Large Eddy Simulation for Turbulent Flows

This thesis is concerned with one of the most promising approaches to the numerical simulation of turbulent flows, the Large Eddy Simulation (LES) in which the large scales of the flow are calculated directly, while the interactions with the small scales are modeled. Specifically, we analyze two new LES models, introduced in [30] and [51]. First, we sketch the derivation of the LES model introduced by Galdi and Layton in [30], pointing out its main improvement over traditional LES models: the modeling of the interactions between small and large scales uses a closure approximation in the Fourier space which attenuates the high wave number components (or equivalently the small scales) of the flow , instead of increasing them. We then present the mathematical analysis of this new model, proving existence, uniqueness and stability of weak solutions. Second, we present three new models for the turbulent fluctuations modeling the interactions between the small scales in the flow. These models, introduced in [51], are based on approximations for the distribution of kinetic energy in the small scales in terms of the mean flow. We also prove existence of weak solutions for one of these models. We then show how this model can be implemented in finite element procedures and prove that its action is no larger than that of the popular Smagorinsky subgrid-scale model. Third, we consider " numerical-errors " in LES. Specifically, for one filtered flow model, we show convergence of the semidiscrete finite element approximation of the model and give an estimate of the error. Finally, we provide a careful numerical assessment and comparison of a classical LES model and the Galdi-Layton model. We are focusing herein on global, quantitative properties of the above models visà vis those of the mean flow variables. Direct numerical simulation (DNS), the classical LES model, two variants of the iii Galdi-Layton LES model and the Smagorinsky model are compared using the two– dimensional driven cavity problem for Reynolds numbers 400 and 10000. iv To my family. v Acknowledgements I thank my advisor, Professor William J. Layton, for everything he has done for me. He spent a lot of time trying to teach me the art of numerical analysis, and he generously let me be part of the " LES adventure ". I also thank Professor Giovanni Center for his kind and generous help in preparing the movie associated with this thesis. Most of Chapter …