Solutions of Two-dimensional Navier-Stokes Equations by a Collocation Method Based on Radial Basis Functions

Abstract

The two-dimensional Navier-Stokes equations consisting of three equations and three unknownshave been solved by conventional methods using the primitive-variable approach and the vorticity-streamfunction approach. A new approach is proposed in this paper. By getting rid of pressure and one of thevelocity components, the Navier-Stokes equations can be reduced to a third-order partial differentialequation with one velocity component as the only unknown. A collocation method based on radial basisfunctions is proposed for solving this equation. Unknown velocity component is approximated as a linearcombination of basis functions. Unknown coefficients are determined by an iterative scheme. Theproposed method is used to solve a test problem, for which exact solution is known. It is found that thenumber of iterations required for a convergence and the accuracy of the solution depends on the freeparameter of basis functions.