Optimal eddy viscosity in closure models for two-dimensional turbulent flows

Abstract

We consider fundamental limitations on the performance of eddy-viscosity closure models for turbulent flows focusing on the Leith model for two-dimensional Large Eddy Simulation (LES). Optimal eddy viscosities depending on the vorticity gradient magnitude are determined subject to minimum assumptions by solving partial-differential-equation-constrained optimization problems defined such that the corresponding optimal LES best matches the filtered Direct Numerical Simulation. Since the optimal eddy viscosities do not converge to a well-defined limit as the regularization vanishes, we conclude that the problem of finding an optimal eddy viscosity does not have a solution and is ill-posed.