We investigate the motion of a thin vortex filament in the presence of buoyancy. The asymptotic model of Moore & Saffman ( Phil. Trans. R. Soc. Lond. A, vol. 272, 1972, pp. 403–429) is extended to take account of buoyancy forces in the force balance on a vortex element. The motion of a buoyant vortex is given by the transverse component of force balance, while the tangential component governs the dynamics of the structure in the core. We show that the local acceleration of axial flow is generated by the external pressure gradient due to gravity. The equations are then solved for vortex rings. An analytic solution for a buoyant vortex ring at a small initial inclination is obtained and asymptotically agrees with the literature.
Axisymmetric contour dynamics for buoyant vortex rings
The present work uses a reduced-order model to study the motion of a buoyant vortex ring with non-negligible core size. Buoyancy is considered in both non-Boussinesq and Boussinesq situations using an axisymmetric contour dynamics formulation. The density of the vortex ring differs from that of the ambient fluid, and both densities are constant and conserved. The motion of the ring is calculated by following the boundary of the vortex core, which is also the interface between the two densities. The velocity of the contour comes from a combination of a specific continuous vorticity distribution within its core and a vortex sheet on the core boundary. An evolution equation for the vortex sheet is derived from the Euler equation, which simplifies considerably in the Boussinesq limit. Numerical solutions for the coupled integro-differential equations are obtained. The dynamics of the vortex sheet and the formation of two possible singularities, including singularities in the curvature and the shock-like profile of the vortex sheet strength, are discussed. Three dimensionless groups, the Atwood, Froude and Weber numbers, are introduced to measure the importance of physical effects acting on the motion of a buoyant vortex ring.
- Award ID(s):
- 1706934
- Publication Date:
- NSF-PAR ID:
- 10189440
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 887
- ISSN:
- 0022-1120
- Sponsoring Org:
- National Science Foundation
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- https://doi.org/10.1017/jfm.2020.26
