The solution of incompressible Navier Stokes equations using the sine collocation method

Abstract

Different kind of numerical approaches have been used in the past to solve the complete set of Navier Stokes equations. The traditional methods that have been used in the past are the finite-difference method, finite-element method and the boundary element method. Multigrid methods have been used recently for solving these complete set of Navier Stokes equations and they help in obtaining a faster rate of convergence of the residual for the solution of these equations. Some of the problems that are faced in the world of numerical methods today are the capacity to handle singularities that occur within or at the boundaries of a computational domain and also the capacity to handle semi-infinite and infinite domains. Sine numerical method has the advantage of handling singularities and semi-infinite domains very effectively. It also provides an exponential convergence rate. This study involves a first step in applying the sine numerical method to the flow within a driven cavity, which requires the solution of the complete two-dimensional Navier Stokes equations. The sine collocation method was applied to the driven cavity problem. The Navier Stokes equations were solved by means of two dimensional sine collocation using the primitive variables method. Simulations were also carried out with the finite-difference method for the same problem and the results were matched with the sine collocation method. Simulations were also carried out by using the commercial CFD code FLUENT. It was seen that the profiles compared well between the different methods.