Transported Probability Density Function 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.

Longitudinal and azimuthal thermo-acoustic instabilities in an industrial gas turbine combustor operating at elevated pressure

This work numerically investigates longitudinal and azimuthal thermo-acoustic instabilities in the swirl-stabilised can-type industrial SGT-100 gas turbine combustor operated at elevated pressures of 3 and 6 bar. Previous experiments have shown that the combustor is susceptible to self-excited flame oscillations sustained by a thermo-acoustic feedback loop at specific operating conditions. In order to gain a better understanding of this feedback loop, a fully compressible large eddy simulation method is applied. The unknown sub-grid scale turbulence-chemistry interactions are modelled via a transported probability density function approach solved by the Eulerian stochastic fields method. First, the reaction zones and global flame topology at both operating pressures are analysed and compared to experimental images providing good qualitative agreement. Radial profiles of time-averaged and root-mean-square quantities furthermore demonstrate good quantitative agreement with the available measurement data. The applied simulation approach is capable of successfully reproducing self-excited thermo-acoustic instabilities in the longitudinal direction. The fundamental frequency of the predicted limit-cycle oscillation matches the experimentally measured frequency with high accuracy. Similar to the experimental observations, the fluctuation amplitudes of the pressure and global heat release rate increase significantly upon increasing the mean operating pressure from 3 to 6 bar. In addition to the dominant longitudinal mode, a high-frequency, low-amplitude azimuthal mode is also identified at both pressures. This azimuthal mode is periodically amplified and attenuated by the superposed longitudinal mode and induces small asymmetric (around the burner circumference) fluctuations of the local fuel and total mixture mass flow rates entering the flame region.

Simulation of turbulent premixed flames with machine learning - tabulated thermochemistry

The numerical integration of the differential equations describing chemical kinetics consumes the majority of computational time in combustion simulations that involve direct coupling of chemistry and flow, such as transported probability density function (PDF) methods, direct numerical simulation (DNS), conditional moment closure (CMC), unsteady flamelet, multiple mapping closure (MMC), thickened flame model, linear eddy model (LEM), partially stirred reactor (PaSR) as in OpenFOAM and laminar flame computation. This step can be accelerated by tabulation, and artificial neural networks (ANNs) have recently emerged as a powerful technique in this domain. To be applicable to a wide family of problems, an ANN tabulation approach must be based on data generated by an abstract process, rather than from the turbulent flame to be simulated. In the present work, the hybrid flamelet/random data and multiple multilayer perceptrons (HFRD-MMLP) method (Ding et al., Combust. Flame 231, 111493, 2021) for non-premixed flames is taken as a basis to develop a thermochemistry tabulation method for premixed flames. In the spirit of maintaining an essentially random data set that still originates in meaningful composition states, a set of one-dimensional premixed flame simulations is employed to generate data that are used as starting points for a random data generation process and subsequently discarded. The approach is applied to large eddy simulations (LES) of the Cambridge/Sandia swirl burner in configurations five and six, with the transported PDF method employed to provide closure for the filtered reaction source terms and the stochastic fields method used for numerical solution. Very good agreement in both major and minor species is observed between the LES-PDF simulations using direct integration of the reaction source term and the ones with the ANNs. Furthermore, the average time taken for reaction source term computations is reduced by fourteen times, while memory requirements constitute only 1.4 MB.

A DNS evaluation of three MMC-like mixing models for transported PDF modelling of turbulent nonpremixed flames

Transported probability density function (TPDF) methods are suitable for modelling turbulent reactive flows. One of the main challenges is to accurately model the molecular mixing terms. In TPDF mixing models, it is desired that the principle of localness is satisfied so that the molecular mixing is performed locally in both physical and composition spaces. The multiple mapping conditioning (MMC) mixing model can ensure mixing localness without violating other desired principles. The present study examines three MMC-like mixing models, including the original MMC (OMMC) mixing model, shadow-position mixing model (SPMM) and a conceptually simplified multiple mapping conditioning (SMMC) mixing model. Three direct numerical simulation (DNS) datasets modelling turbulent nonpremixed ethylene flames with increasing levels of extinction are used for model evaluation. The DNS datasets are also used to provide both initial conditions and inputs needed over the course of TPDF runs to remove the uncertainties caused by turbulence closure, allowing the study to focus on the molecular mixing model. The mixing model coefficients are specified analytically by reference to a canonical mean scalar gradient (MSG) flow in order to achieve specifiable dissipation rate and user-controllable localness. The results show that the MMC-like mixing models yield similar prediction of flame extinction and reignition if the coefficients are properly specified. The MMC-like mixing models can also be tuned to achieve a desired level of localness. For conditional statistics, the MMC-like mixing models can yield correct level of conditional variances. By changing localness, the MMC-like models can yield solutions resembling the interaction by exchange with the mean (IEM) or Euclidean minimum spanning tree (EMST) mixing model for extreme parameter choices. The models differ in their abilities to specify unconditional dissipation rates, at least in the flow considered, and in their ease of implementation.

A pairwise mixing model with kernel constraint and its appraisal in transported PDF simulations of ethylene flames

Modeling micro-mixing remains a primary challenge for the transported probability density function (TPDF) method. In this study, the pairwise mixing with kernel constraint (KerM) model is proposed and evaluated in TPDF simulations of a turbulent non-premixed ethylene flame. The new model features controllable localness by selecting mixing particle pairs with a probability that depends on their distance in composition space, normalized by a characteristic kernel size. To evaluate the model in a controlled setting, TPDF simulations were carried out with the mean velocity, turbulent diffusivity and scalar mixing frequencies extracted from a direct numerical simulation (DNS) dataset. The predictions of the mean statistics and the conditional PDFs illustrate that the KerM model exhibits asymptotic behavior to the modified Curl (MC) and the Euclidean minimum spanning trees (EMST) model, respectively, depending on the kernel size. For the flame considered, results for KerM approach those for EMST when the kernel size is less than 0.01, while they approach those for MC when the kernel size is larger than 1.0. Given a kernel size of 0.1, KerM yields a notably better prediction of the overall combustion process than the conventional MC model. The predictions of the mean temperature and species mass fractions by KerM are similar to those by EMST, while KerM better predicts the conditional PDF of temperature than EMST that overdamps the conditional fluctuations. In addition, its complexity with the number of particles per cell is instead of O(Np2) for EMST. By incorporating different mixing timescales among species, the KerM-DD model better predicts the mean temperature field at the time instant close to local extinction than the KerM and EMST models. This work demonstrates that the KerM model, apart from being simple and robust in terms of implementation, features the advantages of being flexible in localness and being capable of incorporating differential diffusion.

Effects of Hydrogen/Methane on the Thermal Environment of Heavy-Duty Gas Turbine Combustor

Hydrogen is the most promising fuel for reducing carbon emissions, but hydrogen combustion produces higher temperature compared to hydrocarbon fuel. In this paper, a three-dimensional compressible combustion–flow–heat transfer model of combustor was established, and a dry-low-emission combustor was examined by using the realizable model, transported probability density function, and discrete ordinates model combining weighted sum of gray gas model, analyzing the effects of hydrogen/methane blended fuel and thermal boundaries on the combustor thermal environment. The results show that when the fuel hydrogen volume percentage increases from 0 to 75%, the maximum gas temperature and concentration on the central axis of the combustor increase by about 160.8 and 662.9%, respectively; the maximum incident radiant heat flux of the combustor wall increases by about 150%; and the local maximum ratio of the radiant heat transfer to the total heat transfer through the wall increases from about 34 to about 49%. The effect of the boundary conditions varies depending on the hydrogen percentage. At the hydrogen percentage of 75%, the maximum wall-incident radiant heat flux under the adiabatic condition is nearly 180.3 and 77.4% higher than the values at 1370 and 1920 K isothermal boundaries, respectively.

The impact of hydrogen substitution by ammonia on low- and high-temperature combustion

The combustion behaviour of ammonia has attracted intermittent interest with the original patent by Lyon (US3900554A) relating to its use for nitric oxide reduction through selective non-catalytic reduction (SNCR) a focal point. The recent interest in ammonia stems from its use as a hydrogen rich energy carrier with practical use requiring a much wider parameter space. The corresponding challenges (e.g. Kobayashi et al., Proc. Combust. Inst. 37 (2019) 109–133) include different flame dynamics and high emissions of oxides of nitrogen. The current paper explores the complex nature of ammonia oxidation and provides a reduced size reaction mechanism that enables application, without approximation, to the computation of turbulent flames through a joint-scalar transported probability density function (JPDF) method. Comprehensive validation suggests similar accuracy to a reference mechanism (Glarborg et al., Prog. Energy Combust. Sci. 67 (2018) 31–68) and highlights some uncertainties. The selected turbulent flame configuration features auto-ignition stabilised flames supported by a coflow of hot combustion products. The base case features a H2/N2 fuel jet that permits flame stabilisation at 1045 K corresponding to the onset of the SNCR temperature window. The impact of a gradual substitution of hydrogen by ammonia on flame stabilisation, emissions of oxides of nitrogen and the flame structure is quantified. It is shown that ammonia substitution leads to more prevalent local extinction, a more distributed flame structure and requires substantially increased coflow temperatures to achieve a similar flame stabilisation point. A lowering of the coflow temperature to operate within the SNCR regime substantially reduces NOx and leads towards a homogeneous/distributed reaction mode. The reduced fuel reactivity highlights the importance of turbulence-chemistry interactions leading to complexities in the design of practical devices.

Investigation of turbulent mixing using a stochastic hierarchical parcel swapping mixing model

AbstractTurbulent mixing plays an important role in a variety of applications ranging from astrophysics to combustion and even pollutant dispersion. The Direct Numerical Simulation (DNS) that resolves all scales is not feasible for most engineering applications since the flow has a wide range of length and time scales, which yields extremely high resolution requirements. Large Eddy Simulations (LES) overcome this limitation by modeling the sub‐grid scale effects. In transported Probability Density Function (PDF) methods, the key challenge is to develop an accurate mixing model. At this point, the Hierarchical Parcel‐Swapping (HiPS) model, introduced by A.R. Kerstein [J. Stat. Phys. 153, 142‐161 (2013)], is an attractive candidate. It is characterized by a computationally efficient representation of the effects of turbulence on a time‐evolving structure of diffusive scalar fields. The interpretation of the diffusive scalar fields or a state space as a binary tree structure is an alternative approach compared to the most common mixing models. The characteristic feature of HiPS is that every level of the tree corresponds to a specific length and time scale, which is based on turbulence inertial range scaling. The state variables only reside at the base of the tree and are understood as fluid parcels. The effects of turbulent advection are represented by stochastic swaps of sub‐trees at rates determined by prevailing turbulent time scales associated with the sub‐trees. The mixing of adjacent fluid parcels is done either instantaneously or at rates consistent with the corresponding diffusion time scales. In this work, HiPS is detailed for the simulation of passive scalar mixing first. Preliminary results for the scalar power spectra, mean square displacement and scalar dissipation rate are shown and reveal a reasonable agreement with experimental findings. Furthermore, the integrated binary tree structure allows to satisfy a large number of criteria for a good mixing model. Considering the reduced order and computational efficiency, HiPS is an attractive candidate for modeling the mixing in transported PDF methods.

An indirect approach to optimize the reaction rates of thermal NO formation for diesel engines

With stringent emission regulations, it has become more important for modern diesel engine manufacturers to accurately predict engine-out nitrogen oxide (NOx) emissions across a wide range of operating conditions. Thermal NO is the major source of engine-out NOx in modern diesel engines. For thermal NO formation, several earlier studies have recommended the forward and reverse reaction rate coefficients of the rate-limiting reaction (O + N2⇌ NO + N). However, due to deficiencies in sub-models and inadequacies of reduced chemical mechanisms to represent diesel combustion, these recommended values more often than not need to be adjusted in reduced order combustion models to accurately predict engine-out NOx. Hence, in this work a systematic and computationally efficient approach has been proposed to streamline the process of determining the optimum reaction rate coefficients. To develop the optimization approach, four different production diesel engines with different operating conditions in terms of speed, load, and exhaust-gas recirculation have been considered. Numerical simulations have been performed using a detailed zero-dimensional velocity-composition-frequency transported probability density function (0D-VCF-tPDF) model that uses hundreds of notional particles to capture in-cylinder stratification. Four different combinations of hydrocarbon and NOx chemical mechanisms were used to represent chemistry. It was found that for the rate-limiting reaction, the pre-exponent factors (Af1,Ar1) and activation energies (EA,f1,EA,r1) of the forward and reverse reaction rates follow a linear band in Af1−EA,f1 and Ar1−EA,r1 space where predicted engine-out NOx match the measured values closely. By encompassing such bands from different engines and considering constraints on activation energies, a reduced search domain of pre-exponent factors and activation energies was constructed that is expected to be applicable to any diesel engine. Eventually, computationally efficient three-line and one-line search approaches were proposed to determine the optimum values of the pre-exponent factors and activation energies that led to a minimum error between measured and predicted engine-out NOx. Finally, these three-line and one-line NOx optimization approaches were applied to a fifth production diesel engine for which the 0D-VCF-tPDF model showed a very good predictive performance in terms of predicting peak pressure, 50% burn rate, and engine-out NOx when compared to measured and 3D-CFD values.

Zone-adaptive modeling of turbulent flames with multiple chemical mechanisms

A novel zone-adaptive modeling method (AdaCM) with multiple chemical mechanisms is proposed for turbulent flames to achieve high-fidelity, yet computationally efficient, predictions. Specifically, a computational economical species transport model, e.g., the well-mixed model, via finite volume algorithm is employed with a simple mechanism as the base model for the whole computational domain, while the advanced transported probability density function (TPDF) method via Lagrangian particle tracking is employed with a detailed mechanism only for spatial regions with intense turbulence-chemistry interaction (TCI), denoted as the “PDF regions”. The PDF regions are dynamically identified based on local flow and flame characteristics and may evolve with time. A two-way particle/finite volume submodel coupling is formulated to ensure the composition consistency between submodels in the PDF regions and impose the correct interface conditions for composition and mass flow rate on the boundary of the PDF regions. With regard to transformation between different species representations in the mechanisms, a species reconstruction/reduction approach based on constrained chemical equilibrium is proposed to ensure element conservation and an adequate specification of unrepresented species at the model interface. The proposed adaptive modeling method has been applied to the well-known Sandia Flame D, in which the well-mixed combustion model with a 6-species, two-step global mechanism is employed as a base model and the high-fidelity TPDF with a 25-species skeletal mechanism is employed for regions with intense TCI. Results demonstrate the consistency in PDF regions between submodels with two distinct mechanisms. The predictions from the adaptive modeling are almost identical to those of TPDF and agree well with experimental measurements, illustrating the preservation of prediction accuracy in the adaptive method. In addition, the total number of computational particles in AdaCM for Flame D is only 18% of that for the stand-alone TPDF, and the recorded computational speedup is about 2.8.

An extended FGM model with transported PDF for LES of spray combustion

An enhanced flamelet generated manifold (FGM) model for large eddy simulation (LES) of turbulent spray combustion is presented. In the enhanced FGM model, a transported probability density function (TPDF) description of the FGM variables is employed. The TPDF is represented using the Eulerian stochastic fields (ESF) approach, and the method is applied to LES of spray combustion under conditions relevant to internal combustion engines. The new ESF/FGM method achieves an improved accuracy of predictions due to the ESF modelling of the subgrid-scale turbulence-chemistry interaction. It also achieves high computational efficiency due to the FGM tabulation of the chemical kinetic mechanism. The performance of the new ESF/FGM model is assessed by simulation of the Spray-A flames from Engine Combustion Network (ECN) and comparison of the results, firstly, with experimental measurements, and secondly, with conventional FGM model simulation results. It is shown that the ESF/FGM method is capable of predicting both global and local combustion characteristics, i.e., pressure rise, ignition delay time, flame lift-off length and the thermo-chemical structure of the spray flames with improved accuracy compared to the conventional FGM model that is based on the presumed PDF description of FGM variables. The sensitivity of the predictions using ESF/FGM to the number of stochastic fields is examined by varying the number of these fields in the range of 4–128. Furthermore, the influence of different FGM reaction progress variables on the simulations is investigated, and a new reaction progress variable based on the local consumption of oxygen is proposed. The results show that the new progress variable improves predictions of spray combustion, including the prediction of the start of injection, the quasi-steady state liftoff length, the post-injection oxidation, and the pressure evolution.

Cool flame wave propagation in high-pressure spray flames

Cool flame has an important impact on the performance and emission of advanced low-temperature combustion (LTC) internal combustion engines (ICEs) in which the liquid fuel is injected earlier at a lower ambient temperature (Tam) than that in the conventional diesel combustion engines. However, the cool flame characteristics of spray combustion under ICEs conditions are not fully understood, e.g., the effect of cool flame on the spray ignition and flame stabilization is not well studied. In this paper, the so-called cool flame wave propagation (CFWP) in the Engine Combustion Networks Spray A flames at three ambient temperatures Tam (800 K, 900 K and 1000 K) is analyzed using the data from large eddy simulation with an improved Eulerian based transported probability density function sub-grid combustion model. A good agreement between the LES results and experimental data is obtained for the spray liquid penetration length, vapor fuel penetration length, mean pressure rise profile, and flame liftoff length. It is shown that CFWP in the spray ignition process promotes the ignition of the fuel-rich and cold reactant mixture, leading to the most reactive mixture shifting toward fuel-richer locations due to the spatial stratification of temperature and concentration, and turbulent mixing. As a result, the high temperature ignition (HTI) can be shortened compared to the ignition in the homogeneous mixture. At Tam = 800 K, the HTI kernels are consistently formed following the CFWP propagating toward the spray head region. However, as Tam increases, the spatial correlation between HTI kernels and CFWP fronts is weakened. On the other hand, the turbulent mixing (quantified using local scalar dissipation rate) contributes more to the formation of HTI kernels at higher Tam. The present results indicate that CFWP is more profound at lower Tam in the spray ignition process. Finally, it is found that cool flame propagates mainly into pre-reacted fuel-rich mixture in an ignition wave propagation mode from the spray upstream region toward downstream region, whereas the ignition assisted flame mode is found in the spray upstream region where the combustion heat release is negligible.

Consistent submodel coupling in hybrid particle/finite volume algorithms for zone-adaptive modelling of turbulent reactive flows

A hybrid particle/finite volume algorithm has been formulated for zone-adaptive modelling of turbulent reactive flows to achieve both high fidelity in predictions and high computational efficiency. Specifically, a computationally economical species transport model via finite volume algorithm is employed as the base model for the whole computational domain, while the advanced transported probability density function (TPDF) method via Lagrangian particle tracking is employed only for regions with intense turbulence-chemistry interaction. The ‘PDF regions’ can be updated dynamically based on local flow and flame characteristics, and are compatible with complex geometric structures such as separated multi blocks, non-convex, and multi-connected regions. A two-way particle/finite volume submodel coupling is formulated to ensure the composition consistency in submodels in the PDF regions and to impose the correct interface conditions for composition and mass flow rate on the boundary of the PDF regions. The spatial partition and particle algorithms for time-varying PDF regions are demonstrated and the convergence characteristics of the adaptive modelling are investigated specifically for the variation of statistical error and bias with the number of particles per cell. The proposed zone-adaptive hybrid particle/finite volume algorithm has been numerically validated in a turbulent hydrogen/air non-premixed jet flame. It is shown that the predictions from zone-adaptive modelling are almost identical to those of stand-alone TPDF, illustrating the preservation of prediction accuracy but with significantly less computational cost.

Machine learning tabulation of thermochemistry of fuel blends

The objective of the present work is to develop a machine learning tabulation methodology for thermochemistry that accounts for fuel blends. The approach is based on the hybrid flamelet/random data and multiple multilayer perceptrons (HFRD-MMLP) methodology (Ding et al., 2021), the essence of which is to train a set of artificial neural networks (ANNs) using random data so as to anticipate the composition space encountered in turbulent flame simulations. As such, it is applicable to any combustion modelling approach that involves direct coupling of chemistry and flow, such as transported probability density function (PDF) methods, direct numerical simulation (DNS), conditional moment closure (CMC), unsteady flamelet, multiple mapping closure (MMC), thickened flame model, linear eddy model (LEM), partially stirred reactor (PaSR) as in OpenFOAM and laminar flame computation. In this paper, the HFRD approach is further developed to generate data of varying fuel ratios. Furthermore, radiative heat losses are included and it is shown that the ANN-based simulations are able to account for it. The ANNs generated are first tested on 1-D laminar flame simulations and then applied to two turbulent flames with different fuel compositions: a pure methane flame, Sandia flame D, and Sydney flame HM1, which is a methane/hydrogen flame. The results of species mass fraction and temperature are compared between ANN and direct integration, and excellent agreement are achieved. These results indicate that the methodology has great capacity for generalisation and is applicable to a range of blended fuels. Furthermore, a speed-up ratio of 14 to 17 is attained for the reaction step compared with direct integration, which greatly reduces the computational cost of turbulent combustion simulations.

A hybrid stochastic Lagrangian – cellular automata framework for modelling fire propagation in inhomogeneous terrains

A stochastic model motivated by the Lagrangian transported probability density function method for turbulent reacting flows and the cellular automata approach for forest fires was put together to simulate propagation of fires in terrains with inhomogeneous composition. In contrast to the usual cellular automata models for fires where the probability of ignition is prescribed, here the ignition of cells is determined by a random walk that mimics turbulent convection and diffusion of the hot gases and firebrands from upwind and neighbouring fire fronts. Radiation is also included. The model is aimed at speed of computation while approximating the key physics through only a few terrain-related inputs and tunable parameters representing fire intensity, hot gas and ember decay timescales, cell ignition delay and local turbulence. These parameters were calibrated against controlled fire experiments and the model was then used to give reasonable predictions for fires of increasing complexity. The presented framework allows improvements for more accurate representation of the flammable material characteristics, fire-induced flow modifications, and most other phenomena present in fires, hence providing an extendable and simple yet physically-realistic novel modelling approach.

Large Eddy Simulation of a Reacting Kerosene Spray in Hot Vitiated Cross-Flow

The evaporation and combustion characteristics of a kerosene spray injected perpendicularly into a cross-flow of high-temperature vitiated air is investigated. This fundamental flow configuration has wider implications for the future development of ultra-low emission aeronautical combustors, particularly with respect to technologies involving MILD combustion. Large eddy simulations with a Eulerian–Lagrangian framework are performed to investigate the spray evolution and the characteristics of the reaction zone for a range of conditions. For the closure of turbulence-chemistry interactions at the sub-grid scales, a transported probability density function approach solved by the Eulerian stochastic fields method is applied. A configuration based on the use of airblast atomisation is assessed first and compared with experimental observations. The effect of the atomiser air-to-liquid mass flow ratio is studied in greater detail, both in terms of the resulting gas-phase properties and the droplet evaporation process. Then, the effect of ambient pressure on the global spray flame behaviour is examined. For this part of the study, no atomising air is included in the simulation to separate the effects of ambient pressure on the spray from the interaction with the air jet. Analysis of the flame and spray properties at cross-flow operating pressures of 1 atm, 2 bar and 4 bar highlights the strong coupling between the reacting flow and droplet evaporation characteristics, which are highly affected by the penetration of the spray into a flow field characterised by relatively large gradients of temperature. The results reported in this work provide fundamental understanding for the development of novel low-emission combustion technologies and demonstrate the feasibility of applying large eddy simulation with detailed chemistry for the investigation of reacting aviation fuel sprays in hot vitiated cross-flow.

Assessment of critical species for differential mixing in transported PDF simulations of a non-premixed ethylene DNS flame

The difference in the mixing timescale among species, i.e., differential mixing, is a combined result due to the difference in molecular diffusivity and the difference in reaction-induced scalar gradient. Being a key component for micro-mixing, incorporating differential mixing is a desirable feature for a transported probability density function (TPDF) simulation. Nevertheless, the lack of understanding of differential mixing hinders its model development. This paper intends to address that whether there exist critical species for differential mixing, such that considering only the differential mixing of these species can reproduce the prediction with the full differential mixing of all species. To achieve so, TPDF simulations have been performed for a temporally evolving, non-premixed ethylene DNS flame, with the DNS-extracted mean velocity, turbulent diffusivity and species mixing timescales to enable a clean assessment on the critical species for differential mixing. An algorithm based on active subspace (AS) method is proposed to identify the critical species. It is found that the identified 10 critical species are neither the ones having the highest mixing frequency nor the ones causing the largest temperature change. The TPDF simulation with the differential mixing of the 10 critical species can well reproduce the mean and conditional mean quantities of the reference simulation with the full differential mixing of all species, illustrating the effectiveness of the AS method. Compared to the crude single-species trial method, the AS method better identifies the critical species as it accounts for the coupling nature of the chemical reaction system. By analyzing the reaction pathway in 1D opposed jet flames, it is found that the critical species all involve in the controlling reactions for the overall combustion process, thus providing a physical interpretation for the identified critical species. The effects of quantity of interest and micro-mixing model on critical species are also analyzed.

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 mixing timescale model for differential mixing in transported probability density function simulations of turbulent non-premixed flames

The modeling of scalar mixing timescale remains a primary challenge in the transported probability density function (TPDF) method. The variation of scalar mixing timescale among species, i.e., differential mixing, results from the difference in molecular diffusivity and reaction-induced scalar gradient. Nevertheless, the vast majority of TPDF studies on turbulent non-premixed flames simply apply a single mixing timescale determined by the mixture fraction. In this work, a reaction-induced differential mixing timescale (RIDM) model for the mixing timescale of individual species in turbulent non-premixed flames is proposed. The key idea of the RIDM model is to approximate the relative magnitude of the species dissipation rates by using their values in laminar flamelets. A direct numerical simulation dataset of a temporally evolving non-premixed ethylene flame is employed to thoroughly evaluate the model performance via a priori and a posteriori tests. Results show that specifying a single mixing timescale for all species results in a poor prediction of the species dissipation rate and thus the failure to predict the overall combustion process. By accounting for the difference in molecular diffusivity, a slightly better prediction can be obtained, but the improvement is very limited, illustrating that simply modeling the difference due to molecular diffusivities for differential mixing is not sufficient. In comparison, the RIDM model exhibits superior performance in both a priori and a posteriori tests. Moreover, all the components of the RIDM model are readily available in the TPDF method, making the RIDM model a promising candidate employed in practice.

Computation of Entropy Production in Stratified Flames Based on Chemistry Tabulation and an Eulerian Transported Probability Density Function Approach

This contribution presents a straightforward strategy to investigate the entropy production in stratified premixed flames. The modeling approach is grounded on a chemistry tabulation strategy, large eddy simulation, and the Eulerian stochastic field method. This enables a combination of a detailed representation of the chemistry with an advanced model for the turbulence chemistry interaction, which is crucial to compute the various sources of exergy losses in combustion systems. First, using detailed reaction kinetic reference simulations in a simplified laminar stratified premixed flame, it is demonstrated that the tabulated chemistry is a suitable approach to compute the various sources of irreversibilities. Thereafter, the effects of the operating conditions on the entropy production are investigated. For this purpose, two operating conditions of the Darmstadt stratified burner with varying levels of shear have been considered. The investigations reveal that the contribution to the entropy production through mixing emerging from the chemical reaction is much larger than the one caused by the stratification. Moreover, it is shown that a stronger shear, realized through a larger Reynolds number, yields higher entropy production through heat, mixing and viscous dissipation and reduces the share by chemical reaction to the total entropy generated.