MSC 3450, PO Box 30001, Las Cruces, NM 88003, USA
Mechanical Engineering Department, New Mexico State University
Shashikanth B. N., Kidambi R.
The Centroid-Deformation Decomposition for Buoyant Vortex Patch Motion
2021, vol. 26, no. 6, pp. 577-599
The motion of a two-dimensional buoyant vortex patch, i. e., a vortex patch with a uniform density different from the uniform density of the surrounding fluid, is analyzed in terms of evolution equations for the motion of its centroid, deformation of its boundary and the strength distribution of a vortex sheet which is essential to enforce pressure continuity across the boundary. The equations for the centroid are derived by a linear momentum analysis and that for the sheet strength distribution by applying Euler’s equations on the boundary, while the boundary deformation is studied in the centroid-fixed frame. A complicated coupled set of equations is obtained which, to the best of our knowledge, has not been derived before. The evolution of the sheet strength distribution is obtained as an integral equation. The equations are also discussed in the limit of a patch of vanishing size or a buoyant point vortex.
Shashikanth B. N.
Poisson brackets for the dynamically interacting system of a 2D rigid cylinder and $N$ point vortices: the case of arbitrary smooth cylinder shapes
2005, vol. 10, no. 1, pp. 1-14
This paper basically extends the work of Shashikanth, Marsden, Burdick and Kelly  by showing that the Hamiltonian (Poisson bracket) structure of the dynamically interacting system of a 2-D rigid circular cylinder and $N$ point vortices, when the vortex strengths sum to zero and the circulation around the cylinder is zero, also holds when the cylinder has arbitrary (smooth) shape. This extension is a consequence of a reciprocity relation, obtainable by an application of a classical Green's formula, that holds for this problem. Moreover, even when the vortex strengths do not sum to zero but with the circulation around the cylinder still zero, it is shown that there is a Poisson bracket for the system which differs from the previous bracket by the inclusion of a 2-cocycle term. Finally, comparisons are made to the works of Borisov, Mamaev and Ramodanov , , , .