Effects of direction decoupling in flux calculation in finite volume solvers

In a finite volume CFD method for unsteady flow fluxes of mass, momentum and energy are exchanged between cells over a series of small time steps. The conventional approach, which we will refer to as direction decoupling, is to estimate fluxes across interfaces in a regular array of cells by using a one-dimensional flux expression based on the component of flow velocity normal to the interface between cells. This means that fluxes cannot be exchanged between diagonally adjacent cells since they share no cell interface, even if the local flow conditions dictate that the fluxes should flow diagonally. The direction decoupling imposed by the numerical method requires that the fluxes reach a diagonally adjacent cell in two time-steps.To evaluate the effects of this direction decoupling, we examine two numerical methods which differ only in that one uses direction decoupling while the other does not. We examine a generalized form of Pullin’s equilibrium flux method (EFM) [D.I. Pullin, Direct simulation methods for compressible ideal gas flow, J. Comput. Phys. 34 (1980) 231–244] which we have called the true direction equilibrium flux method (TDEFM). The TDEFM fluxes, derived from kinetic theory, flow not only between cells sharing an interface, but ultimately to any cell in the grid. TDEFM is used here to simulate a blast wave and an imploding flow problem on a structured rectangular mesh and is compared with results from direction decoupled EFM. Since both EFM and TDEFM are identical in the low CFL number limit, differences between the results demonstrate the detrimental effect of direction decoupling. Differences resulting from direction decoupling are also shown in the simulation of hypersonic flow over a rectangular body. The computational cost of allowing the EFM fluxes to flow in the correct directions on the grid is minimal.