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.

The Adaptive Mesh-Refinement Philosophy

In computational fluid dynamics, as well as in other problems of physics or engineering, one often encounters the difficulty that the overall accuracy of the numerical solution is deteriorated by local singularities such as, e.g., singularities near re-entrant corners, interior or boundary layers, or shocks. An obvious remedy is to refine the discretization near the critical regions, i.e., to place more grid-points where the solution is less regular. The question then is how to identify these regions automatically and how to guarantee a good balance of the number of grid-points in the refined and un-refined regions such that the overall accuracy is optimal.

Another, closely related problem is to obtain reliable estimates of the accuracy of the computed numerical solution. A priori error estimates, as provided, e.g., by the standard error analysis for finite element or finite difference methods, are in general not sufficient, since they only yield asymptotic estimates and since the constants appearing in the estimates are usually not known explicitly. Morover, they often require regularity assumptions about the solution which, for practical problems, are hardly satisfied.

Therefore, a computational fluid dynamics code should be able to give reliable estimates of the local and global error of the computed numerical solution and to monitor an automatic, self-adaptive mesh-refinement based on these error estimates.

Formation of a vortex sheet

In perfect barotropic fluid, acted upon by conservative forces with a single-valued potential, the first Helmholtz law (§1.5) says that it is not possible to endow a fluid particle with vorticity, and Kelvin's circulation theorem (§1.6) shows that the circulation around a material circuit is zero if initially zero. The question arises whether vorticity can be created without violating these theorems, and without invoking viscosity, non-conservative forces or baroclinic effects. There is no a priori reason why they are not important in subsequent motion if present initially, and so one wishes to know if vorticity can be created without appeal to these effects.

Klein [1910] addressed this question with his Kaffeelöffel experiment. (See also Betz [1950].) The conclusion is that the Helmholtz and Kelvin theorems preclude the generation of piece-wise continuous vorticity, but do not prevent the formation of vortex sheets or the generation of circulation. Consider Klein's experiment. A two-dimensional plate of width 2a is set in motion through a perfect incompressible fluid with velocity U normal to the plate. We introduce the complex potential w(z) = φ+iψ, z = x+iy. The boundary conditions are ψ = Uy on x = 0, |y| < a (the axes are taken to coincide instantaneously with the plate with the y-axis along the plate and the x-axis in the direction of motion), and w ∼ 0 as z → ∞ (circulation at infinity is not allowed).

Vortex jumps

It has been supposed so far that the velocity and vorticity fields are continuous and have continuous derivatives. In a real viscous fluid, these assumptions are generally regarded as appropriate, and the velocity field is assumed analytic everywhere, except possibly at an initial instant o, when the speed of boundaries changes in a non-analytic manner. However, if we go to the the limit of vanishingly small viscosity and consider ideal fluids, to which the Helmholtz laws apply, the mathematics allows non-analytic behaviour and we cannot assert on physical grounds that only continuous fields should be considered. Velocity or vorticity fields with singularities or discontinuities are indeed of considerable importance. The singularities are, of course, not arbitrary and must be consistent with integral forms of the Euler equations or equivalently with the conservation of mass, momentum and energy. In particular, the dynamical constraint that pressure is continuous across a surface of discontinuity must be satisfied unless there are also singularities in the external force fields. We suppose in this chapter that the density is uniform and put it equal to unity unless explicitly stated otherwise. Also, external forces are supposed conservative unless non-conservative forces are explicitly introduced.