Theoretical models for compressible vortex dynamics
Creators: name: Shivani Krishnamurthy, Vikas
Publisher: name: Imperial College London
Year: 2016

description: In this thesis, we study the effects of weak compressibility on staggered vortex streets, which are a ubiquitous phenomenon observed in nature. They are formed by fluid flowing past an obstacle, and can be found at several different length scales ranging from human length scales such as in water flowing under a bridge, to astronomical length scales such as planetary atmospheres, and beyond. They are of fundamental importance in fluid flows of interest to engineers and applied mathematicians alike. Yet, these vortex streets have not been studied as much theoretically. The classical exact solutions of Kármán are more than a century old, and used singular vortex models. The studies by Saffman and co-workers considered a vortex patch street, thus removing the singularity, however this study was numerical. Part of the difficulty in de-singularised models arises because of the unknown shape of the free-boundary of a finite-sized vortex. In particular, the effects of compressibility have not been taken into account in any theoretical investigation from first principles. We undertake a theoretical study of weakly compressible vortex streets, and obtain solutions to first-order for the velocity field. The vortices are modelled as hollow vortices, which are bounded regions of constant pressure with a non-zero circulation around them. For incompressible hollow vortex streets, exact solutions based on a conformal mapping approach were found by Crowdy and Green (2011). We perform a Rayleigh-Jansen (perturbation) analysis on these solutions, assuming isentropic flow of an ideal gas. We develop a novel method involving a rare use of complex analysis in studying compressible flows of this type. We utilise the Imai-Lamla formula [Imai (1942)] for the complex potential, which reduces the solution of the weakly compressible flow to computing analytic functions. Combining this with a conformal mapping approach, we solve the free-boundary problem by obtaining boundary value problems in multiply-connected pre-image domains (an annulus), which are solved through modern applications of classical methods. We utilise the machinery of the Schottky-Klein prime function for this domain, which helps in solving the boundary value problems for the analytic functions. We find that the speed of the vortex street changes due to compressibility, but the details of this change depends importantly on the area of the vortices. The vortex street is found to speed up for smaller areas and slow down at larger areas. The critical area of transition is found to be further dependant on the vortex configuration in the flow. We discuss the relationship of the present results with a few other known results in the context of different flows. We compare our results in the zero vortex size limit, to the results from a weakly compressible point vortex analysis undertaken in Crowdy and Krishnamurthy (2017b) and find agreement. Parts of the results in this thesis are reported as a paper Crowdy and Krishnamurthy (2017a). It is hoped that the present study leads to further studies on related problems such as the stability of vortex streets and vortex wake modelling for both incompressible and compressible flows. Other related problems likely to be influenced by this study include the free compressible vortex pair, which is important from the point-of-view of the theory of vortex sound.