Nitsche method for Navier-Stokes equations with slip boundary conditions: Convergence analysis and VMS-LES stabilization

TL;DR

This paper develops and analyzes a Nitsche's method for stationary Navier-Stokes equations with slip boundary conditions, providing convergence results and a VMS-LES stabilization for high Reynolds number flows.

Contribution

It introduces a robust Nitsche's method for slip boundary conditions in complex domains, with proven convergence and a new VMS-LES stabilization technique for turbulent flows.

Findings

Optimal convergence rates achieved in numerical tests

Stable and accurate simulation of high Reynolds number flows

Validated theoretical results with benchmark problems

Abstract

In this paper, we analyze the Nitsche's method for the stationary Navier-Stokes equations on Lipschitz domains under minimal regularity assumptions. Our analysis provides a robust formulation for implementing slip (i.e. Navier) boundary conditions in arbitrarily complex boundaries. The well-posedness of the discrete problem is established using the Banach Ne\v{c}as Babu\v{s}ka and the Banach fixed point theorems under standard small data assumptions, and we also provide optimal convergence rates for the approximation error. Furthermore, we propose a VMS-LES stabilized formulation, which allows the simulation of incompressible fluids at high Reynolds numbers. We validate our theory through numerous numerical tests in well established benchmark problems.