Concentration Probability Density Function Research Articles

Weekly variability of surface CO concentrations in Moscow

Based on observations of carbon monoxide (CO) concentrations at three Mosekomonitoring stations, we have analyzed the weekly cycle of CO in the surface air of Moscow in 2004–2007. At all stations the minimum long-term mean daily CO values are observed on Sunday. The weekly cycle of CO more clearly manifests itself at the center of Moscow and becomes less clear closer to the outskirts. We have analyzed the reproducibility of the weekly cycle of CO from one year to another, the seasonal dependence, its specific features at different times of day, and the changes in the diurnal cycle of CO during the week. The factors responsible for specific features of the evolution of surface CO concentrations at different observation stations have been analyzed. The empirical probability density functions of CO concentrations on weekdays and at week- end are presented. The regularity of the occurrence of the weekend effect in CO has been investigated and the possible reasons for breaks in weekly cycles have been analyzed. The Kruskal-Wallis test was used to study the statistical significance of intraweek differences in surface CO contents.

A Lagrangian stochastic model for estimating the high order statistics of a fluctuating plume in the neutral boundary layer

We use a Lagrangian stochastic micromixing model to predict the concentration fluctuations of a continuous release in a neutral boundary layer. We present the computational algorithm that implements the interaction by exchange with the conditional mean model and we compare the numerical solutions with the experimental values in order to evaluate the reliability of the model. The influence of the source size on the concentration probability density function in the near and far-field is discussed and some shortcomings of the model are pointed out.

Uncertainty-quantification analysis of the effects of residual impurities on hydrogen–oxygen ignition in shock tubes

This study addresses the influences of residual radical impurities on the computation and experimental determination of ignition times in H2/O2 mixtures. Particular emphasis is made on the often-times encountered problem of the presence of H-atoms in the initial composition of H2/O2 mixtures in shock tubes. Two methods are proposed for quantifying experimentally H-residual impurities in shock tubes, namely, an a priori method that consists of detecting OH traces upon shocking unfueled mixtures, and a posteriori method in which the amount of impurities is inferred by comparing fueled experimental autoignition data with calculations. A stochastic Arrhenius model that describes the amount of H-radical impurities in shock tubes is proposed on the basis of experimental measurements as a function of the test temperature. It is suggested that this statistical model yields a probability density function for the residual concentration of hydrogen radicals in standard shock tubes. Theoretical quantifications of the uncertainties induced by the impurities on autoignition times are provided by using the 5-step short chemistry of Del Álamo et al. [1]. The analysis shows that the relative effects of H-impurities on delay times above crossover become more important as the dilution increases and as the temperature and pressure decrease. Below crossover, the effects of H-impurities on the ignition delay vanish rapidly, and are negligible compared to the departures produced by the non-ideal pressure rise that is seen in some shock-tube experiments at such low temperatures. The influences of kinetic uncertainties on the ignition time are typically negligible compared to the effects of the uncertainties induced by H-impurities when the short mechanism is used, except for air at high temperatures for which kinetic uncertainties dominate. Furthermore, calculations performed with the short mechanism show that correlations between the uncertainties in the rates of branching and termination steps have only some small influences on the ignition-time variabilities near crossover, where a global sensitivity analysis shows an increasing importance of the recombining kinetics. Computational quantifications of uncertainties are carried out by using numerical simulations of homogeneous ignition subject to Monte-Carlo sampling of the concentration of impurities. For the conditions analyzed, these computations show that the variabilities produced in ignition delays by the uncertainties in H-impurities are comparable to the experimental data scatter and to the effects of typical uncertainties of the test temperature when the Stanford chemical mechanism [2] is used. The calculations also unveil that the utilization of two other different chemical mechanisms, namely San Diego [3] and GRI v3.0 [4], yields variations in the ignition delays which are within the range of the uncertainties induced by the H-impurities. Finally, the effects of residual impurities in kinetic-isolation experiments and in supersonic-combustion ramjets are briefly discussed.

Application of Statistical Distribution of PM10 Concentration in Air Quality Management in 5 Representative Cities of China

ObjectiveTo estimate the frequency of daily average PM10 concentrations exceeding the air quality standard (AQS) and the reduction of particulate matter emission to meet the AQS from the statistical properties (probability density functions) of air pollutant concentration. MethodsThe daily PM10 average concentration in Beijing, Shanghai, Guangzhou, Wuhan, and Xi'an was measured from 1 January 2004 to 31 December 2008. The PM10 concentration distribution was simulated by using the lognormal, Weibull and Gamma distributions and the best statistical distribution of PM10 concentration in the 5 cities was detected using to the maximum likelihood method. ResultsThe daily PM10 average concentration in the 5 cities was fitted using the lognormal distribution. The exceeding duration was predicted, and the estimated PM10 emission source reductions in the 5 cities need to be 56.58%, 93.40%, 80.17%, 82.40%, and 79.80%, respectively to meet the AQS. ConclusionAir pollutant concentration can be predicted by using the PM10 concentration distribution, which can be further applied in air quality management and related policy making.

Evaluating and constraining ice cloud parameterizations in CAM5 using aircraft measurements from the SPARTICUS campaign

Abstract. This study uses aircraft measurements of relative humidity and ice crystal size distribution collected during the SPARTICUS (Small PARTicles In CirrUS) field campaign to evaluate and constrain ice cloud parameterizations in the Community Atmosphere Model version 5. About 200 h of data were collected during the campaign between January and June 2010, providing the longest aircraft measurements available so far for cirrus clouds in the midlatitudes. The probability density function (PDF) of ice crystal number concentration (Ni) derived from the high-frequency (1 Hz) measurements features a strong dependence on ambient temperature. As temperature decreases from −35 °C to −62 °C, the peak in the PDF shifts from 10–20 L−1 to 200–1000 L−1, while Ni shows a factor of 6–7 increase. Model simulations are performed with two different ice nucleation schemes for pure ice-phase clouds. One of the schemes can reproduce a clear increase of Ni with decreasing temperature by using either an observation-based ice nuclei spectrum or a classical-theory-based spectrum with a relatively low (5–10%) maximum freezing ratio for dust aerosols. The simulation with the other scheme, which assumes a high maximum freezing ratio (100%), shows much weaker temperature dependence of Ni. Simulations are also performed to test empirical parameters related to water vapor deposition and the autoconversion of ice crystals to snow. Results show that a value between 0.05 and 0.1 for the water vapor deposition coefficient, and 250 μm for the critical diameter that distinguishes ice crystals from snow, can produce good agreement between model simulation and the SPARTICUS measurements in terms of Ni and effective radius. The climate impact of perturbing these parameters is also discussed.

Experimental study of industrial gas turbine flames including quantification of pressure influence on flow field, fuel/air premixing and flame shape

A commercial swirl burner for industrial gas turbine combustors was equipped with an optically accessible combustion chamber and installed in a high-pressure test-rig. Several premixed natural gas/air flames at pressures between 3 and 6bar and thermal powers of up to 1MW were studied by using a variety of measurement techniques. These include particle image velocimetry (PIV) for the investigation of the flow field, one-dimensional laser Raman scattering for the determination of the joint probability density functions of major species concentrations, mixture fraction and temperature, planar laser induced fluorescence (PLIF) of OH for the visualization of the flame front, chemiluminescence measurements of OH* for determining the lift-off height and size of the flame and acoustic recordings. The results give insights into important flame properties like the flow field structure, the premixing quality and the turbulence–flame interaction as well as their dependency on operating parameters like pressure, inflow velocity and equivalence ratio. The 1D Raman measurements yielded information about the gradients and variation of the mixture fraction and the quality of the fuel/air mixing, as well as the reaction progress. The OH PLIF images showed that the flame was located between the inflow of fresh gas and the recirculated combustion products. The flame front structures varied significantly with Reynolds number from wrinkled flame fronts to fragmented and strongly corrugated flame fronts. All results are combined in one database that can be used for the validation of numerical simulations.

Application of a Bivariate Gamma Distribution for a Chemically Reacting Plume in the Atmosphere

The joint concentration probability density function of two reactive chemical species is modelled using a bivariate Gamma distribution coupled with a three-dimensional fluctuating plume model able to simulate the diffusion and mixing of turbulent plumes. A wind-tunnel experiment (Brown and Bilger, J Fluid Mech 312:373–407, 1996), carried out in homogeneous unbounded turbulence, in which nitrogen oxide is released from a point source in an ozone doped background and the chemical reactions take place in non-equilibrium conditions, is considered as a test case. The model is based on a stochastic Langevin equation reproducing the barycentre position distribution through a proper low-pass filter for the turbulence length scales. While the meandering large-scale motion of the plume is directly simulated, the internal mixing relative to the centroid is reproduced using a bivariate Gamma density function. The effect of turbulence on the chemical reaction (segregation), which in this case has not yet attained equilibrium, is directly evaluated through the covariance of the tracer concentration fields. The computed mean concentrations and the O3–NO concentration covariance are also compared with those obtained by the Alessandrini and Ferrero Lagrangian single particle model (Alessandrini and Ferrero, Physica A 388:1375–1387, 2009) that entails an ad hoc parametrization for the segregation coefficient.

Modeling molecular mixing in a spatially inhomogeneous turbulent flow

Simulations of spatially inhomogeneous turbulent mixing in decaying grid turbulence with a joint velocity–concentration probability density function (PDF) method were conducted. The inert mixing scenario involves three streams with different compositions. The mixing model of Meyer [“A new particle interaction mixing model for turbulent dispersion and turbulent reactive flows,” Phys. Fluids 22(3), 035103 (2010)], the interaction by exchange with the mean (IEM) model and its velocity-conditional variant, i.e., the IECM model, were applied. For reference, the direct numerical simulation data provided by Sawford and de Bruyn Kops [“Direct numerical simulation and lagrangian modeling of joint scalar statistics in ternary mixing,” Phys. Fluids 20(9), 095106 (2008)] was used. It was found that velocity conditioning is essential to obtain accurate concentration PDF predictions. Moreover, the model of Meyer provides significantly better results compared to the IECM model at comparable computational expense.

Probability density function of steady state concentration in two‐dimensional heterogeneous porous media

Spatial variability of hydraulic aquifer parameters causes meandering, squeezing, stretching, and enhanced mixing of steady state plumes in concentrated hot‐spots of mixing. Because the exact spatial distribution of hydraulic parameters is uncertain, the spatial distribution of enhanced mixing rates is also uncertain. We discuss how relevant the resulting uncertainty of mixing rates is for predicting concentrations. We develop analytical solutions for the full statistical distribution of steady state concentration in two‐dimensional, statistically uniform domains with log‐hydraulic conductivity following an isotropic exponential model. In particular, we analyze concentration statistics at the fringes of wide plumes, conceptually represented by a solute introduced over half the width of the domain. Our framework explicitly accounts for uncertainty of streamline meandering and uncertainty of effective transverse mixing (defined at the Darcy scale). We make use of existing low‐order closed‐form expressions that lead to analytical expressions for the statistical distribution of local concentration values. Along the expected position of the plume fringe, the concentration distribution strongly clusters at its extreme values. This behavior extends over travel distances of up to tens of integral scales for the parameters tested in our study. In this regime, the uncertainty of effective transverse mixing is substantial enough to have noticeable effects on the concentration probability density function. At significantly larger travel distances, intermediate concentration values become most likely, and uncertainty of effective transverse mixing becomes negligible. A comparison to numerical Monte Carlo simulations of flow and solute transport show excellent agreement with the theoretically derived expressions.

TURBULENT CLUSTERING OF PROTOPLANETARY DUST AND PLANETESIMAL FORMATION

We study clustering of inertial particles in turbulent flows and discuss its applications to dust particles in protoplanetary disks. Using numerical simulations, we compute the radial distribution function (RDF), which measures the probability of finding particle pairs at given distances, and the probability density function of the particle concentration. The clustering statistics depend on the Stokes number, $St$, defined as the ratio of the particle friction timescale, $\tau_{\rm p} $, to the Kolmogorov timescale in the flow. In the dissipation range, the clustering intensity strongly peaks at $St \simeq 1$, and the RDF for $St \sim 1$ shows a fast power-law increase toward small scales, suggesting that turbulent clustering may considerably enhance the particle collision rate. Clustering at inertial-range scales is of particular interest to the problem of planetesimal formation. At these scales, the strongest clustering is from particles with $\tau_{\rm p}$ in the inertial range. Clustering of these particles occurs primarily around a scale where the eddy turnover time is $\sim\tau_{\rm p}$. Particles of different sizes tend to cluster at different locations, leading to flat RDFs between different particles at small scales. In the presence of multiple particle sizes, the overall clustering strength decreases as the particle size distribution broadens. We discuss particle clustering in recent models for planetesimal formation. We point out that, in the model based on turbulent clustering of chondrule-size particles, the probability of finding strong clusters that can seed planetesimals may have been significantly overestimated. We discuss various clustering mechanisms in simulations of planetesimal formation by gravitational collapse of dense clumps of meter-size particles, in particular the contribution from turbulent clustering due to the limited numerical resolution.

Risk characterization for toxic chemicals transported in aquifers

The risk characterization resulting from the introduction of toxic chemicals in a subsurface flow field is presented. The concept of concentration threshold is used to quantify the risk associated with non-carcinogenic chemicals introduced to the population by ingestion of groundwater as the exposure pathway. The risk assessment methodology presented uses the expected mass fraction (EMF) concept with exposure duration to identify the distribution of dosage over different concentrations during the plume migration over the well location. The numerical simulation of the subsurface transport by advection and local diffusion is used to produce the concentration plume that passes different locations of interest. The EMF obtained presents the probability of the expected mass above some concentration threshold found at the location of interest. The risk formulation is defined with the risk reliability (safety) and its complement, the risk exceedance (failure) value. The risk characterization is obtained as a probability for exceeding the human reference dose which is considered uncertain due to the necessary extrapolation between concentration used in toxicological studies and the concentration to which humans could be exposed in nature. The final risk assessment expression is derived in a closed form by coupling the expected mass fraction with the safe human threshold concentration probability density function (pdf) inferred from the toxicological studies. The results indicate the importance of estimating the probability of a concentration mass found at locations of interest together with its exposure duration. The exposure duration was revealed to be an important parameter that needs to be estimated depending on the human concentration threshold selected and the distance from the source. The results in terms of the risk safety and risk failure also indicate the dilution effect on the passing concentration plume in the subsurface as a function of the distance and orientation from the source. Inclusion of uncertainty in the selection of the human concentration threshold shows the important effect on the risk quantification.

Concentration statistics for transport in heterogeneous media due to stochastic fluctuations of the center of mass velocity

We study concentration statistics for tracer transport in the flow through a heterogeneous porous medium. Spatial heterogeneity and temporal variability cause fluctuations in spatial characteristics of the tracer distribution such as the center of mass velocity v( t) and the spreading rate D e ( t), which leads to fluctuations in the concentration. These fluctuations can be quantified statistically by the one-point probability density function (PDF) of concentration and by its simplest characteristics, the concentration mean and variance. We focus on the impact of variability in the center of mass velocity and treat D e ( t) as deterministic. The fluctuating v( t) is modeled as a Gaussian stochastic process. Based on this stochastic description, we derive explicit expressions for the concentration variance and the full one-point concentration PDF. Their behavior is mainly controlled by the ratio of the spatial variance of the tracer distribution κ e ( t) in single realizations and the variance κ m ( t) of the plume centroids between realizations. We explore the behavior of concentration variance and PDF for two velocity correlation models, one that leads to a diffusive increase of centroid variance κ m ( t) and one that yields an asymptotically constant κ m ( t). The obtained results give insight into the dynamics of concentration statistics in heterogeneous media and the mechanisms by which they are controlled.

On the local concentration probability density function of solutes reacting upon mixing

The probability density function (pdf) of solute concentration is a useful tool for modeling transport of contaminants in heterogeneous aquifers that is increasingly used in risk assessment and, more generally, as a mean to quantify uncertainty in transport modeling. In order to be effective the pdfs should be linked in a simple manner to the spatial variability model of hydraulic properties, pore‐scale (local) dispersion, and a suitable parametrization of the geochemical processes. We analyze the pdf and concentration moments of two aqueous species in equilibrium with their precipitate reacting upon mixing in two‐ and three‐dimensional geological formations. The speciation equations, resulting from application of the chromatographic theory, provide the link between concentration pdfs (and moments) of the aqueous species and that of a passive tracer. Within this framework, we investigate the role of pore‐scale dispersion and macrodispersion in enhancing mixing, and thus reaction between the aqueous species, in the case of an instantaneous injection of a water with contrasting concentrations with respect to the ambient water under the constraint that in both waters the two aqueous species are in equilibrium with their precipitate. The main conclusion of our analysis is that for the pore‐scale dispersion typically observed in natural formations, the local concentration pdfs of both species are far from being Gaussian, and therefore the first two concentration moments provide very limited information of the underlying transport dynamics. Instead, the pdfs provide crucial information for applications, such as the probability of exceeding a given concentration, for example, the regulatory limit, at a particular location within the domain of interest. Furthermore, by analyzing the second‐order moments of the concentration, we show that mixing is strongly affected by space dimensionality and that the two‐dimensional approach, often used for computational convenience, may severely underestimate reaction rates in real settings.

Probability density functions for passive scalars dispersed in random velocity fields

Spatial and temporal heterogeneity of ambient natural environments play a significant role in large scale transport phenomena. Uncertainty about spatio‐temporal fluctuations in system parameters (e.g., flow velocity) make deterministic predictions of macroscopic system states (e.g., solute concentration) elusive. Distributions of system states generally exhibit highly non‐Gaussian behavior, which cannot be captured solely by the corresponding mean and variance. Instead, these features of transport are described by the probability density function (PDF) of a system state, e.g., the PDF of concentration at a certain point in space and time. We study the PDF of a passive scalar that disperses in a random velocity field. We derive an explicit map between the velocity distribution and the scalar PDF, and obtain approximate solutions for the PDF of the normalized scalar. These solutions enable one to quantify explicitly the impact of dispersion on the evolution of the passive scalar PDF without recurrence to classical closure approximations used in mixing models.

A joint velocity‐concentration PDF method for tracer flow in heterogeneous porous media

The probability density function (PDF) of the local concentration of a contaminant, or tracer, is an important component of risk assessment in applications that involve flow in heterogeneous subsurface formations. In this paper, a novel joint velocity‐concentration PDF method for tracer flow in highly heterogeneous porous media is introduced. The PDF formalism accounts for advective transport, pore‐scale dispersion (PSD), and molecular diffusion. Low‐order approximations (LOAs), which are usually obtained using a perturbation expansion, typically lead to Gaussian one‐point velocity PDFs. Moreover, LOAs provide reasonable approximations for small log conductivity variances (i.e., σY2 < 1). For large σY2, however, the one‐point velocity PDFs deviate significantly from the Gaussian distribution as demonstrated convincingly by several Monte Carlo (MC) simulation studies. Furthermore, the Lagrangian velocity statistics exhibit complex correlations that span a wide range of scales, including long‐range correlations due to the formation of preferential flow paths. Both non‐Gaussian PDFs and complex long‐range correlations are accurately represented using Markovian velocity processes (MVPs) in the proposed joint PDF method. LOA methods can be generalized to some extent by presuming a certain shape for the concentration PDF (e.g., a β PDF fully characterized by the concentration mean and variance). The joint velocity‐concentration PDF method proposed here does not require any closure assumptions on the shape of the marginal concentration PDF. The Eulerian joint PDF transport equation is solved numerically using a computationally efficient particle‐based approach. The PDF method is validated with high‐resolution MC reference data from Caroni and Fiorotto (2005) for saturated transport in velocity fields, which are stationary in space and time, for domains with σY2 = 0.05, 1, and 2 and Péclet numbers ranging from 100 to 10,000. PSD is modeled using constant anisotropic dispersion coefficients in both the reference MC computations and our PDF method.

An idealised model of turbulent dispersion: two and three spatial dimensions

AbstractIn the idealised model of turbulent dispersion introduced by Zimmerman and Chatwin (1995), the probability density function (pdf) of concentration becomes bimodal at large times, with peaks at the smallest and largest concentrations. That model only has one spatial dimension, but here I extend the model to two and three spatial dimensions. I also extend to two and three dimensions the pdf calculation method that was given by Mole and Yeun (2007). I use this method to derive large‐time analytical solutions for the pdf, and to show that in two and three dimensions the weight of the pdf shifts from extreme concentrations towards the mean concentration. In the typical three‐dimensional case, the large‐time pdf is unimodal with the peak at the mean concentration. Copyright © 2010 John Wiley & Sons, Ltd.

PDF equations for advective–reactive transport in heterogeneous porous media with uncertain properties

We consider advective–reactive solute transport in porous media whose hydraulic and transport properties are uncertain. These properties are treated as random fields, which renders nonlinear advection–reaction transport equations stochastic. We derive a deterministic equation for the probability density function (PDF) of the concentration of a solute that undergoes heterogeneous reactions, e.g., precipitation or dissolution. The derivation treats exactly (without linearization) a reactive term in the transport equation which accounts for uncertainty (randomness) in both flow velocity and kinetic rate constants but requires a closure, such as a Large-Eddy-Diffusivity (LED) approximation used in the present analysis. No closure is required when reaction rates are the only source of uncertainty. We use exact concentration PDFs obtained for this setting to analyze the accuracy of our general, LED-based PDF equations.

Concentration fluctuations in a downtown urban area. Part II: analysis of Joint Urban 2003 wind-tunnel measurements

An analysis of concentration time series measured in a boundary-layer wind tunnel at the University of Hamburg is presented. The measurements were conducted with a detailed aerodynamic model of the Oklahoma City (OKC) central business district (CBD) at the scale of 1:300 and were part of the Joint Urban 2003 (JU2003) project. Concentration statistics, as well as concentration probability density (PDF) and exceedance probability (EDF) functions were computed for street- and roof-level sites for three different wind directions. Taking into account the different length scales and wind speeds in the wind-tunnel (WT) and full-scale experiments, dimensionless concentrations and a dimensionless time scale are computed for the comparison with data from the JU2003 full-scale tracer experiments, conducted in OKC in 2003. Using such dimensionless time, the WT time series cover a ~20 times longer time span than the JU2003 full-scale time series, which are analysed in detail in an accompanying, first part of this paper. The WT time series are thus divided into 20 consecutive blocks of equal length and the statistical significance of parameters based on relatively short records is assessed by studying the variability of the concentration statistics and probability functions for the different blocks. In particular at sites closer to the plume edge, the results for the individual blocks vary significantly and at such sites statistics from short records are not very representative. While the location of three sampling sites in the WT closely matched the sites during the full-scale experiments, the prevailing wind directions during the JU2003 releases were not exactly matched. The comparison between full-scale and WT concentration parameters should thus primarily be interpreted in a qualitative rather than direct quantitative sense. Given the differences in mean wind directions and concerns about the representativeness of full-scale concentration statistics, the WT and full-scale results compared well. The 98 percentile concentrations for almost all full-scale releases analyzed are within the scatter of the percentiles observed in the block analysis of the WT time series. Furthermore, the concentration percentiles appear linearly correlated with the fluctuation intensities and the linear relationships determined in the wind tunnel agree well with full-scale results.

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.

Probability density functions for advective–reactive transport in radial flow

We study transport of a reactive solute in a chemically heterogeneous porous medium whose chemical properties are uncertain. The dissolved substance undergoes a heterogeneous chemical reaction with a solid phase in the presence of advection caused by extraction/injection from a point source. We present semi-analytical solutions for the probability density function of the solute concentration, which allows us to quantify predictive uncertainty associated with uncertain reaction rate constants for both linear and nonlinear reactions. This enables one to compute probabilities of rare events, which are required for quantitative risk analyses.