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  Contour dynamics for Unsteady Vortex Rings

   School of Mathematics

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  Dr M Blyth  No more applications being accepted  Self-Funded PhD Students Only

About the Project

Known as the “sinews of turbulence”, vortices are fundamental to the study of fluid mechanics. A vortex is a volume/region of fluid in which the vorticity (the curl of the velocity field) is non-zero. For a vortex ring, the vorticity is concentrated within a toroidal, or doughnut-shaped, region known as the vortex core. Vortex rings are created when one fluid is forced rapidly through an orifice into another fluid. A smoke ring, for example, is formed when smoke is blown quickly through an opening into clear air. In the axisymmetric case, and when the core vorticity increases linearly from the ring axis, Norbury (1973) showed that there exists a one parameter family of steadily propagating ring solutions with Hill’s spherical vortex at one extreme and a thin-core ring with an approximately circular cross-section at the other extreme. In two dimensions, a family of analogous vortex structures with a constant core vorticity was constructed by Pierrehumbert (1980). In both these scenarios the problem of determining the dynamics of the vortex ring can be reduced to tracking the time evolution of the boundary of the vortex core. Norbury’s solutions can be viewed as a low-order approximation to the more complex vortex ring structures that are encountered in applications and in experiments. In this project we will develop a higher-order approximation protocol that can be used to model vortex rings with an arbitrary, continuous distribution of vorticity in the core. Motivated by the success of well-established contour dynamics methods for two-dimensions and for axisymmetry (see LLewellyn Smith et al. (2018) for a review), we will develop a nested-contour approach (see also O’Farrell & Dabiri 2012) that allows for the computationally efficient prediction of vortex equilibria, calculating their stability, and determining their general time evolution. 

Mathematics (25)

Funding Notes

This PhD project is offered on a self-funding basis. It is open to applicants with funding or those applying to funding sources. Details of tuition fees can be found at
A bench fee is also payable on top of the tuition fee to cover specialist equipment or laboratory costs required for the research. Applicants should contact the primary supervisor for further information about the fee associated with the project.


i) Llewellyn Smith, S., Chang, C., Chu, T., Blyth, M. G., Hattori, Y. & Salman, H., Generalised contour dynamics: a review. Regular and chaotic dynamics, 23, 507-518.
ii) Norbury, J. (1973) A family of steady vortex rings. J. Fluid Mech., 57, 417-431.
iii) O’Farrell, C & Dabiri, J. O. (2012) Perturbation response and pinch-off of vortex rings and dipoles, J. Fluid Mech., 704, 280-300.
iv) Pierrehumbert, R. T. (1980) A family of steady, translating vortex pairs with distributed vorticity, J. Fluid Mech., 99, 129-144.

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