Meshless Local Petrov-Galerkin (MLPG) Method for Incompressible Viscous Fluid Flows

In this paper, the truly Meshless Local Petrov-Galerkin (MLPG) method is extended for computation of unsteady incompressible flows, governed by the Navier–Stokes equations (NSE), in vorticity-stream function formulation. The present method is a truly meshless method based only on a number of randomly located nodes. The formulation is based on two equations including stream function Poisson equation and vorticity advection-dispersion-reaction eq. (ADRE). The meshless method is based on a local weighted residual method with the Heaviside step function and quartic spline as the test functions respectively over a local subdomain. Moving Least Square approximation (MLS) is employed in shape function construction for approximation of a gauss point. Due to dissatisfaction of kronecker delta property in MLS approximation, the penalty method is employed to enforce the essential boundary conditions. In order to overcome instability and numerical errors encountering in convection dominant flows, a new upwinding scheme is used to stabilize the convection operator in the streamline direction (as is done in SUPG). In this upwinding technic, instead of moving subdomains the weight function is shifted in the direction of flow. The efficiency, accuracy and robustness are demonstrated by some test problems, including the standard driven cavity together with the driven cavity flow in an L shaped cavity and flow in a Z shaped channel. The comparison of computational results shows that the developed method is capable of accurate resolution of flow fields in complex geometries.