CFD and Transonic Airfoils

The analysis of transonic flows suffers from the same problems as those for the supersonic blunt body discussed above. Just considering the flow to be inviscid, the governing Euler equations are highly nonlinear for both transonic and hypersonic flows. From the numerical point of view, both flow fields are mixed regions of locally subsonic and super­sonic flows. Thus, the numerical solution of transonic flows originally encountered the same problem as that for the supersonic blunt body problem: whatever worked in the subsonic region did not work in the supersonic region, and vice versa. Ultimately, this problem was solved from two points of view. Historically, the first truly successful CFD solu­tion for the inviscid transonic flow over an airfoil was carried out in
1971 by Earll Murman and Julian Cole of Boeing Scientific Research Laboratories, whose collaborative research began at the urging of Arnold "Bud” Goldburg, then Chief Scientist of Boeing.[776] They treated a simpli­fied version of the Euler equations called the small-perturbation veloc­ity potential equation. This limited their solutions to the flows over thin airfoils at small angles of attack. Nevertheless, Murman and Cole intro­duced the concept of writing the finite differences in the equations such that they reached in both the upstream and downstream directions when in the subsonic region, but they reached in only the upwind direction in the supersonic regions. This is motivated by the physical process that in subsonic flow disturbances propagate in all directions but in a supersonic flow disturbances propagate only in the downstream direction. Thus it is proper to form the finite differences in the supersonic region such that they take only information from the upstream side of the grid point.

Today, this approach in modern CFD is called "upwinding” and is part of many modern algorithms in use for all kinds of flows. In 1971, this idea was groundbreaking, and it allowed Murman and Cole to obtain the first successful numerical solutions of the transonic flow over a body. In addition to the restriction of thin airfoils at small angles of attack, how­ever, their use of the small perturbation velocity potential equation also limited their solutions to isentropic flows. This meant that, although their solution captured the semblance of a shock wave in the flow, the loca­tion and flow changes across a shock wave were not accurate. Because many transonic flows involve shock waves embedded in the flow, this was definitely a bit of a problem. The solution to this problem involved the numerical treatment of the Euler equations, which, as we have dis­cussed early in this article, accurately pertain to any inviscid flow, not just one with small perturbations and free of shocks.

The finest in such CFD solutions were developed by Antony Jameson, then a professor at Princeton University (and now at Stanford), whose work was heavily sponsored by the NASA Langley Research Laboratory. Using the concept of time marching in combination with a Runge-Kutta time integration of the unsteady equations, Jameson constructed a series of outstanding transonic airfoil codes under the general code name of the FLO codes. These codes entered standard use in many aircraft com­panies and laboratories. Once again, NASA had been responsible for a
major advancement in CFD, helping to develop transonic flow codes that advanced the design of many airfoil shapes used today on modern commercial jet transports.[777]