Shape adaptive RBF-FD implicit scheme for incompressible viscous Navier–Strokes equations

Abstract

Meshless methods for solving fluid flow problems have become a promising alternative to mesh-based methods. In this paper, a meshless method based on radial basis functions in a finite difference mode (RBF-FD) has been developed for the incompressible Navier–Stokes (N–S) equations in primitive variable form. Pressure–velocity decoupling has been achieved using a fractional step method whereas time splitting has been done using both explicit and implicit schemes. The RBF-FD implicit scheme shows better accuracy and stability, and is able to accurately capture higher gradients of field variables even at coarser grids; unlike the RBF-FD explicit scheme where loss of accuracy was especially prominent at places with larger gradients. To overcome the ill-conditioning and accuracy problems arising from the use of non-uniform and random node distribution, a novel concept of adaptive shape parameter (ASP) for RBF functions is introduced. The use of ASP allows much finer nodal distribution at regions of interest enabling accurate capturing of gradients and leading to better results. The performance of the Implicit RBF-FD scheme with the ASP strategy is validated against a variety of benchmark problems, including lid driven cavity flow problems, and steady and unsteady laminar flow around circular cylinder at various Reynolds, and is found to be in good agreement with the existing results.