Determination of laminar flame speeds using stagnation and spherically expanding flames: Molecular transport and radiation effects

Combustion and Flame xxx (2014) xxx–xxx

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Combustion and Flame j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m b u s t fl a m e

Determination of laminar flame speeds using stagnation and spherically expanding flames: Molecular transport and radiation effects Jagannath Jayachandran, Runhua Zhao, Fokion N. Egolfopoulos ⇑ Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089-1453, USA

a r t i c l e

i n f o

Article history: Received 19 January 2014 Received in revised form 28 February 2014 Accepted 14 March 2014 Available online xxxx Keywords: Flame propagation Laminar flame speed Experimental uncertainty Flame kinetics

a b s t r a c t The uncertainties associated with the extraction of laminar flame speeds through extrapolations from directly measured experimental data were assessed using one-dimensional direct numerical simulations with focus on the effects of molecular transport and thermal radiation loss. The simulations were carried out for counterflow and spherically expanding flames given that both configurations are used extensively for the determination of laminar flame speeds. The spherically expanding flames were modeled by performing high fidelity time integration of the mass, species, and energy conservation equations. The simulation results were treated as ‘‘data’’ for stretch rate ranges that are encountered in experiments and were used to perform extrapolations using formulas that have been derived based on asymptotic analyses. The extrapolation results were compared then against the known answers of the direct numerical simulations. The fuel diffusivity was varied in order to evaluate the flame response to stretch and to address reactant differential diffusion effects that cannot be captured based on Lewis number considerations. It was found that for large molecular weight hydrocarbons at fuel-rich conditions, the flame behavior is controlled by differential diffusion and that the extrapolation formulas can result in notable errors. Analysis of the computed flame structures revealed that differential diffusion modifies the fluxes of fuel and oxygen inside the flame and thus affect the reactivity as stretch increases. Radiation loss was found to affect notably the extracted laminar flame speed from spherically expanding flame experiments especially for slower flames, in agreement with recent similar studies. The effect of radiation could be eliminated however, by determining the displacement speed relative to the unburned gas. This can be achieved in experiments using high-speed particle image velocimetry to determine the flow velocity field within the few milliseconds duration of the experiment. In general, extrapolations were found to be unreliable under certain conditions, and it is proposed that the raw experimental data in either flame configurations are compared against results of direct numerical simulations in order to avoid potential falsifications of rate constants upon validation. Ó 2014 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

1. Introduction The laminar flame speed, Sou , defined as the propagation speed of a steady, laminar, one-dimensional, planar, adiabatic flame is a fundamental property of any combustible mixture and it is a measure of the mixture’s reactivity, diffusivity, and exothermicity. The accurate knowledge of Sou is essential towards validating kinetic models (e.g., [1]) and constraining uncertainties of rate constants [2]. Furthermore, Sou along with the Markstein length, L, which characterizes the response of laminar flame propagation to stretch, are inputs in turbulent flame models under conditions that the flamelet concept is applicable [3–5]. ⇑ Corresponding author. Fax: +1 213 740 8071.

Measurement of Sou began as early as in the 1920s when Stevens [6,7] studied flame propagation at constant pressure by tracking spherically expanding flames, SEF, in a soap bubble filled with a flammable mixture. Since then, significant progress has been made both in the experimental and numerical determination of Sou . However, notable scatter by as much as 25 cm/s was persistent in published Sou ’s of methane flames [8] until the 1980s when the effect of flame stretch [9] on flame propagation was accounted for and subtracted from the measurements reducing thus the experimental uncertainty notably [10–13]. Despite this progress, due to the relatively low sensitivity of Sou to chemical kinetics [14], there is need for experimental data with even lower uncertainty compared to what is reported currently so that they can be used for kinetic model validation.

E-mail address: [email protected] (F.N. Egolfopoulos). http://dx.doi.org/10.1016/j.combustflame.2014.03.009 0010-2180/Ó 2014 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

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Among the various methods for measuring Sou , the counterflow flame, CFF, and the SEF configurations are well established and widely used, as they are considered to result in reliable data. Despite the fact that considerable effort has been devoted to understanding the intricacies and physics behind each approach, significant discrepancies persist in reported data, even when using the same method. Figure 1 depicts the relative deviation of experimental Sou with a normalized equivalence ratio U  //(1 + /) [15], where / is the equivalence ratio, of n-heptane/air mixtures reported in different studies from the data of Ji et al. [16] that are used as the reference value. One can observe the increasing discrepancy between data obtained using the SEF [17] and CFF [18,19] configurations for off-stoichiometric / > 1 mixtures; corrections of the data reported in Refs. [18,19] to account for the different unburned mixture temperatures, Tu, were made using the recommendation of Wu et al. [20]. It is evident that the disparity between the Sou data sets increases with / and this trend persists for flames of several high molecular weight, MW, fuels [16]. For / > 1 hydrocarbon/air mixtures, air is abundant on both mass and molar basis compared to the fuel. Thus, the thermal diffusivity of the mixture is nearly that of nitrogen, and hence a Lewis number, Le, calculated based on oxygen, being the deficient reactant for / > 1 mixtures, will be close to unity as shown in Fig. 2a regardless of the fuel MW. Yet, a high sensitivity of L to / has been reported, for example, for rich n-butane/air mixtures [21]. These inconsistencies point to possible uncertainties in the experimental determination of Sou , and could be associated with the reactant flow rates, i.e. /, diagnostic equipment, the flow velocity measuring approach, data analysis, and finally data interpretation. In order to tackle uncertainties associated with each experimental approach, detailed understanding of the physics controlling the flame behavior and response to fluid mechanics and loss mechanisms is required. At pressures less than 10 atm, Sou can be measured using the CFF approach in which steady, laminar, and planar flames (e.g., [5,13]) are established. Under such conditions, the only parameter that can be varied for a given set of thermodynamic conditions is the flame stretch, and this effect can be characterized readily using available quasi-one dimensional codes (e.g., [22]). Law and co-workers introduced the CFF approach to determine Sou [5,13,23]. The method involves the determination of the axial velocity profile along the system centerline and subsequently the identification of two distinct observables. A reference flame speed, Su,ref, which is the minimum velocity just upstream of the flame, and a characteristic stretch, K, which is the maximum absolute

Fig. 1. Deviation of experimental Sou ’s of n-heptane/air mixtures at p = 1 atm from that of Ji et al. [16] (Tu = 353 K) represented by the solid line. Data represented by symbols include: ( ) Kelley et al. [17] (Tu = 353 K), ( ) Smallbone et al. [18] (Tu = 350 K) and ( ) Kumar et al. [19] (Tu = 360 K).

Fig. 2. (a) Variation of Le with carbon number for n-alkane/air mixtures at p = 1 atm and Tu = 298 K, for / = 0.7 (dashed line) and / = 1.4 (solid line). (b) Variation of the ratio of oxygen diffusivity to fuel diffusivity with carbon number for / = 1.4 nalkane/air mixtures at p = 1 atm and Tu = 298 K.

value of the axial velocity gradient in the hydrodynamic zone. Thus, by varying Su,ref with K in the experiments, it was proposed [5,13,23] that Sou can be determined by performing a linear extrapolation of the experimental data to zero stretch given that as K ? 0, Su,ref should degenerate to Sou . This approach was used in several studies (e.g., [24–26]) for H2 and C1–C2 hydrocarbon flames. Subsequently, Tien and Matalon [27] demonstrated through asymptotic analysis that the Su,ref vs. K response is non-linear as K ? 0, and that linear extrapolation of Su,ref to K = 0 results in the over-estimation of Sou ; it should be noted that Su,ref is not the stretched flame speed, Su, as it is affected by thermal dilatation and flow divergence effect [13,27,28]. Tien and Matalon [27] produced also a non-linear expression describing the variation Su,ref with K, which subsequently was expressed by Davis and Law [29] in a more compact way as:

Su;ref ¼ Sou f1  ðl  1ÞKa þ Ka ln½ðr  1Þ=Kag:

ð1Þ 2

In Eq. (1), l is the Markstein number, Ka  aK=ðSou Þ the Karlovitz number, a the thermal diffusivity of the mixture, and r  (qu/qb) with qu and qb being the densities of the unburned and burned states at equilibrium respectively. Chao et al. [30] used asymptotic analysis to show that the error introduced by linear extrapolations can be reduced for small Ka and large burner separation distance relative to the flame thickness. Vagelopoulos et al. [31] further showed computationally and experimentally that in order for the linear extrapolation to be accurate, Ka must be of the order of 0.1 for CH4/air, C3H8/air, and lean H2/air flames. Recently, Egolfopoulos and co-workers [16,32,33] introduced a computationally assisted approach in quantifying the non-linear variation of Su,ref with K. Specifically, direct numerical simulations (DNS) of the experiments are carried out with detailed description of molecular transport and chemical kinetics to avoid simplifying assumptions used in asymptotic analysis. Thus, the variation of Su,ref with K is computed and can be used to perform the non-linear extrapolations of the experimental data; indeed the DNS approach reproduces the non-linear behavior of Su,ref with K as predicted by Tien and Matalon [27]. Given that the computed Su,ref vs. K curve may lie over or below the data due to transport and kinetic model uncertainties, it was shown that as long as the discrepancies between data and predictions are not large, say within 30–40%, the shape of the Su,ref vs. K curve is minimally affected and could be translated to best fit the data and derive Sou at K = 0. This was confirmed through DNS in which the rates of main H + O2 ? OH + O branching or CO + OH ? CO2 + H oxidation reactions as well as the diffusion coefficients of the reactants were modified intentionally by as much as 30–40%. It was shown that even under such notable but not excessive modifications of the overall reaction

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rate, the shape of the computed Su,ref vs. K curves are nearly indistinguishable [32]. Ji et al. [16] showed that for the same sets of experimental data of C5–C12 n-alkanes, linear extrapolation yield higher Sou ’s for fuel rich mixtures, as compared to nonlinear extrapolation using the computationally assisted approach. Considering also the results shown in Fig. 1, it is reasonable to assume that the discrepancies between reported Sou ’s for / > 1 mixtures could be attributed partially to the extrapolations. The SEF approach has been used extensively for measuring Sou due to the wide pressure range of applicability, the relative simplicity of diagnostics, and the well defined stretch rate that makes the determination of the burned gas Markstein length, Lb, straightforward (e.g., [34–40]). The commonly used method involves tracking using Schlieren or shadowgraph the flame radius, Rf, of the expanding flame as a function of time, t, [36] during the initial phase of propagation during which the pressure rise is negligible. The flame speed with respect to the burned gas is defined as Sb  dRf / dt, based on the assumption that the burned gas is stationary, and the flame stretch is defined as K  (2/Rf)(dRf/dt) (e.g., [17]). During one experiment, the variation of Sb with K is monitored and the through extrapolation to K = 0 the stretch free Sob value is determined. Subsequently, Sou is determined through the density correction as Sou  Sob ðqb =qu Þ. The majority of the Sou ’s reported in the literature (e.g., [36,37,39,40]) were extracted using linear extrapolations to determine Sob [36]. Kelley and Law [21] identified that for mixtures with Le – 1.0 and Ka relevant to experiments, Sb varies non-linearly with K and proposed a quasi-steady non-linear extrapolation equation, which was derived originally by Ronney and Sivashinksy [41] for flames of mixtures sufficiently far from stoichiometry, constant transport properties, and one-step reaction:



Sb =Sob

2

 2 ln Sb =Sob ¼ 2Lb K=Sob

ð2Þ

Subsequently, Kelley et al. [42] relaxed the quasi-steady assumption and proposed an improved non-linear extrapolation formula; a review of all extrapolation methodologies can be found in Refs. [42,43]. Uncertainties associated with prolonged effects of ignition during the initial stages of flame propagation [38,44] and fluid dynamic effects induced by using a cylindrical rather than a spherical chamber [45] have been studied and accounted for. Mclean et al. [46] carried out a numerical study and determined that the radiation losses in SEF’s can cause a systematic underestimation of Sou . It has been established also from previous studies that radiation affects flame propagation notably only for mixtures at near limit conditions [47]. Chen [48] performed DNS of nearlimit CH4/air mixtures to investigate the effects of radiation loss and reabsorption. It was shown that radiation heat loss from the burned gas could cause an inward flow and that accounting for reabsorption moderates the overall heat loss from the burned gas. Santner et al. [49] showed that Sob of slow flames could be affected significantly by the burned gas cooling. In the same study a correction methodology to obtain accurate Sob values was proposed based on an analytical model for fluid flow coupled with an optically thin limit (OTL) model for radiation. However, this correction method has to be used with caution, especially for highpressure flames for which reabsorption can be important (e.g., [50,51]). Lecordier and coworkers [52,53] performed for the first time direct measurements of the flow velocities in SEF experiments by seeding the flow with silicon oil droplets and using kHz-level particle image velocimetry (PIV). Thus, the displacement velocity with respect to the unburned gas, Un, was determined as Un  Ug  Sb [34], where Ug is the maximum velocity upstream of the flame.

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In these experiments Sb and K are determined readily once the temporal variation of Rf is known; Rf is defined as the location at which the droplets have been vaporized completely. Sou is then obtained by extrapolating Un to K = 0. This approach provides direct measurements and does not require the use of the density correction to obtain Sou , which introduces the questionable assumption of equilibrium in the burned gas that in reality is affected by radiation and its density varies both in time and space. Renou and coworkers (e.g., [54]) extended this technique to flames of liquid fuels and demonstrated that discrepancies exist between Sou values measured using PIV and those determined using the Schlieren or shadowgraph approach. Based on the aforementioned considerations, the main goal of this investigation was to perform DNS of CFF and SEF configurations and assess uncertainties in determining Sou under realistic experimental conditions. The emphasis was on transport and radiation effects. While Le effects have been studied extensively in past pertinent studies, this is not the case for reactant differential diffusion that can be rather important for large MW fuels as evident from Fig. 2b in which the ratio of oxygen to fuel diffusivities is shown to increase with the fuel carbon number for / = 1.4 nalkane/air mixtures. The effect of radiation was assessed also for SEF’s in the context of the various approaches available for deriving the raw experimental data.

2. Numerical approach 2.1. Freely propagating and counterflow flames In order to assess the validity of current practices in determining Sou using the CFF and SEF approaches, DNS of both types of experiments were performed using a variety of codes and detailed description of chemical kinetics and molecular transport. The DNS results were treated as ‘‘data’’ for the range of K’s that are typically used in both types of experiments, and subsequently Eqs. (1) and (2) were used to perform extrapolations. The advantage of this approach is that both the response of flame propagation to K from high to near-zero values and Sou are known so that the merits and shortcomings of Eqs. (1) and (2) can be assessed. Furthermore, the DNS approach allows for the rigorous assessment of reactant differential diffusion effects. A parametric study was performed on the effect of the fuel diffusivity on the response of CH4/air flames to K given the relatively small size of the kinetic model, essential for unsteady DNS of SEF’s, and the fact that diffusivities of CH4 and O2 do not differ substantially. The variation of the CH4 diffusivity was implemented through modification of its Lennard-Jones (L-J) parameters. The unperturbed case is referred to as OD (original diffusivity). ID (increased diffusivity) and DD (decreased diffusivity) refer to the cases in which the L-J parameters of CH4 were replaced with those of H2 and n-C12H26 respectively. This approach ensures that the chemistry is consistent in all computations, and also circumvents the complexities associated with fuel cracking which high MW fuels are susceptible to. The values of Le and ratio of fuel to oxygen diffusivities in the mixture are shown in Table 1 for / = 0.7, 1.0 and 1.4. DNS were performed also for steady n-C12H26/air CFF’s in order to verify the results obtained from CH4/air flames. Sou ’s and variation of Su,ref with K for CFF’s were computed respectively, using the PREMIX code [55,56] and an opposed-jet flow code [57] that was originally developed by Kee and co-workers [22]. Both codes were integrated with the CHEMKIN [58] and the Sandia transport [59,60] subroutine libraries. The H and H2 diffusion coefficients of several key pairs are based on the recently updated set [61]. Both codes have been modified to account for thermal radiation (OTL) of CH4, CO, CO2, and H2O [57,62].

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Table 1 Lewis number, Le and ratio of fuel to O2 diffusivities for the mixtures used in the present study. /

Le

 Dfuel DO2

Original L-J Parameters (OD) 0.7 1.0 1.4

1.0 N/A 1.1

1.14 1.16 1.17

n-C12H26 L-J Parameters (DD) 0.7 1.0 1.4

2.3 N/A 1.0

0.45 0.48 0.51

H2 L-J Parameters (ID) 0.7 1.0 1.4

0.7 N/A 1.2

1.67 1.67 1.68

Sou ’s and the variation of Su,ref with K in CFF’s were computed using the USC-Mech II [63] and JetSurF 1.0 [64] kinetic models for CH4/air and n-C12H26/air flames respectively. The CFF simulations were performed for the twin flame configuration and for a large burner separation distance (10 cm) to avoid conductive heat loss to the burner at very low K’s.

2.2. Spherically expanding flames In order to perform one-dimensional DNS of SEF’s, a transient one-dimensional reacting flow code (TORC) was developed using the PREMIX code [55,56] as the framework. The conservation equations for mass, species, and energy are solved numerically as a function of time in spherical coordinates [38]. The method of lines approach was adopted to solve the stiff non-linear system of equations, which involves the replacement of spatial derivatives with discrete difference approximations, relying on an ordinary differential equation (ODE) solver to perform the time integration. This simplifies the method of solution as the ODE solver takes the burden of time step selection to maintain stability and local error control of the evolving solution. Finite difference approximations were used for spatial discretization. A first order windward scheme was used for the convective terms and a second order central difference scheme for the diffusion terms, similar to the PREMIX code [55,56]. The set of discretized equations form a differential algebraic equation (DAE) system of index 1 [65], with velocity being the only algebraic variable among the primitive variables that are temperature, T, velocity, u, and species mass fractions, Y. Time integration of this DAE system was performed using the DASPK [66–69] solver, which implements a backward-difference formula with adaptive time step and order control. An adaptive grid methodology was developed as computational efficiency is severely compromised if a static mesh is used for a moving flame problem. The spatial domain was divided into five regions of different grid point densities (L1–L5), each having uniform grid spacing. The algorithm ensures that the flame is always located within the region of highest mesh point density (L1) and that the other regions are distributed around the L1 such that the furthest region (L5) will have the least mesh density. Grid restructuring was performed every time the flame separation from the L1 boundary was within a user defined tolerance. In order to overcome the computational overhead of restarting DASPK every time a re-gridding process was completed, a ‘‘flying/warm restart’’ was facilitated by interpolating the solutions at previous time steps on to the new grid [70–72]. Thus, the solver can continue integrating using the higher order multistep method and/or using a larger time step. The restructuring of the mesh was done in such a way

that the grid points located in the flame remain intact to avoid interpolation errors. In order to ensure zero gradients upstream of the flame, R2u where R is the spatial coordinate was chosen as the dependent variable instead of u. This facilitates further reduction of interpolation errors. A monotone cubic Hermite interpolation [73] technique as suggested by Hyman et al. [72] was found to work the best. DASPK requires the initial condition to satisfy the governing DAE system of equations [69]. To obtain the initial condition, the transient terms were discretized using the backward Euler method and the system of equations were integrated in time over a couple of small time steps (depending on the time scales of the problem) using a modified Newton method implemented in the TWOPNT [74] solver. In order to validate TORC, simulations of planar flames were performed and the results were compared against those obtained using PREMIX [55,56]. The kinetic, thermodynamic, transport and radiation calculations were performed similarly to PREMIX [55,56]. In order to reduce the computational cost, DNS of SEF’s were performed only for CH4/air flames and employed two models obtained from USCMech II using the DRG reduction strategy [75]. The reduction was done using an array of PREMIX solutions for lean and rich mixtures separately. For / = 0.7 and 1.0 a model consisting of 17 species and 78 reactions was used, while for / = 1.4 the reduced model included 24 species and 137 reactions. A domain of radius Re = 25 cm was used in the simulations. At t = 0, a stagnant pocket of hot burned gases of radius 2.5 mm surrounded by the unburned combustible gas mixture was used to achieve ignition. It should be noted that while Re = 25 cm is of no relevance to the actual experimental conditions it is large enough radius at which near-zero stretch are reached and at which the extrapolations are carried out. At t > 0 the boundary conditions are:

    dT dY K ¼ 0; uR¼0 ¼ 0; ¼ 0; dR dR R¼0  R¼0   dT dY K ¼ 0; ¼ 0; dR R¼Re dR R¼Re where YK is the mass fraction of species K. Note that only one boundary condition for u is allowed and is specified at the center of the domain. The maximum heat release rate was used as a marker to track Rf as a function of t. Sb is obtained by differentiating a 3rd order polynomial used to fit locally the variation of Rf with t, and as mentioned earlier, K  (2/Rf)(dRf/dt).

3. Results and discussion 3.1. Differential diffusion effects for CFF’s and SEF’s Figure 3 depicts the variation of Sou with / for CH4/air mixtures for various CH4 diffusivities, DCH4 , while in Fig. 4 the logarithmic sensitivity coefficients of Sou to the CH4–N2 and O2–N2 binary diffusion coefficients are shown. Results indicate that the modification of DCH4 has an opposite effect on Sou for / < 1.0 and / P 1.0. Details of the flame structure are shown in Fig. 5, and it can be seen that a change in DCH4 results in a corresponding change in its diffusion length relative to O2. The diffusion length of CH4 for a / = 1.4 CH4/air flame computed with DD (Fig. 5b) is reduced compared to the OD case (Fig. 5a). Thus, Y CH4 and the local equivalence ratio, /local, increase at the location at which the CH4 consumption initiates as shown in Fig. 6, which results in the reduction of reactivity, shown also in Fig. 6. Similar analysis can be used to explain the dependence of Sou on DCH4 for all mixtures shown in Table 1. Furthermore, it is of interest to note that the dependence of Sou

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Fig. 3. Computed Sou ’s of CH4/air flames at p = 1 atm and Tu = 298 K using USC-Mech II. ( ) OD; ( ) DD; ( ) ID.

Fig. 4. Logarithmic sensitivity coefficients of Sou to the CH4–N2 (black) and O2–N2 (grey) binary diffusion coefficients for CH4/air flames at p = 1 atm, Tu = 298 K, and various /’s.

on DCH4 is not captured by the following equation that is based on Le considerations [76]:

pffiffiffiffiffi Sou ðLe – 1Þ ¼ Sou ðLe ¼ 1Þ Le

ð3Þ

The effect of reactant differential diffusion on the propagation of stretched flames was assessed in the CFF configuration. Figures 7– 9 depict the variation of Su;ref =Sou with Ka for / = 0.7, 1.0, and 1.4 mixtures respectively. These figures include also the extrapolation curves using Eq. (1) that fit the DNS results for a range of Ka that

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Fig. 6. Variation of /local with temperature in a / = 1.4 freely propagating CH4/air flame at p = 1 atm, and Tu = 298 K computed using USC-Mech II with OD ( ) and DD ( ), and variation of CH4 consumption rate with temperature with OD ( ) and DD ( ).

are representative of those used in experiments (e.g., [27,29]). Using OD, Eq. (1) predicts closely the DNS results. As DCH4 starts deviating from the oxygen diffusivity, DO2 , for the ID and DD cases, a discrepancy is observed between the extrapolated Sou from its known value by as much as 5% for / = 0.7 with ID and 30% for / = 1.4 with DD. A summary of extrapolation errors resulting from linear and asymptotically derived nonlinear methods is reported in Table S1 in the supplementary material. From Figs. 7–9 it is apparent also that there is a significant change in slope of the Su;ref =Sou vs. Ka curve when DCH4 is modified for the / = 0.7 and 1.4 mixtures. In CFF’s, it is not possible to monitor the modification of the burning intensity with K by simply tracking the variation of Su,ref with K, as Su,ref is affected also by thermal dilatation and flow divergence [13,27,28]. On the other hand, the burning intensity is best described by the total heat release rate per unit area, HRRtot, obtained by integrating the heat release rate over the entire flame. Figures 10–12 depict the variation of HRRtot with Ka for / = 0.7, 1.0 and 1.4 mixtures respectively. The results for the / = 0.7 mixture shown in Fig. 10 can be explained based on Le – 1.0 effects caused by the imbalance of energy loss from and energy gain by the reaction zone [76]. For the / = 1.4 mixture however, even though Le  1.0 for all hydrocarbons, a substantial increase in HRRtot with Ka is seen for the DD

Fig. 5. (a) Normalized mass fraction profiles of CH4 ( ) and O2 ( ), and CH4 consumption rate profile (—) for a / = 1.4 freely propagating CH4/air flame at Tu = 298 K and p = 1 atm, computed using USC-Mech II and OD. (b) Normalized mass fraction profiles of CH4 ( ) and O2 ( ), and CH4 consumption rate profile (—) for a freely propagating flame at Tu = 298 K, p = 1 atm, and / = 1.4, computed using USC-Mech II and DD.

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Fig. 7. Variation of Su;ref =Sou with Ka of a / = 0.7 CH4/air CFF at p = 1 atm and Tu = 298 K computed using USC-Mech II with ID ( ), OD ( ), and DD ( ). ID ( ), ) correspond to fitting using Eq. (1). The full range DNS results OD ( ), and DD ( are shown in hollow symbols, while the DNS results used for fitting Eq. (1) are shown in solid symbols.

Fig. 8. Variation of Su;ref =Sou with Ka of a / = 1.0 CH4/air CFF at p = 1 atm and Tu = 298 K computed using USC-Mech II with ID ( ), OD ( ), and DD ( ). ID ( ), OD ( ), and DD ( ) correspond to fitting using Eq. (1). The full range DNS results are shown in hollow symbols, while the DNS results used for fitting Eq. (1) are shown in solid symbols.

Fig. 10. Variation of HRRtot with Ka of a / = 0.7 CH4/air CFF at p = 1 atm and Tu = 298 K computed using USC-Mech II with ID ( ), OD ( ), and DD ( ).

Fig. 11. Variation of HRRtot with Ka of a / = 1.0 CH4/air CFF at p = 1 atm and Tu = 298 K computed using USC-Mech II with ID ( ), OD ( ), and DD ( ).

Fig. 12. Variation of HRRtot with Ka of a / = 1.4 CH4/air CFF at p = 1 atm and Tu = 298 K computed using USC-Mech II with ID ( ), OD ( ), and DD ( ). Fig. 9. Variation of Su;ref =Sou with Ka of a / = 1.4 CH4/air CFF at p = 1 atm and Tu = 298 K computed using USC-Mech II with ID ( ), OD ( ), and DD ( ). ID ( ), OD ( ), and DD ( ) correspond to fitting using Eq. (1). The full range DNS results are shown in hollow symbols, while the DNS results used for fitting Eq. (1) are shown in solid symbols.

case for which there is a notable difference between DCH4 and DO2 . Thus, the diffusion rate of O2 towards the reaction zone increases compared to CH4 with increasing K, making thus the mixture more stoichiometric and increasing the overall reactivity [76]. For the /

= 1.0 mixture the slope of HRRtot with Ka does not change for the different DCH4 values. This is due to the fact that for near-stoichiometric mixtures there is a minor sensitivity of the overall reactivity to modifications in / as it reaches a maximum value. Figures 13 and 14 depict the variations of /local and the consumption rate of CH4 for a / = 1.4 flame at K = 30 and 200 s1 respectively and computed with OD and DD. The results confirm that as K increases, /local decreases at the locations at which the CH4 consumption begins. As a result, there is a notable increase

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Fig. 13. Variation of /local with temperature in a / = 1.4 CH4/air CFF at p = 1 atm, Tu = 298 K, and K = 30 s1 computed using USC-Mech II with OD ( ) and DD ( ), and variation of CH4 consumption rate with temperature with OD ( ) and DD ( ).

Fig. 14. Variation of /local with temperature in a / = 1.4 CH4/air CFF at p = 1 atm, Tu = 298 K, and K = 200 s1 computed using USC-Mech II with OD ( ) and DD ( ), and variation of CH4 consumption rate with temperature with OD ( ) and DD ( ).

of the CH4 consumption rate as K increases for the DD case compared to OD. More specifically, the maximum CH4 consumption rate is about 40% higher for the DD case for K = 30 s1, and by a factor of 3.5 for K = 200 s1. These results reveal the basis physics that control the dependence of the overall flame reactivity with stretch for rich mixtures of high MW fuels and which need to be accounted for when raw experimental data are interpreted to determine nondirectly measured properties such as Sou . n-C12H26/air CFF’s were computed also in order to verify the findings for CH4/air flames. The diffusivity of n-C12H26/air was modified also by using the L-J parameters of CH4 and this case is referred to as ID given that n-C12H26 becomes more diffusive. Figures 15 and 16 depict the variation of Su;ref =Sou with Ka for / = 0.7 and 1.4 respectively and the behavior is consistent with that observed for CH4/air flames. The / = 0.7 flame computed with OD mixture exhibits lower Su,ref values and the flame extinguishes at lower K compared to the ID case as shown in Fig. 15. Furthermore, the use of Eq. (1) in both OD and ID cases results in Sou ’s that are close to its known value. On the other hand, for the / = 1.4 flame, using Eq. (1) results in the over-prediction of the known Sou value by 9% and 4% for the OD and ID cases respectively. The variations of the peak local equivalence ratio, of (/local)peak and HRRtot are shown in Fig. 17 for a / = 1.4 n-C12H26/air flame computed with

7

Fig. 15. Variation of Su;ref =Sou with Ka of a / = 0.7 n-C12H26/air CFF at p = 1 atm and Tu = 443 K computed using JetSurF 1.0 with ID ( ) and OD ( ). ID ( ) and OD ( ) correspond to fitting using Eq. (1). The full range DNS results are shown in hollow symbols, while the DNS results used for fitting Eq. (1) are shown in solid symbols.

Fig. 16. Variation of Su;ref =Sou with Ka of a / = 1.4 n-C12H26/air CFF at p = 1 atm and Tu = 443 K computed using JetSurF 1.0 with ID ( ) and OD ( ). ID ( ) and OD ( ) correspond to fitting using Eq. (1). The full range DNS results are shown in hollow symbols, while the DNS results used for fitting Eq. (1) are shown in solid symbols.

Fig. 17. Variation of (/local)peak ( ) and HRRtot ( ) with Ka of a / = 1.4 n-C12H26/air CFF at p = 1 atm and Tu = 443 K computed using JetSurF 1.0 with OD.

OD. Similarly to CH4/air flames computed with DD, (/local)peak decreases and HRRtot increases as Ka increases given that the flame becomes more stoichiometric. CH4/air SEF’s were computed for a wide range of Rf’s and the computed Sb values for 1 cm < Rf < 3 cm were used for performing extrapolation using Eq. (2). This Rf range is used typically in experiments (e.g., [39,54]) so that the data are not affected by the

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Fig. 18. Variation of Sb =Sob with Ka of a / = 0.7 CH4/air SEF (ADB) at p = 1 atm and Tu = 298 K computed using a reduced USC Mech II with ID ( ), OD ( ), and DD ( ). ID ( ), OD ( ), and DD ( ) correspond to fitting using Eq. (2). The full range DNS results are shown in hollow symbols, while the DNS results used for fitting Eq. (2) are shown in solid symbols.

Fig. 19. Variation of Sb =Sob with Ka of a / = 1.0 CH4/air SEF (ADB) at p = 1 atm and Tu = 298 K computed using a reduced USC Mech II with ID ( ), OD ( ), and DD ( ). ID ( ), OD ( ), and DD ( ) correspond to fitting using Eq. (2). The full range DNS results are shown in hollow symbols, while the DNS results used for fitting Eq. (2) are shown in solid symbols.

ignition energy and pressure rise [38,45]. The simulations were performed for adiabatic flames (ADB), i.e. without considering radiation, with ID, OD, and DD similarly to the CFF’s, and the variation of Sb =Sob with Ka is shown in Figs. 18–20 for / = 0.7, / = 1.0, and / = 1.4 respectively. With DD, the extrapolation error in Sob is approximately 6% for the lean and rich flames. Extrapolations of the / = 1.4 results obtained with OD and ID result in errors of 4.5% and 10% respectively. It is also of interest to note that for flames with positive Lb, the variation of Sb with Ka is nearly linear despite the high sensitivity of Sb to Ka, and that Eq. (2) always generates a highly non-linear curve for such flames especially when the Sb vs. K slope is steep. In general, mixtures with Le – 1.0 and/or notable differences between the fuel and O2 diffusivities, the magnitude of the extrapolation error increases. Other than the aforementioned conditions, the extrapolation error was determined to be in general below 3%. For SEF’s, Sb represents the flame propagation speed with respect to the stationary burned gas at adiabatic conditions. The variation of HRRtot with Ka is shown in Fig. 21 for / = 1.4 and it can be seen that the behavior computed using ID, OD, and DD is similar to that of Sb =Sob with Ka shown in Fig. 20 indicating that Sb is a good indicator of the overall burning intensity, similar to the results obtained for CFF’s. Figure 22 depicts the variation of Sb =Sob with Ka for / = 0.7, / = 1.0, and / = 1.4 computed with DD that is representative of high MW hydrocarbons, and the results are consistent with

Fig. 20. Variation of Sb =Sob with Ka of a / = 1.4 CH4/air SEF (ADB) at p = 1 atm and Tu = 298 K computed using a reduced USC Mech II with ID ( ), OD ( ), and DD ( ). ID ( ), OD ( ), and DD ( ) correspond to fitting using Eq. (2). The full range DNS results are shown in hollow symbols, while the DNS results used for fitting Eq. (2) are shown in solid symbols.

Fig. 21. Variation of HRRtot with Ka of a / = 1.4 CH4/air SEF at p = 1 atm and Tu = 298 K computed using a reduced USC-Mech II with ID ( ), OD ( ), and DD ( ).

Fig. 22. Variation of Sb =Sob with Ka of CH4/air SEF’s (ADB) at p = 1 atm and Tu = 298 K computed using a reduced USC Mech II with DD for / = 0.7 ( ), / = 1.0 ( ), and / = 1.4 ( ).

experimental results of n-butane/air mixtures [21]. More specifically, the sign of Lb for heavy hydrocarbons changes from positive for / < 1.0 to negative for / > 1.0. For / < 1.0 mixtures of high MW hydrocarbons the flame response is controlled by Le effects, whereas for / > 1.0 differential diffusion is the controlling factor. Kelley et al. [42] used asymptotic analysis to account for differential diffusion effects, and they showed that there is a non-monotonic variation of the flame speed with stretch for near stoichiometric mixtures with contrasting fuel and oxygen diffusivities. Using the formulation obtained in Ref. [42] for extrapolations to obtain accurate flame speeds was not feasible due to the large

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Fig. 23. Temperature profiles for a / = 1.0 CH4/air SEF at p = 5 atm, Tu = 298 K, and Rf = 8 cm computed using a reduced USC Mech II with ADB ( ), OTL ( ), and HOTL ( ).

number of parameters that have to be determined by fitting the extrapolation equation to the experimental data. Hence all extrapolation equations were obtained by invoking the assumption that the reactant differential diffusion effect is negligible (for off-stoichiometric mixtures) and that Le solely governs the flame dynamics. In the present study it was shown that this assumption is not valid for rich mixtures of large MW hydrocarbons and thus using such an equation to extrapolate experimental data can lead to notable errors.

3.2. Radiation effects for SEF’s Figure 23 depicts the spatial temperature profiles for a / = 1.0 CH4/air SEF at p = 5 atm that has propagated to Rf = 8 cm. Recall, that the adiabatic and optically thin limit cases are represented by ADB and OTL respectively. In order to account for potential re-absorption, additional simulations were carried out by using half of the Planck’s mean absorption coefficient represented by HOTL, which is in a way equivalent to reabsorption of half the energy emitted by the burned gas. It is apparent that the presence of radiation results in a notable reduction of the temperature of the burned gases and the assumption of equilibrium does not hold. The spatial variation of the gas velocity for the conditions of Fig. 23 is shown in Fig. 24. The radial inward flow, observed as negative velocities, is a result of density change in the burned gas due to radiative heat loss. It is seen that the extent of this inward flow

Fig. 25. Variation of Sb with K of a / = 1.0 CH4/air SEF at p = 1 atm and Tu = 298 K computed using a reduced USC Mech II with ADB ( ), OTL ( ), and HOTL ( ). ADB ( ), OTL ( ), and HOTL ( ) correspond to fitting using Eq. (2). The full range DNS results are shown in hollow symbols, while the DNS results used for fitting Eq. (2) are shown in solid symbols.

is reduced with reabsorption, e.g. HOTL, as the total heat loss is reduced, which is consistent with the findings of Chen [48]. Figure 25 illustrates the variation of Sb with K, and the discrepancies that can be induced due to extrapolations to K = 0 are apparent. As expected, in the presence of radiation Sb is reduced and as a result the effect of inward flow is augmented at large radii, which is evident also by the equation derived by Santner et al. [49]. However, the reported results for OTL and HOTL may not be entirely physical, given that at very large radii some of the lost energy may be reabsorbed due to large optical thickness. Nevertheless, caution is recommended when SEF raw data are interpreted as Santner et al. [49] has suggested also. Table 2 illustrates the errors induced when the Sb data derived using OTL are used to extract Sob for CH4/air mixtures at various conditions. The percentage difference in Sou calculated using the PREMIX code [55,56] between the OTL and ADB conditions represents the difference that should be obtained after extrapolating Sb at the corresponding conditions to obtain Sob , as they are linearly related through the continuity equation. Results show that the error due to cooling of the burned gas increases for off-stoichiometric mixtures and at high pressures, in agreement with the findings of Santner et al. [49] in that the slower flames are affected most by the radiative heat loss given that more time is available for the burned gases to cool during the duration of the experiment. The alternative approach of Lecordier and coworkers [52,53] was evaluated also using the computed SEF structures from which in addition to Sb the values of Ug were extracted as well so that the displacement velocity Un  Ug  Sb [34] can be evaluated. Figure 26 compares the variations of Sb and Un with K, for the ADB, OTL, and HOTL cases. It is of interest to note that while radiation has a major effect on Sb, its effect on Un is minor as its values computed in all three cases collapse in a single curve. Mathematically, radiation Table 2 Differences in Sou ’s between the ADB and OTL cases as obtained from freely propagating flame simulations, and in Sob ’s obtained by extrapolating the SEF results for the ADB and OTL cases.

Fig. 24. Velocity profiles for a / = 1.0 CH4/air SEF at p = 5 atm, Tu = 298 K, and Rf = 8 cm computed using a reduced USC Mech II with ADB ( ), OTL ( ), and HOTL ( ).

/

Pressure (atm)

% Difference between OTL & ADB Sou ’s from freely propagating flame simulations

% Difference between OTL & ADB extrapolated Sob ’s from SEF data

0.7 1.0 1.4 0.8 1.0

1 1 1 5 5

0.4 0.2 2.4 0.1 0.1

4.0 2.0 6.2 6.0 4.6

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Fig. 26. Variation of Sb (solid) and Un (hollow) with K of a / = 1.0 CH4/air SEF at p = 5 atm and Tu = 298 K computed using a reduced USC Mech II with ADB ( ), OTL ( ), and HOTL ( ).

affects similarly Sb and Ug through the inward flow, so that its effect is subtracted out. From a physical point of view this result is reasonable, as Un is a measure of the stretched flame speed with respect to the unburned gas, which is not affected from radiation for mixtures that are not close to the flammability limits [47]. Radiation calculations involving spectrally resolved emission and absorption are computationally expensive and prone to uncertainty due to simplifying assumptions required to make the numerical simulation feasible [77]. Therefore, by measuring Un the complications associated with accurate reabsorption calculations needed to model the experimental data are circumvented, and enables using the OTL model to simulate SEF’s to reasonable accuracy for mixtures which have limited overlap of spectral bands with the burned products and are far from the flammability limit. Any approach in SEF’s is not applicable for near-limits for which Sou can be of the order of 10 cm/s or less [47], given that buoyancy will dominate the flame behavior and will cause severe distortion of the flame surface in experiments. In order to extrapolate the experimental Un data to obtain Sou , Renou and coworkers [54] used the formula of Kelley and Law [21] derived for the unburned flame propagation speed, defined as the flow velocity relative to the flame upstream of the preheat zone where the temperature rise is negligible. Figure 27 depicts Un as a function of Ka for / = 1.4 CH4/air mixtures computed with ID, OD and DD, the cases examined in Fig. 20 along with the extrapolation curves obtained using the formula of Kelley and Law [21]. It is evident from Figs. 20 and 27 that the trends of variation of Sb and Un with Ka are different and that Un unlike Sb and HRRtot (Fig. 21) is not a proper indicator of the burning intensity. Extracting the unburned flame propagation speed from an unsteady flame in a non-planar flow geometry in not trivial. Dixon-Lewis and Islam [28] simulated a planar steady flame in a quasi-1D diverging flow geometry and showed that the flame speed can be obtained by density correction of the flow velocity at the location of peak reaction rate. It became evident also from the simulations in the current study that this point of maximum velocity (Ug), used to compute Un, is influenced by thermal dilatation as shown in Fig. 28 which is a plot depicting the variation of temperature at the location of maximum velocity with stretch. These observations are indicative of the fact that Un, obtained from Ug, is not the actual unburned stretched flame speed, but a reference flame speed similarly to CFF’s. Using the equation of Kelley and Law [21], which does not account for thermal dilatation and geometric effects, to extrapolate Un could result in substantial error in the extrapolated value of Sou . This increased uncertainty is evident from the difference in extrapolation errors of Un and Sb

Fig. 27. Variation of U n =Sou with Ka for a / = 1.4 CH4/air SEF at p = 1 atm and Tu = 298 K computed using a reduced USC Mech II with ID ( ), OD ( ), and DD ( ). ), OD ( ), and DD ( ) correspond to fitting using the equation derived by ID ( Kelley and Law [21]. The full range DNS results are shown in hollow symbols, while the DNS results used for fitting the extrapolation equation are shown in solid symbols.

Fig. 28. Variation of temperature at location of maximum velocity (Ug) with stretch for a / = 1.4 CH4/air SEF at p = 1 atm and Tu = 298 K.

depicted by Figs. 27 and 20 respectively and the consistent over prediction of the known Sou values in all cases. 3.3. Experimental uncertainties Compared to the DNS results presented in this study, experimental data exhibit larger uncertainty and/or scatter due to several factors. Clearly, in steady state experiments like those of CFF’s the directly measured Su,ref’s can be optimized so that the uncertainties are minimized. In carefully performed CFF experiments, the uncertainty based on 2r, where r is the standard deviation, can be as low as 5% [16]. However, the uncertainty of CFF experiments can be 10% or higher if issues related to the quality of the flow, reactant concentrations especially for / > 1.0 flames of liquid fuels, flow tracer seeding density, the implementation of particle image velocimetry (PIV) or laser Doppler velocimetry (LDV) to measure flow velocities, and interpretation of the raw data are not addressed carefully and rigorously. Performing SEF experiments is by far more challenging. First, in static experiments the equivalence ratio uncertainties can be greater compared to steady state experiments. Second, the experiments last only few milliseconds making thus the implementation of any type of diagnostics with satisfactory temporal resolution challenging. Uncertainties in SEF experiments are expected to be greater for very slow (buoyancy) or very fast (time resolution)

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flames as well as for higher pressures in which flames tend to be unstable. It should be noted that uncertainties in the reported Sou of the order of 10% or higher are not desirable as such data cannot be used effectively for the validation of kinetic models given the relatively low sensitivity of Sou to kinetics. An alternative viable approach in validating kinetic models is to compare the raw experimental data from either CFF or SEF experiments against corresponding DNS results, so that the uncertainties associated with extrapolations are removed.

4. Concluding remarks Direct numerical simulations of counterflow and spherically expanding flames were carried out in order to assess uncertainties stemming from current practices that are used to interpret experimental data and derive the laminar flame speed. The analysis focused on the effects of molecular transport and thermal radiation. The counterflow and freely propagating flames were simulated using established codes. A new code was developed in order to simulate spherically expanding flames by integrating accurately in time the one-dimensional transient conservation equations in spherical coordinates. The results of the simulations were treated as data in the range of stretch rates that are encountered in experiments, and were used to perform extrapolations to zero stretch using formulas that have been derived from asymptotic analyses. The validity of these practices was tested upon comparing the results against the known answers of the direct numerical simulations. The effect of molecular transport was studied by varying the fuel diffusivity. It was concluded that for fuel lean hydrocarbon/ air mixtures, the preferential diffusion of heat or mass as manifested by the Lewis number dominates the flame response to stretch. For fuel rich mixtures, the controlling factor was determined to be the differential diffusion of the reactants into the reaction zone for heavy hydrocarbons. It was found also that using extrapolation equations derived based on asymptotics analysis and simplifying assumptions to obtain the laminar flame speeds, could result in significant errors for rich flames of heavy hydrocarbons. Numerical simulations of spherically expanding flames with radiative heat loss revealed that the standard approach of measuring the flame propagation speed relative to the burned gas using the shadowgraph/Schlieren techniques, could result in a systematic under-prediction of the true laminar flame speed due to an inward flow induced by the density change in the burned gas. Furthermore, it was shown that by simultaneously measuring the maximum velocity upstream of the flame and the burned flame speed and evaluating thus the displacement speed relative to the fresh gases, this error could be avoided. It was determined that there is a negligible effect of the density change in the burned gas due to radiation on the displacement speed relative to the fresh gas.

Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.combustflame. 2014.03.009. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52]

Acknowledgments This material is based upon work supported as part of the CEFRC, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, and Office of Basic Energy Sciences under Award Number DE-SC0001198. The authors would like to thank Dr. Linda Petzold and Dr. Shengtai Li for providing subroutines to implement the ‘‘warm restart’’ for the DASPK solver.

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