Transported Probability Density Function Modeling Research Articles

Graphics processing unit/artificial neural network-accelerated large-eddy simulation of swirling premixed flames

Within the scope of reacting flow simulations, the real-time direct integration (DI) of stiff ordinary differential equations for the computation of chemical kinetics stands as the primary demand on computational resources. Meanwhile, as the number of transport equations that need to be solved increases, the computational cost grows more substantially, particularly for those combustion models involving direct coupling of chemistry and flow such as the transported probability density function model. In the current study, an integrated graphics processing unit-artificial neural network (GPU-ANN) framework is introduced to comply with heavy computational costs while maintaining high fidelity. Within this framework, a GPU-based solver is employed to solve partial differential equations and compute thermal and transport properties, and an ANN is utilized to replace the calculation of reaction rates. Large eddy simulations of two swirling flames provide a robust validation, affirming and extending the GPU-ANN approach's applicability to challenging scenarios. The simulation results demonstrate a strong correlation in the macro flame structure and statistical characteristics between the GPU-ANN approach and the traditional central processing unit (CPU)-based solver with DI. This comparison indicates that the GPU-ANN approach is capable of attaining the same degree of precision as the conventional CPU-DI solver, even in more complex scenarios. In addition, the overall speed-up factor for the GPU-ANN approach is over two orders of magnitude. This study establishes the potential groundwork for widespread application of the proposed GPU-ANN approach in combustion simulations, addressing various and complex scenarios based on detailed chemistry, while significantly reducing computational costs.

Using a flamelet generated manifold method in transported probability density function modeling of soot formation and thermal radiation

The simple semi-empirical precursor soot model of Brookes and Moss based on the soot number density and soot mass concentration is adopted in a transported probability density function (PDF) method for turbulent diffusion flames. The gas phase chemistry is described by a flamelet generated manifold (FGM) based on the mixture fraction, progress variable and enthalpy loss. The accuracy of the FGM method is validated by using flamelet solutions that are not included in the generating set of the FGM. To account for the radiative heat transfer in the flames, we use a non-gray weighted sum of gray gases model for the gas radiation and a gray soot radiation model. Turbulence–radiation interaction is closed at the level of the optically thin fluctuation approximation and the Reynolds averaged radiative transfer equation is solved by means of a discrete transfer method. The proposed modeling approach is applied in simulations of two turbulent non-premixed methane–air flames at one bar and three bar pressure, respectively. Predictions of the mean temperature and mean soot volume fraction are in good agreement with the measurements in the one bar flame. In the higher pressure flame the mean soot volume fraction is over predicted. For this flame, simulation results using the semi-empirical model of Lindstedt provided better agreement with the measurements. The main difference between the Brookes and Moss model and the Lindstedt model is the nine-times increased soot particle agglomeration rate of the latter. When using the same increased agglomeration rate parameter in the Brookes and Moss model the results become virtually identical. The negligible molecular diffusion of the soot was accounted for by neglecting mean molecular diffusion of the soot variables and by greatly reducing their micro-mixing. The effect of this differential soot diffusion on the mean soot volume fraction is found to be small, but it is significant for the variance of the soot volume fraction.

A novel model for incorporation of differential diffusion effects in PDF simulations of non-premixed turbulent flames based on reaction-diffusion manifolds (REDIM)

In this work, reaction-diffusion manifold (REDIM) reduced chemistry is used in the simulation of turbulent non-premixed flames based on a transported-probability density function model. Differential molecular diffusion is applied in the generation of the manifolds. This is the first work to consider the gradients of the reduced variables as additional parameters in the REDIM model, and one-directional gradients are utilized to generate the REDIM reduced chemistry. Hereby, the influence of turbulence on differential molecular diffusion is automatically considered in terms of reduced variable gradients, and the physical transport properties (e.g., diffusion coefficients) are used in a detailed way, without any additional modeling (e.g., unity-Lewis number assumption). Although the scalar gradients appear as multi-directional in a general turbulent reacting flow, previous direct numerical simulation analysis reveals that REDIMs generated from one-directional gradients can accurately describe the system featuring multi-directional gradients, if this one-directional gradient has a major effect on the chemistry. Here, it is proposed to obtain such gradients under the hypothesis that the flame structure is locally one-dimensional at each spatial position. In order to retrieve the gradients of the reduced variables for the interpolation of the thermo-kinetic states from the REDIM table, the sub-grid gradient is evaluated here from the particle fields. The well-known Sandia series of flames is selected to validate the proposed algorithm. The results show that the new algorithm can reproduce the thermo-kinetic quantities with high accuracy for all investigated flames.

LES/TPDF investigation of the effects of ambient methanol concentration on pilot fuel ignition characteristics and reaction front structures

Large-eddy simulations with a transported probability density function model coupled with a finite-rate chemistry is applied to study the ignition process of an n-heptane spray in a constant volume chamber with a premixed methanol-air atmosphere under conditions relevant to reactivity controlled compression ignition (RCCI) engines. Three reacting spray cases with initial methanol-air equivalence ratio (ϕm) ranging from 0 to 0.3 are investigated at an initial temperature of 900 K. The case setup is based on the Engine Combustion Network Spray-H configuration, where n-heptane fuel is used. The effects of the ambient methanol-air equivalence ratio on the ignition characteristics and the reaction front structures in n-heptane/methanol RCCI combustion are studied in detail. It is found that the ambient methanol affects the low temperature chemistry of n-heptane, which results in a change of spatial distribution of key species such as heptyl-peroxide, and therefore the cool flame structure. With the presence of methanol in the ambient mixture cool flame is found in the entire fuel-rich region of the n-heptane jet, while when methanol is absent in the ambient mixture, the cool flame is established only around the stoichiometric mixture close to the n-heptane injector nozzle. In general, both low- and high-temperature ignition stages of n-heptane ignition are retarded by the methanol chemistry. An increase in ϕm leads to a decrease of the peak heat release rate of the n-heptane first-stage ignition. The chemistry of methanol inhibits the n-heptane ignition by decreasing the overall hydroxyl radicals (OH) formation rate and reducing the OH concentration during the transition period from the first-stage ignition to the second-stage ignition. As a result, the transition time between the two ignition stages is prolonged. Under the present lean methanol/air ambient mixture conditions, the impact of methanol on n-heptane ignition has a tendency of reducing the high-temperature, fuel-rich region, which is in favor of soot reduction.

Transported PDF modeling of compressible turbulent reactive flows by using the Eulerian Monte Carlo fields method

Although the transported probability density function (PDF) method has been developed for decades, its application has been mainly focused on the low-Mach number flow problems. This work extends the transported PDF method to compressible flow problems. The Eulerian Monte Carlo fields (EMCF) solution method is employed to solve the transported PDF equation for compressible flow problems. A pseudo stagnation enthalpy is introduced and its stochastic partial differential equation is derived to ensure total energy conservation numerically. A new mixing model called interaction by partial exchange with mean (IPEM) is introduced to expand the available choices of mixing models for the EMCF method. The consistency of the EMCF method is examined for solving the transported PDF equation. Numerical implementation details are discussed, such as the density coupling between the compressible flow solver and the EMCF solver, discretization schemes for the mixing terms and the stochastic terms. The implemented compressible flow solver coupled with the EMCF solver is verified and validated in a series of test cases with increasing level of complexity, ranging from a statistically one-dimensional turbulent mixing layer to a self-excited resonance model rocket combustor. It is observed that in general with the increase of compressibility, there is an increase in the sensitivity of the modeling results to the different models and algorithms. This makes it necessary to develop a thorough understanding of the model sensitivity in order to develop a robust and accurate simulation solver for highly compressible turbulent reactive flows. The thermo-acoustic instability inside the model rocket combustor case is captured reasonably, which demonstrates the overall capability of the developed compressible turbulent combustion solver based on the transported PDF method.

A priori analysis of a power-law mixing model for transported PDF model based on high Karlovitz turbulent premixed DNS flames

Accurate modeling of mixing in large-eddy simulation (LES)/transported probability density function (PDF) modeling of turbulent combustion remains an outstanding issue. The issue is particularly salient in turbulent premixed combustion under extreme conditions such as high-Karlovitz number Ka. The present study addresses this issue by conducting an a priori analysis of a power-law scaling based mixing timescale model for the transported PDF model. A recently produced DNS dataset of a high-Ka turbulent jet flame is used for the analysis. A power-law scaling is observed for a scaling factor used to model the sub-filter scale mixing timescale in this high-Ka turbulent premixed DNS flame when the LES filter size is much greater than the characteristic thermal thickness of a laminar premixed flame. The sensitivity of the observed power-law scaling to the different viewpoints (local or global) and to the different scalars for the data analysis is examined and the dependence of the model parameters on the dimensionless numbers Ka and Re (the Reynolds number) is investigated. Different model formulations for the mixing timescale are then constructed and assessed in the DNS flame. The proposed model is found to be able to reproduce the mixing timescale informed by the high-Ka DNS flame significantly better than a previous model.

LES/PDF modeling of swirl-stabilized non-premixed methane/air flames with local extinction and re-ignition

Turbulent non-premixed flames with local extinction and re-ignition exhibit multiple combustion modes including ignition waves, diffusion flames, partially premixed flames, and ignition-assisted partially premixed flames. The mechanisms of local extinction and re-ignition are not well understood and numerical modeling of multi-mode combustion is a challenging task. In this work, a specially designed swirl-burner was used to study local extinction and re-ignition of non-premixed turbulent methane/air flames. High speed Particle Image Velocimetry (PIV) and laser induced fluorescence of OH radicals (OH-PLIF) measurements along with Large Eddy Simulation (LES) were carried out to investigate the mechanisms of extinction and re-ignition processes in the burner. LES is based on a transported probability density function model within the framework of Eulerian Stochastic Fields (PDF-ESF). It is found that local extinction occurs when the scalar dissipation rate around the stoichiometric mixture fraction is high. The characteristic time scale for local extinction and re-ignition in the present flames is an order of magnitude longer than the characteristic time scale of diffusion/extinction of laminar flamelets. There are two mechanisms for flame hole re-ignition in the present flames. First, under low degree of local extinction conditions (i.e., for small flame holes surrounded by flames) the flame hole re-ignition is due to the mechanism of turbulent flame folding. Second, under high degree of extinction conditions (i.e., with large regions of extinction and lifted flames), re-ignition of the locally extinguished flame is due to the mechanism of ignition assisted partially premixed flame propagation. The results show that the PDF-ESF model is capable of simulating the quenching and re-ignition process found in the experiments.

Effects of ambient pressure on ignition and flame characteristics in diesel spray combustion

This work reports on numerical investigation of effects of ambient pressure (Pam) on spray combustion under engine-like conditions. Three cases with different Pam of 42, 85 and 170 bar at a fixed ambient temperature of 1000 K are considered. Zero-dimensional calculations are first performed for autoignition of stagnant adiabatic homogenous mixtures to evaluate performance of the selected diesel surrogate fuel models and to identify the Pam effects on the most reactive mixture. An Eulerian-based transported probability density function model is then chosen for the three-dimensional computational fluid dynamics study. The results show the predicted ignition delay times and flame lift-off lengths are in reasonably good agreement with experiment, with the relative difference below 28%. The current work reveals that low-temperature reactions occur across a wide range of mixture fraction but a noticeable rise of temperature (>100 K above ambient temperature) is detected first on the fuel-lean side of the stoichiometric line in all three cases. The high-temperature ignition occurs first on the fuel-rich side in the 42 and 85 bar cases, where the igniting mixture appears to be more fuel-rich in the latter case. As Pam is further increased to 170 bar, the igniting mixture becomes more fuel-lean and the high-temperature ignition occurs on the fuel-lean side. The ignition behavior is found to depend on both physical and chemical processes. At 170 bar, the reaction rate increases and the associated transition from low- to high-temperature ignition is relatively fast, as compared to the transport of warmer products from the lean zone into the fuel-rich mixture. Also, within the fuel-rich region, the local temperature is low due to liquid fuel vaporization and the condition is not appropriate for ignition. These collectively cause the high-temperature ignition to occur on the fuel-lean side. Analyses on the quasi-steady spray flame structures reveal that, apart from poorer air entrainment due to reduced lift-off length, the higher rich-zone temperature and lower scalar dissipation rate also lead to a higher peak soot volume fraction at higher Pam.

Transported PDF Modeling of Ethanol Spray in Hot-Diluted Coflow Flame

This paper presents a numerical modeling study of one ethanol spray flame from the Delft Spray-in-Hot-Coflow (DSHC) database, which has been used to study Moderate or Intense Low-oxygen Dilution (MILD) combustion of liquid fuels (Correia Rodrigues et al. Combust. Flame 162(3), 759–773, 2015). A “Lagrangian-Lagrangian” approach is adopted where both the joint velocity-scalar Probability Density Function (PDF) for the continuous phase and the joint PDF of droplet properties are modeled and solved. The evolution of the gas phase composition is described by a Flamelet Generated Manifold (FGM) and the interaction by exchange with the mean (IEM) micro-mixing model. Effects of finite conductivity on droplet heating and evaporation are accounted for. The inlet boundary conditions starting in the dilute spray region are obtained from the available experimental data together with the results of a calculation of the spray including the dense region using ANSYS Fluent 15. A method is developed to determine a good estimation for the initial droplet temperature. The inclusion of the “1/3” rule for droplet evaporation and dispersion models is shown to be very important. The current modeling approach is capable of accurately predicting main properties, including mean velocity, droplet mean diameter and number density. The gas temperature is under-predicted in the region where the enthalpy loss due to droplet evaporation is important. The flame structure analysis reveals the existence of two heat release regions, respectively having the characteristics of a premixed and a diffusion flame. The experimental and modeled temperature PDFs are compared, highlighting the capabilities and limitations of the proposed model.

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.

MS3-4 A CPU Efficient SI In-Cylinder Combustion and Knock Prediction Model Utilizing a Stochastic Reactor Approach, Turbulent Flame Propagation and Detailed Chemistry(MS: Modeling and Simulation,General Session Papers)

An accurate in-cylinder combustion model for SI-engines has been constructed by combining a stochastic reactor model with predictive turbulent flame propagation. The computational time for full cycle calculations is kept low through load balanced parallelization and particle clustering. This paper presents a two-zone zero dimensional transported probability density function model, utilizing a stochastic reactor for solution of the ordinary differential equations in each zone. A turbulent flame propagation model is applied, and laminar flame speed is achieved from a flame speed library built with detailed chemistry. The flame front is tracked with a CPU efficient polygon method. Knock and emissions are calculated using on-line detailed chemistry. The model has previously shown very good predictability of combustion as well as knock. In this work, the model calculation time has been decreased through load balanced distribution of the on-line chemistry calculations on a large number of CPU's. In addition, an algorithm for chemical state based particle clustering in each zone has been developed. Particles are clumped into groups based on user provided dispersion thresholds for any number of cluster tracking parameters. As a consequence, not all particles need to be treated during the chemistry step; only one cluster representative particle needs to be passed to the chemistry solver. The paper presents the clustering algorithm, and its verification. It is shown that a very limited number of clustering parameters can be used, and that the dispersion thresholds can be set high. On average, the clustering yields a factor 3-4 in speedup of chemistry calculations and CPU times of about a minute per cycle are achieved. Since chemistry and in-homogeneities are taken into account this yields a model suitable for e.g. fuel analysis and performance studies.

Conditional statistics of passive-scalar mixing in a confined wake flow

Experimental results for conditional statistics in a confined turbulent wake were compared to the linear model and the transported probability density function (PDF) model for predicting the velocity conditioned on scalar mixture fraction and to a linear model for predicting the scalar mixture fraction conditioned on velocity. For velocity conditioned on scalar mixture fraction, the linear model was found to accurately predict the conditional transverse velocity, but for the conditional streamwise velocity, agreement between the experiments and the linear model was poor. The transverse conditional velocity was accurately predicted by the transported PDF model, but for the streamwise conditional velocity, a modified transported PDF model that incorporated two components of the scalar diffusivity tensor had to be used. Finally, it was found that a linear model for predicting the fluctuating scalar conditioned on velocity performed poorly due to the strongly non-Gaussian scalar PDF at the downstream locations investigated.

A model for the mixing time scale of a turbulent reacting scalar

Current micromixing models in transported PDF (probability density function equation) modeling of initially nonpremixed, turbulent reacting flows ignore the direct influence of chemistry on the dissipation rate of reacting scalar fluctuations. Following mapping closure, a model for the time scale of reactive scalar mixing in the flamelet regime is developed and comparisons made with direct numerical simulations of a turbulent reacting jet. The modeling results show a significant improvement over the generally used estimate of substituting the time scale of the reacting scalar by that of a conserved scalar in traditional transported PDF approaches. The model can readily be applied to many existing mixing submodels used in transported PDF modeling when the time scales of the reacting scalars appear explicitly in the formulation. The submodels allow the transported PDF approach to now encompass the fast as well as slow chemistry regimes.