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FINAL REPORT ON-A-POSR CONTRACT F49620-85-C-0026

Steven A. Orszag, Principal InvestigatorDepartment of Mechanical and Aerospace Engineering

S..... . Princeton UniversityPrinceton, NJ 08544

Volume 1

1 VETESEL. ECLE: D

DISrM.UTIOrl STATEMENT AApproved for public reles

PRINCETON UNIVERSITY

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of fluids; 8) The developrent of efficient rethods to analyze the structure ofstrange attractors in the description of dynnical systems; 9) The analysis ofh"perscale instability as a mec.anisn for destabilization of coherent flow structures.

0p.

.J.

I.

FINAL REPORT ON*AOR CONTRACT F49620-85-C-0026

Steven A. Orszag, Principal InvestigatorDepartment of Mechanical and Aerospace Engineering

Princeton UniversityPrinceton, NJ 08544

Volume 1

DTICELECTE--TSEP 30 1987.[ifi

DISTRIBUT:.F-0 , 5'1 -T A

Approved fox public xelecsel 1Ditribution Unumited

[I

FINAL REPORT ON AFOSR CONTRACT F49620-85-C-0026

Steven A. Orszag, Principal Investigator

Department of Mechanical and Aerospace Engineering

Princeton University

Princeton, NJ 08544

In the attached papers, we summarize work done on this research

project. The major results include:

1. Development and application of the renormalization group

method to the calculation of fundamental constants of turbulence,

the construction of turbulence transport models, and large-eddy

simulations.

2. The application of RNG methods to turbulent heat transfer

through the entire range of experimentally accessible Reynolds

numbers.

3. The discovery that high Reynolds number turbulent flows tend

to act as if they had weak nonlinearities, at least when viewed in

terms of suitable 'quasi-particles.' This discovery suggests that tur-,tp CI

bulence acts as if there were a significant scale separation in the

flows, even though turbulence does not appear to have such scale

separation. These ideas seem to provide some justification for |__

eddy transport ideas that have proven so useful in engineering CJ

descriptions of turbulence.

4. The further analysis of secondary instability mechanisms in

free shear flows, including the role of these instabilities in chaotic,

three-dimensional free shear flows.

-I- !

.w.- ~ V i._1

5. The further development of numerical simulations of turbulent

spots in wall bounded shear flows.

6. The study of cellular automata for the solution of fluid

mechanical problems. In particular, a realistic assessment of the

utility of these new methods for complex high Reynolds number

flow problems has been given.

7. The clarification of the relationship between the hyperscale ins-

tability of anisotropic small-scale flow structures to long-

wavelength perturbations and the cellular automaton description

of fluids.

8. The development of efficient methods to analyze the structure

of strange attractors in the description of dynamical systems. This

includes both the computation of Lyapunov exponents and the

computation of dimensions of attractors.

9. The analysis of hyperscale instability as a mechanism for desta-

bilization of coherent flow structures.

Further details are given in the attached papers.

-2.

*',1 )¢.K9 .' , 4 " * Z. , %( -- . / - %° : 2 -.- % . . * %-.% P -% . ." ":"' ''*';.''%.''"' .,-.'

List of Papers

_-Analogy between Hyperscale Transport and Cellular Automaton Hydro-

dynamics' Phys. Fluids 29, 2025-2027 (1986).

S.A. Orszag, R.B. Pelz and B.J. Bayly, Secondary Instabilities, Coherent

Structures and Turbulence, in Supercomputers and Fluid Dynamics; (ed.

by K. Kuwahara, R. Mendez, S. A. Orszag), Springer (1986).

S.A. Orszag and V. Yakhot, Reynolds Number Scaling of Cellular Auto-

maton Hydrodynamics; Phys. Rev. Lett. 56, 1693-1696 (1986).

V. Yakhot and S.A. Orszag, Renormalization Group Analysis of Tur-

bulence. I. Basic Theory. J. Sci. Comp. 1, 3-51 (1986).

V. Yakhot and S.A. Orszag, Renormalization Group Analysis of Tur-

bulence. Phys. Rev. Lett. 57 14, 1722-1724 (1986).

V. Yakhot, S.A. Orszag, and A. Yakhot, Heat Transfer in Turbulent

Fluids. I. Pipe Flow. Intl. J. Heat and Mass Transfer 30, 15 (1987).

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

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

R.W. Metcalfe, S.A. Orszag;, M.E. Brachet, S. Menon, and J.J. Riley,

Secondary Instability of a Temporally Growing Mixing Layer. J. Fluid

Mech., in press.

V. Yakhot, S.A. Orszag, A. Yakhot, R. Panda, U. Frisch, and R.H.

Kraichnan, Weak Interactions and Local Order in Strong Turbulence,

Phys. Rev. Let. (1986) submitted.

-3-

V. Yakhot and S.A. Orszag, Relation Between the Kolmogorov and

Batchelor Constants, Phys. Fluids 30, 3 (1986).

I. Goldhirsch, S.A. Orszag, and B.K. Maulik, An Efficient Method for

Computing Leading Eigenvalues and Eigenvectors of Large Asymmetric

Matrices, J. Sci. Comp., in press (1987).

M.E. Brachet, R.W. Metcalfe, S.A. Orszag and J.J. Riley, Secondary Ins-

tability of Free Shear Flows. In Progress and Supercomputing in Compu-

tational Fluid Dynamics, (ed. by E. M. Murman and S. S. Abarbanel),

Birkhauser, Boston, 1985.

B.J. Bayly and V. Yakhot, Positive- and Negative-Effective-Viscosity

Phenomena in Isotropic and Anisotropic Beltrami Flows, Phys. Rev.

A34, 381 (1986).

-4-

UL I I tHbThe purpose of thu Letten section is to provide rapid dissemination of important new results In the fields regularly

covered by The Physics of Fluids Results of extended research should not be presented as a series of letters in place ofcomprehensive artictes Letters cannot exceed four printed pages in length, including space allowed for title. figures.

tables. references and an abstract limited to about 100 words.

Analogy between hyperscale transport and cellular automaton fluiddynamics

Victor Yakhot, Bruce J. Bayly, and Steven A. OrszagApplied and Conputational Mathematics, Princeton University, Princeton. New Jersey 08544

(Received 24 February 1986; accepted 16 April 1986)

It is argued that the dynamics of a very large scale (hyperscale) flow superposed on the stationarysmall-scale flow maintained by a force f(x) is analogous to the cellular automatonhydrodynamics on a lattice having the same spatial symmetry as the force f.

While real fluids consist of discrete particles, they can be that in three dimensions none of the space-filling crystallo-

regarded as continuous media at scales that are much larger graphic lattices have sufficient symmetry to guarantee the

than the typical intermolecular distance, and, on these isotropy of the corresonding hydrodynamic equations.

scales, they can be described by the equations of continuum However, icosahedral symmetry would produce isotropic

hydrodynamics. These equations are quite insensitive to the viscosity and nonlinear terms.' Unfortunately, no periodic

details of the molecular dynamics; the microscopic interac- lattice has such symmetry (with the possible exception of

tions affect only the viscosity coefficient. Microscopically some recently conjectured quasicrystal structures ).7

dissimilar fluids can be described by the Navier-Stokes Hydrodynamic equations are derived from the micro-

equations, although the microscopic properties of different scopic equations of motion by averaging over small scales. It

fluids may be reflected in a very wide range of viscosity coef- is natural to pose the following problem: Let us consider a

ficients. viscous fluid driven by a force which generates a stationary

The lack of dependence of the hydrodynamics on the field / on the small-scale . The equation of motion for the

microscopic properties of the fluids is the basis for the recent perturbation us- defined on scales L which are much larger

interest in discrete approximations to molecular dynamics than the scale I of the basic flow can be derived by averaging

or Cellular automata (CA's).'" Cellular automata are dis- over the small-scale velocity field v1. The resulting equations

cretely and locally linked, finite state machines. The "mole- describe hyperscale hydrodynamics. To see that this problem

cules" in a CA fluid move in discrete steps over the lattice arises quite naturally, let us imagine a system of microscopic

sites and interact according to a well defined set of rules that particles driven by an external force f. The molecules partici-

typically conserve momentum and the total number of parti- pate in two kirlds of motion, one related to thermodynamic

cles. The hydrodynamic behavior of the CA fluid is given by noise and the other caused by the external force. Filtering

the evolution of the average macroscopic properties of the out the smallest (thermodynamic) scales, one derives the

system ("slow" modes). Some limitations of CA hydrodyn- Navier-Stokes equation subject to the external force. If we

amics have been discussed in Ref. 5. also average over the scales corresponding to the force f, the

The first lattice model of a fluid was introduced by Har- resulting equation will not necessarily be the Navier-Stokes

dy, de Passis, and Pomeau (HPP).' Recently, new models, equation, but will rather be an equation describing the large-

which are modifications of the HPP ideas, have led to simu- scale (hyperscale) motion that does not explicitly include

lations of two-dimensional fluid motions that appear to be the external force.compatible with experimental data." Hydrodynamic equa- Some examples of hyperscale dynamics have been con-

tions for such a CA fluid can be derived using techniques sidered in Ref. 8. It has been shown that a system of square

based on the Chapman-Enskog expansion, as in the kinetic vortices (eddies) gives rise to the equation of motion for the

theory of gases. It has been found that the form of the contin- velocity perturbation at large scales with an anisotropic vis-

uum equations for a given CA fluid depends strongly on the cosity:symmetry properties of the lattice. In particular, the HIP - slattice gas model based on the two-dimensional square lat- v( I - Re2 - Re2 sin 2d')I

tice leads to anisotropic viscosity and anisotropic nonlinear where v, is the effective viscosity in the direction (cos 4,

terms in the resulting continuum dynamics. l-owever, a re- sin (6). In this and all subsequent equations,. Re denote,% the

gular hexagonal lattice, again in two dimensions, introduced Reynolds number of the small-scale flow: the formulas

by Frisch. Hasslacher. and Pomeau, is symmetric enough to quoted are the lowest-order nontrivial results ofa hirard hy

produce the Navier-Stokes equations with isotropic viscos- of successive smoothing approximations.' A plane parallel

ity and nonlinear terms It has been pointed out by Wc ,fram 6' system of eddies also leads to an anisotropic viscosity ' 1l, 1w.

2025 Pl y; FluIdS 29 f7). Juliv 1986 0031 o171 '86/072025 03$01 90 986 A,*'rca, n,t,o nf Phy.,

ever. a system of triangular vortices having hexagonal sym- the hyperscale flow is therefore unstable, if the small-scale 111C

metry (invariance under rotation by 60') generates an iso- Reynolds number Recl/ v exceeds \2..tropic viscosity coefficient for the hyperscale motion: The so-called ABC flow is obtained when S consists of

= '( I + i Re2). (2) six wave vectt,s located at the vertices of an octahedron:

The analogy with cellular automata is striking: in two f '/dimensions only the triangular lattice and triangular set of S = {Q} 0 ) (4± I4 M

vortices produce an isotropic equation for the large-scale .J ±ivelocity fluctuations. Moreover, it has been shown by Siva- with all amplitudes having the same modulus ;A (Q)I = ci ory t

shinsky' ° that hyperscale hydrodynamics is not Galilean in- N icos

variant because the averaged Navier-Stokes equation with scal wereis tagai the fins tfcle

the forcing term is not. The same holds for the CA hydro- scale equation takes the form tim

dynamics: It has been pointed out2 that the continuum equa- d Vk 2 [ 6., + Re' oin () ] v,, oi

tions following from cellular automaton models are not Ga- at = -

lilean invariant as a result of the discrete lattice underlying where M is a matrix that depends only on the unit vector k in m

the model, the direction of k. The eigenvalues of M are all greater than ti n

The dynamics of hyperscale flows superposed on small. or equal to zero, with equality occurring only if k lies in one ar

scale flows in three dimensions have been studied in Refs. 9 of the coordinate planes. The hyperscale flow is therefore ge

and 11. The analogs of steady cellular flows in two dimen- stable although still somewhat anisotropic.sions are the family of so-called Beltrami flows in three di- A more complex flow with similar structure to the ABC

mensions, defined by finite Fourier sums of the form flow can be obtained by augmenting the wave vector set by mat

(x) Z4A(Q)(n + QXn) el". (3) 2 12±3 3 3 icl

Here S is a finite set of unit-magnitude vectors Q, n is a unit 2 2 1vector perpendicular to Q, and the complex amplitudes --- 3 3 t (5)

A(Q) satisfy the reality condition A( -Q) =A *(Q). 2These flows, like two-dimensional cellular flows, are exactsteady solutions of the inviscid fluid equations, and can be

maintained in viscous fluid by the action of an externally and assigning the corresponding Fourier modes amplitudes 0imposed body force. A(Q) =A(c/,T1), QESI. Here A is a parameter: setting

The form of the equation of motion for hyperscale flow A = 0 recovers the ABC flow with its anisotropic equation ofon the small-scale flow (3) involves the fourth-rank ten- motion, but if A is raised to the special value of 9/(156) '/-sor 9 II then the conditions for isotropy of the tensor N,,k are satis-

fied, and we obtain an isotropic equation of motion for theNu= !A(Q) 12(6bik56 - 26AQQ1 hyperscale modes. This example illustrates the fact that a

W small-scale flow can be constructed to have different large--, Q, Qj, - 2bi, Qk Q, + 4Q, Q, Q, Q), scale properties from a discrete lattice gas with the same

which is isotropic only if spatial symmetry group.Going on to more and more symmetric flows, it turns

3'IA (Q) I'Q, Qj = A60, out that icosahedral or dodecahedralt symmetry in the small-s scale flow gives exact isotropy to the hyperscale dynamics.

Z JIA (Q) 'Q, Q, Qk Qp (6,131 + 1515A~j + 6,k 6 For example, the wave vector set for the icosahedral flow is r

, for some onstats A and I. Clearly the more vectors Q we S or + 1-)_ , -- ,o(6

have in S, the better our chances of being able to select ampli- s -IsS tudesA (Q) of the corresponding components so as to obtain t

an isotropic tensor N. We shall give some examples to illus- where r is the golden ratio (I + -5)/2 and s (5 + a'5)/2.trate the connection between the underlying small-scale flow It is easily checked that the tensor N,,, is isotropic for thisstructures and the resulting hyperscale dynamics. flow, provided that the amplitudes of the modes are all cho-

The simplest Beltrami flow in the family (3) has only sen equal. The hyperscale properties of the icosahedral flow dcltwo Fourier components with the wave vectors appear to be indistinguishable from those of the exactly iso-4Q = ( 1.0,0) and amplitudes A( ± Q,) = c/2 v1 tropic flow obtained as the limiting case of flows with more- c (0,cos x/l, - sin xi). A hyperscale perturbation with and more Fourier components distributed uniformly on the d

the wave vector (0.k.0) then obeys the equation of motion unit sphere.

= - vk 2(I - Re2 )v,, i.=-0. It is interesting to observe that not oiliy does the mo-mentum equation for the hyperscale modes take the clasoical k

i - vk 2(1 + Re')'. form, but so does the hyperscale diffusion equation. The ten-

The effective viscosity for the I-, component is negative, and sor that enters the correction for the effective diflu si tt in

2026 Phys Fluids. Vol 29, No 7. July 1986 If,!?Ars ,'6

';,t.,,. -..+ ' ...,. , ,........ .s - .;, < A, : .. PA ,' . . _. P + -..,.. , .. ,,. .. .. .R., -,. .+..- '.. ,. A: .. ..., '.&., -. ,.. . .. L. .• . . . , - .. ..A:'

"ale the lowest smoothing approximation '" is the sum of the lattice gas oitwiels are equivalent to the Navier-Stoks equa.-"projection tensors" tion with an external fore having the x.rnery h a,-of TO n t:6dQ.

which is isotropic for all the aforementioned flows except the4) simple flow with only two Fourier components. Indeed, it

has been demonstrated independently of the smoothing the-ory that a passive contaminant disperses diffusively in theicosahedral and augmented cubic flows. Simulations of par.

(r ticle dispersion in these flows demonstrate that, for largetimes, almost all particles migrate away from their startingpoint with a finite effective diffusivity.

The analogy between the CA and hyperscale hydrody- 'J. Hardy, 0. de Pazzis. and Y. PomeaU. Phys. Rev. A 13.1949 (1976).

in namic descriptions of the fluids goes even further. The equa- 21. Frsch. B. Hasslacher. and Y. Pomeau Phys. Rev. Let. 56. 1505inin t1986).

tions for the large-scale velocity field derived in Refs. 7-10 is. Wolfram, submitted to Phys. Rev. Lett.are based on the neglect of the higher-order nonlinear terms 'D. d'Humieres, Y. Pomeau. and P. Lallemand. 1Mech. Fluids in press).generated by the scale elimination procedure and thus they *S. A. Orszag and V. Yakhot. Phys. Rev. Lett. 56. 1693 (198o).r e g n e r t ed b y t e s al e eli i n at o n r o c d u r a n d th u th y ' . W o lfra m (p rivate o m m u n ca to n ).are valid when the ratio ul <V. The same holds for the CA ld'Humieres, Lallemand. and Frisch have recently observed that the 3-D

e hydrodynamics:2 the Navier-Stokes equation is an approxi. projection of the 4-D Bravais 24-hedral lattice leads to isotropic 3-D fluidmation valid only when the Mach number Ma = v/v,,, < 1, dynamics at low Mach numbers.where v,, is the velocity of the particles on the lattice. 'G. Sivashinsky and V. Yakho,. Phys. Fluids 28. 1040 (1985).

BB. Bayly and V. Yakhot, submitted to Phys. Rev. A.Based on the analogy between hyperscale hydrodyna.- '°". $ivashinsky, Physica D 17, 243 (1985).mics and the CA description of the fluids, we argue that "V. Yakhot and G. Sivashinsky, submitted to Phys. Fluids.

Long waf t.a canal with a porous plate located at a stepaPeder A. Tyvand

Diartment of Physic and Me . Agriculural University of Nor'wa .1432 Aas.oNLH, Norway

(Received 19 February 1986; acce 10 April 1986)

A linear shallow water theory of waves in al is consider e energy dissipated by a porousplate located at a step where both the depth width of the al changes abruptly is calculated.Itis found that a broadening of the canal, com i ed with plate with optimal properties, is anefficient way of dissipating wave energy. It is also sibl Io have a strong wave damping withoutany reflection.

Lamb' gave the reflection and transmission coefficie s v id in the long-wave limit. Hereg is the gravitational accel-for lon waves at a step in a canal. Both the width and pth era 'on, y, and y. are the surface elevations on each side ofof the canal may change abruptly at the step. The ailed the pte and u is the average horizontal velocity in the sec-analysis of BartholoreUSZ2 verified that Lamb's s tion is tion c ered by the plate:correct to the leading order in a long-wav expsion (see u = /min( l,b/b,)min(l,h2/h,)

I also Mci'). We will generalize Lamb's solution the case ofa thin porous plate located at the step, and di uss the dissi- =u, min(b,/b:,l)min(h,/h,1) . (2)pation of wave energy. Thereby the recent ork of Chwang Here u, and are the average horizontal velocities next toand Dong4 is extended. the plate, for x and x > 0, respectively.

We consider a straight horizont canal of uniform Following La'b,. we calculate the transmission coefli-depthh, andwidth b forx<0wher isa coordinatealong cient T, and the refl ion coefficient R, for waves incidentthecanal.Forx>Othedepth is h thewidthb.Atthe from x < 0:step (x = 0) there is a thin vert al porous plate ofthickness T, 21G -4-- 1 r)(d and permeability K. The kinematic viscoxity of the fluid isT - , . I.....v, but the flow outside the plate is assumed inviscid. Theflow R,= (G-'-+ I - r/(G I + r). (4)

inside the porous plate is assumed to be governed by Darcy's In Eqs. (3) and (4) we have introduced the dinie,',.Ilclaw, which may be stated as parameters

u (Kg/vd)(y 1 - y.) , t I) G - (K /vd(c. rnin(h,/h:.l )ri||(hlh.l )5)

2027 PIys Fluas 29 (71 July 1986 0031.9 ,"I/86,0720Or j2$50 90 ' 986 At, ,cuv vh4! O o1

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Reynolds Number Scaling of Cellular-Automaton Hydrodynamics

Steven A. Orszag and Victor YakhotApplied and Computatiuoal I Alailheipiai,. Prueron Ln,irvLb. Pritnct, ,pi. New Jersc' OS5 44

(Received 25 November 1985)

We argue that the computational requircments for prcscntly envisaged ccltular-autonmatin simu-lations of continuum fluid dynamics are much more severe than for solution of the continuumequations.

PACS numbers: 47.10.+g

It has recently been suggested" that cellular auto- flows. We present three successively more restrictivemata (CAs) [defined as discretely and locally linked, arguments that show that i7/a must grow rapidly withfinite- (and few-) state machinesi may be an effective Reynolds number.way to compute complex fluid flows. These automata Signal-to-noise ratio.-The hydrodynamic velocity inhave the advantage that they may be simply and the CA simulation is calculated by subdividing theperhaps inexpensively constructed with use of specially computational domain into cells with linear dimen-designed parallel hardware. With suitable interaction sions >> a, averaging over the CAs within a (finite)rules, it has been argued," 2 the space-time average cell, and smoothing (filtering) the resulting (noisy)kinetic behavior of the CA system follows the in- velocity field. Thus, the hydrodynamic velocity at acompressible Navier-Stokes dynamical equations. point x is the (space-time) filtered velocity of the CAsWhile the Navier-Stokes equations for continuum in the cell C, centered at x. vH(x) - (v(x)), wherefluids can be calculated efficiently on parallel- the local velocity in C, isarchitecture machines, it is probably easier to make ef- Ificient use of the parallel architecture with CAs. In v(x) - - Iu, (1)this Letter, we wish to point out that there are someconsiderations that require resolution before these where n is the number of occupied sites i within themethods can be considered to be a viable alternative to cel. We assume that the possible velocity values at antraditional continuum mechanical methods for high- occupied CA site are v,- t_ v,, where vi, is the con-Reynolds-number fluid dynamics. stant (thermal) velocity over the CA grid. At low

Let us compare the resolution and work require- Mach numbers, UH << Un,. In this case, the fluctua-ments for a CA simulation of a high-Reynolds- tions in u(x) are of order n- 1/Vh. In order that thenumber flow with those of direct numerical solution of hydrodynamic velocity found in this way may be athe incompressible Navier-Stokes equations. It is well good representation of the continuum hydrodynamics.known '4 that, at Reynolds number NRe, the Kolmo- it is necessary that the noise n - 1/Vh be small com-gorov and Batchelor-Kraichnan theories of three- and pared to the smallest significant hydrodynamic veloci-two-dimensional equilibrium range dynamics, respec- ty. The smallest significant hydrodynamic velocity istively, predict that the range of excited scales is of or- the eddy velocity on scales of order of the dissipationder N4 and N,1/ 2 and the computational work re- scale -q. In three dimensions, - ((e/v 3) - 1/4) andquired to calculate a significant time in the evolution the eddy velocity on the scale of q is v, - O ((q,)e ).of large-scale flow structures is of order V3 , and N " Here v is the viscosity and e is the turbulent energyin three and two dimensions, respectively, dissipation rate per unit mass. Thus, we require that

The suggested evolution rules for CAs to reproduce the number of CA sites n within a cell of size -q be athydrodynamic behavior are based on conservation laws leastof mass, momentum, and energy. Dissipation is n >> V12/ (e,,) / 2. (2)modeled through the thermalization of coherent hy-drodynamic modes. Therefore. the lattice resolution Since e - 0 U 3 /L ) where U is the large-scale rmsof the CA calculation must be much finer than that of fluctuating velocity and L is the associated large-scalethe hydrodynamic simulation, the latter requiring the length of these velocity fluctuations, we find thatretention of only those degrees of freedom describing n >> 'VA... 2, where ,Vi, - L ,' v is the Reynoldsmotions on scales of the dissipation range or larger. number and V/ -/L'/i,, is the Mach number.' SinceThus. the lattice spacing a must be smaller than the the number of cells of size i, within a three-dissipation scale 17 in the turbulent fluid. dimensional turbulent eddy of size L scales as .

We now discuss some conditions that CA models the overall number of CA ics must ncrease at leiaishould satisfy to describe high-Rcynolds-number fluid as .R..

C 1996 The .\merican Phvsical Society I f,)l

'PWLI %-A F.II"r.1% pul TRFM K -WMK X." 17V M/1WVW

O=.L LM , NLMHtii Ib PHYSiCAL REVIEW LET1TERS 21 APRIL !-,,)

Since the elTective evolution time of the fluid sys- for CA (31)),tern is Li'L, while the time step on the CA lattice is S W i

,/lh it follows that the CA simulation icquires it

least L/aM steps in time. Since the computational for Navier-Stokes (3D).work for each site update is of order 1, it follows that livdrodynani" fluctuaons.-The CA system willthe CA simulation requires at least of order (.VKJ yield a sell'-consistent continuum hydrodynamic de-M work. scription only if the thermal energy fluctuations on hy-

In summary, the above signai-to-noise considera- drodynamic spatial scales are small compared to thetions suggest the following lower-bound estimates for energy of the hydrodynamic modes on correspondingthe computer storage S and work W for CA simula- length scales. If the "mass" of an occupied CA site istions of high-Reynolds number, Iow-Mach-number m, its energy is d121n vt in d space dimensions. Then

flows (where, for reference, we include the corre- the fluctuation in total thermal energy over a cell with

sponding estimates for the continuum Navier-Stokes n occupied CAs is v'1m1p,. (We note that in a CAequations): for CA (2D), with velocity states t v,,, energy fluctuations are pro-

r 3/t2 I, 91/4 . portional to density fluctuations.) The correspondingS-N~/M. WN~'M';hydrodynamic energy within a cell of size -n is pyl uj,,

for Navier-Stokes (2D), 6 where VH is the hydrodynamic velocity and p is the by-S - NR,, W =- Na 1/2. drodynamic density. In three dimensions, the dissipa-

SN, W p, tion scale is 71 and the associated hydrodynamic veluci-

for CA (3D), ty is v,. Also, the relation between m and p is nm=,pg3. Thus, for thermal fluctuations to be small, we

S ,= i," /M 2.. W,,=(NR/M)tif; must require that n >> NRC/M 4. In two dimensions,

for Navier-Stokes (3D),7 the corresponding result is n >> NR/M'This argument shows that the storage and work re-

N4/.4 W-N . quired for a self-consistent hydrodynamic descriptionusing CAs is of order

Upper bound jor the Reynolds number.-A more

stringent condition on the Reynolds-number depen- S-N' /M'. W-,N, ,/7

dence of the minimal number of lattice sites in a CAsimulation of hydrodynamics is found as follows. Ifthe discrete velocity of the CAs is = v,, (again, the S - N , W -N.g

thermal velocity or sound speed on the CA lattice) and for Navier-Stokes (2D).the lattice spacing is a, then the kinematic viscosity von the lattice is at least of order va. For the CA to S 1 / 4/.V 4 W - N1I/M19/3R=v e /,, l v ¢1: *,

give a self-consistent hydrodynamic simulation, theviscosity determined on the "molecular" level must for CA (3D),

. equal the viscosity governing the dissipation of the hy- S ,,,v, ; Y -

drodynamic modes. Thus the Reynolds number of the

simulated fluid can be at most UL/v or MLla. Since for Navier-Stokes (3D).

the number N of CA sites in the lattice is of order The CA models approximate fluids that are by their

(L!a )d, where d is the dimension of space, we obtain nature necessarily compressible. This means thit an

the result that N must be at least of order (VR,/M )". equation of state for pressure is needed. Howeer,

As above, the CA simulation of the flow requires at self-consistency requires that thermodynamic pressure

least L/aM steps in time. It follows that the CA simu- fluctuations over dissipation scales be small. Thislation requires at least of order ' memory latter condition leads to results identical to those jus;

,an r /iet order VJtA/ m o obtained by use of energy estimates. Indeed, it is

t, and of order .- R/~ work. knowng that the rms thermodynamic pressure tluctua-

These estimates for storage S and work W based on 3lower bounds for the eifeciive viscosity on the lattice tions in a volume T are (pkTc-/Ti) ' -, where Tis th,:

are of order temperature of the fluid and c is the sound speed. Butthe hydrodynamic pressure fluctuations over length

S ,(.'K/.I[)., - scales of order Yj are of order p(ev) V . Since T

for CA 2D), - m u,, and C==- ,,, the previously given estimu:-,apply.

S - ,\', , - We believe that these pes.misic estimates tor

for Naer-Stokes (2D, Reynolds numbers and low Niah numbers niut t!

overcome heore CAs can be an cliiLne ni(,docL

S - .X 1 At . | , .l tool fur complex fluid flows. This can, pri nt pI ,. c

1692

%!:~ . . . . . ,-... . . . . .: - . . .. ,- . .. . V,, - , - . .. . .. : .. . , - -V , '

done by averaging over the shortest '.ales z --- I inorder to reduce the number of degrees of freedom. "

lowever. it seems that this renormalization can beuseful (in the context of local, few-bit, parallel compu-tations) only if it does not generate nonlocal, complexinteractions in the set of basic rules defining CAs. Un- 1U. Frisch. B. 1I1aslacher. and Y. Ponicau. Phys. Rcv.

fortunately, we do not now understand why this kind Leit. 56. 1505 (1996).

of "turbulence transport" modeling should be either 2J. Salem and S. Wolfram. to be published.easiert orumorce succes fuonr t hodeiCA sh l e ith or S. A. Orszag, J. Fluid Mech. 41. 363-386 (1970).easier or more successful on the CA lattice than for 4j. R. Herring, S. A. Orszag. R. H. Kraichnan. and D. G.the continuum equations or for molecular dynamics. Fox. J. Fluid Mech. 66. 417-444 (1974).

While the above estimates for CA simulations of , 5Notice that the Mach number is bounded in CA simula-turbulence are quite pessimistic, there may be cases in tions of fluid dynamics. With allowed velocities of ± v,,.,which CA simulations of turbulence may be effective. At -_ 0 (1). the upper limit being achieved when the CAsIn a turbulent boundary layer, the local Reynolds exhibit pure streaming, nonhydrodynamic motion.number is 0(1) in the viscous sublayer and is modest 6The Batchelor-Kraichnan theory suggests that there maywithin the buffer layer. A CA model could be erec- be logarithmic corrections to these estimates.

tive in these regions in the modeling of turbulent burst 7intermittency effects may slightly change these estimates.

formation and evolution. However, this application 8L. D. Landau and E. M. Lifschitz, Flud echanics (Per-

requires the development of three-dimensional CA gamon. London. 1959), p. 529.models 9A kind of renormalization using "pseudovertices'" is sug-o and suitable techniques to match the outer re- gested in Ref. 1. The idea is to introduce vertices on subhy-

gions of" the flow. drodynamic scales that are not kept track of explicitly. If

this idea can be successfully implemented, it would reducethe computational storage requirements given above but ap-parently not the computational work. We are grateful toC. H. Bennett for this comment.

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