Micromixing modelling of concentration fluctuations in inhomogeneous turbulence in the convective boundary layer

The micromixing technique, widely used in engineering calculations of mixing and chemical reaction, is extended to atmospheric boundary-layer flows. In particular, a model based on the interaction-by-exchange-with-the-conditional-mean (IECM) micromixing approach is formulated to calculate concentration fluctuation statistics for a line source and a point source in inhomogeneous and non-Gaussian turbulence in the convective boundary layer. The mixing time scale is parameterised as a linear function of time with the intercept value determined by the source size at small times. Good agreement with laboratory data for the intensity of concentration fluctuations is obtained with a value of 0.9 for the coefficient of the linear term in the time-scale parameterisation for a line source, and a value of 0.6 for a point source. Calculation of higher-order moments of the concentration field for a line source shows that non-Gaussian effects persist into the vertically well-mixed region. The cumulative distribution function predicted by the model for a point source agrees reasonably well with laboratory data, especially in the far field. In the limit of zero mixing time scale, the model reduces to a meandering plume model, thus enabling the concentration variance to be partitioned into meandering and relative components. The meandering component is shown to be more persistent for a point source than for a line source.     

A Lagrangian stochastic model for the trajectories of particle pairs and its application to the prediction of concentration variance within plant canopies

A simple Lagrangian stochastic model for the trajectories of particle pairs in high Reynolds-number turbulent flows is presented. In this model, the velocities of particle pairs are initially correlated but subsequently each particle moves independently. The independent single-particle trajectories are simulated using Thomson's model [J. Fluid Mech. 180, 529–556, 1987]. This two-particle model exactly satisfies the well-mixed condition for Gaussian turbulence when length scales, characterizing the two-point Eulerian velocity correlation function, vanish. Temperature variances, due to heat released as a passive scalar from an elevated plane source, within a model plant canopy (Coppin et al. Boundary Layer Meteorol. 35, 167–191, 1986) are shown to be well predicted by the model. It is suggested that for strongly inhomogeneous flows, the two-point Eulerian velocity function is of secondary importance in determining the simulated trajectories of particle pairs compared to the importance of ensuring satisfaction of the two-to-one constraint (Borgas and Sawford. J. Fluid Mech. 279, 69–99, 1994); i.e ensuring that one-particle statistics obtained from the two-particle model are the same as those obtained from the corresponding one-particle model. Limitations of this modelling approach are discussed.     

Comments on the 'Universality of the Lagrangian Velocity Structure Function Constant (C0) ACROSS DIFFERENT KINDS OF TURBULENCE’

Recently Du ( Boundary-Layer Meteorology 83, 207–219, 1997) estimated the value of the Lagrangian velocity structure constant, C0, in the inertial subrange by comparing experimental diffusion data and simulation results obtained with the one-dimensional form of Thomson's model ( J. Fluid Mech. 180, 529–556, 1987). Du reported that for several different flows (grid turbulence, a wind-tunnel boundary layer and the atmospheric surface layer under neutral stratification) the value of C0 is 3.0±0.5. Here, it is shown that optimal model agreement with experimental diffusion data for the wind-tunnel boundary layer is, in fact, obtained when C0=5.0 ± 0.5. It is also shown that accounting for the skewness of velocity statistics and finite Reynolds number effects does not significantly change this estimate for the value of C0. It is suggested that one-dimensional Lagrangian stochastic models are inconsistent with the supposed universality of C0.     

On Trajectory Curvature as a Selection Criterion for Valid Lagrangian Stochastic Dispersion Models

Recently Wilson and Flesch (Boundary-Layer Meteorology, 84, 411-426, 1997) suggested that the average increment d z to the orientation = arctan(w/u) of the Lagrangian velocity-fluctuation vector can be used to distinguish the better Lagrangian stochastic models within the well-mixed class. Here it is demonstrated that the specification of d z constitutes neither a sufficient or universally applicable criterion to distinguish the better Lagrangian stochastic models within the well-mixed class. The hypothesis made by Wilson and Flesch that Lagrangian stochastic models with /P_E irrotational are zero-spin models, having d z=0, is proven     

On The Influence Of The Eulerian Velocity PDF Closure On The Eddy Diffusion Coefficient

The flux-gradient model, often used to describe turbulent dispersion, implicitly defines an eddy diffusion coefficient K that is known to be related to the Eulerian probability density function (pdf) of the turbulent velocity field. In the strict limit of applicability of Fick's law, the relationship between K and the pdf is used to investigate the influence of non-Gaussianity on dispersion in homogeneous turbulence. A bi-Gaussian pdf is used as a closure model that allows for separate studies of skewness and kurtosis variations. The choice of model parameters can have a significant influence on K, especially when the pdf is bimodal. Both arbitrariness of the closure and bimodality are then reduced using the maximum entropy criterion for the selection of the free parameter of the closure scheme, together with the assumption that the model is valid only for those values of the parameters for which a unimodal pdf is possible. The variations of K are found to be sensitive to both skewness and kurtosis showing a more complex behaviour than that found in literature.     

UNIVERSALITY OF THE LAGRANGIAN VELOCITY STRUCTURE FUNCTION CONSTANT (C0) ACROSS DIFFERENT KINDS OF TURBULENCE

In this paper, we evaluate the Lagrangian velocity structure function constant, C0, in the inertial subrange by comparing experimental diffusion data and simulation results obtained with applicable Lagrangian stochastic models. We find in several different flows (grid turbulence, laboratory boundary-layer flow and the atmospheric surface layer under neutral stratification) the value for C0 is 3.0 ± 0.5. We also identify the reasons responsible for earlier studies having not reached the present result.     

Forward Lagrangian stochastic simulation of a transient source in the atmospheric surface layer

We show that a forward Lagrangian stochastic (LS) model simulates well the ensemble-averaged concentration transient due to a short time (5 min) point source in the uniform atmospheric surface layer. In LS models, computational particles, which may not descend below ground level, are necessarily reflected at an imposed (artificial) boundary above ground. Model results were rather insensitive to the placing of the lower reflection boundary, and no definite benefit stemmed from including a parametrization for unresolved delays/displacements beneath the lower boundary.     

A Simple Model of Concentration Fluctuations in Neutrally Buoyant Clouds

This paper presents a new model of concentration fluctuations for neutrally buoyant gas clouds dispersing in a wind tunnel. It is derived from a series of exact results, which apply in the hypothetical case when there is no molecular diffusion, coupled with a probability density function model previously used to describe steady releases of contaminant. A simple self-similar relationship between the evolution of the concentration intensity and mean is established. As a first step the time independent variant of the model, applicable to a continuous plume, is tested against some previously published experimental data for steady wind-tunnel releases. Comparisons of experimental results and model predictions at different downwind positions, heights and source geometry are presented. Then, the results for the time dependent model, applicable to instantaneous releases, are discussed. The experimental evidence presented here supports the self-similar relationship established earlier. The implications for modelling higher moments of concentration and the fixed point probability density function are investigated.     

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.     

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.     

Joint PDF Modelling of Turbulent Flow and Dispersion in an Urban Street Canyon

The joint probability density function (PDF) of turbulent velocity and concentration of a passive scalar in an urban street canyon is computed using a newly developed particle-in-cell Monte Carlo method. Compared to moment closures, the PDF methodology provides the full one-point one-time PDF of the underlying fields containing all higher moments and correlations. The small-scale mixing of the scalar released from a concentrated source at the street level is modelled by the interaction by exchange with the conditional mean (IECM) model, with a micro-mixing time scale designed for geometrically complex settings. The boundary layer along no-slip walls (building sides and tops) is fully resolved using an elliptic relaxation technique, which captures the high anisotropy and inhomogeneity of the Reynolds stress tensor in these regions. A less computationally intensive technique based on wall functions to represent the boundary layers and its effect on the solution are also explored. The calculated statistics are compared to experimental data and large-eddy simulation. The present work can be considered as the first example of computation of the full joint PDF of velocity and a transported passive scalar in an urban setting. The methodology proves successful in providing high level statistical information on the turbulence and pollutant concentration fields in complex urban scenarios.     

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.     

Forward-in-time and Backward-in-time Dispersion in the Convective Boundary Layer: the Concentration Footprint

The turbulence field obtained using a large-eddy simulation model is used to simulate particle dispersion in the convective boundary layer with both forward-in-time and backward-in-time modes. A Lagrangian stochastic model is used to treat subgrid-scale turbulence. Results of forward dispersion match both laboratory experiments and previous numerical studies for different release heights in the convective boundary layer. Results obtained from backward dispersion show obvious asymmetry when directly compared to results from forward dispersion. However, a direct comparison of forward and backward dispersion has no apparent physical meaning and might be misleading. Results of backward dispersion can be interpreted as three-dimensional or generalized concentration footprints, which indicate that sources in the entire boundary layer, not only sources at the surface, may influence a concentration measurement at a point. Footprints at four source heights in the convective boundary layer corresponding to four receptors are derived using forward and backward dispersion methods. The agreement among footprints derived with forward and backward methods illustrates the equivalence between both approaches. The paper shows explicitly that Lagrangian simulations can yield identical footprints using forward and backward methods in horizontally homogeneous turbulence.     

Estimation of the Lagrangian Velocity Structure Function Constant C0 by Large-Eddy Simulation

The inertial subrange Kolmogorov constant C ₀, which determines the effective turbulent diffusion in velocity space, plays an important role in the Lagrangian modelling of pollutants. A wide range of values of the constant are found in the literature, most of them determined at low Reynolds number and/or under different assumptions. Here we estimate the constant C ₀ by tracking an ensemble of Lagrangian particles in a planetary boundary layer simulated with a large-eddy simulation model and analysing the Lagrangian velocity structure function in the inertial subrange. The advantage of this technique is that it easily allows Reynolds numbers to be achieved typical of convective turbulent flows. Our estimates of C ₀ is C ₀=4.3±0.3 consistent with values found in the literature     

A Simple Parameterisation for Flux Footprint Predictions

Flux footprint functions estimate the location and relative importance of passive scalar sources influencing flux measurements at a given receptor height. These footprint estimates strongly vary in size, depending on receptor height, atmospheric stability, and surface roughness. Reliable footprint calculations from, e.g., Lagrangian stochastic models or large-eddy simulations are computationally expensive and cannot readily be computed for long-term observational programs. To facilitate more accessible footprint estimates, a scaling procedure is introduced for flux footprint functions over a range of stratifications from convective to stable, and receptor heights ranging from near the surface to the middle of the boundary layer. It is shown that, when applying this scaling procedure, footprint estimates collapse to an ensemble of similar curves. A simple parameterisation for the scaled footprint estimates is presented. This parameterisation accounts for the influence of the roughness length on the footprint and allows for a quick but precise algebraic footprint estimation.     

A Lagrangian Stochastic Model for Heavy Particle Dispersion in the Atmospheric Marine Boundary Layer

The dispersion of heavy particles and pollutants is often simulated with Lagrangian stochastic (LS) models. Although these models have been employed successfully over land, the free surface at the air-sea interface complicates the implementation of traditional LS models. We present an adaptation of traditional LS models to the atmospheric marine boundary layer (MBL), where the bottom boundary is represented by a realistic wavy surface that moves and deforms. In addition, the correlation function for the turbulent flow following a particle is extended to the anisotropic, unsteady case. Our new model reproduces behaviour for Lagrangian turbulence in a stratified air flow that departs only slightly from the expected behaviour in isotropic turbulence. When solving for the trajectory of a heavy particle in the air flow, the modelled turbulent forcing on the particle also behaves remarkably well. For example, the spectrum of the turbulence at the particle location follows that of a massless particle for time scales approximately larger than the Stokes’ particle response time. We anticipate that this model will prove especially useful in the context of sea-spray dispersion and its associated momentum, sensible and latent heat, and gas fluxes between spray droplets and the atmosphere.     

Comparison of the Langrangian Footprint

We compare flux and concentration footprint estimates of athree-dimensional Lagrangian stochastic dispersion modelapplying backward trajectories with the results of ananalytical footprint model by Kormann and Meixner.The comparison is performed for varying stability regimesof the surface layer as well as for different measurementheights. In general, excellent correspondence is found.     

An approximate analytical solution for the deposition of heavy particles released from an elevated line source

This work re-examines and further develops an analytical solution for the deposition swath of heavy particles released in the atmosphere from an elevated source over uniform terrain, correcting the particle diffusivity for the crossing trajectory effect. The revised (approximate) analytical solution proves to be accurate within 20% over a wide range of micrometeorological conditions and particle size, despite its neglect of the turbulence component of the deposition flux. It compares very satisfactorily with experimental data and with the simulations of a Lagrangian stochastic model, provided the variable U(H)/w _g ≤7 (ratio of the mean horizontal wind speed at source height to the particle settling velocity). In this domain of validity, simple formulae relating the statistics of the deposition swath to U(H)/w _g are derived.