Compressible vortex arrays

Citation

Ardalan, Kayvan (1996) Compressible vortex arrays. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/s4n5-1v32. https://resolver.caltech.edu/CaltechETD:etd-02062008-091358

Abstract

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We construct steady, two dimensional, compressible vortex arrays with specified vorticity distributions. We begin by examining the effects of compressiblity on the structure of a single row of hollow-core, constant pressure vortices. The problem is formulated and solved in the hodograph plane. The transformation from the physical plane to the hodograph plane results in a linear problem that is solved numerically. The numerical solution is checked via a Rayleigh-Janzen expansion. It is observed that for an appropriate choice of the parameters [...], the Mach number at infinity, and the speed ratio, a, transonic shock-free flow exists. Also, for a given fixed speed ratio, a, the vortices shrink in size and get closer as the Mach number at infinity, [...], is increased. In the limit of an evacuated vortex core, we find that all such solutions exhibit cuspidal behaviour corresponding to the onset of limit lines.

The hollow core vortex array corresponds to a vorticity distribution wherein the vorticity is concentrated on the vortex boundary. In the second part of this thesis, we examine vortex arrays with continuous vorticity distributions. In particular, we construct Stuart-type vortices in a channel by requiring the vorticity distribution to be an exponential func-tion of the stream function of the flow. The problem is formulated and solved in the physical plane. The numerical solution is checked via a Rayleigh-Janzen expansion of the unbounded Stuart vortex solution. It is shown that, in the limit of infinite speed of sound (incompressible flow), as the channel walls tend to infinity, [...] , the Stuart vortex solutions are recovered. Furthermore, it is shown that unbounded, compressible Stuart vortices exist and that a generalized Stuart vortex solution is the proper incompressible limit. For a given fixed circulation, [...], Mach number, [...], and [...], (a measure of the compactness of the vorticity distribution) it is shown that the limit of a very narrow channel, [...]0, is a parallel shear flow. Exact analytical solutions for the compressible parallel shear flow are also found in implicit form.

Thesis Files