Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-12-01T10:37:02.414Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-dimensional visualization of the interaction of a vortex ring with a stratified interface

Published online by Cambridge University Press:  10 May 2017

Jason Olsthoorn Open the ORCID record for Jason Olsthoorn [Opens in a new window]  and
Stuart B. Dalziel
Jason Olsthoorn*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WA, UK
Stuart B. Dalziel
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WA, UK
*
†Email address for correspondence: jason.olsthoorn@cantab.net
Get access Rights & Permissions [Opens in a new window]

Abstract

The study of vortex-ring-induced stratified mixing has long played a key role in understanding externally forced stratified turbulent mixing. While several studies have investigated the dynamical evolution of such a system, this study presents an experimental investigation of the mechanical evolution of these vortex rings, including the stratification-modified three-dimensional instability. The aim of this paper is to understand how vortex rings induce mixing of the density field. We begin with a discussion of the Reynolds and Richardson number dependence of the vortex-ring interaction using two-dimensional particle image velocimetry measurements. Then, through the use of modern imaging techniques, we reconstruct from an experiment the full three-dimensional time-resolved velocity field of a vortex ring interacting with a stratified interface. This work agrees with many of the previous two-dimensional experimental studies, while providing insight into the three-dimensional instabilities of the system. Observations indicate that the three-dimensional instability has a similar wavenumber to that found for the unstratified vortex-ring instability at later times. We determine that the time scale associated with this instability growth has an inverse Richardson number dependence. Thus, the time scale associated with the instability is different from the time scale of interface recovery, possibly explaining the significant drop in mixing efficiency at low Richardson numbers. The structure of the underlying instability is a simple displacement mode of the vorticity field.

JFM classification

Geophysical and Geological Flows: Stratified flows Vortex Flows: Vortex instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 820 , 10 June 2017 , pp. 549 - 579
Check for updates
Copyright
© 2017 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

Archer, P. J., Thomas, T. G. & Coleman, G. N. 2008 Direct numerical simulation of vortex ring evolution from the laminar to the early turbulent regime. J. Fluid Mech. 598, 201–226.Google Scholar
Archer, P. J., Thomas, T. G. & Coleman, G. N. 2009 The instability of a vortex ring impinging on a free surface. J. Fluid Mech. 642, 79–94.Google Scholar
Atta, C. W. & Hopfinger, E. J. 1989 Vortex ring instability and collapse in a stably stratified fluid. Exp. Fluids 7 (3), 197–200.CrossRefGoogle Scholar
Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 2160–2163.Google Scholar
Bethke, N. & Dalziel, S. B. 2012 Resuspension onset and crater erosion by a vortex ring interacting with a particle layer. Phys. Fluids 24, 063301.Google Scholar
Bristol, R. L., Ortega, J. M., Marcus, P. S. & Savaş, Ö. 2004 On cooperative instabilities of parallel vortex pairs. J. Fluid Mech. 517, 331–358.CrossRefGoogle Scholar
Camassa, R., Khatri, S., Mclaughlin, R., Mertens, K., Nenon, D., Smith, C. & Viotti, C. 2013 Numerical simulations and experimental measurements of dense-core vortex rings in a sharply stratified environment. Comput. Sci. Disc. 6 (1), 014001.Google Scholar
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 2172–2179.Google Scholar
Dahm, W. J. A., Scheil, C. M. & Tryggvason, G. 1989 Dynamics of vortex interaction with a density interface. J. Fluid Mech. 205, 1–43.CrossRefGoogle Scholar
Feng, H., Kaganovskiy, L. & Krasny, R. 2009 Azimuthal instability of a vortex ring computed by a vortex sheet panel method. Fluid Dyn. Res. 41 (5), 051405.Google Scholar
Harris, D. M. & Williamson, C. H. K. 2012 Instability of secondary vortices generated by a vortex pair in ground effect. J. Fluid Mech. 700, 148–186.CrossRefGoogle Scholar
Haynes, W. M. 2012 CRC Handbook of Chemistry and Physics, 93rd edn. Taylor & Francis.Google Scholar
von Helmholtz, H. 1858 Über Integrale der hydrodynamischen Gleichungen, welche der Wirbelbewegung entsprechen. J. Reine Angew. Math. 55, 25–55.Google Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34 (1), 83–113.Google Scholar
Krutzsch, C.-H., Bolster, D., Hershberger, R. & Donnelly, R. J. 1939 On an experimentally observed phenomenon on vortex rings during their translational movement in a real liquid. Ann. Phys. 523 (5), 360–379.Google Scholar
Leweke, T., Dizs, S. L. & Williamson, C. H. K. 2016 Dynamics and instabilities of vortex pairs. Annu. Rev. Fluid Mech. 48 (1), 507–541.Google Scholar
Linden, P. F. 1973 The interaction of a vortex ring with a sharp density interface: a model for turbulent entrainment. J. Fluid Mech. 60, 467–480.Google Scholar
Maxworthy, T. 1977 Some experimental studies of vortex rings. J. Fluid Mech. 81, 465–495.Google Scholar
Mcdougall, T. J. 1979 On the elimination of refractive-index variations in turbulent density-stratified liquid flows. J. Fluid Mech. 93, 83–96.Google Scholar
Moore, D. W. & Saffman, P. G. 1975 The instability of a straight vortex filament in a strain field. Proc. R. Soc. Lond. A 346 (1646), 413–425.Google Scholar
Munro, R. J., Bethke, N. & Dalziel, S. B. 2009 Sediment resuspension and erosion by vortex rings. Phys. Fluids 21 (4), 046601.Google Scholar
Nomura, K. K., Tsutsui, H., Mahoney, D. & Rottman, J. W. 2006 Short-wavelength instability and decay of a vortex pair in a stratified fluid. J. Fluid Mech. 553, 283–322.Google Scholar
Olsthoorn, J. & Dalziel, S. B. 2015 Vortex-ring-induced stratified mixing. J. Fluid Mech. 781, 113–126.Google Scholar
Orlandi, P. & Verzicco, R. 1993 Vortex rings impinging on walls: axisymmetric and three-dimensional simulations. J. Fluid Mech. 256, 615–646.CrossRefGoogle Scholar
Ortiz, S., Donnadieu, C. & Chomaz, J.-M. 2015 Three-dimensional instabilities and optimal perturbations of a counter-rotating vortex pair in stratified flows. Phys. Fluids 27 (10), 106603.CrossRefGoogle Scholar
Pierrehumbert, R. T. 1986 Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Lett. 57, 2157–2159.Google Scholar
Ponitz, B., Sastuba, M. & Brücker, C. 2015 4D visualization study of a vortex ring life cycle using modal analyses. J. Vis. 19, 1–23.Google Scholar
Saffman, P. G. 1978 The number of waves on unstable vortex rings. J. Fluid Mech. 84, 625–639.Google Scholar
Scase, M. M. & Dalziel, S. B. 2006 An experimental study of the bulk properties of vortex rings translating through a stratified fluid. Eur. J. Mech. (B/Fluids) 25 (3), 302–320.Google Scholar
Shariff, K. & Leonard, A. 1992 Vortex rings. Annu. Rev. Fluid Mech. 24 (1), 235–279.Google Scholar
Stock, M. J., Dahm, W. J. A. & Tryggvason, G. 2008 Impact of a vortex ring on a density interface using a regularized inviscid vortex sheet method. J. Comput. Phys. 227 (21), 9021–9043.Google Scholar
Swearingen, J. D., Crouch, J. D. & Handler, R. A. 1995 Dynamics and stability of a vortex ring impacting a solid boundary. J. Fluid Mech. 297, 1–28.Google Scholar
Tsai, C.-Y. & Widnall, S. E. 2006 The stability of short waves on a straight vortex filament in a weak externally imposed strain field. J. Fluid Mech. 73 (4), 721–733.Google Scholar
Turner, J. S. 1957 Buoyant vortex rings. Proc. R. Soc. Lond. A 239 (1216), 61–75.Google Scholar
Widnall, S. E., Bliss, D. B. & Tsai, C.-Y. 1974 The instability of short waves on a vortex ring. J. Fluid Mech. 66, 35–47.Google Scholar
Widnall, S. E. & Sullivan, J. P. 1973 On the stability of vortex rings. Proc. R. Soc. Lond. A 332 (1590), 335–353.Google Scholar
Widnall, S. E. & Tsai, C.-Y. 1977 The instability of the thin vortex ring of constant vorticity. Phil. Trans. R. Soc. Lond. A 287 (1344), 273–305.Google Scholar