Type of Document	Dissertation
Author	Kaya, Songul
Author's Email Address	sokst20@pitt.edu
URN	etd-12072004-184313
Title	Numerical Analysis of a Variational Multiscale Method for Turbulence
Degree	Doctor of Philosophy
Program	Mathematics
School	School of Arts and Sciences
Advisory Committee	Advisor Name Title William J. Layton Committee Chair Beatrice Riviere Committee Co-Chair Ivan Yotov Committee Member Vivette Girault Committee Member
Keywords	Discontinuous Galerkin Method Finite element method Variational multiscale method Time dependent Navier Stokes Subgrid eddy viscosity Error analysis
Date of Defense	2004-10-21
Availability	unrestricted
Abstract This thesis is concerned with one of the most promising approaches to the numerical simulation of turbulent flows, the subgrid eddy viscosity models. We analyze both continuous and discontinuous finite element approximation of the new subgrid eddy viscosity model introduced in [43], [45], [44]. First, we present a new subgrid eddy viscosity model introduced in a variationally consistent manner and acting only on the small scales of the fluid flow. We give complete convergence of the method. We show convergence of the semi-discrete finite element approximation of the model and give error estimates of the velocity and pressure. In order to establish robustness of the method with respect to Reynolds number, we consider the Oseen problem. We present the error is uniformly bounded with respect to the Reynolds number. Second, we establish the connection of the new eddy viscosity model with another stabilization technique, called Variational Multiscale Method (VMM) of Hughes et.al. [35]. We then show the advantages of the method over VMM. The new approach defines mean by elliptic projection and this definition leads to nonzero fluctuations across element interfaces. Third, we provide a careful numerical assessment of a new VMM. We present how this model can be implemented in finite element procedures. We focus on herein error estimates of the model and comparison to classical approaches. We then establish that the numerical experiments support the theoretical expectations. Finally, we present a discontinuous finite element approximation of subgrid eddy viscosity model. We derive semi-discrete and fully discrete error estimations for the velocity.
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