On the relevance of the type of contraction of the Germano identity in the new integral-based dynamic Smagorinsky model

The 20-year old dynamic model is a well-established procedure in large eddy simulation that allowed us achieving several progresses in the field of numerical simulation of turbulence. The key of the procedure is in the derivation of the exact Germano identity, which is a tensor relation. After introducing the eddy viscosity model, in order to achieve a scalar relation and thereby determine a single value of the model constant, Germano et al. proposed contracting the tensor identity with the resolved strain rate tensor. Then, Lilly observed that this method can be efficient, but does raise the problem of indetermination when the resolved strain rate tensor cancels out. To remedy this problem, Lilly proposed calculating the model constant by a least-squares method, that is by minimizing the Germano identity error, observing that a negative value of the constant results to be consistent. Such a contraction is now a standard in the LES community, however, since the model function is arbitrarily extracted out from filtering, the necessity to adopt suitable averaging, clipping and some other cares to achieve numerically stable solutions are considered to be still open problems. Unlike the original Germano identity, which was developed in the framework of the differential formulation of the filtered equations, the recent integral-based filtered equations were shown to produce a new tensor identity wherein the model function is no longer acted on by filtering and . While the constant model preserves its full three-dimensional character, this new model was proved to produce stable and accurate solutions and is suitable also for complex flows. The aim of this study is to investigate the effects of three different contractions of the new integral-based tensor identity since each one of the projection represents a different minimization of the residual. The new expression of the Germano identity error is introduced to show the resulting accuracy in testing the turbulent flow in a plane channel. Unlike what is reported for the differential form of the dynamic model, the results highlight that the integral-based method is much less sensitive to the type of contraction than expected in the differential-based formulation. This fact confirms the better aspect of the integral-based Germano identity.