Eddy viscosity of cellular lows

The goal of this work is to understand the phenomenon of eddy viscosity in two-dimensions. In the presence of small-scale eddies the transport of large-scale vector quantities can be accompanied with depleted, and even ``negative" diffusion at large Reynolds number. For stationary flows, the eddy viscosity is a tensor in the equation that governs the transport of the slow spatial modulations of highly oscillatory cellular solutions (eddies) of the Navier-Stokes equations.

Using multiscale techniques we derive eddy viscosity of cellular flows - special periodic stationary solutions of the Euler's equations. We justify this derivation using bootstrapping and Littlewood-Paley type arguments. We design a numerical upscaling method, that allows us to compute eddy viscosity of cellular flows when ratio between the scales is small, and compare it with the predictions of the multiscale analysis. For cellular flows with closed streamlines we give rigorous bounds on eddy viscosity at high Reynolds number by means of saddle point variational principles for nonlocal, nonselfadjoint operators.

Refreshments will be served at 2:45 p.m. in the Faculty Lounge, Math Annex (Room 1115).