Colliding particles in highly turbulent flows

We discuss relative velocities and the collision rate of small particles suspended in a highly turbulent fluid. In the limit where the viscous damping is very weak, we estimate the relative velocities using the Kolmogorov cascade principle. PACS numbers: 05.20.Dd Kinetic theory 45.50.Tn Collisions 47.27.-i Turbulent flows 47.57.ESuspensions Introduction. This paper considers collisions of small particles suspended in a highly turbulent gas. The collisions of these particles can facilitate aggregation of the suspended particles. This process may be relevant to the precipitation of rain from turbulent cumulus clouds [1], and to the formation of planets by aggregation of dust particles suspended in the gas surrounding a growing star [2]. The suspended particles are characterised by a dimensionless measure of the importance of inertia, termed the Stokes number: St = 1/γτ , where τ is a correlation time of the flow and γ is the damping rate of the suspended particles (both quantities are defined more precisely below). In [3] we showed how the collision rate increases very rapidly when St exceeds a threshold value, due to fold caustics making the velocity field of the suspended particles multi-valued. In [3], which discussed initiation of rainfall from turbulent clouds, it was sufficient to use a single-scale flow model of the turbulent motion (described by a correlation length η and correlation time τ) because the Stokes number is never very large for particles suspended in terrestrial atmospheric clouds. However, in astrophysical contexts it is necessary to consider flows with large values of St, where the multi-scale aspect of turbulent flow [4] becomes important. (It is hard to study cases where St is large in terrestrial contexts because heavy particles fall out of the fluid). In the following we derive an expression for the collision rate in a highly turbulent flow with large Stokes number. We employ the Kolmogorov cascade principle to deduce an expression for the variance of the relative velocities of colliding particles, which in turn determines the collision rate. Formulation of the problem. We assume that the drag force on a particle is proportional to the difference in velocity between the particle and the surrounding gas, so that the equation of motion is r̈ = γ[u(r, t)− ṙ] (1)