Coda: Transonic Airfoils and SLE

We close by describing two rather unusual applications of complex variables. The details are beyond the scope of this book, but the ideas involved definitely deserve mention. The first area of application is fluid dynamics. It was observed already in the nineteenth century that the equations describing the incompressibility and irrotationality of fluids are just the Cauchy-Riemann equations for the velocity components in two-dimensional flow. Since low velocity flow is nearly incompressible, this made it possible to use analytic functions (more specifically, the theory of conformal mapping) to describe such flows around airfoils and to determine lift and drag. However, for high speed flows, which are compressible, this approach is not available. In high speed flows over airfoils, the flow becomes supersonic over parts of the airfoil. This leads to the formation of shock waves, an undesirable effect since shocks increase drag. Although Cathleen Morawetz proved mathematically that, in general, shock waves occur in partially supersonic flows [M1], [M2], this did not rule out the existence of special airfoils for which shockless flows are possible. In fact, Paul Garabedian and his student David Korn developed a hodograph method based on complex characteristics that enabled them to calculate supercritical wing sections free of shocks at a specified speed and angle of attack [K], [GK1]. However, the extensive trial and error involved in the selection of parameters defining the flow rendered this method impractical. After the preliminary results of [BGK], a completely satisfactory solution of the problem was obtained by Garabedian and Korn in [GK2]. They solve the partial differential equations of two-dimensional inviscid gas dynamics by analytic continuation into the domain of two independent complex characteristic coordinates. After mapping the domain of integration conformally onto the unit disk in the plane of one of these coordinates, they formulate a boundary value problem on that disk for the stream function which is well-posed even in the case of transonic flow. This enables them to give a procedure for calculating an airfoil on which the speed is prescribed as a function of arclength, leading to an exact solution of the problem in the case of subsonic flow and, in the transonic case, generally to a shockless flow which assumes the assigned subsonic values of the speed and approximates the given supersonic values. Truly a tour de force of applied complex analysis.