On tie Non-existence of Limiting Lines in Transonic Flows

We consider flows past airfoils which are irrotationai and subsonic for low Mach numbers at large distances from the airfoil. If the Mach number at infinity is gradually increased, at some particular value a small supersonic region appears in the flow next to the airfoil. The flow appears to remain without shocks. If the Mach number at infinity is increased still more, a definite observable shock wave appears. The question arises as to why this shock wave appears. It is possible and even quite probable, that no mixed flows without shocks exist as transition stages and that in genera! as soon as the supersonic region appears there is a shock wave which is at first so weak as to be unobserved. However, if these continuous mixed flows with varying Mach number do exist, an explanation of why the continuous flow breaks down is needed, and it has been suggested by various authors that a "limiting line" appears in the flow. ToUmien, Ringleb, von Karman, and Tsien have observed that, if flows and their corresponding profiles are constructed using solutions in the hodograph plane which depend continuously on some Mach number, there is always a critical Mach number where the mapping of the hodograph plane into the physical plane breaks down. In fact, the image in the physical plane has a fold in the supersonic region whose edge is known as the "limiting line." The Jacobian of the transformation from the hodograph plane to the physical plane vanishes along the limiting line. Tsien [l] and von Karman have proposed that continuous flow past a fixed airfoil also breaks down because of the appearance of a limiting line. Nikolskii and Taganov [2] have shewn that such a limiting line would have to start en the sonic curve. It was finally shown by Fiiedrichs [3] that limiting lines cannot appear anywhere in analytic flows which depend continuously on the entrance Mach number and that therefore the breakdown of potential flow must be due to other causes. (Friedrichs' proof was challenged in a controversial review by Tsien [4].) Man well [5] has shortened the proof considerably and eliminated the difficult lemma supplied by Flanders in [3]. The proof presented here dispenses with the condition of analyticity and requires only continuous second derivatives of the stream function. In addition we shall show, as in [5], that in the construction of flows past continuously changing profiles, [6,7] etc., the incidence of a limiting line corresponds to a profile of infinite curvature. These two results are formulated in two theorems.