A High-Order Accurate Meshless Method for Solution of Incompressible Fluid Flow Problems
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Meshless solution to differential equations using radial basis functions (RBF) is an alternative to grid based methods commonly used. Since it does not need an underlying connectivity in the form of control volumes or elements, issues such as grid skewness that adversely impact accuracy are eliminated. Gaussian, Multiquadrics and inverse Multiquadrics are some of the most popular RBFs used for the solutions of fluid flow and heat transfer problems. But they have an additional shape parameter that has to be fine tuned for accuracy and stability. Moreover, they also face stagnation error when the point density is increased for accuracy. Recently, Polyharmonic splines (PHS) with appended polynomials have been shown to solve the above issues and give rapid convergence of discretization errors with the degree of appended polynomials. In this research, we extend the PHS-RBF method for the solution of incompressible Navier-Stokes equations. A fractional step method with explicit convection and explicit diffusion terms is combined with a pressure Poisson equation to satisfy momentum and continuity equations. Systematic convergence tests have been performed for four model problems with two of them having analytical solutions. We demonstrate fast convergence both with refinement of number of points and degree of appended polynomials. The proposed method shows promise to solve fluid flow and heat transfer problems in complex domains with high accuracy.
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