On the influence of non-Gaussianity on turbulent transport

Non-Gaussianity effects, first of all the influence of the third and fourth moments of the velocity probability density function, have to be assessed for higher-order closure models of turbulence and Lagrangian modelling of turbulent dispersion in complex flows. Whereas the role and the effects of the third moments are relatively well understood as essential for the explanation of specific observed features of the fully developed convective boundary layer, there are indications that the fourth moments may also be important, but little is known about these moments. Therefore, the effects of non-Gaussianity are considered for the turbulent motion of particles in non-neutral flows without fully developed convection, where the influence of the fourth moments may be expected to be particularly essential. The transport properties of these flows can be characterized by a diffusion coefficient which reflects these effects. It is shown, for different vertical velocity distributions, that the intensity of turbulent transport may be enhanced remarkably by non-Gaussianity. The diffusion coefficient is given as a modification of the Gaussian diffusivity, and this modifying factor is found to be determined to a very good approximation by the normalized fourth moment of the vertical velocity distribution function. This provides better insight into the effect of fourth moments and explains the varying importance of third and fourth moments in different flows.     

Boundedness of Turbulent Temperature Probability Distributions,and their Relation to the Vertical Profile in the Convective Boundary Layer

Higher-order moments, minima and maxima of turbulent temperature and water vapour mixing ratio probability density functions measured with an eddy-covariance system near the ground were related to each other and to vertical boundary-layer profiles of the same scalars obtained through airborne soundings. The dependence of kurtosis on squared skewness showed a kurtosis intercept below the Gaussian expectation, suggesting a compression of the probability density function by the presence of natural boundaries. This hypothesis was corroborated by comparing actual minima and maxima of turbulent fluctuations to estimates obtained from the first four sample moments by fitting a four-parameter beta distribution. The most sharply defined boundaries were found for the minima of temperature datasets during the day, indicating that negative temperature fluctuations at the sensor are limited by the availability of lower temperatures in the boundary layer. By comparison to vertical profiles, it could be verified that the turbulent minimum of temperature near the ground is close to the minimum of potential temperature in the boundary layer. The turbulent minimum of water vapour mixing ratio was found to be equal to the mixing ratio at a height above the minimum of the temperature profile. This height roughly agrees with the top of the non-local unstable domain according to bulk Richardson number profiles. We conclude that turbulence statistics measured near the surface cannot be solely explained by local effects, but contain information about the whole boundary layer including the entrainment zone.     

Simplified expressions for adjusting higher-order turbulent statistics obtained from open path gas analyzers

When density fluctuations of scalars such as CO₂ are measured with open-path gas analyzers, the measured vertical turbulent flux must be adjusted to take into account fluctuations induced by ‘external effects’ such as temperature and water vapour. These adjustments are needed to separate the effects of surface fluxes responsible for ‘natural’ fluctuations in CO₂ concentration from these external effects. Analogous to vertical fluxes, simplified expressions for separating the ‘external effects’ from higher-order scalar density turbulence statistics are derived. The level of complexity in terms of input to these expressions are analogous to that of the Webb–Pearman–Leuning (WPL), and are shown to be consistent with the conservation of dry air. It is demonstrated that both higher-order turbulent moments such as the scalar variances, the mixed velocity-scalar covariances, and the two-scalar covariance require significant adjustments due to ‘external effects’. The impact of these adjustments on the turbulent CO₂ spectra, probability density function, and dimensionless similarity functions derived from flux-variance relationships are also discussed.     

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.     

Dispersion of a Passive Scalar Fluctuating Plume in a Turbulent Boundary Layer. Part III: Stochastic Modelling

We analyze the reliability of the Lagrangian stochastic micromixing method in predicting higher-order statistics of the passive scalar concentration induced by an elevated source (of varying diameter) placed in a turbulent boundary layer. To that purpose we analyze two different modelling approaches by testing their results against the wind-tunnel measurements discussed in Part I (Nironi et al., Boundary-Layer Meteorology, 2015, Vol. 156, 415–446). The first is a probability density function (PDF) micromixing model that simulates the effects of the molecular diffusivity on the concentration fluctuations by taking into account the background particles. The second is a new model, named VP\(\varGamma \), conceived in order to minimize the computational costs. This is based on the volumetric particle approach providing estimates of the first two concentration moments with no need for the simulation of the background particles. In this second approach, higher-order moments are computed based on the estimates of these two moments and under the assumption that the concentration PDF is a Gamma distribution. The comparisons concern the spatial distribution of the first four moments of the concentration and the evolution of the PDF along the plume centreline. The novelty of this work is twofold: (i) we perform a systematic comparison of the results of micro-mixing Lagrangian models against experiments providing profiles of the first four moments of the concentration within an inhomogeneous and anisotropic turbulent flow, and (ii) we show the reliability of the VP\(\varGamma \) model as an operational tool for the prediction of the PDF of the concentration.     

Extension of a fluctuating plume model of tracer dispersion to a sheared boundary layer and to a large array of obstacles

Fluctuating plume models provide a useful conceptual paradigm in the understanding of plume dispersion in a turbulent flow. In particular, these models have enabled analytical predictions of higher-order concentration moments, and the form of the one-point concentration probability density function (PDF). In this paper, we extend the traditional formalism of these models, grounded in the theory of homogeneous and isotropic turbulent flow, to two cases: namely, a simple sheared boundary layer and a large array of regular obstacles. Some very high-resolution measurements of plume dispersion in a water channel, obtained using laser-induced fluorescence (LIF) line-scan techniques are utilised. These data enable us to extract time series of plume centroid position (plume meander) and dispersion in the relative frame of reference in unprecedented detail. Consequently, experimentally extracted PDFs are able to be directly compared with various theoretical forms proposed in the literature. This includes the PDF of plume centroid motion, the PDF of concentration in the relative frame, and a variety of concentration moments in the absolute and relative frames of reference. The analysis confirms the accuracy of some previously proposed functional forms of model components used in fluctuating plume models, as well as suggesting some new forms necessary to deal with the complex boundary conditions in the spatial domain.     

An ensemble of dynamically identical thermals and vertical profiles of turbulent moments in the convective surface layer of atmosphere

An ensemble of convective thermals is distinguished from the surface layer of penetrative turbulent convection over a heated horizontally uniform surface. For an isolated convective element, an integral model of a quasi-stationary spontaneous jet is proposed which admits an exact analytical solution. A simple statistical model is constructed for an ensemble of dynamically identical thermals. In this model, convective thermals ascend in a static environment, their dynamic parameters are described by the equations of an isolated quasi-stationary jet, and their diameters are stochastic. It is shown that the ensemble of thermals rising in a horizontally homogeneous environment forms surface-layer turbulent moments. The analytical relationships for higher turbulent moments of vertical velocity and temperature are compared with experimental data from the second to the forth order inclusive.     

大气边界层风速脉动的分形模拟

应用曲线分数维计算方法计算大气边界层实际风速观测资料的分数维值,并根据分形理论构造出一个理论模型来模拟真实的大气边界层风速时间序列。将模拟和真实数据的一些重要的统计特征,例如方差、风速概率分布、谱密度函数和自相关函数等,进行了比较,结果表明二者具有很好的一致性。     

The Transport Time pdf for a Simple Turbulent Flow

In a turbulent fluid, the time a particle needs to travel from a point source to the observation point, can be considered as a random variable. It is shown that the probability density function (pdf) for this random variable is determined by the Lagrangian particle position pdf. The characteristics of the transport time pdf are discussed for the simple case of a turbulent fluid moving with a constant mean velocity.     

Analysis of oscillations in the stable atmospheric boundary layer using wavelet methods

A wavelet method is used to estimate kinetic energy and fluxes from data collected under stable conditions during the CASES-99 field campaign. Results in the high frequency range are compared with those obtained by the traditional method used to estimate turbulent moments, which is based on the Reynolds decomposition of variables into a mean and a turbulent part. The fact that the wavelet transform performs much better as a filter than the averaging process accounts for most of the disagreements between results. Since the wavelet method can be applied at very different spectral ranges, it is also used to analyse two different coherent structures: a density current and a train of internal gravity waves. The strong burst of turbulence related to the density current reflects the complexity of the first event. The wavelet method discriminates the different scales of motion, which are present in the perturbation, and is therefore an ideal tool for assessing the interactions between them. A method based on the phase difference between wavelet-transformed time series is then applied to the analysis of the horizontal and vertical structure of the gravity waves, and a three-dimensional image of the oscillations is provided.     

Experimental Measurements of Expected Mass Fraction in a Contaminant Plume

The sudden release of a quantity of gas into the atmospheric boundary layer produces a contaminant cloud. The expected mass fraction function provides a relatively simple measure of the contaminant concentration values found within the cloud and represents the ensemble-averaged fraction of the conserved release mass found at the different contaminant concentration intervals as the cloud evolves. The plume generated by a line source in grid turbulence is used to investigate the expected mass fraction function as it applies to scalar concentration values found on a typical line normal to the plume axis. Simultaneous particle image velocimetry and planar laser induced fluorescence are used to measure velocity and concentration fields, respectively. The measured expected mass fraction functions are observed to be approximately self-similar when concentration values are normalized by the centreline mean concentration. The moments of the expected mass fraction function are observed to be simply related to the centreline moments of the probability density function of scalar concentration. Arguments based on a source fluid, non-source fluid decomposition of the scalar probability density function are used to explain these observations. The results are compared with the theoretical and experimental results established for a line source of scalar in grid turbulence.     

Relationships between higher moments of concentration and of dose in turbulent dispersion

Chatwin and Sullivan (1990) proposed simple results for the relationships between moments of scalar fluctuations in self-similar turbulent shear flows. They showed these relationships to be well satisfied by observations from a range of experiments. Here their theory is extended to the skewness, kurtosis and higher order equivalents. It is shown that the relationships between these normalised moments are parameter-free, and are identical to those for zero molecular diffusion. Experimental observations are presented which show a remarkable degree of collapse when these normalised moments are plotted against each other. The agreement with the theoretical results is reasonably good, and better than for some other standard statistical distributions which are commonly applied to such observations. This is true not only for the concentration, but also for generalised doses. It is concluded that the simple theory provides a satisfactory basis for a model of both the concentration and of dose. Furthermore, the results suggest that the concentration and the dose can be modelled through a perturbation to a two-state model.     

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.     

Characteristics of vertical turbulent velocities in the urban convective boundary layer

Turbulence measurements of the vertical velocity component were obtained by an instrumented aircraft under fair weather conditions in the St. Louis, Missouri, metropolitan area. Time series of vertical velocity fluctuations from horizontal flight segments made in the lower part of and near the middle of the convective boundary layer (CBL) over the urban area and surrounding region were subjected to various statistical and objective analyses. Higher order vertical velocity moments, and positive and negative velocity statistics, were computed. The horizontal dimensions of updrafts and downdrafts, and related properties of these turbulent eddies were derived by conditional sampling analysis. Emphasis is on a comparison of the results from urban and selected rural measurements from the lower part of the CBL.The vertical velocity probability density distribution for each segment was positively skewed and the mode was negative. The means and standard deviations of positive and negative velocity fluctuations were greater over the urban area. The urban vertical velocity variance was 50% greater than rural values, and power spectra revealed greater production of vertical turbulent energy in the urban area over a wide frequency range.The mean and maximum widths of downdrafts were generally larger than the corresponding values for updrafts. Differences between urban and rural eddy sizes were not statistically significant. The widths of the largest updraft and downdraft are comparable to the boundary-layer depth, Z _i, and the mean value of the ratio of spectral peak wavelength to Z _iwas about 1.3 and 1.1 for urban and rural areas, respectively. Convective similarity scaling parameters appeared to order both the urban and rural measurements.On assignment from the National Oceanic and Atmospheric Administration, U.S. Dept. of Commerce.     

Probability distributions of concentration fluctuations of a weakly diffusive passive plume in a turbulent boundary layer

Results are presented from an experimental investigation of turbulent dispersion of a saline plume of large Schmidt number (Sc=830) in a turbulent boundary-layer shear flow simulated in a laboratory water channel. The dispersion measurements are obtained in a neutrally buoyant plume from an elevated point source over a range of downstream distances, where both plume meandering and fine-structure variations in the instantaneous plume are important. High-resolution measurements of the scalar fluctuations in the plume are made with a rake of conductivity probes from which probability distributions of concentration at various points throught the plume are extracted from the time series.Seven candidate probability distributions were tested, namely, the exponential, lognormal, clipped normal, gamma, Weibull, conjugate beta, andK-distributions. Using the measured values of the conditional mean concentration, , and the conditional fluctuation intensity,i _p , the Weibull distribution provided the best match to the skewness and kurtosis over all downstream fetches. The skewness and kurtosis were always overpredicted by the lognormal probability density function (pdf), and underpredicted by the gamma pdf. The conjugate beta distribution for which the model parameters are determined using a method of moments based on the fluctuation intensity,i _p , and skewness,S _p , was capable of modeling the distribution of scalar concentration over a wide range of positions in the plume.     

On the Prediction of the Kurtosis of Velocity Derivatives in a Turbulent Field

The prediction of the values of non-dimensional fourth-order moment (kurtosis) of the velocity derivative in a turbulent field is made under the assumption that the values of kurtosis depend on both the turbulence Reynolds number and the intermittency factor. The method consists of modeling a suitable probability density of the variable in a given turbulence Reynolds number and the intermittency factor. A crude model of the probability density function is derived, and the numerical calculations based on the model show excellent agreement with many of the experimental data. The analysis shows that the values of kurtosis depend strongly on the intermittency factor, and that depending on the value of the intermittency factor, it is entirely possible to have values of kurtosis as low as five in a flow with a turbulence Reynolds number of 5000.     

On the prediction of the kurtosis of velocity derivatives in a turbulent field

The prediction of the values of non-dimensional fourth-order moment (kurtosis) of the velocity derivative in a turbulent field is made under the assumption that the values of kurtosis depend on both the turbulence Reynolds number and the intermittency factor. The method consists of modeling a suitable probability density of the variable in a given turbulence Reynolds number and the intermittency factor. A crude model of the probability density function is derived, and the numerical calculations based on the model show excellent agreement with many of the experimental data. The analysis shows that the values of kurtosis depend strongly on the intermittency factor, and that depending on the value of the intermittency factor, it is entirely possible to have values of kurtosis as low as five in a flow with a turbulence Reynolds number of 5000.     

On the moments approximation method for constructing a Lagrangian Stochastic model

The exact Eulerian velocity probability density function (pdf) of a turbulent field is generally unknown, and one normally has available only partial information in the form of low order moments. We compare two alternative Lagrangian Stochastic (LS) approaches formed from this partial information, (i) the moments approximation approach (Kaplan and Dinar, 1993); and (ii) the well-mixed model (Thomson, 1987) that corresponds to the maximum missing information pdf formed from the available information. We show that the moments approximation model does not in general satisfy the well-mixed constraint, and can give an inferior prediction of dispersion.     

夜间稳定边界层中间歇性湍流热通量的 提取及其谱特征

在夜间稳定边界层中, 湍流热通量往往具有间歇性特征。在一阵阵出现的强度较大的湍流热通量之间, 混杂着弱噪声或其他微弱的难以辨识的高频脉动信号。为了研究间歇性湍流热通量的特征, 必须将这些无关信 号剔除, 以提取出干净的湍流热通量。本文提出了一种新的提取间歇性湍流热通量的方法, 该方法通过分析 湍流热通量的概率密度函数, 并与稳定分布进行比较, 湍流热通量的概率密度函数开始偏离稳定分布的位置, 即是间歇性湍流热通量开始出现的阈值。本文通过夜间稳定边界层外场试验数据的验证, 发现利用稳定分布确定 的阈值可有效地提取出间歇性湍流热通量。在此基础之上, 本文对比了提取前后湍流热通量的功率谱, 发现 提取后低频信号的方差所占比重下降, 而高频信号略有上升。此外, 间歇性湍流热通量在高频区的功率谱满足“-7/6”律。     

On the downward transport of turbulent kinetic energy in the surface layer over plant canopies

Downward fluxes of turbulent kinetic energy have been frequently observed in the air layer just above plant canopies. In order to investigate the mechanism for such downward transport, analysis of observational data is attempted. Height-dependency of turbulent kinetic energy flux and turbulence statistics including higher order moments is represented as a function of a non-dimensional height z/H, where z is an observational height and H an average height of plant canopies. Downward fluxes and non-Gaussianity of wind velocity fluctuations are predominant just above plant canopies and decrease with increasing height. The downward flux is closely related to the high intensity of turbulence and the non-Gaussianity of wind velocity fluctuations, especially with a positive skewness in the longitudinal wind and a negative skewness in the vertical wind. The analysis method of conditional sampling and averaging is applied to the present observations. The results show that the predominance of the intermittent inrush phase over the intermittent ejection phase leads to the above-mentioned non-Gaussianity. Finally, a simple explanation is given in order to interpret the turbulent flow structure in the air layer near the plant canopies, which is associated with the downward energy transport process.