Abstract
The link between the shape of a two-dimensional, uniform vortex and self-induced velocities on its boundary is investigated through a contour-dynamics approach. The tangent derivative of the velocity along the boundary is written in a complex form, which depends on the analytic continuation of the tangent unit vector outside the vortex boundary. In this way, a classical analysis in terms of Schwarz's function of the boundary, due to Saffman, is extended to vortices of arbitrary shape. Time evolution of intrinsic quantities (tangent unit vector, curvature and Fourier's coefficients for the vortex shape) is also analyzed, showing that it depends on tangent derivatives of the velocities, only. Furthermore, a spectral method is proposed, aimed at investigating the dynamics of nearly-circular vortices in an inviscid, isochoric fluid. Comparisons with direct numerical simulations are also established.
Similar content being viewed by others
References
1._ H. Lamb, Hydrodynamics. New York: Dover Publications (1932) 738 pp.
M.V. Melander, N.J. Zabusky and A.S. Styczek, A moment model for vortex interactions of the two dimensional Euler equations. Part 1. Computational validation of a Hamiltonian elliptical representation. J. Fluid Mech. 167 (1986) 95–115.
B. Legras and D.G. Dritschel, The elliptical model of two-dimensional vortex dynamics. I: The basic state. Phys. Fluids A 3 (1990) 845–854.
M.V. Melander, N.J. Zabusky and J.C. McWilliams, Symmetric vortex merger in two dimensions: causes and conditions. J. Fluid Mech. 195 (1988) 303–340.
G. Riccardi and R. Piva, Motion of an elliptical vortex under rotating strain: conditions for asymmetric merging. Fluid Dyn. Res. 23 (1998) 63–88.
D.G. Dritschel, Contour dynamics and Contour Surgery: numerical algorithms for extended high-resolution modeling of vortex dynamics in two-dimensional, inviscid, incompressible flows. Comp. Phys. Repts. 10 (1989) 77–146.
N.J. Zabusky, M.H. Hughes and K.V. Roberts, Contour Dynamics for the Euler equations in two dimensions. J. Comp. Phys. 48 (1979) 96–106.
B. Legras and V. Zeitlin, Conformal dynamics for vortex motions. Phys. Lett. A 167 (1992) 265–271.
J. Jimenez, Linear stability of a non-symmetric, inviscid, Karman street of small uniform vortices. J. Fluid Mech. 189 (1988) 337–348.
P.G. Saffman, Vortex Dynamics. Cambridge: Cambridge University Press (1992) 311 pp.
D.G. Crowdy, A class of exact multipolar vortices. Phys. Fluids 11 (1999) 2556–2564.
D.G. Crowdy, Exact solutions for rotating vortex arrays with finite-area cores. J. Fluid Mech. 469 (2002) 209–235.
A.J. Majda and A.L. Bertozzi, Vorticity and Incompressible Flow. Cambridge: Cambridge University Press (2002) 632 pp.
D.I. Pullin, The nonlinear behaviour of a constant vorticity layer at a wall. J. Fluid Mech. 108 (1981) 401–421.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Riccardi, G. Intrinsic dynamics of the boundary of a two-dimensional uniform vortex. Journal of Engineering Mathematics 50, 51–74 (2004). https://doi.org/10.1023/B:ENGI.0000042119.98370.14
Issue Date:
DOI: https://doi.org/10.1023/B:ENGI.0000042119.98370.14