First-order Conditional Moment Closure Research Articles

Effect of the conditional scalar dissipation rate in the conditional moment closure

In the context of modeling turbulent scalar mixing using probability density function (PDF) methods, the treatment of molecular mixing is of paramount importance. The conditional moment closure (CMC) offers a high-fidelity description for molecular mixing in nonpremixed flows. Recent work has demonstrated that first-order CMC can be implemented numerically using the moments of the conditioning variable and first-order joint moments of the scalar of interest. When solving the CMC using, for example, quadrature-based moment methods (QBMM), a functional form must be chosen for the conditional scalar dissipation rate (CSDR) of the conditioning variable. In prior work, the CSDR was chosen to produce a β-PDF for the conditioning variable (mixture fraction) at steady state. This choice has the advantage that the system of moment equations used in QBMM-CMC can be written in closed form. In this work, an alternative choice for the CSDR is investigated, namely, the amplitude mapping closure (AMC). With the AMC, the moment equations can be closed using the quadrature method of moments incorporated into a realizable ordinary differential equation solver. Results are compared with the β-CSDR closure for binary, passive scalar mixing in homogeneous single- and disperse-phase turbulent flows. It is also demonstrated that the moment formulation of CMC provides a straightforward method for modeling the effect of differential diffusion in the context of CMC.

Solution of the first-order conditional moment closure for multiphase reacting flows using quadrature-based moment methods

The quadrature-based semi-analytical solution for the conditional moment closure (SA-CMC) given in (A. D. Ilgun, A. Passalacqua, and R. O. Fox, “A quadrature-based conditional moment closure for mixing-sensitive reactions,” Chem. Eng. Sci., 226, 2020) eliminates the additional conditioning-space discretization in CMC applications by assuming that the mixture-fraction PDF is well represented by a β-PDF. A Gaussian quadrature provides the mixture-fraction abscissae, and the conditional scalar mean is expressed in terms of Jacobi polynomials. Here, by preserving the computational efficiency of SA-CMC, a novel quadrature-based moment method (QBMM-CMC) is developed for CMC applications, which does not assume the form of the mixture-fraction PDF. Remarkably, by solving the closed forms for the micromixing terms from CMC, exact expressions result for the mixture-fraction moments of any order. Thus, QBMM-CMC covers cases where the mixture-fraction PDF cannot be well represented by a β-PDF and can be applied to disperse multiphase flows with mass transfer (e.g., droplet evaporation). For single-phase and multiphase pure-mixing problems, the QBMM-CMC mixture-fraction moments are observed to deviate from the β-PDF. For single-phase mixing with and without dispersed-phase mass transfer, QBMM-CMC predictions for mixing-sensitive competitive-consecutive and parallel reactions are investigated parametrically.

A DNS study on turbulence-chemistry interaction in lean premixed syngas flames

Syngas is a mixture of hydrogen and CO and it is considered as a cleaner alternative fuel for engines. Hydrogen, a component of syngas, is a sub-unity Lewis number (Le) fuel and the influence of Le on pure hydrogen flames has been documented in the past. However, the effect of Lewis number and turbulence on heat release rate (HRR) in a turbulent premixed syngas flames has not received enough attention. DNS of a turbulent premixed syngas flame in a radially expanding flame kernel is performed to investigate the effect of Lewis number, turbulence, and initial curvature on HRR. An equimolar mixture of hydrogen and CO is considered for most of the simulations except for one simulation which had higher mole fraction of CO. Enhancement in HRR at low temperature due to the chain terminating reaction H + O2 + M↔HO2 + M is observed. Higher diffusivity of hydrogen-containing species, viz. H2, H, and OH, is found to be responsible for the enhancement in HRR. Among these species, diffusion of atomic hydrogen is shown to have maximum influence on HRR at low temperature. The influence of differential diffusion of hydrogen is observed to diminish when the mole fraction of CO is raised to 70%. Syngas flame at high temperature and high pressure is also studied to mimic the conditions seen in SI engine. The shift of HRR towards low temperature zone observed in the atmospheric flames is absent under high temperature and pressure conditions. Non-unity Lewis number of hydrogen-containing species increases the conditional fluctuation of mass fraction and HRR, in the progress variable space. A care has to be taken while using first-order conditional moment closure for such flames, where conditional fluctuations cannot be neglected.

Large-eddy simulation of direct injection natural gas combustion in a heavy-duty truck engine using modified conditional moment closure model with low-dimensional manifold method

This article describes the application of a modified first-order conditional moment closure model used in conjunction with the trajectory-generated low-dimensional manifold method in large-eddy simulation of pilot ignited high-pressure direct injection natural gas combustion in a heavy-duty diesel engine. The article starts with a review of the intrinsic low-dimensional manifold method for reducing detailed chemistry and various formulations for the construction of such manifolds. It is followed by a brief review of the conditional moment closure method for modelling the interaction between turbulence and combustion chemistry. The high computational cost associated with the direct implementation of the basic conditional moment closure model was discussed. The article then describes the formulation of a modified approach to solve the conditional moment closure equation, whose reaction source terms for the conditional mass fractions for species were obtained by projecting the turbulent perturbation onto the reaction manifold. The main model assumptions were explained and the resulting limitations were discussed. A numerical experiment was conducted to examine the validity the model assumptions. The model was then implemented in a combustion computational fluid dynamics solver developed on an open-source computational fluid dynamics platform. Non-reactive jet simulations were first conducted and the results were compared to the experimental measurement from a high-pressure visualization chamber to verify that the jet penetration under engine relevant conditions was correctly predicted. The model was then used to simulate natural gas combustion in a heavy-duty diesel engine equipped with a high-pressure direct injection system. The simulation results were compared with the experimental measurement from a research engine to verify the accuracy of the model for both the combustion rate and engine-out emissions.

Predictions of lift-off height of turbulent methane and propane flames issuing in cold surroundings using conditional moment closure coupled with an extinction model

Recently, an extinction model was successfully integrated with a conditional moment closure (CMC) approach to predict lift-off height accurately for the lifted jet diffusion flame discharging to hot surroundings Roy et al. (2014). This method, referred as CMCE, is a low-cost model based on the RANS and first-order CMC approach. The proposed methodology may not be superior compared to involved LES and second-order CMC combination, but enables to simulate industrial problems like gas turbine processes in a reasonable CPU time. Though the credibility of this novel approach was well established by validating its predictions against the experimental data, the applicability of this method and constants involved in this model, for other test cases was questionable. In this brief communication, CMCE model is further validated against lift-off studies of two different fuels issuing in cold surroundings, without changing the model constants used in the earlier work. The results are encouraging and a good match is found for lift-off heights for both the test cases and for varying jet velocities. Large variations are found between the predictions of lift-off heights by CMC and CMCE models.

Soot Formation Modeling of n-Heptane Sprays Under Diesel Engine Conditions Using the Conditional Moment Closure Approach

Numerical simulations of soot formation of n-heptane autoigniting spray in a constant-volume vessel under diesel engine conditions with different ambient densities (14.8 and 30 kg/m3) and ambient oxygen concentrations (8–21% O2 mol fraction) were performed using two-dimensional, first-order conditional moment closure (CMC). Soot formation was modeled with a semiempirical two-equation model that considers simultaneous soot particle inception, surface growth, coagulation, and oxidation by O2 and OH. Soot radiation was accounted for with an optical-thin formulation. Results are compared to experimental data by means of ignition delay time, lift-off length, and soot volume fraction distribution. Good predictions of ignition delay and lift-off for all nine cases are achieved. High volume fraction soot location and semi-quantitative distribution have been well described even with this comparatively simple soot model. The findings suggest that the conditional moment closure approach is a promising framework for soot modeling under diesel engine conditions.

LES-CMC Simulations of Different Auto-ignition Regimes of Hydrogen in a Hot Turbulent Air Co-flow

Large-Eddy Simulation (LES) results in combination with first-order Conditional Moment Closure (CMC) are presented for a hydrogen jet, diluted with nitrogen, issued into a turbulent co-flowing hot air stream. The fuel mixes with the co-flow air, ignites and forms a lifted-like flame. Global trends in the experimental observations are in general well reproduced: the auto-ignition length decreases with increase in co-flow temperature and increases with increase in co-flow velocity. In the experiments, the co-flow temperature was varied, so that different auto-ignition regimes, including low Damkohler number situations, were obtained (no ignition, random spots, flashback and lifted flame). All regimes are recovered in the simulations. Auto-ignition is found to be the stabilizing mechanism. The impact of different detailed chemistry mechanisms on the auto-ignition predictions is discussed. With increasing air temperature, the differences between the mechanisms considered diminish. The evolution of temperature, H2O, H, HO2 and OH from inert to burning conditions is discussed in mixture fraction space.

Investigation of detailed kinetic scheme performance on modelling of turbulent non-premixed sooting flames

This paper presents the results of an application of a first-order conditional moment closure (CMC) approach coupled with a semi-empirical soot model to investigate the effect of various detailed combustion chemistry schemes on soot formation and destruction in turbulent non-premixed flames. A two-equation soot model representing soot particle nucleation, growth, coagulation and oxidation, was incorporated into the CMC model. The turbulent flow-field of both flames is described using the Favre-averaged fluid-flow equations, applying a standard k-ɛ turbulence model. A number of five reaction kinetic mechanisms having 50–100 species and 200–1000 elementary reactions called ABF, Miller-Bowman, GRI-Mech3.0, Warnatz, and Qin were employed to study the effect of combustion chemistry schemes on soot predictions. The results showed that of various kinetic schemes being studied, each yields similar accuracy in temperature prediction when compared with experimental data. With respect to soot prediction, the kinetic scheme containing benzene elementary reactions tends to result in a better prediction on soot concentrations in comparison to those contain no benzene elementary reactions. Among five kinetic mechanisms being studied, the Qin combustion scheme mechanism turned to yield the best prediction on both flame temperature and soot levels.

Modeling evaporation effects in conditional moment closure for spray autoignition

Simulations of an n-heptane spray autoigniting under conditions relevant to a diesel engine are performed using two-dimensional, first-order conditional moment closure (CMC) with full treatment of spray terms in the mixture fraction variance and CMC equations. The conditional evaporation term in the CMC equations is closed assuming interphase exchange to occur at the droplet saturation mixture fraction values only. Modeling of the unclosed terms in the mixture fraction variance equation is done accordingly. Comparison with experimental data for a range of ambient oxygen concentrations shows that the ignition delay is overpredicted. The trend of increasing ignition delay with decreasing oxygen concentration, however, is correctly captured. Good agreement is found between the computed and measured flame lift-off height for all conditions investigated. Analysis of source terms in the CMC temperature equation reveals that a convective–reactive balance sets in at the flame base, with spatial diffusion terms being important, but not as important as in lifted jet flames in cold air. Inclusion of droplet terms in the governing equations is found to affect the mixture fraction variance field in the region where evaporation is the strongest, and to slightly increase the ignition delay time due to the cooling associated with the evaporation. Both flame propagation and stabilization mechanisms, however, remain unaffected.

Direct Quadrature Conditional Moment Closure for Modelling of Turbulent Combustion

We present a method of direct quadrature conditional moment closure (DQCMC) for the treatment of realistic turbulence-chemistry interaction in computational fluid dynamics (CFD) software. The method which is based on the direct quadrature method of moments (DQMOM) coupled with the conditional moment closure (CMC) equations is in simplified form and easily implementable in existing CMC formulation for CFD. The observed fluctuations of scalar dissipation around the conditional mean values are captured by the treatment of a set of mixing environments, each with its pre-defined weight. Unlike the early versions of the DQCMC method the resulting equations are similar to that of the first-order CMC, and the “diffusion” term is strictly positive and no correction factors are used. We present results for two mixing environments where the resulting matrices of the DQCMC can be inverted analytically. We have performed this analysis for a simple hydrogen flame using a multi species chemical scheme containing nine species. The effects of the fluctuations around the conditional means are captured accurately and the predicted results are in very good agreement with observed trends from direct numerical simulations. Furthermore, the differences between the first order CMC and DQCMC are discussed.

Modeling of scalar mixing in turbulent jet flames by multiple mapping conditioning

Multiple mapping conditioning (MMC) combines the probability density function (PDF) and the conditional moment closure (CMC) methods via the application of a generalized mapping function to a prescribed reference space. Stochastic and deterministic formulations of MMC exist, and the deterministic implementation has been applied here to a piloted jet diffusion flame (Sandia Flame D). This paper focuses on the feasibility of MMC and its closures for real (laboratory) flames and a relatively simple one-dimensional reference space that represents mixture fraction has been used. The remaining chemically reactive species are implicitly conditioned on mixture fraction and their fluctuations around the conditional mean are neglected. This work primarily evaluates the ability of the deterministic form of MMC to provide accurate and consistent closures for the mixture fraction PDF and the conditional scalar dissipation which do not rely on presumed shape functions for the PDF such as the commonly used β-PDF. Computed probability distributions agree well with measurements, and a detailed comparison of the modeled conditional and mean scalar dissipation with experimental data and conventional closures demonstrate MMC’s potential. Predictions of reactive species and temperature are in good agreement with experimental data and similar in quality to singly conditioned, first-order CMC predictions. MMC therefore provides an attractive-since consistent-alternative approach for the modeling of scalar mixing in turbulent reacting flows.

Conditional moment closure modelling of soot formation in turbulent, non-premixed methane and propane flames

Presented are results obtained from the incorporation of a semi-empirical soot model into a first-order conditional moment closure (CMC) approach to modelling turbulent, non-premixed methane–air and propane–air flames. Soot formation is determined via the solution of two transport equations for soot mass fraction and particle number density, with acetylene and benzene employed as the incipient species responsible for soot nucleation, and the concentrations of these calculated using a detailed gas-phase kinetic scheme involving 70 species. The study focuses on the influence of differential diffusion of soot particles on soot volume fraction predictions. The results of calculations are compared with experimental data for atmospheric and 3 atm methane flames, and propane flames with air preheated to 323 K and 773 K. Overall, the study demonstrates that the model, when used in conjunction with a representation of differential diffusion effects, is capable of accurately predicting soot formation in the turbulent non-premixed flames considered.

Conditional Moment Closure (CMC) applied to autoignition of high pressure methane jets in a shock tube

Autoignition of non-premixed methane–air mixtures is investigated using first-order Conditional Moment Closure (CMC). Turbulent velocity and mixing fields simulations are decoupled from the CMC calculations due to low temperature changes until ignition occurs. The CMC equations are cross-stream averaged and finite differences are applied to discretize the equations. A three-step fractional method is implemented to treat separately the stiff chemical source term. Two detailed chemical kinetics mechanisms are tested as well as two mixing models. The present results show good agreement with published experimental measurements for the magnitude of both ignition delay and kernel location. The slope of the predicted ignition delay is overpredicted and possible sources of discrepancy are identified. Both scalar dissipation rate models produce comparable results due to the turbulent flow homogeneity assumption. Further, ignition always occurs at low scalar dissipation rates, much lower than the flamelet critical value of ignition. Ignition is found to take place in lean mixtures for a value of mixture fraction around 0.02. The conditional species concentrations are in qualitative agreement with previous research. Homogeneous and inhomogeneous CMC calculations are also performed in order to investigate the role of physical transport in the present autoignition study. It is found that spatial transport is small at ignition time. Predicted ignition delays are shown to be sensitive to the chemical kinetics. Reasonable agreement with previous simulations is found. Improved formulations for the mixing model based on non-homogeneous turbulence are expected to have an impact.

Modelling of autoignition for methane-based fuel blends using Conditional Moment Closure

First-order Conditional Moment Closure (CMC) is applied to investigate the autoignition of high pressure CH4, CH4/C2H6, CH4/H2, and CH4/N2 jets injected into lower pressure heated air over a wide range of pre-combustion temperatures. Detailed and optimized chemical kinetics mechanisms are used for each fuel blend. The turbulent flow field and CMC calculations are decoupled based on the frozen mixing assumption. The standard k–ε turbulence model is used to perform the velocity and mixing fields calculations. The CMC equations are cross-stream averaged due to the small radial dependence of the conditional scalars in shear flows, and are solved elliptically. A three-step fractional method is implemented to treat the stiff chemical source term separately. Two formulations are implemented to model the conditional scalar dissipation term. The numerical results are compared to shock tube experiments. Predicted ignition delays and locations are found to be in reasonable agreement with the experiments. Further analysis is provided.

Diesel Engine Simulations with Multi-Dimensional Conditional Moment Closure

First-order elliptic Conditional Moment Closure (CMC), coupled with a computational fluid dynamics (CFD) solver, has been employed to simulate combustion in a direct-injection heavy-duty diesel engine. The three-dimensional structured finite difference CMC grid has been interfaced to an unstructured finite volume CFD mesh typical of engine modelling. The implementation of a moving CMC grid to reflect the changes in the domain due to the compression and expansion phases has been achieved using an algorithm for the cell addition/removal and modelling the additional convection term due to the CMC cell movement. Special care has been taken for the boundary conditions and the wall heat transfer. An operator splitting formulation has been used to integrate the CMC equations efficiently. A CMC domain reduction of the three-dimensional problem to two- and zero-dimensions through appropriate volume integration of the CMC equation has been explored in terms of accuracy and computational time. Additional considerations have been reported concerning the initialization of the CMC domain in conserved scalar space during transient calculations where the probability density function of the mixture fraction changes drastically with time as during fuel injection. Predictions compare favourably with the experimental pressure traces for tests at full and half load. A balance of terms in the CMC equation allows conjectures on the structure of the flame and its expansion across the spray after autoignition.

Conditional moment closure prediction of soot formation in turbulent, nonpremixed ethylene flames

Results obtained from incorporating a semiempirical soot model into a first-order conditional moment closure (CMC) approach to modeling turbulent nonpremixed flames of ethylene and air are presented. Soot formation is determined via the solution of two transport equations for soot mass fraction and particle number density, with acetylene and benzene employed as the incipient species responsible for soot nucleation, and the concentrations of these species calculated using a detailed gas-phase kinetic scheme involving 463 reactions and 70 species. The study focuses on the influence of differential diffusion of soot particles on soot volume fraction predictions. The results of calculations are compared with experimental data for three sooting ethylene flames and, in general, predictions of mixing and temperature fields within the three flames show good agreement with data. Soot volume fraction predictions are found to be in significantly better accord with data when differential diffusion is accounted for in the CMC-based soot model, supporting the importance of such effects in sooting flames, as previously noted by Kronenburg et al. in relation to methane combustion. Overall, the study demonstrates that the CMC-based soot model, when used in conjunction with a model of differential diffusion effects, is capable of accurately predicting soot formation in turbulent nonpremixed ethylene–air flames.

Analysis of kinetic mechanism performance in conditional moment closure modelling of turbulent, non-premixed methane flames

This paper presents results obtained from the application of a first-order conditional moment closure approach to the modelling of two methane flames of differing geometries. Predictions are based upon a second-moment turbulence and scalar-flux closure, and supplemented with full and reduced chemical kinetic mechanisms, ranging from a simple 12-step to a complex 1207-step mechanism. Alongside analysis of the full kinetic schemes' performance, is an appraisal of the behaviour of their derivatives obtained using mechanism-reduction techniques. The study was undertaken to analyse the practicality of incorporating kinetic models of varying complexity into calculations of turbulent non-premixed flames, and to make comparison of their performance. Despite extensive studies of the predictive ability of such schemes under laminar flame conditions, systematic evaluations have not been performed for turbulent reacting flows. This paper reflects upon the impact that selection of chemical kinetics has upon subsequent calculations and concludes that, although application of reduced schemes is more than adequate to reproduce experimental data, selection of the parent mechanism is of paramount importance to the prediction of minor species. Although widely used schemes are well documented and validated, their performances vary considerably. Thus, careful consideration must be made to their application and origins during the evaluation of combustion models.

Simulations of Turbulent Non-Premixed Counterflow Flames with First-Order Conditional Moment Closure

Simulations of turbulent CH4-air counterflow flames are presented, obtained in terms of zero and two-dimensional first-order Conditional Moment Closure (CMC) to study the flame structure and extinction limits. The CMC equation with detailed chemistry is solved without the need for operator splitting, while the accompanying flow field is determined using a commercial CFD software employing a Reynolds stress turbulence model and additional transport equations for the turbulent scalar flux and for the mean scalar dissipation rate. Two detailed chemical mechanisms and different conditional scalar dissipation rate models have been examined and small differences were found.

Prediction of turbulent non-premixed hydrogen flames using second-order conditional moment closure modelling

The advent of increasingly stringent emissions legislation inevitably leads to the requirement for more accurate modelling of pollutant formation in practical combustion applications. Previous limited success in modelling species such as NO, using first-order conditional moment closure (CMC) models indicates the need for more advanced modelling techniques. Here, a method of including higher-order chemistry within a one-dimensional, parabolic CMC framework is investigated, and applied to the prediction of three hydrogen jets of varying degrees of helium dilution. Interaction of the combustion model with both the k-ε and Reynolds stress turbulence models is examined. Results are encouraging and found to be in line with expectations. Suggestions are made in light of this to account for anomalous predictions of nitrous radical formation.

Conditional filtering method for large-eddy simulation of turbulent nonpremixed combustion

The conditional filtering method is proposed as a subfilter combustion model for large-eddy simulation (LES) of turbulent nonpremixed combustion. The novel method is based on conditional filtering of a reactive scalar field and an extension of conditional moment closure (CMC) for LES. Filtering conditioned on isosurfaces of the mixture fraction is adopted to resolve small-scale mixing and chemical reactions in nonpremixed combustion. The conditionally filtered equations are derived and the closure assumptions are discussed. A priori tests are performed using direct numerical simulation data for reacting mixing layers. The primary closure assumption on the subfilter flux in mixture fraction space is shown to work much better than the corresponding closure for the Reynolds averaged CMC due to resolved large-scale fluctuations of the scalar dissipation rate and of reactive scalars. Results show that first-order closure of the reaction rate performs well except for the boundaries of flame holes. In the boundaries of flame holes, fluctuations of reactive scalars around the conditionally filtered values are large enough for the effects of higher-order correlations to be significant. The accuracy of the first-order closure is less sensitive to the level of local extinction than that of first-order CMC, since large-scale fluctuations of reactive scalars on isosurfaces of the mixture fraction are resolved. This shows that extinction processes occur primarily over length scales comparable to the large scales of the turbulence. The integrated conditional filtering approach is introduced to reduce the computational cost and to resolve the low probability problem in the conditional filtering method. While the assumption of homogeneity in the integration direction is not as good as in the conditional average, the integrated formulation is shown to represent the extinction process caused by large-scale fluctuations of the scalar dissipation rate quite well.