A divergence free fractional step method for the Navier-Stokes equations on non-staggered grids

The Navier--Stokes equations may be discretised using finite-volume schemes on non-staggered or on staggered grids. Non-staggered grid schemes typically require a derivation that leads to an error in the divergence to prevent the occurrence of oscillations in the pressure field with accompanying problems in integration. The staggered grid does not have the problem of pressure oscillations; however, this is at the expense of increased code complexity. Two standard non-staggered schemes are compared to a modified scheme in which the divergence error does not occur. References S. Armfield. Finite difference solutions of the Navier--Stokes equations on staggered and non-staggered grids. Computers and Fluids, 20:1--17, 1991. J. C. Strikewerda. Finite difference methods for the Stokes and Navier--Stokes equations. SIAM J. Sci. Stat. Comput., 5:56--68, 1984. S. Abdallah. Numerical solutions of the pressure Poisson equation with Neumann boundary conditions using a non-staggered grid. 1. J. Comp. Phys., 70:182--192, 1987. G. Schneider and M. Raw. Control volume finite-element method for heat transfer and fluid flow using collocated variables. 1. Computational procedure. Numerical Heat Transfer, 11:363--390, 1987. C. M. Rhie and W. L. Chow. A numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA-82-0988, 1982. S. W. Armfield and R. Street. Fractional step methods for the Navier--Stokes equations on non-staggered grids. ANZIAM J. 42(E):C134--C156, 2000. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/593. N. Williamson, S. W. Armfield and W. Lin. Transition behavior of weak turbulent fountains, J. Fluid Mech., 655:306--326, 2010. R. D. Moser, J. Kim and N. N. Mansour. DNS of turbulent channel flow up to $Re_\tau =590$. Phys. Fluids, 11:943--945, 1999