| Work in chapters two and three is motivated by some earlier calculations of under-resolved turbulent flows. In these calculations at points on the boundary where tangential stresses concentrate, non-physical eddies were generated. In order to eliminate spurious vortices obtained in other studies we relax the strongly imposed no-slip condition by penalty method. When Re number is large the strong no-slip condition is physically sensitive and it should be replaced by Navier's slip law. Penalty imposition of the no-slip condition is equivalent to the Navier slip law.; Imposing the no-penetration condition correctly is a hard problem since it is linked with physically important mechanism of surface roughness. In the first problem to capture surface roughness when the computational boundary is a regularization of a rough boundary we impose no-penetration condition via penalty method. In the second problem we penelized both no-slip and no-penetration. We verify the theoretical results for the second problem by two numerical examples. Weak imposition of boundary condition is not a new idea but it has never progressed beyond the Stokes problem in [17]. Since momentum effects dominate the motivating applications, it was important to extend the validity of the method to NSE (nonlinear) problem. We have accomplished in the problems we improved the rate of convergence significantly.; In the third problem we consider the Navier-Stokes-Alpha model as an approximation of turbulent flows under realistic, non-periodic, boundary conditions. We derive the variational formulation of Navier-Stokes-Alpha model under non-periodic boundary conditions, and prove that it has a unique weak solution. Next we consider finite element approximation of the model. We give semi discretization of the model and prove convergence of the method. |
