Lukas Lundgren: High-order finite element methods for incompressible variable density flow

Abstract:

Subsonic flow with density variations can be described by the variable density incompressible Navier–Stokes equations. In my talk, I will describe our recent work developing novel Taylor–Hood-based finite element methods for this problem. Incompressible flow is characterized by the velocity field satisfying the divergence-free condition. However, numerically satisfying this condition is one of the main challenges in simulating such flows. In practice, this condition is rarely satisfied exactly, which can result in stability and conservation issues in computations. We develop a new practical and useful formulation for variable density flow. This formulation allows Galerkin methods to conserve mass, squared density, kinetic energy, momentum and angular momentum without the divergence-free constraint being strongly enforced, leading to significantly improved accuracy and robustness. In addition to favorable conservation properties, the formulation is shown to make the density field invariant to global shifts.

Another primary difficulty in simulating fluid flows arises from the challenge of accurately representing underresolved flows, where the mesh resolution cannot capture the gradient of the true solution. This leads to stability issues unless appropriate stabilization techniques are used. We develop new high-order accurate artificial viscosity techniques to deal with this issue. Furthermore, we thoroughly investigate the properties of viscous regularizations, ensuring that kinetic energy stability is guaranteed when using artificial viscosity.