Preferential concentration versus clustering in inertial particle transport by random velocity fields

Clustering of particles in turbulence due to phoresis

We demonstrate that diffusiophoretic, thermophoretic, and chemotactic phenomena in turbulence lead to clustering of particles on multifractal sets that can be described using one single framework, valid when the particle size is much smaller than the smallest length scale of turbulence l_{0}. To quantify the clustering, we derive positive pair correlations and fractal dimensions that hold for scales smaller than l_{0}. For scales larger than l_{0} the pair-correlation function is predicted to show a stretched exponential decay towards 1. In the case of inhomogeneous turbulence we find that the fractal dimension depends on the direction of inhomogeneity. By performing experiments with particles in a turbulent gravity current we demonstrate clustering induced by salinity gradients in conformity to the theory. The particle size in the experiment is comparable to l_{0}, outside the strict validity region of the theory, suggesting that the theoretical predictions transfer to this practically relevant regime. This clustering mechanism may provide the key to the understanding of a multitude of processes such as formation of marine snow in the ocean and population dynamics of chemotactic bacteria.

Inhomogeneous distribution of water droplets in cloud turbulence

We consider sedimentation of small particles in the turbulent flow where fluid accelerations are much smaller than acceleration of gravity g. The particles are dragged by the flow by linear friction force. We demonstrate that the pair-correlation function of particles' concentration diverges with decreasing separation as a power law with negative exponent. This manifests fractal distribution of particles in space. We find that the exponent is proportional to ratio of integral of energy spectrum of turbulence times the wave number over g. The proportionality coefficient is a universal number independent of particle size. We derive the spectrum of Lyapunov exponents that describes the evolution of small patches of particles. It is demonstrated that particles separate dominantly in the horizontal plane. This provides a theory for the recently observed vertical columns formed by the particles. We confirm the predictions by direct numerical simulations of Navier-Stokes turbulence. The predictions include conditions that hold for water droplets in warm clouds thus providing a tool for the prediction of rain formation.

Distribution of particles and bubbles in turbulence at a small Stokes number

The inertia of particles driven by the turbulent flow of the surrounding fluid makes them prefer certain regions of the flow. The heavy particles lag behind the flow and tend to accumulate in the regions with less vorticity, while the light particles do the opposite. As a result of the long-time evolution, the particles distribute over a multifractal attractor in space. We consider this distribution using our recent results on the steady states of chaotic dynamics. We describe the preferential concentration analytically and derive the correlation functions of density and the fractal dimensions of the attractor. The results are obtained for real turbulence and are testable experimentally.

Tangling clustering of inertial particles in stably stratified turbulence

We have predicted theoretically and detected in laboratory experiments a tangling clustering of inertial particles in a stably stratified turbulence with imposed mean vertical temperature gradient. In the stratified turbulence a spatial distribution of the mean particle number density is nonuniform due to the phenomenon of turbulent thermal diffusion, i.e., the inertial particles are accumulated in the vicinity of the minimum of the mean temperature of the surrounding fluid, and a nonzero gradient of the mean particle number density, ∇N , is formed. It causes generation of fluctuations of the particle number density by tangling of the large-scale gradient ∇N by velocity fluctuations. In addition, the mean temperature gradient ∇T produces the temperature fluctuations by tangling of the large-scale gradient ∇T by velocity fluctuations. The anisotropic temperature fluctuations contribute to the two-point correlation function of the divergence of the particle velocity field, i.e., they increase the rate of formation of the particle clusters in small scales. We have demonstrated that in the laboratory stratified turbulence this tangling clustering is much more effective than a pure inertial clustering (preferential concentration) that has been observed in isothermal turbulence. In particular, in our experiments in oscillating grid isothermal turbulence in air without imposed mean temperature gradient, the inertial clustering is very weak for solid particles with the diameter of ≈10 μm and Reynolds numbers based on turbulent length scale and rms velocity, Re=250 . In the experiments the correlation function for the inertial clustering in isothermal turbulence is much smaller than that for the tangling clustering in nonisothermal turbulence. The size of the tangling clusters is on the order of several Kolmogorov length scales. The clustering described in our study is found for inertial particles with small Stokes numbers and with the material density that is much larger than the fluid density. Our theoretical predictions are in a good agreement with the obtained experimental results.