Statistical Analysis of Scalar Dissipation Rate Transport in Turbulent Partially Premixed Flames: A Direct Numerical Simulation Study

Statistically planar turbulent premixed and partially premixed flames for different initial turbulence intensities are simulated for global equivalence ratios ??>?=?0.7 and ??>?=?1.0 using three-dimensional Direct Numerical Simulations (DNS) with simplified chemistry. For the simulations of partially premixed flames, a random distribution of equivalence ratio following a bimodal distribution of equivalence ratio is introduced in the unburned reactants ahead of the flame. The simulation parameters in all of the cases were chosen such that the combustion situation belongs to the thin reaction zones regime. The DNS data has been used to analyse the behaviour of the dissipation rate transports of both active and passive scalars (i.e. the fuel mass fraction Y _F and the mixture fraction ξ) in the context of Reynolds Averaged Navier–Stokes (RANS) simulations. The behaviours of the unclosed terms of the Favre averaged scalar dissipation rates of fuel mass fraction and mixture fraction (i.e. \(\widetilde {\varepsilon }_Y =\overline {\rho D\nabla Y_F^{\prime \prime } \cdot \nabla Y_F^{\prime \prime } } /\overline{\rho }\) and \(\widetilde {\varepsilon }_\xi =\overline {\rho D\nabla \xi ^{\prime \prime }\cdot \nabla \xi ^{\prime \prime }} /\overline {\rho })\) transport equations have been analysed in detail. In the case of the \(\widetilde {\varepsilon }_Y \) transport, it has been observed that the turbulent transport term of scalar dissipation rate remains small throughout the flame brush whereas the terms due to density variation, scalar–turbulence interaction, reaction rate and molecular dissipation remain the leading order contributors. The term arising due to density variation remains positive throughout the flame brush and the combined contribution of the reaction and molecular dissipation to the \(\widetilde {\varepsilon }_Y \) transport remains negative throughout the flame brush in all cases. However, the behaviour of scalar–turbulence interaction term of the \(\widetilde {\varepsilon }_Y \) transport equation is significantly affected by the relative strengths of turbulent straining and the straining due to chemical heat release. In the case of the \(\widetilde {\varepsilon }_\xi \) transport, the turbulent transport term remains small throughout the flame brush and the density variation term is found to be negligible in all cases, whilst the reaction rate term is exactly zero. The scalar–turbulence interaction term and molecular dissipation term remain the leading order contributors to the \(\widetilde {\varepsilon }_\xi \) transport throughout the flame brush in all cases that have been analysed in the present study. Performances of existing models for the unclosed terms of the transport equations of \(\widetilde {\varepsilon }_Y \) and \(\widetilde {\varepsilon }_\xi \) are assessed with respect to the corresponding quantities obtained from DNS data. Based on this exercise either suitable models have been identified or new models have been proposed for the accurate closure of the unclosed terms of both \(\widetilde {\varepsilon }_Y \) and \(\widetilde {\varepsilon }_\xi \) transport equations in the context of Reynolds Averaged Navier–Stokes (RANS) simulations.     

Noise Correction and Length Scale Estimation for Scalar Dissipation Rate Measurements in Turbulent Partially Premixed Flames

A recently developed conditional sampling-based method for correcting noise effects in scalar dissipation rate measurements and for estimating the extent of resolution of the dissipation rate is employed to analyze the data obtained in turbulent partially premixed (Sandia) flames. The method uses conditional sampling to select instantaneous fully resolved local scalar fields, which are analyzed to determine the measurement noise and to correct the Favre mean, conditional, and conditionally filtered dissipation rates. The potentially under-resolved local scalar fields, also selected using conditional sampling, are corrected for noise and are analyzed to examine the extent of resolution. The error function is used as a model for the potentially under-resolved local scalar to evaluate the scalar dissipation length scales and the percentage of the dissipation resolved. The results show that the Favre mean dissipation rate, the mean dissipation rate conditional on the mixture fraction, and dissipation rate filtered conditionally on the mixture fraction generally are well resolved in the flames. Analyses of the dissipation rates filtered conditionally on the mixture fraction and temperature show that the length scale increases with temperature, due to lower dissipation rate and higher diffusivity. The dissipation rate is well resolved for temperatures above 1,300 K but is less resolved at lower temperatures, although the probability of very low temperature events is low. To fully resolve these rare events the sample spacing needs to be reduced by approximately one half. The present study further demonstrates the effectiveness of the new noise correction and length scale estimation method.     

Transient Behavior of Turbulent Scalar Transport in Premixed Flames

Transition from gradient to countergradient scalar transport in a statistically planar, one-dimensional, developing, premixed turbulent flame is studied both theoretically and numerically. A simple criterion of the transition referred to is derived from the balance equation for the combustion progress variable, with the criterion highlighting an important role played by flame development. A balance equation for the difference in velocities $\bar{u}_b$ and $\bar{u}_u$ conditioned on burned and unburned mixture, respectively, is numerically integrated. Both analytical and computed results show that; (1) The flux $\overline{\rho u'' c''}$ is gradient during an early stage of flame development followed by transition to countergradient scalar transport at certain instant t _tr . (2) The transition time is increased when turbulence length scale L is increased or when the laminar flame speed S _L and/or the density ratio are decreased. (3) The transition time normalized using the turbulence time scale is increased by u??. Moreover, the numerical simulations have shown that the transition time is increased by u?? if a ratio of u??/S _L is not large. This dependence of t _tr on u?? is substantially affected by (i) the mean pressure gradient induced within the flame due to heat release and (ii) by the damping effect of combustion on the growth rate of mean flame brush thickness. The reasonable qualitative agreement between the computed trends and available experimental and DNS data, as well as the agreement between the computed trends and the present theoretical results, lends further support to the conditioned balance equation used in the present work.     

Scalar Dissipation Rate Transport in the Context of Large Eddy Simulations for Turbulent Premixed Flames with Non-Unity Lewis Number

The effects of global Lewis number Le on the statistical behaviour of the unclosed terms in the transport equation of the Favre-filtered scalar dissipation rate (SDR) Ñ _c have been analysed using a Direct Numerical Simulation (DNS) database of freely propagating statistically planer turbulent premixed flames with Le ranging from 0.34 to 1.2. The DNS data has been explicitly filtered to analyse the statistical behaviour of the unclosed terms in the SDR transport equation arising from turbulent transport T ₁, density variation due to heat release T ₂, scalar-turbulence interaction T ₃, reaction rate gradient T ₄, molecular dissipation (?D ₂) and diffusivity gradients f(D) in the context of Large Eddy Simulations (LES). It Le has significant effects on the magnitudes of T ₁, T ₂, T ₃, T ₄, (?D ₂) and f(D). Moreover, both qualitative and quantitative behaviours of the unclosed terms T ₁, T ₂, T ₃, T ₄, (?D ₂) and f(D) are found to be significantly affected by the LES filter width Δ, which have been explained based on a detailed scaling analysis. Both scaling analysis and DNS data suggest that T ₂, T ₃, T ₄, (?D ₂) and f(D) remain leading order contributors to the SDR \(\tilde {{N}}_{c} \) transport for LES. The scaling estimates of leading order contributors to the SDR \(\tilde {{N}}_{c} \) transport has been utilised to discuss the possibility of extending an existing SDR model for Reynolds Averaged Navier Stokes (RANS) simulation for SDR \(\tilde {{N}}_{c} \) closure in the context of LES of turbulent premixed combustion.     

Statistical Analysis and a-priori Modelling of Flame Surface Density Transport in Turbulent Stratified Flames: A Direct Numerical Simulation Study

Statistically planar turbulent premixed and stratified flames for different initial intensities of decaying turbulence have been simulated for global equivalence ratios <???> = 0.7 and <???> = 1.0 using three-dimensional simplified chemistry based Direct Numerical Simulations (DNS). The simulation parameters are chosen such that the thin reaction zones regime combustion is realised in all cases and a random bi-modal distribution of equivalence ratio ? is introduced in the unburned gas ahead of the flame to account for the mixture inhomogeneity for stratified flames. The modelling of the unclosed terms (i.e. the turbulent transport term T ₁, the tangential strain rate term T ₂, the propagation term T ₃, and the curvature term T ₄) of the generalised FSD transport equation has been addressed in the context of RANS simulations. It has been found that the turbulent transport term T ₁ remains small in comparison to the leading order contributions of the tangential strain rate and curvature terms (i.e. T ₂ and T ₄, respectively) in the globally stoichiometric cases, but T ₁ begins to play a more important role in the globally fuel-lean cases. The strain rate term T ₂ remains positive throughout the flame brush and acts as a leading order source term for all the flames considered in this analysis. It is has been found that the magnitude of T ₂ decreases with decreasing root-mean-square velocity fluctuations u ^′ (<???>) for a given value of <???> (u ^′). The contribution of the propagation term T ₃ remains generally positive towards the unburned gas side of the flame brush but assumes generally negative values towards the burned gas side of the flame brush. Moreover, whilst the order of magnitude of the propagation term T ₃ is comparable in all cases, T ₃ remains small in comparison to the leading order contributors (i.e. T ₂ and T ₄) in the globally stoichiometric cases however it plays a more important role in the globally fuel-lean cases. The curvature term T ₄ acts as a leading order sink term in all cases except towards the unburned gas side of the flame brush in low u ^′ globally stoichiometric (i.e. <???> = 1.0) flames. Furthermore, it has been demonstrated that the magnitude of T ₄ decreases with decreasing u ^′ (<???>) for a given value of <???> (u ^′). Appropriate model expressions have been identified for T ₁, T ₂, T ₃ and T ₄ based on an a-priori analysis of the DNS data.     

Effects of Lewis Number on Head on Quenching of Turbulent Premixed Flames: A Direct Numerical Simulation Analysis

The head on quenching of statistically planar turbulent premixed flames by an isothermal inert wall has been analysed using three-dimensional Direct Numerical Simulation (DNS) data for different values of global Lewis number Le(0.8, 1.0 and 1.2) and turbulent Reynolds number Re_t. The statistics of head on quenching have been analysed in terms of the wall Peclet number Pe (i.e. distance of the flame from the wall normalised by the Zel’dovich flame thickness) and the normalised wall heat flux Φ. It has been found that the maximum (minimum) value of Φ(Pe) for the turbulent Le=0.8 cases are greater (smaller) than the corresponding laminar value, whereas both Pe and Φ in turbulent cases remain comparable to the corresponding laminar values for Le=1.0 and 1.2. Detailed physical explanations are provided for the observed Le dependences of Pe and Φ. The existing closure of mean reaction rate \(\overline {\dot {\omega }}\) using the scalar dissipation rate (SDR) in the near wall region has been assessed based on a-priori analysis of DNS data and modifications to the existing closures of mean reaction rate and SDR have been suggested to account for the wall effects in such a manner that the modified closures perform well both near to and away from the wall.     

A-Priori Validation of Scalar Dissipation Rate Models for Turbulent Non-Premixed Flames

The modelling of scalar dissipation rate in conditional methods for large-eddy simulations is investigated based on a priori direct numerical simulation analysis using a dataset representing an igniting non-premixed planar jet flame. The main objective is to provide a comprehensive assessment of models typically used for large-eddy simulations of non-premixed turbulent flames with the Conditional Moment Closure combustion model. The linear relaxation model gives a good estimate of the Favre-filtered scalar dissipation rate throughout the ignition with a value of the related constant close to the one deduced from theoretical arguments. Such value of the constant is one order of magnitude higher than typical values used in Reynolds-averaged approaches. The amplitude mapping closure model provides a satisfactory estimate of the conditionally filtered scalar dissipation rate even in flows characterised by shear driven turbulence and strong density variation.

     

Transition from Countergradient to Gradient Scalar Transport in Developing Premixed Turbulent Flames

A simple model of turbulent scalar flux developed recently by the present authors is applied to determine the direction of the flux in a statistically planar one-dimensional premixed flame that does not affect turbulence and has self-similar mean structure. Results obtained in the case of statistically stationary turbulence indicate that transition from countergradient to gradient turbulent scalar transport may occur during flame development, as the peak mean rate of product creation moves to the trailing edge of the flame brush. In the case of decaying turbulence, the opposite transition (from gradient to countergradient transport) was simulated in line with available DNS data. In both cases, transition instant depends strongly on turbulence and mixture characteristics. In particular, countergradient transport is suppressed by an increase in the rms turbulent velocity and by a decrease in the laminar flame speed or density ratio, in line with available experimental and DNS data. The obtained results lend qualitative support to the model of turbulent scalar flux addressed in the present work.     

A-Priori Direct Numerical Simulation Modelling of Co-variance Transport in Turbulent Stratified Flames

Three-dimensional Direct Numerical Simulations of statistically planar turbulent stratified flames at global equivalence ratios <???>?=?0.7 and <???>?=?1.0 have been carried out to analyse the statistical behaviour of the transport of co-variance of the fuel mass fraction Y _F and mixture fraction ξ (i.e. $\widetilde{Y_F^{\prime\prime} \xi ^{\prime\prime}}={\overline {\rho Y_F^{\prime\prime} \xi^{\prime\prime}} } \Big/ {\overline \rho })$ for Reynolds Averaged Navier Stokes simulations where $\overline q $ , $\tilde{q} ={\overline {\rho q} } \big/ {\overline \rho }$ and $q^{\prime\prime}= q-\tilde{q}$ are Reynolds averaged, Favre mean and Favre fluctuation of a general quantity q with ρ being the gas density and the overbar suggesting a Reynolds averaging operation. It has been found that existing algebraic expressions may not capture the statistical behaviour of $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ with sufficient accuracy in low Damköhler number combustion and therefore, a transport equation for $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ may need to be solved. The statistical behaviours of $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ and the unclosed terms of its transport equation (i.e. the terms originating from turbulent transport T ₁ , reaction rate T ₄ and molecular dissipation $\left( {-D_2 } \right))$ have been analysed in detail. The contribution of T ₁ remains important for all cases considered here. The term T ₄ acts as a major contributor in <???>?=?1.0 cases, but plays a relatively less important role in <???>?=?0.7 cases, whereas the term $\left( {-D_2 } \right)$ acts mostly as a leading order sink. Through an a-priori DNS analysis, the performances of the models for T ₁ , T ₄ and $\left( {-D_2 } \right)$ have been addressed in detail. A model has been identified for the turbulent transport term T ₁ which satisfactorily predicts the corresponding term obtained from DNS data. The models for T ₄ , which were originally proposed for high Damköhler number flames, have been modified for low Damköhler combustion. Predictions of the modified models are found to be in good agreement with T ₄ obtained from DNS data. It has been found that existing algebraic models for $D_2 =2\overline {\rho D\nabla Y_F^{\prime\prime} \nabla \xi^{\prime\prime}} $ (where D is the mass diffusivity) are not sufficient for low Damköhler number combustion and therefore, a transport equation may need to be solved for the cross-scalar dissipation rate $\widetilde{\varepsilon }_{Y\xi } ={\overline {\rho D\nabla Y_F^{\prime\prime} \nabla \xi^{\prime\prime}} } \big/ {\overline \rho }$ for the closure of the $\widetilde{Y_F^{\prime\prime} \xi^{\prime\prime}}$ transport equation.     

Statistical Analysis of Turbulent Flame-Droplet Interaction: A Direct Numerical Simulation Study

Turbulent combustion of mono-disperse droplet-mist has been analysed based on three-dimensional Direct Numerical Simulations (DNS) in canonical configuration under decaying turbulence for a range of different values of droplet equivalence ratio (?_d), droplet diameter (a_d) and root-mean-square value of turbulent velocity (u^′). The fuel is supplied in liquid phase and the evaporation of droplets gives rise to gaseous fuel for the flame propagation into the droplet-mist. It has been found that initial droplet diameter, turbulence intensity and droplet equivalence ratio can have significant influences on the volume-integrated burning rate, flame surface area and burning rate per unit area. The droplets are found to evaporate predominantly in the preheat zone, but some droplets penetrate the flame front, reaching the burned gas side where they evaporate and some of the resulting fuel vapour diffuses back towards the flame front. The combustion process in gaseous phase takes place predominantly in fuel-lean mode even for ?_d > 1. The probability of finding fuel-lean mixture increases with increasing initial droplet diameter because of slower evaporation of larger droplets and this predominantly fuel-lean mode of combustion exhibits the attributes of low Damköhler number combustion and gives rise to thickening of flame with increasing droplet diameter. The chemical reaction is found to take place under both premixed and non-premixed modes of combustion and the relative contribution of non-premixed combustion to overall heat release increases with increasing droplet size. The statistical behaviours of the flame propagation and mode of combustion have been analysed in detail and detailed physical explanations have been provided for the observed behaviour.     

Direct Numerical Simulation of Inert Droplet Effects on Scalar Dissipation Rate in Turbulent Reacting and Non-Reacting Shear Layers

Three-dimensional direct numerical simulation has been performed to investigate the effects of inert evaporating droplets on scalar dissipation rate χ in temporally-developing turbulent reacting and non-reacting mixing layers with the Reynolds number based on the vorticity thickness up to 8000 and the number of traced Lagrangian droplets up to 10⁷. The detailed instantaneous field analysis and ensemble-averaged statistics reveal complex interactions among combustion, droplet dynamics and evaporation, all of which have a considerable influence on χ. The presence of inert evaporating droplets promotes χ in both non-reacting and reacting mixing layers. In the latter, combustion reduces χ, so when combustion is suppressed by evaporating droplets, χ is enhanced. The transport equation of χ has been analyzed to investigate the various effects on χ in detail. The terms in the equation contain explicitly the evaporation rate and its spatial derivative, acting as a sink and a source for χ, respectively. On the whole, the net effect of the evaporation-rate terms is to promote χ. However, the production and dissipation terms are the dominant source and sink terms, respectively.     

Direct Numerical Simulations of Premixed Turbulent Flames with Reduced Chemistry: Validation and Flamelet Analysis

Direct numerical simulation is a very powerful tool to evaluate the validity of new models and theories for turbulent combustion. In this paper, direct numerical simulations of spherically expanding premixed turbulent flames in the thin reaction zone regime and in the broken reaction zone regime are performed. The flamelet-generated manifold method is used in order to deal with detailed reaction kinetics. The computational results are analyzed by using an extended flame stretch theory. It is investigated whether this theory is able to describe the influence of flame stretch and curvature on the local burning velocity of the flame. It is found that if the full profiles of flame stretch and curvature through the flame front are included in the theory, the local mass burning rate is well predicted. The influence of several approximations, which are used in other existing theories, is studied. When flame stretch is assumed constant through the flame front or when curvature of the flame front is neglected, the theory fails to predict the local mass burning rate. The influence of using a reduced chemistry model is investigated by comparing flamelet simulations with reduced and detailed chemistry.     

Turbulence Statistics and Scalar Transport in Highly-Sheared Premixed Flames

This paper describes recent progress in the analysis of the nature of turbulent premixed flames stabilised behind an axisymmetric baffle which are of fundamental interest in the development of new and cleaner combustion systems. The work includes the use of laser-based diagnostics for velocity and temperature measurements, which are extended to the analysis of turbulence statistics, including the energy spectrum and typical length scales in a reacting shear layer. The results provided experimental evidence of the extension of the flamelet regime beyond the Klimov--Williams criterion. Arguments based on the shape of the weighted-joint-probability distributions of axial velocity and temperature fluctuations show that the counter-gradient nature of heat flux is associated with the preferential deceleration of products of combustion in relation to the cold reactants.     

A Comparison of Strategies for Direct Numerical Simulation of Turbulence Chemistry Interaction in Generic Planar Turbulent Premixed Flames

Three different methods to introduce turbulence in the computational domain of Direct Numerical Simulations (DNS) of statistically planar turbulent premixed flame configurations have been reviewed and their advantages and disadvantages in terms of run time, natural flame development, control of turbulence parameters and convergence of statistics extracted from the simulations have been discussed in detail. It has been found that there is no method, which is clearly superior to the other two alternative methods. An analysis has been performed to explain why Lundgren’s physical space linear forcing results in an integral length scale which is, independent of the Reynolds number, a constant fraction of the domain size. Furthermore, an evolution equation for the integral length scale has been derived, and a scaling analysis of its terms has been performed to explain the evolution of the integral length scale in the context of Lundgren’s physical space linear forcing. Finally, a modification to Lundgren’s forcing approach has been suggested which ensures that the integral length scale settles to a predetermined value so that DNS of statistically planar turbulent premixed flames with physical space forcing can be conducted for prescribed values of Damköhler and Karlovitz numbers.     

A Posteriori Assessment of Algebraic Scalar Dissipation Models for RANS Simulation of Premixed Turbulent Combustion

This paper examines the effects of scalar dissipation rate modelling on mean reaction rate predictions in turbulent premixed flames. The sensitivity of the mean reaction rate is explored by using different closures for scalar dissipation and the sensitivity of the scalar dissipation models themselves is also examined with respect to their defining constants. The influence of different scalar dissipation models on the flame location and mean velocities is reported and compared with experimental results. The predicted reaction rate is found to be sensitive to the choice of closure used for scalar dissipation and also to the respective constants used in the scalar dissipation models. It is also found that the scalar dissipation models involving chemical and turbulent time scales yield a more physically plausible reaction rate when compared with the scalar dissipation models relying only on the turbulent time scale.     

Computation of Conditional Average Scalar Dissipation in Turbulent Jet Diffusion Flames

The modelling of conditional scalar dissipation in locally self-similar turbulent reacting jets is considered. The streamwise dependence in the transport equation of the conserved scalar pdf is represented by a function solely dependent on centreline mixture fraction. This procedure provides a simple model suitable for non-homogeneous flows and ensures positive values for conditional scalar dissipation. It has been tested in pure hydrogen-air jet diffusion flames using a Conditional Moment Closure method with detailed 12species, 23 reactions chemistry. The calculations show good agreement of the averaged scalar dissipation with reference values and the model proves to be superior to previous models based on homogeneous flows if the distribution of the conditional scalar dissipation in mixture fraction space is compared with experimental results. A dependence of NO predictions on the model of conditional scalar dissipation can be observed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Effects of Lewis Number on Scalar Variance Transport in Premixed Flames

The influences of differential diffusion rates of heat and mass on the transport of the variances of Favre fluctuations of reaction progress variable and non-dimensional temperature have been studied using three-dimensional simplified chemistry based Direct Numerical Simulation (DNS) data of statistically planar turbulent premixed flames with global Lewis number ranging from Le?= 0.34 to 1.2. The Lewis number effects on the statistical behaviours of the various terms of the transport equations of variances of Favre fluctuations of reaction progress variable and non-dimensional temperature have been analysed in the context of Reynolds Averaged Navier Stokes (RANS) simulations. It has been found that the turbulent fluxes of the progress variable and temperature variances exhibit counter-gradient transport for the flames with Lewis number significantly smaller than unity whereas the extent of this counter-gradient transport is found to decrease with increasing Lewis number. The Lewis number is also shown to have significant influences on the magnitudes of the chemical reaction and scalar dissipation rate contributions to the scalar variance transport. The modelling of the unclosed terms in the scalar variance equations for the non-unity Lewis number flames have been discussed in detail. The performances of the existing models for the unclosed terms are assessed based on a-priori analysis of DNS data. Based on the present analysis, new models for the unclosed terms of the active scalar variance transport equations are proposed, whenever necessary, which are shown to satisfactorily capture the behaviours of unclosed terms for all the flames considered in this study.     

Finite Rate Chemistry Effects in Highly Sheared Turbulent Premixed Flames

Detailed scalar structure measurements of highly sheared turbulent premixed flames stabilized on the piloted premixed jet burner (PPJB) are reported together with corresponding numerical calculations using a particle based probability density function (PDF) method. The PPJB is capable of stabilizing highly turbulent premixed jet flames through the use of a small stoichiometric pilot that ensures initial ignition of the jet and a large shielding coflow of hot combustion products. Four lean premixed methane-air flames with a constant jet equivalence ratio are studied over a wide range of jet velocities. The scalar structure of the flames are examined through high resolution imaging of temperature and OH mole fraction, whilst the reaction rate structure is examined using simultaneous imaging of temperature and mole fractions of OH and CH₂O. Measurements of temperature and mole fractions of CO and OH using the Raman–Rayleigh–LIF-crossed plane OH technique are used to examine the flame thickening and flame reaction rates. It is found that as the shear rates increase, finite-rate chemistry effects manifest through a gradual decrease in reactedness, rather than the abrupt localized extinction observed in non-premixed flames when approaching blow-off. This gradual decrease in reactedness is accompanied by a broadening in the reaction zone which is consistent with the view that turbulence structures become embedded within the instantaneous flame front. Numerical predictions using a particle-based PDF model are shown to be able to predict the measured flames with significant finite-rate chemistry effects, albeit with the use of a modified mixing frequency.