Finite element formulation of general boundary conditions for incompressible flows
This work provides a unified framework for boundary conditions in incompressible flow simulations, benefiting computational fluid dynamics researchers.
The paper develops a finite element formulation for general boundary conditions in incompressible flows using Nitsche's method, handling both viscous and inviscid limits. Numerical experiments on standard benchmarks demonstrate the method's effectiveness.
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.