Residual dilation and the accuracy of non-staggered grid incompressible flow schemes

This paper considers the issue of mass conservation for non-staggered grid solutions to the incompressible Navier-Stokes equations. The discrete form of the Divergence Theorem is used to show that dilation results from fundamental inconsistencies between the discrete and continuum quantities. The standard Poisson formulation for pressure is not adequate to drive the calculation to a divergence-free solution. A representative non-staggered grid scheme is implemented to study the problem numerically. Three grid resolutions are employed to determine the behavior of both dilation and accuracy as functions of the iteration parameter (CFL number). Results show that dilation decreases as CFL is increased from very small values, a behavior not previously observed. After reaching a minimum, the dilation increases with CFL as predicted by analysis and the results of previous investigators. Accuracy plotted as a function of CFL shows increasing curvature as the grid resolution is increased. This suggests that the range of optimal CFL numbers narrows as the grid is refined. Therefore, a time step sensitivity analysis should also be considered for high resolution calculations. However, this multiplies the number of cases that must be computed and may negate any numerical advantage gained over staggered grid schemes.