A high–order discontinuous Galerkin multiphase flow solver for industrial applications

Abstract

This thesis investigates models and simulation techniques to solve multiphase flows. Applications for multiphase flows encompasses chemical and process technology, power generation, nuclear reactor safety, or oil and gas. We concentrate on developing models that allow for varying flow morphologies (e.g. both a polydisperse bubbly flow and a fully separated flow) as those present in crude oil transport in pipes. Additionally, we want to retain most chemical phenomena as the phase separation of immiscible fluids, i.e., the coalescence and growth of stable separated phases from an initially mixed state, as well as other fundamental properties such as phases (mass) conservation.We model multiphase phenomena using phase field methods. Phase field (or diffuse interface) methods, as opposed to the sharp interface methods, consider a finite, yet thin, thickness of the interface that separates the two fluids. Therefore, the thermodynamic properties (e.g. the density) vary smoothly across the interface. From the available phase field methods, we solve the Cahn-Hilliard equation, which naturally retains the phase separation phenomena from the minimization of a free-energy, and additionally is phase-conserving.The Cahn-Hilliard equation is coupled with the incompressible Navier-Stokes equations. This is a two-way coupling, since the Cahn-Hilliard governs the density of the Navier-Stokes flow, and also introduces capillary forces at the interfaces. Then, the Navier-Stokes equations modify the Cahn-Hilliard equation to transport the phases with its velocity field. To obtain an efficient model, we relax the incompressibility constraint with an artificial compressibility model, that hyperbolizes the system providing an evolution equation for the pressure. In this thesis, we construct the approximation of two multiphase models: an provably stable two-phase NS/CH system, and a (non provably stable) three-phase NS/CH system.We approximate the models with the high-order Discontinuous Galerkin Spectral Element Method (DGSEM). The DGSEM is chosen since the schemes are arbitrarily high-order accurate, they handle unstructured curvilinear meshes, and they have been widely used to solve hyperbolic/parabolic systems. From the different DGSEM variants, we use the DGSEM with Gauss-Lobatto points, since it allows us to construct schemes that are provably stable. The parabolic systems studied in this thesis involve non-linear terms that can easily make the scheme unstable. We include a thourough vii viii study of the errors incurred to construct schemes that are provably stable.To study the stability of the proposed schemes, we first study two simplified cases: provably stable approximations of the incompressible Navier-Stokes equations with variable density and artificial compressibility, and the standalone version (i.e. without fluid motion) of the Cahn-Hilliard equations. Then, we combine the findings to construct a provably stable approximation of the two-phases incompressible Navier-Stokes/Cahn-Hilliard system. Finally, we also construct...