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Turbulent pair dispersion as a continuous-time random walk

Published online by Cambridge University Press:  14 August 2014

Simon Thalabard ,
Giorgio Krstulovic  and
Jérémie Bec
Simon Thalabard
Affiliation:
Laboratoire Lagrange UMR 7293, Université de Nice-Sophia Antipolis, CNRS, Observatoire de la Côte d’Azur, Bd. de l’Observatoire, 06300 Nice, France
Giorgio Krstulovic
Affiliation:
Laboratoire Lagrange UMR 7293, Université de Nice-Sophia Antipolis, CNRS, Observatoire de la Côte d’Azur, Bd. de l’Observatoire, 06300 Nice, France
Jérémie Bec*
Affiliation:
Laboratoire Lagrange UMR 7293, Université de Nice-Sophia Antipolis, CNRS, Observatoire de la Côte d’Azur, Bd. de l’Observatoire, 06300 Nice, France
*
†Email address for correspondence: [email protected]
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Abstract

The phenomenology of turbulent relative dispersion is revisited. A heuristic scenario is proposed, in which pairs of tracers undergo a succession of independent ballistic separations during time intervals whose lengths fluctuate. This approach suggests that the logarithm of the distance between tracers self-averages and performs a continuous-time random walk. This leads to specific predictions for the probability distribution of separations, which differ from those obtained using scale-dependent eddy-diffusivity models (e.g. in the framework of Richardson’s approach). These predictions are tested against high-resolution simulations and shed new light on the explosive separation between tracers.

JFM classification

Mixing: Mixing Turbulent Flows: Turbulent Flows Mixing: Turbulent mixing
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 755 , 25 September 2014 , R4
Copyright
© 2014 Cambridge University Press 

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References

Batchelor, G. K. 1950 The application of the similarity theory of turbulence to atmospheric diffusion. Q. J. R. Meteorol. Soc. 76, 133–146.Google Scholar
Benveniste, D. & Drivas, T. D. 2014 Asymptotic results for backwards two-particle dispersion in a turbulent flow. Phys. Rev. E 89, 041003.Google Scholar
Berg, J., Lüthi, B., Mann, J. & Ott, S. 2006 Backwards and forwards relative dispersion in turbulent flow: an experimental investigation. Phys. Rev. E 74, 016304.Google Scholar
Biferale, L., Boffetta, G., Celani, A., Devenish, B. J., Lanotte, A. & Toschi, F. 2005 Lagrangian statistics of particle pairs in homogeneous isotropic turbulence. Phys. Fluids 17, 115101.Google Scholar
Bitane, R., Homann, H. & Bec, J. 2012 Timescales of turbulent relative dispersion. Phys. Rev. E 86, 045302.CrossRefGoogle Scholar
Bitane, R., Homann, H. & Bec, J. 2013 Geometry and violent events in turbulent pair dispersion. J. Turbul. 14 (2), 23–45.Google Scholar
Boffetta, G. & Sokolov, I. M. 2002 Relative dispersion in fully developed turbulence: the Richardson’s law and intermittency corrections. Phys. Rev. Lett. 88, 094501.CrossRefGoogle ScholarPubMed
Cardy, J., Falkovich, G. & Gawedzki, K. 2008 Non-Equilibrium Statistical Mechanics and Turbulence, London Mathematical Society Lecture Note Series, vol. 355. Cambridge University Press.Google Scholar
Dimotakis, P. E. 2005 Turbulent mixing. Annu. Rev. Fluid Mech. 37, 329–356.Google Scholar
Eyink, G. L. 2011 Stochastic flux freezing and magnetic dynamo. Phys. Rev. E 83, 056405.CrossRefGoogle ScholarPubMed
Eyink, G. L. & Benveniste, D. 2013 Diffusion approximation in turbulent two-particle dispersion. Phys. Rev. E 88 (4), 041001.Google Scholar
Eyink, G. L. & Drivas, T. D.2014 Spontaneous stochasticity and anomalous dissipation for Burgers equation. Preprint, arXiv:1401.5541.Google Scholar
Falkovich, G., Gawȩdzki, K. & Vergassola, M. 2001 Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913–975.Google Scholar
Faller, A. J. 1996 A random-flight evaluation of the constants of relative dispersion in idealized turbulence. J. Fluid Mech. 316, 139–161.CrossRefGoogle Scholar
Hentschel, H. G. E. & Procaccia, I. 1984 Relative diffusion in turbulent media: the fractal dimension of clouds. Phys. Rev. A 29, 1461–1470.CrossRefGoogle Scholar
Hughes, B. D. 1995 Random Walks and Random Environments. vol. 1, Clarendon.Google Scholar
Ilyin, V., Procaccia, I. & Zagorodny, A. 2010 Stochastic processes crossing from ballistic to fractional diffusion with memory: exact results. Phys. Rev. E 81, 030105.Google Scholar
Kurbanmuradov, O. & Sabelfeld, K. 1995 Stochastic Lagrangian models of relative dispersion of a pair of fluid particles in turbulent flows. Monte Carlo Meth. Applic. 1, 101–136.Google Scholar
Obukhov, A. M. 1941 On the distribution of energy in the spectrum of turbulent flow. Izv. Akad. Nauk SSSR, Ser. Geogr. Geofiz. 5, 453–466.Google Scholar
Ott, S. & Mann, J. 2000 An experimental investigation of the relative diffusion of particle pairs in three-dimensional turbulent flow. J. Fluid Mech. 422, 207–223.Google Scholar
Ouellette, N. T., Xu, H., Bourgoin, M. & Bodenschatz, E. 2006 An experimental study of turbulent relative dispersion models. New J. Phys. 8, 109 (1–23).Google Scholar
Rast, M. P. & Pinton, J.-F. 2011 Pair dispersion in turbulence: the subdominant role of scaling. Phys. Rev. Lett. 107, 214501.CrossRefGoogle ScholarPubMed
Richardson, L. F. 1926 Atmospheric diffusion shown on a distance–neighbour graph. Proc. R. Soc. Lond. A 110, 709–737.Google Scholar
Salazar, J. P. L. C. & Collins, L. R. 2009 Two-particle dispersion in isotropic turbulent flows. Annu. Rev. Fluid Mech. 41, 405–432.Google Scholar
Sawford, B. L. 2001 Turbulent relative dispersion. Annu. Rev. Fluid Mech. 33, 289–317.Google Scholar
Sawford, B. L., Yeung, P. K. & Hackl, J. F. 2008 Reynolds number dependence of relative dispersion statistics in isotropic turbulence. Phys. Fluids 20 (6), 065111.Google Scholar
Scatamacchia, R., Biferale, L. & Toschi, F. 2012 Extreme events in the dispersions of two neighboring particles under the influence of fluid turbulence. Phys. Rev. Lett. 109, 144501.CrossRefGoogle ScholarPubMed
Shlesinger, M. F., West, B. J. & Klafter, J. 1987 Lévy dynamics of enhanced diffusion: application to turbulence. Phys. Rev. Lett. 58 (11), 1100–1103.Google Scholar
Sokolov, I. M., Klafter, J. & Blumen, A. 2000 Ballistic versus diffusive pair dispersion in the Richardson regime. Phys. Rev. E 61 (3), 2717–2722.Google Scholar
Thomson, D. J. & Wilson, J. D. 2013 History of Lagrangian stochastic models for turbulent dispersion. In Lagrangian Modeling of the Atmosphere, pp. 19–36. Wiley Online Library.Google Scholar
Yeung, P. K. 1994 Direct numerical simulation of two-particle relative diffusion in isotropic turbulence. Phys. Fluids 6 (10), 3416–3428.Google Scholar