A non-diffusive, divergence-free, finite volume-based double projection method on non-staggered grids

Second-order accurate projection methods for simulating time-dependent incompressible flows on cell-centred grids substantially belong to the class either of exact or approximate projections. In the exact method, the continuity constraint can be satisfied to machine-accuracy but the divergence and Laplacian operators show a four-dimension nullspace therefore spurious oscillating solutions can be introduced. In the approximate method, the continuity constraint is relaxed, the continuity equation being satisfied up to the magnitude of the local truncation error, but the compact Laplacian operator has only the constant mode. An original formulation for allowing the discrete continuity equation to be satisfied to machine-accuracy, while using a finite volume based projection method, is illustrated. The procedure exploits the HelmholtzHodge decomposition theorem for deriving an additional velocity field that enforces the discrete continuity without altering the vorticity field. This is accomplished by solving a second elliptic field for a scalar field obtained by prescribing that its additional discrete gradients ensure discrete continuity based on the previously adopted linear interpolation of the velocity. The resulting numerical scheme is applied to several flow problems and is proved to be accurate, stable and efficient. This paper has to be considered as the companion of: 'F. M. Denaro, A 3D second-order accurate projection-based finite volume code on non-staggered, non-uniform structured grids with continuity preserving properties: application to buoyancy-driven flows. IJNMF 2006; 52(4):393-432. Now, we illustrate the details and the rigorous theoretical framework. Copyright (c) 2006 John Wiley & Sons, Ltd.