G. K. Batchelor

Bio: G. K. Batchelor is an academic researcher. The author has contributed to research in topics: Turbulent diffusion & Diffusion (business). The author has an hindex of 1, co-authored 1 publications receiving 621 citations.

Diffusion in a field of homogeneous turbulence II. The relative motion of particles

Abstract: In an earlier paper the author considered the problem of the turbulent diffusion, relative to a fixed origin, of a cloud of marked fluid whose initial position is given. This was found to be determined by the initial shape of the cloud and the statistical properties of the displacement of a single fluid particle. The present paper is concerned with the relative diffusion of the cloud, i.e. with the tendency to change its shape, or, more precisely, with that part of the relative diffusion which is described by the probability that a given vector y can lie with both its ends in marked fluid at time t. This aspect of the relative diffusion is found to be determined by the initial shape of the cloud and the statistical properties of the separation, at time t, of two fluid particles of given initial separation. The statistical functions introduced to describe the relative diffusion are found to be related to Richardson's distance-neighbour function.The relative diffusion of two particles is a more complex problem than diffusion of a single particle about a fixed origin because the relative diffusion depends on the initial separation. The closer the particles are together, the smaller is the range of eddy sizes that contributes to their relative velocity; for the same reason, relative diffusion is an accelerating process, until the particles are very far apart and wander independently. The hypothesis is made that if the initial separation is small enough, the probability distribution of the separation will tend asymptotically to a form independent of the initial separation, before the particles move independently. This hypothesis permits various simple deductions, some of which make use of Kolmogoroff's similarity theory. The important question of the description of the relative diffusion by a differential equation is examined; Richardson has put forward one suggestion, and another, based on a normal distribution of the separation, is made herein.

Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking.

Abstract: Modern microscopic techniques following the stochastic motion of labelled tracer particles have uncovered significant deviations from the laws of Brownian motion in a variety of animate and inanimate systems. Such anomalous diffusion can have different physical origins, which can be identified from careful data analysis. In particular, single particle tracking provides the entire trajectory of the traced particle, which allows one to evaluate different observables to quantify the dynamics of the system under observation. We here provide an extensive overview over different popular anomalous diffusion models and their properties. We pay special attention to their ergodic properties, highlighting the fact that in several of these models the long time averaged mean squared displacement shows a distinct disparity to the regular, ensemble averaged mean squared displacement. In these cases, data obtained from time averages cannot be interpreted by the standard theoretical results for the ensemble averages. Here we therefore provide a comparison of the main properties of the time averaged mean squared displacement and its statistical behaviour in terms of the scatter of the amplitudes between the time averages obtained from different trajectories. We especially demonstrate how anomalous dynamics may be identified for systems, which, on first sight, appear to be Brownian. Moreover, we discuss the ergodicity breaking parameters for the different anomalous stochastic processes and showcase the physical origins for the various behaviours. This Perspective is intended as a guidebook for both experimentalists and theorists working on systems, which exhibit anomalous diffusion.

Percolation, statistical topography, and transport in random media

Abstract: A review of classical percolation theory is presented, with an emphasis on novel applications to statistical topography, turbulent diffusion, and heterogeneous media. Statistical topography involves the geometrical properties of the isosets (contour lines or surfaces) of a random potential $\ensuremath{\psi}(\mathrm{x})$. For rapidly decaying correlations of $\ensuremath{\psi}$, the isopotentials fall into the same universality class as the perimeters of percolation clusters. The topography of long-range correlated potentials involves many length scales and is associated either with the correlated percolation problem or with Mandelbrot's fractional Brownian reliefs. In all cases, the concept of fractal dimension is particularly fruitful in characterizing the geometry of random fields. The physical applications of statistical topography include diffusion in random velocity fields, heat and particle transport in turbulent plasmas, quantum Hall effect, magnetoresistance in inhomogeneous conductors with the classical Hall effect, and many others where random isopotentials are relevant. A geometrical approach to studying transport in random media, which captures essential qualitative features of the described phenomena, is advocated.

Modeling non‐Fickian transport in geological formations as a continuous time random walk

Abstract: [1] Non-Fickian (or anomalous) transport of contaminants has been observed at field and laboratory scales in a wide variety of porous and fractured geological formations. Over many years a basic challenge to the hydrology community has been to develop a theoretical framework that quantitatively accounts for this widespread phenomenon. Recently, continuous time random walk (CTRW) formulations have been demonstrated to provide general and effective means to quantify non-Fickian transport. We introduce and develop the CTRW framework from its conceptual picture of transport through its mathematical development to applications relevant to laboratoryand field-scale systems. The CTRW approach contrasts with ones used extensively on the basis of the advectiondispersion equation and use of upscaling, volume averaging, and homogenization. We examine the underlying assumptions, scope, and differences of these approaches, as well as stochastic formulations, relative to CTRW. We argue why these methods have not been successful in fitting actual measurements. The CTRW has now been developed within the framework of partial differential equations and has been generalized to apply to nonstationary domains and interactions with immobile states (matrix effects). We survey models based on multirate mass transfer (mobile-immobile) and fractional derivatives and show their connection as subsets within the CTRW framework.

Lagrangian PDF Methods for Turbulent Flows

Abstract: Lagrangian Probability Density Function (PDF) methods have arisen the past 10 years as a union between PDF methods and stochastic Lagrangian models, similar to those that have long been used to study turbulent dispersion. The methods provide a computationally-tractable way of calculating the statistics, of inhomogeneous turbulent flows of practical importance, and are particularly attractive if chemical reactions are involved. The information contained at this level of closure--equivalent to a multi-time Lagrangian joint pdf--is considerably more than that provided by moment closures. The computational implementation is conceptually simple and natural. At a given time, the turbulent flow is represented by a large number of particles, each having its own set of properties--position, velocity, composition etc. These properties evolve in time according to stochastic model equations, so that the computational particles simulate fluid particles. The particle-property time series contain information equivalent to the multi-time Lagrangian joint pdf. But, at a fixed time, the ensemble of particle properties contains no multi-point information: Each particle can be considered to be sampled from a different realization of the flow. (Hence two particles can have the same position, but different velocities and compositions.) It is generally acknowledged (e.g. Reynolds 1990) that many different approaches have important roles to play in tackling the problems posed by turbulent flows. Each approach has its own strengths and weaknesses.