Eddy Viscosity and Diffusivity: Exact Formulas and Approximations

Exact asymptotic expressions for eddy diffusivity and eddy viscosity are ob tained as the lead ing terms of in finite-series representations of integral equations which express the act ion of turbulence on an infinitesimal mean field. The series are transformed term by te rm from Euler ian to Lagrangian form. The latter is more suitable for constructing approximations to the exact asymptotic express ions. T he analysis is prefaced by some qualitative remarks on possible improvements of eddy transport algorithms in turbulence computations. 1. Introd u ction E ddy viscos ity and eddy diffusivity have long been fruitful concepts in turbu lence theory, and the ir use has made possible the computation of turbulent flows at Reynolds numbers too high for fu ll numerical simulation. However, there is a fundamental logical Haw. Mo lecular viscosity is a va lid concept when there is a strong separation of space and time scales between hydrodynamic modes and gas-kinet ic collision processes. In high-Reyno ldsnumber turbu lence, on the other hand , there is typically a continuous range of sign ificantly excited modes between the largest motions and those small motions which are represented by an eddy viscosity. In t he present paper, the lack of clean scale separation of modes is expressed by exact statistical equations in which the interaction between a (large-scale) mean field and a (small-scale) fluctuat ing field is nonlocal in space and t ime. In an asymptotic case of interaction be tween modes whose sp ace and t ime scales are strongly separated, the exact formu las reduce t o ones which are effectively local in space and time and which express what can be termed the distant-interaction eddy viscosity. Even in this asymptotic case, the exact eddy viscosity is not a simple expression, and it is not perfectly reproduced by any approximations that have been proposed. Most of the present paper is devoted to a derivat ion of t he exact expressions for asymptotic eddy viscosity and eddy diffusiv ity, the embedding of "Consultant, Theoretical Division and Center for Nonlinear St udies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA © 1987 Complex Systems Publications, Inc. 806 Robert H. Kraichnan these express ions within infinite-series representati ons of the general nonlocal integral equations, and, particular ly, the t ransformat ion of Eulerian formulas into Lagrangian ones . The Lagran gian representat ion is probably the one in which approximations to the exact eddy viscosity and eddy diffusivity can most successfully be carried out. It is hoped that both the Eu lerian and Lagr angian exact formulas can be of use in analy zing approximations and interpreting various approaches to t he construction of eddy transport coefficients. T his mathemat ical analys is is prefaced by a qualitative d iscuss ion of the more difficult question of improv ing eddy-viscosity and eddy-d iffusivity algor it hms actually used for subgrid-scale representat ion in computations of turbulent flows. The spat ia l an d temporal nonlocalness exhibi ted in the mathematical analysis may here be of some practical sign ificance . The relative success of very cru de eddy-viscosity approximat ions in comput at ions suggests that the dynamics of tur bulence yields robust statistics, with feedback characteristics that somehow partly compensat e for bad app roximat ions. However , exist ing subgr id-scale ap prox imations do no t perform well in the computation of the point-to-point amplit ude st ructure of a largescale flow, as opposed to statistics. It may he that here the incorporation of nonlocal effects is essential. Nonlocality in time means that the su bgrid modes exert reactive as well as resistive forces on the explicit modes, and this may be important in reproducing finite-amplitude ins tabilities and . other properties of the explicit modes. One conse quence of the lack of clean separation of explicit and subgrid mo des is that the latter exert fluctuat ing driving forces on the explicit mo des which are conceptually distinct from eddy viscosity (or even negative eddy viscosity ) [11 . Since the detailed st ructure of the subgrid modes is unknown in a flow comput ation, the fluctuating forces must be treated stat ist ica lly, but the close coupl ing between the two classes of mo des means that the re levant stat istics are not purely random. T he existence of fluctuat ing for ces on the exp licit mo des implies t hat the explicit velocity field in a calculation is not simp ly replaceab le by its stat ist ical mean. Some simple numer ics y ield a strong mot ivat ion for improvement of subg rid representa ti ons. If the smallest spatial scale t reated explicitly in a h igh-Reynolds number flow increases by a factor c, t he computa tional load of the explicit calc ulation decreases by a factor of perhaps c4 • T he precise ratio depends on the method of computation used. If the calculat ion is Eu lerian, and logarithmic factors assoc iated with fast Fourier t ransforms are ignored , the power four used above ar ises from increase in grid mesh size in three dimensions together with increase in the smallest t ime ste p needed , the latter determined by the convection by the large-scale flow. If these cr ude est imates are relevant , an increase of m inimum exp licit sca le by a factor 2 decreases the load by a factor 16, and an increase by a factor 4 decreases the load by a factor 256. This means that an imp rovem ent in subgrid representation that permits a shrinking of t he explicit scale range by a factor 2 may be cost-effect ive , provided that it increases t he computation Eddy Viscosity and Diffusivity 807 size by less t han a factor 16 over a cruder subgrld-scale representation . Caveats should be stated at this point. F irst, a useful subgrid a lgorithm must be practical to program and implement. This implies, among other th ings, t hat it must have a reasonably broad application. Second, a new algorithm must in fact be an improvement. An analytical approximation wh ich includes higher-order effects may actually make things worse rather than better, because the convergence properties of the relevant approximation sequences are subtle and dangerous. It seems unlikely that adding correction terms to the local asymptotic formulas exhibited in the body of the present paper is a valid route to improving eddy-viscosity and eddy-diffusivity representations. And, of course, no eddy-viscosity representation, however good, takes account of the random forces exerted by the subgrid modes. Moreover, the statistical facts about the subgrid scales needed to evaluate even the lowest-order asymptotic formulas are unavailable in a practical flow calculation. One alternative approach is to infer as much as possible about the be. havior of the subgrid mo des by ext rapolation from the dynamics and statistics of the explicitly computed modes. The well-known Smagorinsky eddyviscosity formul a [2J can be v iewed as a simple example of this approach. Here, the effective eddy viscosity is determined from the local rate-of-strain te nso r of t he exp licit ve locity field by appeal to Ko lmogorov inert ia l-range scaling ar guments. It may be worthwhile, however, to extract substant ia lly mo re detailed infor mation from the exp licit ve locity field in order to estimate the dynamical effects of the subgr id scales. Supp ose, for example, the flow calcu lation is sufficiently large t hat a subst antial ran ge of spat ia l scales is included in the explicit veloc ity field . It is then possib le t o analyze the explicit field to extract information abou t the mean transfer of energy between di fferent scale sizes, reactive interactions, and, as well , the fluctuating forces exerted by exp licit mo des of small scale on those of lar ger scale. Such an analysis could be performed individually on each calc u lated flow field as the computation proceeds. However, it m ight be mo re economical to try to build up library tables of results fro m which these quantit ies could be rapidly estimated for a given computa tion using rela tively few measured parameters. In either event, the effects of subgrid mo des on the explicit field could then be estimated by assuming similarity with interscale dynamics within the explicit field. T his wou ld seem a less dras tic assumption than adopting idealized inertial-range dynami cs for the subgrid scales. Of course, the similarity analysis could be modified by taking into account crucial dynamical differences between explicit and subgrid modes-for example, increase of molecular viscosity effects with wavenumber. 2. Exact Eulerian analysis for eddy diffusivity The equation of motion for a passive scalar advected by an incompressible velocity field may be manipulated to yield statistical equations which are 808 Robert H. Kraichnan exact and which display an eddy-diffusivity term acting on scalar modes having very large space and time scales. Let the scalar field o;6 (x , t) obey L(,,)0;6(x,t) + u (x , t) . V 0;6(x , t) = 0 (2.1) where a L(,,) = "v' (2.2) at and V · u(x,t) = O. (2.3) Write o;6(x, t) = o;6"(x, t) + 0;6' (x , t) (2.4) where o;6· ' (x , t) = (0;6 (x, t» (2.5) and ( ) denotes ensemb le average. Assume (u (x , t) = O. (2.6) If, ins tead, u (x, t) has a non zero mean , extra terms appear in the following analys is. Equations (2.1) through (2.6) yield the following equat ions for the mean and fluctuating sca lar fields: L( ,,)0;6·' (x ,t) = Q(x,t) (2.7) IL (,,) + u(x, t) . V)]0;6' (x ,t) = u (x, t) . V0;6·' (x, t) Q(x , t) (2.8)