A pressure correction approach coupled with the MLPG method for the solution of the navier-stokes equations

Abstract

We present a pressure-velocity correction approach for the solution of the Navier-Stokes equations. The Meshless Local Petrov Galerkin (MLPG) method is used to solve the two dimensional, incompressible and steady state viscous fluid flow equations. The weak form of these equations, which are formulated in the Cartesian coordinate system, are integrated in a local standard domain by the Gauss- Lobatto-Legendre quadrature rule. The Moving Least Square (MLS) scheme is used to generate the interpolation shape functions. The pressure-velocity correction approach (segregated solution procedure) follows an iterative process, in which the momentum equations are solved sequentially to obtain the velocities v ** 1 and v ** 2 from initial guessed values for the velocity (v *1 and v *2 ) and pressure (p *) fields. Using the corrected velocities vi = v ** i + v' i and pressure p = p * + p' in the weak form of the continuity and momentum equations, we generate a system of three equations with three unknown variables (a fully implicit method): the velocity corrections (v' 1 and v' 2) and the pressure correction (p'). Using the correction values the pressure is updated and the velocities are corrected to satisfy the continuity equation. The updated values are taken as the new guessed values, and the iterative process continues until convergence. We apply the method for the solution of four (low Rayleigh number and low Reynolds number) fluid flow problems. We conclude that the MLPG method coupled with an implicit procedure to calculate the corrections of pressure and velocities can be used as a reliable methodology for the solution of the Navier-Stokes equations.