Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-29T08:05:14.788Z Has data issue: false hasContentIssue false
  • English
  • Français

Inertial particle trapping in an open vortical flow

Published online by Cambridge University Press:  11 March 2014

Jean-Régis Angilella ,
Rafael D. Vilela  and
Adilson E. Motter
Jean-Régis Angilella*
Affiliation:
Université de Caen et de Basse Normandie, LUSAC, 50130 Cherbourg, France
Rafael D. Vilela
Affiliation:
Centro de Matemática, Computação e Cognição, Universidade Federal do ABC (UFABC), Santo André-SP, 09210-170, Brazil
Adilson E. Motter
Affiliation:
Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA
*
†Email address for correspondence: jean-regis.angilella@unicaen.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Recent numerical results on advection dynamics have shown that particles denser than the fluid can remain trapped indefinitely in a bounded region of an open fluid flow. Here, we investigate this counterintuitive phenomenon both numerically and analytically to establish the conditions under which the underlying particle-trapping attractors can form. We focus on a two-dimensional open flow composed of a pair of vortices and its specular image, which is a system we represent as a vortex pair plus a wall along the symmetry line. Considering particles that are much denser than the fluid, referred to as heavy particles, we show that two attractors form in the neighbourhood of the vortex pair provided that the particle Stokes number is smaller than a critical value of order unity. In the absence of the wall, the attractors are fixed points in the frame rotating with the vortex pair, and the boundaries of their basins of attraction are smooth. When the wall is present, the point attractors describe counter-rotating ellipses in this frame, with a period equal to half the period of one isolated vortex pair. The basin boundaries remain smooth if the distance from the vortex pair to the wall is large. However, these boundaries are shown to become fractal if the distance to the wall is smaller than a critical distance that scales with the inverse square root of the Stokes number. This transformation is related to the breakdown of a separatrix that gives rise to a heteroclinic tangle close to the vortices, which we describe using a Melnikov function. For an even smaller distance to the wall, we demonstrate that a second separatrix breaks down and a new heteroclinic tangle forms farther away from the vortices, at the boundary between the open and closed streamlines. Particles released in the open part of the flow can approach the attractors and be trapped permanently provided that they cross the two separatrices, which can occur under the effect of flow unsteadiness. Furthermore, the trapping of heavy particles from the open flow is shown to be robust to the presence of viscosity, noise and gravity. Navier–Stokes simulations for large flow Reynolds numbers show that viscosity does not destroy the attracting points until vortex merging takes place, while simulation of thermal noise shows that particle trapping persists for extended periods provided that the Péclet number is large. The presence of a gravitational field does not alter the permanent trapping by the attracting points if the settling velocities are not too large. For larger settling velocities, however, gravity can also give rise to a limit-cycle attractor next to the external separatrix and to a new form of trapping from the open flow that is not mediated by a heteroclinic tangle.

JFM classification

Mixing: Chaotic advection Nonlinear Dynamical Systems: Fractals Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 744 , 10 April 2014 , pp. 183 - 216
Copyright
© 2014 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Angilella, J.-R. 2010 Dust trapping in vortex pairs. Physica D 239, 1789–1797.Google Scholar
Angilella, J.-R. 2011 Asymptotic properties of wall-induced chaotic mixing in point vortex pairs. Phys. Fluids 23, 113602.Google Scholar
Antonsen, T. M. & Ott, E. 1991 Multifractal power spectra of passive scalars convected by chaotic fluid-flows. Phys. Rev. A 44, 851–857.Google Scholar
Arnold, V. 1965 Sur la topologie des écoulements stationnaires des fluides parfaits. C. R. Acad. Sci. Paris I 261, 17–20.Google Scholar
Babiano, A., Cartwright, J. H. E., Piro, O. & Provenzale, A. 2000 Dynamics of a small neutrally buoyant sphere in a fluid and targeting in Hamiltonian systems. Phys. Rev. Lett. 84, 5764–5767.CrossRefGoogle Scholar
Balkovsky, E., Falkovich, G. & Fouxon, A. 2001 Intermittent distribution of inertial particles in turbulent flows. Phys. Rev. Lett. 86, 2790–2793.Google Scholar
Barge, P. & Sommeria, J. 1995 Did planet formation begin inside persistent gaseous vortices?. Astron. Astrophys. 295, L1–L4.Google Scholar
Bec, J. 2003 Fractal clustering of inertial particles in random flows. Phys. Fluids 15, L81–L84.Google Scholar
Benczik, I. J., Toroczkai, Z. & Tél, T. 2002 Selective sensitivity of open chaotic flows on inertial tracer advection: Catching particles with a stick. Phys. Rev. Lett. 89, 164501.Google Scholar
Benczik, I. J., Toroczkai, Z. & Tél, T. 2003 Advection of finite-size particles in open flows. Phys. Rev. E 67, 036303.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer-Verlag.Google Scholar
Carton, X., Maze, G. & Legras, B. 2002 A two-dimensional vortex merger in an external strain field. J. Turbul. 3, 045.Google Scholar
Cartwright, J., Feudel, U., Karolyi, G., De Moura, A., Piro, O. & Tel, T. 2010 Dynamics of finite-size particles in chaotic fluid flows. In Nonlinear Dynamics and Chaos: Advances and Perspectives (ed. Thiel, M.), Springer-Verlag.Google Scholar
Cerretelli, C. & Williamson, C. H. K. 2003 The physical mechanism for vortex merging. J. Fluid Mech. 475, 41–77.Google Scholar
Cuzzi, J. N., Hogan, R. C., Paque, J. M. & Dobrovolskis, A. R. 2001 Size-selective concentration of chondrules and other small particles in protoplanetary nebula turbulence. Astrophys. J. 546, 496–508.Google Scholar
Daitche, A. & Tél, T. 2011 Memory effects are relevant for chaotic advection of inertial particles. Phys. Rev. Lett. 107, 244501.Google Scholar
De Lillo, F., Cecconi, F., Lacorata, G. & Vulpiani, A. 2008 Sedimentation speed of inertial particles in laminar and turbulent flows. Europhys. Lett. 84, 40005.Google Scholar
Drossinos, Y. & Reeks, M. W. 2005 Brownian motion of finite-inertia particles in a simple shear flow. Phys. Rev. E 71, 031113.Google Scholar
Drotos, G. & Tél, T. 2011 Chaotic saddles in a gravitational field: the case of inertial particles in finite domains. Phys. Rev. E 83, 056203.Google Scholar
Duncan, K., Mehlig, B., Ostlund, S. & Wilkinson, M. 2005 Clustering by mixing flows. Phys. Rev. Lett. 95, 240602.CrossRefGoogle ScholarPubMed
Falkovich, G., Gawedski, K. & Vergassola, M. 2001 Particles and fields in fluid turbulence. Rev. Mod. Phys. 73, 913–975.Google Scholar
Falkovich, G., Fouxon, A. & Stepanov, M. G. 2002 Acceleration of rain initiation by cloud turbulence. Nature 419, 151–154.CrossRefGoogle ScholarPubMed
Fessler, J. R., Kulick, J. D. & Eaton, J. K. 1994 Preferential concentration of heavy particles in a turbulent channel flow. Phys. Fluids 6, 3742–3749.Google Scholar
Fouxon, I. 2012 Distribution of particles and bubbles in turbulence at a small Stokes number. Phys. Rev. Lett. 108, 134502.Google Scholar
Gatignol, R. 1983 The Faxén formulae for a rigid particle in an unsteady non-uniform Stokes flow. J. Méc. Théor. Appl. 1, 143–160.Google Scholar
Gautero, J. L. 1985 Chaos lagrangien pour une classe d’écoulements de Beltrami. C. R. Acad. Sci. Paris II 301 (15), 1095–1098.Google Scholar
Gelfreich, V. G. 1997 Melnikov method and exponentially small splitting of separatrices. Physica D 101, 227–248.Google Scholar
Grassberger, P. 1986 Estimating the fractal dimensions and entropies of strange attractors. In Chaos (ed. Holden, A. V.), Manchester University Press.Google Scholar
Guckenheimer, J. & Holmes, P. 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer.Google Scholar
Haller, G. & Sapsis, T. 2008 Where do inertial particles go in fluid flows?. Physica D 237, 573–583.Google Scholar
Haller, G. & Sapsis, T. 2010 Localized instability and attraction along invariant manifolds. SIAM J. Appl. Dyn. Syst. 9 (2), 611–633.Google Scholar
Ijzermans, R. H. A. & Hagmeijer, R. 2006 Accumulation of heavy particles in $N$ -vortex flow on a disk. Phys. Fluids 18, 063601.Google Scholar
Liu, S.-J., Wei, H.-H., Hwang, S.-H. & Chang, H.-C. 2010 Dynamic particle trapping, release, and sorting by microvortices on a substrate. Phys. Rev. E 82, 026308.Google Scholar
McLaughlin, J. B. 1988 Particle size effects on Lagrangian turbulence. Phys. Fluids 31, 2544–2553.Google Scholar
Maxey, M. R. 1987a The motion of small spherical particles in a cellular flow field. Phys. Fluids 30, 1915–1928.Google Scholar
Maxey, M. R. 1987b The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields. J. Fluid Mech. 174, 441–465.Google Scholar
Maxey, M. R. & Corrsin, S. 1986 Gravitational settling of aerosol particles in randomly oriented cellular flow fields. J. Atmos. Sci. 43, 1112–1134.2.0.CO;2>CrossRefGoogle Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a non uniform flow. Phys. Fluids 26, 883–889.Google Scholar
Maze, G., Carton, X. & Lapeyre, G. 2004 Dynamics of a 2D vortex doublet under external deformation. Regular Chaotic Dyn. 9, 477–497.CrossRefGoogle Scholar
Medrano, R. O., Moura, A., Tel, T., Caldas, I. L. & Grebogi, C. 2008 Finite-size particles, advection, and chaos: a collective phenomenon of intermittent bursting. Phys. Rev. E 78, 056206.Google Scholar
Mehlig, B., Wilkinson, M., Duncan, K., Weber, T. & Ljunggren, M. 2005 Aggregation of inertial particles in random flows. Phys. Rev. E 72, 051104.Google Scholar
Meiburg, E., Wallner, E., Pagella, A., Riaz, A., Hartel, C. & Necker, F. 2000 Vorticity dynamics of dilute two-way-coupled particle-laden mixing layers. J. Fluid Mech. 421, 185–227.Google Scholar
Meyer, C. J. & Deglon, D. A. 2011 Particle collision modeling - a review. Miner. Eng. 24, 719–730.Google Scholar
Michaelides, E. E. 1997 The transient equation of motion for particles, bubbles, and droplets. J Fluid Eng. T. Asme 119, 233–247.Google Scholar
Olla, P. 2010 Preferential concentration versus clustering in inertial particle transport by random velocity fields. Phys. Rev. E 81, 016305.Google Scholar
Ottino, J. 1989 The Kinematics of Mixing: Stretching, Chaos and Transport. Cambridge University Press.Google Scholar
Pasquero, C., Provenzale, A. & Spiegel, E. A. 2003 Suspension and fall of heavy particles in random two-dimensional flow. Phys. Rev. Lett. 91, 054502.Google Scholar
Pushkin, D. O., Melnikov, D. E. & Shevtsova, V. M. 2011 Ordering of small particles in one-dimensional coherent structures by time-periodic flows. Phys. Rev. Lett. 106, 234501.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.Google Scholar
Rubin, J., Jones, C. K. R. T. & Maxey, M. 1995 Settling and asymptotic motion of aerosol particles in a cellular flow field. J. Nonlinear Sci. 5, 337–358.CrossRefGoogle Scholar
Sapsis, T. & Haller, G. 2008 Instabilities in the dynamics of neutrally buoyant particles. Phys. Fluids 20, 017102.Google Scholar
Sapsis, T. & Haller, G. 2009 Inertial particle dynamics in a hurricane. J. Atmos. Sci. 66, 2481–2492.CrossRefGoogle Scholar
Sapsis, T. & Haller, G. 2010 Clustering criterion for inertial particles in two-dimensional time-periodic and three-dimensional steady flows. Chaos 20, 017515.Google Scholar
Shaw, R. A. 2003 Particle-turbulence interactions in atmospheric clouds. Annu. Rev. Fluid Mech. 35, 183–227.Google Scholar
Squires, K. D. & Eaton, J. K. 1991 Preferential concentration of particles by turbulence. Phys. Fluids 3, 1169–1178.Google Scholar
Stommel, H. 1949 Trajectories of small bodies sinking slowly through convection cells. J. Mar. Res. 8, 24–29.Google Scholar
Vilela, R. D. & Motter, A. E. 2007 Can aerosols be trapped in open flows?. Phys. Rev. Lett. 99, 264101.Google Scholar
Wallner, E. & Meiburg, E. 2002 Vortex pairing in two-way coupled, particle laden mixing layers. Intl J. Multiphase Flow 28, 325–346.Google Scholar
Wilkinson, M., Mehlig, B., Ostlund, S. & Duncan, K. P. 2007 Unmixing in random flows. Phys. Fluids 19, 113303.Google Scholar
Wilkinson, M., Mehlig, B. & Gustavsson, K. 2010 Correlation dimension of inertial particles in random flows. Europhys. Lett. 89, 50002.CrossRefGoogle Scholar
Zahnow, J. C. & Feudel, U. 2009 What determines size distributions of heavy drops in a synthetic turbulent flow?. Nonlinear Process. Geophys. 16, 677–690.Google Scholar