Introduction

This volume contains 14 papers on computational fluid dynamics written between 1967 and 1982, in particular papers on vortex methods and the projection method, as well as papers on the numerical solution of problems in kinetic theory, combustion theory, and gas dynamics. A great deal of practical experience and theoretical understanding has accumulated in these fields in recent years, and a systematic exposition of current knowledge is difficult to write and would be difficult to read. I believe that some of the ideas in this field are easier to learn if they are presented in the simpler garb that preceded the development of sophisticated implementations and of a general mathematical theory, and this is the motivation for publishing the present collection. These papers explain, among other topics, how one might set up a discrete approximation of the Navier–Stokes equations for an incompressible fluid, build a vortex method, or solve a Riemann problem. Some of these papers are by now quite old, especially by the standards of a rapidly changing field, and the reader should be aware of the existence of a large literature that is more up-to-date. In the next few paragraphs I would like to present a short summary of what the papers collected here contain and make some suggestions for further reading. These suggestions reflect my own interests and do not provide a complete bibliography of computational fluid mechanics or even of the topics covered in this book.

The general theme of these papers is the numerical solution of the equations of fluid mechanics in circumstances where the viscosity is small; the big problem that is looming beyond the specific applications is the problem of turbulence. It is generally understood that turbulence in fluids is dominated by the mechanics of vorticity, and many of the methods are based on vortex representations of the flow. A number of them employ random numbers in some form; the major motivation for studying random algorithms is the belief that they may lead to effective methods for sampling a turbulent flow field. The sequence of papers on vortex methods ends here with a paper that represents a possible starting point for speculations on the structure of turbulent flow. The belief that turbulence can be best understood in physical space, by considering the interactions between physical structures, rather than in wave-number space, where one considers interactions between Fourier modes, is not universally held. The reader interested in turbulence theory should find additional sources of information.

The first paper in the book presents the artificial compressibility method for finding steady-state solutions of the Navier–Stokes equations for an incompressible fluid, with applications to thermal convection. Related ideas have been introduced by Temam [T1] and Yanenko. A recent application with a bibliography can be found in [R2]. The main idea is to find a compressible system that has the same steady state as the given incompressible system, but has the property that its steady state can be reached with less labor. More generally, since the limit of infinite sound speed that leads from a compressible to an incompressible flow is well behaved [K2], one can try to approximate incompressible flow by an appropriate compressible flow even far from a steady state. These ideas are closely related to the penalty method, which is more natural in the context of finite element methods [B9].

Paper 2 presents the projection method for incompressible flow. The idea is that the equation of continuity can be viewed as a constraint, and one can solve the equations step-by-step by first ignoring the constraint and then projecting the result on the space of incompressible flows. Some subtlety is required to formulate the boundary conditions for the intermediate step. This construction was partially inspired by the existence theory for the Navier–Stokes equations presented in [F1] and has by now become standard (and indeed, in [C5] I showed that any consistent and stable approximation to the Navier–Stokes equations in pressure–-velocity variables is essentially equivalent to the projection method). The correct formulation of the projection solves the problem of finding numerical boundary conditions for the pressure. Standard references in which this method is discussed include [C13], [G9], [P2], and [T1]. In the paper the projection was implemented by relaxation with a checkerboard pattern, which was a reasonable methodology for the time; however, faster Laplace solvers can be adapted for this purpose, see, e.g., [A4]. As a sidelight, I would like to mention that this paper contains a discussion of the relations between the DuFort–Frankel scheme, checkerboard relaxation and matrix condition (A). An interesting second-order variant of the projection method is presented in [B7]. Further discussion and finite element versions can be found, e.g., in [G9] and [T1].

Paper 3 contains a convergence proof for the projection method. Further work can be found, e.g., in [T1]. The analysis in the paper has two elements that are still of current interest: It shows that convergence and stability depend on the approximations for the gradient and for minus the divergence being at least approximately adjoint, and it contains a model of error growth which says basically that the error accumulates slowly as long as it is small, but if it ceases to be small all hell can break loose. A fixed-point theorem is needed to get convergence in a strong sense. This analysis shows that it is important to obtain error bounds rather than merely weak convergence results. Furthermore, there is a large recent literature in which it is claimed that very poor approximations can provide qualitative models of turbulence; the analysis in this paper suggests that such claims should be viewed with great caution. There is an interesting proof of the validity of a time-discretized projection in [E1].

Paper 4 is a numerical study of Boltzmann’s equation by a method that can be viewed as a numerical generalization of Grad’s thirteen moment approximation. I am not sure that this is a very good method of solution (but I do not know what is). The method is also an early example of a combined expansion–collocation method (as in the pseudo-spectral method), it provides some insight into the usefulness of some of the standard analytical approximation for Boltzmann’s equation, and it provides an interesting contrast to the lack of convergence of the seemingly related Wiener–Hermite expansions of turbulence theory (see, e.g., [C6]).

Paper 5 is the original paper on the random vortex method. An incompressible flow can be approximated by a collection of vortex elements. The main ideas in this paper are: the use of vortices with finite core to improve convergence, the use of a random walk to approximate diffusion, and vorticity creation at boundaries to represent the no-slip boundary condition. For reviews of the applications of this method, see, e.g., [A6], [G10], [M3], and [M4]. The method has been greatly improved in the years since this paper appeared and has been the object of a great amount of outstanding theoretical analysis. The generalizations to three space dimensions and more accurate boundary conditions will be discussed below. I would like to provide here references to some of the major developments:

Paper 6 (with P. Bernard) describes a calculation of the motion of a vortex sheet represented by a collection of vortex blobs (i.e. vortex elements smoothed by a finite cut-off). Later papers that expanded and developed this approach include [A1], [C17], and [K4]. These latter exceptionally fine calculations have also led to important theoretical developments [D1].

Papers 7 and 8 describe applications of the random choice method to problems in compressible gas dynamics and...