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Weak–strong clustering transition in renewing compressible flows

Published online by Cambridge University Press:  25 November 2014

Ajinkya Dhanagare ,
Stefano Musacchio  and
Dario Vincenzi
Ajinkya Dhanagare
Affiliation:
Laboratoire J. A. Dieudonné, CNRS, Université Nice Sophia Antipolis, 06100 Nice, France
Stefano Musacchio
Affiliation:
Laboratoire J. A. Dieudonné, CNRS, Université Nice Sophia Antipolis, 06100 Nice, France
Dario Vincenzi*
Affiliation:
Laboratoire J. A. Dieudonné, CNRS, Université Nice Sophia Antipolis, 06100 Nice, France
*
†Email address for correspondence: [email protected]
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Abstract

We investigate the statistical properties of Lagrangian tracers transported by a time-correlated compressible renewing flow. We show that the preferential sampling of the phase space performed by tracers yields significant differences between the Lagrangian statistics and its Eulerian counterpart. In particular, the effective compressibility experienced by tracers has a non-trivial dependence on the time correlation of the flow. We examine the consequence of this phenomenon on the clustering of tracers, focusing on the transition from the weak- to the strong-clustering regime. We find that the critical compressibility at which the transition occurs is minimum when the time correlation of the flow is of the order of the typical eddy turnover time. Further, we demonstrate that the clustering properties in time-correlated compressible flows are non-universal and are strongly influenced by the spatio-temporal structure of the velocity field.

JFM classification

Mixing: Chaotic advection Mixing: Mixing
Type
Papers
Information
Journal of Fluid Mechanics , Volume 761 , 25 December 2014 , pp. 431 - 442
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© 2014 Cambridge University Press 

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References

Abraham, E. R. 1998 The generation of plankton patchiness by turbulent stirring. Nature 391, 577–580.CrossRefGoogle Scholar
Alexakis, A. & Tzella, A. 2011 Bounding the scalar dissipation scale for mixing flows in the presence of sources. J. Fluid Mech. 688, 443–460.Google Scholar
Antonsen, T. M. Jr, Fan, Z., Ott, E. & Garcia-Lopez, E. 1996 The role of chaotic orbits in the determination of power spectra of passive scalars. Phys. Fluids 8, 3094–3104.Google Scholar
Bec, J., Gawȩdzki, K. & Horvai, P. 2004 Multifractal clustering in compressible flows. Phys. Rev. Lett. 92, 224501.Google Scholar
Benettin, G., Galgani, L., Giorgilli, A. & Strelcyn, J.-M. 1980 Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems. A method for computing all of them. Part 2: Numerical application. Meccanica 15, 21–30.Google Scholar
Boffetta, G., Davoudi, J., Eckhardt, B. & Schumacher, J. 2004 Lagrangian tracers on a surface flow: the role of time correlations. Phys. Rev. Lett. 93, 134501.CrossRefGoogle ScholarPubMed
Boffetta, G., Davoudi, J. & De Lillo, F. 2006 Multifractal clustering of passive tracers on a surface flow. Europhys. Lett. 74, 62–68.Google Scholar
Boffetta, G., Celani, A., De Lillo, F. & Musacchio, S. 2007 The Eulerian description of dilute collisionless suspension. Europhys. Lett. 78, 140001.Google Scholar
Cardy, J., Gawȩdzki, K. & Falkovich, G. 2008 Non-equilibrium statistical mechanics and turbulence (ed. Nazarenko, S. & Zaboronski, O. V.), London Mathematical Society Lecture Note Series, vol. 355. Cambridge University Press.Google Scholar
Chaves, M., Gawȩdzki, K., Horvai, P., Kupiainen, A. & Vergassola, M. 2003 Lagrangian dispersion in Gaussian self-similar velocity ensembles. J. Stat. Phys. 113, 643–692.CrossRefGoogle Scholar
Chertkov, M., Kolokolov, I. & Vergassola, M. 1998 Inverse versus direct cascades in turbulent advection. Phys. Rev. Lett. 80, 512–515.Google Scholar
Cressman, J. R., Davoudi, J., Goldburg, W. I. & Schumacher, J. 2004 Eulerian and Lagrangian studies in surface flow turbulence. New J. Phys. 6, 53.CrossRefGoogle Scholar
Csanady, G. T. 1973 Turbulent Diffusion in the Environment. Springer.Google Scholar
Ducasse, L. & Pumir, A. 2008 Intermittent particle distribution in synthetic free-surface turbulent flows. Phys. Rev. E 77, 066304.Google Scholar
Elperin, T., Kleeorin, N., Rogachevskii, I. & Sokoloff, D. 2000 Passive scalar transport in a random flow with a finite renewal time: mean-field equations. Phys. Rev. E 61, 2617–2625.Google Scholar
Elperin, T., Kleeorin, N., L’vov, V. S., Rogachevskii, I. & Sokoloff, D. 2002 Clustering instability of the spatial distribution of inertial particles in turbulent flows. Phys. Rev. E 66, 036302.Google Scholar
Falkovich, G., Gawȩdzki, K. & Vergassola, M. 2001 Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913–975.Google Scholar
Falkovich, G. & Martins Afonso, M. 2007 Fluid–particle separation in a random flow described by the telegraph model. Phys. Rev. E 76, 026312.Google Scholar
Falkovich, G., Musacchio, S., Piterbarg, L. & Vucelja, M. 2007 Inertial particles driven by a telegraph noise. Phys. Rev. E 76, 026313.CrossRefGoogle ScholarPubMed
Gilbert, A. D. & Bayly, B. G. 1992 Magnetic field intermittency and fast dynamo action in random helical flows. J. Fluid Mech. 241, 199–214.Google Scholar
Gustavsson, K. & Mehlig, B. 2013a Distribution of velocity gradients and rate of caustic formation in turbulent aerosols at finite Kubo numbers. Phys. Rev. E 87, 023016.Google Scholar
Gustavsson, K. & Mehlig, B. 2013b Lyapunov exponents for particles advected in compressible random velocity fields at small and large Kubo numbers. J. Stat. Phys. 153, 813–827.CrossRefGoogle Scholar
Kraichnan, R. H. 1968 Small-scale structure of a scalar field convected by turbulence. Phys. Fluids 11, 945–953.Google Scholar
Larkin, J. & Goldburg, W. I. 2010 Decorrelating a compressible turbulent flow: an experiment. Phys. Rev. E 82, 016301.Google Scholar
Larkin, J., Goldburg, W. & Bandi, M. M. 2010 Time evolution of a fractal distribution: particle concentrations in free-surface turbulence. Physica D 239, 1264–1268.CrossRefGoogle Scholar
Le Jan, Y. 1984 Exposants de Lyapunov pour les mouvements browniens isotropes. C. R. Acad. Sci. Paris Ser. I 299, 947–949.Google Scholar
Le Jan, Y. 1985 On isotropic Brownian motions. Z. Wahrscheinlichkeitstheor. verw. Gebiete 70, 609–620.Google Scholar
Lovecchio, S., Marchioli, C. & Soldati, A. 2013 Time persistence of floating-particle clusters in free-surface turbulence. Phys. Rev. E 88, 033003.Google Scholar
Maxey, M. R. 1987 The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields. J. Fluid Mech. 174, 441–465.Google Scholar
Musacchio, S. & Vincenzi, D. 2011 Deformation of a flexible polymer in a random flow with long correlation time. J. Fluid Mech. 670, 326–336.Google Scholar
Pandit, R., Perlekar, P. & Ray, S. S. 2009 Statistical properties of turbulence: an overview. Pramana – J. Phys. 73, 157–191.Google Scholar
Perez-Munuzuri, V. 2014 Mixing and clustering in compressible chaotic stirred flows. Phys. Rev. E 89, 022917.Google Scholar
Pergolizzi, B.2012 Étude de la dynamique des particules inertielles dans des écoulements aléatoires. PhD thesis, Université Nice Sophia Antipolis.Google Scholar
Pierrehumbert, R. T. 1994 Tracer microstructure in the large-eddy dominated regime. Chaos, Solitons Fractals 4, 1091–1110.Google Scholar
Schumacher, J. & Eckhardt, B. 2002 Clustering dynamics of Lagrangian tracers in free-surface flows. Phys. Rev. E 66, 017303.Google Scholar
Sommerer, J. C. & Ott, E. 1993 Particles floating on a moving fluid: a dynamically comprehensible physical fractal. Science 259, 335–339.Google Scholar
Thiffeault, J.-L., Doering, C. R. & Gibbon, J. D. 2004 A bound on mixing efficiency for the advection–diffusion equation. J. Fluid Mech. 521, 105–114.Google Scholar
Toschi, F. & Bodenschatz, E. 2009 Lagrangian properties of particles in turbulence. Annu. Rev. Fluid Mech. 41, 375–404.Google Scholar
Vucelja, M., Falkovich, G. & Fouxon, I. 2007 Clustering of matter in waves and currents. Phys. Rev. E 75, 065301.Google Scholar
Wilkinson, M. 2011 Lyapunov exponent for small particles in smooth one-dimensional flows. J. Phys. A 44, 045502.Google Scholar
Young, W. R. 1999 Stirring and Mixing: Proceedings of the 1999 Summer Program in Geophysical Fluid Dynamics (ed. Thiffeault, J.-L. & Pasquero, C.), Woods Hole Oceanographic Institution.Google Scholar
Zel’dovich, Ya. B., Ruzmaikin, A. A., Molchanov, S. A. & Sokoloff, D. D. 1984 Kinematic dynamo problem in a linear velocity field. J. Fluid Mech. 144, 1–11.Google Scholar