Stationarity of Lagrangian Velocity¶in Compressible Environments

We study the transport of a passive tracer particle by a random d-dimensional, Gaussian, compressible velocity field. It is well known, since the work of Lumley, see [13], and Port and Stone, see [20], that the observations of the velocity field from the moving particle, the so-called Lagrangian velocity process, are statistically stationary when the field itself is incompressible. In this paper we study the question of stationarity of Lagrangian observations in compressible environments. We show that, given sufficient temporal decorrelation of the velocity statistics, there exists a transformation of the original probability measure, under which the Lagrangian velocity process is time stationary. The transformed probability is equivalent to the original measure. As an application of this result we prove the law of large numbers for the particle trajectory. Received: 1 May 2001 / Accepted: 4 December 2001     

Statistics of particle pair relative velocity in the homogeneous shear flow

Small scale clustering of inertial particles and relative velocity of particle pairs have been fully characterized for statistically steady homogeneous isotropic flows. Depending on the particle Stokes relaxation time, the spatial distribution of the disperse phase results in a multi-scale manifold characterized by local particle concentration and voids and, because of finite inertia, the two nearby particles have high probability to exhibit large relative velocities. Both effects might explain the speed-up of particle collision rate in turbulent flows. Recently it has been shown that the large scale geometry of the flow plays a crucial role in organizing small scale particle clusters. For instance, a mean shear preferentially orients particle patterns. In this case, depending on the Stokes time, anisotropic clustering may occur even in the inertial range of scales where the turbulent fluctuations which drive the particles have already recovered isotropy. Here we consider the statistics of particle pair relative velocity in the homogeneous shear flow, the prototypical flow which manifests anisotropic clustering at small scales. We show that the mean shear, by imprinting anisotropy on the large scale velocity fluctuations, dramatically affects the particle relative velocity distribution even in the range of small scales where the anisotropic mechanisms of turbulent kinetic energy production are sub-dominant with respect to the inertial energy transfer which drives the carrier fluid velocity towards isotropy. We find that the particles’ populations which manifest strong anisotropy in their relative velocities are the same which exhibit small scale clustering. In contrast to any Kolmogorov-like picture of turbulent transport these phenomena may persist even below the smallest dissipative scales where the residual level of anisotropy may eventually blow-up. The observed anisotropy of particle relative velocity and spatial configuration is suggested to influence the directionality of the collision probability, as inferred on the basis of the so-called “ghost collision” model.     

Brownian motion with active fluctuations

We study the effect of different types of fluctuation on the motion of self-propelled particles in two spatial dimensions. We distinguish between passive and active fluctuations. Passive fluctuations (e.g., thermal fluctuations) are independent of the orientation of the particle. In contrast, active ones point parallel or perpendicular to the time dependent orientation of the particle. We derive analytical expressions for the speed and velocity probability density for a generic model of active Brownian particles, which yields an increased probability of low speeds in the presence of active fluctuations in comparison to the case of purely passive fluctuations. As a consequence, we predict sharply peaked Cartesian velocity probability densities at the origin. Finally, we show that such a behavior may also occur in non-Gaussian active fluctuations and discuss briefly correlations of the fluctuating stochastic forces.     

Turbulentlike fluctuations in quasistatic flow of granular media

We analyze particle velocity fluctuations in a simulated granular system subjected to homogeneous quasistatic shearing. We show that these fluctuations share the following scaling characteristics of fluid turbulence in spite of their different physical origins: (i) scale-dependent probability distribution with non-Gaussian broadening at small time scales; (ii) spatial power spectrum of the velocity field showing a power-law decay, reflecting long-range correlations and the self-affine nature of the fluctuations; and (iii) superdiffusion of particles with respect to the mean background flow.     

Weakly driven anomalous diffusion in non-ergodic regime: an analytical solution

We derive the probability density of a diffusion process generated by nonergodic velocity fluctuations in presence of a weak potential, using the Liouville equation approach. The velocity of the diffusing particle undergoes dichotomic fluctuations with a given distribution ψ(τ) of residence times in each velocity state. We obtain analytical solutions for the diffusion process in a generic external potential and for a generic statistics of residence times, including the non-ergodic regime in which the mean residence time diverges. We show that these analytical solutions are in agreement with numerical simulations.     

First-Passage and Extreme-Value Statistics of a Particle Subject to a Constant Force Plus a Random Force

We consider a particle which moves on the x axis and is subject to a constant force, such as gravity, plus a random force in the form of Gaussian white noise. We analyze the statistics of first arrival at point x ₁ of a particle which starts at x ₀ with velocity v ₀. The probability that the particle has not yet arrived at x ₁ after a time t, the mean time of first arrival, and the velocity distribution at first arrival are all considered. We also study the statistics of the first return of the particle to its starting point. Finally, we point out that the extreme-value statistics of the particle and the first-passage statistics are closely related, and we derive the distribution of the maximum displacement m=max  _t [x(t)].     

Experimental investigation of particle velocity distributions in windblown sand movement

With the PDPA (Phase Doppler Particle Analyzer) measurement technology, the probability distributions of particle impact and lift-off velocities on bed surface and the particle velocity distributions at different heights are detected in a wind tunnel. The results show that the probability distribution of impact and lift-off velocities of sand grains can be expressed by a log-normal function, and that of impact and lift-off angles complies with an exponential function. The mean impact angle is between 28° and 39°, and the mean lift-off angle ranges from 30° to 44°. The mean lift-off velocity is 0.81–0.9 times the mean impact velocity. The proportion of backward-impacting particles is 0.05–0.11, and that of backward-entrained particles ranges from 0.04 to 0.13. The probability distribution of particle horizontal velocity at 4 mm height is positive skew, the horizontal velocity of particles at 20 mm height varies widely, and the variation of the particle horizontal velocity at 80 mm height is less than that at 20 mm height. The probability distribution of particle vertical velocity at different heights can be described as a normal function. Supported by the National Natural Science Foundation of China (Grant No. 10532030) and the National Basic Research Program of China (Grant No. G2000048702)     

The renewal equation for persistent diffusion

Persistent diffusion in one dimension, in which the velocity of the diffusing particle is a dichotomic Markov process, is considered. The flow is non-Markovian, but the position and the velocity together constitute a Markovian diffusion process. We solve the coupled forward Kolmogorov equations and the coupled backward Kolmogorov equations with appropriate initial conditions, to establish a generalized (matrix) form of the renewal equation connecting the probability densities and first passage time distributions for persistent diffusion.     

On the Asymptotic Behavior of an Ornstein-Uhlenbeck Process with Random Forcing

We study the asymptotic behavior of an inertial tracer particle in a random force field. We show that there exists a probability measure, under which the process describing the velocity and environment seen from the vantage point of the moving particle is stationary and ergodic. This measure is equivalent to the underlying probability for the Eulerian flow. As a consequence of the above we obtain the law of large numbers for the trajectory of the tracer. Moreover, we prove also some decorrelation properties of the velocity of the particle, which lead to the existence of a non-degenerate asymptotic covariance tensor. The research of both authors was supported by the Polish Committee for Scientific Research (KBN) grant No. 2PO3A03123.     

Random walk near the surface

The random walk of a particle on a three-dimensional semi-infinite lattice is considered. In order to study the effect of the surface on the random walk, it is assumed that the velocity of the particle depends on the distance to the surface. Moreover it is assumed that at any point the particle may be absorbed with a certain probability. The probability of the return of the particle to the starting point and the average time of eventual return are calculated. The dependence of these quantities on the distance to the surface, the probability of absorption and the properties of the surface is discussed. The method of generating functions is used.     

CFD-DEM simulation of three-dimensional aeolian sand movement

A three-dimensional CFD-DEM model is proposed to investigate the aeolian sand movement.The results show that the mean particle horizontal velocity can be expressed by a power function of heights.The probability distribution of the impact and lift-off velocities of particles can be described by a log-normal function,and that of the impact and lift-off angles can be expressed by an exponential function.The probability distribution of particle horizontal velocity at different heights can be described as a lognormal function,while the probability distribution of longitudinal and vertical velocity can be described as a normal function.The comparison with previous two-dimensional calculations shows that the variations of mean particle horizontal velocity along the heights in two-dimensional and three-dimensional models are similar.However,the mean particle density of the two-dimensional model is larger than that in reality,which will result in the overestimation of sand transportation rate in the two-dimensional calculation.The study also shows that the predicted probability distributions of particle velocities are in good agreement with the experimental results.     

Adiabatic approximation on the impact-parameter model

In this paper we study the impact-parameter model for the scattering of a light particle by two heavy ones in the case when the coupling constants of the potentials acting on the light particle due to the presence of the two heavy ones are the same. We study the asymptotic behavior of the transition probability when the relative velocity of the heavy particles goes to zero. We show that the probability of a transition can be arbitrarily close to the one of no transition.     

Asymptotic solution of the Wang-Uhlenbeck recurrence time problem

A Langevin particle is initiated at the origin with positive velocity. Its trajectory is terminated when it returns to the origin. In 1945, Wang and Uhlenbeck posed the problem of finding the joint probability density function (PDF) of the recurrence time and velocity, naming it "the recurrence time problem". We show that the short-time asymptotics of the recurrence PDF is similar to that of the integrated Brownian motion, solved in 1963 by McKean. We recover the long-time t(-3/2) decay of the first arrival PDF of diffusion by solving asymptotically an appropriate variant of McKean's integral equation.     

Rotational intermittency and turbulence induced lift experienced by large particles in a turbulent flow

The motion of a large, neutrally buoyant, particle freely advected by a turbulent flow is determined experimentally. We demonstrate that both the translational and angular accelerations exhibit very wide probability distributions, a manifestation of intermittency. The orientation of the angular velocity with respect to the trajectory, as well as the translational acceleration conditioned on the spinning velocity, provides evidence of a lift force acting on the particle.     

Derivation of the probability distribution of velocities of a detector-recorded brownian particle

The probability distribution of velocities in the given space region (detector) is found for particles of a passive admixture in a stream of external gas. Since direct calculation of the above probability density involves significant difficulties, the solution is based on the classical problem of the probability distribution of coordinates and velocity of a Brownian particle at a fixed time. Analyzing dependence of the solution on the parameters of the initial problem, we obtain conditions under which the assumptions on the character of particle motion hold true. State University, Nizhny Novgorod, Russia. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 41, No. 10, pp. 1301–1313, October 1998.     

Clustering of the low-inertia particle number density field in random divergence-free hydrodynamic flows

We consider the diffusion of the low-inertia particle number density field in random divergence-free hydrodynamic flows. The principal feature of this diffusion is the divergence of the particle velocity field, which results in clustering of the particle number density field. This phenomenon is coherent, occurs with a unit probability, and must show up in almost all realizations of the process dynamics. We calculate the statistical parameters that characterize clustering in three-dimensional and two-dimensional random fluid flows and in a rapidly rotating two-dimensional random flow. In the former case, the vortex component of the random divergence-free flow generates a vortex component of the low-inertia particle velocity field, which generates a potential component of the velocity field through advection. By contrast, in the case of rapid rotation, a potential component of the velocity field is generated directly by the vortex component of the random divergence-free flow (linear problem).     

气固两相流中颗粒-颗粒随机碰撞新模型

本文提出一种气固两相流中计算颗粒－颗粒碰撞新模型．该模型提出一种碰撞概率新概念,与已有模型相比,新模型对碰撞概率的思考另辟蹊径,由所得公式可以确定影响碰撞概率大小的因素．此外,该模型提出在所研究控制体中选择虚拟颗粒的方法,特别是对虚拟颗粒的粒径、速度和随机数之间提出相关性准则．它弥补了以往模型的缺陷．经过计算,验证了新模型的合理性．     

Some remarks concerning Nambu mechanics

The quantum probability flux of a particle integrated over time and a distant surface gives the probability for the particle crossing that surface at some time. We prove the free flux-across-surfaces theorem, which was conjectured by Combes, Newton and Shtokhamer (Phys. Rev. D. 11 (1975), 366), and which relates the integrated quantum flux to the usual quantum mechanical formula for the cross-section. The integrated quantum flux is equal to the probability of outward crossings of surfaces by Bohmian trajectories in the scattering regime.     

Statistics of Lagrangian velocities in turbulent flows

We present a generalized Fokker-Planck equation for the joint position-velocity probability distribution of a single fluid particle in a turbulent flow. Based on a simple estimate, the diffusion term is related to the two-point two-time Eulerian acceleration-acceleration correlation. Dimensional analysis yields a velocity increment probability distribution with normal scaling v approximately t(1/2). However, the statistics need not be Gaussian.