Reduction of the Germano-identity error in the dynamic Smagorinsky model

Abstract

We revisit the Germano-identity error in the dynamic modeling procedure in the sense that the current modeling procedure to obtain the dynamic coefficient may not truly minimize the error in the mean and global sense. A “corrector step” to the conventional dynamic Smagorinsky model is proposed to obtain a corrected eddy viscosity which further reduces the error. The change in resolved velocity due to the coefficient variation as well as nonlocal nature of the filter and flow unsteadiness is accounted for by a simplified suboptimal control formalism without resorting to the adjoint equations. The objective function chosen is the Germano-identity error integrated over the entire computational volume and pathline. In order to determine corrected eddy viscosity, the Fréchet derivative of the objective function is directly evaluated by a finite-differencing formula in an efficient predictor-corrector-type framework. The proposed model is applied to decaying isotropic turbulence and turbulent channel flow at various Reynolds numbers and resolutions to obtain noticeable reduction in the Germano-identity error and significantly improved flow statistics. From channel flow large-eddy simulation, it is shown that conventional dynamic model underestimates subgrid scale eddy viscosity when the resolution gets coarse, and this underestimation is responsible for increased anisotropy of predicted Reynolds stress. The proposed model raises both the overall and near-wall subgrid scale eddy viscosity to reduce exaggerated Reynolds stress anisotropy and yield significantly improved flow statistics.