Solution of the Stationary Three-dimensional Navier-Stokes Equations by Using Radial Basis Functions and Reducing the Number of Governing Equations

Abstract

The stationary three-dimensional Navier-Stokes equations consisting of four equations and four unknowns have been solved by conventional methods using the primitive-variable approach. By getting rid of pressure and one of the velocity components, the Navier-Stokes equations can be reduced to two higher-order partial differential equations with the remaining two velocity components as the only two unknowns. In this paper, a meshless method based on a radial basis function, known as multiquadrics, is proposed to solving such equations. Unknown velocity components are approximated as linear combinations of multiquadrics centered at domain nodes and boundary nodes. Unknown coefficients are solved by a Picard iterative scheme. The proposed method is used to solve a test problem, for which exact solution is known. It is found that the number of iterations required for a converged solution and the accuracy of the solution depend on the shape parameter of multiquadrics. A small value of the shape parameter results in a low number of required iterations, but the resulting solution may not be accurate. On the other hand, a large value of the shape parameter can yield a very accurate solution provided that a converged solution is obtained. There appears to be an upper limit to the value of the shape parameter for which the proposed method is capable of yielding a converged solution.