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The merger of two-dimensional radially stratified high-Froude-number vortices

Published online by Cambridge University Press:  14 June 2007

ENSICA, 1 Place Émile Blouin, 31056 Toulouse, France
Mathematical Institute, University of St Andrews, KY16 9SS, St Andrews, UK


We investigate the influence of density inhomogeneities on the merger of two corotating two-dimensional vortices at infinite Froude number. In this situation, buoyancy effects are negligible, yet density variations still affect the flow by pure inertial effects through the baroclinic torque. We first re-address the effects of a finite Reynolds number on the interaction between two identical Gaussian vortices. Then, by means of direct numerical simulations, we show that vortices transporting light fluid in a heavier counterpart merge from further distances than vortices in a uniform density medium. On the other hand, heavy vortices only merge from small separation distances. We measure the critical distance a/b0 of the vortex radii to their initial separation distance. It departs from the homogeneous threshold of 0.22 in response to increasing density contrasts between the vortices and their surroundings. An analysis of the contribution of the baroclinic vorticity to the dynamics of the flow is detailed and explains the observed behaviour. This analysis is completed by a simple model based on point vortices that mimics the flow. It is concluded that vortices carrying light fluid are more likely to generate large-scale structures than heavy ones in an inhomogeneous fluid.

Copyright © Cambridge University Press 2007

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Bell, J. B., Colella, P. & Glaz, H. 1989 A 2nd-order projection method for the incompressible Navier–Stokes equations. J. Comput. Phys. 85, 257283.CrossRefGoogle Scholar
Bird, R., Stewart, W. & Lightfoot, E. 1960 Transport Phenomena. Wiley.Google Scholar
Bretonnet, L., Joly, L. & Chassaing, P. 2002 Direct numerical simulation of inertia sensitive turbulence. In Eur. Turbulence Conf. 9. Southampton, UK.Google Scholar
Cardoso, O., Gluckmann, B., Parcollet, O. & Tabeling, P. 1996 Dispersion in a quasi-two-dimensional-turbulent flow: an experimental study. Phys. Fluids 8 (1), 209214.CrossRefGoogle Scholar
Cerretelli, C. & Williamson, C. 2003 The physical mechanism for vortex merging. J. Fluid Mech. 475, 4177.CrossRefGoogle Scholar
Chorin, A. J. 1969 On the convergence of discrete approximations to the Navier–Stokes equations. Maths Comput. 23, 341353.CrossRefGoogle Scholar
Dritschel, D. G. 1995 A general theory for two-dimensional vortex interactions. J. Fluid Mech. 293, 269303.CrossRefGoogle Scholar
Dritschel, D. G. & Waugh, D. W. 1992 Qualification of the inelastic interaction of unequal vortices in two-dimensional vortex dynamics. Phys. Fluids A 4, 17371744.CrossRefGoogle Scholar
Jiménez, J., Moffatt, H. K. & Vasco, C. 1996 The structure of the vortices in freely decaying two-dimensional turbulence. J. Fluid Mech. 313, 209222.CrossRefGoogle Scholar
Joly, L. 2002 Inertia effects in variable density flows. Habilitation à diriger des recherches, INPT, Toulouse.Google Scholar
Joly, L., Fontane, J. & Chassaing, P. 2005 The Rayleigh–Taylor instability of two-dimensional high-density vortices. J. Fluid Mech. 537, 415431.CrossRefGoogle Scholar
Joseph, D. 1990 Fluid dynamics of two miscible liquids with diffusion and gradient stresses. Eur. J. Mech. B/Fluids 9, 565596.Google Scholar
Kurzweil, Y., Livne, E. & Meerson, B. 2003 Vorticity production and turbulent cooling of ‘hot channels’ in gases: three-dimensions versus two dimensions. Phys. Fluids 15 (3), 752762.CrossRefGoogle Scholar
Legras, B., Dritchel, D. & Caillol, P. 2001 The erosion of a distributed two-dimensional vortex in a background straining flow. J. Fluid Mech. 441, 369398.CrossRefGoogle 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 (8), 27572766.CrossRefGoogle Scholar
Meunier, P., Le Dizès, S., & Leweke, T. 2005 Physics of vortex merging. C. R. Phys. 6, 431450.Google Scholar
Nicoud, F. 1998 Numerical study of a channel flow with variable properties. Center for Turbulence Res., Annu. Res. Briefs, pp. 289310.Google Scholar
Paoli, R., Laporte, F. & Cuenot, B. 2003 Dynamics and mixing in jet/vortex interactions. Phys. Fluids 15 (7), 18431860.CrossRefGoogle Scholar
Peters, N. 2000 Turbulent Combustion. Cambridge University Press.CrossRefGoogle Scholar
Pradeep, D. & Hussain, F. 2004 Effects of boundary conditions in numerical simulations of vortex dynamics. J. Fluid Mech. 516, 115124.CrossRefGoogle Scholar
Rogberg, P. & Dritschel, D. 2000 Mixing in two-dimensional vortex interactions. Phys. Fluids 12 (12), 32853288.CrossRefGoogle Scholar
Saffman, P. G. & Szeto, R. 1980 Equilibrium shapes of a pair of equal uniform vortices. Phys. Fluids 23 (12), 23392342.CrossRefGoogle Scholar
Sandoval, D. 1995 The dynamics of variable-density turbulence. PhD thesis, University of Washington.CrossRefGoogle Scholar
Waugh, D. W. 1992 The efficiency of symmetric vortex merger. Phys. Fluids} A 4 (8), 17451758.CrossRefGoogle Scholar