Eddy viscosity of cellular flows

We analyse modulational (large-scale) perturbations of stationary solutions of the two-dimensional incompressible Navier-Stokes equations. The station ary solutions are cellular flows with stream function phi = sin y(1) sin y( 2) + delta cos y(1) cos y(2), 0 less than or equal to delta less than or eq ual to 1. Using multiscale techniques we derive effective coefficients, inc luding the eddy viscosity tensor, for the (averaged) modulation equations. For cellular flows with closed streamlines we give rigorous asymptotic boun ds at high Reynolds number for the tensor of eddy viscosity by means of sad dle-point variational principles. These results allow us to compare the lin ear and nonlinear :modulational stability of cellular flows with no channel s and of shear flows at high Reynolds number. We find that the geometry of the underlying cellular flows plays an important role in the stability of t he modulational perturbations. The predictions of the multiscale analysis a re compared with direct numerical simulations.