Analysis of Navier-Stokes equations by a BC/GE embedded local meshless method

Abstract

A meshless numerical model is developed for the simulation of two-dimensional incompressible viscous flows. We directly deal with the pressure–velocity coupling system of the Navier–Stokes equations. With an efficient time marching scheme, the flow problem is separated into a series of time-independent boundary value problems (BVPs) in which we seek the pressure distribution at discretized time instants. Unlike in conventional works that need iterative time marching processes, numerical results of the present model are obtained straightforwardly. Iteration is implemented only when dealing with the linear simultaneous equations while solving the BVPs. The method for solving these BVPs is a strong form meshless method which employs the local polynomial collocation with the weighted-least-squares (WLS) approach. By embedding all the constraints into the local approximation, i.e. ensuring the satisfaction of governing equation at both the internal and boundary nodes and the satisfaction of the boundary conditions (BCs) at boundary points, this strong form method is more stable and robust than those just collocate one boundary condition at one boundary node. We innovatively use this concept to embed the satisfaction of the continuity equation into the local approximation of the velocity components. Consequently, their spatial derivatives can be accurately calculated. The nodal arrangement is quite flexible in this method. One can set the nodal resolution finer in areas where the flow pattern is complicated and coarser in other regions. Three benchmark problems are chosen to test the performance of the present novel model. Numerical results are well compared with data found in reference papers.