Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T08:45:22.127Z Has data issue: false hasContentIssue false
  • English
  • Français

‘H-states’: exact solutions for a rotating hollow vortex

Published online by Cambridge University Press:  01 March 2021

D.G. Crowdy Open the ORCID record for D.G. Crowdy [Opens in a new window] ,
R.B. Nelson Open the ORCID record for R.B. Nelson [Opens in a new window]  and
V.S. Krishnamurthy Open the ORCID record for V.S. Krishnamurthy [Opens in a new window]
D.G. Crowdy*
Affiliation:
Department of Mathematics, Imperial College London, LondonSW7 2AZ, UK
R.B. Nelson
Affiliation:
Department of Earth Science and Engineering, Imperial College London, LondonSW7 2AZ, UK
V.S. Krishnamurthy
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090Vienna, Austria
*
†Email address for correspondence: d.crowdy@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Exact solutions are found for an $N$-fold rotationally symmetric, steadily rotating hollow vortex where a continuous real parameter governs its deformation from a circular shape and $N \ge 2$ is an integer. The vortex shape is found as part of the solution. Following the designation ‘V-states’ assigned to steadily rotating vortex patches (Deem & Zabusky, Phys. Rev. Lett., vol. 40, 1978, pp. 859–862) we call the analogous rotating hollow vortices ‘H-states’. Unlike V-states where all but the $N=2$ solution – the Kirchhoff ellipse – must be found numerically, it is shown that all H-state solutions can be written down in closed form. Surface tension is not present on the boundaries of the rotating H-states but the latter are shown to be intimately related to solutions for a non-rotating hollow vortex with surface tension on its boundary (Crowdy, Phys. Fluids, vol. 11, 1999a, pp. 2836–2845). It is also shown how the results here relate to recent work on constant-vorticity water waves (Hur & Wheeler, J. Fluid Mech., vol. 896, 2020, R1) where a connection to classical capillary waves (Crapper, J. Fluid Mech., vol. 2, 1957, pp. 532–540) is made.

JFM classification

Vortex Flows: Vortex dynamics
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 913 , 25 April 2021 , R5
Check for updates
Copyright
© The Author(s), 2021. Published by 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

REFERENCES

Ardalan, K., Meiron, D.I. & Pullin, D.I. 1995 Steady compressible vortex flows: the hollow-core vortex array. J. Fluid Mech. 301, 1–17.CrossRefGoogle Scholar
Baker, G.R., Saffman, P.G. & Sheffield, J.S. 1976 Structure of a linear array of hollow vortices of finite cross-section. J. Fluid Mech. 74, 469–476.CrossRefGoogle Scholar
Batchelor, G.K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Crapper, G. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2, 532–540.CrossRefGoogle Scholar
Crowdy, D.G. 1999 a Circulation-induced shape deformations of drops and bubbles: exact two-dimensional models. Phys. Fluids 11, 2836–2845.CrossRefGoogle Scholar
Crowdy, D.G. 1999 b Exact solutions for steady capillary waves on a fluid annulus. J. Nonlinear Sci. 9, 615–640.CrossRefGoogle Scholar
Crowdy, D.G. 2000 A new approach to free surface Euler flows with capillarity. Stud. Appl. Maths 105, 35–58.CrossRefGoogle Scholar
Crowdy, D.G. 2001 Steady nonlinear capillary waves on curved sheets. Eur. J. Appl. Maths 12, 689–708.CrossRefGoogle Scholar
Crowdy, D.G. 2005 Quadrature domains and fluid dynamics. In Quadrature Domains and Their Applications (ed. P. Ebenfelt, B. Gustafsson, D. Khavinson & M. Putinar). Birkhauser.Google Scholar
Crowdy, D.G. 2020 Solving Problems in Multiply Connected Domains. Society for Industrial and Applied Mathematics.CrossRefGoogle Scholar
Crowdy, D.G. & Green, C.C. 2011 Analytical solutions for von Kármán streets of hollow vortices. Phys. Fluids 23, 126602.CrossRefGoogle Scholar
Crowdy, D.G. & Krishnamurthy, V.S. 2017 The effect of core size on the speed of compressible hollow vortex streets. J. Fluid Mech. 836, 797–827.CrossRefGoogle Scholar
Crowdy, D.G., Llewellyn Smith, S.G. & Freilich, D.V. 2013 Translating hollow vortex pairs. Eur. J. Mech. (B/Fluids) 37, 180–186.CrossRefGoogle Scholar
Crowdy, D.G. & Nelson, R.B. 2010 Steady interaction of a vortex street with a shear flow. Phys. Fluids 22, 096601.CrossRefGoogle Scholar
Crowdy, D.G. & Roenby, J. 2014 Hollow vortices, capillary water waves and double quadrature domains. Fluid Dyn. Res. 46, 031424.CrossRefGoogle Scholar
Deem, G.S. & Zabusky, N.J. 1978 Vortex waves: stationary ‘V-states’, interactions, recurrence and breaking. Phys. Rev. Lett. 40, 859–862.CrossRefGoogle Scholar
Freilich, D.V. & Llewellyn Smith, S.G. 2017 The Sadovskii vortex in strain. J. Fluid Mech. 825, 479–501.CrossRefGoogle Scholar
Hur, V.M. & Dyachenko, S.A. 2019 a Stokes waves with constant vorticity: folds, gaps and fluid bubbles. J. Fluid Mech. 878, 502–521.Google Scholar
Hur, V.M. & Dyachenko, S.A. 2019 b Stokes waves with constant vorticity: I. Numerical computation. Stud. Appl. Maths 142, 162–189.Google Scholar
Hur, V.M. & Vanden-Broeck, J.-M. 2020 A new application of Crapper's exact solution to waves in constant vorticity flows. Eur. J. Mech. (B/Fluids) 83, 190–194.CrossRefGoogle Scholar
Hur, V.M. & Wheeler, M.H. 2020 Exact free surfaces in constant vorticity flows. J. Fluid Mech. 896, R1.CrossRefGoogle Scholar
Kinnersley, W. 1977 Exact large amplitude capillary waves on sheets of fluid. J. Fluid Mech. 77, 229–241.CrossRefGoogle Scholar
Lamb, H. 1994 Hydrodynamics. Cambridge University Press.Google Scholar
Llewellyn Smith, S.G. & Crowdy, D.G. 2012 Structure and stability of hollow vortex equilibria. J. Fluid Mech. 691, 178–200.CrossRefGoogle Scholar
Meunier, P., Ehrenstein, U., Leweke, T. & Rossi, M. 2002 A merging criterion for two-dimensional co-rotating vortices. Phys. Fluids 14 (8), 2757–2766.CrossRefGoogle Scholar
Michell, J.H. 1890 On the theory of free stream lines. Phil. Trans. R. Soc. Lond. A 181, 389–431.Google Scholar
Morikawa, G.K. & Swenson, E.V. 1971 Interacting motion of rectilinear geostrophic vortices. Phys. Fluids 14, 1058.CrossRefGoogle Scholar
Nelson, R.B., Krishnamurthy, V.S. & Crowdy, D.G. 2020 The corotating hollow vortex pair: steady merger and break-up via a topological singularity. J. Fluid Mech. 907, A10.CrossRefGoogle Scholar
Newton, P.K. 2001 The N–Vortex Problem. Analytical Techniques. Springer.CrossRefGoogle Scholar
Overman, E.A. 1986 Steady-state solutions of the Euler equations in two dimensions II: local analysis of limiting V-states. SIAM J. Appl. Maths 46, 765–800.CrossRefGoogle Scholar
Pocklington, H.C. 1895 The configuration of a pair of equal and opposite hollow straight vortices of finite cross-section, moving steadily through fluid. Proc. Camb. Phil. Soc. 8, 178–187.Google Scholar
Pullin, D.I. 1992 Contour dynamics methods. Annu. Rev. Fluid Mech. 24, 89–115.CrossRefGoogle Scholar
Saffman, P.G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Saffman, P.G. & Szeto, R. 1980 Equilibrium shapes of a pair of equal uniform vortices. Phys. Fluids 23, 2339.CrossRefGoogle Scholar
Shapiro, H.S. 1992 The Schwarz Function and Its Generalization to Higher Dimensions. John Wiley & Sons.Google Scholar
Tanveer, S. 1996 Some analytical properties of solutions to a two-dimensional steadily translating inviscid bubble. Proc. R. Soc. A 452, 1397–1410.Google Scholar
Wegmann, R. & Crowdy, D.G. 2000 Shapes of two-dimensional bubbles deformed by circulation. Nonlinearity 13, 2131–2141.CrossRefGoogle Scholar
Zannetti, L., Ferlauto, M. & Llewellyn Smith, S.G. 2016 Hollow vortices in shear. J. Fluid Mech. 809, 705–715.CrossRefGoogle Scholar