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 Regul. Chaotic Dyn., 2018, Volume 23, Issue 5, Pages 507–518 (Mi rcd341)

Generalized Contour Dynamics: A Review

Stefan G. Llewellyn Smithab, Ching Changb, Tianyi Chub, Mark Blythc, Yuji Hattorid, Hayder Salmanc

a Scripps Institution of Oceanography, UCSD, 9500 Gilman Drive, La Jolla CA 92093-0213, USA
b Department of Mechanical and Aerospace Engineering, Jacobs School of Engineering, UCSD, 9500 Gilman Drive, La Jolla CA 92093-0411, USA
c School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK
d Institute of Fluid Science, Tohoku University, 2-1-1 Katahira, Aoba, Sendai Japan 980-8577

Abstract: Contour dynamics is a computational technique to solve for the motion of vortices in incompressible inviscid flow. It is a Lagrangian technique in which the motion of contours is followed, and the velocity field moving the contours can be computed as integrals along the contours. Its best-known examples are in two dimensions, for which the vorticity between contours is taken to be constant and the vortices are vortex patches, and in axisymmetric flow for which the vorticity varies linearly with distance from the axis of symmetry. This review discusses generalizations that incorporate additional physics, in particular, buoyancy effects and magnetic fields, that take specific forms inside the vortices and preserve the contour dynamics structure. The extra physics can lead to time-dependent vortex sheets on the boundaries, whose evolution must be computed as part of the problem. The non-Boussinesq case, in which density differences can be important, leads to a coupled system for the evolution of both mean interfacial velocity and vortex sheet strength. Helical geometry is also discussed, in which two quantities are materially conserved and whose evolution governs the flow.

Keywords: vortex dynamics, contour dynamics, vortex patch, vortex sheet, helical geometry

 Funding Agency Grant Number National Science Foundation CBET-1706934 Tohoku University J16R004J17R004 Part of this research was supported by NSF Award CBET-1706934. Support from Collaborative Research Project 2017 and 2018, Institute of Fluid Science, Tohoku University, Project Codes J16R004 and J17R004 is also acknowledged.

DOI: https://doi.org/10.1134/S1560354718050027

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MSC: 76B47, 76W05
Accepted:22.08.2018
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Citation: Stefan G. Llewellyn Smith, Ching Chang, Tianyi Chu, Mark Blyth, Yuji Hattori, Hayder Salman, “Generalized Contour Dynamics: A Review”, Regul. Chaotic Dyn., 23:5 (2018), 507–518

Citation in format AMSBIB
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