A Stochastic Time Series Model for Threshold Crossing Statistics of Concentration Fluctuations in non-Intermittent Plumes

A numerical stochastic model is developed for the upcrossing rate across a specified threshold concentration. The model assumes that the concentration time series at a given spatial point within a dispersing plume can be approximated as a first-order Markovian process designed to be consistent with a given time-invariant concentration probability density function (pdf). The model requires only the specification of a concentration pdf with a given mean and variance and a concentration fluctuation integral time scale. Predicted upcrossing rates are compared with atmospheric plume concentration data obtained from a point source near the ground. For this data set, a log-normal pdf is found to give better estimates of the threshold crossing rate than a gamma pdf.     

A Study of Concentration Fluctuations in Instantaneous Clouds Dispersing in the Atmospheric Surface Layer for Relative Turbulent Diffusion: Basic Descriptive Statistics

A series of tracer experiments studying the statistical properties of concentration fluctuations in clouds dispersing in the atmospheric surface layer is described and analyzed. Experiments were conducted at downwind fetches between about 200 and 1200 m, under a wide range of atmospheric conditions ranging from very unstable to moderately stable stratification. The present experiments have addressed basic requirements not met by past field experiments involving instantaneously released clouds; namely, the experiments provided repeat realizations of instantaneously released clouds measured with high-resolution concentration detectors, accompanied by the contemporaneous acquisition of high-quality meteorological and turbulence measurements.Extensive analyses are performed on the cloud concentration data in the framework of relative diffusion. Ensembles of cloud concentration realizations have been constructed. From these ensembles, crosswind and time profiles of the ensemble-mean concentration, concentration variance, ensemble-mean dosage, and dosage variance are obtained. The behaviour of the time profiles of the integral time scale of cloud concentration fluctuations is studied. The use of surface-layer similarity theory for the analysis of the downwind variation of a number of cloud quantities (e.g., cloud size and duration, cloud centre ensemble-mean concentration and dosage, cloud centre concentration and dosage variance, cloud centre integral time scale) is shown to be an effective basis for ordering these quantities. Furthermore, a number of approximate universal relationships describing the behavior of these cloud quantities has been derived. Finally, it is shown that the scaled crosswind and time profiles of ensemble-mean concentration and concentration variance as well as the scaled time profiles of the concentration fluctuation integral time scale exhibit self-similar forms that are independent of atmospheric stratification and downwind fetch.     

An Analytical Model For Threshold Crossing Rates Of Concentration Fluctuations In Dispersing Plumes

A simple analytical model is developed for the meanupcrossing rate of plume concentration fluctuations assuming that thisprocess can be well approximated by a lognormal process. The resultingexpression requires only the specification of the in-plume fluctuationintensity and in-plume Taylor micro-time scale and, hence, does notexplicitly involve the joint probability density function of theconcentration and its derivative. The analytical model provides agood fit to some field measurements of the mean upcrossing rate ina dispersing plume.     

A Concentration pdf for the Relative Dispersion of a Contaminant Plume in the Atmosphere

Observations of the dispersion of a contaminant plume in theatmospheric boundary layer, obtained using a Lidar, are analysedin a coordinate frame relative to the instantaneous centre of massof the plume. To improve the estimates of relative dispersionstatistics, maximum entropy inversion is used to remove noise fromthe Lidar concentration profiles before carrying out the analysis.A parametric form is proposed for the probability density function(pdf) of concentration, consisting of a mixture of a betadistribution and of a generalised Pareto distribution (GPD). Thispdf allows for the possibility of a unimodal or bimodaldistribution, and is shown to give a satisfactory fit toobservations from a range of positions relative to the source. Thevariation of the fitted parameters with crossplume location isanalysed, and the maximum possible concentration is found todecrease away from the plume centre.     

Spatial Resolution in the Measurement of Concentration Fluctuations

In a turbulent flow, a miscible contaminant is confined to sheets and strands of the very thin Batchelor conduction cut-off length. This fact has been surmised for some time and has recently been observed directly through high precision measurements. This fine-scaled texture of the contaminant concentration field makes it an extraordinary challenge to achieve (experimentally) adequate continuum scale resolution, particularly in important environmental flows such as the atmospheric boundary layer. In this paper, an extrapolation scheme is proposed whereby the systematic measurement (with known sample volumes) of the lower-order moments of the concentration fluctuations are used to approximate the true, perfectly resolved values, and hence to approximate the true probability density function. Such a scheme relies on empirical data, and so the need for more experiments designed to investigate the effects of spatial resolution cannot be over-emphasized.     

Some Simple Statistical Models For Relative And Absolute Dispersion

Observations of the dispersion of a contaminant plume in the atmospheric boundary layer, obtained using a Lidar, are analysed in the coordinate frame relative to the instantaneous centre of mass of the plume, as well as the absolute (or fixed) coordinate frame. The study extends the work presented in a previous article, which analysed the structure of the probability density function (pdf) of concentration within the relative coordinate frame. Firstly, the plume displacement component, or plume meander, is analysed and a simple parametric form for the pdf of the plume centreline position is suggested. This is then used to analyse the accuracy and applicability of absolute framework statistical quantities obtained by a convolution of the relative frame statistical quantity with the plume centreline pdf.     

Probabilistic Model for Concentration Fluctuations in Compact-Source Plumes in an Urban Environment

A comprehensive model for the prediction of concentration fluctuations in plumes dispersing in the complex and highly disturbed wind flows in an urban environment is formulated. The mean flow and turbulence fields in the urban area are obtained using a Reynolds-averaged Navier-Stokes (RANS) flow model, while the standard k-ϵ turbulence model (k is the turbulence kinetic energy and ϵ is the viscous dissipation rate) is used to close the model. The RANS model provides a specification of the velocity statistics of the highly disturbed wind flow in the urban area, required for the solution of the transport equations for the mean concentration and concentration variance (both of which are formulated in the Eulerian framework). A physically-based formulation for the scalar dissipation time scale t _d , required for the closure of the transport equation for , is presented. This formulation relates t _d to an inner time scale corresponding to “internal” concentration fluctuation associated with relative dispersion, rather than an outer time scale associated with the entire portion of the fluctuation spectrum. The two lowest-order moments of concentration ( and ) are used to determine the parameters of a pre-chosen functional form for the concentration probability density function (clipped-gamma distribution). Results of detailed comparisons between a water-channel experiment of flow and dispersion in an idealized obstacle array and the model predictions for mean flow, turbulence kinetic energy, mean concentration, concentration variance, and concentration probability density function are presented.     

Simulation of Time Series of Concentration Fluctuations in Atmospheric Dispersion Using a Correlation-distortion Technique

Short-duration fluctuations in the concentration of airborne substances can be important in a variety of atmospheric dispersion problems, especially when assessing the risks posed by harmful materials. This paper discusses a simulation technique for generating individual realisations of fluctuating concentration time series in dispersing plumes based on target probability distributions and spectral functions. The scheme uses a correlation-distortion approach to simulate these time series. Gaussian processes with modified spectral characteristics are generated and then transformed to yield non-Gaussian processes with the desired spectral characteristics. The simulation scheme is initially developed for a single receptor, and is then extended to model pairs of correlated time series at two receptors. In fact, the modelling technique can be generalised to an arbitrary number of receptors and this provides, in principal, an approach that is applicable to a wide class of similar problems (such as the modelling of instantaneous puff releases or the response of line-of-sight detection systems). The simulation technique is illustrated using observations made during recent field experiments, conducted both in the United Kingdom and in the U.S.A., investigating the short-range dispersion of a passive tracer.     

Computational Fluid Dynamical Modelling of Concentration Fluctuations in an Idealized Urban Area

Turbulent mixing induces variability in concentration that is important in many applications, such as reactive plumes, risk assessments or odour impact analyses (when the effects can have time scales on the order of a second). In urban canopies, the variability may be modified by the presence of buildings. Our purpose is to study concentration fluctuation variance in built-up areas using an Eulerian approach. We performed numerical simulations with the computational fluid dynamics model Mercure_Saturne, which is a three-dimensional model adapted to atmospheric flow and pollutant dispersion. We use a k − ϵ turbulence closure and predict the concentration variance with a transport equation model. The model performance is evaluated with the near-full scale experiment MUST (Mock Urban Setting Test), a field experiment conducted in Utah’s West Desert Test Center. The modelled root-mean-square of the concentration fluctuations is compared to measurements for 20 of the MUST trials. The model shows good agreement with the measurements, with the fraction of predictions within a factor of two of observations of 60.1%, with better results for horizontal lines of detectors than for the detectors on vertical masts (with fractions of predictions within a factor of two of observations of respectively 66.4% and 52.6%). The influence of different parameters on the fluctuation variance is also studied and we show the importance of taking into account the stability of the stratification when modelling the turbulent kinetic energy.     

A Comparison Of The Detailed Structure In Dispersing Tracer Plumes Measured In Grid-Generated Turbulence With A Meandering Plume Model Incorporating Internal Fluctuations

A meandering plume model that explicitly incorporatesinternal fluctuations has been developed and used to model the evolutionof concentration fluctuations in point-source plumes in grid turbulenceobtained from a detailed water-channel simulation. This fluctuating plumemodel includes three physical parameters: the mean plume spread in fixedcoordinates, which represents the outer plume length scale; the meaninstantaneous plume spread in coordinates attached to the instantaneousplume centroid, which represents the inner plume length scale; and, theconcentration fluctuation intensity in the meandering reference frame,which represents the in-plume fluctuation scale. These parameters arespecified in terms of a set of coupled dynamical equations that modeltheir development with downstream distance from the source. Explicitexpressions for the concentration moments of arbitrary integral orderand the concentration probability density function have been obtainedfrom the fluctuating plume model. Detailed comparisons of model predictionsagainst water-channel measurements for the first four concentrationmoments and the concentration probability distributions generally showvery good overall quantitative agreement. Exact quantitative conditions,expressed in terms of the physical parameters of the fluctuating plumemodel, have been derived for the emergence of off-centreline peaks inthe concentration variance profile. These quantitative conditions havebeen illustrated in terms of a diagram of states of the dispersing plume,and the qualitatively different regimes of plume concentration variancebehaviour on this state diagram have been identified and characterized.     

COMPARISON AMONG EMPIRICAL PROBABILITY DENSITY FUNCTIONS OF THE VERTICAL VELOCITY IN THE SURFACE LAYER BASED ON HIGHER ORDER CORRELATIONS

Vertical velocity fluctuations were measured in theatmospheric surface layer by means of an ultrasonic anemometer andhigher order correlations were calculated on two time series, recordedin unstable and neutral conditions, and selected for the wholemeasurement period on the basis of the inversion test (stationaritytest). Comparisons have been made between observed and predictedcorrelations by considering Gaussian joint-PDF and Gram-Charlierseries expansions truncated to the fourth and sixth order as doneearlier by Frenkiel and Klebanoff. A bi-Gaussian PDF, given by amixture of two Gaussian PDFs, has also been considered. This lasthas been constructed assuming that either the first three or the firstfour moments are given, and the relationships between correlationfunctions of different order are derived. The departure from Gaussianbehaviour in both stability conditions is derived. Though Gram-Charlier series expansions show a good correspondence toexperimental reality, their use as non-Gaussian probabilitydistributions cannot be suggested in theoretical approaches andshould be considered with care in practical applications, due topossible occurrences of small negative probabilities. The resultsshown in this paper support the applicability of the bi-Gaussian PDFcreated using up to the fourth moment.     

The Interference of Higher-Order Statistics of the Concentration Field Produced by Two Point Sources According to a Generalized Fluctuating Plume Model

The higher-order correlation functions for the concentrationfluctuations arising from a two-point-source configuration have beencalculated analytically within the context of the phenomenology of afluctuating plume model (viz., a meandering plume model that explicitlyincorporates internal fluctuations). Explicit expressions for thesecond-, third-, and fourth-order correlationfunctions between the concentrationfluctuations produced by two point sources are given in terms of the sourceseparation d and the five physically based parameters that define thegeneralized fluctuating plume model: namely, the absolute plume dispersion,_a, which determines the outer plume length scale; the relative plume dispersion, _r, which determines the inner plume length scale; the fluctuation intensity, i_r, in relative coordinates, which determines the internal concentration fluctuation level; the correlation coefficient, r,between the positions of the centroids of the two interfering plumes; and,the correlation coefficient, r^*, between the concentration fluctuationsof the two plumes in relative coordinates, which determines the degree ofinternal mixing of the two scalars. Furthermore, the form of the totalconcentration probability density function arising from the interferenceproduced by two point sources is presented. Predictions for the second-ordercorrelation function, , and for the total concentration probabilitydensity function have been compared with some new experimental data fora two-point-source configuration in grid turbulence generated in awater-channel simulation. These results are in good agreement with the dataand suggest that the analytical model for the second-order correlationfunction and the total concentration probability density function canreproduce many qualitative trends in the interaction of plumes from twosources.     

The Relationship between Skewness and Kurtosis of A Diffusing Scalar

It has been demonstrated that in turbulent dispersion, there exists a quadratic relationship between the skewness (S) and kurtosis (K) statistics obtained from continuous, elevated sources of scalar contaminant released into both convective and stable atmospheric boundary layers. Specifically, one observes that where A and B are empirically fitted constants that depend on the flow. For two reasons, this is potentially useful information in regard to modelling the probability density function (PDF) of a diffusing scalar. First, since many PDFs have a signature relationship between their skewness and kurtosis, candidate models can immediately be either accepted or rejected depending upon whether they conform to the quadratic curve that is observed experimentally. Second, if one intends to model the PDF by inverting a limited number of moments, the task is reduced when there is a functional relationship between the standardized third and fourth moments. The aforementioned relationship has been corroborated by others who have examined data over a wide range of experimental configurations. However, from one flow to another, there appears to be a non-negligible variability in the two fitting constants of the quadratic curve. In this paper we put forth a framework to help explain this phenomenon, and we also attempt to predict how these parameters vary in space and/or time. Our point is illustrated with well-resolved data from a wind-tunnel, grid-turbulence, plume experiment.     

Effect of Vertical Wind Shear on Concentration Fluctuation Statistics in a Point Source Plume

Measurements of concentration fluctuation intensity, intermittency factor, and integral time scale were made in a water channel for a plume dispersing in a well-developed, rough surface, neutrally stable, boundary layer, and in grid-generated turbulence with no mean velocity shear. The water-channel simulations apply to full-scale atmospheric plumes with very short averaging times, on the order of 1–4 min, because plume meandering was suppressed by the water-channel side walls. High spatial and temporal resolution vertical and crosswind profiles of fluctuations in the plume were obtained using a linescan camera laser-induced dye tracer fluorescence technique. A semi-empirical algebraic mean velocity shear history model was developed to predict these concentration statistics. This shear history concentration fluctuation model requires only a minimal set of parameters to be known: atmospheric stability, surface roughness, vertical velocity profile, and vertical and crosswind plume spreads. The universal shear history parameter used was the mean velocity shear normalized by surface friction velocity, plume travel time, and local mean wind speed. The reference height at which this non-dimensional shear history was calculated was important, because both the source and the receptor positions influence the history of particles passing through the receptor position.     

The High Concentration Tails Of The Probability Density Function Of A Dispersing Scalar In The Atmosphere

The distribution function for concentrations of a scalar pollutant dispersing in the turbulent atmosphere has a finite domain that is bounded above and below. Three methods, based on extreme value statistics, are used to obtainestimates for the upper bound and to describe the high concentration tailbehaviour of the distribution; all three methods are applied to concentrationdata obtained from experimental atmospheric releases. Quantile quantile (QQ)plots are used to assess the goodness of fit of the resulting estimates of thedistribution, and also to compare the performance of the three methods. Thepredicted values for the upper bound are orders of magnitude less than thesource concentration, illustrating that molecular diffusion has a large effecton the high concentrations.     

基于Weierstrass-Mandelbrot函数的分形风速脉动仿真

实际风速脉动普遍具有自相似分形特征,而传统的谐波合成法和线性滤波法仿真的风速脉动均不具有自相似分形特征.因此,基于随机型Weierstrass-Mandelbrot函数,设计了一种能够仿真自相似风速脉动的方案.其中,表征风速脉动自相似特征的重要参数分形维度可与湍流惯性区能谱的幂指数建立联系.将该方案仿真的风速脉动与实际风速脉动一些重要的统计特征,如功率谱和概率密度函数等,进行了比较,结果表明提出的新方案能有效仿真风速脉动的中高频变化及其概率分布特征.     

Modelling of concentration fluctuations in canopy turbulence

The knowledge of the concentration probability density function (pdf) is of importance in a number of practical applications, and a Lagrangian stochastic (LS) pdf model has been developed to predict statistics and concentration pdf generated by continuous releases of non-reactive and reactive substances in canopy generated turbulence. Turbulent dispersion is modelled using a LS model including the effects of wind shear and along-wind turbulence. The dissipation of concentration fluctuations associated with turbulence and molecular diffusivity is simulated by an Interaction by Exchange with the Conditional Mean (IECM) micromixing model. A general procedure to obtain the micromixing time scale needed in the IECM model useful in non-homogeneous conditions and for single and multiple scalar sources has been developed. An efficient algorithm based on a nested grid approach with particle splitting, merging techniques and time averaging has been used, thus allowing the calculation for cases of practical interest. The model has been tested against wind-tunnel experiments of single line and multiple line releases in a canopy layer. The approach accounted for chemical reactions in a straightforward manner with no closure assumptions, but here the validation is limited to non-reacting scalars.