Skip to main content

A quantitative assessment of viscous asymmetric vortex pair interactions

  • Patrick J. R. Folz (a1) and Keiko K. Nomura (a1)

The interactions of two like-signed vortices in viscous fluid are investigated using two-dimensional numerical simulations performed across a range of vortex strength ratios, $\unicode[STIX]{x1D6EC}=\unicode[STIX]{x1D6E4}_{1}/\unicode[STIX]{x1D6E4}_{2}\leqslant 1$ , corresponding to vortices of circulation, $\unicode[STIX]{x1D6E4}_{i}$ , with differing initial size and/or peak vorticity. In all cases, the vortices evolve by viscous diffusion before undergoing a primary convective interaction, which ultimately results in a single vortex. The post-interaction vortex is quantitatively evaluated in terms of an enhancement factor, $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D6E4}_{end}/\unicode[STIX]{x1D6E4}_{2,start}$ , which compares its circulation, $\unicode[STIX]{x1D6E4}_{end}$ , to that of the stronger starting vortex, $\unicode[STIX]{x1D6E4}_{2,start}$ . Results are effectively characterized by a mutuality parameter, $MP\equiv (S/\unicode[STIX]{x1D714})_{1}/(S/\unicode[STIX]{x1D714})_{2}$ , where the ratio of induced strain rate, $S$ , to peak vorticity, $\unicode[STIX]{x1D714}$ , for each vortex, $(S/\unicode[STIX]{x1D714})_{i}$ , is found to have a critical value, $(S/\unicode[STIX]{x1D714})_{cr}\approx 0.135$ , above which core detrainment occurs. If $MP$ is sufficiently close to unity, both vortices detrain and a two-way mutual entrainment process leads to $\unicode[STIX]{x1D700}>1$ , i.e. merger. In asymmetric interactions and mergers, generally one vortex dominates; the weak/no/strong vortex winner regimes correspond to $MP<,=,>1$ , respectively. As $MP$ deviates from unity, $\unicode[STIX]{x1D700}$ decreases until a critical value, $MP_{cr}$ is reached, beyond which there is only a one-way interaction; one vortex detrains and is destroyed by the other, which dominates and survives. There is no entrainment and $\unicode[STIX]{x1D700}\sim 1$ , i.e. only a straining out occurs. Although $(S/\unicode[STIX]{x1D714})_{cr}$ appears to be independent of Reynolds number, $MP_{cr}$ shows a dependence. Comparisons are made with available experimental data from Meunier (2001, PhD thesis, Université de Provence-Aix-Marseille I).

Corresponding author
Email address for correspondence:
Hide All
Brandt, L. K. & Nomura, K. K. 2006 The physics of vortex merger: further insight. Phys. Fluids 18 (5), 051701.
Brandt, L. K. & Nomura, K. K. 2007 The physics of vortex merger and the effects of ambient stable stratification. J. Fluid Mech. 592, 413446.
Brandt, L. K. & Nomura, K. K. 2010 Characterization of the interactions of two unequal co-rotating vortices. J. Fluid Mech. 646, 233253.
Carnevale, G. F., McWilliams, J. C., Pomeau, Y., Weiss, J. B. & Young, W. R. 1991 Evolution of vortex statistics in two-dimensional turbulence. Phys. Rev. Lett. 66 (21), 27352737.
Carnevale, G. F., McWilliams, J. C., Pomeau, Y., Weiss, J. B. & Young, W. R. 1992 Rates, pathways, and end states of nonlinear evolution in decaying two-dimensional turbulence: scaling theory versus selective decay. Phys. Fluids A 4 (6), 13141316.
Cerretelli, C. & Williamson, C. H. K. 2003 The physical mechanism for vortex merging. J. Fluid Mech. 475, 4177.
Dritschel, D. G., Scott, R. K., Macaskill, C., Gottwald, G. A. & Tran, C. V. 2008 Unifying scaling theory for vortex dynamics in two-dimensional turbulence. Phys. Rev. Lett. 101 (9), 094501.
Dritschel, D. G. & Waugh, D. W. 1992 Quantification of the inelastic interaction of unequal vortices in two-dimensional vortex dynamics. Phys. Fluids A 4 (8), 17371744.
Folz, P. J. R. & Nomura, K. K. 2014 Interaction of two equal co-rotating viscous vortices in the presence of background shear. Fluid Dyn. Res. 46 (3), 031423.
Gerz, T., Schumann, U. & Elghobashi, S. E. 1989 Direct numerical simulation of stratified homogeneous turbulent shear flows. J. Fluid Mech. 200, 563594.
Huang, M. J. 2005 The physical mechanism of symmetric vortex merger: a new viewpoint. Phys. Fluids 17 (7), 074105.
Huang, M. J. 2006 A comparison between asymmetric and symmetric vortex mergers. WSEAS Trans. Fluid Mech. 1 (5), 488496.
Hunt, J. C. R., Wray, A. A. & Moin, P.1988 Eddies, streams, and convergence zones in turbulent flows. Center for Turbulence Research Report CTR-S88.
Jing, F., Kanso, E. & Newton, P. K. 2012 Insights into symmetric and asymmetric vortex mergers using the core growth model. Phys. Fluids 24 (7), 073101.
Kimura, Y. & Herring, J. R. 2001 Gradient enhancement and filament ejection for a non-uniform elliptic vortex in two-dimensional turbulence. J. Fluid Mech. 439, 4356.
Le Dizès, S. & Laporte, F. 2002 Theoretical predictions for the elliptical instability in a two-vortex flow. J. Fluid Mech. 471, 169201.
Le Dizès, S. & Verga, A. 2002 Viscous interactions of two co-rotating vortices before merging. J. Fluid Mech. 467, 389410.
Mariotti, A., Legras, B. & Dritschel, D. G. 1994 Vortex stripping and the erosion of coherent structures in two-dimensional flows. Phys. Fluids 6 (12), 39543962.
Maze, G., Carton, X. & Lapeyre, G. 2004 Dynamics of a 2D vortex doublet under external deformation. Regular Chaotic Dyn. 9 (4), 477497.
McWilliams, J. C. 1990 The vortices of two-dimensional turbulence. J. Fluid Mech. 219, 361385.
Melander, M. V., McWilliams, J. C. & Zabusky, N. J. 1987a Axisymmetrization and vorticity-gradient intensification of an isolated two-dimensional vortex through filamentation. J. Fluid Mech. 178, 137159.
Melander, M. V., Zabusky, N. J. & McWilliams, J. C. 1987b Asymmetric vortex merger in two dimensions: which vortex is victorious? Phys. Fluids 30 (9), 26102612.
Melander, M. V., Zabusky, N. J. & McWilliams, J. C. 1988 Symmetric vortex merger in two dimensions: causes and conditions. J. Fluid Mech. 195, 303340.
Meunier, P.2001 Etude expérimentale de deux tourbillons corotatifs. PhD thesis, Université de Provence-Aix-Marseille I.
Meunier, P., Ehrenstein, U., Leweke, T. & Rossi, M. 2002 A merging criterion for two-dimensional co-rotating vortices. Phys. Fluids 14 (8), 27572766.
Orlandi, P. 2000 Fluid Flow Phenomena: A Numerical Toolkit. Fluid Flow Phenomena: A Numerical Toolkit, vol. 1. Springer.
Overman, E. A. II & Zabusky, N. J. 1982 Evolution and merger of isolated vortex structures. Phys. Fluids 25 (8), 12971305.
Perrot, X. & Carton, X. 2010 2D vortex interaction in a non-uniform flow. Theoret. Comput. Fluid Dyn. 24 (1–4), 95100.
Riccardi, G., Piva, R. & Benzi, R. 1995 A physical model for merging in two-dimensional decaying turbulence. Phys. Fluids 7 (12), 30913104.
Saffman, P. G. 2001 Vortex Dynamics. Cambridge University Press.
Saffman, P. G. & Szeto, R. 1980 Equilibrium shapes of a pair of equal uniform vortices. Phys. Fluids 23 (12), 23392342.
Sire, C., Chavanis, P. H. & Sopik, J. 2011 Effective merging dynamics of two and three fluid vortices: application to two-dimensional decaying turbulence. Phys. Rev. E 84 (5), 056317.
Tabeling, P. 2002 Two-dimensional turbulence: a physicist approach. Phys. Rep. 362 (1), 162.
Trieling, R. R., Dam, C. E. C. & van Heijst, G. J. F. 2010 Dynamics of two identical vortices in linear shear. Phys. Fluids 22 (11), 117104.
Trieling, R. R., Velasco Fuentes, O. U. & van Heijst, G. J. F. 2005 Interaction of two unequal corotating vortices. Phys. Fluids 17 (8), 087103.
Waugh, D. W. 1992 The efficiency of symmetric vortex merger. Phys. Fluids A 4 (8), 17451758.
Yasuda, I. & Flierl, G. R. 1997 Two-dimensional asymmetric vortex merger: merger dynamics and critical merger distance. Dyn. Atmos. Oceans 26 (3), 159181.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


Full text views

Total number of HTML views: 22
Total number of PDF views: 301 *
Loading metrics...

Abstract views

Total abstract views: 453 *
Loading metrics...

* Views captured on Cambridge Core between 14th September 2017 - 20th July 2018. This data will be updated every 24 hours.