Collection of abstracts 13th GAMM-Seminar Kiel on Numerical Treatment of Multi-Scale Problems January 24th to 26th, 1997.

Friday, January 24th, 1997

Three-dimensional multigrid based pressure solver for the computation of the flow around the HSVA Tanker

We consider three-dimensional numerical methods for the incompressible Reynolds averaged Navier-Stokes equations discretised by finite difference techniques on non-staggered grids in boundary fitted coordinates. A segregated approach is used to solve the pressure-velocity coupling problem, whose resolution of pressure Poisson equation -a crucial task especially for three-dimensional complex geometries- is handled by a multigrid based solver.

We present the numerical treatment and implementation aspects of this solver and show its performance on the computation of the turbulent viscous flow around the HSVA tanker at high Reynolds number (five millions) [1]. An elliptic pressure equation with highly varying coefficients mainly involved by the topology of the mesh (96 x 48 x 32, non-orthogonal and stretched near the hull) is to be solved at each non-linear iteration. The linear cell-centered multigrid employs the Galerkin Coarse Grid Approximation to build the matrices on coarser grids [2]. The cell-centered layout avoids the use of matrix-dependent operators. The uniform operators (linear interpolation for the prolongation, arithmetical average for the restriction) ensures the optimal sparsity of the stencil (7 points) on each grid and simultaneously allows to keep the M-matrix property for the pressure operator on each grid. As smoother, we employ the Strongly Implicit Procedure of Stone [3].

Using this multigrid method as a preconditioner for the BICGSTAB algorithm [4] results in a CPU-time speed up ratio greater than two with regard to other conventional acceleration techniques (MILU preconditioned Krylov subspace methods) on this problem. According to the aforementioned choices of transfer operators and smoothing strategy, this multigrid based solver is the starting point for a parallelizable tool well-suited to a multiblock approach.

1. J. Piquet and M. Visonneau: Computation of the flow past shiplike hulls, Computers and Fluids 19--2, pp. 183--215, 1991.
2. M. Khalil: Analysis of linear multigrid methods for elliptic differential equations with discontinous and anisotropic coefficients, Doctor thesis, University of Technology,Department of Mathematics and Informatics, Delft, 1991.
3. H. Stone: Iterative solution of implicit approximations of multidimensional partial differential equations, SIAM J.Num.Anal 5, pp. 530--558, 1968.
4. H. Van Der Vorst: BICGSTAB: a fast and smoothly converging variant of BiCG for the solution of nonsymmetric linear systems, SIAM Journal of Scientifical and Statistical Computation 13, pp. 631--644, 1992.