Non‐existence of transonic flow past a profile

Abstract

In the study of transonic flow, one of the most illuminating theorems to prove would be:Given an airfoil projle and a continuous two-dimensional irrotational transonic compressible inuiscidflow past it with some given speed at i n j n i g , there does not exist a corresponding Jlow with a slightly dtjfeerent speed at infinity.Although this theorem was first formulated in 1954, on the basis of conjectures of Frank1 and Guderley, see [l], it has not yet been established. Strong evidence that the theorem is true and proof that smooth transonic flows do not exist generally are given by non-existence theorems in which the profile is varied in the supersonic region and the speed at infinity kept fixed.In [2], Part I, there is such a %on-existence" theorem for continuous transonic flows which are considered as disturbances about a given smooth flow. Except for considering only a linear perturbation this theorem is quite general and complete but the proof is tediously long. In [2], Part 11, allowance is made for the non-linearity at the expense of still further complication. It seems worthwhile to present here a "non-existence" theorem which covers the physically interesting situation and which is fairly simpIe to prove. The proof will be made even more elementary by the addition of a few assumptions on the pressure-density relation.In Section 1 we describe the unperturbed flow and the assumptions, in Section 2 the perturbation flow, in Section 3 the non-existence theorem, in Section 4 the underlying uniqueness theorem. We begin by discussing the flow which is to be varied by varying the airfoil profile.

Transonic shock in a nozzle I: 2D case

“…Various interesting wave phenomena involving this model had been pointed out and investigated by Courant and Friedrichs in [10] and numerous rigorous mathematical theories (mainly on two-dimensional smooth flows) have been surveyed by Bers in [4]. Particularly, Morawetz has used the potential equation and its variants to treat many important problems in transonic flows; see [27,28,29,30,31] and the references therein.…”

“…Phenomena involving transonic flows and transonic flows with shocks are fundamental to fluid dynamics, especially gas dynamics, and have been studied extensively in the literature [4,10,11,13,23,24,25,27,28,29,30,31] and the references therein). Profound understanding has been achieved both physically and mathematically by Morawetz [27,28,30] and others [4,10] on smooth transonic flows. Most previous studies of transonic flows with shocks involve either experimental or numerical simulations or analysis of special wave patterns [4,10,12,16], except the rigorous results on the existence and stability of the quasi one-dimensional transonic shocks; see [11,23].…”

A convergent finite element formulation for transonic flow

“…It is elliptic in the subsonic and hyperbolic in the supersonic region. Although many people have worked for quite a long time on the existence and uniqueness question (see Morawetz [17][18][19][20], Ne~as and Feistauer [9] and Ne~as [22]), so far no general existence result is known. Numerical computations indicate in agreement with physics that the solution of the full potential equation can develop shocks in the case of transonic flow.…”

A Dual Variational Approach to a Class of Nonlocal Semilinear Tricomi Problems

“…Even for linear problems, the techniques employed often involve pasting together solutions found independently in elliptic and hyperbolic regions; with notable exceptions such as works based upon the positive symmetric systems technique of Friedrichs (cf. [12], [16]). Variational tools would provide another approach which is independent of type, with some added ability to interpret the results.…”