# A semi-implicit method for all speeds and all grid aspect ratios

## Description

An AUSM-type discretisation is proposed with stabilisation for incompressible fluid flow by a pressure gradient term added to the mass flux in the continuity equation. For stabilisation of low Mach number flow of a compressible fluid, a pressure gradient term is added to the mass flux in the energy equation and a combined pressure gradient and temperature gradient term to the mass flux in the continuity equation, or the combined term is added to the mass flux in all equations. The terms are properly scaled such that accurate solutions are obtained. For high-speed flows, the method is blended with the AUSM+ algorithm. Steady flow solutions on high aspect ratio structured grids are obtained by time stepping of preconditioned time-depending equations. The time discretisation is explicit for the convective part and line-implicit for the acoustic and diffusive parts of the equations. The lines are chosen in the direction of the grid points with the shortest connection. Convergence penalisation due to stiffness from acoustic terms and viscous terms is avoided.

## Publications

- VIERENDEELS J., RIEMSLAGH K., DICK E.: A multigrid semi-implicit line-method for viscous incompressible and low-Mach number flows on high aspect ratio grids. J. of Computational Physics, 54 (1999), 310-341.
- VIERENDEELS J., MERCI B., DICK E.: Blended AUSM+ method for all speeds and all grid aspect ratios. AIAA-Journal, 39 (2001), 2278-2282.
- VIERENDEELS J., MERCI B., DICK E.: Numerical study of natural convective heat transfer with large temperature differences. Int. J. of Numerical Methods for Heat and Fluid Flow, 11 (2001), 329-341.
- VIERENDEELS J., MERCI B., DICK E.: Benchmark solutions for the natural convective heat transfer problem in a square cavity with large horizontal temperature differences. Int. J. of Numerical Methods for Heat and Fluid Flow, 13 (2003), 1057-1078.
- VIERENDEELS J., MERCI B., DICK E.: A multigrid method for natural convective heat transfer with large temperature differences. J. of Computational and Applied Mathematics, 168 (2004), 509-517.