Date of Award

Summer 2016

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Computational Analysis and Modeling

First Advisor

Dong Liu

Abstract

In this dissertation, two numerical methods with high order accuracy, Spectral Element Method (SEM) and Discontinuous Galerkin Finite Element Method (DG-FEM), are chosen to solve problems in Computational Fluid Dynamics (CFD). The merits of these two methods will be discussed and utilized in different kinds of CFD problems. The simulations of the micro-flow systems with complex geometries and physical applications will be presented by SEM. Moreover, the numerical solutions for the Hyperbolic Flow will be obtained by DG-FEM. By solving problems with these two methods, the differences between them will be discussed as well.

Compressible Navier-Stokes equations with Electro-osmosis body force and slip boundary conditions are solved to simulate two independent models. The third order Adams-Bashforth method on time splitting, and up to the eighth order SEM on space analysis are utilized in our cases of the electro-osmosis flow (EOF). To solve the body force caused by EOF, simplification of the Poisson-Boltzmann is discussed in details. Results show that SEM can clearly simulate the electric double layers in EOF. Compared with the finite element method, which uses h-refinement to increase resolution, SEM has obvious advantages by using hp-refinement.

The other case for SEM is the simulations of drug delivery through the micro needle. The drug flowing inside the needle is treated as a micro-flow system with complex geometry, while the process of drug fluxing in human skin is developed as in the case of CFD problem in porous media. Incompressible Navier-Stokes equations and Darcy-Brinkman equations are solved to simulate the drug flowing inside the needle and diffusing in human skin, respectively. Results are compared with COMSOL simulation, experimental data, and numerical solutions from Smoothed Particle Hydrodynamics (SPH). The high order DG-FEM method is chosen to do research on Hyperbolic Flow.

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