Eddy viscosity of cellular flows

We analyse modulational (large-scale) perturbations of stationary solutions of the two-dimensional incompressible Navier-Stokes equations. The stationary solutions are cellular flows with stream function φ = sin y₁ sin y₂ + δ cos y₁ cos y₂, 0 ≤ δ ≤ 1. Using multiscale techniques we derive effective coefficients, including the eddy viscosity tensor, for the (averaged) modulation equations. For cellular flows with closed streamlines we give rigorous asymptotic bounds at high Reynolds number for the tensor of eddy viscosity by means of saddle-point variational principles. These results allow us to compare the linear and nonlinear modulational stability of cellular flows with no channels 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 the modulational perturbations. The predictions of the multiscale analysis are compared with direct numerical simulations.