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Kyler Booker | Applied Mathematics, University of Waterloo
H(div)-conforming Discontinuous Galerkin Methods for Multiphase Flow
Computational fluid dynamics (CFD) is concerned with numerically solving and visualizing complex problems involving fluids with numerous engineering applications. Mathematical models are derived from basic governing equations using assumptions of the initial conditions and physical properties. CFD is less costly than experimental procedures while still providing an accurate depiction of the phenomenon. Models permit to test different parameters and sensitivity quickly, which is highly adaptable to solving similar conditions; however, these problems are often computationally costly, which necessitates sophisticated numerical methods.
Modeling multiphase flow problems involving two or more fluids of different states, phases, or physical properties. Boilers are an example of bubbly flows where accurate models are relevant for operation safety or contain turbulence, resulting in reduced efficiency. Bubbly flows are an example of continuous-dispersed phase flow, modeled using the Eulerian multiphase flow model. The dispersed phase is considered an interpenetrating continuum with the continuous phase.
In the two-fluid model, a phase fraction parameter varying from zero to one is used to describe the fraction of fluid occupying each point in space. This model is ill-posed, non-linear, non-conservative, and non-hyperbolic, which affects the stability and accuracy of the solution. There have been methods allowing the model to be well-posed to obtain stability and uniqueness, but this raises questions regarding the physicality of the solution. Approaches to increasing the well-posedness of the model include additional momentum transfer terms, virtual mass contributions, dispersion terms, or inclusion of momentum flux. There is division among which methods are valid for an accurate description of the phenomena, and more research is required to examine these effects.
While finite difference schemes are often simple to implement, they do not scale well to problems with complicated geometries or difficult boundary conditions. Numerical methods may also add ad-hoc terms that compromise the physicality of the solution. The choice of numerical method results from a time versus accuracy trade-off. In industry, efficient performing schemes have become standard; however, this might sacrifice physical properties of the natural phenomenon.
$H$(div)-conforming finite element spaces contain vector functions where both the function and its divergence are continuous on each element. Examples of $H$(div)-conforming spaces include Raviert Thomas and Brezzi-Marini-Douglas spaces. These spaces allow for the velocity vector function to be pointwise divergence-free with machine precision and being pressure-robust.
In this thesis, a two-fluid model is presented, which depends on the mixture velocity and the velocity of the dispersed phase. An $H$(div)-conforming finite element space is used to create a discontinuous Galerkin weak formulation where the mixture velocity is pointwise divergence-free. Simulations of two-dimensional channel flow validate the $H$(div)-conforming method.