Presumed Probability Density Function Model Research Articles

Development of a filtered reaction rate model for reactive gas–solid flows based on fine‐grid simulations

AbstractThe filtered reaction rate for the coarse‐grid simulations of reactive gas–solid flows can be obtained via a correction to its microscopic reaction rate. The correction term is defined as the mesoscale effectiveness factor ηΔ, which is the ratio of the reaction rate obtained from the fine‐grid simulations to that obtained from the coarse‐grid simulations. The considered reaction is an isothermal, solid‐catalyzed surface reaction with a power law reaction rate model. The simulation results show that the mean ηΔ is almost invariant with the reaction orders at the same Damköhler number. A closure correlation for the mean ηΔ is formulated. However, the standard deviation of ηΔ is found to be quite large. A presumed probability density function model is proposed to capture the fluctuating properties of ηΔ. The predictability of the closure correlations are evaluated via performing the filtered two fluid model simulations in circulating fluidized beds of ozone decomposition.

Presumed Joint-PDF Modelling for Turbulent Stratified Flames

Turbulence–chemistry interaction models such as the conditional moment closure and various flamelet models require a presumed Probability Density Function (PDF) model of the conditioning variables. In turbulent stratified flames, a joint-PDF model for a reaction progress variable and the mixture fraction is required. The joint-PDF is often modelled by two beta functions using the first and second moments of the two conditioning variables. In this work, the performance of a joint-PDF model based on the flamelet PDF approach is compared to the double beta-PDF model. The conditional PDF of the reaction progress variable conditioned on mixture fraction is modelled with the flamelet PDF. Unlike the beta PDF, the flamelet functional form for the conditional PDF varies with the local mixture fraction. The two PDF models are coupled with a two-dimensional premixed flamelet-generated manifold chemistry model. Two stratified flames of the Cambridge–Sandia bluff-body burner are simulated using a Reynolds-Averaged Navier–Stokes (RANS) approach. The results indicate that the flamelet PDF model has a superior, albeit marginal, predictions of the temperature, major and minor species mass fractions.

Deep learning for presumed probability density function models

In this work, we use machine learning (ML) techniques to develop presumed probability density function (PDF) models for large eddy simulations (LES) of reacting flows. The joint sub-filter PDF of mixture fraction and progress variable is modeled using various ML algorithms and commonly used analytical models. The ML algorithms evaluated in the work are representative of three major classes of ML techniques: traditional ensemble methods (random forests), deep learning (deep neural network (DNN)s), and generative learning (conditional variational autoencoder (CVAE)). The first two algorithms are supervised learning algorithms, and the third is an unsupervised learning algorithm. Data from direct numerical simulation (DNS) of the low-swirl burner[1] are used to develop training data for sub-filter PDF models. Models are evaluated on predictions of the sub-filter PDF as well as predictions of the filtered reaction rate of the progress variable, computed through an integral of the product of the sub-filter PDF and the conditional means of the reaction rate. This a-priori modeling study demonstrates that deep learning models for presumed PDF modeling are three times more accurate than analytical β–β PDF models and linear regression models. These models are as accurate as random forest models while using five times fewer trainable parameters and being 25 times faster for inference. In this work, conditional unsupervised learning did not present additional advantages beyond supervised learning with a feed-forward neural network. We illustrate how models generalize to other regions of the flow and develop criteria based on the Jensen–Shannon divergence to quantify the performance of a model on new data.

OpenCrys: Open-Source Software for the Multiscale Modeling of Combined Antisolvent and Cooling Crystallization in Turbulent Flow

The open-source software, called openCrys, is provided for the multiscale simulation of antisolvent and combined antisolvent-cooling crystallization. It simulates the macro- and micromixing scales, and the complete energy and population balance equations during crystal nucleation and growth. The model is based on the Reynolds-Averaged-Navier–Stokes equation, coupled with a three-environment presumed probability density function model, and the spatially varying population balance equation semidiscretized using a high resolution finite-volume method. openCrys is implemented in C++ object oriented programming language using the open-source CFD package OpenFOAM. The software is used to compare the performance of dual impinging jet, coaxial, and radial crystallizers. It is shown that improving the micromixing does not necessarily result in a narrower crystal size distribution when temperature effects are taken into account. The complex interplay of crystallizer kinetics and momentum, mass, and heat transfer makes the selection of the best mixer for a particular application to be nonobvious, which motivates the development and application of high-fidelity multiscale simulations for the design of antisolvent crystallizers.

A parametric study of microjet assisted methane/air turbulent flames

A parametric study of microjet assisted methane/air turbulent flames characteristics is numerically investigated. The Presumed Probability Density Function model and the Discrete Ordinates model are respectively considered for combustion and radiation modeling. The k-epsilon Standard model with Pope Correction is adopted as a turbulence closure model. The two step Tesner model is used to quantify the soot particle production in the flame configuration. Comparison with our previous work using the k-epsilon Realizable model shows that the k-epsilon Standard model with Pope Correction ensures better predictions. The microjet velocity and diameter effects on thermal field, mixing process and soot emission are then discussed. Numerical findings show that the microjet can be used as an efficient tool controlling methane/air turbulent flames. On the one hand, it is shown that the microjet creates an inner flame in the vicinity of the central nozzle exit but does not globally alter the methane/air flame shape. On the other hand, mixing process can be enhanced for high microjet Reynolds number either by increasing the microjet velocity or by decreasing its nozzle diameter for a constant microjet mass flow rate. Soot production can be consequently reduced for low microjet diameter and high velocity values.

Simulation of swirl-stabilized turbulent partially premixed combustion

ABSTRACTProperly describing turbulence-complex reaction interactions is crucial for predicting turbulent combustion and pollutant emission. A presumed probability density function model for temperature fluctuation is adopted in the present paper. It incorporates a 25-step skeletal mechanism for methane combustion. The gas turbulent transport is simulated with the algebraic Reynolds stress model for turbulence-swirl interactions. These models are applied to the simulation of swirl-stabilized turbulent partially premixed jet flame. The calculated gas velocities, fluctuating velocities, Reynolds shear stresses, temperature, and species mass fractions are in agreement with the measured test data.

Large eddy simulation of turbulent premixed combustion using tabulated detailed chemistry and presumed probability density function

ABSTRACTA method of chemistry tabulation combined with presumed probability density function (PDF) is applied to simulate piloted premixed jet burner flames with high Karlovitz number using large eddy simulation. Thermo-chemistry states are tabulated by the combination of auto-ignition and extended auto-ignition model. To evaluate the predictive capability of the proposed tabulation method to represent the thermo-chemistry states under the condition of different fresh gases temperature, a-priori study is conducted by performing idealised transient one-dimensional premixed flame simulations. Presumed PDF is used to involve the interaction of turbulence and flame with beta PDF to model the reaction progress variable distribution. Two presumed PDF models, Dirichlet distribution and independent beta distribution, respectively, are applied for representing the interaction between two mixture fractions that are associated with three inlet streams. Comparisons of statistical results show that two presumed PDF models for the two mixture fractions are both capable of predicting temperature and major species profiles, however, they are shown to have a significant effect on the predictions for intermediate species. An analysis of the thermo-chemical state-space representation of the sub-grid scale (SGS) combustion model is performed by comparing correlations between the carbon monoxide mass fraction and temperature. The SGS combustion model based on the proposed chemistry tabulation can reasonably capture the peak value and change trend of intermediate species. Aspects regarding model extensions to adequately predict the peak location of intermediate species are discussed.

Evaluation of Presumed Probability-Density-Function Models in Non-Premixed Flames by using Large Eddy Simulation

Four kinds of presumed probability-density-function (PDF) models for non-premixed turbulent combustion are evaluated in flames with various stoichiometric mixture fractions by using large eddy simulation (LES). The LES code is validated by the experimental data of a classical turbulent jet flame (Sandia flame D). The mean and rms temperatures obtained by the presumed PDF models are compared with the LES results. The β-function model achieves a good prediction for different flames. The predicted rms temperature by using the double-δ function model is very small and unphysical in the vicinity of the maximum mean temperature. The clip-Gaussian model and the multi-δ function model make a worse prediction of the extremely fuel-rich or fuel-lean side due to the clip at the boundary of the mixture fraction space. The results also show that the overall prediction performance of presumed PDF models is better at mediate stoichiometric mixture fractions than that at very small or very large ones.

Systematically reduced rate mechanisms and presumed PDF models for premixed turbulent combustion

The use of reduced kinetic mechanisms in models for turbulent premixed flames, at large but finite Damköhler numbers, is described. Taking as examples the two cases of hydrogen–air and methane–air systems for which there exist reduced kinetic schemes characterized by two independent scalar variables only, it is shown that the mean chemical production rate can be in a general way expressed as the product of a mixing factor, depending on the segregation of species due to turbulence, with a reaction rate factor. In this latter factor the probability density function (PDF) of the two scalars appears only as a constant integral quantity. The extension of this analysis to systems involving more than two independent scalars is envisaged. The general result is applied to the two specific cases of interior PDF’s modelled by (i) a single Dirac delta function and (ii) a laminar flamelet.

Large-eddy simulation of turbulent combustion using different combustion models

The second-order moment (SOM) combustion model proposed by the present authors is compared with eddy-break-up (EBU) and presumed probability density function (PDF) combustion models in large-eddy simulation of jet diffusion combustion, swirling diffusion combustion and premixed combustion behind a bluff body. The statistical results for time-averaged and RMS fluctuation temperatures are validated by experimental results. It is seen that the SOM model can always give good statistical results in LES of both non-premixed and premixed combustion, whereas the EBU and presumed PDF models cannot always do. The instantaneous results indicate that organized large vortex and thin flame surface structures are observed in jet diffusion combustion and bluff-body stabilized premixed combustion, whereas no organized vortex and thin flame surface structures are observed in swirling diffusion combustion.

Conditional Source-Term Estimation as a Method for Chemical Closure in Premixed Turbulent Reacting Flow

A Conditional Source-term Estimation (CSE) model is used to close the mean reaction rates for a turbulent premixed flame. A product-based reaction progress variable is introduced as the conditioning variable for the CSE method. Different presumed probability density function (PDF) models are studied and a modified version of a laminar flame-based PDF model is proposed. Improved predictions of the variable distribution are obtained. The conditional means of reactive scalars are evaluated with CSE and compared to the direct numerical simulation (DNS). The mean reaction rates in a turbulent premixed flame are evaluated with the CSE model and the presumed PDFs. Comparison of the CSE closure method to DNS shows promising results.

E233 乱流拡散火炎の確率的モデリング手法の比較検討(乱流燃焼)

A comparative study of stochastic modeling of turbulent combustion has been conducted in the configuration of jet diffusion flame. Flamelet model, multivariate presumed probability density function (PDF) model, and PDF model are evaluated in terms of numerical accuracy. Fuel is a mixture of hydrogen and nitrogen. A fast chemistry is assumed in the flamelet model, a one-step irreversible reaction in the multivariate presumed PDF, and a detailed chemistry in the PDF model. The flamelet model overestimates the reaction than the others but shows an agreement with experiment results. This is attributable to turbulent models employed here. Therefore, the accuracy of combustion modeling strongly depends on the turbulent model of the flow field.

Eulerian transported probability density function sub-filter model for large-eddy simulations of turbulent combustion

Reactive flow simulations using large-eddy simulations (LES) require modelling of sub-filter fluctuations. Although conserved scalars like mixture fraction can be represented using a beta-function, the reactive scalar probability density function (PDF) does not follow an universal shape. A one-point one-time joint composition PDF transport equation can be used to describe the evolution of the scalar PDF. The high-dimensional nature of this PDF transport equation requires the use of a statistical ensemble of notional particles and is directly coupled to the LES flow solver. However, the large grid sizes used in LES simulations will make such Lagrangian simulations computationally intractable. Here we propose the use of a Eulerian version of the transported-PDF scheme for simulating turbulent reactive flows. The direct quadrature method of moments (DQMOM) uses scalar-type equations with appropriate source terms to evolve the sub-filter PDF in terms of a finite number of delta-functions. Each delta-peak is characterized by a location and weight that are obtained from individual transport equations. To illustrate the feasibility of the scheme, we compare the model against a particle-based Lagrangian scheme and a presumed PDF model for the evolution of the mixture fraction PDF. All these models are applied to an experimental bluff-body flame and the simulated scalar and flow fields are compared with experimental data. The DQMOM model results show good agreement with the experimental data as well as the other sub-filter models used.

On the Comparison between Presumed and Full PDF Methods for Turbulent Precipitation

Turbulent precipitation is an important topic in chemical reaction engineering because of its numerous industrial applications. Several modeling approaches have been used in the past, but in recent years, computational fluid dynamics (CFD) coupled with micromixing models has been successfully applied to predict the influence of mixing on the crystal size distribution (CSD). The micromixing model is generally based on the presumed probability density function (PDF) approach, such as finite-mode PDF or beta PDF, and the aim of this work is to compare presumed PDF predictions and full PDF predictions with experimental data. The experimental data were obtained from a tubular reactor in which turbulent precipitation of barium sulfate is carried out from aqueous solutions of barium chloride and sodium sulfate. The implementation of the presumed PDF model was done using FLUENT user-defined subroutines, whereas the full PDF calculations were carried out with an in-house code based on Monte Carlo methods using the flow field prediction from FLUENT.

Identification of local extinction topology in axisymmetric bluff-body diffusion flames with a reactedness-mixture fraction presumed probability density function model

The effects of finite-rate chemistry, such as partial extinctions and re-ignitions, are investigated in turbulent non-pre-mixed reacting flows stabilized in the wake of an axisymmetric bluff-body burner. A two-dimensional large-eddy simulation procedure is employed that uses a partial equilibrium/two-scalar reactedness mixture fraction probability density function (PDF) combustion sub-model, which is applied at the sub-grid scale (SGS) level. An anisotropic sub-grid eddy–viscosity and two equations for the SGS turbulence kinetic and scalar energies complete the SGS closure model. The scalar covariances required in the joint PDF formulation are obtained from an extended scale-similarity assumption between the resolved and the sub-grid fluctuations. Extinction due to strong turbulence/chemistry interactions is recognized with the help of a ‘critical’, locally variable, turbulent Damkohler number criterion, while transient localized extinctions and re-ignitions are treated with a Lagrangian transport equation for a reactedness progress variable. Comparisons with available experimental data suggested that the formulated approach was capable of identifying the effects of large-scale vortex structure activity, which were inherent in the reacting wake and dominant in the counterpart isothermal flows that otherwise would have been obscured if a standard time-averaged procedure had been used. Additionally, the post-extinction and re-ignition behaviour and its time-varying interaction with the large-scale structure dynamics were more appropriately addressed within the context of the present time-dependent method. Copyright © 2001 John Wiley & Sons, Ltd.

Simulation of NOx Formation in Methane‐Air Turbulent Combustion Using an Improved Second‐Order Moment Model

AbstractThe NOx formation in methane‐air turbulent combustion is simulated using an improved second‐order moment (NSOM) model of turbulence‐chemistry interaction. The predicted results from three different versions of the presumed probability density function (PDF) model are compared with the experimental data from the literature. The results show that the NSOM model with algebraic expression of second‐order moments is as good as the proposed PDF model using transport equations of second‐order moments, but the former needs much less computation time than the latter. Therefore, the NSOM model can be used to simulate turbulent combustion and NOx formation with detailed known chemistry, while the proposed PDF model is more difficult to apply.

A Study of Partial Extinction and Reignition Effects in Turbulent Non-Premixed Jet Flames of CH4 and CO/H2/N2 with a Two-Scalar Reactedness-Mixture Fraction Presumed PDF Model

A method for modeling the effects of finite-rate chemistry such as partial extinctions and reignitions is developed and used to compute turbulent CO/H2/N2 or CH4-air-piloted jet diffusion flames close to extinction. The method is combined with a two-dimensional Large Eddy Simulation procedure employing a partial equilibrium/two-scalar exponential Probability Density Function (PDF) combustion submodel applied at the subgrid scale (SGS) level. Subgrid motions are modeled with a first-order closure utilizing an anisotropic subgrid eddy-viscosity and two equations for the subgrid scale turbulent kinetic and scalar energies. Statistical independence of the joint PDF scalars is here avoided and the required moments are obtained from an extended scale-similarity assumption. Extinction is determined by comparing the local Damkohler number against a ‘critical’ local limit related to the Gibson scalar scale and the reaction zone thickness. The post-extinction regime is treated with a reactedness progress variable computed from a Lagrangian transport equation and a two-scalar reacledness-mixture fraction presumed PDF. Comparisons with well documented turbulent flame data suggested the ability of the presented methodology to describe many experimental trends and variations of the momentum and reactive scalar mixing fields at flame conditions close to extinction.