A global meshless collocation particular solution method for solving the two-dimensional Navier-Stokes system of equations

The two-dimensional Navier-Stokes system of equations for incompressible fluids is solved by the method of approximate particular solutions (MAPS) in its global formulation. The fluid velocity and pressure fields are approximated by a linear superposition of particular solutions of a Stokes non-homogeneous system of equations with multiquadric (MQ) radial basis function as the source term. The nonlinear convective terms of the momentum equations are linearly approximated by using a guess value of the velocity field, and the resulting linear system of equations is solved by a simple direct iterative scheme (Picard iteration), with the velocity guess given by the solution at the previous iteration. Although the continuity equation is not explicitly imposed in the resulting formulation, the scheme is mass conservative because the particular solutions exactly satisfy the mass conservation equation. The proposed numerical scheme is validated by comparison of the obtained numerical results with the corresponding analytical solution of the Kovasznay flow problem at different Reynolds numbers, Re. From this analysis, it is observed that the MAPS results are stable and accurate for a wide range of shape parameter values. In addition, lid-driven cavity flow problems in rectangular and triangular domains up to Re=3200 and Re=1000, respectively, and the backward-facing step at Re=800 are solved, and the results obtained are compared with corresponding benchmark numerical solutions, showing excellent agreement.