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

Abstract

SUMMARYSecond-order accurate projection methods for simulating time-dependent incompressible flows on cellcentred 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.

THREE-DIMENSIONAL SOLARWINDMODELING FROM THE SUN TO EARTH BY A SIP-CESE MHD MODEL WITH A SIX-COMPONENT GRID

“…Usually, the approximate projection operator does not depress high-frequency modes standing for a local decoupling of the magnetic field as occurred for the decoupling of the velocity field in numerical solutions of incompressible flows (Aprovitola & Denaro 2007;Rider 1998). For robust consideration, filtering is necessary, especially for long-term integrations.…”

Colocated pressure-velocity coupling in finite difference methods

“…This removes the spurious oscillations due to odd-even decoupling and produces results very similar to those obtained when using a staggered grid (Ferziger and Peric, 2002;Peric et al, 1988). Many authors point out that the Rhie and Chow interpolation method results in a scheme that does not conserve kinetic energy (Aprovitola and Denaro, 2007;Shashank et al, 2010;Felten and Lund, 2006) although the ramification of this shortcoming seem to be a major problem only in LES turbulence simulations of high accuracy. In commercial software, it seems to be the dominant type of interpolation method.…”

“…Projection methods are a popular choice for solving the incompressible Navier-Stokes equations (Chorin, 1968). For projection methods on colocated grids, there are both exact projection methods (EPM) and approximate projection methods (APM), both with advantages and drawbacks (Aprovitola and Denaro, 2007). The main difference between the two methods is that the discrete divergence criteria can be satisfied to machine precision for the EPM, whereas the APM takes on a more relaxed divergence criteria constraint.…”

Comparative Study of the Accuracy of the Fundamental Mesh Structures for the Numerical Solution of Incompressible Navier-Stokes Equations in the Two-Dimensional Cavity Problem

“…Within this methodology, the cell-centered option was adopted from the pioneering works of Chorin [3], Peric et al [4], and Majumdar [5], to the recent contributions [6][7][8][9][10][11][12][13][14][15][16][17][18], while finite-volume applications of vertexcentered meshes are reported by [19][20][21][22]. Collocated meshes are known to generate spatially oscillating, checkerboard-like pressure fields, which can be eliminated through the so-called momentum interpolation, due to Rhie and Chow [23], by which continuity is no longer satisfied in its discretized form, but only in the limit of refinement, according to the order of the scheme.…”