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. 2022 Jun 27;380(2226):20210052.
doi: 10.1098/rsta.2021.0052. Epub 2022 May 9.

Equations of motion for weakly compressible point vortices

Stefan G Llewellyn Smith  1   2 , T Chu  1 , Z Hu  3
Affiliations

Equations of motion for weakly compressible point vortices

Stefan G Llewellyn Smith et al. Philos Trans A Math Phys Eng Sci. .

Abstract

Equations of motion for compressible point vortices in the plane are obtained in the limit of small Mach number, M, using a Rayleigh-Jansen expansion and the method of Matched Asymptotic Expansions. The solution in the region between vortices is matched to solutions around each vortex core. The motion of the vortices is modified over long time scales [Formula: see text] and [Formula: see text]. Examples are given for co-rotating and co-propagating vortex pairs. The former show a correction to the rotation rate and, in general, to the centre and radius of rotation, while the latter recover the known result that the steady propagation velocity is unchanged. For unsteady configurations, the vortex solution matches to a far field in which acoustic waves are radiated. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Keywords: Mach number; Rayleigh–Jansen expansion; matched asymptotic expansions; point vortices.

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Conflict of interest statement

We declare we have no competing interests.

Figures

Figure 1.
Figure 1.
Schematic illustrating the different regions of the flow.
See this image and copyright information in PMC

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