Navier-Stokes CFD Solutions

As described earlier in this article, the Navier-Stokes equations are the full equations that govern a viscous flow. Solutions of the Navier-Stokes equations are the ultimate in fluid dynamics. To date, no general analyt­ical solutions of these highly nonlinear equations have been obtained. Yet they are the equations that reflect the real world of fluid dynamics. The only way to obtain useful solutions for the Navier-Stokes equations is by means of CFD. And even here such solutions have been slow in coming. The problem has been the very fine grids that are necessary to define certain regions of a viscous flow (in boundary layers, shear layers, separated flows, etc.), thus demanding huge numbers of grid point in the flow field. Practical solutions of the Navier-Stokes equations had to wait for supercomputers such as the Cray X-MP and Cyber 205 to come on the scene. NASA became a recognized and emulated leader in CFD solu­tions of Navier-Stokes equations, its professionalism evident by its hav­ing established the Institute for Computer Applications in Science and Engineering (ICASE) at Langley Research Center, though other Centers as well, particularly Ames, shared this interest in burgeoning CFD. [778] In particular, NASA researcher Robert MacCormack was responsible for the development of a Navier-Stokes CFD code that, by far, became the most popular and most widely used Navier-Stokes CFD algorithm in the last quarter of the 20th century. MacCormack, an applied math­ematician at NASA Ames (and now a professor at Stanford), conceived a straightforward algorithm for the solution of the Navier-Stokes equa­tions, simply identified everywhere as "MacCormack’s method.”

To understand the significance of MacCormack’s method, one must understand the concept of numerical accuracy. Whenever the derivatives in a partial differential equation are replaced by algebraic difference
quotients, there is always a truncation error that introduces a degree of inaccuracy in the numerical calculations. The simplest finite differences, usually involving only two distinct grid points in their formulation, are identified as "first-order” accurate (the least accurate formulation). The next step up, using a more sophisticated finite difference reaching to three grid points, is identified as second-order accurate. For the numer­ical solution of most fluid flow problems, first-order accuracy is not sufficient; not only is the accuracy compromised, but such algorithms frequently blow up on the computer. (The author’s experience, however, has shown that second-order accuracy is usually sufficient for many types of flows.) On the other hand, some of the early second-order algo­rithms required a large computation effort to obtain this second-order accuracy, requiring many pages of paper to write the algorithm and a lot of computations to execute the solution. MacCormack developed a predictor-corrector two-step scheme that was second-order accurate but required much less effort to program and many fewer calculations to execute. He introduced this scheme in an imaginative paper on hyper­velocity impact cratering published in 1969.[779]

MacCormack’s method broke open the field of Navier-Stokes solu­tions, allowing calculation of myriad viscous flow problems, beginning in the 1970s and continuing to the present time, as was as well (in this author’s opinion) the most "graduate-student friendly” CFD scheme in existence. Many graduate students have cut their CFD teeth on this method and have been able to solve many viscous flow problems that otherwise could not have attempted. Today, MacCormack’s method has been supplanted by several very sophisticated modern CFD algorithms, but even so, MacCormack’s method goes down in history as one of NASA’s finest contributions to the aeronautical sciences.