On the non-existence of continuous transonic flows past profiles I

✍ Scribed by Cathleen S. Morawetz

• Publisher: John Wiley and Sons
• Year: 1956
• Tongue: English
• Weight: 981 KB
• Volume: 9
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/cpa.3160090104

✦ Synopsis

In this paper we shall show that the perturbation problem belonging to a two-diniensional steady transonic flow past an obstacle is not correctly posed. This theorem has been proposed in various forms1 [I, 2, 3, 41 as an explanation for the breakdown of continuous transonic flow. No rigorous proof has been given before. The statement of the theorem may be found as Conjectured Theorem C in [I]. The basic plausibility arguments were developed by Busemann [2], Frank1 [3] and Guderley [4].

The perturbation problem may be described roughly as follows. Suppose for some Mach number at infinity, M , < I, there is a steady continuous symmetric transonic flow, past a profile given by y = 5 Y ( x ) , with continuously differentiable potential p and stream function y. Analytic expressions describing such flows are known, see for example the work of Lighthill, Cragg and Goldstein, or Tomatika and Tamada. 9 and y form a solution of a boundary value problem for a pair of nonlinear elliptic-hyperbolic equations of first order and y vanishes on y = IfI Y ( x ) .

Consider anyneighboringprofile P+ 6Pgiven by y = & (Y (2) + 6Y (x)) .