Conditional Moment Closure Equations Research Articles

Large eddy simulation and finite-volume conditional moment closure modelling of a turbulent lifted H2/N2 flame

Large eddy simulations with three-dimensional finite-volume Conditional Moment Closure (CMC) model are performed for a hydrogen/nitrogen lifted flame with detailed chemical mechanism. The emphasis is laid on the influences of mesh resolution and convection scheme of finite-volume CMC equations on predictions of reactive scalars and unsteady flame dynamics. The results show that the lift-off height is underestimated and the reactive scalars are over-predicted with coarser CMC mesh. It is also found that further refinement of the CMC mesh would not considerably improve the results. The time sequences of the most reactive and stoichiometric hydroxyl radical mass fractions indicate that finer CMC mesh can capture more unsteady details than the coarser CMC mesh. Moreover, the coarse CMC mesh has lower conditional scalar dissipation rate, which would promote the earlier auto-ignition of the flame base. Besides, the effects of the convection scheme for the CMC equations (i.e., upwind, central differencing and their blends) on the lifted flame characteristics are also investigated. It is shown that different convection schemes lead to limited differences on the time-averaged temperature, mixture fraction and species mass fractions. Moreover, the root-mean square values of hydrogen and hydroxyl mass fractions show larger deviation from the measurements with hybrid upwind and central differencing scheme, especially around the flame base. Furthermore, the distributions of the numerical fluxes on the CMC faces also show obvious distinctions between the upwind and blending schemes. The budget analysis of the individual CMC terms shows that a sequence of CMC faces has comparable contributions with upwind scheme. However, with the hybrid schemes, the instantaneous flux is dominantly from limited CMC faces. The reactivity of a CMC cell is more easily to be affected by its neighbors when the upwind scheme is used.

Conditional moment closure method with tabulated chemistry in adiabatic turbulent nonpremixed jet flames

The conditional moment closure (CMC) method with tabulated chemistry is proposed to reduce the computational load for integration of stiff reaction steps in the conventional CMC methods. It is based on the assumption that temperature and all species except CO and CO2 adjust fast enough so that they can be uniquely determined by the single reaction progress variable given as the sum of CO and CO2 mass fractions. The CMC equations are solved for the conditional reaction progress variable, Qpv, whereas the other species mass fractions and the reaction rates including that for Qpv are retrieved from lookup tables. A steady homogeneous form of the conditional variance and covariance equations are solved to construct the second-order closure table to be consistent with Steady Laminar Flamelet Model (SLFM)-based first-order closure. Good agreement is shown between the results by integrated and tabulated chemistry in both the first-order and the second-order CMC for Sandia Flames D, E and F. The overall computational time is reduced to about 0.5% of that by the conventional CMC method involving integration of the detailed reaction mechanism for all species and temperature.

Assessment of Tabulated Premixed Conditional Moment Closure Model Using Direct Numerical Simulation

ABSTRACT The Conditional Moment Closure (CMC) is a proven methodology to model non-premixed flames, but an emerging technique for analyzing turbulent premixed flames. However, solving a full set of CMC equations is a time-consuming process. The Tabulated Premixed CMC (TPCMC) model is a modification of the premixed CMC model that seeks to reduce the time consumed in solving equations by making the premixed CMC model amenable to the tabulation of chemistry. TPCMC model considers the effect of small scale mixing of reactants and products using the scalar dissipation rate and effect of large scale turbulence on the flame using the beta probability density function. The validation of TPCMC model using Direct Numerical Simulation (DNS) data has been attempted in this paper. DNS of growth of an ignition kernel in a turbulent medium was performed for this purpose. Two sets of simulations, namely 3D DNS with three-step reduced chemistry and 2D DNS with multi-step chemistry, have been carried out in this study using PENCIL code. The Favre averaging of this DNS data was performed on the surfaces of constant radius to assess the accuracy of TPCMC model. The variations in turbulence intensity and integral length scale were considered to generate DNS database. TPCMC model accurately predicted the variation of fuel mass fraction and temperature for all the cases considered in this study and captured the qualitative trends in the CO mass fraction where a negligible deviation from the actual values was observed. The influence of initial parameters related to turbulence considered in this study was negligible on the accuracy of predictions obtained from the model.

OpenFOAM based conditional moment closure (CMC) model for solving non-premixed turbulent combustion: Integration and validation

This article presents a direct integration of conditional moment closure (CMC), an advanced combustion model with a computational fluid dynamics (CFD) package, OpenFOAM (Open-source Field Operation And Manipulation) to solve non-premixed turbulent combustion. Earlier our group (Roy and Sreedhara, 2016) had coupled CMC code with FLUENT to model non-premixed turbulent combustion, but it was through manual two-way coupling method where CFD and CMC codes were sequentially solved one after the other until the steady state is reached. The extensible and flexible framework of OpenFOAM encouraged us to build a CFD-CMC solver called cmcFoam. This fully-coupled solver solves CMC equations internally in OpenFOAM along with flow-field equations. Major advantages of this solver are (1) it is built in the open-source framework, (2) it can predict the transient behavior of flames, and (3) one can benefit from all features of OpenFOAM. The validity of the proposed solver has been thoroughly demonstrated by accurately predicting experimental data published for the Sandia syngas (CO/H2/N2) flame. Two-dimensional axisymmetric simulations were performed with a modified k-ε turbulence model and using a chemical kinetic scheme having 21 species and 84 chemical reactions (DRM 19). Results predicted by the solver showed an excellent agreement with the published data, both in mixture fraction and physical space. This benchmarked coupled code may be useful to analyze fairly complex industrial problems.

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.

LES/CMC Simulations of Swirl-Stabilised Ethanol Spray Flames Approaching Blow-Off.

Large Eddy Simulations (LES) with the Conditional Moment Closure (CMC) combustion model of swirling ethanol spray flames have been performed in conditions close to blow-off for which a wide database of experimental measurements is available for both flame and spray characterization. The solution of CMC equations exploits a three-dimensional unstructured code with a first order closure for chemical source terms. It is shown that LES/CMC is able to properly capture the flame structure at different conditions and agrees reasonably well with the measurements both in terms of mean flame shape and dynamic behaviour of the flame evaluated in terms of local extinctions and statistics of the lift-off height. Experimental measurements of the overall (liquid plus gaseous) mixture fraction, performed using the Laser-Induced Breakdown Spectroscopy technique, are also included allowing further assessment and validation of the numerical method. The sensitivity of the simulation results to the various boundary conditions is discussed.

Detailed chemistry LES/CMC simulation of a swirling ethanol spray flame approaching blow-off

A swirling ethanol spray flame in conditions close to blow-off has been simulated using Large Eddy Simulation (LES) and the Conditional Moment Closure (CMC) combustion model aiming to further validate the capability of the LES/CMC approach to capture local extinctions in turbulent spray flames. A detailed chemical mechanism was used and a transport equation of the mixture fraction sub-grid variance, with spray interaction terms included, was solved. Numerical results are in good agreement with the experiment in terms of both instantaneous and mean flame shape and droplet velocity and size. Local extinctions were detected in the region around the bluff-body, resulting in a fluctuating lift-off of the flame there, and the probability density function of the lift-off height was in very good agreement with the experiment, suggesting that the degree of local extinction is captured quantitatively. Analysis of the CMC equation suggested that local extinction was influenced by both transport in physical space and high scalar dissipation rate. The modelling of the latter needs development in areas where the spray evaporation is strong enough to increase significantly the sub-grid mixture fraction fluctuations and their small-scale gradients, possibly leading to deviations from the present usual approach of relating the sub-grid scalar dissipation to the sub-grid mixture fraction variance.

Conditional statistics in a planar liquid jet with a second-order chemical reaction

Conditional statistics, conditioned on mixture fraction, are experimentally investigated in a planar liquid jet with a second-order isothermal chemical reaction A+B→R. Reactants A and B are contained in the jet and ambient flows, respectively, and are supplied into the test section under the non-premixed condition. Streamwise velocity, mixture fraction and concentrations of all reactive species are simultaneously measured by using an I-type hot-film probe and an optical fibre probe based on light absorption spectrometry. The cross-streamwise profiles of conditional mean concentrations and conditional mean reaction rate show that the conditional mean concentrations and the conditional mean reaction rate near the jet exit change with the cross-streamwise position, whereas they are independent of the cross-streamwise position in the downstream regions. On the jet centreline, the conditional mean reaction rate has a peak value on the condition that the conditional mean concentration of product R has a peak value. A comparison between the conditional mean reaction rate and the covariance of conditional concentration fluctuations shows that the covariance of conditional concentration fluctuations is so small that a first moment closure model for the conditional mean reaction rate is valid. The conditional mean streamwise velocity is almost linear to the mixture fraction for small mixture fraction fluctuations. The conditional scalar dissipation rate is calculated from the budget of the conditional moment closure equation. The results show that the conditional scalar dissipation rate has a single peak value in the upstream region, whereas in the downstream region, it has two peaks for the large and small mixture fraction values. The conventional mean scalar dissipation rate is calculated from the probability density function of the mixture fraction and the conditional scalar dissipation rate. The conventional mean scalar dissipation rate on the jet centreline decreases in the downstream direction as (x∗)-2.9 (where x∗ is the distance from the virtual origin), which is almost the same as that expected from scaling arguments.

A Conditional Moment Closure (CMC) formulation including differential diffusion applied to a non-premixed hydrogen–air flame

A Conditional Moment Closure (CMC) formulation of the species transport equation including differential diffusion is presented. Further, the CMC equation for conditional enthalpy is also derived. The objectives of the present study are to (1) derive the CMC equations including the differential effects for both the species and enthalpy in the context of hydrogen combustion, (2) implement the new CMC equations in RANS in two dimensions without any cross-stream averaging and (3) evaluate the benefits and drawbacks of the present CMC calculations applied to a non-premixed hydrogen–air flame. The current study is focused on improving the NO concentration near the nozzle. Predictions of the flow and mixing fields and CMC results for the temperature, major chemical species, and NO concentrations are compared with the experimental data. A budget of the terms in the CMC equations for the differentially diffusing chemical species is investigated. The model is shown to have a prominent effect on NO predictions, with an improvement in the regions close to the nozzle. Some possible sources of modeling discrepancy are identified.

Assessment of implementation variants of conditional scalar dissipation rate in LES-CMC simulation of auto-ignition of hydrogen jet

In this paper the large eddy simulation (LES) and conditional moment closure (CMC) combustion model have been applied for modelling of auto-ignition of hydrogen jet issuing into a hot ambient co-flow. Most of the attention was devoted to modelling aspects of the conditional scalar dissipation rate which is a key quantity of the CMC model. Two models are compared with emphasis on differences in distributions in mixture fraction space. Analysis of mutual relations between the terms of CMC equations confirms importance of the conditional scalar dissipation rate. It is also shown that model constants are crucial from the point of view of an auto-ignition location and a flame lift off height. The numerical results are compared with experimental data and both the mean and the root mean square fluctuating values of the temperature and species mass fraction agree well with measurements.

Complex chemistry DNS of n-heptane spray autoignition at high pressure and intermediate temperature conditions

Direct Numerical Simulations (DNS) of turbulent n-heptane sprays autoigniting at high pressure (P=24bar) and intermediate air temperature (Tair=1000K) have been performed to investigate the physical mechanisms present under conditions where low-temperature chemistry is expected to be important. The initial turbulence in the carrier gas, the global equivalence ratio in the spray region, and the initial droplet size distribution of the spray were varied. Results show that spray ignition exhibits a spotty nature, with several kernels developing independently in those regions where the mixture fraction is close to its most reactive value ξMR (as determined from homogeneous reactor calculations) and the scalar dissipation rate is low. Turbulence reduces the ignition delay time as it promotes mixing between air and the fuel vapor, eventually resulting in lower values of scalar dissipation. High values of the global equivalence ratio are responsible for a larger number of ignition kernels, due to the higher probability of finding regions where ξ=ξMR. Spray polydispersity results in the occurrence of ignition over a wider range of mixture fraction values. This is a consequence of the inhomogeneities in the mixing field that characterize these sprays, where poorly mixed rich spots are seen to alternate with leaner ones which are well-mixed. The DNS simulations presented in this work have also been used to assess the applicability of the Conditional Moment Closure (CMC) method to the simulation of spray combustion. CMC is found to be a valid method for capturing spray autoignition, although care should be taken in the modelling of the unclosed terms appearing in the CMC equations.

Scalar Dissipation Rate in a Planar Liquid Jet with a Chemical Reaction (Experimental Study on the Evaluation of the Conditional Moment Closure Equation)

Conditional scalar dissipation rate <N | ξ = η> conditioned on a mixture fraction ξ and mean scalar dissipation rate <N> in a planar liquid jet are experimentally investigated. Streamwise velocity and reactive concentrations are simultaneously measured in the planar liquid jet with a second-order chemical reaction A+B→R. <N | η> on the jet centerline is obtained from a conditional moment closure (CMC) equation and measured conditional mean values. To calculate <N | η>, we derive the equations which represent conditional mean concentrations by assuming that the ratio of the conditional mean concentration of product R and the conditional mean reaction rate does not depend on the mixture fraction. The conditional dissipation rate is obtained from the budget of CMC equation. The results show that <N | η> becomes small in the downstream direction. It is also shown that the profiles of <N | η> change in the downstream direction; in the downstream region, <N | η> has a local minimum value at η ≈ ξS, where ξS is the stoichiometric ratio in mixture. <N> can be obtained from <N | η> and the probability density functions of mixture fraction. It is observed that <N> on the jet centerline decays in the downstream (x) direction as x-2.9, which is close to the -3 power expected from scaling arguments.

Investigation of the “TECFLAM” Non-premixed Flame Using Large Eddy Simulation and Proper Orthogonal Decomposition

The confined turbulent swirling non-premixed TECFLAM S09c flame has been investigated using Large Eddy Simulation and a pre-calculated mixture-fraction-based flamelet model in a preliminary step to simulate this flame. The simplified formulation, despite the fact that it cannot capture localized extinctions, is found to reproduce the experimentally-observed vortex breakdown and the results agree well with available experimental data for velocity and mixture fraction. The unsteady flow features before the burner exit and inside the combustion chamber are analyzed with spectral analysis, correlations, and Proper Orthogonal Decomposition. The results show the presence of longitudinal vortices whose axes rotate with the swirl and which cause separation inside the inlet pipe. With combustion, some of the structures are damped and the spectral peaks shift to higher frequencies. Finally, simulations with the full 3D Conditional Moment Closure equation, which allows spatial and temporal fluctuations of the conditionally-filtered reacting scalars, reproduce successfully the flame lift-off observed experimentally.

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.

Large Eddy Simulation of a Turbulent Lifted Flame using Conditional Moment Closure and Rate-Controlled Constrained Equilibrium

The objective of this paper is the simulation of a turbulent flame by employing the Rate-Controlled Constrained Equilibrium (RCCE) approach for the chemistry reduction, and Large-Eddy Simulation (LES) coupled with Conditional Moment Closure (CMC) for the turbulence-chemistry interaction modelling. RCCE is a systematic method for mechanism reduction, based on the concept that certain species characterized by faster time scales are in a constrained equilibrium state, determined by the concentration of the species controlled by the chemical kinetics. A general system of differential equations can be derived, independent on the selection of the fast and slow species (which appears as a parameter). The RCCE system is used to compute the conditional source term in the CMC equation. The flame simulated here is a methane flame issuing into a vitiated co-flow formed by hot combustion products, the “Cabra” flame, which is controlled by auto-ignition and is therefore sensitive to the chemical mechanism. The results show an influence of the chosen chemistry in the ignition length.

Large Eddy Simulations of forced ignition of a non-premixed bluff-body methane flame with Conditional Moment Closure

Large Eddy Simulations (LES) of forced ignition of a bluff-body stabilised non-premixed methane flame using the Conditional Moment Closure (CMC) turbulent combustion model have been performed. The aim is to investigate the feasibility of the use of CMC/LES for ignition problems and to examine which, if any, of the characteristics already observed in related experiments could be predicted. A three-dimensional formulation of the CMC equation was used with simple and detailed chemical mechanisms and sparks with different parameters (location, size) were used. It was found that the correct pattern of flame expansion and overall flame appearance were predicted with reasonable accuracy with both mechanisms, but the detailed mechanism resulted in expansion rates closer to the experiment. Moreover, the distribution of OH was predicted qualitatively accurately, with patches of high and low concentration in the recirculation zone during the ignition transient, consistent with experimental data. The location of the spark relative to the recirculation zone was found to determine the pattern of the flame propagation and the total time for the flame stabilisation. The size was also an important parameter, since it was found that the flame extinguishes when the spark is very small, in agreement with expectations from experiment. The stabilisation mechanism of the flame was dominated by the convection and sub-grid scale diffusion of hot combustion products from the recirculation zone to the cold gases that enter the burner, as revealed by analysis of the CMC equation.

“Derivation of the conditional moment closure equations for spray combustion” [Combust. Flame Vol. 155, Issue 3

“Derivation of the conditional moment closure equations for spray combustion” [Combust. Flame Vol. 155, Issue 3

Analysis of second-order conditional moment closure applied to an autoignitive lifted hydrogen jet flame

The timing and location of autoignition can be highly sensitive to turbulent fluctuations of composition. Second-order Conditional Moment Closure (CMC) provides transport equations for conditional (co)variances in turbulent reacting flows. CMC equations accounting for compressibility and differential diffusion are analyzed using data from direct numerical simulation of an autoignitive lifted turbulent hydrogen jet flame [C.S. Yoo, R. Sankaran, J.H. Chen, Three-dimensional direct numerical simulation of turbulent lifted hydrogen/air jet flame in a heated coflow. Part 1. J. Fluid. Mech., (2008)]. At the flame base, second-order moments were required to accurately model the conditional reaction rates. However, over 80% of the second-order reaction rate component was obtainable with a small subset (16%) of the species-temperature covariances. The balance of the second-order CMC equation showed that turbulent transport across spatial composition gradients initiates generation of conditional variances.

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.