Clustering of particles in weakly divergent flows

Abstract

The clustering of particles in turbulent flows is an important mechanism in both natural and industrial processes. It appears across a diverse range of natural phenomena such as rain initiation and growth of rain drops, plankton dispersion in the ocean, and planetesimal formation as a result of aggregation of dust grains. In particular, turbulent particle laden flows in the ocean lead to the clustering of inorganic particulate matter, a topic of active interest with increasing concerns about microplastic and other particulate pollutants.

Divergence in the particle velocity field is the cause of clustering of particles. The divergence is not necessarily a product of the compressibility of the carrier fluid and can occur from indirect causes. This thesis examines the clustering that results from two such indirect causes. The first is the inertia of particles, which makes the particle velocity differ from the fluid velocity and leads to clustering. We examine this process using the Maxey–Riley equation for inertial particle motion in the limit of small Stokes number St ≪ 1. In this limit, the dynamics in the full position–velocity phase space can be reduced to simple advection in position space, but with a new ‘effective’ velocity that is weakly divergent even though the fluid flow is incompressible. We study the statistics of clustering in this set up using a simple two-dimensional kinematic model in which the fluid velocity is random in time.

The second indirect cause of divergence that is studied concerns particles floating at the surface of the ocean. These particles experience a two-dimensional divergent velocity field, namely the horizontal part of the full three-dimensional non-divergent fluid velocity. We model this horizontal flow using the SQG+1 model. This is an improved-accuracy version of the surface quasigeostrophic (SQG) model which captures the first-order corrections in the Rossby number (Ro) including the weak (horizontal) divergence.

Particles in divergent flows converge to a fractal attractor in the position–velocity iii phase space. The projection of this attractor in physical space forms a fractal set.

For floating particles, we make use of techniques from chaos theory to quantify the dimension of this set and consider in particular the information dimension D1 and correlation dimension D2. For D1, we rely on the Kaplan-Yorke conjecture which expresses D1 in terms of Lyapunov exponents. D2 is computed as a correlation sum.

The dimension of the attractor is shown to scale quadratically in Ro for both estimates.

For both inertial and floating particles, we examine in detail the rate of clustering as measured by large-deviations statistics of the particle density distribution. A Lagrangian approach is taken to overcome the challenge arising from the finite resolution of Eulerian models. We consider both the rate function describing the long time behaviour of this distribution and its Legendre dual, the free energy, describing the long time behaviour of the moments of the density.

We develop and implement an importance sampling procedure to estimate the free energy accurately. This is necessary to capture the statistics of high moments of the density which are dominated by rare trajectories.

We examine the applicability of the Kraichnan limit of short correlation time of the velocity for both inertial and floating particles. In this limit, we predict a parabolic shape for the free energy. The prediction is shown to be robust even when applied to experiments not strictly in the short correlation time limit.