chemical species, is responsible for certain fluid mechanical phenomena, such as turbulence, that have no electrodynamic analog and that complicate solution of the conservation equations.

Analytical solutions (e.g., obtained by eigenfunction expansion, Fourier transform, similarity transform, perturbation methods, and the solution of ordinary differential equations for one-dimensional problems) to the conservation equations are of great interest, of course, but they can be obtained only under restricted conditions. When the equations can be rendered linear (e.g., when transport of the conserved quantities of interest is dominated by diffusion rather than convection) analytical solutions are often possible, provided the geometry of the domain and the boundary conditions are not too complicated. When the equations are nonlinear, analytical solutions are sometimes possible, again provided the boundary conditions and geometry are relatively simple. Even when the problem is dominated by diffusive transport and the geometry and boundary conditions are simple, nonlinear constitutive behavior can eliminate the possibility of analytical solution.

Consequently, numerical solution of the equations of change has been an important research topic for many decades, both in solid mechanics and in fluid mechanics. Solid mechanics is significantly simpler than fluid mechanics because of the absence of the nonlinear convection term, and the finite element method has become the standard method. In fluid mechanics, however, the finite element method is primarily used for laminar flows, and other methods, such as the finite difference and finite volume methods, are used for both laminar and turbulent flows. The recently developed lattice-Boltzmann method is also being used, primarily in academic circles. All of these methods involve the approximation of the field equations defined over a continuous domain by discrete equations associated with a finite set of discrete points within the domain and specified by the user, directly or through an automated algorithm. Regardless of the method, the numerical solution of the conservation equations for fluid flow is known as computational fluid dynamics (CFD).

CFD was initially done without automation because the need to solve these equations (e.g., in aircraft design) preceded the development of electronic computers by several decades. With the advent of electronic computers, more ambitious numerical calculations became possible. Initially, CFD codes were written for specific problems. It was natural to generalize these codes somewhat, and eventually, particularly as computational resources became more readily available, general-purpose CFD codes were developed. It was then recognized that a business could be built upon the development and licensing of these codes to industrial, academic, and government users. Today, many of the general-purpose commercial codes are quite sophisticated, cost a tiny fraction of their development cost, and are probably the mainstay of the industrial application of CFD.

Four steps are required to apply a general-purpose CFD code to an industrial problem. First, the domain must be defined. This amounts to constructing the