In this thesis, we investigate the stochastic three dimensional Navier-Stokes-∝ model and the stochastic three dimensional Leray-∝ model which arise in the modelling of turbulent flows of fluids. We prove the existence of probabilistic weak solutions for the stochastic three dimensional Navier-Stokes-∝ model. Our model contains nonlinear forcing terms which do not satisfy the Lipschitz conditions. We also discuss the uniqueness. The proof of the existence combines the Galerkin approximation and the compactness method. We also study the asymptotic behavior of weak solutions to the stochastic three dimensional Navier-Stokes-∝ model as ∝ approaches zero in the case of periodic box. Our result provides a new construction of the weak solutions for the stochastic three dimensional Navier-Stokes equations as approximations by sequences of solutions of the stochastic three dimensional Navier-Stokes-∝ model. Finally, we prove the existence and uniqueness of strong solution to the stochastic three dimensional Leray-∝ equations under appropriate conditions on the data. This is achieved by means of the Galerkin approximation combines with the weak convergence methods. We also study the asymptotic behavior of the strong solution as alpha goes to zero. We show that a sequence of strong solution converges in appropriate topologies to weak solutions of the stochastic three dimensional Navier-Stokes equations. All the results in this thesis are new and extend works done by several leading experts in both deterministic and stochastic models of fluid dynamics. / Thesis (PhD)--University of Pretoria, 2010. / Mathematics and Applied Mathematics / unrestricted

In this thesis, I examine filtering based stabilization methods to design new regularized reduced order models (ROMs) for under-resolved simulations of unsteady, nonlinear, convection-dominated systems. The new ROMs proposed are variable delta filtering applied to the evolve-filter-relax ROM (V-EFR ROM), variable delta filtering applied to the Leray ROM, and approximate deconvolution Leray ROM (ADL-ROM). They are tested in the numerical setting of Burgers equation, a nonlinear, time dependent problem with one spatial dimension. Regularization is considered for the low viscosity, convection dominated setting. / Master of Science / Numerical solutions of partial differential equations may not be able to be efficiently computed in a way that fully captures the true behavior of the underlying model or differential equation, especially if significant changes in the solution to the differential equation occur over a very small spatial area. In this case, non-physical numerical artifacts may appear in the computed solution. We discuss methods of treating these calculations with a goal of improving the fidelity of numerical solutions with respect to the original model.