  Meshless solution to differential equations using radial basis functions
(RBF) is an alternative to grid based methods commonly used. Since the meshless
method 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 additional shape parameters that have 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 the momentum and continuity equations. Systematic
convergence tests have been performed for five 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 method
is further applied to solve problems such as lid-driven cavity and vortex
shedding over circular cylinder. We have also analyzed the performance of this
approach for solution of Euler equations. The proposed method shows promise to
solve fluid flow and heat transfer problems in complex domains with high
accuracy.
