Unsteady Transonic Flows in an Axial Flow Compressor*

Introduction. This paper deals with some transonic flow problems for the supersonic axial flow compressor: these arise in connection with flutter analysis for fan blading. Very detailed solutions for unsteady plane transonic flows through a cascade of flat plates have been given by Goldstein Braun and Adamczyk (1977) and Nagashima and Whitehead (1977), see also Goldstein (1975), Kurosaka (1974), Mani and Horvay (1970), Miles (1959), Morfey and Fisher (1970) and Verdon and McCune (1975). The present paper treats the subject from the point of view of boundary value problems for the reduced wave equations of supersonic and subsonic flow, with particular reference to (a) the rotor in a finite duct and (b) a non-uniform incident stream. In Sees. (1) and (2) we solve the supersonic entry problem for a = 0, tt by Fredholm theory. If the frequencies are small we can show that the solutions depend on the duct length, in some cases very significantly. The uniqueness proof of Sec. (3) requires strong conditions on the incident stream and, see App. (B), we give a counter-example for a differential equation of the same general class. If we could strengthen this result to do the same for a specific case of entry solutions in the two-dimensional model of the ducted rotor, (i.e. for Eq. (3.1) with given ip) then we would have a basis for a strict analytical discussion of the 'buzz-saw' phenomenon, cf. Morfey and Fisher loc cit. Sec. (4) shows that the exit problems for the subsonic stream ought likewise to be set in a bounded region. The above analysis of the differential equations has some interesting points of contact with the physical arguments of Ferri (1940), (1964) and Kantrowitz (1950). Ferri envisaged a supersonic biplane such that the external shock wave system might be eliminated. Kantrowitz, having developed a working supersonic compressor, explained its operation in terms of a two-dimensional model which is now standard, c.f. Levine (1957), Stratford and Sansome (1959), Hutton (1974) and Lichtfuss and Starken (1974), from an extensive literature.