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

Abstract

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).

On multiphase turbulence models for collisional fluid–particle flows

“…For inertial particles (i.e. finite Stokes number) the correlations in α A result from the preferential concentration of particles (see, e.g., Sundaram & Collins 1999;Ahmed & Elghobashi 2000), particle trajectory crossing (see, e.g., Simonin 1996;IJzermans et al 2010) and to particle clustering (see, e.g., Agrawal et al 2001).…”

“…In the KT of granular flow, the velocity distribution represents the velocities present in an instantaneous flow field. In other words, at a given time and location in the flow, it is possible to find particles with different velocities (see, e.g., Février et al 2005;IJzermans et al 2010). In contrast, p.d.f.…”

Droplet growth in warm turbulent clouds

“…This also enables one to calculate the particle concentration (the rate of change of which is proportional to the divergence of the particle velocity field) along a particle trajectory and in turn the statistical moments of the volume-averaged particle concentration. IJzermans et al (2009IJzermans et al ( , 2010 and Reeks (2010, 2011) report measurements of the moments at small St and St = O(1) in simple periodic flows, kinematic simulations (with a distribution of energy and length scales typical of real turbulent flows) and in DNS of homogeneous isotropic turbulence. For small St the moments grow exponentially and smoothly with time, consistent with the behaviour predicted by Balkovsky et al (2001).…”

Mitigation of preferential concentration of small inertial particles in stationary isotropic turbulence using electrical and gravitational body forces

“…Dynamical systems theory has been applied to the motion of particles, first by Sommerer and Ott 15 and in turbulent flows by Bec 16 and Wilkinson et al 17 who showed that periods in time exist where particle trajectories may cross leading to particle collisions. Relaxing an assumption of Wilkinson et al, 17 that the typical correlation time of the carrier flow is small, Reeks and co-workers have deployed the full Lagrangian method (FLM) of Ostiptsov 18 first in synthetic flows 19 and recently in isotropic turbulence 20 to quantify non-uniformities and singularities in the spatial distribution of particles more generally. The advantage of FLM being that the analysis takes place on infinitesimally small scales and does not rely on defining a box size for counting purposes and that a small Stokes number limit is not implicit in the method.…”