Concentration Probability Density Function Research Articles

Evolution of pore-scale concentration PDFs and estimation of transverse dispersion from numerical porous media column experiments

Knowing local concentration distributions is important for transport and mixing, particularly in porous media, yet a comprehensive understanding of them remains a challenge. Computing advancements have enabled high-resolution pore-scale simulations, offering an unprecedented opportunity for in-depth investigation of mixing. In this study we use simulation data to examine concentration distributions at the pore scale in the context of longitudinal (pseudo-one-dimensional) solute transport through a porous column. These distributions arise in a single column from heterogeneous flow at the pore-scale, which gets averaged out when upscaled and are not with reference to statistics across multiple random realizations. To measure these distributions, we first devise a semi-analytical approach to estimate the mean effective transport velocity profile for a non-uniform Darcy-scale fluid velocity, which unavoidably occurs due to the presence of lateral boundaries. This development allows sampling micro-scale concentrations over a moving surface that possesses a well defined Darcy-scale mean concentration, enabling empirical computation of the local concentration distribution. As an added benefit we find that our approach allows for the estimation of transverse dispersion coefficients, which is not typical in traditional column experiments. The implemented approach can estimate it via inverse modeling, and it agrees closely with previously published experimental data across the range of Peclet numbers we studied. We found that the measured pore-scale concentration probability density functions are best represented by a beta distribution, thus validating this longstanding hypothesis with direct evidence. Furthermore, we propose a model to describe the temporal and spatial evolution of the local concentration pdf, as well as its Péclet number dependence.

Concentration fluctuations and flammability of cryo-compressed hydrogen and methane jets

Compressed hydrogen stored at cryogenic temperatures has a much higher density than room-temperature storage, which enables large-scale hydrogen storage and transport. An understanding of the release of cryogenic hydrogen from pressurized vessels is needed to evaluate the risk and safety concerns with the use of this fuel. The present work extends the analysis of previous experimental studies that measured the gas concentrations of cryo-compressed hydrogen jets and methane jets using a laser Raman scattering diagnostic system. Since the Raman signals are very small, a denoising algorithm was applied to significantly reduce the noise to enable statistical analysis of the data. The transient features of the turbulent jets were characterized by their concentration intermittencies and probability density functions (PDFs). A two-part PDF was developed to predict the bimodal features of the jet concentration distributions. Then, the flammability factors of the cryogenic jets were calculated based on the intermittency and the PDF.

Spatial heterogeneity of air pollution statistics in Europe

Air pollution is one of the leading causes of death globally, and continues to have a detrimental effect on our health. In light of these impacts, an extensive range of statistical modelling approaches has been devised in order to better understand air pollution statistics. However, the time-varying statistics of different types of air pollutants are far from being fully understood. The observed probability density functions (PDFs) of concentrations depend very much on the spatial location and on the pollutant substance. In this paper, we analyse a large variety of data from 3544 different European monitoring sites and show that the PDFs of nitric oxide (NO), nitrogen dioxide (NO_2) and particulate matter (PM_{10} and PM_{2.5}) concentrations generically exhibit heavy tails and are asymptotically well approximated by q-exponential distributions with a given width parameter lambda. We observe that the power-law parameter q and the width parameter lambda vary widely for the different spatial locations. For each substance, we find different patterns of parameter clouds in the (q, lambda ) plane. These depend on the type of pollutants and on the environmental characteristics (urban/suburban/rural/traffic/industrial/background). This means the effective statistical physics description of air pollution exhibits a strong degree of spatial heterogeneity.

Real-time probabilistic sediment concentration forecasting using integrated dynamic network and error distribution heterogeneity

Sediment forecasting at a dam site is important for the operation and management of water and sediment in a reservoir. However, the forecast results generally have some uncertainties, which may hinder the operation of the dam. In this study, a real-time sediment concentration probabilistic forecasting model is proposed based on a dynamic network model. Under this framework, the Elman neural network (ENN) and nonlinear auto-regressive with exogenous inputs (NARX) neural network models were established for sediment concentration forecasting with different lead times. A hybrid algorithm, which combined the Levenberg–Marquardt algorithm and real-time recurrent learning, was used to train the model. Using the aforementioned method, the sediment concentration was forecast for at the Sanmenxia Dam, China, and, subsequently, the forecast results were evaluated. Among the selected lead time, the results at 5 h exhibited the highest accuracy and practical significance. Compared with the ENN model, the sediment concentration peak error using the NARX neural network was reduced by 4.5%, and the sediment yield error was reduced by 0.043%. Therefore, the NARX neural network was selected as the deterministic sediment forecasting model. Additionally, the probability density function of the sediment concentration was derived based on the heterogeneity of the error distribution, and the sediment concentration interval, with different confidence levels, expected values, and median values, was forecast. The Nash–Sutcliffe coefficient of efficiency for the sediment concentration, as forecasted based on the median value, was the highest (0.04 higher than that using a deterministic model), whereas the error of the sediment concentration peak and sediment yield remained unaltered. These results indicated the accuracy and superiority of the proposed real-time sediment probabilistic forecasting hybrid model.

Stochastic switches of eutrophication and oligotrophication: Modeling extreme weather via non-Gaussian Lévy noise.

Disturbances related to extreme weather events, such as hurricanes, heavy precipitation events, and droughts, are important drivers of evolution processes of a shallow lake ecosystem. A non-Gaussian α-stable Lévy process is esteemed to be the most suitable model to describe such extreme events. This paper incorporates extreme weather via α-stable Lévy noise into a parameterized lake model for phosphorus dynamics. We obtain the stationary probability density function of phosphorus concentration and examine the pivotal roles of α-stable Lévy noise on phosphorus dynamics. The switches between the oligotrophic state and the eutrophic state can be induced by the noise intensity σ, skewness parameter β, or stability index α. We calculate the mean first passage time, also referred to as the mean switching time, from the oligotrophic state to the eutrophic state. We observe that the increased noise intensity, skewness parameter, or stability index makes the mean switching time shorter and thus accelerates the switching process and facilitates lake eutrophication. When the frequency of extreme weather events exceeds a critical value, the intensity of extreme events becomes the most key factor for promoting lake eutrophication. As an application, we analyze the available data of Lake Taihu (2014-2018) for monthly precipitation, phosphorus, and chlorophyll-a concentrations and quantify the linkage among them using the Lévy-stable distribution. This study provides a fundamental framework to uncover the impact of any extreme climate event on aquatic nutrient status.

Dilution of Reactive Plumes: Evolution of Concentration Statistics Under Diffusion and Nonlinear Reaction

Concentration fields of solutes in porous media often exhibit large fluctuations, driven by physical and chemical heterogeneity from the pore to the Darcy scale. For many applications, ranging from reactive transport modeling to toxicology, the knowledge of mean concentrations is not sufficient, and quantifying concentration variability is necessary. The probability density function (PDF) of concentration quantifies the frequency of occurrence of concentration values throughout a spatial domain. While evolution equations and analytical solutions for the concentration PDF exist for conservative solutes, less is known about its evolution under the joint action of transport and reaction. In this work, we investigate how dilution of a reactive plume by diffusion affects the statistics of concentrations. While mixing has no effect on first-order reactions, its coupling with nonlinear reactions leads to non-trivial effective kinetics relevant for a broad range of reactive transport problems. We study the evolution of the concentration PDF under diffusion and nonlinear reaction in one spatial dimension, which represents a critical step toward further coupling with heterogeneous advection. We show that the dependence of the scalar dissipation rate on concentration encodes the impact of diffusive transport on the concentration PDF and derive a dynamical equation for its time evolution. Using a weak-coupling approximation for the reaction and diffusion dynamics, we derive analytical predictions for the concentration PDF and its moments. Our results provide new insights into how diffusion and reaction control concentration variability and open new opportunities for coupling mixing models with chemical reactions.

Large Eddy simulation of tracer gas dispersion in a cavity

This paper assesses the prediction of inert tracer gas dispersion within a cavity of height (H) 1.0 m, and unity aspect ratio, using large Eddy simulation (LES). The flow Reynolds number was 67 000, based on the freestream velocity and cavity height. The flow upstream of the cavity was laminar, producing a cavity shear layer which underwent a transition to turbulence over the cavity. Three distinct meshes are used, with grid spacings of (coarse), (intermediate), and (fine) respectively. The Smagorinsky, WALE, and Germano-Lilly subgrid-scale models are used on each grid to quantify the effects of subgrid-scale modelling on the simulated flow. Coarsening the grid led to small changes in the predicted velocity field, and to substantial over-prediction of the tracer gas concentration statistics. Quantitative metric analysis of the tracer gas statistics showed that the coarse grid simulations yielded results outside of acceptable tolerances, while the intermediate and fine grids produced acceptable output. Interrogation of the fluid dynamics present in each simulation showed that the evolution of the cavity shear layer is heavily influenced by the grid and subgrid scale model. On the coarse and intermediate grids the development of the shear layer is delayed, inhibiting the entrainment and mixing of the tracer gas into the shear layer, reducing the removal of the tracer gas from the cavity. On the fine grid, the shear layer developed more rapidly, resulting in enhanced removal of the tracer gas from the cavity. Concentration probability density functions showed that the fine grid simulations accurately predicted the range, and the most probable value, of the tracer gas concentration towards both walls of the cavity. The results presented in this paper show that the WALE and Germano-Lilly models may be advantageous over the standard Smagorinsky model for simulations of pollutant dispersion in the urban environment.

The effect of tree planting on traffic pollutant dispersion in an urban street canyon using large eddy simulation with a recycling and rescaling inflow generation method

The influence of various tree planting configurations on tracer gas dispersion in an urban street canyon is studied numerically, using OpenFOAM Large Eddy Simulation (LES). A recycling and rescaling turbulent inflow generation method is implemented and validated against a canonical backwards-facing step case. The street canyon simulations are based on the CODASC experiments, where the tracer gas was emitted from line sources at street level. The effect of tree stand density on the pollutant dispersion is studied for the first time using LES. Performance metric analysis of the CODASC simulations shows that the influence of tree crown porosity, and tree stand density, on the concentration field within the canyon can be captured by LES. The simulations show that tree crowns reduce the effectiveness of the canyon vortex at ventilating the street, and enhance the mean tracer gas concentration at pedestrian level on the leeward wall. A novel feature of the research arises from concentration probability density functions at pedestrian height in the canyon. Pedestrians can be exposed to pollutant concentrations almost three times the mean value at the leeward wall. The results have implications for urban greening strategies in city streets with heavy traffic.

Modeling solute transport and mixing in heterogeneous porous media under turbulent flow conditions

We develop and test a modeling approach to quantify turbulence-driven solute transport and mixing in porous media. Our approach addresses two key elements: (a) the spatial variability of the effective diffusion coefficient which is typically documented in the presence of a sediment–fluid interface and (b) the need to provide a model that can yield the complete distribution of the concentration probability density function, not being limited only to the mean concentration value and thus fully addressing solute mixing. Our work is motivated by the importance of solute transport processes in the hyporheic zone, which can have strong implications in natural attenuation of pollutants. Our approach combines Lagrangian schemes to address transport and mixing in the presence of spatial variability of effective diffusion. An exemplary scenario we consider targets a setup constituted by a homogeneous (fully saturated) porous medium underlying a clear water column where turbulent flow is generated. Solute concentration histories obtained through a model based solely on diffusive transport are benchmarked against an analytical solution. These are then compared against the results obtained by modeling the combined effects of diffusion and mixing. A rigorous sensitivity analysis is performed to evaluate the influence of model parameters on solute concentrations and mixing, the latter being quantified in terms of the scalar dissipation rate.

Scalar mixing in homogeneous isotropic turbulence: A numerical study

We investigate the dynamics of turbulent dispersion by means of direct numerical simulations of a passive tracer released in a homogeneous isotropic turbulent flow. We focus on the link between the probability density function (PDF) of the passive scalar concentration and its mixing properties. In particular, we show how the gamma distribution can be used as a suitable model for the PDF, as has been previously verified in wind tunnel experiments in wall-bounded turbulent flows. Finally, we develop a simple mixing model to estimate the time scale that regulates the decay rate of the intensity of concentration fluctuations.

An integrated approach for analyzing air quality monitoring data: a case study in Hanoi, Vietnam

In this research, we propose an integrated approach to retrieve the information of air quality monitoring data for air quality assessment. Used data is the hourly concentrations of PM10, PM2.5, SO2, NO, NO2, O3, and CO measured by an ambient air quality monitoring station in Hanoi, Vietnam, in the period of 2010–2018. First, the dataset was preprocessed to remove outliers by a simple removing method. Then, the missing values were filled up using a deep learning technique. After that, the preprocessed data went through the process of analysis to infer the probability distributions. After the probability density functions (pdf) of the hourly concentrations of these air pollutants in the year were determined, the information of air quality data in each year including the annual mean and median, the maximum concentration, and the number of exceedances of hourly thresholds for the period of 2010–2018 was obtained. High level of PM2.5, exceeding its national annual standard, was observed implying the potential of health risk to the people. Trend and temporal variations of these air pollutants were also analyzed. It is found out that the effect of direct emission from local traffic activity on the ambient levels of SO2 and PM2.5 is not dominant.

Concentration Fluctuations from Localized Atmospheric Releases

We review the efforts made by the scientific community in more than seventy years to elucidate the behaviour of concentration fluctuations arising from localized atmospheric releases of dynamically passive and non-reactive scalars. Concentration fluctuations are relevant in many fields including the evaluation of toxicity, flammability, and odour nuisance. Characterizing concentration fluctuations requires not just the mean concentration but also at least the variance of the concentration in the location of interest. However, for most purposes the characterization of the concentration fluctuations requires knowledge of the concentration probability density function (PDF) in the point of interest and even the time evolution of the concentration. We firstly review the experimental works made both in the field and in the laboratory, and cover both point sources and line sources. Regarding modelling approaches, we cover analytical, semi-analytical, and numerical methods. For clarity of presentation we subdivide the models in two groups, models linked to a transport equation, which usually require a numerical resolution, and models mainly based on phenomenological aspects of dispersion, often providing analytical or semi-analytical relations. The former group includes: large-eddy simulations, Reynolds-averaged Navier–Stokes methods, two-particle Lagrangian stochastic models, PDF transport equation methods, and heuristic Lagrangian single-particle methods. The latter group includes: fluctuating plume models, semi-empirical models for the concentration moments, analytical models for the concentration PDF, and concentration time-series models. We close the review with a brief discussion highlighting possible useful additions to experiments and improvements to models.

Side ratio effects on flow and pollutant dispersion around an isolated high-rise building in a turbulent boundary layer

Large-eddy simulations were performed to investigate side ratio (SR) effects on flow and dispersion around an isolated high-rise building. Four cross sections were considered with SR ranging from 0.25 to 1.5. Three pollutants were released from the ground; the first source (P1) was upwind of the building, and the other two (P2 and P3) were downwind. Reattachments on the side and top surfaces were observed for SR = 1.5. As SR decreased, the recirculation area behind the building increased and vortex shedding strengthened. The dispersion area of pollutant P1 nearly coincided with the primary separation region at ground level because its source location was inside the primary separation region. More of pollutant P2 accumulated near the leeward wall because the discharge hole was located inside the recirculation region behind the building for all simulations, and large SR values had high P2 concentrations because of the weak vortex shedding strength. Pollutant P3 (whose source was far from the building) exhibited different distribution patterns because of different recirculation sizes; it was blown directly downstream for SR = 1.5 and 1.0, whereas more was transported back toward the leeward wall for SR = 0.25. An analysis of the probability density functions of instantaneous concentration indicated that positive fluctuations of the concentration occurred more frequently near source locations; the opposite held in relatively low-concentration areas. The extremely high positive fluctuations reached 15 times the standard deviation for the second mode.

On the Convergence and Capability of the Large-Eddy Simulation of Concentration Fluctuations in Passive Plumes for a Neutral Boundary Layer at Infinite Reynolds Number

Large-eddy simulation (LES) experiments have been performed using the Parallelized LES Model (PALM). A methodology for validating and understanding LES results for plume dispersion and concentration fluctuations in an atmospheric-like flow is presented. A wide range of grid resolutions is shown to be necessary for investigating the convergence of statistical characteristics of velocity and scalar fields. For the scalar, the statistical moments up to the fourth order and the shape of the concentration probability density function (p.d.f.) are examined. The mean concentration is influenced by grid resolution, with the highest resolution simulation showing a lower mean concentration, linked to larger turbulent structures. However, a clear tendency to convergence of the concentration variance is observed at the two higher resolutions. This behaviour is explained by showing that the mechanisms driving the mean and the variance are differently influenced by the grid resolution. The analysis of skewness and kurtosis allows also the obtaining of general results on plume concentration fluctuations. Irrespective of grid resolution, a family of Gamma p.d.f.s well represents the shape of the concentration p.d.f. but only beyond the peak of the concentration fluctuation intensity. In the early plume dispersion phases, the moments of the p.d.f. are in good agreement with those generated by a fluctuating plume model. To the best of our knowledge, our study demonstrates for the first time that, if resolution and averaging time are adequate, atmospheric LES provides a trustworthy representation of the high order moments of the concentration field, up to the fourth order, for a dispersing plume.

An Investigation of the Precipitation Net Effect on the Particulate Matter Concentration in a Narrow Valley: Role of Lower-Troposphere Stability

AbstractPrecipitation modulates the pollutant concentration in the atmosphere directly and indirectly through different mechanisms. Wet deposition, including in-cloud scavenging and below-cloud scavenging, is one of the main drivers of particulate matter removal from the atmosphere. Our focus is to assess the net effect of precipitation on the pollutant concentration using observations, a nonparametric conditional analysis, and the signed overlapping coefficient complement index. The study considers not only the direct near-surface aerosol below-cloud scavenging, but also the indirect effect associated with the induced lower-troposphere stability limiting the vertical dispersion, leading to a pollutant concentration increase. We consider the precipitation effects on fine and coarse particulate matter concentration. Probability density functions of the pollutant concentration during precipitation events and under dry conditions are analyzed for every hour of the day. The net effect of precipitation on the fine and coarse particulate matter concentration is strongly dependent on the atmospheric stability. During the nighttime and before midmorning, the atmosphere is stable; a precipitation event during this time induces below-cloud scavenging, reducing the concentration of particulate matter in the atmosphere. During the afternoon, unstable atmospheric conditions are predominant, and aerosols disperse vertically; a precipitation event during this period stabilizes the atmosphere, generating early stabilization that, together with the continued anthropogenic emissions during the day, leads to near-surface pollutant accumulation, offsetting the washout effect of the precipitation.

On the electrical conductivity of composites with a polymeric matrix and a non-uniform concentration of carbon nanotubes

We present a multiscale modelling approach to explore the effects of a non-uniform concentration of carbon nanotubes (CNTs) on the electrical conductivity of CNT-polymer composites. Realistic three-dimensional representative volume elements (RVEs) are generated from a two-dimensional CNT concentration map, obtained via microscopy techniques. The RVEs capture the measured probability density function of the CNT concentration and include a length-scale to represent the details of the spatial distribution of the concentration. The homogenized conductivity of the RVEs is computed via multiscale FE analyses for different values of such length-scale, and it is compared to measurements. The modelling strategy is then used to explore the effects of the microstructural features of these materials on their electrical conductivity.

Large Eddy Simulation of lifted turbulent flame in cold air using Doubly Conditional Source-term Estimation

The present study is focused on Large Eddy Simulation with Doubly Conditional Source-term Estimation (DCSE) applied to a series of lifted turbulent flames in cold air without autoignition. DCSE is formulated for partially-premixed turbulent combustion. Predictions for time-averaged methane concentration and its root mean square (rms), Probability Density Function (PDF), flame width, lift-off height and some qualitative observations are presented and compared with experimental data. The predicted methane concentrations are in good agreement with the experiments. Discrepancies for the rms are observed and several sources of discrepancies are identified. PDF of methane concentration and flammability region are also in good agreement. The lift-off heights are predicted very well. Some edge flame characteristics are visible in the simulated flames. The progress variable scalar dissipation rate model does not have a significant impact on methane concentration predictions, while it has a larger effect on predicted lift-off height and temperature. Inclusion of the non-premixed mode in the closure of the progress variable transport equation has a small effect on the current lift-off height predictions, but has a larger impact on the temperature profiles.

OH planar laser-induced fluorescence measurements with high spatio-temporal resolution for the study of auto-ignition.

For the detailed understanding of transient combustion processes, in particular, of auto-ignition, quantitative measurements with high spatio-temporal resolution are desirable. These can, for instance, serve as validation data for time-resolved numerical simulations and in particular for the combustion models used in those simulations. In the current study, a jet-in-hot-coflow (JHC) burner, developed at the German Aerospace Center (DLR), the DLR JHC, was used to inject a turbulent methane jet into the hot exhaust gas of a lean hydrogen/air flame, and a steady state jet flame was established. In addition, fuel could be injected in a transient manner. Here, an auto-igniting jet was observed. The flame stabilization of the steady state jet flame and the auto-ignition during transient fuel injection were studied using high-speed laser-based and optical measurements. A strategy for quantifying high-speed OH planar laser-induced fluorescence is presented, and the measurement uncertainties are evaluated. The flame stabilization mechanism in steady state jet flames was assessed using probability density functions of the OH concentration at different axial and radial locations. The formation of auto-ignition kernels during transient fuel injection is evaluated based on time series of the OH concentration. It is shown how the OH concentration levels and PDF shapes can be used to characterize the chemical state of the reacting flow and to distinguish between auto-ignition and flame propagation.

Assessment of Puff-Dispersion Variability Through Lagrangian and Eulerian Modelling Based on the JU2003 Campaign

In the framework of the Urban Dispersion International Evaluation Exercise (UDINEE) project coordinated by the European Commission’s Joint Research Centre, a case study was conducted of the Joint Urban 2003 (JU2003) experimental campaign in the central area of Oklahoma City, USA. The UDINEE project concerned the cases of puff dispersion of the JU2003 campaign, which are of special interest to scenarios related to security studies, such as explosions of radiological dispersal devices. Starting from the fact that puff-dispersion variability is substantial, especially in complex urban areas, even for puffs released under similar meteorological conditions, a methodology is presented for assessing this variability, which is applied to the dispersion of puffs in two of the intensive operation periods of the JU2003 campaign. Lagrangian and Eulerian dispersion models are applied for the simulations. For the Lagrangian model, variability is assessed by repeating the computations a large number of times. For the Eulerian model, variability is assessed by constructing probability density functions of concentrations on the basis of the dispersion-model results. Peak concentrations, dosages, puff-arrival times and puff durations are considered. Percentiles calculated by the Lagrangian model for all the above quantities and by the Eulerian model for peak concentrations and dosages are compared with the measurements. The results are encouraging since in several cases the measured and computed ranges of values overlap.

Effect of boundary layer thickness on transverse sonic jet mixing in a supersonic turbulent crossflow

The surface boundary through which a sonic jet in supersonic cross flow is injected is shown to have a significant effect on the size, penetration, and mixing characteristics of the jet plume. A circular, high-pressure, sonic jet is injected into a M = 3.4 supersonic crossflow through a well-characterized turbulent boundary layer of two different thicknesses (δ/d = 0.6 and 6.1), with variable momentum ratios (J = 1.2, 2.6, and 5). Planar laser Mie scattering of condensed ethanol droplets is used to quantitatively image the injected fluid concentration in both side and end-views at multiple downstream locations. Jet penetration, plume area, and characteristic size and location of regions of intense mixing are compared. The jets injected through the thicker boundary layer are shown to have significantly enhanced jet penetration (∼50%), spread (∼100%), and mixing intensity (∼100%, especially in the near-field) over a wider area of the jet plume. Additionally, characterization of mixing is examined using the variance in the concentration field as well as probability density functions of concentration determined along contours of constant jet fluid concentration. From these results, the jet injections associated with the thicker boundary layer transition from shear dominated mixing zones on the windward side to more distributed mixing zones throughout the plume at earlier downstream locations and show influence of interactions between boundary layer vorticity and vortical structures within the jet leading to larger lateral expansion.