On the convergence of a fully discrete scheme of LES type to physically relevant solutions of the incompressible Navier-Stokes

TL;DR

This paper proves that a fully discrete numerical scheme combining implicit Euler and Fourier-Galerkin methods converges to physically relevant weak solutions of the incompressible Navier-Stokes equations, aiding turbulence simulation validation.

Contribution

It establishes convergence of a fully discrete LES-type scheme to physically relevant solutions of the Navier-Stokes equations, bridging numerical approximation and physical realism.

Findings

Convergence of the scheme to weak solutions of Navier-Stokes.

The solutions satisfy the natural local entropy condition.

The method effectively approximates turbulent flow properties.

Abstract

Obtaining reliable numerical simulations of turbulent fluids is a challenging problem in computational fluid mechanics. The Large Eddy Simulations (LES) models are efficient tools to approximate turbulent fluids and an important step in the validation of these models is the ability to reproduce relevant properties of the flow. In this paper we consider a fully discrete approximation of the Navier-Stokes-Voigt model by an implicit Euler algorithm (with respect to the time variable) and a Fourier-Galerkin method (in the space variables). We prove the convergence to weak solutions of the incompressible Navier-Stokes equations satisfying the natural local entropy condition, hence selecting the so-called physically relevant solutions