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

Quantitative Prediction of Sling Events in Turbulence at High Reynolds Numbers

Collisional growth of droplets, such as occurring in warm clouds, is known to be significantly enhanced by turbulence. Whether particles collide depends on their flow history, in particular on their encounters with highly intermittent small-scale turbulent structures, which despite their rarity can dominate the overall collision rate. Here, we develop a quantitative criterion for sling events based on the velocity gradient history along particle paths. We show by a combination of theory and simulations that the problem reduces to a one-dimensional localization problem as encountered in condensed matter physics. The reduction demonstrates that the creation of slings is controlled by the minimal real eigenvalue of the velocity gradient tensor. We use fully resolved turbulence simulations to confirm our predictions and study their Stokes and Reynolds number dependence. We also discuss extrapolations to the parameter range relevant for typical cloud droplets, showing that sling events at high Reynolds numbers are enhanced by an order of magnitude for small Stokes numbers. Thus, intermittency could be a significant ingredient in the collisional growth of rain droplets.

Efficient mate finding in planktonic copepods swimming in turbulence

Zooplankton live in dynamic environments where turbulence may challenge their limited swimming abilities. How this interferes with fundamental behavioral processes remains elusive. We reconstruct simultaneously the trajectories of flow tracers and calanoid copepods and we quantify their ability to find mates when ambient flow imposes physical constrains on their motion and impairs their olfactory orientation. We show that copepods achieve high encounter rates in turbulence due to the contribution of advection and vigorous swimming. Males further convert encounters within the perception radius to contacts and then to mating via directed motion toward nearby organisms within the short time frame of the encounter. Inertial effects do not result in preferential concentration, reducing the geometric collision kernel to the clearance rate, which we model accurately by superposing turbulent velocity and organism motion. This behavioral and physical coupling mechanism may account for the ability of copepods to reproduce in turbulent environments.

Clustering of floating tracers in weakly divergent velocity fields

This work deals with buoyant tracers floating at the ocean surface, where the geostrophic velocity component is two dimensional and rotational (nondivergent) and the ageostrophic component can contain rotational and potential (divergent) contributions that are comparable in size. We consider a random kinematic flow model and study the process of clustering, that is, aggregation of the floating tracer in localized spatial patches. In the long-time limit and in the cases of strongly and weakly divergent flows, the existing analytical theory predicts the process of exponential clustering, which is the emergence of spatial singularities containing all the available tracer. Here we confirm this analytical prediction, in numerical model solutions spanning different combinations of rotational and potential surface velocity components, and report that exponential clustering persists even in weakly divergent flows, however at significantly slower rates. For a wide range of parameters, we analyze not only the exponential clustering but also the other type of tracer aggregation, referred to as fragmentation clustering, as well as the coarse-graining effects on clustering. For the presented analysis we consider ensembles of Lagrangian particles and introduce and apply the statistical topography methodology.

Density and tracer statistics in compressible turbulence: Phase transition to multifractality

We study the statistics of fluid (gas) density and concentration of passive tracer particles (dust) in compressible turbulence. As Ma increases from small or moderate values, the density and the concentration in the inertial range go through a phase transition from a finite continuous smooth distribution to a singular multifractal spatial distribution. Multifractality is associated with scaling, which would not hold if the solenoidal and the potential components of the flow scaled differently, producing transport which is not self-similar. Thus, we propose that the transition occurs when the difference of the scaling exponents of the components, decreasing with Ma, becomes small. Under the smallness assumption, the particles' volumes obey a power-law evolution. That, by the use of conservation of the total volume of the flow, entails the volumes' shrinking to zero with probability 1 and formation of a singular distribution. We discuss various concepts of multifractality and propose a way to calculate fractal dimensions from numerical or experimental data. We derive a simple expression for the spectrum of fractal dimensions of isothermal turbulence and describe limitations of lognormality. The expression depends on a single parameter: the scaling exponent of the density spectrum. We demonstrate that the pair-correlation function of the tracer concentration has the Markov property. This implies applicability of the compressible version of the Kraichnan turbulence model. We use the model to derive an explicit expression for the tracer pair correlation that demonstrates their smooth transition to multifractality and confirms the transition's mechanism. The obtained fractal dimension explains previous numerical observations. Our results have potentially important implications for astrophysical problems such as star formation as well as technological applications such as supersonic combustion. As an example, we demonstrate the strong increase of planetesimal formation rate at the transition. We prove that finiteness of internal energy implies vanishing of the sum of Lyapunov exponents in the dissipation range. Our study leads to the question of whether the fluid density which is an active field that reacts back on the transporting flow and the passive concentration of tracers must coincide in the steady state. This is demonstrated to be crucial both theoretically and experimentally. The fields' coincidence is provable at small Mach numbers; however, at finite Mach numbers, the assumption of mixing is needed, which we demonstrate to be not self-evident because of the possibility of self-organization.

Gyrotactic phytoplankton in laminar and turbulent flows: A dynamical systems approach

Gyrotactic algae are bottom heavy, motile cells whose swimming direction is determined by a balance between a buoyancy torque directing them upwards and fluid velocity gradients. Gyrotaxis has, in recent years, become a paradigmatic model for phytoplankton motility in flows. The essential attractiveness of this peculiar form of motility is the availability of a mechanistic description which, despite its simplicity, revealed predictive, rich in phenomenology, easily complemented to include the effects of shape, feedback on the fluid and stochasticity (e.g., in cell orientation). In this review we consider recent theoretical, numerical and experimental results to discuss how, depending on flow properties, gyrotaxis can produce inhomogeneous phytoplankton distributions on a wide range of scales, from millimeters to kilometers, in both laminar and turbulent flows. In particular, we focus on the phenomenon of gyrotactic trapping in nonlinear shear flows and in fractal clustering in turbulent flows. We shall demonstrate the usefulness of ideas and tools borrowed from dynamical systems theory in explaining and interpreting these phenomena.

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.

Relative velocity distribution of inertial particles in turbulence: A numerical study

The distribution of relative velocities between particles provides invaluable information on the rates and characteristics of particle collisions. We show that the theoretical model of Gustavsson and Mehlig [K. Gustavsson and B. Mehlig, J. Turbul. 15, 34 (2014)], within its anticipated limits of validity, can predict the joint probability density function of relative velocities and separations of identical inertial particles in isotropic turbulent flows with remarkable accuracy. We also quantify the validity range of the model. The model matches two limits (or two types) of relative motion between particles: one where pair diffusion dominates (i.e., large coherence between particle motion) and one where caustics dominate (i.e., large velocity differences between particles at small separations). By using direct numerical simulation combined with Lagrangian particle tracking, we assess the model prediction in homogeneous and isotropic turbulence. We demonstrate that, when sufficient caustics are present at a given separation and the particle response time is significantly smaller than the integral time scales of the flow, the distribution exhibits the same universal power-law form dictated by the correlation dimension as predicted by the model of Gustavsson and Mehlig. In agreement with the model, no strong dependency on the Taylor-based Reynolds number is observed.

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.

Phytoplankton's motion in turbulent ocean

We study the influence of turbulence on upward motion of phytoplankton. Interaction with the flow is described by the Pedley-Kessler model considering spherical microorganisms. We find a range of parameters when the upward drift is only weakly perturbed or when turbulence completely randomizes the drift direction. When the perturbation is small, the drift is either determined by the local vorticity or is Gaussian. We find a range of parameters where the phytoplankton interaction with the flow can be described consistently as diffusion of orientation in effective potential. By solving the corresponding Fokker-Planck equation we find exponential steady-state distribution of phytoplankton's propulsion orientation. We further identify the range of parameters where phytoplankton's drift velocity with respect to the flow is determined uniquely by its position. In this case, one can describe phytoplankton's motion by a smooth flow and phytoplankton concentrates on fractal. We find fractal dimensions and demonstrate that phytoplankton forms vertical stripes in space with a nonisotropic pair-correlation function of concentration increased in the vertical direction. The probability density function of the distance between two particles obeys power law with the negative exponent given by the ratio of integrals of the turbulent energy spectrum. We find the regime of strong clustering where the exponent is of order one so that turbulence increases the rate of collisions by a large factor. The predictions hold for Navier-Stokes turbulence and stand for testing.

Clustering of vertically constrained passive particles in homogeneous isotropic turbulence

We analyze the dynamics of small particles vertically confined, by means of a linear restoring force, to move within a horizontal fluid slab in a three-dimensional (3D) homogeneous isotropic turbulent velocity field. The model that we introduce and study is possibly the simplest description for the dynamics of small aquatic organisms that, due to swimming, active regulation of their buoyancy, or any other mechanism, maintain themselves in a shallow horizontal layer below the free surface of oceans or lakes. By varying the strength of the restoring force, we are able to control the thickness of the fluid slab in which the particles can move. This allows us to analyze the statistical features of the system over a wide range of conditions going from a fully 3D incompressible flow (corresponding to the case of no confinement) to the extremely confined case corresponding to a two-dimensional slice. The background 3D turbulent velocity field is evolved by means of fully resolved direct numerical simulations. Whenever some level of vertical confinement is present, the particle trajectories deviate from that of fluid tracers and the particles experience an effectively compressible velocity field. Here, we have quantified the compressibility, the preferential concentration of the particles, and the correlation dimension by changing the strength of the restoring force. The main result is that there exists a particular value of the force constant, corresponding to a mean slab depth approximately equal to a few times the Kolmogorov length scale η, that maximizes the clustering of the particles.

Gravity-driven enhancement of heavy particle clustering in turbulent flow

Heavy particles suspended in a turbulent flow settle faster than in a still fluid. This effect stems from a preferential sampling of the regions where the fluid flows downward and is quantified here as a function of the level of turbulence, of particle inertia, and of the ratio between gravity and turbulent accelerations. By using analytical methods and detailed, state-of-the-art numerical simulations, settling is shown to induce an effective horizontal two-dimensional dynamics that increases clustering and reduce relative velocities between particles. These two competing effects can either increase or decrease the geometrical collision rates between same-size particles and are crucial for realistic modeling of coalescing particles.

Turbulent fluid acceleration generates clusters of gyrotactic microorganisms

The motility of microorganisms is often biased by gradients in physical and chemical properties of their environment, with myriad implications on their ecology. Here we show that fluid acceleration reorients gyrotactic plankton, triggering small-scale clustering. We experimentally demonstrate this phenomenon by studying the distribution of the phytoplankton Chlamydomonas augustae within a rotating tank and find it to be in good agreement with a new, generalized model of gyrotaxis. When this model is implemented in a direct numerical simulation of turbulent flow, we find that fluid acceleration generates multifractal plankton clustering, with faster and more stable cells producing stronger clustering. By producing accumulations in high-vorticity regions, this process is fundamentally different from clustering by gravitational acceleration, expanding the range of mechanisms by which turbulent flows can impact the spatial distribution of active suspensions.

Distribution of velocity gradients and rate of caustic formation in turbulent aerosols at finite Kubo numbers

In a one-dimensional model for a turbulent aerosol (inertial particles suspended in a random flow) we compute the distributions of particle-velocity gradients and the rate of caustic formation at finite but small Kubo numbers, Ku, for arbitrary Stokes numbers, St. Our results are consistent with those obtained earlier in the limit Ku→0 and St→∞ such that Ku(2)St remains constant. We show how finite-time correlations and nonergodic effects influence the inertial-particle dynamics at finite but small Kubo numbers.