Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-25T23:16:54.587Z Has data issue: false hasContentIssue false

Steady point vortex pair in a field of Stuart-type vorticity

Published online by Cambridge University Press:  10 July 2019

Vikas S. Krishnamurthy*
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
Miles H. Wheeler
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
Darren G. Crowdy
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK
Adrian Constantin
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
*
Email address for correspondence: vikas.krishnamurthy2@gmail.com

Abstract

A new family of exact solutions to the two-dimensional steady incompressible Euler equation is presented. The solutions provide a class of hybrid equilibria comprising two point vortices of unit circulation – a point vortex pair – embedded in a smooth sea of non-zero vorticity of ‘Stuart-type’ so that the vorticity $\unicode[STIX]{x1D714}$ and the stream function $\unicode[STIX]{x1D713}$ are related by $\unicode[STIX]{x1D714}=a\text{e}^{b\unicode[STIX]{x1D713}}-\unicode[STIX]{x1D6FF}(\boldsymbol{x}-\boldsymbol{x}_{0})-\unicode[STIX]{x1D6FF}(\boldsymbol{x}+\boldsymbol{x}_{0})$, where $a$ and $b$ are constants. We also examine limits of these new Stuart-embedded point vortex equilibria where the Stuart-type vorticity becomes localized into additional point vortices. One such limit results in a two-real-parameter family of smoothly deformable point vortex equilibria in an otherwise irrotational flow. The new class of hybrid equilibria can be viewed as continuously interpolating between the limiting pure point vortex equilibria. At the same time the new solutions continuously extrapolate a similar class of hybrid equilibria identified by Crowdy (Phys. Fluids, vol. 15, 2003, pp. 3710–3717).

Type
JFM Rapids
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Constantin, A. & Krishnamurthy, V. S. 2019 Stuart-type vortices on a rotating sphere. J. Fluid Mech. 865, 10721084.Google Scholar
Crowdy, D. G. 1997 General solutions to the 2D Liouville equation. Intl J. Engng Sci. 35, 141149.Google Scholar
Crowdy, D. G. 2003 Polygonal N-vortex arrays: a Stuart model. Phys. Fluids 15, 37103717.Google Scholar
Crowdy, D. G. 2004 Stuart vortices on a sphere. J. Fluid Mech. 498, 381402.Google Scholar
Goodman, J., Hou, T. Y. & Lowengrub, J. 1990 Convergence of the point vortex method for the 2-D Euler equations. Commun. Pure Appl. Maths 43, 415430.Google Scholar
Haslam, M. C. & Mallier, R. 2003 Vortices on a cylinder. Phys. Fluids 15, 20872088.Google Scholar
Loutsenko, I. 2004 Equilibrium of charges and differential equations solved by polynomials. J. Phys. A 37, 13091321.Google Scholar
Mallier, R. 1995 Stuart vortices on a beta-plane. Dyn. Atmos. Oceans 22, 213238.Google Scholar
Mallier, R. & Maslowe, S. A. 1993 A row of counter-rotating vortices. Phys. Fluids A 5, 10741075.Google Scholar
Meiron, D. I., Moore, D. W. & Pullin, D. I. 2000 On steady compressible flows with compact vorticity; the compressible Stuart vortex. J. Fluid Mech. 409, 2949.Google Scholar
Morikawa, G. K. & Swenson, E. V. 1971 Interacting motion of rectilinear geostrophic vortices. Phys. Fluids 14, 10581073.Google Scholar
Newton, P. K. 2001 The N-Vortex Problem: Analytical Techniques. Springer.Google Scholar
O’Neil, K. 2006 Minimal polynomial systems for point vortex equilibria. Physica D 219, 6979.Google Scholar
O’Reilly, G. & Pullin, D. I. 2003 Structure and stability of the compressible Stuart vortex. J. Fluid Mech. 493, 231254.Google Scholar
Saffman, P. G. 1992 Vortex dynamics. Cambridge University Press.Google Scholar
Sakajo, T. 2019 Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus. Proc. R. Soc. Lond. A 475, 20180666.Google Scholar
Shusser, M. 2004 Comment on vortices on a cylinder. Phys. Fluids 16, 3506.Google Scholar
Stuart, J. T. 1967 On finite amplitude oscillations in laminar mixing layers. J. Fluid Mech. 29, 417440.Google Scholar
Tur, A. & Yanovsky, V. 2004 Point vortices with a rational necklace: new exact stationary solutions of the two-dimensional Euler equation. Phys. Fluids 16, 28772885.Google Scholar
Tur, A., Yanovsky, V. & Kulik, K. 2011 Vortex structures with complex points singularities in two-dimensional Euler equations. New exact solutions. Physica D 240, 10691079.Google Scholar