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Stirring, stretching and transport generated by a pair of like-signed vortices

Published online by Cambridge University Press:  30 March 2011

F. RIZZI  and
L. CORTELEZZI
F. RIZZI
Affiliation:
Department of Physics, University of Udine, 33100 Udine, Italy
L. CORTELEZZI*
Affiliation:
Department of Mechanical Engineering, McGill University, Montreal, Quebec H3A 2K6, Canada
*
‡Email address for correspondence: [email protected]
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Abstract

We consider a pair of like-signed, initially elliptical vortices with uniform vorticity distribution embedded in an incompressible, inviscid fluid occupying a two-dimensional, infinite domain. We characterize this finite-time, aperiodic, dynamical system in terms of its fixed points and separatrices, which divide the flow into inner core, inner recirculation, outer recirculation regions and outer flow. We numerically simulate the time evolution of the vortex pair using a contour dynamics algorithm. The rotational and co-rotational motion of the vortices perturb the separatrices, which undergo to deformations, yielding a tangle whose complexity increases as the amplitude of the perturbation increases. We analyse the dynamics of the tangle and explain the transport of fluid between different regions. We use two diagnostics to quantify stirring: stretching of the interface and the mix-norm. These two diagnostics characterize stirring in contradicting ways and present different sensitivity to the parameters considered. We find that stretching is dominated by the chaotic advection induced within the inner core and inner recirculation regions, whereas the mix-norm is dominated by the laminar transport induced within the outer recirculation regions. For pairs of vortices of small aspect ratio, stretching is piecewise linear and the mix-norm does not decrease monotonically. We show that these two effects are strongly coupled and synchronized with the rotational motion of the vortices. Since the nominal domain is unbounded, we quantify stirring on three concentric, circular domains. One domain nearly encloses the outer separatrices of the vortex pair, one is smaller and one larger than the first one. We show that the mix-norm is very sensitive to the size of the domain, while stretching is not. To quantify the sensitivity of stirring to the geometry of the initial concentration field, we consider, as an initial scalar field, two concentrations delimited by a straight-line interface of adjustable orientation. We show that the interface passing through the centroids of the vortices is the one most efficiently stretched, while the initial concentration field with an orthogonal interface is the most efficiently stirred. Finally, we investigate the effects of the angular impulse on the stirring performance of the vortex pair. Stretching is very sensitive to the angular impulse, while the mix-norm is not. We show that there is a value of the angular impulse which maximizes stretching and argue that this is due to two competing mechanisms.

JFM classification

Mixing: Mixing Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Vortex Flows: Vortex Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 674 , 10 May 2011 , pp. 244 - 280
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Aguirre, R. C., Nathman, J. C. & Catrakis, H. C. 2006 Flow geometry effects on the turbulent mixing efficiency. J. Fluids Engng 128, 874–879.CrossRefGoogle Scholar
Aref, H. 1984 Stirring by chaotic advection. J. Fluid Mech. 143, 1–21.CrossRefGoogle Scholar
Beigie, D., Leonard, A. & Wiggins, S. 1991 Chaotic transport in the homoclinic and heteroclinic tangle regions of quasiperiodically forced two-dimensional dynamical systems. Nonlinearity 4, 775–819.CrossRefGoogle Scholar
Beigie, D., Leonard, A. & Wiggins, S. 1994 Invariant manifold templates for chaotic advection. Chaos, Solitons Fractals 4, 749–868.CrossRefGoogle Scholar
Cerretelli, C. & Williamson, C. H. K. 2003 a The physical mechanism for vortex merging. J. Fluid Mech. 475, 41–77.CrossRefGoogle Scholar
Cerretelli, C. & Williamson, C. H. K. 2003 b A new family of uniform vortices related to vortex configurations before merging. J. Fluid Mech. 493, 219–229.CrossRefGoogle Scholar
Chakravarthy, V. S. & Ottino, J. M. 1996 Mixing of two viscous fluids in a rectangular cavity. Chem. Engng Sci. 51, 3613–3622.CrossRefGoogle Scholar
Deneve, J. A., Fröhlich, J. & Bockhorn, H. 2009 Large eddy simulation of a swirling transverse jet into a crossflow with investigation of scalar transport. Phys. Fluids 21 (1), 015101.CrossRefGoogle Scholar
Dimotakis, P. E. 2001 Experiments, phenomenology, and simulations of turbulence and turbulent mixing. APS Meeting Abstracts, p. 2004.Google Scholar
Dimotakis, P. E. 2005 Turbulent mixing. Annu. Rev. Fluid Mech. 37, 329–356.CrossRefGoogle Scholar
Dritschel, D. G. 1985 The stability and energetics of corotating uniform vortices. J. Fluid Mech. 157, 95–134.CrossRefGoogle Scholar
Dritschel, D. G. & Zabusky, N. J. 1996 On the nature of vortex interactions and models in unforced nearly-inviscid two-dimensional turbulence. Phys. Fluids 8, 1252–1256.CrossRefGoogle Scholar
Eckart, C. 1948 An analysis of the stirring and mixing processes in incompressible fluids. J. Mar. Res. 7, 265–275.Google Scholar
Elhmaidi, D., Provenzale, A. & Babiano, A. 1993 Elementary topology of two-dimensional turbulence from a Lagrangian viewpoint and single-particle dispersion. J. Fluid Mech. 257, 533–558.CrossRefGoogle Scholar
Estevadeordal, J. & Kleis Stanly, J. 2002 Influence of vortex-pairing location on the three-dimensional evolution of plane mixing layers. J. Fluid Mech. 462, 43–77.CrossRefGoogle Scholar
Flohr, P. & Vassilicos, J. C. 1997 Accelerated scalar dissipation in a vortex. J. Fluid Mech. 348, 295–317.CrossRefGoogle Scholar
Franjione, J. G. & Ottino, J. M. 1987 Feasibility of numerical tracking on material lines and surfaces in chaotic flows. Phys. Fluids 30, 3641–3643.CrossRefGoogle Scholar
Fuentes, O. U. V. 2001 Chaotic advection by two interacting finite-area vortices. Phys. Fluids 13, 901–912.CrossRefGoogle Scholar
Fuentes, O. U. V. 2005 Vortex filamentation: its onset and its role on axisymmetrization and merger. Dyn. Atmos. Oceans 40, 23–42.CrossRefGoogle Scholar
Haller, G. 2001 Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D 149 (4), 248–277.CrossRefGoogle Scholar
Huang, M.-J. 2005 The physical mechanism of symmetric vortex merger: a new viewpoint. Phys. Fluids 17 (7), 074105.CrossRefGoogle Scholar
Josserand, C. & Rossi, M. 2007 The merging of two co-rotating vortices: a numerical study. Eur. J. Mech. B: Fluids 26, 779–794.CrossRefGoogle Scholar
Khakhar, D. V. & Ottino, J. M. 1986 Fluid mixing (stretching) by time periodic sequences for weak flows. Phys. Fluids 29, 3503–3505.CrossRefGoogle Scholar
Khakhar, D. V., Rising, H. & Ottino, J. M. 1986 Analysis of chaotic mixing in two model systems. J. Fluid Mech. 172, 419–451.CrossRefGoogle Scholar
Le Dizes, S. & Verga, A. 2002 Viscous interactions of two co-rotating vortices before merging. J. Fluid Mech. 467, 389–410.CrossRefGoogle Scholar
Leonard, A., Rom-Kedar, V. & Wiggins, S. 1987 Fluid mixing and dynamical systems. Nucl. Phys. B. 2, 179–190.CrossRefGoogle Scholar
Leonard, A. D. & Hill, J. C. 1992 Mixing and chemical reaction in sheared and nonsheared homogeneous turbulence. Fluid Dyn. Res. 10, 273–297.CrossRefGoogle Scholar
Mathew, G., Mezic, I., Grivopoulos, S., Vaidya, U. & Petzold, L. 2007 Optimal control of mixing in Stokes fluid flows. J. Fluid Mech. 580, 261–281.CrossRefGoogle Scholar
Mathew, G., Mezic, I. & Petzold, L. 2005 A multiscale measure for mixing. Physica D 211 (1–2), 23–46.CrossRefGoogle Scholar
McWilliams, J. C. 1990 The vortices of two-dimensional turbulence. J. Fluid Mech. 219, 361–385.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, 303–340.CrossRefGoogle Scholar
Melander, M. V., Zabusky, N. J. & Styczek, A. S. 1986 A moment model for vortex interactions of the two-dimensional euler equations. Part 1. Computational validation of a Hamiltonian elliptical representation. J. Fluid Mech. 167, 95–115.CrossRefGoogle Scholar
Meunier, P., Le Dizès, S. & Leweke, T. 2005 Physics of vortex merging. C. R. Phys. 6, 431–450.Google Scholar
Meunier, P. & Villermaux, E. 2003 How vortices mix. J. Fluid Mech. 476, 213–222.CrossRefGoogle Scholar
Muzzio, F. J., Swanson, P. D. & Ottino, J. M. 1991 The statistics of stretching and stirring in chaotic flows. Phys. Fluids A 3 (5), 822–834.CrossRefGoogle Scholar
Nybelen, L. & Paoli, R. 2009 Direct and large-eddy simulations of merging in corotating vortex system. Am. Inst. Aeronaut. Astronaut. J. 47, 157–167.CrossRefGoogle Scholar
Ottino, J. M. 1982 Description of mixing with diffusion and reaction in terms of the concept of material surfaces. J. Fluid Mech. 114, 83–103.CrossRefGoogle Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.Google Scholar
Ottino, J. M. 1990 Mixing, chaotic advection and turbulence. Annu. Rev. Fluid Mech. 22, 207–253.CrossRefGoogle Scholar
Padberg, K., Thiere, B., Preis, R. & Dellnitz, M. 2009 Local expansion concepts for detecting transport barriers in dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 14, 4176–4190.CrossRefGoogle Scholar
Poje, A. C. & Haller, G. 1999 Geometry of cross-stream Lagrangian mixing in a double gyre ocean model. J. Phys. Oeanogr. 29, 1649–1665.2.0.CO;2>CrossRefGoogle Scholar
Pozrikidis, C. 2001 Fluid Dynamics: Theory, Computation, and Numerical Simulation. Kluwer.CrossRefGoogle Scholar
Provenzale, A. 1999 Transport by coherent barotropic vortices. Annu. Rev. Fluid Mech. 31, 55–93.CrossRefGoogle Scholar
Pullin, D. I. 1992 Contour dynamics methods. Annu. Rev. Fluid Mech. 24, 89–115.CrossRefGoogle Scholar
Rasmussen, J. J., Nielsen, A. H. & Naulin, V. 2001 Dynamics of vortex interactions in two-dimensional flows. Phys. Scr. T 98, 29–33.CrossRefGoogle Scholar
Rizzi, F. 2009 Characterization of stirring generated by a pair of elliptic vortices. Master's thesis, University of Udine, Italy.Google Scholar
Rom-Kedar, V., Leonard, A. & Wiggins, S. 1990 An analytical study of transport, mixing and chaos in an unsteady vortical flow. J. Fluid Mech. 214, 347–394.CrossRefGoogle Scholar
Rom-Kedar, V. & Wiggins, S. 1990 Transport in two-dimensional maps. Arch. Rat. Mech. Anal. 109, 239–298.CrossRefGoogle Scholar
Rom-Kedar, V. & Wiggins, S. 1991 Transport in two-dimensional maps: concepts, examples, and a comparison of the theory of Rom-Kedar and Wiggins with the Markov model of Mackay, Meiss, Ott and Percival. Physica D 51 (1–3), 248–266.CrossRefGoogle Scholar
Sau, R. & Mahesh, K. 2007 Passive scalar mixing in vortex rings. J. Fluid Mech. 582, 449–461.CrossRefGoogle Scholar
Shadden, S. C., Katija, K., Rosenfeld, M., Marsden, J. E. & Dabiri, J. O. 2007 Transport and stirring induced by vortex formation. J. Fluid Mech. 593, 315–331.CrossRefGoogle Scholar
Slessor, M. D., Bond, C. L. & Dimotakis, P. E. 1998 Turbulent shear-layer mixing at high Reynolds numbers: effects of inflow conditions. J. Fluid Mech. 376, 115–138.CrossRefGoogle Scholar
Smyth, W. D., Moum, J. N. & Caldwell, D. R. 2001 The efficiency of mixing in turbulent patches: inferences from direct simulations and microstructure observations. J. Phys. Oceanogr. 31, 1969–1992.2.0.CO;2>CrossRefGoogle Scholar
Vainchtein, D. 2005 Private communication.Google Scholar
Vassilicos, J. C. 2002 Mixing in vortical, chaotic and turbulent flows. Phil. Trans. R. Soc. Lond. A 360, 2819–2837.CrossRefGoogle ScholarPubMed
Villermaux, E. & Duplat, J. 2003 Mixing as an aggregation process. Phys. Rev. Lett. 91 (18), 184501.CrossRefGoogle ScholarPubMed
VonHardenberg, J. Hardenberg, J., McWilliams, J. C., Provenzale, A., Shchepetkin, A. & Weiss, J. B. 2000 Vortex merging in quasi-geostrophic flows. J. Fluid Mech. 412, 331–353.Google Scholar
Warhaft, Z. 2000 Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32, 203–240.CrossRefGoogle Scholar
Wiggins, S. 1987 Chaos in the quasiperiodically forced duffing oscillator. Phys. Lett. A 124, 138–142.CrossRefGoogle Scholar
Witt, A., Braun, R., Feudel, F., Grebogi, C. & Kurths, J. 1999 Tracer dynamics in a flow of driven vortices. Phys. Rev. E 59, 1605–1614.CrossRefGoogle Scholar
Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A. 1985 Determining Lyapunov exponents from a time series. Physica D 16, 285–317.CrossRefGoogle Scholar
Yasuda, I. 1997 Two-dimensional asymmetric vortex merger: merger dynamics and critical merger distance. Dyn. Atmos. Oceans 26, 159–181.CrossRefGoogle Scholar
Zabusky, N. J., Hughes, M. H. & Roberts, K. V. 1979 Contour dynamics for the Euler equations in two dimensions. J. Comput. Phys. 30, 96–106.CrossRefGoogle Scholar