Clustering and collisions of heavy particles in random smooth flows

Abstract

Finite-size impurities suspended in incompressible flows distribute inhomogeneously, leading to a drastic enhancement of collisions. A description of the dynamics in the full position-velocity phase space is essential to understand the underlying mechanisms, especially for polydisperse suspensions. These issues are here studied for particles much heavier than the fluid by means of a Lagrangian approach. It is shown that inertia enhances collision rates through two effects: correlation among particle positions induced by the carrier flow and uncorrelation between velocities due to their finite size. A phenomenological model yields an estimate of collision rates for particle pairs with different sizes. This approach is supported by numerical simulations in random flows.

Stochastic suspensions of heavy particles

“…In the smooth case (h = 1), the limit value Γ decreases from Γ = 1 for St = 0, which corresponds to a differentiable particle velocity field, to Γ = 0 for St → ∞, which means that particles move with uncorrelated velocities [16]. The fact that Γ < 1 is due to the contribution of caustics appearing in the particle velocity field [15,18,24,25,26] (see Sect. V for a discussion in d = 1).…”

“…For D 2 < d, the second power law can be interpret also as the contribution of caustics [18,26]: With non-zero probability, particles may be very close to each other with quite different velocities, see Section V. Once projected onto physical space, caustics appear as spots of uncorrelated particles, and hence, the correlation dimension is locally D 2 = d. The validity of (44) as well as of the projection formula (43) was confirmed in Ref. [16], .…”

“…(15) where Θ denotes the Heaviside function and the average is defined on the Lagrangian trajectories. As detailed in [18], κ(r) is related to the binary collision rate in the framework of the so-called ghost collision scheme [23]. Within this approach collision events are counted while allowing particles to overlap instead of scattering.…”

“…This quantity enters models for the collision kernel [17]. Following [10,13,16,18], we consider a different, but related way to characterize particle clustering. Instead of the radial distribution function we evaluate the correlation dimension D 2 of the set formed by the particles.…”

“…Indeed, in order to evaluate the collision rate, one needs to know not only the probability that the particles are close to each other, but also their typical velocity difference. Here, following [18], we study the approaching rate κ(r) defined as the flux of particles that are separated by a distance less than r and approach each other, i.e. (15) where Θ denotes the Heaviside function and the average is defined on the Lagrangian trajectories.…”