Michael S. Borgas

A family of stochastic models for two-particle dispersion in isotropic homogeneous stationary turbulence

A family of Lagrangian stochastic models for the joint motion of particle pairs in isotropic homogeneous stationary turbulence is considered. The Markov assumption and well-mixed criterion of Thomson (1990) are used, and the models have quadratic-form functions of velocity for the particle accelerations. Two constraints are derived which formally require that the correct one-particle statistics are obtained by the models. These constraints involve the Eulerian expectation of the ‘acceleration’ of a fluid particle with conditioned instantaneous velocity, given either at the particle, or at some other particles position. The Navier-Stokes equations, with Gaussian Eulerian probability distributions, are shown to give quadratic-form conditional accelerations, and models which satisfy these two constraints are found. Dispersion calculations show that the constraints do not always guarantee good one-particle statistics, but it is possible to select a constrained model that does. Thomsons model has good one-particle statistics, but is shown to have unphysical conditional accelerations. Comparisons of relative dispersion for the models are made.

A skewed meandering plume model for concentration statistics in the convective boundary layer

The meandering plume technique, which assumes that the total plume dispersion can be split into independent meander and relative dispersion components, is especially suited for modelling concentration (fluctuation) statistics in the convective boundary layer (CBL) with its large-scale turbulent motions. We develop a simple and practical meandering plume model for CBL applications that accounts for the skewed and inhomogeneous turbulence characteristics of the convective flow. The meander component is derived from a one-particle Lagrangian stochastic dispersion model by requiring that the meander and relative dispersion components correctly balance the first two total dispersion moments. Balancing of the third total moment implies a skewed relative dispersion, for which a bi-Gaussian distribution is used. The relative dispersion variance is parameterised with an extended asymptotic formulation for travel times much smaller than the Lagrangian integral time scale. For large travel times, the relative dispersion variance approaches the total dispersion variance. The in-plume fluctuations in the relative coordinate system are accounted for via the gamma probability density function. Laboratory data and large-eddy simulation results on total, relative and meander spreads are used to examine the model parameterisations and results. A requirement of the meandering plume model, that it should give the same mean concentration distribution as that obtained by the one-particle Lagrangian approach, is virtually fulfilled. Comparison of the model predictions of the concentration fluctuation intensity with existing laboratory data highlights the important contribution of in-plume fluctuations, which are normally neglected in meandering plume models. The paper also describes limitations of the new model and indicates the scope for further refinements.

Relative dispersion in isotropic turbulence. Part 2. A new stochastic model with Reynolds-number dependence

A new model for Lagrangian particle-pair separation in turbulent flows is developed and compared with data from direct numerical simulations (DNS) of isotropic turbulence. The model formulation emphasizes (i) non-Gaussian behaviour in Eulerian and Lagrangian statistics, (ii) the occurrence of large separation velocities, (iii) the role of straining and streaming flow structure as recognized in kinematic simulations of turbulence, and (iv) the role of conditionally averaged accelerations in stochastic modelling of turbulent relative dispersion. Previous stochastic models of relative dispersion have produced unrealistic behaviour, particularly in the dissipation subrange where viscous effects are important, which have led to questions on the adequacy of stochastic modelling. However, this failure can now be recognized as inadequate detail in formulation, which is explained and rectified in this paper. The model is quasi-one-dimensional in nature, and is focused on the statistics of particle-pair separation distance and its rate of change, referred to as the separation speed. Detailed comparisons are presented at several Reynolds numbers using the DNS database reported in a companion paper (Part 1). Up to fourth-order moments for these quantities are examined, as well as the separation-distance probability density function, which is discussed in the context of recent claims of Richardson scaling in the literature. The model is able to account for the spatial representation of straining regions as well as incompressibility of the flow, and is shown to reproduce strong non-Gaussianity and intermittency in the Lagrangian separation statistics observed in DNS. Comparisons with recent physical experiments are also remarkably consistent. This work demonstrates that stochastic models when properly formulated are effective and efficient representations of the dispersion process and this general class of models therefore possess great utility for calculations of both one-particle and two-particle dispersion. The techniques developed in this paper will facilitate such general model development.

Conditional and unconditional acceleration statistics in turbulence

In this paper we study acceleration statistics from laboratory measurements and direct numerical simulations in three-dimensional turbulence at Taylor-scale Reynolds numbers ranging from 38 to 1000. Using existing data, we show that at present it is not possible to infer the precise behavior of the unconditional acceleration variance in the large Reynolds number limit, since empirical functions satisfying both the Kolmogorov and refined Kolmogorov theories appear to fit the data equally well. We also present entirely new data for the acceleration covariance conditioned on the velocity, showing that these conditional statistics are strong functions of velocity, but that when scaled by the unconditional variance they are only weakly dependent on Reynolds number. For large values of the magnitude u of the conditioning velocity we speculate that the conditional covariance behaves like u6 and show that this is qualitatively consistent with the stretched exponential tails of the unconditional acceleration proba...

THE SMALL-SCALE STRUCTURE OF ACCELERATION CORRELATIONS AND ITS ROLE IN THE STATISTICAL THEORY OF TURBULENT DISPERSION

Some previously accepted results for the from of one- and two-particle Langrangian turbulence statistics within the inertial subrange are corrected and reinterpreted using dimensional methods and kinematic constraints. These results have a fundamental bearing on the statistical theory of turbulent dispersion.

Relative dispersion in isotropic turbulence. Part 1. Direct numerical simulations and Reynolds-number dependence

The relative dispersion of fluid particle pairs in isotropic turbulence is studied using direct numerical simulation, in greater detail and covering a wider Reynolds number range than previously reported. A primary motivation is to provide an important resource for stochastic modelling incorporating information on Reynolds-number dependence. Detailed results are obtained for particle-pair initial separations from less than one Kolmogorov length scale to larger than one integral length scale, and for Taylor-scale Reynolds numbers from about 38 to 230. Attention is given to several sources of uncertainty, including sample size requirements, value of the one-particle Lagrangian Kolmogorov constant, and the temporal variability of space-averaged quantities in statistically stationary turbulence. Relative dispersion is analysed in terms of the evolution of the magnitude and angular orientation of the two-particle separation vector. Early-time statistics are consistent with the Eulerian spatial structure of the flow, whereas the large-time behaviour is consistent with particle pairs far apart moving independently. However, at intermediate times of order several Kolmogorov time scales, and especially for small initial separation and higher Reynolds numbers, both the separation distance and its rate of change (called the separation speed) are highly intermittent, with flatness factors much higher than those of Eulerian velocity differences in space. This strong intermittency is a consequence of relative dispersion being affected by a wide range of length scales in the turbulent flow as some particle pairs drift relatively far apart. Numerical evidence shows that substantial dispersion occurs in the plane orthogonal to the initial separation vector, which implies that the orientation of this vector has, especially for small initial separation, only limited importance.

A comparison of intermittency models in turbulence

A variety of turbulent intermittency models from the literature is listed and directly contrasted. The multifractal notation [J. Feder, Fractals (Plenum, New York, 1988)] is used to present these results. Two of the models in particular, Yamazaki’s [J. Fluid Mech. 219, 181 (1990)] B‐model and a prediction from renormalization‐group analysis (Yakhot et al.22 [Phys. Fluids A 1, 289 (1989)]), are closely examined and inherent flaws are pointed out.

Turbulent dispersion with broken reflectional symmetry

We consider dispersion in axisymmetric turbulence which lacks reflectional symmetry. A stochastic equation for the Lagrangian evolution of the velocity of a fluid particle, which is appropriate for infinite Reynolds number turbulence, is used to model the dispersion. Such equations are now common as Lagrangian dispersion models for atmospheric transport problems, but are only strictly well founded for isotropic homogeneous turbulence. It is the minimalist variation from this state of affairs that is considered here. Axisymmetry is the most highly symmetric turbulence that can be suitably analysed by these techniques, spherical symmetry being equivalent to full isotropy in the class of models considered. This simple relaxation of full symmetry leads to oscillations of the Lagrangian velocity autocorrelation, oscillatory growth of the dispersion, significant reduction of dispersion for fixed turbulence kinetic energy and dissipation rate, spiralling fluid-particle trajectories, and tracer fluxes orthogonal to gradients (skew diffusion). The mean fluid-particle angular momentum is an important parameter.

An efficient Lagrangian stochastic model of vertical dispersion in the convective boundary layer

We consider the one-dimensional case of vertical dispersion in the convective boundary layer (CBL) assuming that the turbulence field is stationary and horizontally homogeneous. The dispersion process is simulated by following Lagrangian trajectories of many independent tracer particles in the turbulent flow field, leading to a prediction of the mean concentration. The particle acceleration is determined using a stochastic di⁄erential equation, assuming that the joint evolution of the particle velocity and position is a Markov process. The equation consists of a deterministic term and a random term. While the formulation is standard, attention has been focused in recent years on various ways of calculating the deterministic term using the well-mixed condition incorporating the Fokker—Planck equation. Here we propose a simple parameterisation for the deterministic acceleration term by approximating it as a quadratic function of velocity. Such a function is shown to represent well the acceleration under moderate velocity skewness conditions observed in the CBL. The coeƒcients in the quadratic form are determined in terms of given turbulence statistics by directly integrating the Fokker—Planck equation. An advantage of this approach is that, unlike in existing Lagrangian stochastic models for the CBL, the use of the turbulence statistics up to the fourth order can be made without assuming any predefined form for the probability distribution function (PDF) of the velocity. The main strength of the model, however, lies in its simplicity and computational eƒciency. The dispersion results obtained from the new model are compared with existing laboratory data as well as with those obtained from a more complex Lagrangian model in which the deterministic acceleration term is based on a bi-Gaussian velocity PDF. The comparison shows that the new model performs well. ( 1999 Elsevier Science Ltd. All rights reserved.

Horton's Laws and the fractal nature of streams

Stream networks and catchment basins are characterized by numerous fractal dimensions, the traditional one being the mainstream length as a function of length scale which we denote by d. We consider an ideal Horton system and define fractal dimensions based on (1) the subcatchment length-area relationship, which gives D1; (2) the total length of streams as a function of scale, which gives a network similarity dimension D2; (3) the mainstream length-area relationship (D3), and (4) the total area of streams as a function of areal scale (D4). These fractal dimensions are related to the values of the bifurcation ratio, the stream length ratio, and the stream area ratio. Their value should be indirectly estimated from these Horton parameters. An improved understanding of the relationships between the fractal dimension and Hortons laws is obtained by defining an eigenarea of a stream of a particular Horton order. This is the area of the drainage basin which is not drained by streams of immediately lower order. The variation of eigenareas with scale controls the value of D1, which equals D3 for hydrologically realistic systems. In such cases D4 = 2. This paper assumes that river systems are Hortonian in nature and that measured departures from Hortonianity are due to statistical variability. Estimates of the Horton parameters and of the fractal dimensions are therefore subject to uncertainty. Two sources of statistical variability are considered herein: The first source of statistical variability is due to the fluctuations in stream numbers, stream lengths, and stream areas. These fluctuations affect the values of the Horton parameters, which are traditionally estimated using a graphical technique. We show that an analytical technique (which corresponds to a constrained graphical technique) is preferable. The second source of variability that we consider arises from the fact that estimates of Hortons parameters and fractal dimensions are always made on truncated systems because the particular scales of measurement determine the order of the catchment. In a Horton system it is straight forward to estimate D1, which is defined exactly at all scales. Direct attempts to estimate D2 and D3 (as opposed to indirect attempts via the Horton parameters) are difficult because their true value is approached asymptotically in the limit of infinite-order catchments.