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Segregation of particles in incompressible random flows: singularities, intermittency and random uncorrelated motion


The results presented here are part of a long-term study in which we analyse the segregation of inertial particles in turbulent flows using the so called full Lagrangian method (FLM) to evaluate the ‘compressibility’ of the particle phase along a particle trajectory. In the present work, particles are advected by Stokes drag in a random flow field consisting of counter-rotating vortices and in a flow field composed of 200 random Fourier modes. Both flows are incompressible and, like turbulence, have structure and a distribution of scales with finite lifetime. The compressibility is obtained by first calculating the deformation tensor Jij associated with an infinitesimally small volume of particles following the trajectory of an individual particle. The fraction of the initial volume occupied by the particles centred around a position x at time t is denoted by |J|, where J ≡ det(Jij) and Jij ≡ ∂xi(x0, t)/∂x0,j, x0 denoting the initial position of the particle. The quantity d〈ln|J|〉/dt is shown to be equal to the particle averaged compressibility of the particle velocity field 〈∇ · v〉, which gives a measure of the rate-of-change of the total volume occupied by the particle phase as a continuum. In both flow fields the compressibility of the particle velocity field is shown to decrease continuously if the Stokes number St (the dimensionless particle relaxation time) is below a threshold value Stcr, indicating that the segregation of particles continues indefinitely. We show analytically and numerically that the long-time limit of 〈∇ · v〉 for sufficiently small values of St is proportional to St2 in the flow field composed of random Fourier modes, and to St in the flow field consisting of counter-rotating vortices. If St > Stcr, however, the particles are ‘mixed’. The level of mixing can be quantified by the degree of random uncorrelated motion (RUM) of particles which is a measure of the decorrelation of the velocities of two nearby particles. RUM is zero for fluid particles and increases rapidly with the Stokes number if St > Stcr, approaching unity for St ≫ 1. The spatial averages of the higher-order moments of the particle number density are shown to diverge with time indicating that the spatial distribution of particles may be very intermittent, being associated with non-zero values of RUM and the occurrence of singularities in the particle velocity field. Our results are consistent with previous observations of the radial distribution function in Chun et al. (J. Fluid Mech., vol. 536, 2005, p. 219).

Corresponding author
Present address: Royal Dutch Shell plc, Shell Technology Centre Amsterdam, Grasweg 31, 1031 HW Amsterdam, The Netherlands. Email address for correspondence:
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Balkovsky E., Falkovich G. & Fouxon A. 2001 Intermittent distribution of inertial particles in turbulent flows. Phys. Rev. Lett. 86, 27902793.
Bec J. 2005 Multifractal concentrations of inertial particles in smooth random flow. J. Fluid Mech. 528, 255277.
Bec J., Biferale L., Cencini M., Lanotte A., Musacchio S. & Toschi F. 2007 Heavy particle concentration in turbulence at dissipative and inertial scales. Phys. Rev. Lett. 98, 084502.
Chen L., Goto S. & Vassilicos J. C. 2006 Turbulent clustering of stagnation points and inertial particles. J. Fluid Mech. 553, 143154.
Chong M. S., Perry A. E. & Cantwell B. J. 1990 A general classification of three-dimensional flow fields. Phys. Fluids 2 (5), 756777.
Chun J., Koch D. L., Rani S. L., Ahluwalia A. & Collins L. R. 2005 Clustering of aerosol particles in isotropic turbulence. J. Fluid Mech. 536, 219251.
Crowe C. T., Chung J. N. & Troutt T. R. 1993 Particle dispersion by organized turbulent structures. In Particulate Two-Phase Flow (ed. Roco M. C.), vol. 626, chap. 18, p. 1. Heinemann.
Elperin T., Kleeorin N. & Rogashevskii I. 1996 Turbulent thermal diffusion of small inertial particles. Phys. Rev. Lett. 76 (2), 224227.
Falkovich G., Fouxon A. S. & Stepanov M. G. 2002 Acceleration of rain initiation by cloud turbulence. Nature 419, 151154.
Falkovich G. & Pumir A. 2004 Intermittent distribution of heavy particles in a turbulent flow. Phys. Fluids 16 (7), L47L50.
Falkovich G. & Pumir A. 2007 Sling effect in collisions of water droplets in turbulent clouds. J. Atmos. Sci. 64, 44974505.
Fessler J. R., Kulick J. D. & Eaton J. K. 1994 Preferential concentration of heavy particles in a turbulent channel flow. Phys. Fluids 6 (11), 37423749.
Février P., Simonin O. & Squires K. D. 2005 Partitioning of particle velocities in gas–solid turbulent flows into a continuous field and a spatially uncorrelated random distribution; theoretical formalism and numerical study. J. Fluid Mech. 553, 146.
Frisch U. 1995 Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press.
Fung J. C. H., Hunt J. C. R., Malik N. A. & Perkins R. J. 1992 Kinematic simulation of homogeneous turbulence by unsteady random Fourier modes. J. Fluid Mech. 230, 281318.
Healy D. P. & Young J. B. 2005 Full Lagrangian methods for calculating particle concentration fields in dilute gas–particle flows. Proc. R. Soc. Lond. A: 461, 21972225.
Hentschel H. G. E. & Procaccia I. 1983 The infinite number of generalized dimensions of fractals and strange attractors. Physica D 8, 435444.
Hunt J. C. R., Buell J. C. & Wray A. A. 1987 Big whorls carry little whorls. Tech. Rep. CTR–S87. NASA.
IJzermans R. H. A. 2007 Dynamics of dispersed heavy particles in swirling flow. PhD thesis, University of Twente, The Netherlands.
IJzermans R. H. A., Hagmeijer R. & van Langen P. J. 2007 Accumulation of heavy particles around a helical vortex filament. Phys. Fluids 19, 107102.
Kraichnan R. H. 1970 Diffusion by a random velocity field. Phys. Fluids 13, 2231.
Luo K., Fan J. & Cen K. 2007 Pressure-correlated dispersion of inertial particles in free shear flows. Phys. Rev. E 75, 046309.
Marcu B., Meiburg E. & Newton P. K. 1995 Dynamics of heavy particles in a Burgers vortex. Phys. Fluids 7 (2), 400410.
Maxey M. R. 1987 The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields. J. Fluid Mech. 174, 441465.
Maxey M. R. & Riley J. J. 1983 Equation of motion for a small rigid sphere in a non-uniform flow. Phys. Fluids 26 (4), 883889.
Osiptsov A. N. 2000 Lagrangian modelling of dust admixture in gas flows. Astrophys. Space Sci. 274, 377386.
Ott E. 1994 Chaos in Dynamical Systems. Cambridge University Press.
Press W. C., Flannery B. P., Tucholsky S. A. & Vetterling W. T. 1992 Numerical Recipes in Fortran 77. Cambridge University Press.
Reeks M. W. 1977 On the dispersion of small particles suspended in an isotropic turbulent fluid. J. Fluid Mech. 83 (3), 529546.
Reeks M. W. 2004 Simulation of particle diffusion, segregation, and intermittency in turbulent flows. In Proc. IUTAM Symposium on Computational Modelling of Disperse Multiphase Flow (ed. Balachandar S. & Prosperetti A.), pp. 2130. Elsevier.
Shaw R. A. 2003 Particle-turbulence interactions in atmospheric clouds. Annu. Rev. Fluid Mech. 35, 183227.
Sommerer J. C. & Ott E. 1993 Particles floating on a moving fluid: a dynamically comprehensible physical fractal. Science 259, 335339.
Spelt P. D. M. & Biesheuvel A. 1997 On the motion of gas bubbles in homogeneous isotropic turbulence. J. Fluid Mech. 336, 221244.
Squires K. D. & Eaton J. K. 1991 Measurements of particle dispersion obtained from direct numerical simulations of isotropic turbulence. J. Fluid Mech. 226, 135.
Sundaram S. & Collins L. R. 1997 Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulations. J. Fluid Mech. 335, 75109.
Wang L. P. & Maxey M. R. 1993 Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence. J. Fluid Mech. 256, 2768.
Wang L. P., Wexler A. S. & Zhou Y. 1998 On the collision rate of small particles in isotropic turbulence. II. Finite inertia case. Phys. Fluids 10 (10), 12061216.
Wilkinson M., Mehlig B., Östlund S. & Duncan K. P. 2007 Unmixing in random flows. Phys. Fluids 19, 113303.
Zaichik L. I. & Alipchenkov V. M. 2003 Pair dispersion and preferential concentration of particles in isotropic turbulence. Phys. Fluids 15 (6), 17761787.
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Journal of Fluid Mechanics
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