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Eddy viscosity of three-dimensional flow

Published online by Cambridge University Press:  26 April 2006

A. Wirth ,
S. Gama  and
U. Frisch
A. Wirth
Affiliation:
CNRS, Observatoire de Nice, B.P. 229, 06304 Nice Cedex 4, France
S. Gama
Affiliation:
CNRS, Observatoire de Nice, B.P. 229, 06304 Nice Cedex 4, France FEUP, Universidade do Porto, R. Bragas, 4099 Porto Codex, Portugal
U. Frisch
Affiliation:
CNRS, Observatoire de Nice, B.P. 229, 06304 Nice Cedex 4, France
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Abstract

Detailed theoretical and numerical results are presented for the eddy viscosity of three-dimensional forced spatially periodic incompressible flow.

As shown by Dubrulle & Frisch (1991), the eddy viscosity, which is in general a fourth-order anisotropic tensor, is expressible in terms of the solution of auxiliary problems. These are, essentially, three-dimensional linearized Navier–Stokes equations which must be solved numerically.

The dynamics of weak large-scale perturbations of wavevector k is determined by the eigenvalues – called here ‘eddy viscosities’ – of a two by two matrix, obtained by contracting the eddy viscosity tensor with two k-vectors and projecting onto the plane transverse to k to ensure incompressibility. As a consequence, eddy viscosities in three dimensions, but not in two, can become complex. It is shown that this is ruled out for flow with cubic symmetry, the eddy viscosities of which may, however, become negative.

An instance is the equilateral ABC-flow (A = B = C = 1). When the wavevector k is in any of the three coordinate planes, at least one of the eddy viscosities becomes negative for R = 1/v > Rc [bsime ] 1.92. This leads to a large-scale instability occurring for a value of the Reynolds number about seven times smaller than instabilities having the same spatial periodicity as the basic flow.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 288 , 10 April 1995 , pp. 249 - 264
Copyright
© 1995 Cambridge University Press

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