Lagrangian Large Eddy Simulations via Physics Informed Machine Learning

Abstract

High Reynolds Homogeneous Isotropic Turbulence (HIT) is fully described within the Navier-Stocks (NS) equations, which are notoriously difficult to solve numerically. Engineers, interested primarily in describing turbulence at reduced but sufficiently large range of resolved scales, have designed heuristics, known under the name of Large Eddy Simulation (LES). In the usual picture, LES is described in terms of the evolving in time Eulerian velocity field defined over the points of a spatial grid with the meanspacing correspondent to the resolved scale. This classic Eulerian LES depends on assumptions about effects of sub-grid scales on the resolved scales.In this manuscript, we take an alternative approach and design novel LES heuristics stated in terms of Lagrangian particles moving with the turbulent flow. Our Lagrangian LES, thus L-LES, is described by equations that generalize the weakly compressible Smooth Particle Hydrodynamics (SPH) formulation with extended parametric and functional freedom which is then resolved/fixed via Machine Learning (ML) training on Lagrangian data from a Direct Numerical Simulation (DNS) of the NS equations. The L-LES framework includes parameters which are explainable in clear physical terms, e.g. parameters describing effects of eddy-diffusivity and smoothing kernels, and Neural Networks (NN) to represent effects of smaller (unresolved) scales and relations between