| In this thesis, we study a number of issues related to the stability of fluid flow. In the first part of the thesis, the iterative grid redistribution method (IGR) is used to solve the Prandtl equation whose solution may develop singularity in some cases. The IGR method enables us to study the singular structures of the blow up solutions. The behavior of the solution near the singularity is shown to be self-similar in the inviscid case and near self-similar in the viscous case.;In the second part of the thesis, we study numerically the solution to the incompressible Navier-Stokes equation with Navier boundary condition. In particular, we are interested in the effect of replacing the no-slip boundary condition by the Navier slip boundary condition on the stability of the solution. We employ an unconditionally stable time discretization which is implicit in viscosity and explicit in both pressure and convection terms and finite difference discretization with local pressure boundary condition. In the case of lid-driven cavity flow, our results show Navier slip boundary condition removes the corner singularity and also increases the critical Reynolds number for Hopf bifurcation. The behavior of the solution near the corner is also studied analytically. We also study the linearized stability of the shear flow solutions of N-S Equations. The Navier boundary condition at the wall again is found to increases the critical Reynolds number for stability, in contrast to some of the results in the literature.;In the last part of the thesis, we introduce a two level preconditioned conjugate gradient method to solve the elliptic type system, which is discretized on the distorted and structured grid that is generated by the IGR method. This two level method is based on multigrid method and use quadratic smoothing and linear correction. We show that the new approach greatly decreases the condition number of the stiff matrix and improves the speed of convergence. |
