The Response of Transient Inhomogeneous Flames to Pressure Fluctuations and Stretch: Planar and...

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The Response of TransientInhomogeneous Flames to PressureFluctuations and Stretch: Planar andOutwardly Propagating Methane/AirFlamesNadeem A. Malik a b & R. P. Lindstedt ba Department of Mathematics and Statistics, King Fahd University ofPetroleum and Minerals, Dhahran, Kingdom of Saudi Arabiab Department of Mechanical Engineering, Imperial College London,London, EnglandAccepted author version posted online: 18 Jul 2012.Version ofrecord first published: 01 Nov 2012.

To cite this article: Nadeem A. Malik & R. P. Lindstedt (2012): The Response of TransientInhomogeneous Flames to Pressure Fluctuations and Stretch: Planar and Outwardly PropagatingMethane/Air Flames, Combustion Science and Technology, 184:10-11, 1799-1817

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THE RESPONSE OF TRANSIENT INHOMOGENEOUSFLAMES TO PRESSURE FLUCTUATIONS ANDSTRETCH: PLANAR AND OUTWARDLY PROPAGATINGMETHANE/AIR FLAMES

Nadeem A. Malik1,2 and R. P. Lindstedt21Department of Mathematics and Statistics, King Fahd University ofPetroleum and Minerals, Dhahran, Kingdom of Saudi Arabia2Department of Mechanical Engineering, Imperial College London, London,England

Transient outwardly propagating premixed methane/air flames subjected to joint pressure

and equivalence ratio oscillations are investigated using an implicit direct method that

couples the compressible flow with realistic chemistry and multicomponent transport

properties. Chemistry is a detailed 30 species C1–C2 mechanism. The chemical kinetic–

transport coupling is compared with results from hydrogen–air flames obtained in earlier

work (Malik and Lindstedt, 2010).

Increasing flow divergence (positive stretch) decreases the relaxation time sCH4R (the time

that a flame takes to return to equilibrium conditions after initial disturbance) from 6.0 ms

(planar), to 4.5 ms (cylindrical) and 4.0 ms (spherical)—these are about six times longer

than sH2R from the hydrogen/air flames. However, the nondimensional relaxation numbers

nR ¼ sR=tRL (where tRL is a characteristic flame time scale based upon the thickness of the fuel

consumption rate profile) are similar in both types of flames, nR� 4.4(P), 3.3(C), and

3.0(S) indicating a similar level of stability to applied disturbances. nR deceases by about

40% with increasing positive flow divergence in both flame types under the conditions studied,

demonstrating the strong coupling of the heat release to the flame stretch and geometry.

Spectral analysis of the pressure fluctuations shows that the spectra scale approximately

like Ep�x�3 in the frequency range 2–10kHz, and a weak spectral broadening in the

spherical case. This is a much weaker kinetic-pressure coupling than in hydrogen/air flames

where Ep�x�2. It was found that increasing flow divergence (from planar to spherical

flames) shifts the range of excited frequencies to higher frequencies.

It is confirmed that individual chemical species show markedly different responses to the

disturbances. Species formed in thin reaction layers are generally not disturbed except with

respect to their peak concentration. By contrast, species with broader profiles are generally

significantly affected. These findings suggest that combustion processes may be usefully

viewed as a spectrum of length scales rather than as just a single global scale.

Received 23 October 2011; revised 1 February 2012; accepted 24 March 2012.

Published as part of the 23rd International Colloquium on the Dynamics of Explosions and

Reactive Systems (ICDERS) Special Issue with Guest Editor Derek Dunn-Rankin.

Address correspondence to Nadeem A. Malik, King Fahd University, Department of Mathematics

and Statistics, P.O. Box 5046, Dhahran 31261, Kingdom of Saudi Arabia. E-mail: [email protected]

Combust. Sci. Technol., 184: 1799–1817, 2012

Copyright # Taylor & Francis Group, LLC

ISSN: 0010-2202 print=1563-521X online

DOI: 10.1080/00102202.2012.693426

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Although the flame adjusts to the local unsteady conditions, some evidence of a time lag or

‘‘memory effect’’ between fuel consumption and the rate of heat relase induced by positive

flow divergence is observed.

Keywords: Chemical kinetics; Curvature; Direct simulations; Flame; Hydrogen; Implicit; Methane;

Multicomponent transport; Premixed; Pressure; Stretch

INTRODUCTION AND AIMS

In the limit of high Karlovitz numbers (K¼ tc=tv� 1), where K is the ratio ofthe chemical timescale (tc) to the convective (turbulent) timescale (tv), velocity andscalar fluctuations can penetrate inside the internal flame structure making itunsteady. Pressure fluctuations arise in a number of practical situations relating togas turbine combustion chambers operating in a LPP mode, and the potential ampli-fication of such disturbances via a coupling to the heat release can lead to uncon-trolled amplifications, see Malik and Lindstedt (2010). Furthermore, in diffusionflames processes such as soot and particulate emissions are typically subject to slowchemistry and therefore sensitive to transient effects. The coupling between the flamestructure and unsteady flow fields is therefore of fundamental importance to thetheory of combustion and to practical flames of all types.

A critical issue, therefore, in flame theory is the impact of combined pressureand equivalence ratio oscillations on flames. In this study we consider such oscilla-tions coupled also to variations in flame stretch through different flame geometriesin methane–air flames. Detailed computational studies to date are comparativelyfew and then usually under some simplifying assumptions. (Bradley et al., 1996;Lindstedt et al., 1998; Meyer, 2001). The sensitivity of flames to parametric varia-tions in properties such as the equivalence ratio, (Marzouk et al., 2000; Sankaranand Im, 2002), induced pressure fluctuations (Teerling et al., 2005), and harmonicvelocity fluctuations (Lieuwen, 2005) have also been explored. Flames are oftencurved and subject to strain (e.g., Dixon-Lewis, 1990) and spherical flames have beenstudied experimentally by Dowdy et al. (1990), Taylor (1991), Kwon et al. (1992),Tseng et al. (1993), Karpov et al. (1997), Gu et al. (2000), Baillot et al. (2002),and others. ‘‘Memory effects’’ may result in extended flammability limits or a greaterability to resist strain (e.g., Christiansen and Law, 2002).

The need for more realistic chemistry in combustion simulations has beenrecognized (e.g., Lu and Law, 2009; Malik, 2012; Malik and Lindstedt, 2010; Maliket al., 2011). The progress in numerical methods and computer technology now offerssuch a possibility, and implicit methods in particular have been receiving increasedattention. Smooke and coworkers, for example, have developed implicit compactscheme solvers for 2D nonreacting flows and for 1D reacting flows with a simplifiedglobal reaction source term (see Novstok and Smooke, 2005; Novstok et al., 2007).Malik has developed an implicit finite volume Eulerian method, called transientadvection reaction diffusion implicit simulations (TARDIS; Malik, 2012; Malikand Lindstedt, 2010) featuring the coupling of the compressible flow to realisticchemistry and multicomponent transport properties. This method resolves all theconvective and chemical length and time scales present in a reacting flow systems.

1800 N. A. MALIK AND R. P. LINDSTEDT

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In the present study, we investigate the response of methane=air flames sub-jected to simultaneous pressure and equivalence ratio fluctuations. The configurationis similar to that in Malik and Lindstedt (2010) for hydrogen=air flames. The aim isto explore methane=air flames subjected simultaneously to different types ofunsteady oscillations and to compare the findings to those obtained from hydro-gen=air flames. The comparison with the hydrogen=air system is important, as thereare significant differences in molecular transport properties and thermochemistrythat play an important role in the flame response to physical and chemical distur-bances. Accordingly, stoichiometric methane=air flames are here studied in the con-text of planar flames and outwardly propagating cylindrical and spherical flames inorder to shed light on the underlying physics of thermoacoustic interactions and onthe thermochemical flame structure under conditions of inhomogeneous fuel distri-bution and for different levels of stretch. The impact of different definitions of lengthand time scales on the interpretation of computed results is also explored.

In our previous work, we analyzed the case of hydrogen, a highly reactive fuel,and in the current work we are considering methane, a fuel located at the other endof the reactivity spectrum. The simulation parameters were chosen to provide wavecharacteristics (frequencies and wavelengths) that lead to a potential coupling to thereactive–diffusive structure of the flames. For this reason, we use the same frequen-cies for both fuels as we are seeking to relate our work to practical combustors whereinstability frequencies typically are influenced by the combustor geometry leading tovalues in the range we cover in our study.

METHOD

The current work features an implicit method, TARDIS (Malik, 2012; Malikand Lindstedt, 2010), in which the balance equations of mass, momentum, energy,and chemical species, together with the equation of state for an ideal gas are solvedin an implicit framework on a staggered Eulerian grid arrangement. This is a directsimulation method for one-dimensional symmetrical flames; the governing balanceequations are functions of time, t, and one spatial coordinate (x or the radius r) only.The governing equations and models for the various fluxes are given in Malik andLindstedt (2010). The diffusive fluxes include Soret thermal–diffusion for light spe-cies (H and H2). Viscosities and binary diffusivities are evaluated using the theoryof Chapman and Enskog (see Reid et al., 1987) and thermal conductivities usingthe approach of Mason and Monchick (1962). Mixture properties are evaluated fromWilkes formula (see Reid et al., 1987). Thus, all transport properties are evaluatedlocally and are functions of the local temperature and composition. The specificLewis numbers are, therefore, non-unity (Lek 6¼ 1) and also functions of the localtemperture that allows the transport properties to be unsteady and strongly inhomo-geneous, which gives the method an extra level of realism.

The chemistry is modeled as a comprehensive system of elementary reactionswhose rates are given by

_RRk ¼XNr

j¼1

ðn00kj � n0kjÞ kfjYNs

l¼1

/n0jl

l � krjYNs

l¼1

/n00jl

l

" #ð1Þ

FLAME RESPONSE TO PRESSURE FLUCTUATIONS AND STRETCH 1801

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where /l is the molar concentration of the lth chemical species, and n 0 and n00 are thestoichiometric coefficients of reactants and products, respectively. The forwardreaction rate constant in reaction number j is given by Arrhenius’ law

kfj ¼ AjTbj expð�DEj=RTÞ ð2Þ

where the constants Aj, bj, and the energy barrier DEj are given as part of thechemical mechanism. A C1–C2 chemistry set featuring 30 species and 148 reactionsas outlined by Lindstedt (2000) is used in this study. The reactions include pressuredependencies, and the thermodynamic data is computed using JANAF polynomials.

Transmissive boundary conditions were imposed on the mean velocity andpressure at the open boundaries, and the fluctuating components of velocity andpressure were reflected back into the domain. For the curved flames, we haveu¼ 0 at the center (r¼ 0). The balance equations were solved implicitly using a frac-tional step finite volume method with central differencing and a second-order vanLeer flux limiter for high cell Peclet number. The energy equation was solved first,and then the momentum and continuity equations were solved using a PISO-typemethod by setting up a Poisson equation for the pressure field. Finally, all the speciesmass fractions were solved simultaneously using a Newton–Raphson linearization ofthe chemical source term, which yields a block tri-diagonal system of algebraic equa-tions. The process was iterated until the full equation set converged before thetime-step was advanced. The solution method is second-order accurate in both timeand space. The implicit nature of the numerical scheme provides the required stab-ility to resolve all the spatial and time scales inside the flame kernel. The grid size wasDx� 10�6m and time step was Dt� 10�8s (see Al-Khateeb et al., 2010; Powers andAslam, 2006.)

TARDIS has been validated by computing the laminar flame velocities forH2=air and CH4=air flames at stoichiometry and atmospheric pressure, and they werefound to be close to experimentally determined values (see Malik, 2012). As there areno adjustable parameters in direct simulations (except the given chemical model andtransport models for the various fluxes), this is an important validation for the method.

In order to simulate conditions inside practical devices sush as engines, it is des-riable to produce unforced pressure fluctuations subject to natural decay due to theaction of viscosity, which is typical of conditions after ignition in an engine. This wasachieved by running a precalculation consisting of releasing a set of square pressureramps of large amplitude inside a domain of length L; these pressure ramps then dif-fuse, convect, and reflect back from the ends of the domain until random pressurefluctuations of small amplitude with a range of frequencies are obtained. Whenthe fluctuations have decayed down to a suitable level (about 2% of atmosphericpressure), the main simulations were started (t¼ 0) by introducing the fuel inhomo-geneity as a sinusoidal oscillation cycle in the equivalence ratio just ahead of theflame front viz /(r)¼ 1.0þA/sin (2pr=k/), where k/ is the lengthscale of variationand A/ is the initial amplitude.

The pressure fluctuations continue to decay, as shown in Figure 1, from theplanar flame simulations. In the case shown, the fluctuations have died out afterabout 4ms (but not in the curved flames as we will see in the Section called PressureSpectra and Burning Velocities.


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The domain size was chosen so as to produce a desirable set of frequencies thathave the potential to couple to the flame structure. Short domain size produces toohigh frequencies that do not couple to the flame and are simply dissipated (seefigure 4 in Malik and Lindstedt, 2010). But too low frequencies are merely transla-tional. It was found that random pressure waves with frequencies in the desiredrange, 200–1000Hz, required a domain size of L¼ 400mm.

Four cases of moderate inhomogeneity A/�1 for each flame were consideredwith A/¼�0.2, �0.1, 0.1, 0.2. Figure 2 shows a schematic of the configuration justas the flame is about to propagate into the equivalence ratio oscillation cycle.

Figure 1 The decay of pressure fluctuations p0 in the domain of a planar flame simulation against time

t[ms]. p0 is given as a per centage of the atmospheric pressure. (Figure is provided in color online.)

Figure 2 A schematic sketch of the start of the simualtions as the flame is about to propagate through the

fuel oscillation cycle.


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RESULTS

Flame Length and Time Scales

As we are investigating the potential coupling of pressure disturbances tothe reactive–diffusive structure of different types of flames, in order to compareH2=air and CH4=air flames it is essential that estimates of flame thickness dL bedetermined. It is useful first to discuss some of the different definitions currentlyin use. Some definitons focus on the inner reaction layer while other definitionsare of an integral nature.

A common estimate of the laminar flame thickness is the ‘‘momentum’’ thick-ness, dnL � n=sL, where n is the kinematic viscosity, and the time scale is thentnL � dL=sL. One can also define a thermal thickness as dDL � Dth=sL where Dth isthe thermal diffusivity, and hence tDL � dDL=sL. These definitions provide only orderof magnitude estimates because the diffusivities are strongly temperature dependentand can differ significantly across the flame structure. Yet an other estimate is basedon the distance between isotherms of flame temperature, designated here as dTL andthe associated time scale as tTL . (However, the choice of isotherm is arbitrary.)

Gottgens et al. (1992) derived an analytic approximation for the flame thick-ness for a number of lean hydrocarbon and hydrogen flames. An inner reaction layertemperature (T0) was identified and interpreted as the temperature at which the reac-tion rates peak. The location in the flame structure coincides with the maximumgradient (m0) of the temperature distribution across the flame. The intersection ofthe straight line with this slope through the point (x0, T0) with the horizontal lineat T¼Tu cuts a line between the intersection point at x0 and the intersection pointat x1. This distance was obtained analytically as

lf ¼ðk=CpÞT0

qusLð3Þ

If the preheat zone is inert, then this estimate should be equal to the geometri-cal distance D1 obtained directly from the sketch of the temperature profile. Gottgenset al. (1992) found that D1=lf was close to unity to within a few percent for hydro-carbon flames, but not for hydrogen flames because the condition of an inert preheatzone was not valid for this case. Hence, an empirical relationship for D1=lf wasobtained for H2=air. lf corresponds to roughly half the distance though the flame.We will therefore compare other estimates of the flame thickness with 2lf and 2D1.

As the current computational method includes realistic chemistry and multi-component transport properties, it provides a means for assessing the differentestimates of length and time scales. From our simulations, we have found thatT0� 1000K in stoichiometric hydrogen–air flames and T0� 1000K also inmethane–air flames. The flame thickness obtained using Equation (3) are 2lf� 0.17mmmm for H2=air and 0.58mm for CH4=air flames. The geometric flame thicknessesobtained directly from the plots of the temperature profiles were found to be 2D1�0.4mm and 0.6mm, respectively, which gives 2D1=2lf� 2.35 and 1.04, respectively.[The empirical estimate by Gottgens et al. (1992) for H2=air gives 3.9.] The 5–95% per-centile flame thicknesses, dTL , were found to be quite different: 3.5mm for the H2=airflame and 2.5mm for the CH4=air flame.


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The profiles of the fuel consumption rate _RRF also provide an estiamte for flamethickness, dRL . From the current simulations using TARDIS, we have obtaineddRL � 0:5 mm for both H2=air and CH4=air flames. It is striking that the two differ-ent types of flames possess approximately the same flame thickness using thisdefinition.

The estimates of flame thickness are summarized in Table 1 and the correspon-dong time-scales based upon tL� dL=sL are shown in Table 2. The estimates dRL basedupon fuel consumption rates are consistent with the geometrically determinedestimates 2D1: 2D1=d

RL � 0:8 for the hydrogen flame and 1.2 for the methane

flame—both the hydrogen and the hydrocarbon flames are fairly well estimated.The time scales based upon tRL � dRL=sL in methane are about six times larger thanin hydrogen=air flames.

Pressure Spectra and Burning Velocities

Figure 3 shows the log–log plots of the nondimensionlaized spectra of pressurefluctuations Ep=EP(0) from the three flame geometries. The forcing frequencies are inthe range 200–1000Hz. The spectral response scales are approximately as � x�3 inthe range 1<x< 5000Hz in the planar and cylindrical flames. Comparing the threegeometries, a progessive shift of the excited frequencies to higher frequencies isobserved. The excited range of frequencies, due to the flame pressure coupling,appears as a spectral broadening, or a plateau, in that range of frequencies. Fromthe planar, cylindrical, and spherical spectra in Figure 3, the spectral broadeningis centered around, respectively, x� 2000Hz, 5000Hz, and 8000Hz.

The spectra in all three geometries fall off rapidly at higher frequencies x�10 kHz. The observed scalings differ significantly to those from the hydrogen=airflames where a consistent � x�2 scaling was observed in all three flame geometries(Malik and Lindstedt, 2010). The methane flame thus produces weaker amplituderesponse and over shorter ranges of frequencies than in hydrogen flames, as wewould expect from a less reactive fuel where the flame–pressure coupling is weaker.

Table 1 Flame length scales [mm] estimated from different methods mentioned in the text (The bottom

line gives the ratio of the methane to hydrogen flame length scales)

Flame type 2lf 2D1 2D1=2lf dTL dRL

H2/air 0.17 0.4 2.35 3.5 0.5

CH4/air 0.58 0.6 1.04 2.5 0.5

[CH4]=[H2] 3.4 1.5 – 0.7 1

Table 2 Time scales [ms] estimated from different methods mentioned in the text, tL� dL=sL(The bottom line gives the ratio of the methane to hydrogen flame time scales)

Flame type t fL tTL tRL

H2/air 0.08 1.67 0.24

CH4/air 1.57 6.77 1.35

[CH4]=[H2] 19 4.1 5.7


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Note that because of the much longer time scale in the methane=air flames, thenondimensionalized frequencies, tLx, in the methane=air flames are about six timeshigher than in the hydrogen=air flames.

Figure 4 shows burning velocities (integral fuel consumption rates) against timet [ms]. The following definition of the burning velocity sn is used:

sn ¼1

quðYuF � Yb

F Þ

Z þ1

�1_RRFMFdr ð4Þ

where F is the fuel (CH4),MF is the molecular weight of the fuel, and qu is the densityof the unburned gases. Yu

F is the unburned fuel mass fraction, and YbF is the burned

fuel mass fraction (which is close to zero for stoichiometric and lean mixtures).The response of the flames to the fluctuating local conditions is evident in the

variation of sn. As in the H2=air flames, the upper and lower bounds, shown as hori-zontal lines, are generally not approached due to the actions of viscosity and trans-port processes that smooth out all gradients, bringing the fuel concentrations

Figure 3 Log–log of the normalized pressure fluctuation spectrum Epi(x)=Ep(0) against the frequency

x[Hz]. Top to bottom, from planar, cylindrical, and spherical flames, respectively.


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towards the mean stoichiometric value—in the planar flames the pressure fluctua-tions have died out after about 4ms. The lower bound at /¼ 0.8 is not approachedat all; sL¼ 0.36m=s at /¼ 0.8 (Chemkin-Pro, 2008). The upper bound at /¼ 1.2appears to be reached, but this is probably coincidental because the burning velocityat /¼ 1.2 is similar to its stoichiometric value: sL¼ 0.37m=s at /¼ 1, andsL¼ 0.37m=s at /¼ 1.2, and the maximum is sL¼ 0.4m=s at /¼ 1.15.

The relaxation time sR, defined in Malik and Lindstedt (2010), is the time takenfor the flame to return to the mean equilibrium conditions after a disturbance. Anondimensional number, the relaxation number nR is obtain by scaling with a flametime scale

nR ¼ sRtL

ð5Þ

Here, the mean equilibrium conditions refers to the undisturbed flame propa-gation mode at atmospheric pressure and stoichiometry. nR(T, P, p

0, l, /,. . .) must bea function of state variables such as temperature and pressure, and also upon otherphysical properties, such as fluctuation level, length scale, equivalence ratio, andflame speed. nR must also be sensitive to molecular and convective transport

Figure 4 Flame burning velocity sn[m=s] against time t[ms] from the three flame geoemtries as indicated.

Four cases considered in each plot: A/¼ 0.2 (red), 0.1 (green), �0.1 (blue), �0.2 (violet). (Figure is

provided in color online.)


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processes—stable flames will have smaller nR values because they return faster toequilibrium after disturbance.

All of the quantities that affect nR must return to their equilibrium values, soany one of them can be used to estimate sR (and hence nR). Here, we have used theflame speed from Figure 4 to estimate sR; we define sR to be the time at which theflame speed has returned to within 5% of its equilibrium value.

Increasing positive stretch reduces the relaxation times in the methane=airflames: sCH4

R � 6:0 ms (planar), 4.5 ms (cylindrical), and 4.0 ms (spherical). Thesevalues are a factor of six larger than in the hydrogen=air flames—which were foundto be 1.1 ms, 0.8 ms, and 0.65 ms, respectively. This is close to the ratio 5.7 betweenthe respective flame time scales based upon tRL � dRL=sL shown in Table 2.

Using tL ¼ tRL ms, from Table 2, in Equation (5) gives the relaxation numbers,nCH4

R � 4:4ðPÞ, 3.3(C), and 3.0(S) in the respective geometries. These are similar inhydrogen=air flames nH2

R � 4:6ðPÞ, 3.3(C), and 2.7(S), shown in Table 3. The simi-larity between the two sets of relaxation numbers indicates that H2=air and CH4=air possess similar levels of stability to applied disturbances under the conditionsof the simulations.

The effect of increasing positive flow divergence (or stretch) on variations in nRis also similar in both flame types; the relaxation numbers deceases by about 30%and 40% from the planar flame to the cylindrical and spherical flames, respectively:nCR=n

PR � 0:7 and nSR=n

PR � 0:6.

Fluctuations in the flame speed sn induced by pressure flcutuations in the pla-nar flames have all but died out at around 4.0 ms (see Figure 4a). However, in thecurved flames (Figures 4b and 3c), the fluctuations are persistent even at 5 ms, indi-cating that positive flame curavture plays an important role in sustaining pressurefluctuations. The flames have been placed close to the centre at around r� 10�20mm, so even small pressure fluctuations can build up to large amplitudes, whichappear in Figure 4 as sharp fluctuations in sn. The largest amplitude fluctuations in snshow a periodicity that is due the periodic reflection of pressure waves from thecurved boundary of the domain.

Thermochemical Flame Structure

Figure 5 shows 3D profiles of six selected species in the methane=air system, H,OH, HO2, CH, CO, and CH2O. Each species is characterized by a thickness, lk, andthe general trend in the response to perturbations is similar to that observed in thehydrogen=air system in Malik and Lindstedt (2010). The response is determined lar-gely by the relative thickness of the species profiles compared to the wavelength, k, of

Table 3 Relaxation times sR and the relaxation numbers nR for hydrogen=air and methane=air flames

sR [ms] nR

Propagation mode Planar Cylindrical Spherical Planar Cylindrical Spherical

H2/air 1.10 0.80 0.65 4.6 3.3 2.7

CH4/air 6.0 4.5 4.0 4.4 3.3 3.0


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the physical disturbance, which determines a spectrum of relative length scales

fSkg ¼ flk=kg; k ¼ 1; . . . ;Ns ð6Þ

Species with thin reaction layers such that Sk< 1 adjust their peak values to thelocal conditions (e.g., CH and CH2O), while species with broader reaction zones,Sk> 1, not only adjust their peak values but are also disturbed across their entirestructures (e.g., OH and CO).

The major reactants and products (CH4, O2, CO2, and H2O) are an exceptionand show a minimal response in their structures; their levels of concentration aredetermined by the local fuel content in the equivalence ratio oscillation cycle, whichsmooths out comparatively quickly due to the action of diffusive and convectivetransport processes, as is evident in the CH4 and O2 profiles. Figure 6a shows themole-fraction profiles of selected species (corresponding to the same species as inthe hydrogen=air system) at regular intervals of 0.05 ms from the planar flamegeometry. Figure 6b shows the mole-fraction profiles of other selected species atthe same intervals. Figures 7a and 7b show corresponding mole-fraction profilesfrom the cylindrical flames and Figures 8a and 8b from the spherical flames. In

Figure 5 Mole-fraction profiles at 0.05 ms intervals of indicated species against time t[ms] and position

x[mm]. From the planar flame.


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Figure 6 (a) Mole-fraction profiles at 0.05 ms intervals of indicated species (corresponding to the hydro-

gen=air subsystem) against position x[mm]. From the planar flame. (b) Mole-fraction profiles at 0.05 ms

intervals of other species against position x[mm]. From the planar flame.


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Figure 7 (a) Simialar to Figure 5a, but from the cylinder flame. (b) Simialar to Figure 5b, but from the

cylinder flame.


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Figure 8 (a) Simialar to Figure 5a, but from the spherical flame. (b) Simialar to Figure 5b, but from the

spherical flame.


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the two curved flames, we observe some very high fluctuations in the minor speciesmole-fraction due to high pressure wave fluctuations impacting directly on thereaction layer.

Time Lag

Figure 9 shows the profiles of the rate of heat release ( _hh) and of themole-fractions of H, O, and CH3 in methane=air planar flames at regular intervals

Figure 10 Simialar to Figure 8, but from the cylindrical flame.

Figure 9 Profiles at 0.0125 ms intervals of, top left clockwise: the rate of heat release h, and the mole-

fractions of H, O and CH3, from the planar flame.


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of 0.0125 ms. Figures 10 and 11 are the corresponding figures from the cylindricaland spherical flames. The planar flames, Figure 9, display well-correlated profiles—peaks and troughs in the species profiles occur at the peaks and troughs of _hh—indicating little or no time-lag between the rate of heat release _hh and the speciesconcentrations in this stretch-free system.

The stretched cylindrical and spherical flame profiles in Figures 10 and 11 areless well-correlated, and some high levels of _hh do not produce corresponding adjust-ments in the species concentrations. Positive flow divergence appears to induce atime-lag, or memory loss, between the rate of heat release and the chemical balanceas the flame thermochemistry attempts to adjust to the local transient conditions.

DISCUSSION AND CONCLUDING REMARKS

The transient response of outwardly propagating methane=air flames subjectedto stretch and to disturbances in pressure and fuel distribution at frequencies andlength scales that can couple to the flame reaction layer has been investigated usingthe powerful implicit direct method, TARDIS (Malik and Lindstedt, 2010), anEulerian method that includes realistic chemistry and multicomponent transportproperties coupled to the fully compressible flow and resolves all the temporal andspatial scales in the system. The results from the weakly reacting methane=air flamestudy have been compared to results from outwardly propagating hydrogen=airflames (Malik and Lindstedt, 2010), which is a highly reactive fuel.

In order to compare the different fuels, it was necessary to obtain estimates offlame length scales dL; these were reviewed, and it was found that a flame thicknessdL ¼ dRL based on the thickness of fuel reaction rate profiles gave the most consistentvalues. Both the H2=air and CH4=air flames possess approximately the same length

Figure 11 Simialar to Figure 8, but from the spherical flame.


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dRL � 0:5 mm using this definition under stoichiometric conditions at atmosphericpressure.

The flame time scales based on tL dRL=SL were found to be 5.7 times larger for

methane than for hydrogen under the conditions of the simulations.The pressure spectra scaled like Ep�x�3, and a progressive shift in the spectral

broadening to higher frequencies was observed, indicating the strong role played byincreasing positive flow divergence on flame dynamics. This is a weaker responsethan in H2=air flames, where a consistent scaling of � x�2 was observed over a widerrange of frequencies, clearly refecting the stronger pressure-thermo-chemicalcoupling in H2=air flames.

The relaxation time sR in the methane=air system is about six times larger thanin the hydrogen=air systems. However, the relaxation numbers nR ¼ sR=tRL are simi-lar, nCH4

R � 4:4ðPÞ; 3:3ðCÞ; 3:0ðSÞ, and nH2R � 4:6ðPÞ; 3:3ðCÞ; 2:7ðSÞ.

The effect of stretch reduces nR significantly, by about 40% in both types offlames. Positive stretch appears to enhance the dissipative processes that smoothout scalar gradients. The relaxation number is clearly sensitive to the flamegeometry, and as such it is an indicator of the coupling of the heat release to flamestretch and to pressure disturbances. It is a quantity that could also be indicative ofstability levels of different flames under different conditions.

The relaxation number must be a function of many variables, such as tempera-ture, pressure, fluctuation level, and equivalence ratio; nR¼ nR(T, P, p

0, l, w,. . .).Here we have examined one set of conditions and shown that nR it is a functionof flow diveregence (or stretch) and oscillation lengthscale. A systemtic study ofnR under different conditions and fuel types is the subject of future studies.

Individual species responses to pressure and fuel oscillations were similar tothose in H2=air flames. Species with a reaction layer thickness lk less than the lengthscale of physical disturbance k are too thin for the shape of their profiles to bealtered, although their peak values adjust in repsonse to the local conditions. Specieswith broad reaction zones, on the other hand, are distorted along their entire lengths.Major reactants and products show little response overall.

These findings strengthen Malik and Lindstedt (2010) observation that it maybe more appropriate to analyze flames in terms of a spectrum of species length scalesflkg rather than just a single global scale.

Finally, planar methane flames are well-correlated between the rate of heatrelease _hh and the species mole-fractions. Positive stretch appears to decrease this cor-relation by altering the thermochemical structure inside the flame kernel as the flameattempts to adjust to the local unsteady conditions. This increases the time-lag, ormemory loss, between heat release and chemical balance. However, we expect thispicture to be influenced by the local Lewis numbers Lek number, and this picturecould change under different conditions.

ACKNOWLEDGMENTS

The author wishes to thank SABIC for funding this work through grantnumber SB101018, and also the Information Technology Center at King FahdUniversity of Petroleum andMinerals for providing High Performance Computingresources.


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The Response of Transient Inhomogeneous Flames to Pressure Fluctuations and Stretch: Planar and...

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