The effect of core size on the speed of compressible hollow vortex streets

Abstract

The effect of weak compressibility on the speed of steadily translating staggered vortex streets of hollow vortices in isentropic subsonic flow is studied. A small-Machnumber perturbation expansion about the incompressible solutions for staggered streets of hollow vortices found recently by Crowdy & Green (Phys. Fluids, 2011, vol. 23, 126602) is carried out; the latter solutions provide a desingularization of the classical point vortex streets of von Kármán. The first-order compressible flow correction is calculated. We employ a novel scheme based on a complex variable formulation of the compressible flow equations (the Imai-Lamla method) combined with conformal mapping theory to track the vortex shape in this free boundary problem. The analysis to find the perturbed streamfunction and compressible vortex shapes is greatly facilitated by exploiting a calculus based on use of the Schottky-Klein prime function of a conformally equivalent parametric annulus. It is found that, for a vortex street of specified aspect ratio comprising vortices of specified circulation, the vortex core size is a key determinant of whether compressibility increases or decreases the steady propagation speed (relative to the incompressible street with the same parameters) and that both eventualities are possible. We focus attention on streets with aspect ratios around 0.28, which is close to the neutrally stable case for incompressible flow, and find that a critical vortex core size exists at which compressibility does not affect the speed of the street at first order in the (squared) Mach number. Streets comprising vortices with core size below the critical value speed up due to compressibility; larger vortices slow down.

Speed of a von Kármán point vortex street in a weakly compressible fluid

“…It is clearly of interest to perform a Rayleigh-Jansen analysis for small Mach numbers about the hollow vortex street solutions of Crowdy and Green [39] to gain insights into how the finite size and structure of the vortex cores can affect the speed of such steady compressible streets. The results, which require much more detailed consideration of the vortex core structure than needed here for small vortices, will be presented elsewhere [41].…”

Liouville chains: new hybrid vortex equilibria of the two-dimensional Euler equation

“…The classical Kármán point vortex street has been extended to a weakly compressible flow recently by Crowdy & Krishnamurthy (2017), who also discuss and clarify the general force-free condition required for a point vortex to be in equilibrium in weakly compressible flows. The weakly compressible counterpart of the hollow vortex street (Crowdy & Green 2011) has been studied by Crowdy & Krishnamurthy (2018). This recent activity on weakly compressible flows with embedded vortices gives analytical and numerical evidence for the existence of smooth transonic flows.…”

The corotating hollow vortex pair: steady merger and break-up via a topological singularity

“…It must be true that in order to avoid a term in which would alter the circulation of the vortices which has already been fixed by the choice of the contribution to the complex potential. While the integral formula (4.9) is explicit, it is often more convenient to use another method based on equating coefficients in a Laurent series representation of the unknown function; this method is described in appendix D of Crowdy & Krishnamurthy (2018); see also Crowdy (2020). Either way, in order for a solution for to exist, a solvability condition must be satisfied and this takes the form (Crowdy 2008, 2020) Since on both and then, for example, on , and hence, on differentiation with respect to , with the same deduction holding for .…”