Gregory Falkovich

Particles and fields in fluid turbulence

The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e. to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in non-equilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scale-invariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo.

Acceleration of rain initiation by cloud turbulence

Vapour condensation in cloud cores produces small droplets that are close to one another in size. Droplets are believed to grow to raindrop size by coalescence due to collision. Air turbulence is thought to be the main cause for collisions of similar-sized droplets exceeding radii of a few micrometres, and therefore rain prediction requires a quantitative description of droplet collision in turbulence. Turbulent vortices act as small centrifuges that spin heavy droplets out, creating concentration inhomogeneities and jets of droplets, both of which increase the mean collision rate. Here we derive a formula for the collision rate of small heavy particles in a turbulent flow, using a recently developed formalism for tracing random trajectories. We describe an enhancement of inertial effects by turbulence intermittency and an interplay between turbulence and gravity that determines the collision rate. We present a new mechanism, the ‘sling effect’, for collisions due to jets of droplets that become detached from the air flow. We conclude that air turbulence can substantially accelerate the appearance of large droplets that trigger rain.

Bottleneck phenomenon in developed turbulence

It is shown how viscosity increases turbulence level in the inertial interval by suppressing turbulent transfer.

Normal and anomalous scaling of the fourth-order correlation function of a randomly advected passive scalar.

Advection of a passive scalar θ in d = 2 by a large-scale velocity field rapidly changing in time is considered. The Gaussian feature of the passive scalar statistics in the convective interval was discovered in [1]. Here we examine deviations from the Gaussianity: we obtain analytically the simultaneous fourth-order correlation function of θ. Explicit expressions for fourth-order objects, like 〈(θ1 − θ2) 〉 are derived.

The laminar-turbulent transition in a fibre laser

Studying the transition from a linearly stable coherent laminar state to a highly disordered state of turbulence is conceptually and technically challenging, and of great interest because all pipe and channel flows are of that type. In optics, understanding how a system loses coherence, as spatial size or the strength of excitation increases, is a fundamental problem of practical importance. Here, we report our studies of a fibre laser that operates in both laminar and turbulent regimes. We show that the laminar phase is analogous to a one-dimensional coherent condensate and the onset of turbulence is due to the loss of spatial coherence. Our investigations suggest that the laminar-turbulent transition in the laser is due to condensate destruction by clustering dark and grey solitons. This finding could prove valuable for the design of coherent optical devices as well as systems operating far from thermodynamic equilibrium.

Lessons from hydrodynamic turbulence

Turbulent flows, with their irregular behavior, confound any simple attempts to understand them. But physicists have succeeded in identifying some universal properties of turbulence and relating them to broken symmetries.

Intermittent distribution of heavy particles in a turbulent flow

The retardation of weakly inertial particles depends on the acceleration of the ambient fluid, so the particle concentration n is determined by the divergence of Lagrangian acceleration which we study by direct numerical simulations. We demonstrate that the second moment of the concentration coarse-grained over the scale r behaves as an approximate power law: 〈nr2〉∼rα. We study the dependencies of the exponent α on the Reynolds number, of the Stokes number, and on the settling velocity. We find numerically that the theoretical lower bound previously suggested [Falkovich et al., Nature 419, 151 (2002)] correctly estimates the order of magnitude (within a factor 2 to 4) as well as the dependencies on the Reynolds, Stokes, and Froude numbers. The discrepancy grows with the Reynolds number and the Froude number. We analyze the possible physical mechanism responsible for that behavior.The retardation of weakly inertial particles depends on the acceleration of the ambient fluid, so the particle concentration n is determined by the divergence of Lagrangian acceleration which we study by direct numerical simulations. We demonstrate that the second moment of the concentration coarse-grained over the scale r behaves as an approximate power law: 〈nr2〉∼rα. We study the dependencies of the exponent α on the Reynolds number, of the Stokes number, and on the settling velocity. We find numerically that the theoretical lower bound previously suggested [Falkovich et al., Nature 419, 151 (2002)] correctly estimates the order of magnitude (within a factor 2 to 4) as well as the dependencies on the Reynolds, Stokes, and Froude numbers. The discrepancy grows with the Reynolds number and the Froude number. We analyze the possible physical mechanism responsible for that behavior.

Sling Effect in Collisions of Water Droplets in Turbulent Clouds

Abstract The effect of turbulence on the collision rate between droplets in clouds is investigated. Because of their inertia, water droplets can be shot out of curved streamlines of the turbulent airflow. The contribution of such a “sling effect” in the collision rate of the same-size water droplets is described and evaluated. It is shown that already for turbulence with the dissipation rate 103 cm2 s−3, the sling effect gives a contribution to the collision rate of 15-μm droplets comparable to that due to the local velocity gradient. That may explain why the formulas based on the local velocity gradient consistently underestimate the turbulent collision rate, even with the account of preferential concentration.

Instantons and Intermittency

We describe the method for finding the non-Gaussian tails of the probability distribution function (PDF) for solutions of a stochastic differential equation, such as the convection equation for a passive scalar, the random driven Navier-Stokes equation, etc. The existence of such tails is generally regarded as a manifestation of the intermittency phenomenon. Our formalism is based on the WKB approximation in the functional integral for the conditional probability of large fluctuation. We argue that the main contribution to the functional integral is given by a coupled field-force configuration\char22{}the instanton. As an example, we examine the correlation functions of the passive scalar u advected by a large-scale velocity field \ensuremath{\delta} correlated in time. We find the instanton determining the tails of the generating functional, and show that it is different from the instanton that determines the probability distribution function of high powers of u. We discuss the simplest instantons for the Navier-Stokes equation. \textcopyright{} 1996 The American Physical Society.

Statistics of a Passive Scalar Advected by a Large-Scale 2D Velocity Field: Analytic Solution

Steady statistics of a passive scalar advected by a random two-dimensional flow of an incompressible fluid is described in the range of scales between the correlation length of the flow and the diffusion scale. That corresponds to the so-called Batchelor regime where the velocity is replaced by its large-scale gradient. The probability distribution of the scalar in the locally comoving reference frame is expressed via the probability distribution of the line stretching rate. The description of line stretching can be reduced to a classical problem of the product of many random matrices with a unit determinant. We have found the change of variables that allows one to map the matrix problem onto a scalar one and to thereby prove the central limit theorem for the stretching rate statistics. The proof is valid for any finite correlation time of the velocity field. Whatever the statistics of the velocity field, the statistics of the passive scalar (averaged over time locally in space) is shown to approach gaussianity with increase in the Peclet number