On the transonic controversy
The problem of the existence of a transonic flow without a shock wave is still unsolved, notwithstanding even very recent research work, including the fundamental work done byMorawetz andManwell: all these researches consider the flow on the odographic plane, and the essential observation that may be made concerning them is that they show the non-existence of a continuous flow in the case of profiles with a contour that isnot regular. In this commentary, the problem is tackled in a different way, namely, by considering the flow directly on the physical plane and agreeing to represent the velocity on the contour of the profile by means of the integral equation ofOswatitsch. With an appropriate modification of this it is possible to construct a «model» of transonic flow, through which one may recognize with all simplicity what happens when the speed uA on the contour exceeds the speed of sound.
To be precise, the integral equation is transformed into a non-linear differential equation of the second degree in a variable Y: the uA is a function of the Y, which presents a branching point for the uA value equal to the critical speed. If the flow is totally subsonic, the conditions at the contour are of the necessary and sufficient number to determine the integration constants, and the solution exists. If the flow is transonic, if one admits that the solutionpasses through the branching point twice, and so the flow is continuous, a supplementary condition is introduced that is generally incompatible with the conditions at the contour, and therefore generally no solution exists. If one admits instead that the flow is discontinuous, and therefore the solution passes through the branching point only once, the solution that satisfies the conditions at the contour exists.
It is also demonstrated that, as regards the fundamental question of the non-existence of a continuous flow, the results obtained are valid also when the model is perfected in such a way as to make it nearer the real phenomenon.
KeywordsDifferential Equation Shock Wave Mechanical Engineer Integral Equation Civil Engineer
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