The equivalence of the Lagrangian-averaged Navier-Stokes-α model and the rational large eddy simulation model in two dimensions

In the large eddy simulation (LES) framework for modeling a turbulent flow, when the large scale velocity field is defined by low-pass filtering the full velocity field, a Taylor series expansion of the full velocity field in terms of the large scale velocity field leads (at the leading order) to the nonlinear gradient model for the subfilter stresses. Motivated by the fact that while the nonlinear gradient model shows excellent a priori agreement in resolved simulations, the use of this model by itself is problematic, we consider two models that are related, but better behaved. The rational LES model that uses a sub-diagonal Pade approximation instead of a Taylor series expansion, and the Lagrangian averaged Navier-Stokes-α model that uses a regularization approach to modeling turbulence. In this article, we show that these two latter models are identical in two dimensions.