Reduced order modeling of turbulent flows using statistical coarse-graining

Traditional sub-grid models for Large Eddy Simulations (LES) of turbulence rely on theoretical and physical arguments and are known to be inadequate in many applications. This work investigates the use of the Mori-Zwanzig formalism, a concept that originates from non-equilibrium statistical mechanics, to develop sub-grid models for LES. The mechanics of the generalized Langevin equation (GLE) are considered and a methodology for directly solving the orthogonal dynamics equation is presented. This process is shown to provide insight into the form of the GLE and the effects of coarse-graining. This knowledge is used to derive a class of sub-grid models that are capable of consistently representing non-local memory effects. The models are applied to the viscous Burgers equation and homogeneous isotropic turbulence. The models are shown to be in good agreement with high resolution simulations and outperform Smagorinsky-type models.