Probability Density Function Transport Equation Research Articles

On distributions of velocity random fields in turbulent flows

AbstractThe present paper aims to derive a partial differential equation (PDE) for the single‐time single‐point probability density function (PDF) of the velocity field of a turbulent flow. The PDF PDE is a highly non‐linear parabolic‐transport equation, which depends on two conditional statistical numerics of important physical significance. The PDF PDE is a general form of the classical Reynolds mean flow equation (O. Reynolds, Philos. Trans. R. Soc. Lond. Ser. A 186 (1895), 123–164. http://doi.org/10.1098/rsta.1895.0004) and is a precise formulation of the PDF transport equation (S. B. Pope, Turbulent flows, Cambridge University Press, Cambridge, 2000. https://doi.org/10.1017/CBO9780511840531.). The PDF PDE provides us with a new method for modelling turbulence. An explicit example is constructed. Though the example is seemingly artificial, it demonstrates the PDF method based on the new PDF PDE.

Fully consistent Eulerian Monte Carlo fields method for solving probability density function transport equations in turbulence modeling

The probability density function (PDF) transport equation method is a sophisticated model for the closure of turbulent mixing and turbulent reactive flows. An efficient solution approach for solving the PDF transport equation has been vital for the method to be widely used in applications. The Eulerian Monte Carlo fields (EMCF) method has been developed to solve the PDF transport equation efficiently for decades. A recent work by Wang et al. [Phys. Fluids 30, 115106 (2018)] revealed a serious issue of the EMCF method for not being fully consistent with the PDF transport equation for which the method is designed to solve. This work advances the state of the art by introducing fully consistent EMCF methods for solving the PDF transport equation. The fully consistent EMCF formulations are derived for two different PDF equation forms. The consistency of the EMCF formulations is mathematically confirmed by examining the derived moment transport equations from the EMCF formulations and from the PDF transport equation. The method of manufactured solutions is employed to further verify the consistency and convergence of the different EMCF formulations numerically. The newly introduced EMCF formulations bring the EMCF method to full consistency with the PDF transport equations for the first time.

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.

Transported PDF simulation of auto-ignition of a turbulent methane jet in a hot, vitiated coflow

In the current work, the auto-ignition of a turbulent round methane jet is studied numerically by means of a transported probability density function (PDF) method. The methane jet is issued into a hot, vitiated coflow, where it ignites to form a steady lifted flame. For this flame, experimental data of hydroxyl, temperature and mixture fraction are provided in the area where the fuel auto-ignites. To model this experiment, the transport equation for the thermochemical PDF is solved using a hybrid finite volume / Lagrangian Monte-Carlo method. Turbulence is modelled using the k-ϵ turbulence model including a jet-correction. Computational results are compared to experimental data in terms of mean quantities, variances and lift-off height. Moreover, the structure of the one-point, one-time marginal PDF of temperature is analysed and compared to experimental data which are provided in this work. It is found that the transported PDF method in conjunction with the k-ϵ model is capable of reproducing these statistical data very well. In particular the effect of ignition on the marginal PDF of temperature can be well reproduced with this approach. To further analyse the relevant processes in the evolution of the temperature PDF, a statistically homogeneous system is studied both numerically and analytically.

LES of the Sandia flame series D-F using the Eulerian stochastic field method coupled with tabulated chemistry

In this paper, the Eulerian Stochastic Field (ESF) model in the Transported Probability Density Function (TPDF) class model is combined with the Flamelet Generated Manifolds (FGM) model. This method solves the joint probability density function transport equation by ESF method that considers the interaction mechanism between flame and turbulence with high precision. At the same time, by making use of the advantage of the FGM model, this model is able to incorporate the detailed chemical reaction mechanism (GRI 3.0) with acceptable computational cost. The new model has been implemented in the open source CFD suite-OpenFOAM. Validation of the model has been carried out by simulating the Sandia flame series (three turbulent piloted methane jet flames) issued by the National Laboratory of the United States. The accuracy and advancement of the ESF/FGM turbulent combustion model are verified by comparing the LES results of the new model with the rich experimental data as well as the RANS results. The results demonstrate that the model has a strong ability in capturing combustion phenomena such as extinction and re-ignition in turbulent flame, which is essential in the accurate prediction of the combustion process in real combustion devices, for example, aircraft engines.

The stochastic fields method applied to a partially premixed swirl flame with wall heat transfer

Large eddy simulations of a partially premixed flame are performed with the purpose of predicting the reacting flow in a swirl-stabilised, low emissions industrial gas turbine combustor. The corresponding sub-grid scale turbulence–chemistry interactions are modelled using a probability density function transport equation, which is solved by the stochastic fields method. A 15-step reduced, but accurate, methane mechanism including 19 species is employed for the description of all chemical reactions. The test case involves a combustor with complex geometry and simulations are carried out for two different combustor operating conditions. Overall, results of the velocity, temperature and species mass fractions (including carbon monoxide) as well as the instantaneous thermochemical properties are shown to be in good agreement with experimental data, demonstrating the capabilities of the applied stochastic fields method. The inclusion of wall heat transfer in the combustion chamber is found to improve temperature and species predictions, especially in the near-wall regions. Comparisons between an oscillating and a ‘stable’ flame case furthermore highlight the influence of experimentally observed thermo-acoustic instabilities on the scalar fluctuations near the combustor centreline. None of the default model parameters were adjusted and the results showcase the accuracy and flexibility of the present large eddy simulation method for an application to complex, partially premixed combustion problems; this being particularly important for the designers of new generation low emission gas turbine combustors.

Consistency and convergence of Eulerian Monte Carlo field method for solving transported probability density function equation in turbulence modeling

The transported probability density function (PDF) method is an attractive model for the closure of turbulent mixing and turbulent reactive flows. The transported PDF method suffers from the curse of dimensionality and an efficient numerical implementation of the method has always been a research topic of great importance. The Eulerian Monte Carlo field (EMCF) method (also termed as the stochastic field method) has been proposed as an efficient solution approach for solving the PDF transport equation for two decades. In this work, we revisit the EMCF method and examine its mathematical consistency analytically and numerically. It is found that the EMCF method is not mathematically consistent with the PDF transport equation that the method intends to solve. This creates a serious inconsistency issue and causes uncertainties in the yielded numerical solutions by EMCF. It is imperative to examine the effect of the inconsistency. We evaluate the effect of the inconsistency in a simplified turbulent mixing layer test case. Corrections to remedy the mathematical inconsistency are proposed and examined. The effectiveness of the corrections is demonstrated numerically through convergence testing. The effect of the Reynolds number on the inconsistency is explored. The impact of the inconsistency is also investigated in a thermal wake behind a line source in grid turbulence to assess the importance of the issue in real turbulence problems.

Developing theory of probability density function for stochastic modeling of turbulent gas-particle flows

Turbulent gas-particle flows are studied by a kinetic description using a probability density function (PDF). Unlike other investigators deriving the particle Reynolds stress equations using the PDF equations, the particle PDF transport equations are directly solved either using a finite-difference method for two-dimensional (2D) problems or using a Monte-Carlo (MC) method for three-dimensional (3D) problems. The proposed differential stress model together with the PDF (DSM-PDF) is used to simulate turbulent swirling gas-particle flows. The simulation results are compared with the experimental results and the second-order moment (SOM) two-phase modeling results. All of these simulation results are in agreement with the experimental results, implying that the PDF approach validates the SOM two-phase turbulence modeling. The PDF model with the SOM-MC method is used to simulate evaporating gas-droplet flows, and the simulation results are in good agreement with the experimental results.

Turbulent Mixing Simulation via a Quantum Algorithm

Probability density function (PDF) methods have been very useful in describing many physical aspects of turbulent mixing. In applications of these methods, modeled PDF transport equations are commonly simulated via classical Monte Carlo techniques, which provide estimates of moments of the PDF at arbitrary accuracy. In this work, recently developed techniques in quantum computing and quantum enhanced measurements (quantum metrology) are used to construct a quantum algorithm that accelerates the computation of such estimates. This quantum algorithm provides a quadratic speedup over classical Monte Carlo methods in terms of the number of repetitions needed to achieve the desired precision. This paper illustrates the power of this algorithm by considering a binary scalar mixing process modeled by means of the coalescence/dispersion (C/D) closure. The equation is first simulated using classical Monte Carlo methods, where error estimates for the computation of central moments are provided. Then the quantum algorithm for this problem is simulated by sampling from the same probability distribution as that of the output of a quantum computer, and it is shown that significantly fewer resources are required to achieve the same precision. The results demonstrate potential applications of future quantum computers for simulation of turbulent mixing, and large classes of related problems.

Analysis of Mixing Models for use in Simulations of Turbulent Spray Combustion

The present study concerns the investigation of different mixing models for use in the transported probability density function (PDF) modeling of turbulent (reacting) spray flows. The modeling of the turbulent mixing and other characteristic scalar variables such as gas enthalpy using transported (joint) PDFs has become an important method to describe turbulent (reacting) spray flows since the evaporation process causes the PDF of the mixture fraction to deviate from the widely used β function, which is typically used in models for turbulent gas flows. In the PDF transport equation, the molecular mixing does not appear in closed form so that modeling strategies are required. For gas combustion, the interaction-by-exchange-with-the-mean (IEM) model, the modified Curl (MC) model, and the Euclidean minimum spanning tree (EMST) models are used. More recently, a new mixing model, the PSP model, which is based on parameterized scalar profiles has been developed. The present study focuses on the use and analysis of the IEM, MC and PSP models for turbulent spray flames. For this purpose, the models are reconsidered with respect to the evaporation process that must be included and evaluated if spray combustion is considered. For model evaluation, turbulent ethanol/air spray flames are simulated, and the results are compared to experimental data by A. Masri, University of Sydney, Australia.

A multiple mapping conditioning mixing model with a mixture-fraction like reference variable. Part 1: Model derivation and ideal flow test cases

Probability density function (PDF) methods have significant advantages in modeling turbulent combustion, in particular because the highly non-linear chemical source terms appear in closed form. The micromixing term in the PDF transport equations, representing diffusion in composition space, is however unclosed. Physically, micromixing occurs between regions of fluid having similar compositions. It is expected and generally shown that models that account for this localness of mixing perform better. In this work, a novel variation of the multiple mapping conditioning (MMC) mixing model is proposed. MMC makes use of reference variables to localise the mixing and was originally formulated for stochastic reference variables with standard Gaussian distributions. Here we follow the original interpretation but modify it by using a reference variable that has an evolving distribution, according to an Ornstein–Uhlenbeck process, with statistics that are similar to those of a physical scalar. In the present application of the model to nonpremixed combustion, mixture fraction is considered as the scalar; extensions of the model may be envisaged for other combustion modes by considering a different scalar variable. While the original version of MMC is mathematically elegant, it is conceptually complex and evaluation of model coefficients is difficult. The new model is both physically intuitive and its coefficients are easy to determine according to desired principles in canonical mixing situations. In Part I of this two-paper set, the model is studied analytically for homogeneous and mean scalar gradient flows without chemical reaction leading to an approach to set the model parameters that controls the unconditional scalar dissipation rate and delivers tunable localness. The behaviour of the model is also examined numerically in simple homogeneous mixing situations with and without chemical reaction. In Part II, the new MMC mixing model is implemented in the RANS context and validated against experimental data for the Sandia D–F flame series.

Numerical investigation of the ignition of diethyl ether/air and propane/air mixtures by hot jets

Electrical equipment for use in hazardous areas has to be specially designed and tested to avoid an explosion. Typical ignition sources like arcing contacts can e.g. be enclosed by flameproof enclosures. An internal explosion may lead to jetting of hot exhaust gases through inevitable gaps like joint clearance or other gaps into the surrounding, which may result in an accidental explosion. The initiation of an explosion in a premixed mixture either of diethyl ether and air or of propane and air caused by hot exhaust gas jets is examined in this work by means of numerical simulations. A PDF-PM algorithm is used to examine the ignition processes. The algorithm is based on a Lagrangian PDF (probability density function) method, in conjunction with a projection method (PM) to calculate the mean pressure for transient flows. The transport equation for a joint velocity-turbulent frequency-scalar PDF is solved by a Monte Carlo/particle method. The reaction-diffusion manifold (REDIM) technique is used to reduce the computational cost concerning the calculation of chemical reactions. The chemical time scales of diethyl ether and propane are nearly identical and they have nearly identical maximum experimental safe gap (MESG) values. Therefore, the ignition processes of both gases are compared to evaluate the used numerical model.

Effects of molecular transport in LES/PDF of piloted turbulent dimethyl ether/air jet flames

We report large-eddy simulation (LES)/probability density function (PDF) modeling of piloted turbulent dimethyl ether (DME)/air jet flames with a skeletal chemical mechanism. The modeled PDF transport equation with three different implementations of molecular transport in the interaction-by-exchange-with-the-mean (IEM) mixing model is solved using the NGA/HPDF code in order to assess the a posteriori performance of these LES/PDF methodologies applied to the DME flames D and F proposed in the TNF Workshop. The three implementations considered are the classical random-walk model (IEM-RW), the mean-drift model with a single molecular diffusivity (IEM-MD), and the mean-drift model with differential diffusion (IEM-DD). Better quantitative agreement with experimental data for the mean and root-mean-square profiles of mixture fraction, temperature and major species mole fractions and conditional means on mixture fraction are obtained with the IEM-MD model than with the IEM-RW model. In LES of the DME flame series, the molecular diffusivity dominates the turbulent diffusivity in the centerline region, so the accurate modeling of molecular transport plays a significant role in LES/PDF of these jet flames. Nevertheless, the IEM-DD model shows only minor effects on the model predictions compared to IEM-MD in these DME flames.

Analysis of High Concentration Spikes Appearing in Mass Plume in Nearly Homogeneous Turbulent Flow Based on the PDF Transport Equation

To evaluate the pollutant dispersion in background turbulent flows, most researches focus on statistical variation of concentrations or its fluctuations. However, those time-averaged quantities may be insufficient for risk assessment, because there emerge many high-intensity pollutant areas in the instantaneous concentration field. In this study, we tried to estimate the frequency of appearance of the high concentration areas in a turbulent flow based on the Probability Density Function (PDF) of concentration. The high concentration area was recognized by two conditions based on the concentration and the concentration gradient values. We considered that the estimation equation for the frequency of appearance of the recognized areas consisted of two terms based on each condition. In order to represent the two terms with physical quantities of velocity and concentration fields, simultaneous PIV (Particle Image Velocimetry) and PLIF (Planar Laser-Induced Fluorescence) measurement and PLIF time-serial measurement were performed in a quasi-homogeneous turbulent flow. According to the experimental results, one of the terms, related to the condition of the concentration, was found to be represented by the concentration PDF, while the other term, by the streamwise mean velocity and the integral length scale of the turbulent flow. Based on the results, we developed an estimation equation including the concentration PDF and the flow features of mean velocity and integral scale of turbulence. In the area where the concentration PDF was a Gaussian one, the difference between the frequencies of appearance estimated by the equation and calculated from the experimental data was within 25%, which showed good accuracy of our proposed estimation equation. Therefore, our proposed equation is feasible for estimating the frequency of appearance of high concentration areas in a limited area in turbulent mass diffusion.

Effects of soot absorption coefficient–Planck function correlation on radiative heat transfer in oxygen-enriched propane turbulent diffusion flame

The main objective of this article is to quantify the influence of the soot absorption coefficient–Planck function correlation on radiative loss and flame structure in an oxygen-enhanced propane turbulent diffusion flame. Calculations were run with and without accounting for this correlation by using a standard k–ε model and the steady laminar flamelet model (SLF) coupled to a joint Probability Density Function (PDF) of mixture fraction, enthalpy defect, scalar dissipation rate, and soot quantities. The PDF transport equation is solved by using a Stochastic Eulerian Field (SEF) method. The modeling of soot production is carried out by using a flamelet-based semi-empirical acetylene/benzene soot model. Radiative heat transfer is modeled by using a wide band correlated-k model and turbulent radiation interactions (TRI) are accounted for by using the Optically-Thin Fluctuation Approximation (OTFA). Predicted soot volume fraction, radiant wall heat flux distribution and radiant fraction are in good agreement with the available experimental data. Model results show that soot absorption coefficient and Planck function are negatively correlated in the region of intense soot emission. Neglecting this correlation is found to increase significantly the radiative loss leading to a substantial impact on flame structure in terms of mean and rms values of temperature. In addition mean and rms values of soot volume fraction are found to be less sensitive to the correlation than temperature since soot formation occurs mainly in a region where its influence is low.

3D Simulation of the Filtration and Dust Retention Process of a Fuel Filter

AbstractA model for the description of the dust retention process inside a filter was developed and successfully applied to a detailed 3D numerical simulation of a Bosch automotive diesel filter. Specifically, the variation of the pressure drop due to the clogging process in the filter paper could be well predicted. For this approach, an Eulerian‐Eulerian two‐phase flow formulation was applied. For the dispersed phase, the Eulerian description was made by means of a probability density function transport equation. The simulations also provide complete spatial information about the flow both in the clean and dirty sides of the filter and the particle distribution, size, and velocity. The intention of the newly developed numerical tool is to help in the design of filtering devices.

Lagrangian Mixing Models for Turbulent Combustion: Review and Prospects

Compared to other simulation approaches utilized for modeling turbulent reacting flows with detailed chemistry, probability density function (PDF) methods offer several advantages. This is because the changes in fluid composition due to convection and reaction processes can be treated exactly. PDF methods require however closure models for the mixing process representing the transport of the PDF owing to molecular diffusion. Over the years several mixing models with different degrees of complexity have been developed. A review of the main Lagrangian mixing models for turbulent combustion developed so far is presented in this work. This review includes models where the composition of a particle changes (i) essentially independently of the composition associated with the other particles, and (ii) through direct interaction with other particles. The main advantages and shortcomings of the mixing models reviewed are highlighted accordingly. Because Lagrangian Monte Carlo techniques are usually used for solving PDF transport equations, a particular emphasis is put on their corresponding particle implementation. The mixing models review is preceded by a section highlighting the mathematical formulation associated with the use of PDF methods for turbulent reacting flows. In the last part of the article both comparative results of mixing models performance and prospects for the mixing models are discussed. Despite the effort that has been devoted to the development of more capable mixing models, currently there is no mixing model presenting all desirable characteristics. Even more, there are significant differences in the results obtained when different mixing models are utilized, especially when higher order scalar statistics are accounted for. Therefore work still needs to be carried out in order to develop a mixing model satisfying all desirable characteristics expected from these models. Several avenues can be further explored in order to achieve this goal. These potential routes include those accounting for spatial scalar structures and both scalar length and turbulent frequency scales distributions. Other approaches based on competitive mixing, manifold-based features and Lagrangian coherent structures have also the potential to further improve upon existing mixing models. The development of a sound mixing model will allow eventually removing one of the largest sources of modeling uncertainty in PDF-based computations.

Transported PDF modeling of pulverized coal jet flames

A transported composition probability density function (PDF) method is developed for pulverized coal combustion. A consistent hybrid Lagrangian particle/Eulerian mesh algorithm is used to solve the modeled PDF transport equation for the gas phase, with finite-rate gas-phase chemistry. The model includes k–ε turbulence, gradient transport for scalars, and a Euclidean minimum spanning tree (EMST) mixing model. A separate Lagrangian description is used to solve for the coal particle phase, including particle tracking, coal devolatilization and surface reaction models. Interphase coupling models are developed to handle the interaction between the gas phase and the solid phase. Radiative heat transfer is modeled by a P1 model for a gray absorbing emitting and scattering gas–particle system. Two independent laboratory-scale pulverized coal jet flames (“flame A” and “flame B”) are simulated using the new model. For flame A, the baseline model reproduces the measured mean and rms particle axial velocity reasonably well. Some discrepancies are found in particle temperature and gas-phase concentrations, which may in part be due to the uncertainties in the experimental data. Sensitivities of model results to coal-related model variations, turbulence–chemistry interactions, different interphase coupling strategies, and finite-rate chemistry are explored to establish sensitivities and to determine which aspects of the models are most important. The same model is applied to a second flame (flame B), the only change being in parameters related to the different coal composition. It is found that experimental standoff heights cannot be reproduced for three different stoichiometric ratios using a single model. Time scales for chemical reactions, devolatilization and turbulence are extracted and compared, to study the level of turbulence–chemistry–particle interactions in flame A and to test the popular assumption of equilibrium chemistry in coal combustion modeling. Mixture-fraction statistics for flame B are explored to test assumptions that have been proposed for mixture-fraction-based coal models. While the usual assumption of Beta distributions is found to be appropriate, assumptions of statistical independence of two mixture fractions are not valid.

PDF modeling and simulations of pulverized coal combustion – Part 1: Theory and modeling

A transported probability density function (PDF) method is developed for pulverized coal combustion. Two separate PDF transport equations, one for the gas phase (joint velocity-composition PDF) and one for the coal particle phase, are solved by means of stochastic Lagrangian methods. The gas composition is described by two mixture fractions (one that tracks the devolatilization products and one that tracks the char combustion products) and the total specific enthalpy. Two different models are proposed for the unclosed terms in the gas-phase PDF transport equation due to the mass transfer from the solid to the gas phase through devolatilization and char combustion. To close the transport equation for the coal particle PDF, the gas phase velocity and composition sampled along the coal particle trajectories are modeled. Radiative heat transfer is modeled by a discrete transfer method (DTM) to solve for the Reynolds averaged radiative heat transfer equation for a gray absorbing emitting and scattering gas–particle mixture. The turbulence–radiation interaction of the emission is given in closed form in the proposed two-phase PDF approach.

PDF modeling and simulations of pulverized coal combustion – Part 2: Application

A transported probability density function (PDF) method developed for pulverized coal combustion is applied in simulations of a semi-industrial scale furnace. It is found that the simulation results are sensitive to the chosen model for unclosed terms in the gas-phase PDF transport equation due to the mass transfer from the solid to the gas phase through devolatilization and char combustion. A model which leads to a generation of mixture fraction fluctuations provides results which are in close agreement with the measurements. Two different models for the gas phase composition seen by the coal particles are investigated. One model is based on the local mean values of the gas composition and one model provides time correlated fluctuations of the seen composition. Small differences between the two model results are observed in the predictions of the char burnout. The simulation results are further used to analyze commonly adopted model assumptions in presumed PDF methods. It is found that both, the char combustion mixture fraction and the enthalpy are not statistically independent from the volatile mixture fraction. It is further shown that the marginal PDF of the volatile mixture fraction is not well approximated by a beta function PDF in regions of rapid devolatilization. The marginal enthalpy PDF indicates significant enthalpy fluctuations which are usually neglected in presumed PDF methods.