EDDY VISCOSITY OF 3-DIMENSIONAL FLOW

Detailed theoretical and numerical results are presented for the eddy viscosity of three-dimensional forced spatially periodic incompressibl e flow. As shown by Dubrulle & Frisch (1991), the eddy viscosity, whic h is in general a fourth-order anisotropic tenser, is expressible in t erms of the solution of auxiliary problems. These are, essentially, th ree-dimensional linearized Navier-Stokes equations which must be solve d numerically. The dynamics of weak large-scale perturbations of wavev ector k is determined by the eigenvalues - called here 'eddy viscositi es' - of a two by two matrix, obtained by contracting the eddy viscosi ty tenser with two k-vectors and projecting onto the plane transverse to k to ensure incompressibility. As a consequence, eddy viscosities i n three dimensions, but not in two, can become complex. It is shown th at this is ruled out for flow with cubic symmetry, the eddy viscositie s of which may, however, become negative. An instance is the equilater al ABC-flow (A = B = C = 1). When the wavevector k is in any of the th ree coordinate planes, at least one of the eddy Viscosities becomes ne gative for R = 1/nu > R(c) similar or equal to 1.92. This leads to a l arge-scale instability occurring for a value of the Reynolds number ab out seven times smaller than instabilities having the same spatial per iodicity as the basic flow.