Local Variational Germano Identity for Dynamic Large Eddy Simulations based on Finite Elements

In this talk we will present a localized formulation of the variational Germano identity (VGI) that is applicable to inhomogeneous turbulent flows. This is done in the context of stabilized finite element methods where a combined subgrid-scale (SGS) model is employed. In particular, the combined SGS model uses the residual-based variational multiscale (RBVMS) approach along with the Smagorinsky eddy-viscosity model. The RBVMS model is used to represent the cross-stress terms while the eddy-viscosity model is used for the Reynolds stresses. The unknown parameter related to the eddy-viscosity model is computed using a dynamic procedure based on the local VGI. The overall idea of the local VGI is to compare the numerical solution at two different levels of discretization/grid in a local fashion. The local VGI based dynamic procedure is made practical by employing suitable assumptions. Further, the procedure is made robust by employing a Lagrangian averaging scheme along with a local spatial averaging. This is equivalent to averaging along local pathtubes and maintains the applicability of the current methodology to inhomogeneous turbulent flows. We demonstrate the current LES methodology on a variety of problems including flow over cylinders and surging airfoils.