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Paper # 070LT-0267Topic: Turbulent Flames8th US National Combustion MeetingOrganized by the Western States Section of the Combustion Instituteand hosted by the University of UtahMay 19-22, 2013.A Priori Model for the Effective Lewis Numbers in PremixedTurbulent FlamesBruno Savard1Guillaume Blanquart21Graduate Aerospace Laboratories,California Institute of Technology, 1200 E California Blvd, Pasadena, CA 911252Mechanical Engineering,California Institute of TechnologyA simple a priori model for the effective Lewis numbers in a premixed turbulent flame is presented.This a priori analysis is performed using data from a series of direct numerical simulations (DNS) oflean (φ 0.4) premixed turbulent hydrogen flames, with Karlovitz number ranging from 10 to 1562 [1].Those simulations were chosen such that the transition from the thin reaction zone to the broken reactionzone is captured. The conditional mean of various species mass fraction ( Yi T ) vs temperatureprofiles are evaluated from the DNS and compared to equivalent unstretched laminar premixed flameprofiles. The turbulent flame structure is found to be different from the laminar flame structure. However, the turbulent flame can still be mapped onto a laminar flame with an appropriate change in Lewisnumbers. Those effective Lewis numbers were obtained by minimizing the error between the DNSresults and predictions from unstretched laminar premixed flames. A transition from “laminar” Lewisnumbers to unity Lewis numbers as the Karlovitz number increases is clearly captured - equivalently, asthe turbulent Reynolds number increases, given that the ratio of the integral length scale to the laminarflame thickness is fixed throughout the series of DNS. Those results suggest the importance of usingeffective Lewis numbers that are neither the “laminar” Lewis numbers nor unity in tabulated chemistry models without considering the impact of the turbulent Reynolds number or Karlovitz number. Amodel for those effective Lewis numbers with respect to the turbulent Reynolds number was also developed. The model is derived from a Reynolds Averaged Navier-Stokes formulation of the reactive scalarbalance equations. The dependency of the effective Lewis numbers to the Karlovitz number insteadof the Reynolds number was studied and is discussed in this paper. These changes in effective Lewisnumbers have significant impacts. First, the laminar flame speed and laminar flame thickness vary by afactor of two through the range of obtained effective Lewis numbers. Second, the regime diagram [2]changes because a unique pair of laminar flame speed and laminar flame thickness cannot be used. Adependency on the effective Lewis numbers have to be introduced.1IntroductionTurbulent premixed flames relevant to industrial applications often belong to the thin reactionzone or even the broken reaction zone. In the thin reaction zone, the smallest eddies are smallenough to penetrate the flame preheat zone, but too big to enter the reaction zone. In the brokenreaction zone, the smallest eddies are small enough to penetrate both the preheat zone and thereaction zone [2]. The relevant non-dimensional parameter is the Karlovitz number defined as theratio of the flame and Kolmogoroz time scales Ka tF /tη . The Karlovitz number can also be1
8th US Combustion Meeting – Paper # 070LT-0267Topic: Turbulent Flameswritten as Ka lF2 /η 2 , where lF is the laminar flame thickness and η the Kolmogoroz lengthscale. Considering that the reaction zone is approximately 10 times smaller than the laminar flamethickness, scaling arguments suggest that 1 Ka 100 defines the thin reaction zone andKa 100 the broken reaction zone, as long as the premixed flame is turbulent.A large body of work has been done on simulating and modeling turbulent premixed flames up tothe broken reaction zone [3–11]. However, most of them consider methane as the fuel. Methane hasthe characteristic that its species diffusivity equals its thermal diffusivity, i.e. it has a unity Lewisnumber. This has the convenient effect of greatly simplifying the equations to be solved. Fuelsfound in industrial applications rarely have unity Lewis numbers. As an example, the Lewis number of n-dodecane, a surrogate for kerosene, is approximately 3.5 in air. Hydrogen is an interestingexample of an alternative fuel for ground-based applications such as gas turbines for electricitygeneration. The Lewis number of hydrogen in a lean hydrogen-air mixture is approximately 0.3,also far from unity.It is obvious that there is great interest in being able to model and simulate accurately turbulentpremixed flames with non-unity Lewis numbers, especially at high Karlovitz numbers. Aspden etal. [1] recently performed direct numerical simulations (DNS) of lean (φ 0.4) premixed hydrogen flames at Karlovitz numbers (Ka) ranging from 10 to 1526. Their results clearly show thatthe flame structure varies significantly between the lowest and the largest Ka flames, unlike turbulent methane flames. They observed that the largest Ka flame had a structure comparable to theone of a methane flame, i.e. the flame behaved as an effective unity Lewis number flame. Highlyturbulent non-unity Lewis number diffusion flames were experimentally observed to behave verysimilarly to unity Lewis number diffusion flames [12]. To the best of the authors’ knowledge, thereis no premixed flame experiment equivalent to those conducted in [12] in the literature. At highturbulence levels, dissipation of species and temperature is dominated by turbulent mixing, resulting in a unity effective Lewis number. From this argument, the same results should be observedexperimentally for premixed flames.Most premixed turbulent combustion models are developed for the corrugated flamelet regime(Ka 1) and extended to higher Karlovitz regimes. In several of those models, tabulated chemistry assumes that the flame structure can be mapped into a corresponding laminar unstretchedflamelet. The results from Aspden et al. seem to suggest that tabulated chemistry is inadequate fornon-unity Lewis numbers premixed turbulent flames. This is one of the motivations for the objectives pursued in this work: 1) to analyze the structure of turbulent premixed lean hydrogen flamesat Karlovitz numbers ranging from 10 to 1526 to determine if a mapping with a correspondinglaminar unstretched flamelet can still be found allowing modifications to the Lewis numbers, 2) toderive an a priori model for the effective Lewis numbers in premixed turbulent flames.Section 2 presents the flame structure obtained from the DNS data of Aspden et al. and comparesit to corresponding laminar unstretched flamelets. Section 3 derives a simple a priori model fromsimplified species and temperature balance equations. Section 4 compares the model against theeffective Lewis numbers computed from the DNS. Section 5 presents the impacts of effectiveLewis numbers on laminar flame speed and flame thickness, the effective Karlovitz number andthe regime diagram.2
8th US Combustion Meeting – Paper # 070LT-02672Topic: Turbulent FlamesFlame structureIn this work, the flame structure from the series of DNS simulations performed by Aspden etal. [1] is compared to the structure of laminar unstretched flamelets. The complete set of parametersused for the DNS cases can be found in [1] and are summarized in Table 1. The reactants are a lean(φ 0.4) hydrogen-air mixture. GRI-2.11 was used as the chemical mechanism (9 species, 27reactions for hydrogen combustion). Soret and Dufour transports as well as radiation are neglectedin the DNS simulations [1]. As confirmed in [1], the transition from the thin reaction zone to thebroken reaction/distributed burning zone is covered by the simulation cases.The laminar flame counterparts are simulated using FlameMaster [13]. The equivalence ratio isfixed to 0.4, the GRI-2.11 chemical mechanism is used and the Soret and Dufour transports andradiation are ignored.CaseEquivalence ratio (φ)Laminar flame speed (sL ) (m/s)Laminar flame thickness (lF ) (mm)Length ratio (l/lF )Velocity ratio (u0 /sL )Turbulent Reynolds numberbased on viscosity of unburnt gases (ReT )Karlovitz number (Ka)ABCD0.40.40.40.40.224 0.224 0.224 0.2240.629 0.629 0.629 0.6290.50.50.50.53.69 17.1 32.9 106.814.2 65.8 126.1 410.7101002661526Table 1: Parameters for series of turbulent premixed hydrogen flame DNS performed in [1].The conditional mean of the mass fractions with respect to temperature Yi T are calculatedfor an instantaneous snapshot of the established (statistically steady) flame. Figure 1 shows theconditional mean of the hydrogen mass fraction as a function of temperature for cases A through D.The temperature is considered here as a progress variable. Are also shown in Fig. 1, the profiles forlaminar unstretched flamelets with 1) full transport, 2) unity Lewis numbers. The “full transport”and “unity Lewis numbers” laminar flames correspond to limiting cases of purely laminar and fullyturbulent premixed flames respectively. A clear trend is observed: the DNS profiles gradually movefrom the purely laminar towards the fully turbulent limiting cases. As mentioned in Section 1, anobjective of this work is to model the transition between the purely laminar to the fully turbulentflame structure.The presence of hot spots, characteristic of thermodiffusive instabilities, are revealed by theextension of the DNS profiles to temperatures higher than the adiabatic flame temperature ( 1, 400 K). The thermodiffusive instabilities are beyond the scope of this work. This observationjustifies the choice of computing the conditional mean as Yi T instead of T Yi .3
8th US Combustion Meeting – Paper # 070LT-0267Topic: Turbulent FlamesFigure 1: Conditional mean of the hydrogen mass fraction as a function of temperature for casesDNS A through D and unstretched laminar flames with full transport and unity Lewis numbers.33.1Model proposedEffective Lewis NumbersNeglecting the Soret and Dufour effects, the Reynolds Averaged Navier-Stokes (RANS) speciesand temperature balance equations, assuming equal thermal and species eddy diffusivities andconstant heat capacity, can be written as: hi ρỸi · ρũỸi · ρ (Di DT ) Ỹi ω̇i ,(1) t hi ρT̃(2) · ρũT̃ · ρ (α DT ) T̃ ω T , twhere ρ is the Reynolds-averaged density, ũ, Ỹi and T̃ , the Favre-averaged velocity, species massfraction, and temperature respectively, ω̇i , ω T , the averaged species mass fraction and temperaturesource terms. α and Di are the thermal and species molecular diffusivities and DT the eddydiffusivity. Note that Eq. 1 uses a simplified Fick’s law of diffusion (species molecular weightsare assumed to be the same). When using Eq. 1, numerical codes usePa correction velocity in thediffusion flux such that there is no net total mass diffusion flux, i.e.ji 0. The diffusion fluxbecomes ji ρ Di Yi Yi vc ,(3)Pwith vc Dj Yj . Once filtered,Xji ρ (Di DT ) Ỹi ρYiDj Yj .(4)jOne can make DT appear in the second term of the flux as follows, using the fact that species massfractions sum up to 1:Xji ρ (Di DT ) Ỹi ρYi(Dj DT ) Yj .(5)j4
8th US Combustion Meeting – Paper # 070LT-0267Topic: Turbulent FlamesIt is obvious from Eq. 5 that the concept of effective Lewis number remains valid even whenconsidering a correction velocity in the diffusion flux. An effective Lewis number can be obtainedfor each species:1 DαTα DT 1Lei,ef f .(6)Di DT DαTLei3.2 A priori model testingThe Lewis numbers of all 9 species are first obtained through a full transport simulation. Theyare then set constant to those values in a second simulation. The Lewis numbers are modified toeffective Lewis numbers for subsequent simulations according to a one-parameter transformation,corresponding to Eq. 6:Lei,ef f 1 γ.1 γLei(7)Figure 2 compares the hydrogen mass fraction versus temperature profiles obtained from DNSand from laminar unstretched flamelets varying the free parameter γ. The Lewis number being aratio of mass diffusion to heat diffusion, Fig. 2 is perfectly adapted to assess its effects on the flamestructure. Two observations can be made from Fig. 2. Firstly, full transport and constant Lewisnumbers simulations show similar profiles. This justifies the assumption of constant Lewis numbers through the flame, assumption necessary to simulate the effective Lewis numbers flamelets.Secondly, effective Lewis numbers flamelet profiles agree very well with the DNS profiles. Thisresult suggests that the turbulent flame can still be mapped onto a laminar flame with an appropriate change in Lewis numbers. Note that for the highest value of γ, all Lewis numbers are veryclose to unity.Figure 2: Comparison of H2 mass fraction profiles between DNS conditional mean and flameletswith modified Lewis numbers using γ as a fitting parameter.5
8th US Combustion Meeting – Paper # 070LT-0267Topic: Turbulent FlamesFigure 3 shows the same results for species O2 , H2 O and OH. For O2 and H2 O the variations inprofiles from case A to D are marginal. The same trend is observed with the laminar flames. TheDNS profiles for OH show globally the same trend and magnitude as the laminar profiles, but lessaccurately than for H2 . However, it is less likely to get the radicals right and the influence of hotspots on the radicals may be far from negligible as opposed to hydrogen. As a consequence, H2mass fraction is the best candidate to describe the flame structure.Figure 3: Comparison of O2 , H2 O and OH mass fraction profiles between DNS conditional meanand flamelets with modified Lewis numbers using γ as a fitting parameter.3.3Turbulent modelUsing a k model for the eddy viscosity [14], the eddy diffusivity can be written asDT νTCµ k 2 .P rTP rT (8)Using the same definition for the integral length scale as in [1],u0RM S 3l (9)32k u0RM S ,2(10)and the turbulent kinetic energyone obtains9 Cµ u0RM S l.(11)4 P rTUsing the definition of the Prandtl number, the ratio of the turbulent to the thermal diffusivities canbe written asDT9 Cµ u0RM S lP r9Pr CµReT .(12)α4νP rT4 P rTThe species effective Lewis number becomesDT Lei,ef f 1 49 Cµ PPrrT ReT1Lei 94 Cµ PPrrT ReT6.(13)
8th US Combustion Meeting – Paper # 070LT-0267Topic: Turbulent FlamesFrom the DNS parameters, ReT is a known quantity. P r is a function of temperature and mixturecomposition and can be obtained from a flamelet simulation. Cµ 0.09 from standard k model [14]. Finally, Lei is known for all species (can be obtained from a flamelet simulation).However, the turbulent (eddy) Prandtl number P rT is not known. Since the thermal and specieseddy diffusivities are assumed equal, it is of the same order of approximation to assume that theeddy viscosity is also equal to the eddy diffusivities. P rT is therefore set to unity.44.1Validation of the modelSensitivity analysis to species Lewis numbersThe laminar unstretched flamelet profiles are not equally sensitive to all species’ effective Lewisnumber. Species like O2 have a “laminar” Lewis number close to unity ( 1.1 for O2 ) and thereforeas their effective Lewis numbers progress towards unity, very small changes in the laminar flameprofiles should be expected. Therefore, a sensitivity analysis is performed first.One species Lewis number at the time is modified, the other ones being fixed to the species Lewisnumbers obtained from the full transport simulation. 100 flamelets are simulated for each modifiedspecies, their Lewis number varying from their “laminar” value towards unity. The resulting H2and OH mass fraction versus temperature profiles are compared with the DNS profiles presentedpreviously in Fig. 2-3. As discussed in section 2, the profiles for O2 and H2 O are not suitablefor this kind of analysis. Figure 4 shows the results for H2 (top) and OH (bottom) mass fractionsfor modified Lewis numbers of H2 , H, and all species but H2 and H, respectively from left toright. Modifying the Lewis number of H2 has by far the most influence, whereas modifying theLewis number of H has small effect on H2 mass fraction and small but non negligible effect onOH mass fraction. Modifying the other species’ Lewis number has negligible influence on theflame structure. It is obvious that the Lewis numbers of H2 and H have the most influence. Twoexplanations are suggested and cannot be differentiated in this study: 1) H2 and H are the specieswith Lewis numbers the furthest from unity and 2) H2 being the fuel and H the most importantradical, their diffusion controls the flame structure.4.2Computing the effective Lewis numbersThe range of variations in the flame structure due to changes in the H2 Lewis number is sufficiently wide and encompasses all the DNS data. As a result, it is expected to be able to find a bestset of Lewis numbers to compare to the DNS. Unfortunately, because of the small sensitivity of theresults on the Lewis numbers of species other than H2 (and H to a lesser extent), such a set cannotbe found for these species. Therefore, the effective Lewis numbers for H2 are computed for casesA through D. Only the Lewis number of H2 is modified. The L2-norm of the error between theDNS and the laminar profiles is minimized to obtain the hydrogen effective Lewis number, i.e., for7
8th US Combustion Meeting – Paper # 070LT-0267Topic: Turbulent FlamesFigure 4: Comparison of H2 , and OH mass fraction profiles between DNS conditional mean andflamelets with modified H2 (left), H (center), and all species but H2 and H (right) Lewis numbers.each DNS case (A through D):LeH2 ,ef f arg min N1Le H2NPi 1 2SYHlam(Le H2 , Ti ) YHDN(Ti )22(14)subject to Le H2 [LeH2 , 1] ,Swhere YHDN(Ti ) corresponds to the interpolated value of YH2 T (from the DNS) at T Ti ,2lamand YH2 (Le H2 , Ti ) corresponds to the value of YH2 obtained from the laminar unstretched flameletsimulation with LeH2 Le H2 , interpolated at T Ti . The temperature is discretized uniformlyfrom the minimum temperature in the domain Tu to the adiabatic flame temperature Tad such thati(Tad Tu ) (the hot spots are avoided). Table 2 presents the effective Lewis numbersTi Tu N 1obtained and Fig. 5 compares them against the model presented in Section 3. Note that the Prandtlnumber is fixed to the unburnt value and a fitting coefficient of 0.5 is used in front of Cµ in Eq. 13.This fitting coefficient is of the order of the relative uncertainties on the coefficient in front of ReTin Eq. 13. In particular, the fitting coefficient does not change the slope in the semilog plot ofFig. 5. The results are in very good agreement with the model. Since the model does not make adifference between a species or another, it is expected that each species should follow the modelpresented. Therefore, each Lewis number is modified according to Eq. 7 in a subsequent series offlamelet simulations. Defining Leeff as the vector containing all species effective Lewis numbers,8
8th US Combustion Meeting – Paper # 070LT-0267Topic: Turbulent Flamesthis vector is obtained, equivalently to Eq. 14, as follows:Leeff arg min N1Le NPi 1subject toSYHlam(Le , Ti ) YHDN(Ti )22 Lej 2(15)1 γ, γ1Lejγ [0, ) .The hydrogen effective Lewis numbers obtained are presented in Table 2. They are also comparedagainst the model in Fig. 5. The agreement with the model remains relatively good, but the slopeshown by the model seems to be slightly off, which suggests that the power of ReT in Eq. 13 maybe incorrect.CaseLeH2 ,ef f obtained through Eq. 14 (only LeH2 modified)LeH2 ,ef f obtained through Eq. 15 (all Lei modified)ABCD0.43 0.66 0.77 0.880.44 0.73 0.85 0.99Table 2: Parameters for series of turbulent premixed hydrogen flame DNS performed in [1].Figure 5: Effective Lewis numbers versus turbulent Reynolds number obtained by L2-norm minimization (DNS cases A through D) for only H2 and all Lewis numbers modified in flamelet simulations.4.3Reynolds versus Karlovitz numberThe series of simulations performed in [1] have a fixed l/lF ratio. Consequently, the effects ofKa cannot be differentiated from those of ReT , as described by the following relation:2/3ReT Kaef f ,9(16)
8th US Combustion Meeting – Paper # 070LT-0267Topic: Turbulent Flamesusing the effective Karlovitz number Kaef f based on the effective laminar flame thickness. Kaef fis discussed in greater details in Section 5. Based on the observation made in subsection 4.2, ReTis replaced by Kaef f in Eq. 13 to form a non-physical model. This model is compared against theLewis numbers presented in Table 2 (obtained from Eq. 15), as depicted in Fig. 6. Here, no fittingcoefficient is used. The agreement with the DNS values is better than for the ReT -based model.This better agreement with the non-physical Kaef f -based model raises interesting questions. Itseems reasonable that the effective Lewis numbers would depend on laminar flame characteristics,which is not the case with the ReT -based model. The assumption that all thermophysical propertiesare constant through the flame hides the flame characteristics. A more complete model is necessaryto verify if a Ka dependency is physical. However, in the DNS analyzed the largest relevantturbulent scale is smaller than the flame thickness and its corresponding turnover velocity is u0 .Supposing that DT uT lT it is not clear that a Ka dependence should have been observed withthe series of DNS studied.Figure 6: Effective Lewis numbers versus Karlovitz number obtained by L2-norm minimization (DNScases A through D) for all species Lewis numbers modified in flamelet simulations.55.1Effective Ka and regime diagramImpact on sL and lFAs the effective Lewis number of the fuel changes with turbulent Reynolds number, the laminarflame speed and flame thickness are expected to change significantly. Since the turbulent flamecan be mapped onto a laminar unstretched flamelet with appropriate effective Lewis numbers, thereference laminar flame speed and flame thickness for the turbulent flame must be recalculatedconsequently. From a one-step reaction, matched asymptotic expansion with only the fuel Lewisnumber different than unity, the following ratios are obtained [15]: 0.5 0.5sL,2LeF,2lF,2LeF,1(17) , ,sLelLeL,1F,1F,1F,2where sL,i and lF,i are the laminar flame speed and flame thickness of a flame with correspondingfuel Lewis number LeF,i . Note that the only different parameter between flames 1 and 2 is the fuelLewis number. Using the ReT -based model, the laminar flame speed and flame thickness obtained10
8th US Combustion Meeting – Paper # 070LT-0267Topic: Turbulent Flamesfrom Eq. 17 are plotted as a function of the turbulent Reynolds number in Fig. 7. The analytic solutions are compared to the flamelet calculation results using the effective Lewis numbers accordingto the ReT -based model. The agreement between the curves is relatively good. The discrepanciescan be due to the numerous simplifying assumptions made in the matched asymptotic expansion.However, it is clear that both the laminar flame speed and flame thickness vary from almost afactor of 2 between the “laminar” Lewis numbers flamelet and the unity Lewis numbers flamelet(hydrogen premixed flame with φ 0.4).Figure 7: Effective laminar flame speed and flame thickness versus turbulent Reynolds number.5.2Impact on regime diagramThe result from subsection 5.1 suggests that the Karlovitz number should take into account theeffective flame thickness or flame speed. Defining an effective Karlovitz number asKaef f2lF,eff, 2ηit can be related to the “traditional” Karlovitz number as follows:LeFKaef f Ka.LeF,ef f(18)(19)Recall the relation derived from Peters in [2] from which the iso-Ka are obtained in the premixedregime diagram: 1/3u0l2/3 Ka.(20)sLlFFollowing the results shown in this paper, the relevant quantity to differentiate whether the smallest eddies will penetrate the preheat zone or the reaction zone is the effective Karlovitz number.Therefore, Eq. 20 can be rewritten as 2/3 1/3u0LeF,ef fl2/3 Kaef f.(21)sLLeFlF11
8th US Combustion Meeting – Paper # 070LT-0267Topic: Turbulent FlamesUsing the Re-based model presented in this paper, the ratio of effective to “laminar” fuel Lewisnumbers can be expressed as01 β suL lFlLeF,ef f ,(22)0LeF1 LeF β su l lL F9C P r sL lF4 µ P rT νwith β a constant. Hence, fixing the effective Karlovitz number to relevant valuesas 1 or 100 [2], one obtains an implicit equation (combining Eq. 21 and Eq. 22) which can be solvednumerically to obtain a modified regime diagram as shown in Fig. 8. The shifts observed for thedelimiting lines are non-negligible. DNS simulations A through D are positioned in Fig. 8. Aspdenet al. found in [1] that the two lowest Ka (10 and 100) DNS simulations showed a thin reactionzone behavior, whereas the highest Ka (1526) simulation was clearly in the broken reaction ordistributed burning zone and the last simulation (Ka 266) was a transition case, whereas scalingarguments [2] suggest that the transition should be around Ka 100. This is not inconsistent withthe scaling. However, predicting more accurately where transition occurs would find importantapplications. These observations from Aspden et al. agree very well with Fig. 8, even though theauthors do not claim to be able to explain the transition Ka found in [1]. Aspden et al. also foundan influence of the equivalence ratio on the transition Ka (from the thin reaction zone to the brokenreaction or distributed burning zone). The effective Lewis number however has a non-negligibleimpact on the transition Ka for the l/lF ratio studied (0.5).Figure 8: Regime diagram taking into account the effective Karlovitz number.6ConclusionsThe average structure of turbulent premixed lean hydrogen flames (φ 0.4), obtained throughDNS simulations [1], is found to vary considerably with Karlovitz number ranging from 10 to1562. These flames cover the transition from the thin reaction zone to the broken reaction/distributedburning zone. The turbulent flames are shown to have the same structure in average as laminar unstretched flamelets with appropriate effective Lewis numbers. The fact that flames in those regimescan be mapped onto laminar unstretched premixed flamelets with an appropriate change in Lewis12
8th US Combustion Meeting – Paper # 070LT-0267Topic: Turbulent Flamesnumbers has important consequences. First, this means that the tabulated chemistry approach remains valid in the thin reaction and the broken reaction/distributed burning zones as long as thetabulated flamelets are simulated with appropriate Lewis numbers corresponding to the parameters of the turbulent flame (Ka or ReT or both). Second, effective laminar flame speed and flamethickness have to be considered. Indeed those quantities vary by almost a factor 2 between the“laminar“ Lewis numbers and the unity Lewis numbers for premixed hydrogen flamelet (φ 0.4)simulations. In light of these results, the effective Karlovitz number Kaef f also has to be considered. The regime diagram proposed by Peters [2] should therefore be adapted to the fuel consideredwhenever the fuel has a Lewis number far from unity.An a priori model for the effective Lewis numbers involving ReT is derived from RANS equations. Considerable simplifications of the physics are implied. Nevertheless, very good agreementwith the effective Lewis numbers obtained from the series of DNS is found. Quantification ofuncertainties has yet to be performed and the model should be seen as a first attempt to describethe transition from “laminar” Lewis numbers to unity effective Lewis numbers. Also, more datapoints are needed to fully validate the model. Flames with fuel Lewis number greater than unityhave yet to be compared to the model.AcknowledgmentsThe authors gratefully acknowledge funding from the Air Force Office of Scientific Research(Award FA9550-12-1-0144) under the supervision of Dr. Chiping Li, and from the Natural Sciences and Engineering Research Council of Canada (NSERC Postgraduate Scholarship D). Thiswork was also made possible by a collaboration with Dr. Andy Aspden, from the University ofPortsmouth, who kindly shared with the authors the DNS data presented in [1].References[1] A.J. Aspden, M.S. Day, and J.B. Bell. J. Fluid Mech., 680 (2011) 287–320.[2] N. Peters. Turbulent Combustion. Cambridge University Press, Cambridge, 2000.[3] O. Colin, F. Ducros, D. Veynante, and T. Poinsot. Physics of Fluids, 12 (2000) 1843–1863.[4] J.P. Legier, T. Poinsot, and D. Veynante. Proc. CTR Summer Program, (2000) 157–168.[5] M.S. Anand and S.B. Pope. Comb. Flame, 67 (1987) 127–142.[6] R.P. Lindstedt and E.M. Vaos. Comb. Flame., 145 (2006) 495–511.[7] O. Gicquel, N. Darabiha, and D. Thévenin. Proc. Combust. Inst., 28 (2000) 1901–1908.[8] J.A. van Oijen, F.A. Lammers, and L.P.H. de Goey. Comb. Flame, 127 (2001) 2124–2134.[9] H. Pitsch and L. Duchamp de Lageneste. Proc. Comb. Inst., 29 (2002) 2001–2008.[10] E. Knudsen and H. Pitsch. Comb. Flame, 156 (2009) 678–696.[11] A.Y. Poludnenko and E.S. Oran. Comb. Flame, 157 (2010) 995–1011.[12] J.C. Ferreira. Flamelet modelling of stabilization in turbulent non-premixed combustion. Dissertation, ETHZürich, 1996.[13] H. Pitsch. A c computer program for 0d combustion and 1d laminar flame calculations. Technical report,University of Technology (RWTH), Aachen, 1998.13
8th US Combustion Meeting – Paper # 070LT-0267Topic: Turbulent Flames[14] D.C. Wilcox. Turbulence Modeling for CFD. DCW Industries, Anaheim, 2000.[15] N. Peters. Fifteen lectures on laminar and turbulent combustion. Ercoftac Summer School, September 1992.14
The dependency of the effective Lewis numbers to the Karlovitz number instead . ( 0:4) hydrogen-air mixture. GRI-2.11 was used as the chemical mechanism (9 species, 27 reactions for hydrogen combustion). Soret and Dufour transports as well as radiation are neglected . and "unity Lewis numbers" laminar flames correspond to limiting .
