# H(div)-conforming Discontinuous Galerkin Methods for Multiphase Flow

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## Date

2021-05-05

## Authors

Booker, Kyle

## Advisor

Rhebergen, Sander

Abukhdeir, Nasser

Abukhdeir, Nasser

## Journal Title

## Journal ISSN

## Volume Title

## Publisher

University of Waterloo

## Abstract

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 the 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 Raviart--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.
This thesis presents a discontinuous Galerkin H(div)-conforming method for the two-fluid model. Instead of solving the dispersed and continuous phase velocities, the dispersed and mixture velocities are solved, allowing us to easily apply our pressure-robust scheme to the divergence constraint of the two-fluid model. The viscous term numerical flux is derived from a standard interior penalty discontinuous Galerkin method flux, and the convective flux is calculated using the local Lax--Friedrichs flux.
Simulations of two-dimensional channel flow are performed using the H(div)-conforming method. While we can qualitatively assess the approximate velocity, pressure, and phase fraction solutions, there still needs to be work done to use this method for actual applications. The mixture velocity is calculated to be divergence-free within machine-precision. Limitations of the numerical scheme are discussed, and possible areas for further research.