High order numerical methods for the Navier-Stokes equations

A second order numerical method based on central difference approximations for derivatives is developed to solve the time dependent Navier Stokes equations written in velocity-pressure form. The method is first tested on a bench mark problem known as the square cavity flow problem. With the success of the method on the square cavity flow problem it is applied to the problem of flow of a viscous homogeneous fluid in the gap between a rotating circular cylinder inside a fixed square cylinder with the necessary modifications. This method is used to determine the height and shape of the free surface of the fluid when the gap between two cylinders is not completely filled with the fluid.;Because of slow convergence of this method for solving time dependant Navier Stokes equations towards the solution need for a faster method was apparent. Thereby, we were motivated to develop a second order method based on central difference approximations for derivatives to solve the steady state Navier Stokes equations. This second order method converges towards the solution very quickly compared to the method for solving time dependant Navier Stokes equations and produced accurate results. We extend this method to a fourth order method in order to increase the accuracy of the solution. This fourth order method is tested on the square cavity flow problem and later applied to the problem of flow of a viscous homogeneous fluid in the gap between a rotating circular cylinder inside a fixed square cylinder. This method is also used to determine the height and shape of the free surface of the fluid when the gap between two cylinders is not completely filled with the fluid.