Regularity analysis of an incompressible Navier-Stokes algorithm for non-staggered grids

Regularity analysis of an incompressible Navier-Stokes algorithm recently proposed by the authors is performed. Such an analysis is used to determine if the discrete set of equations retains the elliptic nature of the continuum equations. The analysis is especially important for the incompressible equations because the treatment of the velocity-pressure coupling in a given algorithm affects the ellipticity of the discrete equations. The present algorithm uses a pressure correction approach, but employs a non-staggered grid instead of the usual staggered grid. A Poisson equation for pressure is derived with the relevant compatibility constraint and the momentum equations are integrated with an Euler-explicit scheme and second-order accurate finite volume discretization. The discrete equations are examined using a discrete Fourier transform. Results show that the algorithm remains elliptic for moderate Reynolds numbers and that the ellipticity of the discrete equations is not a function of the time integration step as is the case with several related algorithms. This behavior has the desirable effect of decoupling the dynamic stability considerations of the time integration scheme for the momentum equations from those of the Poisson equation. Time step limits can therefore be estimated purely with the standard Von-Neumann stability analysis. Application of the algorithm to shear-driven cavity flow provides numerical confirmation of the analysis for moderate Reynolds numbers.