Forward and inverse problem solvers for Reynolds-averaged Navier–Stokes equations with fractional Laplacian

Abstract

It has recently been demonstrated that turbulent flow could be described by the fractional Laplacian Reynolds-averaged Navier–Stokes equations fL-RANS equations, (Epps and Cushman-Roisin, 2018). In this paper, we propose a numerical approach for solving the equations, and then provide a deep-learning based approach for inferring the unknown parameters of the equations. First, we construct a lattice Boltzmann model with BGK operator (LBGK model) for solving the fL-RANS equations by leveraging the fractional centered difference scheme we proposed. Through Chapman–Enskog analysis, the macroscopic equations can be recovered from the LBGK model. Second, we couple the physics-informed neural networks with the fractional centered difference scheme to infer the fractional differential order of the fL-RANS equations. The resulting approach is called fractional Laplacian physics-informed neural networks (fL-PINNs). We provide numerical examples to validate our LBGK model and fL-PINNs.