Manifold method coupled velocity and pressure for Navier–Stokes equations and direct numerical solution of unsteady incompressible viscous flow

Abstract

Numerical manifold method (NMM) application to direct numerical solution for unsteady incompressible viscous flow Navier–Stokes (N–S) equations was discussed in this paper, and numerical manifold schemes for N–S equations were derived based on Galerkin weighted residuals method as well. Mixed covers with linear polynomial function for velocity and constant function for pressure was employed in finite element cover system. The patch test demonstrated that mixed covers manifold elements meet the stability conditions and can be applied to solve N–S equations coupled velocity and pressure variables directly. The numerical schemes with mixed covers have also been proved to be unconditionally stable. As applications, mixed cover 4-node rectangular manifold element has been used to simulate the unsteady incompressible viscous flow in typical driven cavity and flow around a square cylinder in a horizontal channel. High accurate results obtained from much less calculational variables and very large time steps are in very good agreement with the compact finite difference solutions from very fine element meshes and very less time steps in references. Numerical tests illustrate that NMM is an effective and high order accurate numerical method for unsteady incompressible viscous flow N–S equations.