Cell vertex methods for inviscid and viscous flows

The cell vertex formulation of the finite volume method is now proving its worth for both the Navier-Stokes and Euler equations. We begin by reviewing the key advantages of the scheme for the Euler equations, and present some new algorithms for shock recovery and for modifying the update algorithm for cells crossed by recovered shocks. Applications will be shown for flow through a row of turbine blades. However, most attention is now focused on the use of the method for the Navier-Stokes equations. On a multiblock mesh it is important that both viscous and inviscid fluxes are approximated on the same cell boundaries; there are alternative ways in which this may be done, and great care has to be taken in regions of high aspect ratio meshes. Typical results will be shown to demonstrate the capability of the method to model boundary layers accurately on coarse and distorted meshes, by showing detailed results on a variety of meshes. Error analysis for the method is based on setting the individual cell residuals to zero. This is difficult to attain, however, and practical algorithms presently drive to zero certain nodally-based combinations of residuals. Thus, the demonstrated accuracy is still subject to this compromise.