VOLUME 58, NUMBER 6 PHYSICAL REVIEW LETTERS 9 FEBRUARY 1987 Analysis of Subgrid-Scale Eddy Viscosity with Use of Results from Direct Numerical Simulations J. Andrzej Domaradzki, Ralph W. Metcalfe, Robert S. Rogallo, ' and James J. Riley Flo~ Research Company, Kent, Washington 98032 (Received 15 September 1986) Without resort to any modeling, subgrid-scale eddy viscosity is computed from the results of high- resolution (643 and 1283 grid points) direct numerical simulations of three-dimensional homogeneous, isotropic, decaying turbulence. In these simulations the eddy viscosity peaks sharply at the cutoff' wave number, in rough agreement with the results of Kraichnan. In addition, in the low-wave-number range the eddy viscosity may be negative, contrary to the generally accepted concept of a subgrid-scale eddy viscosity. Some possible explanations of this behavior are discussed. PACS numbers: 47. 25. — c, 47. 25.Cg, 92. 10.Lq, 92.60.Ek As a result of the large number of excited modes in high-Reynolds-number turbulent flows and to the in- herent limitations of even the fastest modern computers, it is possible to solve the Navier-Stokes equations numer- ically only for low- to moderate- Reynolds-number flows, Rq~ 0(10 ), where Rz is the Reynolds number based on the Taylor microscale. For higher Reynolds numbers, subgrid-scale modeling is often used with the large-scale motions (i.e. , for wave numbers I k I ( k„k, being the cutoft' wave number) computed explicitly from the Navier-Stokes equations and the eAect of small scales (with wave numbers I k I & k, ) approximated by a subgrid-scale eddy viscosity model. ' The eddy viscosi- ty models the process of the energy transfer from large (I k I ( k, ) to small (I k I & k, ) scales by increased dis- sipation of the large scales. The purpose of this work is to investigate the concept of an eddy viscosity using re- sults of direct numerical simulations of homogeneous iso- tropic turbulence. The Navier-Stokes equations for the velocity field u„ ! in spectral form, [(r)/6t)+ vk ]u„(k) =( — i/2)P„, „(k)„d pu, (p, t)u„(k — p, t), ik„u„=0, lead to an equation for energy amplitudes, r) I u(k) I /Bt = — 2vk I u(k) I +Im u„*(k)P„, „(k)„"dpu, (p)u„(k — p) where I u(k) I =u„(k)u„*(k), P„, „(k) =k„(6„, — k„k, /k )+k, (6„„— k„k„/k ), (2) and the summation convention is assumed. The bracketed term represents nonlinear energy transfer and can be written as follows: T(k) = [T(k) — T (k)]+ T (k), (4) where T(k) =Im u„*P„, „(k)„d'p u, (p)u„(k — p), T (k) =1m u„*P„, „(k) "d'p u, (p) u„(k — p), I k I, I p I, I k — p I «, . (s) (6) T(k I k, ) — = 4trk (T(k) — T (k)), (II a) E(k)— : 4trk' —, ' (Iu(k) I ~). (gb) (. .. ) denotes averaging over thin spherical shells of ra- dius k, and v(k I k, ) is the eddy viscosity. (In the re- stricted sense defined here, the term "eddy-viscosity modeling" used in this paper is also denoted as T(k I k, ) =— 2v(k I k, )k E(k), k ~ k„ (7) In Eq. (6) all wave numbers lie below the prescribed cutoft' wave number k, . Since T (k) represents energy where transfer from the mode k to all modes with wave num- bers less than k„T(k) — T (k) represents energy ex- change between the mode k and two other modes such that at least one of them lies above the cutoA k, . In an eddy-viscosity approach, one attempts to model this term by the following formula: 1987 The American Physical Society 547