| The present work is concerned with the development of numerical methods for the computation of viscous incompressible flows as described by the Navier-Stokes equations of classical fluid mechanics. The emphasis is strictly on finite difference schemes implemented on non-staggered Cartesian grids, which when coupled with a high order explicit time stepping procedure result in very simple and efficient methods.;In chapter 2 a second order method based on the primitive variable formulation of the Navier-Stokes equations is presented. The scheme is suited for the computation of both low and high Reynolds number flows. The novelty of the scheme lies primarily in a simple, consistent, and accurate numerical approximation of the Neumann boundary condition for the pressure Poisson equation. Its use avoids the need for both fractional-step time discretizations and staggered grids traditionally required of the most popular numerical methods based on the primitive variable formulation. The resulting method achieves clean second order accuracy when applied to 1D and 2D test problems, and performs equally well as a second order vorticity-stream function based scheme when used to compute the canonical cavity flow. We note that the scheme can be easily extended to compute 3D flows.;In chapter 3 new developments are presented for a class of fourth order Essentially Compact methods (EC4), originally developed by E and Liu, for solving unsteady viscous incompressible flows in the vorticity-stream function formulation. A novel fine grid patch mesh refinement technique, which is easily incorporated into the original EC4 scheme, is outlined. Its use results in a dramatic increase in computational efficiency, particularly for high Reynolds number flows. In addition, we present results on the use of a very effective far-field boundary condition for the stream function. As a detailed illustration of an application of the above mentioned methodologies, we present high resolution benchmark quality simulations of the impulsively started flow past a circular cylinder at Reynolds numbers ranging from 1,000 to 100,000. Even at the considerably high Reynolds number of 100,000 the flow is completely resolved. |
