A multigrid procedure for three‐dimensional flows on non‐orthogonal collocated grids

Abstract

SUMMARYThe development of a multigrid solution algorithm for the computation of three-dimensional laminar fully-elliptic incompressible flows is presented. The procedure utilizes a non-orthogonal collocated arrangement of the primitive variables in generalized curvilinear co-ordinates. The momentum and continuity equations are solved in a decoupled manner and a pressure-correction equation is used to update the pressures such that the fluxes at the cell faces satisfy local mass continuity. The convergence of the numerical solution is accelerated by the use of a Full Approximation Storage (FAS) multigrid technique. Numerical computations of the laminar flow in a 90" strongly curved pipe are performed for several finite-volume grids and Reynolds numbers to demonstrate the efficiency of the present numerical scheme. The rates of convergence, computational times, and multigrid performance indicators are reported for each case. Despite the additional computational overhead required in the restriction and prolongation phases of the multigrid cycling, the superior convergence of the present algorithm is shown to result in significantly reduced overall CPU times.

Multigrid Calculation of Steady, Viscous Flow in a Triangular Cavity

“…The checker-board split in the pressure field that arises in such equal-order interpolation is avoided, by requiring a different set of velocities (~, ~), located at the cell interfaces, to satisfy mass continuity. This practice is similar to the momentum interpolation concept used in collocated finite volume schemes for structured grids [15][16][17].…”

“…Restriction and prolongation operators for structured rectangular and curvilinear grids are now well established [7,15,21]. For an arbitrarily generated sequence of unstructured grids the integrid transfers must be performed through systematic interpolations using appropriate geometric coordinates of the variable locations [25].…”

Survey of Numerical Methods

“…A number of researchers have developed more strongly coupled approaches [34,35,[54][55][56]. Among the earliest of these attempts is [34].…”

“…Coarse-level meshes are created by grouping fine-level cells together while retaining a structured Cartesian mesh. In an alternative approach, one of the SIMPLE family of algorithms is used as the relaxation sweep in a FAS multigrid scheme in [35,55]. Nonorthogonal body-fitted meshes in a colocated formulation are addressed in [55], whereas unstructured meshes are used in [35]; here, the coarsest mesh is generated first, and progressively fine meshes are created by subdivision.…”

Calculation of three-dimensional turbulent flow with a finite volume multigrid method

“…Recent works, which are based on the SIMPLE pressure correction scheme, mostly deal with laminar flow (Becker et al [1], Smith and Vanka [2], Orth and Schö nung [3]). Depending on the complexity of the discretized geometry, speed-ups of two orders of magnitude for two-dimensional cases are reported compared with single-grid method.…”