A 6th order staggered compact finite difference method for the incompressible Navier–Stokes and scalar transport equations

In a previous paper we have developed a staggered compact finite difference method for the compressible Navier–Stokes equations. In this paper we will extend this method to the case of incompressible Navier–Stokes equations. In an incompressible flow conservation of mass is ensured by the well known pressure correction method [7] and [21]. The advection and diffusion terms are discretized with 6th order spatial accuracy. The discrete Poisson equation, which has to be solved in the pressure correction step, has the same spatial accuracy as the advection and diffusion operators. The equations are integrated in time with a third order Adams–Bashforth method. Results are presented for a 1D advection–diffusion equation, a 2D lid driven cavity at a Reynolds number of 1000 and 10,000 and finally a 3D fully developed turbulent duct flow at a bulk Reynolds number of 5400. In all cases the methods show excellent agreement with analytical and other numerical and experimental work.