Date of Award

8-2014

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Legacy Department

Mathematical Science

Advisor

Dr. Leo Rebholz

Committee Member

Dr. Chris Cox

Committee Member

Dr. Lea Jenkins

Committee Member

Dr. Hyesuk Lee

Abstract

This thesis studies regularization models as a way to approximate a flow simulation at a lower computational cost. The Leray model is more easily computed than the Navier-Stokes equations (NSE), and it is more computationally attractive than the NS-α regularization because it admits a natural linearization which decouples the mass/momentum system and the filter system, allowing for efficient and stable computations. A major disadvantage of the Leray model lies in its inaccuracy. Thus, we study herein several methods to improve the accuracy of the model, while still retaining many of its attractive properties. This thesis is arranged as follows. Chapter 2 gives notation and preliminary results to be used in subsequent chapters. Chapter 3 investigates a nonlinear filtering scheme using the Vreman and Q-criteria based indicator functions. We define these indicator functions, prove stability and state convergence of the scheme to the NSE, and provide several numerical experiments which demonstrate its effectiveness over NSE and Leray calculations on coarse meshes. Chapter 4 investigates a deconvolution-based indicator function. We prove stability and convergence of the resulting scheme, verify the predicted convergence rates, and provide numerical experiments which demonstrate this scheme's effectiveness. Chapter 5 then extends this scheme to the magnetohydrodynamic equations. We prove stability and convergence of our algorithm, and verify the predicted convergence rates. Chapter 6 provides a study of the Leray-α model. We prove stability and convergence for the fully nonlinear scheme, prove conditional stability for a linearized and decoupled scheme, and provide a numerical experiment which compares our scheme with the usual Leray-α model. Specifically, we show that choosing β < α does indeed improve accuracy in computations. Chapter 7 investigates the Leray model with fine mesh filtering. We prove stability and convergence of the algorithm, then verify the increased convergence rate associated with the finer mesh, as predicted by the analysis. Finally, we present a benchmark problem which demonstrates the effectiveness of filtering on a finer mesh.

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