Conditional Moment Closure Model Research Articles

Derivation of the conditional moment closure equations for spray combustion

In this work we derive the fundamental equations for conditional moment closure (CMC) modelling of individual phases set in a two-phase flow. The derivation is based on the instantaneous transport equations for the single phase that involve a level set/indicator function technique for accounting for interfaces. Special emphasis is put on spray combustion with the CMC equations formulated for the gas phase. The CMC equations are to be viewed as an adjunct to existing methods for the modelling of the dynamics of sprays: they provide a refinement of the modelling of chemical reactions in the gas phase. The resulting CMC equations differ significantly from those already in use in the literature. They contain, of course, unclosed terms that need to be modelled. Investigation of the unclosed terms associated with evaporation at the droplet surface is well beyond the capabilities of laboratory measurement or direct numerical simulation. It is proposed that modelling of these terms be based on the well-established ‘laws’ of similarity between heat and mass transfer: an example is detailed for one example of the general modelling of the spray dynamics. Other unclosed terms are important throughout the gas phase. Models used for these terms in single-phase flows are reviewed and it is proposed that any modifications needed for these models be investigated by DNS of suitable model problems having good resolution of the flow and mixing in the inter-droplet space. It is proposed that a spray analogue of the scalar mixing layer that has been widely studied in single-phase flows be used as the model problem for such DNS studies and also for LES and RANS modelling.

Conditional moment closure and transient flamelet modelling for detailed structure and NO x formation characteristics of turbulent nonpremixed jet and recirculating flames

This study has been mainly motivated to assess computationally and theoretically the conditional moment closure (CMC) model and the transient flamelet model for the simulation of turbulent nonpremixed flames. These two turbulent combustion models are implemented into the unstructured grid finite volume method that efficiently handles physically and geometrically complex turbulent reacting flows. Moreover, the parallel algorithm has been implemented to improve computational efficiency as well as to reduce the memory load of the CMC procedure. Example cases include two turbulent CO/H2/N2 jet flames having different flow timescales and the turbulent nonpremixed H2/CO flame stabilized on an axisymmetric bluff-body burner. The Lagrangian flamelet model and the simplified CMC formulation are applied to the strongly parabolic jet flame calculation. On the other hand, the Eulerian particle flamelet model and full conservative CMC formulation are employed for the bluff-body flame with flow recirculation. Based on the numerical results, a detailed discussion is given for the comparative performances of the two combustion models in terms of the flame structure and NO x formation characteristics.

Direct numerical simulations of the double scalar mixing layer: Part II: Reactive scalars

The reacting double scalar mixing layer (RDSML) is investigated as a canonical multistream flow and a model problem for simple piloted diffusion flames. In piloted diffusion flames, the reacting fuel and oxidizer streams are initially separated by a central pilot stream at stoichiometric composition. The primary purpose of this pilot is to delay the mixing of the pure streams until a stable flame base can develop. In such multistream systems, the modeling of turbulent scalar mixing is complicated by the multiple feed streams, leading to more complex fine-scale statistics, which remain as yet an unmet modeling challenge compared to the simpler two-feed system. In Part I we described how multimodal mixture fraction probability density functions (PDFs) and conditional scalar dissipation rates can be modeled with a presumed mapping function approach. In this work we present an efficient and robust extension of the modeling to a general multistream reacting flow and compare predictions to three-dimensional direct numerical simulations (DNS) of the RDSML with a single-step reversible chemistry model and varying levels of extinction. With high extinction levels, the interaction with the pilot stream is described. Additionally, state-of-the-art combustion modeling calculations including conditional moment closure (CMC) and stationary laminar flamelet modeling (SLFM) are performed with the newly developed mixing model. Excellent agreement is found between the DNS and modeling predictions, even where the PDF is essentially a triple-delta shape near the flame base, so long as extinction levels are moderate to low. The suggested approach outlined in this paper is strictly valid only for flows that can be described by a single mixture fraction. For these flows the approach should provide engineers with fine-scale models that are of accuracy comparable to those already available for binary mixing, at only marginally higher complexity and cost.

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.

Lagrangian Stochastic Modelling of Chemical Reaction in a Scalar Mixing Layer

Statistics of conserved and reactive scalar concentration in a scalar mixing layer are calculated using a Lagrangian particle model coupled to a micro-mixing model and using conserved scalar theory for the chemistry. Good agreement is obtained with laboratory experiments for temperature statistics in a thermal mixing layer. For conserved scalar statistics in a reacting mixing layer, good agreement is obtained with direct numerical simulations, but enhanced mixing is necessary in order to account approximately for the effects of a flow instability in the experiments. With this enhanced mixing, good agreement is obtained with laboratory data for reactive scalars on the edges of the mixing layer using conditional moment closure, but discrepancies remain in the middle of the layer. Calculations show that in conditional moment closure modelling, the scalar dissipation conditioned on the scalar concentration can be replaced by its unconditional counterpart with negligible error.

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.

Simulations of spray autoignition and flame establishment with two-dimensional CMC

The unsteady two-dimensional conditional moment closure (CMC) model with first-order closure of the chemistry and supplied with standard models for the conditional convection and turbulent diffusion terms has been interfaced with a commercial engine CFD code and analyzed with two numerical methods, an “exact” calculation with the method of lines and a faster fractional-step method. The aim was to examine the sensitivity of the predictions to the operator splitting errors and to identify the extent to which spatial transport terms are important for spray autoignition problems. Despite the underlying simplifications, solution of the full CMC equations allows a single model to be used for the autoignition, flame propagation (“premixed mode”), and diffusion flame mode of diesel combustion, which makes CMC a good candidate model for practical engine calculations. It was found that (i) the conditional averages have significant spatial gradients before ignition and during the premixed mode and (ii) that the inclusion of physical-space transport affects the calculation of the autoignition delay time, both of which suggest that volume-averaged CMC approaches may be inappropriate for diesel-like problems. A balance of terms in the CMC equation before and after autoignition shows the relative magnitude of spatial transport and allows conjectures on the structure of the premixed phase of diesel combustion. Very good agreement with available experimental data is found concerning ignition delays and the effect of background air turbulence on them.

Modelling of species in hood fires by conditional moment closure

Carbon monoxide, the chief killer in fires, and other species are modelled for a series of enclosure fires. The conditions emulate building fires where CO is formed in the rich, turbulent, nonpremixed flame and is transported frozen to lean mixtures by the ceiling jet which is cooled by radiation and dilution. Conditional moment closure modelling is used and computational domain minimisation criteria are developed which reduce the computational cost of this method. The predictions give good agreement for CO and other species in the lean, quenched-gas stream, holding promise that this method may provide a practical means of modelling real, three-dimensional fire situations.

05/02237 First-order conditional moment closure modeling of turbulent, nonpremixed methane flames

05/02237 First-order conditional moment closure modeling of turbulent, nonpremixed methane flames

Conditional moment closure modelling of turbulent jet diffusion flames of helium-diluted hydrogen

First-order conditional moment closure (CMC) modelling of NO in non-premixed flames has met with limited success due to the need to consider turbulence influences on the conditional production rate of chemical species. This paper presents results obtained using a second-order approach where such effects are incorporated through solution of a transport equation for the conditional variance. In contrast to earlier work, second-order chemistry is implemented using a more robust numerical technique, with predictions obtained using a Reynolds stress turbulence model. First-order CMC and k–ϵ turbulence model predictions are presented for comparison purposes. For the hydrogen flames examined, results demonstrate small differences between first- and second-order calculations of major species and temperature, although second-order corrections reduce NO and OH levels. Additionally, variations occur between results for these species derived using the two turbulence models due to differences in conditional variance predictions. This and the numerical solution method employed are responsible for deviations with earlier results. It is concluded that while the higher-order CMC model does not significantly improve NO predictions, agreement with OH data is superior. Physical space predictions are sufficiently accurate for assessing flame characteristics, with the Reynolds stress model providing superior results.

Modeling of turbulent, two-dimensional nonpremixed CH 4/H 2 flame over a bluffbody using first- and second-order elliptic conditional moment closures

First- and second-order conditional moment closure (CMC) models are applied to a turbulent CH 4/H 2 flame anchored on a bluffbody. Significant extinction is involved as the jet velocity increases for the three test flames: HM1, HM2, and HM3 (in a turbulent nonpremixed flame (TNF) database). The flow and mixing field are calculated with the k – ɛ model with a modified constant, C ɛ 1 . Elliptic CMC formulation is employed to calculate reacting scalar fields with chemical kinetic mechanisms, GRI Mech 2.11 and 3.0. Second-order corrections are limited to three rate-limiting steps to avoid solving a huge number of variance and covariance equations. Even though first-order CMC captures temperature and major species within agreeable limits, the predictions of OH and NO are considerably improved with second-order CMC in both mixture fraction and real space. Results show that the equilibrium assumption performs better than the cold mixture as the conditional inflow boundary condition. There is no noticeable conditional cross-stream dependence downstream, while care needs to be taken in interpretation of the upstream results near the nozzle.

Conditional moment closure modeling of a lifted turbulent flame

Results obtained using conditional moment closure (CMC) approach to modeling a lifted turbulent hydrogen flame are presented. Predictions are based onk-e-g turbulent closure, a 23-step chemical mechanism and a radially averaged CMC model. The objectives are to find out how radially averaged CMC can represent a lifted flame and which mechanism of flame stabilization can be described by this modeling method. As a first stage of the study of multi-dimensional CMC for large eddy simulation (LES) of the lifted turbulent flames, the effect of turbulence upon combustion is included, the high-order compact finite-difference scheme (Pade) is used and previously developed characteristic-wave-based boundary conditions for multicomponent perfect gas mixtures are here extended to their conditional forms but the heat release due to combustion is not part of the turbulent calculations. Attention is focused to the lift-off region of the flame which is commonly considered as a cold flow. Comparison with published experimental data and the computational results shows that the lift-off height can be accurately determined, and Favre averaged radial profiles of temperature and species mole fractions are also reasonably well predicted. Some of the current flame stabilization mechanisms are discussed.

Conditional moment closure modeling of turbulent nonpremixed combustion in diluted hot coflow

The conditional moment closure (CMC) model is applied to predict flame structures and NO formation in the moderate and intense low oxygen dilution combustion mode. The effects of oxygen concentration in a hot diluted oxidant stream are investigated in the experimental condition of Dally et al. [Proc. Combust. Inst. 29 (2002) 1147–1154]. The GRI 2.11 Mech is used for description of chemical reaction including NO x chemistry. The conditional scalar dissipation rate, which describes the effect of turbulent mixing on finite chemistry, is calculated by integrating the transport equation for probability density function (PDF). A new PDF is proposed to describe three stream mixing in terms of a single mixture fraction. The conditional mean predictions of temperature, and CO, OH, and NO mass fractions are in good agreement with measurements. The unconditional Favre mean predictions of CO and NO mass fractions are also in reasonable agreement. Upstream underprediction of OH and NO in the low oxygen concentration case may be attributed to uncertainty in low temperature reaction mechanism and mixing prediction. Differential diffusion effects are shown to be nonnegligible in the present flames. The CMC model is an attractive choice for simulation of MILD combustion in which conditional fluctuations of reactive scalars are small enough for first-order closure of conditional mean reaction rates to remain valid.

Conditional moment closure modeling of a lifted turbulent flame

Conditional moment closure modeling of a lifted turbulent flame

Conditional moment closure modeling of extinction and re-ignition in turbulent non-premixed flames

The conditional moment closure approach is used to describe extinction and re-ignition events in turbulent non-premixed hydrocarbon combustion. It is well known that single conditional moment closure cannot correctly capture these phenomena and a second conditioning variable, sensible enthalpy, needs to be introduced to account for significant fluctuations around the singly conditioned mean. As a first step, the key closure assumptions are validated with the aid of direct numerical simulation with four-step chemistry. Simulations with an increasing degree of local extinction are used to assess the performance of first order closures for the highly non-linear chemical source terms. The results obtained with the doubly conditioned moment closure method are excellent for all cases, and conditional moment closure is capable of accurately predicting local extinction, the onset of re-ignition, and the occurrence of global extinction. Next, it is shown that the conditioned reaction rate terms can be modelled using presumed shape functions for the conditional moments in sensible enthalpy space, thus leading to a new closure for the reaction rate terms in the singly conditioned moment closure equations. Full closure can be based on a presumed β-distribution for the conditional PDF of sensible enthalpy, and the solution of an additional variance equation is needed.

화염편 및 조건평균법 모델을 이용한 재순환 비예혼합 난류 화염장의 해석

The conditional moment closure(CMC) model has been Implemented in context with the unstructured-grid finite-volume method which efficiently handle the physically and geometrically complex turbulent reacting flows The validation cases include a turbulent nonprermixed CO/H₂/N₂ Jet flame and a turbulent nonpremixed H₂/CO flame stabilized on an axisymmetric bluff-body burner. In terms of mean flame field, minor species and NO formation, numerical results has the overall agreement with experimental data The detailed discussion has been made for the turbulence-chemistry interaction and NOx formation characteristics as well as the comparative performance for CMC and flamelet model.

Second-order conditional moment closure modeling of turbulent piloted Jet diffusion flames

A second-order CMC model for a detailed chemical mechanism is presented and applied to turbulent piloted jet diffusion flames, Sandia Flames D, E, and F. Second-order corrections are made for three rate-limiting steps of methane–air combustion, while first-order closure is employed for all the other reaction steps. The effect of a pilot flame is well predicted by the conditional scalar dissipation rate derived from a trimodal PDF for fuel, air, and pilot streams. Results show significantly improved predictions of conditional mean temperature and mass fractions of major and stable intermediate species on the fuel-rich side. Conditional mean OH and NO mass fractions are also in good agreement, although underestimated extinction seems to be responsible for overpredicted NO downstream for Flame E. Results for Flame D are in better agreement than those for Flame E which involves more significant local extinction. Second-order correction of only the chain-branching step, H + O 2 → OH + O, is shown to capture most of the improvements by correction of the three rate-limiting steps. The predictions for Flame F do not seem to be satisfactory due to underpredicted conditional variances and covariances. Further study may be needed to clarify major causes of the deviation and to suggest possible remedies for more accurate prediction of Flame F.

A second-order conditional moment closure model for the simulation of a lifted turbulent flame

This paper presents the results of a second-order closure Conditional Moment Closure model applied to a lifted turbulent H 2 -air flame. The objective is to find out how a second-order closure will modify the lift-off height compared to first-order results. A 10-step chemical mechanism is used to predict rates of reaction in hydrogen-air mixtures but the second-order correction terms are calculated using one-step chemistry. First a progress variable for this reduced mechanism is defined and an additional equation for its conditional variance is derived. The level of the conditional fluctuation is found to be around 30% of its maximum value for stoichiometric mixtures in the flame stabilisation area. The second-order closure has a distinct effect on the flame position determined in the simulations.

First-order conditional moment closure modeling of turbulent, nonpremixed methane flames

Results obtained using a first-order conditional moment closure (CMC) approach to modeling turbulent, nonpremixed flames of methane are presented. Predictions are based on both k– ε and second-moment turbulence closures and use the GRI-Mech 2.11 kinetic scheme involving 279 reactions and 49 species and the GRI-Mech 3.0 scheme applying 325 reactions and 53 species. To provide a comprehensive assessment of the ability of these methods to predict a wide range of flames, comparisons are made with experimental data on three piloted and two unpiloted flames, the majority of which exhibit no local extinction effects. It is concluded that first-order CMC modeling is capable of yielding reliable predictions for flames with little or no extinction effect, with the only exception being NO, which is overpredicted at most locations and stoichiometries. Results derived using the two turbulence closures are in general in close accord, although real space predictions demonstrate the superiority of results derived on the basis of the second-moment turbulence closure. Calculation of flames that exhibit extinction effects demonstrate a gradual deterioration of results with increasing Re, pointing to the requirement for second-order CMC modeling as such effects become more significant. Anomalies are observed in regard to NO, which is significantly overpredicted for the piloted flames, but less so for the unpiloted flames. These results demonstrate a requirement not only to explore second-order CMC modeling approaches for such flames, but also to examine the application of different kinetic schemes, including a detailed investigation of the mechanisms and rates employed for NO chemistry.

An investigation of the performance of turbulent mixing models

Modeling of partially stirred reactors (PaSR) by stochastic Monto Carlo simulations is carried out to investigate the relative performance of three existing turbulent mixing models: the interaction by exchange with the mean model, the modified Curl mixing model, and the Euclidean minimum spanning tree model. A detailed mechanism for hydrogen oxidation, which involves 9 species and 19 reactions, is incorporated into the simulations using the in situ adaptive tabulation algorithm. Numerical simulations are performed for a wide range of residence and mixing times revealing the significant differences in the PDFs of mixture fraction, in the scatter plots, and in the extinction behaviors between the three different mixing models. The conditional mean scalar dissipation implied by each model is deduced analytically or numerically in the PaSR, but it is shown not to have the same significance as in the conditional moment closure and flamelet models.