All Dissertations

8-2012

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematical Science

Rebholz, Leo G

Ervin , Vincent

Committee Member

Jenkins , Eleanor

Committee Member

Yoon , Jeong-Rock

Abstract

This thesis studies novel physics-based methods for
simulating incompressible fluid flow described by the Navier-Stokes equations (NSE) and
magnetohydrodynamics equations (MHD).
It is widely accepted in computational fluid dynamics (CFD) that numerical schemes which are more
physically accurate lead to more precise flow simulations especially over long time intervals.
A prevalent theme throughout will be the inclusion of as much
physical fidelity in numerical solutions as efficiently possible. In algorithm design, model
selection/development, and element choice, subtle changes can provide better physical accuracy,
which in turn provides better overall accuracy (in any measure). To this end we develop and study
more physically accurate methods for approximating the NSE, MHD, and related systems.
Chapter 3 studies extensions of the energy and helicity preserving scheme for the 3D NSE
developed in \cite{Reb07b}, to a more general class of problems. The scheme
is studied together with stabilizations of grad-div type in order to mitigate the effect of the Bernoulli pressure
error on the velocity error. We prove stability, convergence, discuss conservation properties,
and present numerical experiments that demonstrate the advantages of the scheme.
In Chapter 4, we study a finite element scheme for the 3D NSE
that globally conserves energy and helicity and, through the use of Scott-Vogelius elements,
enforces pointwise the solenoidal constraints for velocity and vorticity. A complete numerical analysis is given,
including proofs for conservation laws, unconditional stability and optimal convergence.
We also show the method can be efficiently computed by exploiting a connection between this method, its
associated penalty method, and the method arising from using grad-div stabilized Taylor-Hood elements.
Finally, we give numerical examples which verify the theory and demonstrate the effectiveness of the scheme.
In Chapter 5, we extend the work done in \cite{CELR10} that proved, under mild restrictions,
grad-div stabilized Taylor-Hood solutions of Navier-Stokes problems
converge to the Scott-Vogelius solution of that same problem. In \cite{CELR10}
even though the analytical convergence rate was only shown to be $\gamma^{-\frac{1}{2}}$
(where $\gamma$ is the stabilization parameter),
the computational results suggest the rate may be improvable
$\gamma^{-1}$. We prove herein the analytical rate is indeed $\gamma^{-1}$,
and extend the result to other incompressible flow problems including Leray-$\alpha$ and MHD. Numerical
results are given that verify the theory.
Chapter 6 studies an efficient finite element method for the NS-$\omega$ model, that uses van Cittert approximate
deconvolution to improve accuracy and Scott-Vogelius elements to provide pointwise mass conservative solutions and remove
the dependence of the (often large) Bernoulli pressure error on the velocity error. We provide a complete numerical analysis of
the method, including well-posedness, unconditional stability, and optimal convergence. Several numerical experiments are
given that demonstrate the performance of the scheme, and how the use of Scott-Vogelius elements can
dramatically improve solutions.
Chapter 7 extends Leray-$\alpha$-deconvolution modeling to the incompressible MHD.
The resulting model is shown to be well-posed, and have attractive limiting behavior
both in its filtering radius and order of deconvolution.
Additionally, we present and study a numerical scheme for the model, based on an extrapolated
Crank-Nicolson finite element method. We show the numerical scheme is unconditionally stable,
preserves energy and cross-helicity, and optimally converges to the MHD solution.
Numerical experiments are provided that verify convergence rates, and test the scheme on
benchmark problems of channel flow over a step and the Orszag-Tang vortex problem.

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