrinetoti Vviw.t ity, P'riiic4 ,hi. XJ (A-140.

107

,'.%

.1

Ir

N4

108

(1.1) to

(1.7) l,(.,, t) = 1 .(F) in a.,

and replaing (1.5) by the evolution equation

(1.8) - (V)

with the initial conditions a,,(O) = f. (n = 1,. ).The spectral approximation (1.7), (1.8) to (1.1)-(1.3) is an excee(lingly good

approximation for any time t greater than zero as N-) oo. In fact, the erroru(x, t) - u.v(x, t) satisfies

(1.9) U(X,t)-tN(rt) = -A f0 exp [- Kn2t] sinx = O(exp [- KN2 t])

for any I > 0. In contrast to (1.9), finite-difference approximations to the heat ,equation using N grid points in x lead to errors that decay only algebraically 1.,I

with N as N -) 00. Furthermore, this spectral method for the solution of theheat equation is efficiently implementable by the fast Fourier transform (FET)in O(V log N) operations.

There are several significant difficulties in extending the simple spectralmethod employed for (1.1)-(1.3) to more general problems. Among these dif-ficulties are those caused by imposition of nontrivial boundary conditions, Cnonlinear and nonconstant coefficient terms, and complex geometries. These

difficulties and their solutions will be discussed below (see also [1, 2]). %The Fourier series (1.4) converges fast if u(x, t) is infinitely differentiable

and u(x, f) satisfies the boundary conditions

(1.10) 0 (t(x,0, E)

for all nonnegative integers n. Under these conditions, the error after N teri.s

e.%(x, t) = u(.r, t) - a,(t) sin nrx

got,, to z'ro unif'rlily ill . fl ster than any )ON er of I 'N as N-- 0 . Oil the

ot her halld. if V(.r, f) is lit)t i nfinitely differenti.le or if any of tli(. conditions(1.111) is viil: ,ted , tl l) : 0(1,',N" ) 'IS N flt ?,ollt' fiiilt j). For

% %••V-VrV - I* S. p ~%%,S5*' ~ v

% i

it ll-l' (ON At-I, l -, 1,l1111l1 I'M II It ! I II I \E-1; ' l 1 10 "',

s'i X llpl (', 1

but tle error invurred by truncatijig after N terns is of order /AN for ili vfixed x, 0<.r< t. Fturtherinore, the convergence of (1.11) is not uniformin r; (1.11) exhibits Gibbs' phenomenon, namely

1(/0) (M- oo, $ fixed)

For any fixed N, there are points x at which the error after N terms of (1.11)is not small. The poor convergence of (1.11) is due to the violation of (1.10)for n = 0.

More generally, most eigenfunction expansions of a function f(x) convergefaster than algebraically (i.e. the error incurred by truncating after N termsgoes to zero faster than any finite power of 1/N as N-* oo) only if !(x) is infi-nitely differentiable and I(x) satisfies an infinite number of special boundaryconditions. For example, the Fourier-Bessel expansion

fix) = a.Jo(A.x) (o <.r <1) ,- -o

where A,, is the n-th smallest root of J,(A) 0 0, converges faster than algebrai.cally only if / is infinitely differentiable and

(1.12) ._X f(X) =O at X- 1

for k = 0,1, 2,...

When a spectral expansion converges only algebraically fast, spectralmethods based on these eigenfunction expansions cannot offer significantadvantages over more conventional (finite-difference, finite-element) methods.Eigenfunction expansions of this kind should not normally be used unless %

the boundary conditions of the problem imply all the extra boundary con-

straints like (1.10) or (1.12). For example, if periodic boundary conditionsare compatible with the differential equation to be solved, complex Fourierseries are suitable to develop efficient spectral approximations.

In the development of spectral methods for general problems, it is importantthat the rate of convergence of the eigenfunction expansion being used doesnot depend on special properties of the eigenfunctions, like boundary conditions,but rather depend only on the smoothness of the function being expanded. Ofcourse, if the solution to the problem being solved is not smooth, one shouldnot exl)ect errors that decrease faster than algebraically with 11-Y when global

I.

110 S. A. 01:SA%,,..

eijrelfuili(.tioll V xpalgilo s art u.scd. Fa.ter tli'n algebrtic Iateti (of coliv.Ir4i'l ,-.

way be achieved for these probleis by either at.hitl.. esoltitioll ;it dis.cn-tilluitie. or pIe- anid post -proc(ts i- f the stoltioll (,A(e [-]).

There ix at easy way to ei sure that the rate of convergene of a Npectral

expansion of a function j(.x) depends only on the smoothless of /(x), not on it',

boundary properties. The idea is to expand in terms of suitable chsies oforthogon.1 llynonmials, including Chebyshev and Legendre polyiicimials forall those problems in which constraints like (1.10) and (1.12) are unrealistic.

These polynomial expansions avoid all difficulties asscciated with the Gibbsphenomenon provided the solution /(x) is smcoth.

From the mathematical point of view, the classical orthogonal polynomi-als

are eigenfunctions of singular Sturm-Liouville problems. It is not hard toshow 1l that expansions using eigenfunctions of such singular Sturm-Liouvilleproblems converge at a rate that depends only on the smoothness of f(x), incontrast to eigenfunction expansions based on nonsingular Sturm-Liouvilleproblems that lead to additional boundary constraints like (1.10) on /(x).

These results for orthogonal polynomial expansions are easily demonstratedin the case of Chebyshev polynomial expansions. The n-th-degree Chebyshev

polynomial T,(x) is defined by

(1.13) T.(cgs 0) = cos nO.

Therefore, if

(1.14) /(X) = I a.T.(x)),

then

(1.15) g(O) /(cos 0) - a, cos nO.

Thus the Chebyshev polynomial expansion coefficients a. of j(x) are just theFourier cosine expansion coefficients of the even, periodic function g(O). A simple

integration-by-parts argument then shows that

V'a,-- 0 (n ---> oo) ,

provided g(O) (or, equivalently, /(x)) has p continuous derivatives. Since

1f(r) - a,.T(x)I< 2: Oa, (r[1)U-0 U-X1-1 -.

it follows that the rate of convergence of (1.14) is faster than algebraic if Iis smooth.

%% P r e %p , '. . . ' '.. -

LDI'l" INLA' O I I t ' MI L I 11 1 1 litl I[ l It L~L.\4 '. 1 I' l I % Il,,1 .S I li

IiI sulnlniarv, spectral expalioII. shotuld e nI llt i.Nin series (if orthogon-al."P

polynomials unhls. the boi I %ihy conditiou. of the pr'ldti-. :re fully .omlItibh.l_

with somie olher class of egeim.flliol.s. In prac J tice. 'h1h(v and ],gelidrepolynomial exiansions ire re((nmeh4d for noust alil uations , ,mipplemened

by Fourier series and surface hnrmonic series when boundary conditions permit.Another difficulty with general kinds of spectral methcds is their application

to problems with nonlinear and nonconstant (oeflicent terms. Before explainingthe solution to this problem, let us illustrate the difficulty.

Suppose we wish to solve the partial differential equation

T- = 'r(U U) + YoU ,

where u = u(x, 1) and X4/ is a bilinear (nonlinear) operator that involves only

spatial derivatives and .Y is a linear operator that involves only spatial deriva-tives. The operators .X/ and .2' may depend on both x and f. A spectral methodfor the solution of (1.16) is obtained by seeking the solution as a finite spectralexpansion:

(.17) u(x, t) = .()

where we assume for now that vp.(x) (1 < n < o) are a complete set of orthogonalfunctions. If we introduce the re-expansion coefficients c.., and d.. so that

X. ,,, ',,) = '' V..Cl,.

and equate coefficients of V.(x) (n = ,..., .Y) in (1.16), we obtain

(1.18) da. -v jrf ,tata') "I-- =M - M V 11 + d ,( ~ ,( ) ( = , .. )

Equations (1.18) are the spectral evolution equations for the solution of-'

(1.16). They have one very serious drawback. In general c.., and d,.. arenonzero for typical n, i, p, so that evaluation of da/di from (1.18) for all .

n =-- 1, ... , N" requires O(Y') arithmetic operations for the bilinear term and ?

O(N t ) operations for the linear term. Thus solution of (1.18) requires order .X3 operations per time step. Since operational spectral calculations now involve ..

N> 100, the computational cost of the direct solution of (1.18) is prohibitive '

(even if only linear terms are present).

'

I k . olle 11f 4, -4 - - -N

huthlod. for 014, SollitiouH of (1.16) on N grid poiit. iiUy reijlire r|Ivl ('d1-Tli

A, opraitio.i)I per tille t'p. If tile .iwcr:i uIn tlio d really reqlllii . ol-l .'N

olra tionls per timle step, it eaill liot cumpete ,\1Iel! N is large. iAnother example iluistritiig ile conmputatioml Complexity of .Spectraj ,

inetliods is given ly the inonline;r difflision equation

tr1) 2?(!.19>) -- exp [11.] (X, t)

CyS

If we seek tihe solution as

x

(1.20) u(r, t) = V,.(t),W(X)

in terms of the orthonormal functions V.(x), then

(1.21) da. =Po(x)exp X,(tM .(X)1 a, V(x) d.rilt J

for n 1, ... , N. These evolution equations for {a.(t)} have an exponentialdegree of computational complexity as they are expressed as an integral func-tional of {.(t)}.

The solution to the problem of computational complexity is to use theauthor's transform methods. Let us illustrate the technique for a pseudo-

spectral (or collocation) approximation to (1.19). First, we introduce S suitablecollocation points x,, x., ..., x~v lying within the computational domain. Thenthe approximate solution (1.20) is forced to satisfy the partial differential

equation (1.19) (or its boundary conditions) exactly at these discrete points

at every time f. 'More specifically, the following three steps are done at eachtime step t:

i) Determine N coefficients a.(t) (it - 1, ... , N) so that

(1.22) Ut(x,, t) = W a(t)o,(xj) (j = 1r... .

ii) Evaluate ut(ix, t) by

(1.23) U,.(.-,, t) = ,,a(t)V,'(Xj) (j = 1 ... , N).az-I

iii) Finally, evaluate u(x,, t),?t by

t exp [u(.r,, )]G (,, )( = 1 ...\'

and march forw;ird to tlie iiext time si-p.

OFI

.4

L

I..'1( ES 0% V IA.A|'A 1. %l IIIl ( R I('I|II'TE%(CI C 0114 C1*.k.I I OX 113

o'hi ilI:I (Of tk i)se(iloslH-ttral trainsformin fll('oho(l (all he resla el !,s follim.w:,,

Tr;I II IofnII freel ' N.between i Iv.hsical (.r,) a nl spectral (a.) r Iresentatioins,, eva I;It-

ig ecit eh tern in witever representation that termn is most accurately, and

sitnily. evaluated. Titus, in (1.2 1), we evaluate exp [u] ill the physical represen-tationt while we Poflipute U,, ill tile spectral representation by (1.22) because it

is lio-st aceurately done there.

It should be apparent to the reader that pseudospectral transform methods

can be applied to any problem that can be treated by finite-difference methods

regardless of the technical complexity of nonlinear and nonconstant coefficient

tel .

For the expressions of interest, computation of derivatives of a N-term

spectral expansion requires order -V arithmetical operations. For the Fourier

series (1.7), this fact is obvious:

VI

d a. sin nx n, a, cos nx,

d,." a. sin nx -- - n'a. sin ytx

For the Chebyshev polynomial expansion (1.14), the computational complexityof differentiation is a little less apparent. Since T.(cos 0) = cos nO,

T".() - _.(x) =1 >o),

n+ 1 "-I C,

where co= 2, c,= 1 (n>1) and T- T' " 0. Therefore, ifT.-p

- a. T.(x) = b. T.(r) ,

theu

.+1 2 .- , +" a.r T(G) T . cb. - -- - --- = I [r._,b,-- b,,,] Tr(.r),',.

Equating coefficients of T,'(x) for X + 1, ... , N gives the recurrence re-lation

c.b_- b.+I 2naft (l ' < ),(1 .2_5)., b. = 0 (if >IV).

The ohlttion of (1.25) for b. given a, requires only order N arithmetic operation-.

Sirnihlor re(urrente relations can be obtained for differentiation ,f Slcvtr'1serit'4 Nted oil other set. of ,,rtiotonal polynomnians ad functions.

[ - I9 .,hro.fi.F .XXXVII"

[, I'

[: ,i.'p?,, ','J¢. ,',,.."..?%% '..,.. ". . ,- .. ,.- 'y S t, 0 J...f. a."...%% .. .. ,\';.*- ;,1, % % , % . %-

114 MA'

In the ' se of Fourier series, the tra iiform (I.7) and it, iinenr.e 'ai ll,10Iinputed ill 0(N\ log N) operatiolis if N = 2' g the ;tt Fouirier Iralls-

form. ]l4mcver, nuost if tle eomlputational (.lici,,ncy of tra,.forin inethodscomes not from the FFT but from the separability of multidimnionllil trans- Vforms. Thus a three-dimensional discrete Fourier transform (;iln be ex)re:,,edas three one-dineisiol 1 Fourier transforms h

J0. ,.0 i,/-0

J-I I -I L-I

- exp [2I.ijm,'J] 2 exp [2riku'hKJ V a(j, k., 1) exp [2.i~p/L]..,J-0 k-0 1-0

a V,

The left-hand side of (1.26) requires roughly (JKL) operations to evaluateat all the points O<m<J, O<n< K, O<p<L. On the other hand, evenwithout the FFT, the right-hand side of (1.26) requires only about (JKL)-

* (J ± K - L) operations to evaluate at all the points. When the EFT is appliedto the one-dimensional transforms on the right-hand side of (1.26), the numberof operations necessary to evaluate (1.26) is reduced further to (JKL).

• (log 2J + log, K + logL) if J, K, L are powers of 2.Spectral approximations to general boundary-value problems lead to full

N x matrix equations for the Y expansion coefficients a.. It would seemthat solution of these equations requires O(N,) arithmetic operations, while

storage of the matrix requires O(NI) memory locations. Since typical problemsnow involve Ks- 10', the direct solution (or even the direct formulation) ofsuch problems would seem unworkable now.

Consider the solution of a general linear differential equation Iu =/. Leta X-term spectral approximation to this problem be given by

(1.27)

where f.v is a suitable N-term approximation to f. As mentioned several timesearlier, the matrix representation of (1.27) is generally a full Nx N matrix, sothat direct solution of (1.27) by Gauss elimination methcds would require 01order y2 storage (for the matrix representation of .,) and order Y3: arithmeticoperations.

Here we shall describe a method that permits the solution of (1.27) using

order N storage locations with the number of arithmetic operations of orderthe larger of N log N and the number of operations requircd to solve Lit = fby a first-order finite-difference method. The important conclusion is thatspectral methods for general problems in general geometries ran be implmented

efficiently cith operation costs and storage not much larger than that of the simplestfinite-dillerener approximation to the problem trith the sepne aiimber of degrees offrrlodm. Since spectril inetlods rc.quire nlwiy feweer (egre.s of frvedenhm to

*!p

1.1.4 I 1 1 . is -'.ll INMA . )1i. 11il, s |. - It l|A N ito t Il s ,, 1.1 i'l I ki1,-\'- N| ,a

,igh1 4- 1 :t1cI la ': 1 for a giNcll n)II il:ci " (if 4hg'ct.' 44 fn'4,4.1I'1ll) th::n n' 1 I(411 r i14 "41 b.y

i,,nvies result from t l:e Imew hmthod. NIP

''hle idea of t he iterattion iiictliod is ;-.s f,,llh4 \ i Sili \ c '.re able )t1 (oil-

sti1,. an alpr1.Xinmatiol L* to the .pcrtl A olerator 1., that 1;:s tl:e folk lgproprmt ies

i) L., has a sparse matrix representation so that it call be reprelltedusing only O(N) storage locations.

ii) L, is efficiently invertible in the sense that the equation

is solvable as efficiently as a first-order finite-differemae approximation to the

problem.

iii) L., approximates L, in the sen.e that

(1.29) 0<m<L-L M<

for suitable constants rn, X as Y -- 0 0. Roughly speaking, (1.29) requires

that the eigenvalues of L-'L,, be bounded from above and below as N -* oo.

We propose to construct L., from L., by changing the discretization operator

either in addition to or in place of approximating the differential operator.Thus we construct L,, by a suitable lote-order finite-differevce approximation to L.

A simple example is given by the second-order differential equation

(1.30) Lu = f(x)u"(x) * g(x)u'(x) + h(x)u(x) = r(.r) (0<.r < 2:)

with periodic bound-.ary conditions u(x + 2:?) = r(x) and f(r) > 0. A sreetralapproximation is approximately sought as the finite Fourier Serics

(1.3;) t(.r) a, ci~e p [ik,,'].

If the Fourier ecefficients of f(x), g(x), h(x), r(x) are den(,tAcd L g,, b. rk.

respectively, then the spectral (Galeikin) equ.tions for e, me

(1 .32) ,. u = [- ))2_ - ipgf., + , ='I. .g

('learl.v, these equattins have, in general, a full matrix represent;ition that

rvqllires O(K2) :tolage lcatif Ils iild O(K ) (,ptniitits tl imet.

e5

116 S. . ,.I-z7 t.

A.,,litallh. :kqlpoxinm te opl-ritor 1,., is cow.trinvtl.d Owh h codllm.:Itiol

poilt .r, =-- 2.rj/N (j - o, .1 ... N - 1), \\here N- 2K. II the I hYwivu:iI poc.rlt)4 res litatioll, e l1se the finite-diflirlce ahilh|t(xillt;Ilill

(I .: :€)~ ~~ 1(,', --- ifr V, , )2 u - I, _ + - - , +,_

(Ar), - ' A,' -

Wv'here U, = U(x,) and A.r = 2 /r\N. Obviously. L,, is qpars'e awd eflicienity iii-vertible. To verify (1.29) we use the folloxiing elementary arumIlent (thatmay be made more rigorous but no more correct by more involved WKU-likearguments). If A is an eigenvalue of L-L. , then there exists a function ii(.r)such that

(1.34) L.u = ).L.Pu

If u(x) is a smooth function of x (in the limit N-- oc). then both L.,v and

L.,u should be good approximations to Ls(x), so (1.34) implies A. - 1. On theother hand, if u(.r) is a highly oscillatory function of r (in the lirit X- c),then

(1.35) ' u (W > -c).

Therefore,

(1.36) L. i -f(xr,) *A2u+(Ax)l

and, if transform (pseudospectral) methods are used to evaluate L.,w

(1.37) , Iu(,) , (- kl)a, exp [ikx,]

so (6.18) gives

(1.3S) .r (- k2)a exp [ikx4 ;- 2(.r,) 111- 2 u , if,-,

The eigenfunctions of (1.38) are

it, ex ) [iq.r,] (q' li)

and the associated eigenvdlhe i.,

(q..Ax.rI

I sin't q A I-

.NpN

44- * - I , . ...

I , 11 11 l I 11 h I.|E .- . 11-1,1 1 kill-\- 117

q'w

ThI s (L 21i holds Mith 1-- and _ - -r2; I 2.5.1ire are si-veI1 (,Xt'lti sioli'X of tlie above lneothod for ,li st"i1uti i- 1.,

that aie important in lpiactie. First, in the ca.e of Clebyshcv spectral nllIhs,it is appropriate to uoiistru(.t L.V using finite-difference approxiniatiois ba!i(-d

oil the eollocat ion points .',= Cos rj/-V. In this ease, the operator bouds

(1.29) continue to hold w\ith M= 2.5, m = 1 for a wide variety of operators L.

Second, higher-order equations are best treated by writing them as a system of

lower-order equations. Thus direct construction of X., for L V' gives

However, if we introduce v V-,it and define the second-order operator K by

t

then direct construction of K., as a finite-differen(e operator gives

Third, odd-order operators, initial-value problems and problems of mixed typeare best treated by constructing .,, on a grid that is roughly 50% finer than

that used in construction of L., by collocation. In this case the spectral bounds(1.29) with X<2.5 continue to hold for most problems. For example, theoperator / x with periodic boundary conditions has spectrum ik. while itscentered finite-difference approximation has spectrum i sin (kAx)/Ax, so

AV, 0(kr '/sin kAx),

which is unbounded for IkAxI<;, but bounded by 4-rr3V3 2.4 if IkAx!< 2,/3.

2. - Applications.

2'1. lntroductiopi. - Over the last few years, there has been progress in

understan(ling fundamental jonlinear processes ili sheer flows. In this section,

I shall survey some results that have emerged from numerical studies of tran-

1P

jill A.''I:-, .- w o

.Nil iol :i ld I Im' ielef.. I 1 1 l evhi(%% fo N-( I thIIIr 1ev d i cI t , ;1,114ctI,, (If I I 1.-.

prio bhI, ns. Firs,,t, I s~imIl ,,|lllll.'1 rize-Ilt, onl Ih]w lmsie i ,il ilit ie, tha 11A,,ve.III lbe r. plnsileh for" tile ollset off ehloi ill Itle.e lIII) s. 11wl,. istahili i,,s aplo-11 1to hel |lliVersiil ill t|ai .ttr timay xpliin llally of the 1umiifyin featmues oftransition. Sevond. I shall give sone (x:iniies of prog'.s in the t n rica l F7

sillllati,,n of high-flcyoolds-nnnbr lows. Finally, I will give a .synopsis of %new ideas for smbgrid scale closures of hug,-leyuohls-nmlibr torltimip.

Full details of the ideas discussed here are given inl the referem'(ic.

2"2. A transitional instability,. - The processes by which laminar flows under-go transition to turbulence remain basically unsolved. However, recent numeri-cal studies have provided some insights into transition, including:

2*2.1. 'Nonclassical character 'of transitional instabilities.The primary linear (exponential) instability of classical plane parallel shearflows with noninflectional velocity profiles, as described by the Orr-Sommerfeld(or related) equations, is much too weak to describe transition. For ex-ample, linear instability of plane Poiseuille flow (U(z) = 1 - zI, jz < 1) oc-

curs for Reynolds numbers R. > 5778, while Squire's theorem implies thatthe critical disturbance is two-dimensional. The fact that this instability isinduced by a subtle interplay of viscosity and shear implies that its growthrates are quite small on convective time scales. For example, the most rapidlygrowing exponential mode of the Orr-Sommerfeld equation is obtained atRo,= 48000; its growth rate is only 0.0076; it is so feeble that perturbationsgrow by a factor 10 in a time of about 300, in which time a point on the center-line moves about 150 channel widths. In contrast, transition is observed tooccur explosively over a few channel widths at Reynolds numbers as low as Iroughly 1000. A transitional instability that affects noninflectional planeparallel shear flows must have a characteristic convective time scale.

2"2.2. Three dimensionality of transition. Two-dimensional fluidsdo not appear to exhibit the kind of strong chaos that is characteristic of tur-bulent shear flows. In thermal convection, CvnRY et al. [3] show that two-dimensional flows do not appear to act in a strongly chaotic way, but three-dimensional flows may be strongly chaotic at large enough Reynolds number.Even for inflectional free shear flows, in which there are strong inviscid two-dimensional instabilities, BRACHET and ORSZAG [4] show that the flows thatdevelop from two-dimensional finite-amplitude disturbances are not stronglychaotic, in contrast to the flows that develop three-dimensionally.

2"2.3. Instability of two-dimensional nonlinear travellingwaves. Perhaps the simplest instability that has the character of a transi-tional instaluilitv is the linear three-dimensional instabilitv of two-dimensionalfllniit(-aflhnp]itide flows. ORSZmAG and KFLLS 151 and OJImSZAG and 'ATERA Ii]O

% %

-,c. ~ .-- **-~.%**~. . . ~% 9 . ' --- 9.---. ,9

-, P ~P ~ ?~ '* ~ '. v - . d~ d V ~ d~*~ * -* .'-. *j

S..

z

Fig. 1.- Streamlines of the steady (stable) finite-amiplitude two-dimensional travel.ling wave for plane Poiseuillc flow- at R = 4000, plotted in the rest frame of the wave -(from [6]).

show howv such an instability fits the basic features of transition in classicalshear flows, including their convective growth rates, inherent t hree- dime ns ion-ality, onset at Reynolds numbers in a-ccord with experimental observationsand flow features in accord with early transitional flows. These instabilitieshave been analyzed both by direct numerical simulation of the evolving three-dimensional flow and by a linear perturbation analysis of the nonparallel two-dimensional (nonlinear travelling wave) flow. In ing. 1, we show the streami-lines of a typical two-dimensional base state (here for plane Poisenille flow atBR 4000). The noniparallel, character of the base flow leads to considerablecomplication in its linear stability analysis (see [6] for the formulation of theselarge-matrix eigenvalue problems). A topic of mnuch current research interestis the development of efficient numerical methods for finding eig-envalis ofthe very large zwitrices encountered in p~roblemns of this sort. In fig. 2. wegive a stability diagrai for tliis trarlsitioiAl inrstability h ere wIe plot contouirsof constant growthl rAe ;Is a funlctionl of the amplitude of thle two-dinmciilionialbase state anad the Revcnolds jnmber. The griiwthi rates of this instabilit v are

+2ordlers of ma giiittiiik ;i rger t lani t hose of Ori-S on eifeld incdcis. The

4-30

1.006

0.10 0 03300

0 2000 40

Fig. 2. - Contours of constant growth rate (labelled by growth rate) as a functionof 1R and the amplitude of the background two-dimensional nonlinear wave (see right-hand scale).

development of this three-dimensional secondary instability seems to be con-sistent 'with available experimental data on early tranbitional flows. In fig. 3,we compare contours of the x velocity at the so-called one-spike stage of tran-sition in plane Poiseuille flow obtained a) experimentally by NNxSxiOKs, IIDA

and KAN'BAYA8I [ 7] and b) numerically by KLns8Rx and Scru-x4~xY [8]. The

U+U1(X=-O.7.

z~rnrI)

0.30 0.40 05L2! 1277......-.va e

b ) ... .. . . ..

Fig. 3. -Contouirs of z velocity in the (x, y)-plane at the one-spike stage in the labora-tory experiments of Yishioka el al. (a)) and in the numerical simulation of Kleiser andSchumann (b)) (from [8]).

'N

- .. . INt Iinll i it ii l l il it a m \ 4 11 1 44 -11 14 it 114 t 22 24 I4 1 t~ d I; i i i i r1 1 !.d % -.I i

jIipt. ]144jo ,lli liI lim. (Sev. it; ) ;1 141 ill fl o i lvl r~ I li ;f (,q c ~ I I j).

2*2.. Cominion b41letweeni Ollo-dii liellslllla41iil hgil let *

dliniellsjioln I i I Ist I h )i itI it's. I I I lectioiinlI free shcar. f21awl. Ilike hIix IlI,- l.1.\ el '

andi jct . ore iliviscidly lUt141 to tuo-dtlii(Iitia~il 1iisturlt ies. N1tljil

theoreezn impnjlies that licee iilsta bili ties are str4igest lienvi t itl-dimfeLisonl:

wilell these;( t wo-dimnsional ilistabilit jes evolve iii ti Ine, t heyN sat ili-ate I into

ordered latninar-lowv states characterized by large-stale vortieal flow struet lne.

These vortical flows niav themiselves be un4tale to subharnionic (patiriuigj I

instabilities, ill whic-h two (or miore) vortices are pinre-d and g(-nerate a1 iewlarger-scale vortex miotion [9]. Ini these flow* . tile three-dimiensionail iiistalifit x-discussed above is also present [10], but it it; not ilecessarily stronger thian thepairing instability. However, the three-dimensional secondary instability iseffective at much smaller spanwise spatial scales than is the inviscid primiryinstability a~nd seemi to leaid directly to chaotic flows [4].

1 3C 27x7

1101I

Vi.4.Tm evlio f th ore opnn ,o h eprtr:fedill

ri.4.-Teeolution otrese the Fours isperc oetb 2 o h tmeauefedi

122 :.. . , . -,- .

T2.5. Splriol.,, (ii Ii in eit' alI) il ir}i i '.ivilv . Cu, ur ci al.C M 0 I. low

11:11. uI, ] -ol-flr,r iy~ h';.] .'is , ,,tieyc bns (: ,hi .'r ii J r)uoxinl;,fioi o

1t I1 t\\ -liiici h,.i l Itheisi ('(blatio ' mu;5 I V Xhihit o1'] utic moltioll.,

t Ii~ Ihoots I yjiul di 4)isappears dS le ,liIItIisioli of thi lt l(~ pteiura

(Set, fl"..4). 'inlilarly, i t , O .as \il by (}zIz.\ la ELIS u [5jth t u5lder-'Sresoh'ed numericl cahlulati s of tiUansitionl pla.nar shear flows liy be spu- I.riosl , v chaotic. Utaher-resolved computations do not h'ave (ht-g-recs of freedom n

associated %\ith small sp:!tial scales available to act as an eddy viscosity onwell-resolved large scales.

2"3. Computer simulations of turbulence. - In this subsection, I shall givethree examples of numerical simulations of turbulent flows. The first twoexamples, turbulent channel flow and the simulation of a turbulent spot, are

of the nature of numerical experiments in which the numericist uses the computer

in much the same way as the experimentalist uses the laboratory, namely as asource of data about flows in a controlled environment. The final example,the Taylor-Green vortex, is an example in which the computer is being usedto try to uncover fundamental physical laws of turbulence.

2"3.1. Turbulent channel flow. Turbulent channel flows have been

simulated numerically three ways: a) large-eddy simulation with a subgridscale turbulence closure for eddies outside the wall layer and a heuristic boundarycondition applied at the edge of the viscous sublayer by DEARDORFF [11] and

SCV'MAN N [12], b) large-eddy simulation with a subgrid scale turbulence clo-sure applied to eddies of all scales including those in the wall layer by Moi"and Kim [13] and c) full numerical solution of the Navier-Stokes equations byORsZAG and PATERA [14]. The really crucial differences are, as we again notein subsect. 2"4 below, between a) and b)-c). Simulations of type a) have much

smaller computational requirements at a given Reynolds number R than eitherof types b) or c), the latter requiring asymptotically similar computational work

at large R. The deficiency of simulations of type a) is that they require model-ling of wall layer effects in terms of an over-simplified boundary condition; thedeficiencies of types b) and c) are that, with currently available computer reso-

lution (say 64 x 64 x 65 on a Cray-1 computer), Reynolds numbers are limitedto about 10000 (type b)) or 5000 (type c)). For simulations of types b) or c),the computational work scales as R3, so future increases in computer powerdo little to increase the effective Reynolds number of the computations.

Nevertheless, it is possible to achieve interesting results with full numericalsolutions of the XNavier-Stokes equations. In fig. 5, we plot the mean velocity

profile found in the channel flow computations of Orsz.,g and P.:tera [14].The fit to a logarithmic wall layer velocity profile is only marginal, but the

resulting von K6trmiin constant 0.45 is within experimental bounds, so thiscalculation does give the first computation of a wall layer from the basic prin-

": ,S.'

l~~t.*'I' .l , 11: %II 1H\ . All,III,. , - I, M II IZ I; I I ' , t J A I I ' II \ 12 3

20

@p

100

0 .

0 2.5Log z.

Fig. 5. - Mean turbulent profile obtained by full numerical simulation of plane Poi.seuille flow at R = 5000 using a 64 x 64 x 65 spectral simulation. Note the viscoussublayer, buffer region and logarithmic layer of 8-9 data points (from [14)).

25.0 conv-,aaion

15.0 L.-

_--s. .-. ". ,. "\ , . ",

Cormos

-25.0 Y.-a~ ' Sch~oen'tf ~ 4.. ,, -

-35.0-

-15.0 -10.0 - 5.0 0 5.0 10.0 15.010 L-g F

Fig. 6. - A plot of the turbulent wall pressure spectrum as a function of frequency(from [15]).

ciples of fluid dynamics. Another more recent result from computations of

this type is given in fig. 6, in which we plot the wall pressure spectrum in a

moderate-resolut ion (32 x.32 x33) run compared with available experimental

data (see [15]). Despite the moderately low Reynolds number (R = 5000) (if

the simulation, agreement is nrhievrd because flow features that do not depend

explicitly on the boundary wall layer structure tend to be Reynolds numberindteendnt. .

.%

..

,45

- 4-- - 4Pl I - - : '7 1 k --i W p..P4 i r . + +- . . . ."' " = " + * ,' -' - . . .+ 4*o .+. .' ,

4%~~~~12 1,. . , .- u

2 .2. 'T il' lt I. "l' ,,t q. l I ,iti aN bevI Iii IIIti 44 I t i Iv ll 1 -11 i iL(v%\ oll i-z of hoc;lized sp, , ;. # ill I Il- h111- t Ilo%% s (.t,. I If; 1). 1l'h4. lit 1! 1111n114. ico I

'illl 1.Iioll 4f :1 1 I lnnh'm ,pot ;s ielvorle d by v i 17 , % Ill I I' t .-dilliellsJ;,lw, \'ol'l I I li it-iwll tevhiliIlles to vullipilh, lit, (ill i..l ill) 1li . . I -I v

re'cent Iv,,%14, have bt-'rllll :1 .;tul y\ of . pot. Il~ilig full ljlllll(-'ri(,al (-,]ti i ,., 4Jf It.

Navier-Stokes eijiatiol.s at Iioderate le ,ynols ininiiers I S. Tie latter inm-hitiolls are ltper'criiled by folfvilg fIhl ilitial flh%\ ilsilg a hcealizvd fclj ( to ilii.ma jet of fluid vertically, then allolling the disti,, 1r1b ce to evolve t '.:liv.

If 1ig. n. and , we plot contours of nmaxiniini v4-rtiv;A: velhcity ill the (.,,, y/)and k.r,.-) planes at va-rious times of evolution of plaile 14oiseill 1o1i\. The

ech'.ru'cter of this spot evolution is similar to tlt observcd experiniviitally:

the spot seems to spread in the spanwise direction by # tran.verMe contaiiii-

naitioti *, in agreement with the dye injection experiments of Gad-el-ll:i0i al. [19]; the greatest turbulent activity is near the edges of the spot, the* spreading * angle of the spot relative to its source is about 10, in agreement

with the channel flow experiments of Carlson et al. [20]; the vertical structureof the spot is in qualitative agreement with that observed experiment.ly.

Further numerical experiments are under way that should elucidate det;:il.s of

the flow ii spots and the surrounding fluid.°T 127.fi2 T =8I

Fig. 7. - Contours of the maximum -.velocity in the (x, y)-plane at t-- 12, 18 afterinitializing a turbulent-spot computation by imposed vertical forcing. These computa-tions are performed using a spectral code with 128 (x) x 32(y) x 32(z) resolution. Fourierseries are used in x and y; Chebyshev polynomial expansions are used in z. HereR 6000.

7 = 2 r 5

Fig. 8. - Same as fig. 7, except x-. contours of maxii.i

4.

2'3.3. Taylor-Green vortex. In order to gJn understanding of the

A . basic lihysics of the generation of small-scale turbulent flow feitures, a ltce

.,

sc~b'.. The r(, 'v ortex lia, twen ul.ed to stildy Stichud ena us ions 16

thet t,:iliiitniiit oif vorticitv lI)\ %-rteX Jill( stret(.1jiily, tile :ijiltro~(Iih to isitroply

of tilt-1111 ScaIles. possible4 SnilIa r lhliav ior of t lie Eu i ei co-quations, for niat i( II

oif mli hiertial ranrge and awil vsis oif thle geonli-trY a iid litert mitt ency of Ii igli.

vorticity reazios. Vie TG flow is idatgosfor these studies because its

sleal symnleir bas allowed the development of numerical algorithms thlat

are( it factor 6-1 more efficient in both memory and stoy rage than convent ionalpeiiodie-geonletry spectral methods. For a tlirc-diien~sional flow%\, this factor61 tinla~tes inito :z factor 4 increased range- of Ipatiitl seales-it is now pcs.,ible

to comipute the TG vortext flow wvith 51 x5I2 x5I2 Fourier modes for eachvelocity component on the Cray-I computer (or more than 4 06 effective

deL'rees of freedom !).

10

10

a2.

To0 "/2

ig. 9. - A pilot of thle distrihitt ion of large.vort icity regionot ill thle TG. vortex tlowag: finict iou of tillie t anrd dista nce d awa y ftrl t lie sidc-wal 18 of thle imnipernmieable

elihe ill 1i0ilil the flow% t akeK pl ace. litelrv ]to%% viort icity V v'ldoes ill hriwaiqi tlktveitfil (if ft-e tilte loet itt-to 4 ;intl 1 ( furil

it,

tS

.f126 1-

0114. of till-t Iou e t h XI Il ila 1-1111 to , 4- 4 IIhI( JII tii hit 4. Lt. I '

Siiiall-s;dq tiilbilteiitt. 1144t jillt . ti jg ;t :1 51uk ilf ttiliiiltilit hii iic~

Indteed, N\4. find that tOw (i-lltpiait (if thle t urbiulet itm , evinN Ili it-iil%iscoitvl to li ilotlle iii~tabilities of vortic;Il irticfiultIs ill 1" huh the iliftii l ge

Scale iuttitiirhitiit ti itNot v i iittio a~ll exjplf.sive redist liitjtsi illii oc(set, hg . 9). Thetse visccsit --induIsct1 izistahjiil ic us re robald'yvfiective becalt~viscosity ahloms vortex line !44(i11iii54t ioII jirtibliteit ill illiid( flow. Ninihui

dilfusional i sta buit jes haive now beeni shio%%i to be responisible for the general ion

of small-scale struct ureg ini tuo-diflwusioual i gito vdid ia ue[23] andt

.0

eddes-,reoforder 10 times smller than calirgvN-cviAtuiining eddies. Icuiithe effect of this range of spatial scales onl the allowable tinie htep inl a numericalsolution of the XNavier-Stokes equations gives the estiniate that order R1 oper-ations are required to sirnul,,te a turbulent flow. This is the reatson for interestin the large-eddy simulation method in %hIh excitaticns8 on scale smle

than those resolvable numerically are modelled, usually by an eddy viscosity.9 coefficient (see [11, 12]). The basic action of an eddy viscosity on large eddies

is reasonable, although it cannot reproduce the random character of the actionof small-scale eddies. However, in order to model properly wall turbulence,it is necessary to extend the subgrid modelling idea-s of Deardorff and Schumann

-~~~ ~and treat the turbulente llte way iup to a ii4al as iN r 0n wokb:.4 MoIN and Ki [11I]. Unfirtunately, in order for _IoiN and Ki to reeolve

motions down to the scaile of turbulent bur)sts. which is nece'sary in order tocapture the nieclianikm producing the turbulence, the wcrk restriction 0( R 3

reainq ThsteRyod-numnber restrictions are ,imdir for largeed, nfull numierical .soluthins oif thle XNavier-Stokes equations that attempt to inte-grate all the way throltgh tfie ~l layer region.

-In ~e( !It work, YAXHOT and ORSZAG [251 have used dynamic renormali-zation group (RY'G) methods to treat wall-bounded turbulence. The idea of thleinfra-red RNG mnetbcd is to use perturbationi methods based oni the direct-interaction approximation [261 to eliminate :ll smiall spatial scales up to the(

ii. W resolv-ahle gridl scale froin the Navier- Stokes equ.ations. This is done perturh::-tively by eliminating narrow bands of wave vectcrs from the dvncmis- ( ce

IVfig. 10), renormalizing the resulting- reduced dynamical equation to th:~e liform oif thle Nivicr-Stokies equation with iniihed viscosity : nd randcru forcinigtermns, anad theun 1(*jaeating tile prociess iteral i vel~ unit il :1l] the requircdl imllI

tllod*tl.

('lii 1. 1~t~~ 1 1MIII 1ii 1 ~ l hIt I, I U I 1. ilti. l li 1 1 14 12a hi7l ia ~le

SonuLM ANN anudi MorN 111(1 Il~ ill tilt- \N-ll regills, ill Mhich theris . i!1terfe14'14iL v

beit \\ii'ii the' vdlil . anullloil(('Ilar viscosiis. This iInterfilrenl( v flith 5i1114 ki-Yti ) 4)hta i li I g . a fit ))f u 14 reseiit :I t ii I f th 1wa-ll 11 reglioll . Also. thei illili 4-d

0 kk Aexp-L A

Fig. 10. - A schlema~t ic representation of the niodal structure of the dyiinmic retiortual.ization group. Here k, represents wave Itiumbers within the energy-conltainill rallige.m-hili A givest the high-wvave-numrber (viscous) cut-.off. 'Modes inl the hiatched handare remuovedl at each step of the RNG procedure.

randonm force is, large in the buffer Myer between the viscous sublayer and thelogarithmic layer, giving a turbulence source in this region. Further work isnow under way applying these RNG-based closures to both large-eddy Himula-tions of turbulent ,hear flows and to the derivation of new (la;Sses of turbulencetransport (Reynolds averaged) equations that Aiould be useful in engrineeringapplications.

3. - Conclusion.

I have reviewed several areas of activity in the numeltical simulation oftransition and turbulence in which I have been intimately involved recently.In this short space, it has not been possible to do justice to all of the large numberof researchers involved in these fields; the references do a more complete job ofsurveying the literature. The principal conclusions from our studies are:

i) Numerical methods now provide essential information complementaryto that available from experiment and mathematical analysis.

ii) Computational fluid mechanics has now matured, so that tbure aretechniiques that can be reliably applied to the nicst difficult of fluid-miechanical

problems. In contrast to 10 years ago, it is no longer mainly- a question ofN.' how to compute a complicated flow, rather, now, it is a question of which flow

to compute iin order to extract the most useful information.

iii) It is crucial, especially in our studies of transitionnl flows, that wehave used spectral numerical methods (see sect. 1 above). Spectral methodsi

are so accurate for these Iroblems that we (,all conlfiden1tly conclude that prop-A erly tested niumeical results are true fluid-nieehanical results. In contrast to

fin it e-dihii'remue cr fillite-element methouds in which anil increase in spat il resoilu-

tiont byv a factor 2 leads to an error' 41e4 ree14 ly :I fac-tor -1 or S oir so, iiithi spectrl~t

12-

%

Im etflil, :I |; f -fo ' 2 ilicr'v;-, q- ill I'v -(dlltilI t ry lic[*; % 4 1(q'( r .... s 114~.<l ' I'- 1 v (.1'a -I

(IlIt i,- of Ili;! -l-ntlde. lhi, plvt'iiiit, : ccil'rtt, t-ri i(atiiII of 'esIls. F ir exI: I I Iph.,

ini revelit sllldi .S lif trll.itih.n ill circiflar ('o i .tte floN%, M.\I1cv it Ell. 127 li-,1V,):Ct'.s ["i have bee.n a1ble to achieve a tit l h.t E-dlFinl-lil igrenILitwith oileimnt, wave speeds. The confidence ill these results ha.: perimitted .

I(.\% ;tlml tival inl"igits into the iir:it .er of the olset of w.'avy inst:bilities (ofT alylor v'ortic es ill Co llette flowv 121]. ,

iv) 'Xv% , g l'a liolis (if bi-,qr ailld fister comlllputttrs (' Inliost plroltita ly,be uNiAl to extend the range of appli.,tion of computational fluid dynamics. Il..Transition ond turbulence prolenis in complex geometries with complexphysics, like multiphase flows, will surely be the subject of studies in the nearfuture.

This work was supported by the Office of N-aval Research under ContractsNoo014-82-C-045t and N00014-83-K-0227, by the Air Force Office of Scientific]search under Contract F49620-83-C-0064 and by the National Science Foun-dation under Grants ATM-8310210 and MEA-8215695.

REFERENCES %

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[4] 31. E. BRACHET and S. A. ORPZAG: Seeondary instability of free-shear /loirs, slb.mitted to J. Fluid Mech.

[5] S.A. ORsz. and L.C. KELLS: J. Fluid Mech., 96, 159 (1980).[6] S.A. Onsztvu and A. T. PATERA: J. Fluid Mech., 128, 347 (1983).[7] M. XIS1IIOKA, S. IDA and S. KAxBAYASHI: An experimental investigation of the

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!'.

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- *~'A'~~ P4,d ~ ~ 4 ?, ~., er 4\. .e~.4. 4.~f"

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lamitDor bowndary layer, in Turbulenft She,,r Floerm, II, edited by L. J. S. |lluu,%.BtRy et al. (Berlin, 1981), p. 67.

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simulation, submitted to J. Fluid Xech.[26] R. 11. KR pICHNA: J. Fluid 1ech., 5, 497 (1959).[27) P, S. MARCUS, S. A. ORSZAG and A. T. PATERA: Simulation of circular Gouctte

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[28] 1P. S. MARCUs: to be published.[29] B.J. BATLY and P. 8. MARCUS: Analytic computation of Taylor-Costie rave

speeds. to be publiahed.

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