Skip to main content Accessibility help
×
Home
Hostname: page-component-65dc7cd545-8rn5k Total loading time: 0.218 Render date: 2021-07-25T13:52:58.943Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

The corotating hollow vortex pair: steady merger and break-up via a topological singularity

Published online by Cambridge University Press:  18 November 2020

R. B. Nelson
Affiliation:
Department of Earth Science and Engineering, Imperial College London, London SW7 2AZ, UK
V. S. Krishnamurthy
Affiliation:
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
D. G. Crowdy
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Corresponding
E-mail address:

Abstract

The shapes of two steadily rotating, equal circulation, two-dimensional hollow vortices are determined and their properties examined. By means of a numerical scheme that accounts for the doubly connected nature of the fluid domain, it is shown that a one-parameter family of solutions exists that is a continuation of a corotating point-vortex pair. With $b=2$ set as the distance between the vortex centroids, we find that each vortex reaches a maximum possible area of $0.796$ corresponding to $a/b=0.260$ where $a$ is a measure of the vortex core radius proposed by Meunier et al. (Phys. Fluids, vol. 14, 2002, pp. 2757–2766). Results are compared to those of a previous study by Saffman & Szeto (Phys. Fluids, vol. 23, 1980, pp. 2339–2342) in which two corotating patches of uniform vorticity are considered in place of the hollow vortices studied here. The general behaviour of the two systems is seen to be similar but some differences are highlighted, especially when the vortices become close to touching due to the accumulation of vorticity in thin extended fingers emanating from each of the vortices. The numerical scheme captures the family of equilibria very close to a critical configuration where these fingers tend to touch at the centre of rotation corresponding to $a/b \approx 0.283$. By a simple adaptation of the numerical scheme to compute $2$-fold rotationally symmetric equilibria for a single rotating hollow vortex we then show that its limiting configuration is one where a thin waist forms leading to two separate parts of its single boundary drawing close together. We give evidence that the limit of this single vortex configuration coincides with the limit of the two-vortex configuration. The limiting configuration itself turns out not to be physically admissible, leading to what we refer to as a topological singularity since no physical quantities blow up, indeed they appear to be continuous as the limiting state is approached from the two topologically distinct directions.

Type
JFM Papers
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below.

References

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, 469476.10.1017/S0022112076001894CrossRefGoogle Scholar
Batchelor, G. K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.10.1017/CBO9780511800955CrossRefGoogle Scholar
Crowdy, D. G. 1999 Circulation-induced shape deformations of drops and bubbles: exact two-dimensional models. Phys. Fluids 11 (10), 28362845.10.1063/1.870142CrossRefGoogle Scholar
Crowdy, D. G. 2008 The Schwarz problem in multiply connected domains and the Schottky-Klein prime function. Complex Var. Elliptic 53, 221236.10.1080/17476930701682897CrossRefGoogle Scholar
Crowdy, D. G. 2010 A new calculus for two-dimensional vortex dynamics. Theor. Comput. Fluid Dyn. 24, 924.10.1007/s00162-009-0098-5CrossRefGoogle Scholar
Crowdy, D. G. 2020 Solving Problems in Multiply Connected Domains. Society for Industrial and Applied Mathematics.10.1137/1.9781611976151CrossRefGoogle Scholar
Crowdy, D. G. & Green, C. C. 2011 Analytical solutions for von Kármán streets of hollow vortices. Phys. Fluids 23, 126602.10.1063/1.3665102CrossRefGoogle Scholar
Crowdy, D. G. & Krishnamurthy, V. S. 2018 The effect of core size on the speed of compressible hollow vortex streets. J. Fluid Mech. 836, 797827.10.1017/jfm.2017.821CrossRefGoogle Scholar
Crowdy, D. G. & Marshall, J. S. 2004 Growing vortex patches. Phys. Fluids 16, 31223129.10.1063/1.1767771CrossRefGoogle Scholar
Crowdy, D. G., Llewellyn Smith, S. G. & Freilich, D. V. 2013 Translating hollow vortex pairs. Eur. J. Mech. B/Fluids 37, 180186.10.1016/j.euromechflu.2012.09.007CrossRefGoogle Scholar
Dritschel, D. G. 1985 The stability and energetics of corotating uniform vortices. J. Fluid Mech. 157, 95134.CrossRefGoogle Scholar
Freilich, D. V. & Llewellyn Smith, S. G. 2017 The Sadovskii vortex in strain. J. Fluid Mech. 825, 479501.10.1017/jfm.2017.401CrossRefGoogle Scholar
Leweke, T., Dizés, S. L. & Williamson, C. H. K. 2016 Dynamics and instabilities of vortex pairs. Annu. Rev. Fluid Mech. 48, 507541.10.1146/annurev-fluid-122414-034558CrossRefGoogle Scholar
Llewellyn Smith, S. G. & Crowdy, D. G. 2012 Structure and stability of hollow vortex equilibria. J. Fluid Mech. 691, 178200.10.1017/jfm.2011.467CrossRefGoogle Scholar
Melander, M. V., Zabusky, N. J. & McWilliams, J. C. 1988 Symmetric vortex merger in two dimensions: causes and conditions. J. Fluid Mech. 195, 303340.CrossRefGoogle Scholar
Meunier, P., Ehrenstein, U., Leweke, T. & Rossi, M. 2002 A merging criterion for two-dimensional co-rotating vortices. Phys. Fluids 14, 27572766.10.1063/1.1489683CrossRefGoogle 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, 178187.Google Scholar
Sadovskii, V. S. 1971 Vortex regions in a potential stream with a jump of Bernoulli's constant at the boundary. Z. Angew. Math. Mech. 809, 729735.10.1016/0021-8928(71)90070-0CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Saffman, P. G. & Schatzman, J. C. 1981 Properties of a vortex street of finite vortices. SIAM J. Sci. Stat. Comput. 2 (3), 285295.10.1137/0902023CrossRefGoogle Scholar
Saffman, P. G. & Szeto, R. 1980 Equilibrium shapes of a pair of equal uniform vortices. Phys. Fluids 23, 23392342.CrossRefGoogle Scholar
Tanveer, S. 1986 A steadily translating pair of equal and opposite vortices with vortex sheets on their boundaries. Stud. Appl. Maths 74, 139154.10.1002/sapm1986742139CrossRefGoogle Scholar
Telib, H. & Zannetti, L. 2011 Hollow wakes past arbitrarily shaped obstacles. J. Fluid Mech. 669, 214224.10.1017/S0022112010006154CrossRefGoogle Scholar
Wegmann, R. & Crowdy, D. G. 2000 Shapes of two-dimensional bubbles deformed by circulation. Nonlinearity 13, 21312141.10.1088/0951-7715/13/6/313CrossRefGoogle Scholar
Zannetti, L., Ferlauto, M. & Llewellyn Smith, S. G. 2016 Hollow vortices in shear. J. Fluid Mech. 809, 705715.10.1017/jfm.2016.697CrossRefGoogle Scholar
1
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

The corotating hollow vortex pair: steady merger and break-up via a topological singularity
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

The corotating hollow vortex pair: steady merger and break-up via a topological singularity
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

The corotating hollow vortex pair: steady merger and break-up via a topological singularity
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *