Skip to main content Accessibility help
×
Home

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

  • Jason Olsthoorn (a1) and Stuart B. Dalziel (a1)

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.

Copyright

Corresponding author

Email address for correspondence: jason.olsthoorn@cantab.net

References

Hide All
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, 201226.
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, 7994.
Atta, C. W. & Hopfinger, E. J. 1989 Vortex ring instability and collapse in a stably stratified fluid. Exp. Fluids 7 (3), 197200.
Bayly, B. J. 1986 Three-dimensional instability of elliptical flow. Phys. Rev. Lett. 57, 21602163.
Bethke, N. & Dalziel, S. B. 2012 Resuspension onset and crater erosion by a vortex ring interacting with a particle layer. Phys. Fluids 24, 063301.
Bristol, R. L., Ortega, J. M., Marcus, P. S. & Savaş, Ö. 2004 On cooperative instabilities of parallel vortex pairs. J. Fluid Mech. 517, 331358.
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.
Crow, S. C. 1970 Stability theory for a pair of trailing vortices. AIAA J. 8, 21722179.
Dahm, W. J. A., Scheil, C. M. & Tryggvason, G. 1989 Dynamics of vortex interaction with a density interface. J. Fluid Mech. 205, 143.
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.
Harris, D. M. & Williamson, C. H. K. 2012 Instability of secondary vortices generated by a vortex pair in ground effect. J. Fluid Mech. 700, 148186.
Haynes, W. M. 2012 CRC Handbook of Chemistry and Physics, 93rd edn. Taylor & Francis.
von Helmholtz, H. 1858 Über Integrale der hydrodynamischen Gleichungen, welche der Wirbelbewegung entsprechen. J. Reine Angew. Math. 55, 2555.
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34 (1), 83113.
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), 360379.
Leweke, T., Dizs, S. L. & Williamson, C. H. K. 2016 Dynamics and instabilities of vortex pairs. Annu. Rev. Fluid Mech. 48 (1), 507541.
Linden, P. F. 1973 The interaction of a vortex ring with a sharp density interface: a model for turbulent entrainment. J. Fluid Mech. 60, 467480.
Maxworthy, T. 1977 Some experimental studies of vortex rings. J. Fluid Mech. 81, 465495.
Mcdougall, T. J. 1979 On the elimination of refractive-index variations in turbulent density-stratified liquid flows. J. Fluid Mech. 93, 8396.
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), 413425.
Munro, R. J., Bethke, N. & Dalziel, S. B. 2009 Sediment resuspension and erosion by vortex rings. Phys. Fluids 21 (4), 046601.
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, 283322.
Olsthoorn, J. & Dalziel, S. B. 2015 Vortex-ring-induced stratified mixing. J. Fluid Mech. 781, 113126.
Orlandi, P. & Verzicco, R. 1993 Vortex rings impinging on walls: axisymmetric and three-dimensional simulations. J. Fluid Mech. 256, 615646.
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.
Pierrehumbert, R. T. 1986 Universal short-wave instability of two-dimensional eddies in an inviscid fluid. Phys. Rev. Lett. 57, 21572159.
Ponitz, B., Sastuba, M. & Brücker, C. 2015 4D visualization study of a vortex ring life cycle using modal analyses. J. Vis. 19, 123.
Saffman, P. G. 1978 The number of waves on unstable vortex rings. J. Fluid Mech. 84, 625639.
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), 302320.
Shariff, K. & Leonard, A. 1992 Vortex rings. Annu. Rev. Fluid Mech. 24 (1), 235279.
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), 90219043.
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, 128.
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), 721733.
Turner, J. S. 1957 Buoyant vortex rings. Proc. R. Soc. Lond. A 239 (1216), 6175.
Widnall, S. E., Bliss, D. B. & Tsai, C.-Y. 1974 The instability of short waves on a vortex ring. J. Fluid Mech. 66, 3547.
Widnall, S. E. & Sullivan, J. P. 1973 On the stability of vortex rings. Proc. R. Soc. Lond. A 332 (1590), 335353.
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), 273305.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed