Numerical approximation of statistical solutions of the incompressible Navier-Stokes Equations

TL;DR

This paper develops a numerical method to approximate statistical solutions of the 2D incompressible Navier-Stokes equations, demonstrating convergence and introducing a new algorithm for structure functions on unstructured meshes.

Contribution

It presents a novel Monte Carlo and H(div)-FEM based approach for approximating statistical solutions of NSE with convergence analysis and a new structure function algorithm.

Findings

Convergence of statistical solutions and observables demonstrated in turbulent flows.

New algorithm for structure functions on unstructured meshes.

Numerical method effective for high Reynolds number flows.

Abstract

Statistical solutions, which are time-parameterized probability measures on spaces of square-integrable functions, have been established as a suitable framework for global solutions of incompressible Navier-Stokes equations (NSE). We compute numerical approximations of statistical solutions of NSE on two-dimensional domains with non-periodic boundary conditions and empirically investigate the convergence of these approximations and their observables. For the numerical solver, we use Monte Carlo sampling with an H(div)-FEM based deterministic solver. Our numerical experiments for high Reynolds number turbulent flows demonstrate that the statistics and observables of the approximations converge. We also develop a novel algorithm to compute structure functions on unstructured meshes.