Eddy viscosity of cellular flows by upscaling

The eddy viscosity is the tensor in the equation that governs the transport of the large-scale (modulational) perturbations of small-scale stationary flows. As an approximation to eddy viscosity the effective tensor, that arises in the limit as the ratio between the scales E → 0, can be considered. We are interested here in the accuracy of this approximation. We present results of computational investigation of eddy viscosity, when the small-scale flows are cellular, special periodic stationary flows with the stream function φ = sin y₁ sin y₂ + δ cos y₁ cos y₂, y = x/E, 0 ≤ δ ≤ 1. For small E we used a numerical upscaling method. We designed this method so that it captures the modulational perturbations for any E with O(E²) accuracy and independent of E complexity.