Equations of motion for weakly compressible point vortices

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 O ( M 2 log ⁡ M ) and O ( M 2 ) . 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)’.

Editorial: Mathematical problems in physical fluid dynamics: part II

“…Equations of motion for two-dimensional compressible point vortices are derived in the limit of small Mach number by Llewelyn Smith et al . [ 3 ]. Ohkitani [ 4 ] constructs and analyses, using asymptotic techniques, self-similar solutions to the one-dimensional Burgers equation and the two-dimensional and three-dimensional Navier–Stokes systems, which complement previously known solutions of this type.…”