Summary

We describe a series of algorithms for the numerical simulation of incompressible flows. These algorithms are obtained by following a rational path from a list of design goals for practical incompressible flow solvers to their ultimate realization. Along the way, the important identity of artificial viscosity and pairs of different trial spaces for velocities and pressures is shown rigourously for the mini-element. Several numerical examples demonstrate the accuracy and versatility of the algorithms developed.

Introduction

The applications that require numerical simulations of incompressible flows may be grouped into two families:

• Engineering design and optimization: here the basic physics governing the flows to be simulated are relatively well understood, and the main requirement on the numerical methods employed is versatility, ease of use, and speed. Many configurations have to be simulated quickly, in order to develop or improve a new product. This implies that the whole process of simulating incompressible flow past an arbitrary, new configuration must take at most several days. Usually, the engineer desires a global figure, like lift and drag, as the end-product of a simulation.

• Study of basic physics’, in this case, numerical simulations are used to obtain new insight into basic physical phenomena, like vortex merging and breakdown, or the transition to turbulence. The main requirement placed on the numerical methods employed is accuracy. The geometries for which these calculations are carried out are typically very simple (boxes, channels), and the time required to perform such a simulation plays a secondary role. Some of the runs performed to date have required hundreds of CRAY-hours. Usually, the physicist desires statistical data as the end-product of such a simulation.