One-stage Method of Fundamental and Particular Solutions (MFS-MPS) for the steady Navier–Stokes equations in a lid-driven cavity

Abstract

The coupled nonlinear steady state Navier–Stokes (N–S) equations in the stream function–vorticity form for a lid-driven cavity are solved by a one-stage Method of Fundamental Solutions (MFS) and the Method of Particular Solutions (MPS). This method has been earlier used for linear Poisson-type problems and has not been applied to coupled nonlinear equations. In this method the steady state N–S equations are first put in the form of two nonlinearly coupled Poisson equations and the solution is sought as the sum of their respective homogeneous and particular solutions. The homogeneous solution is obtained using the MFS and the particular solution is found with the help of Radial Basis Functions (RBFs). Both the operations are accomplished in a single stage. The nonlinear coupling of the N–S equations is tackled by iteration and successive relaxation. We find that the method is easy and effective when compared with the boundary element method (BEM) or the two-stage MFS-MPS, due to its meshless, singular integration free qualities and the single stage operation. The results are obtained for the moderate Reynolds numbers by varying the relaxation parameter. The convergence of MFS-MPS scheme for the present nonlinear problem is numerically demonstrated.