Multigrid Methods for Low Mach Number Viscous Flows on Adaptive Unstructured Grids

D. O. Ofengeim
SoftImpact Ltd., St. Petersburg, Russia

S. K. Kochuguevbr> St. Petersburg State University, Russia

A. I. Zhmakin
A. F. Ioffe Physical Technical Institute
194021 St.Petersburg, Russia


The low-Mach number Navier-Stokes equations seems to be the most adequate model for flows with low (compared to the sound speed) velocities and large temperature variations. These equations provide the results identical to the full compressible Navier-Stokes computations while reducing greatly CPU time.

The advantage of unstructured grids is the relative ease with which complex geometry can be treated. This approach needs the minimum input description of the domain to be discretized and does not tied closely to its topology in contrast to a block-structured grid. The required CPU time to attain the prescribed accuracy may be less than for the block-structure approach due to the much lesser total number of grid cells as a direct sequence of the second advantage of unstructured grids - the easiness of adaptive mesh refinement.

Finite volume methods for viscous flow computations are based on body-fitted non-orthogonal grid systems. There are several possibility to discretize equations on structured grid, the extreme ones being: 1) contravariant velocity component on a staggered grid and 2) Cartesian velocity components on a co-located grid. In case of unstructured grid one has to use Cartesian velocities. One can use either semi-staggered grid (the velocity components are stored in the vertices while all other variables - temperature, dynamic pressure, species concentrations - in cells' centers) or a collocated one. The latter seems to be more convenient for multilevel methods and is accepted in the present work.

The aim of the paper is to compare different multigrid approaches to real-life internal viscous flow problems. Among others, algebraic multigrid methods and global solution adaptation coupled to the regular grid refinement are considered. Examples of computations of flow and deposition in epitaxial reactors used for growth of semiconductor materials are presented.