Finite element formulation of general boundary conditions for incompressible flows

TL;DR

This paper develops a finite element formulation for general boundary conditions in incompressible flows, applicable across viscosity regimes including Euler equations, by using Nitsche's method and spectral decomposition.

Contribution

It introduces a unified finite element approach for boundary conditions in incompressible flows that works for all viscosity levels, including the inviscid limit, using a novel spectral decomposition technique.

Findings

Validated with standard viscous flow benchmarks

Successfully applied to inviscid flow cases

Demonstrates stability and accuracy across viscosity regimes

Abstract

We study the finite element formulation of general boundary conditions for incompressible flow problems. Distinguishing between the contributions from the inviscid and viscid parts of the equations, we use Nitsche's method to develop a discrete weighted weak formulation valid for all values of the viscosity parameter, including the limit case of the Euler equations. In order to control the discrete kinetic energy, additional consistent terms are introduced. We treat the limit case as a (degenerate) system of hyperbolic equations, using a balanced spectral decomposition of the flux Jacobian matrix, in analogy with compressible flows. Then, following the theory of Friedrich's systems, the natural characteristic boundary condition is generalized to the considered physical boundary conditions. Several numerical experiments, including standard benchmarks for viscous flows as well as inviscid flows are presented.