EDDY VISCOSITY AND THE STATISTICAL-THEORY OF TURBULENCE

We examine, for the case of stationary turbulence, the correlation bet ween fully resolved turbulence (in the sense of Fourier modes) and a s ystem composed of the same equations on a reduced wavenumber span. Thi s 'reduced system' is subjected to a wavenumber-dependent eddy viscosi ty contrived so that the full and reduced systems have the same energy spectrum, on the reduced span. The particular numerical problem studi ed is isotropic turbulence, and the model of turbulence is a Langevin representation of the test-field model (Kraichnan (1971, J. Fluid Mech ., 47: 513-524). The correlation between the full and reduced systems is surprisingly small, a fact we attribute to be related to the unpred ictability of the underlying Navier-Stokes equations. The (qualitative ) form of the eddy viscosity introduced by such modeling is also discu ssed, and it is noted that at large Reynolds numbers, there exists an optimal cut-off wavenumber which permits an elimination of explicit em piricism from this form of large eddy simulation (LES). Our computatio ns permit a spectral estimate of the magnitude of the backscatter term in LES, along the lines recently proposed by Schumann (1995, Proc. R. Soc. London, Ser. A, 451: 293-318). (C) 1997 Elsevier Science B.V. (C ) 1997 Elsevier Science B.V.