Clustering of heavy particles in random self-similar flow

Heavy inertial particles in rotating turbulence: Distribution of particles in flow and evolution of Lagrangian trajectories

We revisit the problem of heavy particles suspended in homogeneous box turbulence flow subjected to rotation along the vertical axis, which introduces anisotropy along the vertical and horizontal planes. We investigate the effects of the emergent structures due to rotation, on the spatial distribution and temporal statistics of the particles. The distribution of particles in the flow are studied using the joint probability distribution function (JPDFs) of the second and third principle invariants of the velocity gradient tensor, Q and R. At high rotation rates, the JPDFs of Lagrangian Q-R plots show remarkable deviations from the well-known teardrop shape. The cumulative probability distribution functions for times during which a particle remains in vortical or straining regions show exponentially decaying tails except for the deviations at the highest rotation rate. The average residence times of the particles in vortical and straining regions are also affected considerably due to the addition of rotation. Furthermore, we compute the temporal velocity autocorrelation and connect it to the Lagrangian anisotropy in presence of rotation. The spatial and temporal statistics of the particles are determined by a complex competition between the rotation rate and inertia of the particle.

Point-source dispersion of quasi-neutrally-buoyant inertial particles

We analyze the evolution of the distribution, both in the phase space and in the physical space, of inertial particles released by a spatially-localized (punctual) source and advected by an incompressible flow. The difference in mass density between fluid and particles is assumed to be small, and represents the basic parameter for a regular perturbative expansion. By means of analytical techniques such as Hermitianization, we derive a chain of equations of the advection-diffusion-reaction type, easily solvable at least numerically. Our procedure provides results also for finite particle inertia, away from the over-damped limit of quasi-tracer dynamics.

Scale-similar clustering of heavy particles in the inertial range of turbulence

Heavy particle clustering in turbulence is discussed from both phenomenological and analytical points of view, where the -4/3 power law of the pair-correlation function is obtained in the inertial range. A closure theory explains the power law in terms of the balance between turbulence mixing and preferential-concentration mechanism. The obtained -4/3 power law is supported by a direct numerical simulation of particle-laden turbulence.

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.

Polymer stretching in the inertial range of turbulence

We study the deformation of flexible polymers whose contour length lies in the inertial range of a homogeneous and isotropic turbulent flow. By using the elastic dumbbell model and a stochastic velocity field with nonsmooth spatial correlations, we obtain the probability density function of the extension as a function of the Weissenberg number and of the scaling exponent of the velocity structure functions. In a spatially rough flow, as in the inertial range of turbulence, the statistics of polymer stretching differs from that observed in laminar flows or in smooth chaotic flows. In particular, the probability distribution of polymer extensions decays as a stretched exponential, and the most probable extension grows as a power law of the Weissenberg number. Furthermore, the ability of the flow to stretch polymers weakens as the flow becomes rougher in space.

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.

Concentration fluctuations of large Stokes number particles in a one-dimensional random velocity field

We analyze the behavior of an ensemble of inertial particles in a one-dimensional smooth Gaussian velocity field, in the limit of large inertia, but considering a finite correlation time for the random field. The amplitude of the concentration fluctuations is characterized by slow decay at large inertia and a much larger correlation length than that of the random field. The fluctuation structure in velocity space is very different from predictions from short-time correlated random velocity fields, with only few particle pairs crossing at sufficiently small relative velocity to produce correlations. Concentration fluctuations are associated with depletion of the relative velocity variance of colliding particles.

Heavy particle concentration in turbulence at dissipative and inertial scales

Spatial distributions of heavy particles suspended in an incompressible isotropic and homogeneous turbulent flow are investigated by means of high resolution direct numerical simulations. In the dissipative range, it is shown that particles form fractal clusters with properties independent of the Reynolds number. Clustering is there optimal when the particle response time is of the order of the Kolmogorov time scale tau(eta). In the inertial range, the particle distribution is no longer scale invariant. It is, however, shown that deviations from uniformity depend on a rescaled contraction rate, which is different from the local Stokes number given by dimensional analysis. Particle distribution is characterized by voids spanning all scales of the turbulent flow; their signature in the coarse-grained mass probability distribution is an algebraic behavior at small densities.