Combustion and Flame 145 (2006) 663–674www.elsevier.com/locate/combustflame

Substantiating a fractal-based algebraic reaction closureof premixed turbulent combustion for high pressure

and the Lewis number effects

Naresh K. Aluri a, S.P. Reddy Muppala a,1, Friedrich Dinkelacker a,b,∗

a Lehrstuhl für Technische Thermodynamik, Universität Erlangen-Nürnberg, Am Weichselgarten 8, 91058 Erlangen, Germanyb Institut für Fluid- und Thermodynamik, Universität Siegen, Paul-Bonatz-Str. 9, 57068 Siegen, Germany

Received 29 April 2005; received in revised form 30 August 2005; accepted 8 February 2006

Available online 24 March 2006

Abstract

A comprehensive set of nearly 100 atmospheric and high-pressure flame data of Kobayashi et al. are a goodsource for numerical analysis to address two main aspects in premixed turbulent combustion—high-pressure influ-ence and effects of fuel type on the reaction rate. The present work deals with the lucid and realizable fractal-basedreaction rate closure from Lindstedt and Váos (LV model) for premixed flames in the thin-flame limit. In this study,the reaction source term is customized on the eddy viscosity closure of turbulent transport, for practical reasons.Computed results from the LV model show the right qualitative trends with the experimental findings, as a functionof turbulence. However, quantitative predictions of the original model are partly too low, and preclude the effectsof pressure and fuel type on the reaction rate. With an extensive parametric study, based on numerical findingsas well as on theoretical argumentation, the LV model is substantiated for these two effects. Results from theproposed tuned LV model are found to be in very good agreement with most of the measured data.© 2006 The Combustion Institute. Published by Elsevier Inc. All rights reserved.

Keywords: Turbulent premixed combustion; Numerical reaction rate modeling; High-pressure flames; Turbulence scales; Lewisnumber

1. Introduction

In modeling premixed turbulent combustion, twomajor challenges are to be met: turbulence modelingand reaction rate modeling [1]. In complex flow situ-

* Corresponding author. Fax: +49 271 740 2360.E-mail address: dinkelacker@ift.mb.uni-siegen.de

(F. Dinkelacker).1 Now with Département de Mécanique, TERM, Uni-

versité Catholique de Louvain, Place du Levant, 2, 1348Louvain-la-Neuve, Belgium.

0010-2180/$ – see front matter © 2006 The Combustion Institute.doi:10.1016/j.combustflame.2006.02.004

ations, such as swirling flows and recirculating flows,interaction of turbulence and reaction is a nontrivialquestion. The present study encounters a relativelysimple flow situation of Bunsen flames with exit ve-locities mostly in the range of a few m/s, withoutstrong flow gradients in the flame region. Therefore,in this paper, we consolidate investigation of the fun-damental turbulent combustion processes, focusingmainly on reaction-rate modeling. On the other hand,operating pressures beyond atmospheric levels in pre-mixed turbulent combustion are rewarding, with ap-plication to large-scale industrial devices such as gas-turbine combustors and internal combustion engines.

Published by Elsevier Inc. All rights reserved.

664 N.K. Aluri et al. / Combustion and Flame 145 (2006) 663–674

Nomenclature

A fractal areaAT turbulent flame surface areaA averaged flame surface areaCR a preconstant (reaction rate parameter)c Favre-averaged reaction progress vari-

ablec Reynolds-averaged reaction progress

variableCP,Le a prefactor for pressure and Lewis num-

ber effectsD fractal dimensionDa Damköhler numberEA activation energyKa Karlovitz numberk turbulent kinetic energyl3 volume of flame elementlλ Taylor length scale, lx Re−0.5

tlk Kolmogorov length scale, (ν3/ε)1/4

lx integral length scaleLe Lewis number of the fuel–air mixturep operating pressurePR probability of occurrence of reactionRet turbulent Reynolds number, u′lx/ν

sL unstretched laminar flame speedsT turbulent flame speedU mean inlet flow velocityu′ r.m.s. turbulent velocityui Favre-averaged component of the gas

flow velocity u

u′′i

fluctuating component of the gas flowvelocity u

u′′c′′ turbulent fluxVK Kolmogorov velocity, (ν0ε)1/4

wc mean chemical reaction rate

Greek letters

γ unburned to burned density ratioδL laminar flame thicknessδT turbulent flame brush thicknessεi inner cut-off scaleεo outer cut-off scaleε dissipation rateμ molecular dynamic viscosityμt turbulent viscosityν molecular kinematic viscosityνu (unburned) molecular kinematic viscos-

ityνt turbulence exchange coefficientν∗ normalized pressure-variant kinematic

viscosity, ν(p)/ν(p0)

ρ density of gasρu (unburned) density of premixed mixtureρb (burned) density of reaction productsΣ flame surface densityσc turbulent Schmidt numberφ equivalence ratio

Notwithstanding its very significant role, the influ-ence of pressure on turbulent flame speed/reactionrate is barely attended due to the associated difficul-ties in performing defined experiments. Evolution ofsuitable measurement data for numerical validationhas therefore been slow albeit promising [2–6]. Thepresent focus is on two important subjects, not muchregarded in the past: the influence of pressure and fueleffects on reaction closure. In general, a numericalmodel should be able to account for both these ef-fects, to claim generality of the model. However, quiteoften, numerical models are validated over a smallrange of conditions, mainly restricted to atmosphericmethane flames.

In the last years, our attempt has been to inves-tigate some of the well-known existing models forturbulent premixed combustion for the influence ofpressure and fuel type. This paper shows results fromone of a series of similar studies, which are based onthis broad set of experimental data of Kobayashi et al.Different models for turbulent premixed flames havebeen tested recently [7–9] based on turbulent flame-

speed closure [10], BML-type models [11,12], andflame surface density models [13–15]. These modelsrespond only weakly to the aforesaid effects. In a re-cent study, a three-parameter algebraic reaction rateclosure for the flame-wrinkling ratio AT/A was de-veloped with the reaction progress variable gradientapproach [16], by including explicit high-pressure ef-fects and an empirical 1/Le dependency to describethe influence of nonunity Lewis number. The im-portance of the Lewis number to turbulent reactionrates was discussed elaborately in a recent review byLipatnikov and Chomiak [17], including similar Le-dependency2 findings from detailed numerical simu-

2 Very recently, Chen and Bilger [18] reported observa-tions on the differences of the curvature distribution, on theconditional mean scalar dissipation rate, and on the extinc-tion behavior between lean propane–air and methane–airflames, which they attribute to difference in the Lewis num-bers between the two. Experimental studies by Renou et al.[19] and by Brutscher et al. [20] show alternative relationsbetween the Markstein length and the flame behavior.

N.K. Aluri et al. / Combustion and Flame 145 (2006) 663–674 665

lation [21], on the importance of molecular transporteffects associated to highly turbulent flames.

In [9], two algebraic reaction rate models fromLindstedt and Váos (LV) [12] and Bray–Moss–Libby(BML) [11] were numerically evaluated with the ex-perimental data of Kobayashi, for nonunity Le flamesat 1 bar. The predictive capabilities of the two mod-els are well preserved for the methane/air flames bytuning the model preconstant in the LV closure andby slightly altering the exponent value of the wrin-kling length scale Ly in the BML-model. To includethe influence of different fuels, the mean reaction rateclosures were substantiated by inclusion of an ex-plicit 1/Le factor, for improved numerical findings.Prediction of flame shape and brush thickness, how-ever, suggested a preference for the LV closure [9],supported by the qualitative shapes of the experimen-tal flames of Kobayashi. We therefore restricted to thefurther investigation of the simple-structured Lindst-edt and Váos reaction closure, with a special interestto the influence of pressure and fuel type.

The present numerical analysis is facilitated bythe extensive set of Kobayashi flame data [2,22,23]at high pressure and varied degree of turbulence onthree gaseous hydrocarbon fuels. These data were ob-tained for operations up to 30 bar, involving morethan 100 flames [24]. Nozzle exit velocities (of thepremix mixture) ranged between 0.86 and 8.86 m/s,with geometrical Reynolds numbers (based on nozzleexit diameter) up to 115,000; turbulence r.m.s. veloc-ity and transverse integral length scale up to 2.06 m/sand 1.90 mm, respectively. The data set involvedthree gaseous hydrocarbon fuels, with lean methane/air mixtures of φ = 0.9 for 1, 5, 10, 20, and 30 bar,ethylene/air with φ = 0.5, 0.7, and 0.9 for 1, 5, and10 bar, and propane/air φ = 0.9 for 1 and 5 bar. Theexperimental averaged shape of a Bunsen flame wasdetermined using the schlieren technique from the en-semble average of 50 instantaneous images for everyflame. Typical average cone angle was determined foreach flame from the location with a 50% probabil-ity of finding burned gas. In Fig. 1, the experimentaldata are shown in a premixed turbulent regime dia-gram. While some of the flames (especially at weakpressure) fall into the classical flamelet regime C,the high-pressure flames are expected to take cor-rugated or thickened flame fronts, following oldertheories. However, both direct numerical simulations[25,26] and evaluation of some of the detailed exper-iments [27–29] and theoretical evaluations [29–31]show an increased spread of the thin flame regime,especially if the turbulent Reynolds number is not toohigh. Within this thin flame regime, reaction is as-sumed to occur in asymptotically thin layers with awell-defined inner structure corresponding to that ofstretched laminar flame. The instantaneous composi-

Fig. 1. Full set of experimental data of Kobayashi et al.[2,22,23], along with the boundary of flamelet quenching,and thick flame in the modified phase diagram of turbu-lent premixed combustion. Methane (1), ethylene (P), andpropane (!).

tion field is postulated to consist of regions of eitherunburned or completely burned gas, separated by thinwrinkled or corrugated interfaces. And, as sL is astrong decreasing function of pressure, higher ratiosof turbulence–chemistry interactive term u′/sL reach-ing partly up to at elevated pressures.

Increase of pressure is known to influence bothturbulence as well as laminar premixed flame char-acteristics. Experimentally observed flames are morewrinkled at increased pressure [2–4], where espe-cially smaller turbulence scales (Taylor scale, Kol-mogorov scale) decrease, while the turbulent inte-gral scale remains nearly unaffected by pressure. Thiscan be understood from the classical turbulence the-ory, based on decrease of kinematic viscosity ν =μ/ρ with pressure. Thus, increased pressure induceshigher turbulent Reynolds number, Ret = u′lx/ν (ifthe mean velocity is held constant, typically the turbu-lence intensity u′ is also only weakly affected by pres-sure). The small scales of turbulence depend inverselyon the turbulent Reynolds number (Taylor scale lλ ∝lx/Re0.5

t ; Kolmogorov scale lk ∝ lx/Re0.75t ; lx is the

integral length scale). Correspondingly, measured en-ergy spectra on turbulence show a shift to higher fre-quency regions [2]. On the other hand, the structure oflaminar flames also depends on pressure. It is knownfrom detailed laminar flame calculations that the lam-inar flame speed decreases with increase of pressure.It is approximated for methane as sL ∝ p−0.5, and forethylene and propane as sL ∝ p−0.25. The laminarflame thickness also decreases with increasing pres-sure, depending on the details of the local transportand reaction processes, where asymptotic theoriespredict a dependency such as δL ∝ α/sL ∝ ν/sL [32].The kinematic viscosity can thus be featured as afundamental parameter relating the influence of pres-sure on flame characteristics. It is of particular inter-

666 N.K. Aluri et al. / Combustion and Flame 145 (2006) 663–674

est to note that the mean reaction rate of premixedturbulent flames increases with pressure, despite de-crease in sL. This is the result of pressure effectson turbulence-induced flame wrinkling and laminarflame and is discussed in more detail in the frame ofthis study.

With a short introduction to the LV reactionmodel, the work is organized as follows. Numericalresults of lean methane/air flames (Le = 1) followedby lean ethylene and propane/air flames (Le > 1) at1 bar are convened. In a successive step, the influ-ence of high pressure is investigated for three-fuelflame data. Based on comparisons between calcu-lated and experimental flame angles, the LV modelis “tuned” to include the influence of pressure andfuel. For the latter, a simple 1/Le-relation found ina previous study is exchanged with the more elab-orate dependent term. Some theoretical discussionfor the devised “tuned” Lindstedt–Váos (tLV) modelwith its pressure dependencies is given based onthe KPP-analysis of the integrated reaction rate inthe frame of turbulent flame speed. In a support-ive study, the cross-influence of the turbulent vis-cosity model is investigated for some representativeflames for two Schmidt number values and for thebehavior of the reaction closure under spatially uni-form turbulence fields. Finally, advantages as well aslimits of the substantiated reaction closure are dis-cussed.

2. Numerical reaction model

2.1. Reaction progress variable

As turbulent premixed flames are often charac-terized by thin reaction zones [33], the mass frac-tions of the species and temperatures may be ex-pressed as a function of a single reduced progressvariable, c (c = 0 within fresh reactants and c = 1within burned products). The transport equation forthe Favre-averaged (mass-weighted) progress vari-able c takes the form

(1)∂

∂xj(ρuj c) + ∂

∂xj

(ρ˜u′′

jc′′) = ∇(ζ ) + wc,

where wc is the mean chemical reaction rate. Thecontribution of molecular diffusion ζ is usually ne-glected for high-Reynolds-number flows. The sec-ond term describes the turbulent flux, which is mod-eled using the classical gradient-transport assump-tion, ρu′′c′′ = −(ρνt/σc)∇ c, where σc is the turbu-lent Schmidt number. Thus, for steady state condi-tions, Eq. (1) reads as

(2)∂

∂xj(ρuj c) = ∂

∂xj

(ρ

νt

σc

∂c

∂xj

)+ wc.

2.2. The reaction rate model

Assuming that reaction occurs in thin flame sheetsseparating unburned and burned gases, wc may be ex-pressed as the product of the flame surface area perunit volume Σ and the laminar flame speed sL:

(3)wc = ρu · sL · Σ.

One major advantage of this approach is in decou-pling chemistry from the flame–turbulence interac-tion described by Σ . Of many possibilities in model-ing this complex term Σ (e.g., [10,14,30,34–38]), thefractal concept from which the LV reaction model wasderived is discussed in the following. This LV modelbelongs to the genre of algebraic models. It was devel-oped on the assumption that the flame surface geom-etry is fractal [34], following a self-similarity powerlaw between an inner and an outer cut-off scale. Thefractal theory was applied to evaluate the increasein flamelet surface area due to turbulent eddies. Themean flame surface density 〈Σ〉 is

(4)〈Σ〉 ≡ A/l31

l

(ε

l

)2−D

,

with 2 < D < 3. For its compatibility with the dif-fusive/dissipative characteristics of passive or reac-tive scalars, a finite limit for the surface area is es-tablished, with an inner cut-off εi introduced suchthat lx � ε � εi. Gouldin and Dandekar [39] haveargued for identifying the inner cut-off scale as theKolmogorov length scale (i.e., εi = lk). Similarly,to accommodate for the geometrical constraints, thelargest self-similar scale of wrinkling is related tothe lx , with outer cut-off εo ∼= lx , so that l � εo � ε �εi. To ensure isotropicity and for l to be at least equalto the expected largest scale of wrinkles, l = εo = lx .Thus,

(5)〈Σ〉 ∝ 1

lx

(lk

lx

)2−D

PR,

where PR is the probability of reaction occurringwithin the volume under consideration. For the proba-bility of reaction, following [40], Lindstedt and Váosused the empirical relation satisfying the extremumflame boundary conditions c = 0 and c = 1 across theflame front for the flamelet regime of combustion:

(6)PR = c(1 − c).

Lindstedt and Sakthitharan [41] proposed D equalto 7/3. Substituting this value into Eq. (6), with the in-troduction of Kolmogorov velocity VK and assuminglx ∝ k3/2/ε, the Lindstedt–Váos (LV) reaction modelis [12]

(7)wc = CRρusL ε

c(1 − c),

VK k

N.K. Aluri et al. / Combustion and Flame 145 (2006) 663–674 667

where CR is the model constant.The critical assumption implicit in the derivation

of the above expression is that vortices of all sizes be-tween the integral and the inner cut-off Kolmogorovlength scales [34] contribute to the wrinkling of theflame surface. Gülder et al. [42] found from other ex-periments the fractal dimension D to be 2.2, ratherthan 2.33, used in the LV model (see also [43,44]).Other expressions found in literature relate the in-ner cut-off to the Gibson scale [33] or the laminarflame thickness δL [45]. The quantity sL/VK is statedto represent the relation between reacting (laminarflame propagating with sL) and passive scalars (turbu-lent mixing). Lindstedt and Váos set the reaction rateparameter CR = 2.6 to reach quantitative agreementwith counterflow experiments [12]. The exact valueseemed to depend on the flame geometry [46]. Addi-tionally, it should be noted that Lindstedt and Váosmodeled the turbulent flux term with a second mo-ment closure, while in the following study a simplereddy viscosity approach is used. Váos investigated thecross influence between the turbulent flux model andthe reaction model [46]. For an increased CR value,the simple eddy viscosity approach gave reasonableresults (CR = 3.25 for the eddy viscosity approachcompared to CR = 1.5 for the second moment closurefor the discussed experimental data [47]). We con-clude that the eddy viscosity closure for turbulent fluxin the combustion progress variable transport equa-tion is an acceptably practiced approach, at least aslong as the prediction of flame brush thickness is notthe central focus.

Together with the transport equation for c, this re-action model is implemented via subroutines into thecommercial finite-volume-based computational fluiddynamics code [48], solving for the Favre-averagedNavier–Stokes equations. Pressure–density couplingis based on the SIMPLE algorithm, and turbulenceis modeled with the standard formulation of the k–ε model. The following relations link the theoreticalturbulence properties and the calculated turbulencequantities k and ε,

(8)u′ =√

23 k, lx = c

3/4μ

k3/2

ε, νt = cμ

k2

ε,

with cμ = 0.09 and νt the turbulent kinematic viscos-ity.

With the thin flame assumption, the followingmean density relation is shown to be valid [49]:

(9)1

ρ(c)= 1 − c

ρu+ c

ρb.

Both ρu and ρb are fed to the solver as input, assum-ing adiabatic flame conditions. In a post processingstep, calculated c is transformed to c using

(10)c = (1 + τ )c,

1 + τ c

with the heat release parameter τ = (ρu/ρb − 1) [11].This conversion is necessitated for direct comparisonwith the experimental data available in Reynolds-averaged form. Note that the difference betweenReynolds-averaged and Favre-averaged flames isremarkable for a Bunsen flame [9]. In an earlierstudy [50], the grid dependency was investigated forone typical flame. Only a weak influence of the gridresolution (by varying the number of cells per cm be-tween 5 and 50 corresponding to a number of totalgrid volumes slightly above 60,000) on the calculatedflame cone angle was found. Therefore, in the fol-lowing study also a grid resolution of 25 cells/cm isused on a two-dimensional axisymmetric computa-tional domain, corresponding to 75 × 200 grid points.

3. Investigation of the fractal-based reactionmodel from Lindstedt and Váos

Calculations are performed for a wide range ofdata—three fuels (CH4, C2H4, and C3H8), varied tur-bulence levels (u′/sL as high as 25), and pressures upto as high as 30 bar. Simulated results obtained fromthe LV model (Eq. (7)) are compared with the exper-imental counterparts, and the merits and limitationsof the model are elucidated. Comparisons are madefor the flame cone angle, estimated through the op-timal tangent drawn over the c = 0.5 contour of theReynolds-averaged reaction progress variable. Usingthe flame angle method, a turbulent flame speed maybe used as a representative of the flame cone angle,sT = U sin(θ/2). In the following, the flame anglesare mostly presented in the nondimensional sT/sLform. This term also may be interpreted as an approx-imation of the ratio between turbulent and laminarreaction rates.

3.1. Influence of fuel

In Fig. 2, the calculated flame cone angles ob-tained from the LV model of Eq. (7) with CR =2.6 are compared with the corresponding measureddata at 1 bar. This clearly shows that the model isable to predict the flame speed variation qualitativelywith turbulence. Comparing the methane–air flameswith the increase of u′/sL from 0.3 to 1.35, devi-ations enlarge, an indication of the nonapplicabilityof the original constant CR = 2.6. Additionally, thisfigure shows that all calculated flames of methane–air, ethylene–air (equivalence ratio φ = 0.7 and 0.9;Le = 1.2), and propane–air (φ = 0.9; Le = 1.62) es-sentially fall onto a straight line, while the experi-mental data differ significantly. While the reactionrate (being proportional to sT/sL) is underpredicted

668 N.K. Aluri et al. / Combustion and Flame 145 (2006) 663–674

Fig. 2. Comparison of calculated flame angle in sT/sL fromthe LV model with the experimentally measured data ofKobayashi for methane (φ = 0.9), ethylene (φ = 0.7 and0.9), and propane (φ = 0.9) flames at 1 bar.

Fig. 3. Methane–air flames (φ = 0.9) at 1 bar from the tLVmodel. Also included are results from the LV model for com-parison.

for methane flames, and to a lesser extent for eth-ylene flames, it is slightly overpredicted in the caseof propane flames. Obviously the numerical model isinsufficient to elucidate the fuel effects in this form.Before overseeing the expansion of the LV model,(Section 4, with results in Figs. 3 and 4), the influ-ence of pressure on the LV model is discussed.

3.2. High-pressure influence

The influence of pressure on turbulent flame speedis also studied, being of practical interest such as togas-turbine combustors. It was evident from experi-ments that as pressure rises (from 5 to 30 bar), sT/sLincrease with u′/sL. Calculations performed for pres-sures 5, 10, 20, and 30 bar consist of 5, 4, 10, and7 methane–air flames, respectively. For the 20 and30 bar cases, due to unavailability of the measuredintegral length scale, a constant lx = 1.17 mm is as-sumed. With pressure rise from 1 to 10 bar, experi-ments showed a significant increase in sT/sL, espe-

Fig. 4. Flame angles in sT/sL from the tLV model for CH4(φ = 0.9), C2H4 (φ = 0.7), and C3H8 (φ = 0.9) flames at1 bar.

Fig. 5. Measured and calculated flame angle in sT/sL (LVand tLV) for methane flames (φ = 0.9) at 5 and 10 bar.

cially up to 5 bar, whereas the behavior of the LV re-action model remains passive, yielding a single low fitcurve (see Fig. 5; note that calculations from the mod-ified model (tLV), which appears in the next section,are already included in the following figures). Clearly,pressure effects are missing in the LV reaction model.For methane flames at still higher pressures, 20 and30 bar, again a clear difference between experimentand simulation is found (Fig. 6). The ethylene flames,shown in Fig. 7, cover a broad spectrum of exper-imental data based on a blend of three equivalenceratios (0.5, 0.7, and 0.9) for two different pressures,5 and 10 bar, with u′/sL ranging up to 24.0 and tur-bulent Reynolds numbers as high as Ret = 1200.The versatility of the LV model in yielding respon-sive qualitative fits over this studied range is shown.However, as was found in the aforementioned sec-tion, the LV closure conceals the embedded pressureeffects, including the variation in fuel type (Fig. 7).The simulated data on a set of more than 60 flamesexhibits too low reaction rates collapsing to a singlecurve. For propane flames at 5 bar, the model showsquantitative differences, although giving a fairly good

N.K. Aluri et al. / Combustion and Flame 145 (2006) 663–674 669

Fig. 6. Methane flame angles (φ = 0.9) at 20 and 30 bar fromthe LV and tLV reaction models.

Fig. 7. Ethylene flame angles for three equivalence ratios,φ = 0.5, 0.7, and 0.9 (not distinguished separately), for pres-sures 5 and 10 bar from the LV and tLV reaction models.

Fig. 8. Propane flame angles (φ = 0.9) at 5 bar using theLV and tLV (calculated using the exponential Lewis numberrelation, Eq. (11)) models. The square symbols show the tur-bulent flame speed evaluated using the 1/Le, relation whichover predicts at increased turbulence level.

trend, shown in Fig. 8. So far, this comparative studyshows that the LV closure gives qualitatively accept-able trends for varied turbulence conditions, witness-ing unfavorable quantitative yields, not accommodat-ing the effects of fuel–air mixture and pressure. In the

following, this closure is further addressed in its mod-ified form.

4. The tuned Lindstedt–Váos (tLV) reactionclosure

Principal objective of this part of the work refersto the LV model in its substantiated form—whichwe call the tuned Lindtstedt–Váos model (tLV)—byinfusing the fuel–air mixture effect and the high-pressure influence. Recall that in the previous section,the LV model in its original form has been shownto underrate the experimental findings, and insistson tuning of the preconstant accordingly. In an ear-lier study [9], a new preconstant CR = 4.0 for thefour measured atmospheric methane–air flame data(Fig. 3) was found, resulting in close proximity tothe experimental data. This new value remains asthe principal constant for the rest of the investigateddata. Interestingly, Gouldin et al. [40] have also pro-posed CR = 4.0 in their numerical investigation of anoblique flame. Indeed, Lindstedt and Váos assigned arange of values CR = 3.25 to 4.5, if the eddy viscosityturbulent diffusion closure is used [46,51], supportingour choice of the model constant.

4.1. Substantiating the LV model by inclusion of theLewis number

In practice, fuels of higher molecular weights(usually of high Lewis numbers) than methane haveapplication in spark ignition engines and partly in gas-turbine combustors, so that study of the fuel effects onthe flame characteristics at high pressure is of signif-icant importance. A recent experimental and theoret-ical study on high-pressure flames by Soika et al. [4]affirms the causative important Lewis number effect,highlighting that both flame-generated vorticity andflame instability behavior depend strongly on ther-mophysical properties of the premixed flame, i.e.,effects caused by the density jump and differentialdiffusive fluxes. It further finds that the Lewis num-ber of the fuel–air mixture has a substantial impacton the extent of flame curvature in the given turbu-lent flow field. In addition, nonunity Le influence isone of the key parameters in proper understandingof the flame–turbulence interaction [18]. In a recentstudy by Muppala et al. [16], an explicit Lewis num-ber effect was found for an algebraic reaction rateclosure, developed on the concept of turbulent flame–wrinkling ratio. Computed results on the nonunityLe ethylene– and propane–air mixture data using thisreaction rate parameter show that there exists an addi-tional difference between experiment and simulation.To reach cooperative agreement for the atmospheric

670 N.K. Aluri et al. / Combustion and Flame 145 (2006) 663–674

flame data, the influence of fuel type has been inter-preted as a Le effect. Testing several CR(Le) func-tions (not produced here) it is found that an exponen-tial Le term,

(11)CR,Le = 4.0

eLe−1,

results in very good agreement with the measure-ments. Fig. 4 shows this impressively for the 40 dif-ferent atmospheric flames for all the three fuels.

In a previous work, an approximated CR ∝ 1/Ledependency was used [9]. The difference is small forthe methane– and ethylene–air flames, but it is sig-nificant for the propane–air flames with Le = 1.62,as can be seen in Fig. 8. The 1/Le relation overpre-dicts, especially at increased pressure or turbulencelevel. The empirically found exponential dependencyis consistent within the leading point model conceptsdiscussed in a recent review by Lipatnikov and Cho-miak [17, pp. 38–48]. They outline the following:(1) Premixed turbulent flame propagation is consid-ered to be controlled by the flamelets that advance far-thest into the unburned mixture (the so-called leadingpoints). (2) These leading flamelets are assumed tohave an inner structure same as a critically perturbedlaminar flame independently on turbulence charac-teristics. (3) And, a critically curved laminar flameis invoked to model the inner structure of the lead-ing flamelets. Accordingly, turbulent flame speed iscontrolled by the characteristics of a critically curvedlaminar flame, rather than by the characteristics ofan unperturbed planar laminar flame. To calculate theformer characteristics, Lipatnikov and Chomiak [17]have invoked the well-known theoretical solution thatpredicts that the burning rate in a stationary flame ball(a critically curved laminar flame) is higher than theburning rate in the unperturbed planar laminar flameby a factor of exp(1 − Le) [52, pp. 327–331; 53]. Toelucidate the direct presence and significant impact ofthe Lewis number, it is of interest to compare typi-cal cases of methane and propane flames with nearlyidentical flow and turbulence conditions. This is donefor a pair of flames (Table 1). Calculated flame coneangles from the LV and tLV models are tabulatedin Table 2, showing that the latter model is in goodagreement with experiment. For the unity Lewis num-ber flame, the LV model differs by as much as 21.5◦with the measured value, with Eq. (11) simplified toCR = 4.0. For the nonunity Lewis number (Le = 1.6)flame this factor results in CR = 2.15, which is mar-ginally close to the original preconstant CR = 2.6. Itis worth emphasizing here that fuel–air mixtures char-acterized by very weak turbulence are not specificallydistinguished from the other data here, as sL remainsa strong variant of pressure, but a majority of theseflames can be easily classified based on the u′/sLrange.

Table 1Methane and propane flame data under nearly identicalflow and turbulence conditions (pressure in bar, length scalein mm, velocities in m/s)

Fuel φ p Le U u′ lx sL u′/sL

CH4 0.9 1 1.0 2.36 0.46 1.25 0.34 1.35C3H8 0.9 1 1.62 2.25 0.51 0.9 0.395 1.29

Table 2Full flame cone angle θ and normalized turbulent flamespeed sT/sL for the cases described in Table 1

Model/fuel LV Exp tLV LV Exp tLVθ◦ θ◦ θ◦ sT/sL sT/sL sT/sL

CH4 40 61.5 60 2.37 3.55 3.47C3H8 45.6 38 43 2.21 1.85 2.09

4.2. Intermediate discussion: KPP analysis

In this section, the influence of pressure on re-action rate, interpreted in sT, is illustrated. Thesebelow-discussed relations are not used in the numeri-cal simulations, but only serve to physically interpretthe numerical and experimental observations. Forthis purpose the classical KPP analysis (see [54])is applied. Here, the balance equation of a one-dimensional steady propagating flame is combinedwith the LV reaction closure with the assumptions thatsT is equal to the magnitude of the incoming mean ve-locity, and that turbulence is not affected by the flame:

(12)ρusT∂c

∂x= ρ

νt

σc

∂2c

∂x2+ CRρu

sL

VK

ε

kc(1 − c).

Assuming that the leading edge of the flame (small c)determines the dynamics of the flame (see also [17]),the last term may be expanded (c(1 − c) → c), lead-ing to an ordinary differential equation. Following theKPP theorem, this has a physical solution for sT, if itsdiscriminant is zero [54], leading to

(13)sT = 2

√νt

σcCR

sL

ν0.25u

ε0.75

k.

Assuming turbulence parameters (k, ε, and νt) to beindependent of pressure (a suitable first-order approx-imation), and with the pressure-dependent quantitiesνu ∝ p−1.0 and sL ∝ p−0.5 (for methane–air flames),the overall pressure influence on sT shrinks to

(14a)sT ∝ p−0.125.

This analysis implies that sT (or reaction rate) woulddecrease with pressure, unlike theoretical [17] and ex-perimental findings [2,4–6]. For the two nonunity Leflames (with approximately sL ∝ p−0.25),

(14b)sT ∝ p0.

N.K. Aluri et al. / Combustion and Flame 145 (2006) 663–674 671

Following Figs. 5–8 for methane, ethylene, andpropane flames, a large gap has been found be-tween experimental and calculated values, with thedifferences growing larger with pressure rise. Also,computed results based on the corrected factor ofCR = 4.0 could not account for the influence of pres-sure (these intermediate results are not presented hereexplicitly). Thus, both theory and comparative stud-ies using the simulation and measured data necessitatethat an additional (pressure) influence be accommo-dated into the model.

4.3. Substantiating the LV model for pressureinfluence

A set of nine methane flames for two high pres-sures, 5 and 10 bar, are simulated and analyzed inde-pendently, with the aim of unveiling the influence ofpressure. For 5 bar, with an additional multiplicativecorrection factor of 2.2 (thus CR = 4.0 × 2.2 = 8.8),calculated angles are found to be near the measuredones. Similarly, for 10 bar, following a few itera-tive trials a reasonable but a still bigger factor, CR =4.0 × 3.1 = 12.4, is realized. These additional correc-tion factors fit rather well to a supplemental pressure-dependent factor,

(15a)wc ∝(

p

p0

)0.5,

where p is the operating pressure, with p0 = 1 bar.These calculated flame angles are plotted in Fig. 5(tLV calculation). As can be seen, experiment and thetLV model are in good agreement with each other,with few exceptions at high turbulence.

In terms of the KPP analysis, a pressure depen-dency of the turbulent flame speed is thus

(15b)sT ∝ p0.125

for lean methane–air flames. This value is relativelynear to the experimentally found exponent of 0.07(see Table 3). This explicit pressure influence maybe related to a more fundamental quantity, the mole-cular kinematic viscosity ν(∝ 1/p), as it is scaledclosely to the small scales of turbulence and lam-inar flame thickness. Therefore a correction factorν∗ = ν/ν0(= p0/p) is interpreted with the modified

Table 3Pressure dependency of turbulent flame speed: LV model,Experiment (Kobayashi), and tLV model

sT ∝ px , x = LV Experiment tLV

CH4–air (Le = 1.0) −0.25 0.07 0.125C2H4–air (Le = 1.2) 0.01 0.24 0.26C3H8–air (Le = 1.62) −0.01 0.25 0.25

reaction source term as wc ∝ 1/√

ν∗, with subscript0 being the corresponding atmospheric value. Withthe pressure-dependent term Eq. (15a), calculationsare performed for the other 17 high-pressure 20- and30-bar data. Though the results are not so favorable inretaining the correct quantitative trends (see Fig. 6),they seem promising in a first-order approximation,and certainly give much better results than the ac-tual values. For ethylene and propane flames, the LVmodel pronounces a pressure-independent reactionrate (see Figs. 7 and 8 and Eq. (14b)), whereas experi-ments place an sT ∝ p0.24–0.26 dependency. Here, thepressure-dependent reaction rate (Eq. (15a)) from thetheoretical KPP analysis leads to

(16)sT ∝ p0.25.

In Table 3 these theoretically derived pressure depen-dencies for the three fuels are compared with the ex-perimental fits between 1 and 10 bar. The exponentsof the tLV model are fairly near to the experimentalones.

Combining this pressure dependency (Eq. (15a))with the earlier discussed Lewis number effect (ofEq. (11)), the simulated flame data show reasonableagreement to the experimental data for the large set ofdata points from ethylene and propane flames (Figs. 7and 8) under varied turbulence conditions. For theethylene flames, the calculations for 5 bar cases arevery near to the measurements, while somewhat un-derestimating for 10 bar (Fig. 7). For the propaneflames at 5 bar (Fig. 8) the tLV model gives rathergood results in conjunction with the mentioned ex-ponential Lewis number dependency, leading only toa slight overprediction at higher turbulence intensity.Following these studies, we conclude that this tunedLV model is found to give fairly good quantitativeresults so far for the broad set of nearly 100 ex-perimental flames. These numerical results are wellsupported via theoretical argumentation. In summary,the tuned Lindstedt–Váos (tLV) reaction model isgiven as

wc = CRCP,Le ρus0L

VK

ε

kc(1 − c), with CR = 4.0

(17)and CP,Le =√

p

p0

(1

eLe−1

),

including the explicit influence of both fuel type (viaLewis number) and pressure.

Two supportive verification studies are carried outand are described briefly in the following. The firstis to investigate the influence of the turbulent fluxmodel. In the second, the behavior of the reactionclosure in spatially uniform turbulence fields is stud-ied.

672 N.K. Aluri et al. / Combustion and Flame 145 (2006) 663–674

Table 4Inlet flow and flame conditions and experimental and calculated flame angles (sT/sL) for varied Schmidt number (tLV model)

Fuel φ p U u′ sL lx u′/sL sT/sLExp

sT/sLtLV(σc = 0.7)

sT/sLtLV(σc = 1.0)

CH4 0.9 1 2.36 0.46 0.340 1.25 1.35 3.55 3.47 2.880.9 5 2.21 0.40 0.152 1.15 2.63 8.38 8.52 7.640.9 10 2.11 0.36 0.108 1.10 3.35 12.34 11.54 11.6

C2H4 0.5 5 6.95 1.64 0.086 1.40 19.17 15.55 15.85 12.640.7 5 7.53 1.75 0.243 1.50 7.19 12.32 12.08 10.85

Fig. 9. The weak influence of the Schmidt number σc (= 0.7and 1.0, respectively, for left and right flames) on the tur-bulent flame speed is shown for a lean methane–air mixture(φ = 0.9, u′/sL = 2.63), using the tLV model.

4.4. Eddy viscosity turbulent diffusion closure

In the original work of Lindstedt and Váos, the tur-bulent flux term has been treated with second momentbalance equations [12], while in the current study it isrestricted to the use of gradient closure of ρu′′c′′ withthe standard eddy viscosity approach, with turbulentdiffusion coefficient νt/σc. Váos [46] has investigatedthe difference between the two approaches, as men-tioned in the beginning. Here, we varied σc in orderto understand the influence of diffusion coefficient onthe computed sT. The value of σc is usually taken aseither 0.7 or 1.0 (our standard calculations were basedon σc = 0.7). For a set of six test flames with dif-ferent fuels and pressures, the calculated as well asexperimental sT values are given in Table 4 for thetwo Schmidt numbers σc. A decreasing influence ofσc on sT with turbulence is observed. Fig. 9 shows atypical calculated methane flame in reaction progresscontours for σc = 0.7 and 1.0 at 5 bar from Table 4.

This study reveals that the flame cone angle showsno visible variation with σc at moderate turbulence,and that its effect decreases with u′/sL. But it is likelyto influence the flame brush thickness marginally.This analysis supports so far the usage of simplifiededdy viscosity closure for turbulent scalar flux.

4.5. Behavior of the reaction closure in spatiallyuniform turbulence field

An open question that deserves further inves-tigation is the influence of the numerically calcu-

lated flame on the modeled turbulence in the vari-able density case. In order to get some indicationof this influence, we compared in a supplementalstudy few numerical flame angles and correspondingmean velocities with those obtained, for spatially uni-form turbulent flow fields. A comparison is shown inFig. 10 for the three methane flames in Table 4. Themean flame shapes are essentially similar comparedto the cases of variable densities. A slight differencein flame brush, broadening along the entire flameregion for the high-pressure case, is seen. The com-puted turbulent flame speed from both the turbulentflow situations remains more or less unchanged. Themean velocity magnitude is affected within the mar-ginally acceptable level. In overall, the cross influencebetween reaction and turbulence model is not verysignificant.

4.6. Delimits of the tLV model

Applying the tuned LV model, it is found that sim-ulated flame data differ from measurements at veryweak turbulence level at elevated pressures. For a fewsuch flames falling in the range u′/sL < 1 at andbeyond 5 bar, the calculated flames are partly sup-pressed to a flat flame at the very exit of the noz-zle (similar to “numerical flashback” due to overpre-dicted reaction rate). On the other hand, the propaneflames at 5 bar have been found with underpredictedflame angles. It should be noted that for very low tur-bulence intensities the rather complex phenomenon oflaminar flame instabilities, as well as possible mis-modeling of eventually relaminarized turbulence maybe of additional importance. Lindstedt and Váos [12]proposed an empirical procedure, formally adoptedfrom Sreenivasan [55], for low-turbulent flows. Thismethod evaluates the reaction rate via an exclusiveexpression for the outer cut-off (integral length) scale,lx = CLu′3/ε, where CL(Reλ), in order to reconcilecalculations and experiments. Allowance of this ap-proach to the current data has not benefited, so far(not shown here) that the results were found inconsis-tent over the span of very weak turbulence data.

N.K. Aluri et al. / Combustion and Flame 145 (2006) 663–674 673

Fig. 10. Influence of the numerical effect on flame-generated turbulence. Three cases: methane–air at 1, 5, and 10 bar (from leftto right). Top row: reaction progress variable. Bottom row: mean velocity magnitude. Left block: spatially uniform turbulencefield. Right block: variable-density nonuniform turbulence field. White regions in the velocity magnitude in the second and thirdcontours from left correspond to velocities higher than the shown maximum scale of 4.5 m/s.

It is therefore necessary that special care be takenin simulating flames at very low turbulence, whichseems to be the case with any other existing reactionmodel when used in conjunction with the standardk–ε turbulence model. Whether or not the proposedfuel influence is sufficient also for other fuels (orfor stoichiometric or even fuel-rich mixtures) has re-mained beyond the scope of the current work.

5. Conclusions

A formally simple algebraic reaction rate closurederived from fractal flame analysis proposed by Lind-stedt and Váos (LV model) was extensively stud-ied for a broad set of turbulent premixed Bunsenflames. The comparison was based on data of nearly100 flames, consisting of lean methane–, ethylene–,and propane–air mixtures for pressures up to 10 bar(partly 30 bar) and for a wide range of turbulence andflow conditions, u′/sL � 24. For comparison, essen-tially the mean flame position of the Bunsen flames,described as the mean flame cone angle, or in dimen-sionless form as sT/sL, was used.

With the original fractal reaction closure (LVmodel), for model constant CR = 2.6, the calculatedatmospheric methane–air flames showed underpre-diction of reaction rates. Here, tuning to CR = 4.0was found essential. With this new value, however,ethylene and propane flames were overpredicted,with the differences being fuel-dependent. Follow-ing comparison with the experimental set of data aswell as based on theoretical arguments, these differ-ences were attributed to the nonunity Lewis number(Le > 1) of the fuels. A new term was successfully in-troduced, modeling the reaction rate as proportionalto exp(1 − Le).

In case of high-pressure flames, the LV modelalways underpredicted the mean reaction rate, withgrowing differences with pressure. Through a system-atic semiempirical approach based on the set of 5-and 10-bar methane–air flame data, the multiplicativepressure term

√p/p0 substantiated the model signif-

icantly. This relation yielded acceptable trends evenat higher pressures (20 and 30 bar of methane–airflames).

For other fuels (ethylene and propane, besidesmethane) at varied pressures, the combined prefactorexp(1 − Le)

√p/p0 added to the LV closure has been

found to be in very good terms for most of the investi-gated flames under varied flow and turbulence levels.

This tuned LV reaction model includes now twoimportant phenomena of technical relevance—theLewis number effects for different fuels and the in-fluence of high pressure on the reaction rate.

Acknowledgments

We are grateful to Hideaki Kobayashi (TohokuUniversity, Japan) for providing us with his ex-perimental flame data. This work was funded inthe frame of the Bavarian Research CooperationFORTVER, hosted by the Arbeitsgemeinschaft Bay-erischer Forschungsverbünde abayfor.

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