Computational Fluid Dynamics: What It Is, What It Does

What constitutes computational fluid dynamics? The basic equations of fluid dynamics, the Navier-Stokes equations, are expressions of three fundamental principles: (1) mass is conserved (the continuity equation), (2) Newton’s second law (the momentum equation), and (3) the energy equation (the first law of thermodynamics). Moreover, these equations in their most general form are either partial differential equations (as we have discussed) or integral equations (an alternate form we have not discussed involving integrals from calculus).

The partial differential equations are those exhibited at the NASM. Computational fluid dynamics is the art and science of replacing the partial derivatives (or integrals, as the case may be) in these equations with discrete algebraic forms, which in turn are solved to obtain num­bers for the flow-field values (pressure, density, velocity, etc.) at discrete points in time and or space.[765] At these selected points in the flow, called grid points, each of the derivatives in each of the equations are simply replaced with numbers that are advanced in time or space to obtain a solution for the flow. In this fashion, the partial differential equations are replaced by a large number of algebraic equations, which can then be solved simultaneously for the flow variables at all the grid points.

The end product of the CFD process is thus a collection of numbers, in contrast to a closed-form analytical solution (equations). However, in the long run, the objective of most engineering analyses, closed-form or otherwise, is a quantitative description of the problem: that is, num­bers. Along these lines, in 1856, the famous British scientist James Clerk Maxwell wrote: "All the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact
science is to reduce the problems of nature to the determination of quantities by operations with numbers.”[766] Well over a century later, it is worth noting how well Maxwell captured the essence of CFD: operations with numbers.

Note that computational fluid dynamics results in solutions for the flow only at the distinct points in the flow called grid points, which were identified earlier. In a CFD solution, grid points are either initially dis­tributed throughout the flow and/or generated during the course of the solution (called an "adaptive grid”). This is in theoretical contrast with a closed-form analytical solution for the flow, where the solution is in the form of equations that allow the calculation of the flow variables at any point of one’s choosing, that is, an analytical solution is like a con­tinuous answer spread over the whole flow field. Closed-form analyti­cal solutions may be likened to a traditionalist Dutch master’s painting consisting of continuous brush strokes, while a CFD solution is akin to a French pointillist consisting of multicolored dots made with a brush tip.

Generating a grid is an essential part of the art of CFD. The spacing between grid points and the geometric ways in which they are arrayed is critical to obtaining an accurate numerical CFD solution. Poor grids almost always ensure poor CFD solutions. Though good grids do not guarantee good CFD solutions, they are essential for useful solutions. Grid genera­tion is a discipline all by itself, a subspecialty of CFD. And grid generation can become very labor-intensive—for some flows over complex three­dimensional configurations, it may take months to generate a proper grid.

To summarize, the Navier-Stokes equations, the governing equations of fluid dynamics, have been in existence for more than 160 years, their creation a triumph of derivative insight. But few knew how to analyti­cally solve them except for a few simple cases. Because of their complex­ity, they thus could not serve as a practical widely employed tool in the engineer’s arsenal. It took the invention of the computer to make that pos­sible. And because it did so, it likewise permitted the advent of computa­tional fluid dynamics. So how did the idea of numerical solutions to the Navier-Stokes equations evolve?