High-order finite element methods for incompressible variable density flow

Abstract: The simulation of fluid flow is a challenging and important problem in science and engineering. This thesis primarily focuses on developing finite element methods for simulating subsonic two-phase flows with varying densities, described by the variable density incompressible Navier-Stokes equations. These equations are commonly used to model a wide range of phenomena, including aerodynamic forces around vehicles, climate and weather prediction, combustion and the spread of pollution.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. Moreover, enforcing the divergence-free condition is a primary computational bottleneck for incompressible flow solvers. To improve computational efficiency, we explore and develop artificial compressibility techniques, which regularize this constraint. Additionally, we develop a new practical and useful formulation for variable density flow. This formulation allows Galerkin methods to enhance conservation properties when the divergence-free condition is not strongly enforced, leading to significantly improved accuracy and robustness.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. In this thesis, 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.