Numerical Analysis of a Variational Multiscale Method for Turbulence
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 , , .
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. . 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.
Advisor:William J. Layton; Beatrice Riviere; Ivan Yotov; Vivette Girault
School:University of Pittsburgh
School Location:USA - Pennsylvania
Source Type:Master's Thesis
Date of Publication:02/04/2005