A Family of Orthogonalised Nonlinear LES Models Based on the Velocity Gradient: Discretisation and Analysis

Abstract

Large Eddy Simulation (LES) aims at computing local spatial averages of solutions to the Navier–Stokes equations. The nonlinearity of the Navier–Stokes equations results in the the need for a LES model to obtain a closed set of equations for the spatially averaged velocity field. Existing LES models, e.g., the Smagorinsky and Gradient model, are often derived from the velocity gradient. In this report we generalise this by considering a general form of a nonlinear LES model which is derived from the velocity gradient. This general form of a LES model encompasses dissipative as well as transport terms. We are interested in the former as well as the latter, and therefore propose the explicit separation of dissipative and transport terms by means of orthogonalisation.

Provided with this framework of LES models we develop a Finite Volume discretisation in which we preserve the favorable properties of the analytical equations, like the vanishing sub-grid dissipation due to the LES model in case of a non-dissipative model. Since we base our discretisation on a symmetry preserving discretisation, this results in preserving the energy equality at the discrete level.

The a posteriori analysis of this general framework is done by considering one non-dissipative term in particular. Using the simulation of decaying Homogeneous and Isotropic Turbulence (HIT) as a test case, we perform numerous simulations to characterise this term in terms of both its energy transport, as well as several statistical correlation and structure functions. Our simulations show that using this non-dissipative term in combination with an eddy viscosity model allows for less eddy viscosity being used while obtaining a similar decay of kinetic energy. The correlation functions also show good agreement with experimental data. The proposed general framework of LES models therefore yields promising results, showing that we can indeed model non-dissipative effects in a LES while maintaining good agreement to experimental data. The obtained results provide opportunities for future research.