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 ε→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=siny1siny2+δcosy1cosy2, y=x/ε, 0⩽δ⩽1. For small ε we used a numerical upscaling method. We designed this method so that it captures the modulational perturbations for any ε with O(ε2) accuracy and independent of ε complexity.