Modeling and simulation of turbulence not at statistical equilibrium

Standard eddy viscosity models, while popular, cannot represent backscatter and have severe difficulties with complex turbulence not at statistical equilibrium.~In this work, we give a derivation of eddy viscosity models from an equation for the evolution of variance in a turbulent flow showing how to correct eddy viscosity models. We show the corrected models preserve important features of the true Reynolds stresses and give algorithms for their discretization including a minimally invasive modular step to adapt an eddy viscosity computational~methodology to the extended models. A numerical test is given with the usual and over diffusive Smagorinsky model. The consistency and the predictive capability of the model are established by conducting LES of a three-dimensional compressible mixing layer, and comparison with DNS data. The correction does~exhibit intermittent backscatter.