Analytical and numerical study of certain sub-grid scale alpha-models of turbulence

The global regularity of the three-dimensional Navier-Stokes equations and Euler equations is a major challenging problem in applied analysis. This thesis is concerned with the analytical and computational study of a class of three-dimensional analytic sub-grid scale models of turbulence, known as alpha-models of turbulence. It consists of three parts:; In the first part, the thesis proposes a new analytical sub-grid scale turbulence alpha-model called the modified-Leray-alpha model. The explicit steady state solution to the modified-Leray-alpha in channels and pipes matches the empirical data in infinite channels and pipes for a wide range of large Reynolds numbers. This remarkable result is shared by a family of turbulence models known as alpha-models, which include the Navier-Stokes-alpha model, the Clark-alpha model, the Leray-alpha model, and the simplified Bardina model. The global well-posedness of the modified-Leray-alpha model is established in this study along with the existence of a global attractor with finite Hausdorff and fractal dimensions. Our explicit estimates for the dimension of the global attractor, in terms of the relevant physical parameters, are compatible with physical heuristic arguments for the number of degrees of freedom in turbulent flows.; In the second part, global well-posedness of another sub-grid scale alpha-model, the simplified Bardina model, is established. Inspired by the work of Bardina on sub-grid scale models of turbulence, Layton and Lewandowski introduced the simplified Bardina model and proved its global well-posedness. In this thesis the global well-posedness of the viscous three-dimensional simplified Bardina model is being established for a weaker class of initial conditions than those studied by Layton and Lewandowski. Global existence and uniqueness of solutions of various alpha sub-grid scale turbulence models has been established when viscosity is nonzero. However, a remarkable result for the simplified Bardina model is the global existence and uniqueness of solutions to the inviscid model. No such result is known for the other three-dimensional inviscid alpha-models.; The global existence and uniqueness results established for the inviscid simplified Bardina model has an important consequence in computational fluid dynamics when this inviscid model is used as an approximation/regularization to the three-dimensional Euler equations. The inviscid simplified Bardina model is a globally well-posed model which regularizes the Euler equations without adding any hyperviscous terms. Note that the addition of hyperviscous terms in the numerical simulation can produce unrealistic results because it runs the risk of damping too much energy in the smaller scales.; In the third part, a computational study of the two-dimensional Navier-Stokes-alpha model is presented. The main focus of the computational study was to measure the scaling of the energy spectra in the wave number regime consisting of scales smaller than the filter width alpha. This spectral scaling determines the characteristic time scale of eddies smaller than the filter width alpha. In addition, it is also shown, analytically, that the two-dimensional Navier-Stokes-alpha model have similar transfer and cascade behavior as the two-dimensional turbulence, i.e. two-dimensional Navier-Stokes equations.