Large-eddy simulation of a lifted methane jet flame in a vitiated coflow

Combustion and Flame 152 (2008) 415–432 www.elsevier.com/locate/combustflame

Large-eddy simulation of a lifted methane jet flame in a vitiated coflow P. Domingo a , L. Vervisch a,∗ , D. Veynante b a LMFN, CORIA—CNRS, Institut National des Sciences Appliquées de Rouen, France b EM2C—CNRS, Ecole Centrale Paris, France

Received 2 March 2007; received in revised form 20 June 2007; accepted 10 September 2007 Available online 14 November 2007

Abstract The impact of burned gases on flame stabilization is analyzed under the conditions of a laboratory jet flame in vitiated coflow. In this experiment, mass flow rate, temperature, and the exact chemical composition of hot products mixed with air sent toward the turbulent flame base are fully determined. Autoignition and partially premixed flame propagation are investigated for these operating conditions from simulations of prototype combustion problems using fully detailed chemistry. Using available instantaneous species and temperature measurements, a priori tests are then performed to estimate the prediction capabilities of chemistry tabulations built from these archetypal reacting flows. The links between autoignition and premixed flamelet tables are discussed, along with their controlling parameters. Using these results, large-eddy simulation of the turbulent diluted jet flame is performed, a new closure for the scalar dissipation rate of reactive species is discussed, and numerical predictions are successfully compared with experiments. © 2007 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Large-eddy simulation; Detailed chemistry tabulation; Autoignition; Partially premixed turbulent combustion

1. Introduction Recirculating burned gases exist in most combustion systems where fuel and oxidizer are mixed with combustion products before they enter the main combustion zone. These recirculating hot products carry thermal energy that may help flame stabilization. They are also useful to dilute the reaction zones, which homogenizes the mixture and avoids the high temperature levels responsible for NOx emissions. In gas turbine combustion chambers, hot recirculat* Corresponding author.

E-mail address: [email protected] (L. Vervisch).

ing combustion products contribute to flame stabilization downstream of the swirled injection [1]. In car engines, exhaust gas recirculation (EGR) is one of the options for reducing NOx emissions in usual engines and for controlling ignition in new homogeneous charge compression ignition (HCCI) engines, where the chemical composition of the burned gases plays a key role [2]. Burned gases are also present in furnaces, where they contribute to flame stability. For a high level of flame dilution by burned gases, the moderate or intense low-oxygen dilution (MILD) [3] or the flameless regime is observed [4]. In the latter, very high dilution by burnt gases leads to a strong decrease of temperature gradients followed by a dramatic reduction of NOx emission.

0010-2180/$ – see front matter © 2007 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.combustflame.2007.09.002

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Accordingly, almost no practical combustion systems exist where recirculating burned gases are not intimately involved in the process and control part of the behavior of the turbulent reaction zones. The accurate control of recirculating hot products is now the main target of many design studies devoted to real combustion chambers. Nevertheless, so far, only a few studies have discussed very high dilution by burnt gases (MILD or flameless) in the context of turbulent combustion modeling [5]. A variety of combustion regimes can be observed downstream of a nonpremixed injection of reactants within recirculating burnt products. The high temperature of the gases can favor autoignition, the partial premixing of fuel with oxidizer can lead to the development of partially premixed flame propagation, and, further downstream, diffusion flame burning is likely to develop. Chemistry tabulation is discussed for these various regimes in the context of large-eddy simulation (LES) of turbulent combustion, which is a rapidly growing field [6–12]. Links between the a priori distinct autoignition, premixed, and diffusion flamelets’ low-dimensional manifolds are evidenced for the operating conditions of the lifted methane– air jet flame in a vitiated coflow experiment reported by Cabra et al. [13]. This experiment does not operate in the MILD or flameless regimes, but it constitutes a first step in the study of flame stabilization by burnt gases. A priori tests of chemistry tabulation are organized apart from any flow simulation. The control parameters of chemistry tabulation are obtained from the measurements, and species concentrations extracted from the tables are compared with their measured counterparts. Subsequently, this chemistry tabulation is combined with presumed conditional moment (PCM) subgrid scale (SGS) modeling [11,14], to perform LES of the turbulent lifted jet flame in a vitiated stream. A new closure for the scalar dissipation rate of reactive species is discussed and included in the LES. Time averages of filtered means and filtered conditional means in mixture fraction space of temperature and species are then compared with measurements, and the dynamics of the turbulent flame base is analyzed.

2. Autoignition and flame propagation in a vitiated jet flame 2.1. Conditions of the Cabra et al. experiment The exact chemical composition and mass flow rate of recirculating burnt gases are often difficult to calibrate and measure accurately in real combustion chambers. To gain greater understanding of flame stabilization in environments where burnt gases domi-

nate, Cabra et al. [13] designed a laboratory vitiated burner in which the injection conditions of reactants and hot products are fully determined. In addition to detailed scalar measurements, Reynolds-averaged Navier–Stokes (RANS) calculations were also reported [13,15–17]. The burner consists of a round rich premixed fuel jet issuing into a coflow of hot combustion products. This vitiated stream is obtained from a previous hydrogen/air lean premixed combustion (equivalence ratio φ = 0.4) mainly composed of H2 O and air flowing at 1350 K. The central fuel jet mixture is composed of 33% CH4 and 67% air, by volume (equivalence ratio φ = 4.4). The bulk velocity of the fuel jet and of the coflow velocity are of the order of 100 and 5.4 m/s, respectively. The geometry of the burner and the operating conditions considered in the present work are summarized in Table 1. This burner configuration was first used to study H2 /N2 jet flame in vitiated coflow [18–20] and LES of this hydrogen flame was reported [21]. More details concerning the burner and measurements techniques may be found in [13]. 2.2. Autoignition and flame propagation in diluted combustion Various flame propagation scenarios have been discussed to explain experimental and numerical observations of lifted flame stabilization in the case of gaseous or spray injection, with or without swirl [22–25]. Flame stabilization was attributed to the existence of distorted partially premixed fronts interacting with turbulence and propagating against the incoming flow. In the presence of hot products, the reactants are partially premixed with gases that bring additional energy that can help with flame stabilization, mainly through thermal effects that reduce the autoignition delay, increase the flame speed, and diminish the flame sensitivity to quenching. These effects may be counterbalanced when too much diluent is mixed with the reactants; then the reaction zones development is reduced because of the presence of an additional thermal ballast and the lack of reactivity of the highly diluted mixture. In the Cabra et al. experiment, the fresh reactants issuing from the central jet are mixed with hot product (H2 O) and air upstream of the turbulent flame base. The lifted flame starts at a streamwise position located somewhere between X/D = 30 and 40. X denotes the vertical axis of the burner and D is the diameter of the fuel jet; X = 0 at the burner lips (Table 1). Upstream of the flame base, scalars ϕi (temperature, species concentrations) follow a response close to ϕi (Z) = ZϕiF,o + (1 − Z)ϕiO,o ,

(1)

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Table 1 Conditions for the lifted methane–air jet flame in a vitiated coflow

Fuel jet Coflow

Re

D (mm)

U (m/s)

T (K)

X O2

X N2

X H2 O

XCH4

28,000 23,300

4.57 210

100 5.4

320 1350

0.15 0.12

0.52 0.73

0.0029 0.15

0.33 0.0003

Note: Re: Reynolds number, D: diameter, T : temperature, X: mole fraction, from Cabra et al. [13].

where Z is the mixture fraction (a passive scalar that equals unity in the fuel jet and that vanishes in the vitiated coflow), and the subscripts F,o and O,o denote the ϕi value in fuel jet or vitiated streams, respectively (ZF,o = 1 and ZO,o = 0). The exact local response of species and temperature upstream of the flame base is in fact more complex, because it is influenced by the physical properties of the fluid (differential diffusion and heat capacities), which may lead to a deviation from the nonlinear response versus mixture fraction. Nevertheless, the measured scatterplot of temperature upstream of the flame base suggests that, on the average, the temperature deviation from the linear behavior stays moderate [26], and Eq. (1) is therefore adopted in this study. The stoichiometric point is located at Zs = 0.1769. For every mixture fraction ranging between the flammability limits, a set of diluted reactants ϕi are generated. Ignition delays and freely propagating premixed flames are then computed for these conditions, chemistry is described with the 49 species detailed in the GRI 3.0 methane–air mechanism [27], and SENKIN and PREMIX [28] softwares have been used with complex transport properties. For a fixed value of the mixture fraction (or equivalence ratio), the species mass fractions YiAI and YiPF , describing autoignition and premixed flamelets, respectively, are controlled by the balance equations ρ

∂YiAI ∂t

ρo SL

= ω˙ iAI ,

∂YiPF ∂x



=

PF 

∂Y ∂ ρDi i ∂x ∂x

+ ω˙ iPF ,

(a)

(b)

(2)

Fig. 1. T (φ, Yc ) for various equivalence ratios φ, whose value is indicated at the end of the trajectory. (a) Autoignition table. (b) Premixed flame table.

(3)

flame propagation are tabulated as

SL is the flame speed and usual notations are otherwise adopted [29]. The results are collected in two chemical tables, YiAI (Z, t) for the perfectly mixed reactors, used to compute autoignition delays, and YiPF (Z, x) for premixed flamelets; t is the time coordinate in the reactor and x the position through the premixed flame. The flame prolongation of intrinsic low-dimensional manifolds (FPI) method is adopted [30]. The tables are reorganized with the progress of reaction Yc (Z, ξ ) = YCO (Z, ξ ) + YCO2 (Z, ξ ), where ξ denotes either time or position. By eliminating the ξ coordinate between Yi (Z, ξ ) and Yc (Z, ξ ), twodimensional manifolds capturing autoignition and

YiAI (Z, Yc );

YiPF (Z, Yc ).

(4)

Fig. 1 shows temperature trajectories for various values of the equivalence ratio φ; the two tables provide temperature trajectories that are almost similar. Chemical source terms are also available in the form ω˙ iPF (Z, Yc ), or ω˙ iAI (Z, Yc ), as displayed in Fig. 2, where ω˙ Yc , the source of Yc , is plotted as a function of Z and Yc . The similarities and differences between the autoignition and premixed flamelets database are further discussed below. The chemical look-up tables used in this study contain 200 × 200 points in the Z–Yc space, with a refined mixture fraction mesh close to the stoichiometry.

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Fig. 4. Flame speed versus equivalence ratio. Fig. 2. Distribution of ω˙ YPF (Z, Yc ) in s−1 . c

Fig. 3. Ignition delay versus equivalence ratio.

Fig. 3 shows ignition delays versus equivalence ratio, for lean mixtures (φ  1). A minimum value of o = 4 ms is found on the this delay of the order of τig very lean side at φ o ≈ 0.02 (Z o ≈ 0.00427), in agreement with observations reported in [13]. Autoignition delays then rapidly increase with equivalence ratio. Autoignition is more likely to contribute to stabilization on the very lean side of the axisymmetric mixing layer. This imposes serious constraints for the numerical simulations of such vitiated combustion problems. For fixed boundary conditions (chemical gas compositions and vitiated stream temperature) and fixed chemical scheme, small variations in the description of mixing on the lean side would lead to large variations of ignition delay, which may strongly impact the turbulent flame base streamwise location. It is therefore crucial in the simulations to carefully capture the mixing up to mixture fraction values well below the stoichiometric one. The LES mesh resolution needs to be kept sufficiently high when approaching the vitiated coflow stream and the overall accuracy of the numerical scheme used to transport scalars must be

high enough to avoid any spurious numerical diffusion. The flame speed response to equivalence ratio, SL (φ), is plotted in Fig. 4. It features a usual shape with a maximum level of SLo = 3.6 m/s in the vicinity of the stoichiometric condition, as discussed in [13]. On the lean side, premixed flames exist till φ = 0.24 (Z ≈ 0.05). Below this equivalence ratio, the mixture evolution is governed by fast autoignition and a socalled freely propagating flame cannot be observed because of rapid autoignition of the fresh mixtures. Due to flow divergence ahead of curved flame fronts, the displacement speed of partially premixed flames is known to be higher than SLo , the flame speed of the stoichiometric mixture. Previous studies on laminar and partially premixed flames stabilization [22, 31] have shown that  these reactionzones can sustain velocities U o ≈ (ρo /ρb )SLo ≈ (Tb /To )SLo ; here the subscripts o and b denote fresh and burnt gases for the stoichiometric condition, leading to U o ≈ 5 m/s. Since the coflow maximum velocity (5.4 m/s) is on the same order as this characteristic partially premixed flame speed, it is anticipated that flame propagation can also promote flame stabilization on the stoichiometric isosurfaces. This flame propagation may occur when autoignition was not successful due to local flow conditions, or just after autoignition has initiated combustion. Studies on lifted flames with a 300K coflow have shown that it was difficult to stabilize flames when the coflow velocity approaches U o [32]. In the Cabra et al. experiment, flames were stabilized with coflow velocity up to 6.5 m/s, which is higher than U o ; thus propagation cannot be the only driving mechanism. From these observations, it is expected that both autoignition and flame propagation can contribute to the stabilization of the turbulent flame in the vitiated coflow. The reaction zone thickness, δL , may be evaluated for each equivalence ratio from the maximum gradient of Yc in the unstretched laminar flame δL =

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419

(a) Fig. 5. Characteristic flame thickness versus φ, the equivalence ratio. Eq

Yc / max(dYc /dx) and is shown in Fig. 5. The dilution by hot products contributes to increase the characteristic reaction zone thickness on the lean side. With chemistry tabulated from relations (4), to fully resolve in a direct numerical simulation (DNS) the Yc signal describing the reaction zones, the minimal mesh resolution h ≈ δL /10 would need to be between h < 0.03 mm and h < 0.2 mm, depending on the local equivalence ratio. Notice that much thinner (more than an order of magnitude smaller) radical layers are imbedded within the Yc signal. Those radicals should be resolved in fully detailed chemistry DNS of this jet flame. In the LES discussed below, the Yc signal cannot be fully resolved and subgrid scale effects are modeled.

(b)

2.3. A priori tests of chemistry tabulation from experiments Single-point species concentration measurements are available for the Cabra et al. experiment [26]. The instantaneous measured values of Yc = YCO + YCO2 (using CO LIF) and Z are used to construct chemical flame structures from YiAI (Z, Yc ) and YiPF (Z, Yc ), the autoignition and premixed flamelet chemical tables (Eq. (4)). From every single-point measurement, Z and Yc are extracted and then entered into the tables to get YiAI and YiPF . For every point of the measurements, three values of Yi are then available, one from the measurements and two from the tables, YiAI for autoignition and YiPF for premixed flames freely propagating. Conditional means are computed as averages of those points for a given bin in Z. The comparison of YiAI |Z ∗  and YiPF |Z ∗ , the conditional means versus mixture fraction obtained from look-up tables, with their measured values provide a direct a priori test of such chemistry tabulation. The results are shown in Figs. 6 and 7, where autoignition and premixed flamelet tables are found to be very close. The flame structure in mixture fraction space is

(c) Fig. 6. Conditional mean versus mixture fraction, a priori tests from measurements. (a) CO, (b) CO2 , (c) CH4 mass fractions. Symbols: measurements. Solid line: FPI database. Dashed line: autoignition database. From bottom to top, streamwise position X/D = 40, 50, and 70.

globally well captured at the flame base (X < 70D), specifically on the lean side. On the rich side, the tabulated responses deviate slightly from the measurements. Part of this deviation may be attributed to the occurrence of diffusion flame effects leading to fluxes in mixture fraction space that are not included in the chemical tables, as discussed in [33]. Far downstream (X/D = 70), where diffusion flame burning may be expected to dominate, the deviation

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much further downstream of its stabilization zone. In the performed a priori tests, the chemical source of Yc in the autoignition and premixed flamelet tables does not play any role, since Yc is taken from the measurements. A more careful analysis of the interlinks between chemistry tabulation and flame combustion regimes is now conducted to enlighten the differences between the chemical sources of autoignition and premixed flamelet databases.

(a)

(b)

(c) Fig. 7. Conditional mean versus mixture fraction, a priori tests from measurements. (a) O2 , (b) H2 O mass fractions, (c) temperature (K). Symbols: measurements. Solid line: FPI database. Dashed line: autoignition database. From bottom to top, streamwise position X/D = 40, 50, and 70.

from measurements becomes also apparent on the lean side. These preliminary a priori tests constitute a very severe evaluation of chemistry tabulation. The results indicate that, in a first approach, the turbulent flame base can be approximated from autoignition and premixed flamelet chemical tables. Nevertheless, a more careful analysis of diffusion flame burning would be required to capture the flame development

3. Linking autoignition, flame propagation, and diffusion flames in chemistry tabulation In the chemistry tabulation discussed above, it was implicitly assumed that any species mass fraction Yi can be expressed as a simple function of Yc , a measure of the reaction progress, and Z, the mixture fraction. This is a strong hypothesis that is now further discussed. Equations (2) and (3) are specialized in determining the specific behavior of Yi when the mixture undergoes autoignition without straining or freely propagating premixed flame burning. A more generic balance equation (i.e., not specialized for a given combustion mode), which controls the evolution of Yi in the Yc –Z, space may be derived to better understand the links between the autoignition- and premixed-flamelet-based chemical tables and to help combine them in SGS modeling for the lifted flame in vitiated coflow. The balance equations for the two control parameters of chemistry, Yc and Z, that have been chosen read ∂ρZ + ∇ · (ρuZ) = ∇ · (ρD∇Z), (5) ∂t ∂ρYc + ∇ · (ρuYc ) = ∇ · (ρD∇Yc ) + ω˙ Yc (6) ∂t and the primitive balance equation for the i species mass fraction Yi ( x, t) is ∂ρYi + ∇ · (ρuYi ) = ∇ · (ρD∇Yi ) + ω˙ i . (7) ∂t For simplicity, in this part Lewis and Schmidt numbers are assumed constant. To seek out the evolution of Yi in the Z–Yc space, the time and space evolutions of Yi need to be expressed as functions of Z and Yc , so that Yi = Yi (Z, Yc ) is known at once. The following relations are verified: ∇Yi =

∂Yi ∂Yi ∇Z + ∇Yc , ∂Z ∂Yc

∇ 2 Yi =

(8)

∂ 2 Yi ∂Yi 2 ∂ 2 Yi ∇ Z+ |∇Z|2 + |∇Yc |2 2 ∂Z ∂Z ∂Yc2 +

∂Yi 2 ∂ 2 Yi ∇ Yc + 2 ∇Z · ∇Yc . ∂Yc ∂Z∂Yc

(9)

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When they are introduced into Eq. (7) and combined with Eqs. (5) and (6), the following relation is obtained, ω˙ Yc

∂Yi ∂ 2 Yi ∂ 2 Yi = ρχZ + ρχYc 2 ∂Yc ∂Z ∂Yc2 + 2ρχZ,Yc

∂ 2 Yi + ω˙ i , ∂Z∂Yc

(10)

where the scalar dissipation rates χZ = D|∇Z|2 , χYc = D|∇Yc |2 and the cross-scalar dissipation rate χZ,Yc = D∇Z · ∇Yc have been introduced. These scalar dissipation rates depend on the spatial distributions of Z and Yc , which are controlled by Eqs. (5) and (6). This may be cast in a form where three Damkhöher numbers, DaDF = ω˙ Yc /ρχZ , DaPF = ω˙ Yc /ρχYc , and DaPPF = ω˙ Yc /ρχZ,Yc appear (DF: diffusion flame, PF: premixed flame, PPF: partially premixed flame), leading to 1 ∂ 2 Yi 1 ∂ 2 Yi ∂Yi = + ∂Yc DaDF ∂Z 2 DaPF ∂Yc2 +

1 ω˙ i ∂ 2 Yi + . DaPPF ∂Z∂Yc ω˙ Yc

(11)

Varying these numbers, well-known model combustion problems are recovered: (i) autoignition in a homogeneous mixture, (ii) freely propagating premixed flame, (iii) diffusion flamelet, and (iv) partially premixed flame. (i) When chemistry is much faster than transport in physical space, all the Damköhler numbers go to infinity and Eq. (11) becomes 

∂Yi ∂Yc

AI

 =

ω˙ i ω˙ Yc

This last relation can also be derived from equations similar to Eq. (3) characterizing a freely propagating premixed flame. Equation (13) is recovered when the relation (3) is applied to both Yi and Yc to be then combined with the relations (8) and (9). Equation (13) is thus the budget representative of a freely propagating premixed flame. The flame speed SL is not found explicitly in Eq. (13), which expresses the evolution of Yi in the composition space limited to Yc . In this one-dimensional premixed flame limit, Yi is sensitive to flame thickness and flame speed variation via DaPF , which is determined from the distribution of Yc . Indeed, the flame speed explicitly appears in the balance equation for Yc , which takes the form of Eq. (3). Yc becomes in this context the unique control parameter of Yi , along with χYc , which is related to δL ≈ [max(|∇Yc |)]−1 = max(D/χYc )1/2 , a measure of the flame thickness. When chemistry is tabulated by unstrained premixed flamelets, as in FPI [30] or in flamelet generated manifold (FGM) [34] approaches, Eq. (13) then governs the species trajectories in the Yc reaction progress variable space. (iii) In the case of a laminar steady diffusion flamelet, all quantities are unique functions of Z and its gradient |∇Z|. There is no explicit dependence of Yi versus Yc , since both are related to Z and χZ . Thus, Eq. (11) becomes  2 DF   1 ∂ Yi ω˙ i DF (14) + = 0. ω˙ Yc DaDF ∂Z 2 Multiplying this equation by ω˙ Yc , the well-known steady flamelet equation [35] is recovered: ρχZ

AI .

(12)

This corresponds to the time evolution of an homogeneous mixture. A relation that can also be obtained after eliminating time between the balance equations verified by Yi and Yc during an autoignition process (Eq. (2)). In this limit situation, all points of the mixture in physical space can be seen as isolated homogeneous reactors, since the gradients of Z and Yc in physical space play a negligible role. In a turbulent flow, according to so-called “turbulent combustion diagrams,” this limit also requires a large Reynolds number [29]. (ii) The freely propagating premixed flame limit is recovered when the equivalence ratio of the mixture is fixed, therefore when Z does not evolve in space. In this case, where DaDF and DaPPF go to infinity and the problem is controlled by DaPF , Eq. (11) reads   2 PF    1 ∂ Yi ω˙ i PF ∂Yi PF = + . (13) ∂Yc ω˙ Yc DaPF ∂Yc2

421

∂ 2 Yi + ω˙ i = 0. ∂Z 2

(15)

For a sufficiently small value of DaDF , the diffusion flame part of the problem described with Eq. (11) can thus be expected to quench. (iv) The last case is a combination of premixed and diffusion burning regimes; it is controlled by the value of DaPPF , which is inversely proportional to χZ,Yc = D∇Z · ∇Yc . The behavior and the role of this cross-scalar dissipation rate were previously discussed in a different context [36,37]. χZ,Yc is proportional to the projection of the gradient of the progress of the reaction in the direction where the mixture fraction evolves. It therefore characterizes equivalence ratio variations across the flame front and brings in Eq. (11) a direct measure of the degree of partial premixing of the burning zone. All these regimes are expected in the Cabra experiment. At the turbulent flame base, nevertheless, autoignition and premixed flame propagation are likely to dominate and contribute to flame stabilization. Also, the above a priori tests of chemistry tabulation from experiments have shown that these regimes

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(a)

(a)

(b) (b) Fig. 8. Reaction rates of various species plotted versus the progress variable mass fraction Yc = YCO + YCO2 : ω˙ c (—); ω˙ CH4 (—); ω˙ O2 (· · ·); ω˙ CO2 (- - - -); ω˙ CO (– – –); ω˙ H2 O (–··–); ω˙ OH (–·–). (a) Autoignition. (b) Freely propagating one-dimensional premixed flame. φ = 0.8. Reaction rates are given in s−1 . At the location where χYc reaches its maximum, DaPF ≈ 200 (see Eq. (11)).

provide a fair reproduction of the flame structure in mixture fraction space. Equations (12) and (13) are now used to seek out, more carefully, the differences and an eventual similitude between the chemical tables YiAI and YiPF , in order to determine how they can be combined in subgrid scale turbulent combustion modeling. Fig. 8 shows reaction rates versus Yc in both chemical look-up tables for φ = 0.8 (other equivalence ratios have also been examined, leading to similar results). The autoignition and premixed flamelet reaction rate responses differ in Yc -space, in both fresh and burnt gas regions. Because of molecular diffusion, absent in homogeneous autoignition, prod-

Fig. 9. Comparison of ∂Yk /∂Yc in the autoignition (—) and the propagation (—) databases for methane. (a) CH4 . (b) CO. ∂Yk /∂Yc is plotted versus Yc mass fraction. Same conditions as Fig. 8.

ucts and heat diffuse toward fresh gases in premixed flames, modifying the reaction rate distribution. Note also that the OH reaction rate is always positive for autoignition, whereas this radical, transported by molecular diffusion, is consumed on the fresh gas side of premixed flames. The major differences between autoignition and premixed flamelet modeled problems can be quantified from the slopes ∂Yi /∂Yc measured in the two look-up tables; they are shown versus Yc in Fig. 9 for methane and carbon monoxide. The shapes of the curves are quite similar for both prototype combustion problems, but some differences in magnitude are observed. According to Eqs. (12) and (13), those tables would feature similar responses in the limit of large

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423

values of DaPF , leading to 

   ∂Yi PF ∂Yi AI ≈ . ∂Yc ∂Yc

(16)

The differences between the chemical tables can thus be further investigated by introducing in Eqs. (12) and (13) the assumption that the slopes of species concentrations remain very close in Yc space (Eq. (16)) and by measuring the error on the burning rate estimation resulting from this hypothesis. Equations (12), (13), and (16) provide 

 ω˙ i PF 1 ∂ 2 Yi + ω˙ Yc DaPF ∂Yc2 PF ω˙ YAI  ∂ 2 Yi = PFc ρχYc + ω ˙ . i ω˙ Y ∂Yc2 c

ω˙ iAI = ω˙ YAIc

(17) (a)

(18)

In this last relation, the premixed flame reaction rate response is corrected by the scalar dissipation rate, ρχYc , to give an approximation of the autoignition source. Fig. 10 shows the comparison between autoignition sources estimated from their direct tabulation and from Eq. (18). The comparison is done here assuming a single average value of ρD. An interesting agreement is obtained up to Yc = 0.04 for the chemical source of CH4 and up to Yc = 0.03 for CO. In the intense burning zone and toward the burnt gases, however, autoignition and flame propagation responses differ slightly. At the turbulent flame base, combustion may start following two scenarios associated with autoignition or partially premixed flame propagation. In the following, it is assumed that when combustion starts, DaPF is sufficiently large so that Eq. (16) is verified, meaning that autoignition can be used to tabulate the behavior of very first ignition points in the turbulent jet. Very rapidly after combustion has started, autoignition is less likely to be the major controlling phenomenon, and the premixed flame tabulation is retained as a more appropriate choice. It is explained in the next section how a progress variable based on Yc is used to combine these two look-up tables. Much further downstream, diffusion combustion may be foreseen as the major combustion mode controlling the turbulent flame, but this regime will not be considered in the reported LES focused on flame stabilization mechanisms. Notice that LES using only the autoignition lookup table (Eq. (12)) was attempted. It was found that only the very leading edge of the flame base was then properly captured and that the complementary premixed mode was mandatory to fully reproduce flame stabilization in LES, in accordance with the above analysis of flame propagation in the vitiated mixture.

(b) Fig. 10. Comparison of autoignition (—), propagation (—), and reconstructed ignition (· · ·, Eq. (18)) reaction rates versus progress of reaction. (a) CH4 . (b) CO. Same conditions as Fig. 8.

4. LES of the turbulent vitiated jet flame The SGS flame structure is approximated using presumed conditional moment (PCM), a closure based on beta-shape presumed filtered probability density function that was initially formulated for nonpremixed turbulent flames [14], and then extended to LES of premixed turbulent combustion [11]. Filtered scalars and source terms are cast in the form 1 ϕ( ˜ x, t) =

(ϕ|Z ∗ ; x, t)P˜ (Z ∗ ; x, t) dZ ∗ .

(19)

0

(ϕ|Z ∗ ; x, t) is the filtered conditional mean of ϕ for a given value Z ∗ of the mixture fraction and P˜ (Z ∗ ; x, t) is the filtered probability density function of Z. (ϕ|Z ∗ ; x, t) describes the filtered chemical flame structure in mixture fraction space. Before autoignition, the filtered conditional mean fol-

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lows a mixing-type response (Eq. (1)) (ϕ|Z ∗ ; x, t) = ϕo (Z ∗ ) = Z ∗ ϕF,o + (1 − Z ∗ )ϕO,o . After ignition, and for a very large global Damköhler number, a response close to equilibrium chemistry would be observed: (ϕ|Z ∗ ; x, t) ≈ ϕ Eq (Z ∗ ). Between these limits, (ϕ|Z ∗ ; x, t) evolves according to the progress of the reaction. The analysis of autoignition and flame propagation properties in the mixture formed by turbulent mixing upstream of the flame base suggests that combustion is likely to begin with autoignition, which can be combined with, or followed by, flame propagation. Also, when autoignition cannot initially occur, flame propagation may be the main stabilizing mechanism, as discussed above. The conditional filtered means (ϕ|Z ∗ ; x, t) must be modeled to capture, in mixture fraction space, this peculiar structure of the lifted flame in vitiated coflow. Accounting for these observations, in a first attempt, (ϕ|Z ∗ ; x, t) is approximated from autoignition and premixed flamelet tables according to the simple linear decomposition ¯ AI (Z ∗ , c∗ ) + cϕ ¯ PF (Z ∗ , c∗ ), ϕ(Z ∗ , c∗ ) = (1 − c)ϕ (20) leading to (ϕ|Z ∗ ; x, t) = (1 − c)

1

ϕ AI (Z ∗ , c∗ )P (c∗ ; x, t) dc∗

0

1 + c¯

ϕ PF (Z ∗ , c∗ )P (c∗ ; x, t) dc∗ ,

0

(21) Eq where c = Yc /Yc is a progress variable. P (c∗ ; x, t)

is the filtered probability density function of c presumed with a beta-shape [38]. The progress variable, c, is assumed to be independent of the mixture fracEq ¯ leading to (Yc |Z ∗ ) = cY ¯ c (Z ∗ ) tion, (c|Z ∗ ) ≈ c, and, after integrating with the mixture fraction filtered Eq probability density function, Y˜c = c¯Y˜c . It is important to note that the hypothesis of statistical independence with Z applies to c only, which is a normalized quantity, but not to the progress of the reaction Yc or to any other quantity ϕi extracted from the tables [11]. This hypothesis of statistical independence cannot be strictly exact, but it was found reasonable after direct tests from measurements in a nonpremixed turbulent jet flame (see Fig. 13 of [11]), where the combination of CO and CO2 was found to provide the best compromise to construct the progress variable. Under ¯ the SGS variance of the this hypothesis, cv = cc − c¯c, Eq2 Eq 2 /Y progress variable reads cv = Y − (Y˜c /Yc )2 . c c ∗ In Eq. (19), P˜ (Z ; x, t) is also assumed to have a shape close to a beta-function. The variables nec-

˜ Zv essary to presume the two beta-shape pdfs, Z,  2 ∗ ˜ ˜ for P (Z ; x, t) and Yc , Yc (giving c¯ and cv ) for P (c∗ ; x, t), used in Eq. (21), are obtained from their modeled balance equations: ∂ρ Z˜ ˜ + ∇ · ρ u˜ Z˜ = −∇ · τ Z + ∇ · (ρD∇ Z), (22) ∂t ∂ρZv ˜ v = −∇ · τ Zv + ∇ · (ρD∇ Z˜ v ) + ∇ · ρ uZ ∂t (23) − 2τ Z · ∇ Z˜ − 2s χZ , ∂ρ Y˜c + ∇ · ρ u˜ Y˜c = −∇ · τ Yc + ∇ · (ρD∇ Y˜c ) ∂t (24) + ω˙ Yc ,  2 ∂ρ Yc 2 = −∇ · τ + ∇ · (ρD∇ Y 2 ) + ∇ · ρ u˜ Y c c Yc2 ∂t − 2ρχ Yc + 2Yc ω˙ Yc . (25) A single diffusive coefficient D is assumed for all quantities, with a Schmidt number of 0.72. The SGS ˜ τ Y = ρuYc − turbulent fluxes, τ Z = ρuZ − ρ u˜ Z, c  ρ u˜ Y˜c , and τ 2 = ρuYc2 − ρ u˜ Yc2 , are expressed usYc

ing a gradient transport hypothesis with an SGS eddy viscosity obtained from the filtered structure function closure [39], which is detailed in Appendix A. An SGS turbulent Schmidt number of 0.9 is used for the scalars. The filtered sources of progress of reaction, ω˙ Yc , Yc ω˙ Yc , and of energy are expressed from the relations (19) and (21). Notice that ω˙ Yc is actually proportional to density, ω˙ Yc = ρ ω˙ Y∗ , and the c ∗ filtered mean may be written ω˙ Yc = ρ ω˜˙ Yc , which may be computed from the presumed sub-grid scale mixture fraction filtered probability density function parameterized by mass-weighted averaged quantities. ˜ Y˜c , SZ = Those sources are tabulated in terms of Z, ˜ − Z)) ˜ and Sc = cv /(c(1 ¯ − c)); ¯ these last Zv /(Z(1 two variables SZ and Sc are the unmixedness (normalized variance) of the mixture fraction and of the progress variable, respectively. Filtered look-up tables for every useful chemical quantity are then available as ˜ SZ , Y˜c , Sc ). ϕ˜ i = ϕ˜ iPCM (Z, (26) The filtered tables for autoignition and flame propagation contain 100 × 20 × 100 × 20 points in ˜ Z –Y˜c –Sc space. Sc also plays an important role Z–S in the scalar dissipation rate modeling of the progress of reaction, as discussed now. 4.1. SGS scalar dissipation rates The scalar dissipation rate of the mixture fraction may be written ˜ 2 + s χZ , ρ χ˜ Z = ρD|∇Z|2 = ρD|∇ Z|

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where s χZ , found in Eq. (23), is the unresolved part of ρ χ˜ Z , which is modeled assuming a linear relaxation of the variance within the subgrid: Zv s χZ = ρ 2 . Δ /νT

(28)

Δ is the characteristic filter size and νT is the SGS eddy viscosity given by the filtered structure function model [39]. The SGS scalar dissipation rate of the progress of reaction deserves a careful treatment [40]. As for the mixture fraction, it may be decomposed into resolved and SGS parts: ρ χ˜ Yc = ρD|∇Yc |2 = ρD|∇ Y˜c |2 + s χYc .

(29)

Yc is a reactive species; its source ω˙ Yc modifies ∇Yc and therefore the scalar dissipation rate χ˜ Yc is sensitive to chemistry. Two different closures are tested for s χYc , the unresolved part of χ˜ Yc (Eq. (29)). The first is the direct transposition of the linear relaxation hypothesis (LRH) to the reactive species Yc , leading to Yc s χYc = ρ 2 v , (30) Δ /νT 2 − Y˜ Y˜ is the SGS variance of Y . where Ycv = Y c c c c The second closure for s χYc is an attempt to improve the scalar dissipation rate modeling of Yc , taking into account that it is a reactive species. The derivation follows a bimodal-limit (BML) approach of turbulent flames described with a progress variable c [41]. The balance equation for c(1 − c) may be written [38] 

 ∂

ρc(1 − c) + ∇ · ρuc(1 − c) ∂t



 = ∇ · ρD∇ c(1 − c) + 2ρD|∇c|2 − 2cω˙ c + ω˙ c ,

(31)

where the fluxes of c in mixture fraction space, which are present in the original balance equation for c, have been neglected [36,37]. The upper asymptotic limit of the scalar dissipation rate is obtained when the c spatial distribution takes the form of a telegraphic signal across flame fronts, i.e., when c = 0 and c = 1 are the only values observed for c; the filtered probability density function of c is then bimodal and Sc → 1 [41]. Under these conditions, Eq. (31) reduces to 2ρD|∇c|2 = 2cω˙ c − ω˙ c , transposed in terms Eq of Yc = cYc (Z ∗ ), and after filtering, this expression becomes Eq

2ρ χ˜ Yc = 2Yc ω˙ Yc − Yc ω˙ Yc ,

the progress variable gradients, and therefore the dissipation rate. The linear relaxation closure for the scalar dissipation rate applied to a reactive species (Eq. (30)) is more likely to be valid for small values of Sc , since it is exact for a quasi-Gaussian pdf featuring a weak level of Sc [42], whereas the BML expression (Eq. (32)), by definition [41], holds for large levels of Sc . A closure for the SGS part of the progress variable may be proposed that combines both expressions, to account for the local flame regime, Yc s χYc = (1 − Sc )ρ 2 v Δ /νT

Eq + Sc −ρD|∇ Y˜c |2 + Yc ω˙ Yc − Yc ω˙ Yc /2 . (33) Equation (33) provides a way to automatically adjust the SGS scalar dissipation rate of the progress of the reaction, in agreement with some of the flow features. The chemical source terms found in Eq. (33) are computed from the chemical tables combined with the presumed filtered probability density functions through relations (19) and (21). Both expressions for s χYc (Eqs. (30) and (33)) have been tested and the obtained results are compared with experiments below. When not indicated, the new closure for the scalar dissipation rate (Eq. (33)) has been used. Recent RANS studies have derived similar procedures based on BML hypothesis to express scalar dissipation rates [43]. The set of Navier–Stokes equations in their fully compressible form are solved, together with the above equations. A fourth-order finite-volume skewsymmetric-like scheme [44] is adopted for the spatial derivatives. This scheme was specifically developed for LES and is combined here with second-order Runge–Kutta explicit time stepping. Acoustic waves are included in the fully compressible formulation and Navier–Stokes characteristic boundary conditions are retained [45]. The mesh is composed of 2,150,000 nodes with a characteristic mesh size h, 0.3 mm < h < 2.5 mm. Its spanwise and streamwise lengths are 28D × 28D and 90D, respectively. Turbulence is generated in the inlet plane using the forcing procedure of Klein et al. [46]. The solver is fully parallel and vectorial, and the filtered chemical table is loaded on every of the eight processors of a NEC SX-8. A typical run that makes it possible to collect statistical properties of the turbulent flame takes about 10 h cpu.

(32)

Eq leading to s χYc = −ρD|∇ Y˜c |2 + Yc ω˙ Yc − Yc ω˙ Yc /2. Eq

425

This relation, in which Yc varies with mixture fraction, is valid in the limit of chemistry fully controlling

4.2. LES results The development of the turbulent jet is shown in Fig. 11. The Q-criterion [47] is used to extract

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Fig. 11. Snapshot of iso-Q colored by burning rate (WE, W/m3 ) for Q = (0.5UF /D)2 ; UF is the fuel jet bulk velocity. Frame: Zoom in on the turbulent flame base.

the zone of intense flow rotation from LES. Q = 0.5(Ωij Ωij − Sij Sij ), where Sij = 0.5(∂ui /∂xj + ∂uj /∂xi ) and Ωij = 0.5(∂ui /∂xj − ∂uj /∂xi ) are the symmetrical and antisymmetrical parts of the velocity gradient tensor, respectively. Positive values of Q locate zones where flow rotation dominates over straining. The iso-Q surface for the value Q = 0.5(UF /D)2 is seen in Fig. 11, where UF is the fuel jet bulk velocity. The coherent-vortex dynamics typical of a turbulent round jet is observed upstream of the flame base [39]. The so-called alternate pairing of the Kelvin–Helmholtz toroidal structures is rapidly observed. This pairing results from the subharmonic mode interacting with the Kelvin–Helmholtz structures. The initially toroidal vortices are displaced right and left with reference to the jet axis, and every displaced vortex ring is convected at a streamwise velocity that is reduced, compared to the inner-jet velocity. The resulting torque tends to incline the structures till the edges of two consecutive rings merge, leading to the coherent structures observed in Fig. 12. The zoom within the flame base in Fig. 11 shows that combustion at leading edge-flames mainly avoids zones of intense rotation. The stoichiometric isosurface colored by temperature is shown in Fig. 13 at a given simulation time. Intense turbulent mixing is observed upstream of the flame. Starting from the jet exit, and after the development of the large eddies, the almost cylindrical iso-Zs

Fig. 12. Zoom at X ≈ 40D of iso-Q showing the typical coherent structures resulting from the merging of two consecutive Kelvin–Helmholtz toroidal vortices (Q = 0.5(UF /D)2 ).

surface rapidly transitions toward a highly strained and curved surface. Mixing is there very intense: fresh fuel mixes with burnt products and combustion starts between 30 and 40 diameters downstream of the jet exit. The adiabatic equilibrium temperature (T Eq ≈ 2200 K) is not immediately reached after the first ignition spots. Pockets where temperature is of the order of T Eq appear soon after the location of primary ignition, but they are first separated by flow zones where temperature stays below 1800 K. This intermediate regime is sustained in a zone located approximatively

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427

Fig. 13. Stoichiometric isomixture fraction surface colored by temperature.

between 40 and 60 diameters. After X = 60D, combustion is fully developed. The vitiated coflow temperature is 1350 K; the isotemperature surface T ∗ = 1400 K may thus be used to locate points where combustion has just started. Fig. 14 shows the mixture fraction field and that iso-T ∗ = 1400 K. Chemically frozen flow mixing exists till X = 40D, with few isolated pockets of very first ignition that are observed between X = 30D and X = 40D. Between 40D and 45D, a well-developped turbulent flame base is found. The isomixture fraction surface Z o = 0.00427, corresponding to the shortest ignition delay discussed above, is also shown in Fig. 14. From the burner exit, this Z o mixture fraction surface is bordering the fuel jet. At X ≈ 40D and for the particular instant in time presented in Fig. 14, the iso-Z o has penetrated up to the core of the jet because of turbulent mixing. It is seen that this surface perfectly follows the flame base located by the iso-T ∗ surface. In the simulations, the continuous unsteady movement of the turbulent edge-flame is organized according to the position of the iso-Z o , which may be located at various radial positions. Hence, the position of the ignition front actually moves with turbulent fluctuations over quite a large zone, from the radial

Fig. 14. Cut in the centerline vertical plane. Color: isomixture fraction. Dashed double dotted line: Z o = 0.00427 (corresponds to the shortest ignition delay in Fig. 3). Red line: isotemperature T ∗ = 1400 K (the inlet coflow temperature is T = 1350 K).

leading edge of the fuel/oxidizer stream mixing layer, up to the center of the jet. Transverse cuts of the instantaneous LES heat release rate at X = 33D and X = 45D are shown in Fig. 15, with two isomixture fraction lines corresponding to the position of the shortest ignition delay in the mixture (Z = Z o ) and to Zs , the stoichiometric condition. As expected, the maximum level of heat release rate is always located at the stoichiometric surface. The comparison of these two heat release maps helps to visualize the spreading of combustion within the reactants, when moving downstream of the first ignition point. Combustion is first located in the lean region of the jet having Z o < Z < Zs (X = 33D in Fig. 15), also corresponding to isolated pockets of iso-T ∗ observed in Fig. 14. From these first burning points, the reaction zones extend to the inner and fuel-richer part of the flow to develop the main stabilization zone of the turbulent flame (X = 45D in Fig. 15).

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(a)

(b) Fig. 15. Transverse cut at X = 33D (a) and X = 45D (b). Dashed line: isomixture fraction surface for Z o = 0.00427 and Z = Zs = 0.17. Flood: burning rate (W/m3 ). Axis lengths are in m.

Even though this global picture persists in time, the turbulent flame base is strongly unsteady, and the simulations suggest that the concepts of “timeaveraged location” or “mean behavior” of a turbulent flame base is in fact quite difficult to envisage, since it is difficult to pack in a single averaged view physical behaviors that correspond to very different burning fluid parcel histories. The global behavior of the lifted flame in the vitiated coflow is illustrated in Fig. 16, which shows the jet centerline time average, T˜ , and the corresponding variance, Tv = T˜ T˜  − T˜ T˜ , extracted from LES. These last quantities have been averaged over 4τX , with τX = 90D/UF , a jet-flow time measured from the fuel jet bulk velocity and streamwise length of the computational domain. (This time was found sufficient to obtain converged statistics for the present case, but depending on the flow, the required time may need to be approximated differently.) When compared

Fig. 16. Profile of average and RMS temperature on the centerline of the jet. Cross: measurements of averaged temperature. Square: measurements of RMS temperature. Solid line: LES with s χc given by Eq. (30). Dashed line: LES with s χc given by Eq. (33).

to measurements, the centerline temperature distribution is well captured by the simulation. The mean flame structure features some of the main properties of the instantaneous picture discussed above. Starting from the jet exit, the temperature first increases due to turbulent mixing between the room temperature fuel jet and the hot vitiated coflow. At approximatively X = 40D, there is a change in mean temperature slope; from this zone the averaged temperature increases till X = 60D, where the flame is fully developed. The order of magnitude of temperature fluctuations is also reproduced. Nevertheless, this result has to be tempered in the light of the recent analysis of comparison procedures between LES results and experimental data on reacting flows [48], which concluded that it is not completely exact to directly compare Tv , obtained from LES fields, with its experimental counterpart. The sensitivity of the flame to the modeling of s Yc , the SGS scalar dissipation rate of Yc , is not that visible on the temperature centerline distribution (Fig. 16). The impact of s Yc modeling is, however, noticeable on the radial spreading of the turbulent flame-base and on the species profiles. Fig. 17 shows isolines of mean temperature for the two SGS models. The BML-type closure (Eq. (33)) provides a flame base that is more developed radially, with a flame edge that is more pronounced in the direction normal to the flow. Centerline distributions of averaged mass fractions of H2 O, CO2 , CO, O2 , OH, and NO are compared to experimental data in Fig. 18. They have been computed by time integration over the duration 4τX , as was done for the temperature. These species profiles are also sensitive to the expression retained for modeling s χYc . The sensitivity to chemistry of the BML-type scalar dissipation rate closure (Eq. (33)) leads to species profiles having a slope through the turbulent flame base that is larger than

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429

(a)

Fig. 17. Isoline of mean temperature. Left: LES with s χc given by Eq. (33). Right: LES with s χc given by Eq. (30).

(b) Fig. 19. Radial profiles of mixture fraction (a) and temperature (b). Symbol: measurements. Solid line: LES with s χc given by usual linear relaxation hypothesis (Eq. (30)). Dashed line: LES with s χc given by new formulation (Eq. (33)). From bottom to top in every graph, streamwise position X/D = 30, 40, 50, and 70. (a)

(b) Fig. 18. Centerline distribution of averaged H2 O, CO2 , and O2 (a), OH, NO, and CO (b) mass fractions. Symbol: measurements. Solid line: LES with s χc given by usual linear relaxation hypothesis (Eq. (30)). Dashed line: LES with s χc given by new formulation (Eq. (33)).

the one obtained with the linear relaxation hypothesis (Eq. (30)). Radial profiles of mixture fraction, temperature, and mass fractions of CO2 , O2 , OH, H2 O, and CO2 are shown in Figs. 19, 20, and 21. In addition to time integration, these profiles are obtained after averaging in space using the axisymmetric character of the mean flow. The radial temperature profiles better reproduce experimental data when s χYc is modeled by Eq. (33). Overall, the comparison with measurements suggest that some of the major statistical properties of the turbulent flame base are captured by the performed simulations. Major features observed in the a priori tests of chemistry tabulation are recovered, for instance, the underestimation of CO2 at X = 70D. To further investigate this point, conditional means in mixture fraction space, (Yi |Z ∗ )(X), are constructed from LES results for the streamwise position X for temperature and CO mass fraction. This is done using a relation to evaluate conditional means over a span-

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(a) (a)

(b) (b) Fig. 21. Radial profiles of H2 O (a) and CO (b) mass fractions. Symbol: measurements. Solid line: LES with s χc given by usual linear relaxation hypothesis (Eq. (30)). Dashed line: LES with s χc given by new formulation (Eq. (33)). From bottom to top in every graph, streamwise position X/D = 40, 50, and 70.

(c) Fig. 20. Radial profiles of CO2 (a), O2 (b), and OH (c) mass fractions. Symbol: measurements. Solid line: LES with s χc given by usual linear relaxation hypothesis (Eq. (30)). Dashed line: LES with s χc given by new formulation (Eq. (33)). From bottom to top in every graph, streamwise position X/D = 40, 50, and 70.

where (Yi |Z ∗ )(X, r, t) is obtained from Eq. (21), for every radial position r of the X streamwise plane, and P˜ (Z ∗ ; X, r, t) is the SGS-filtered probability density function for that given point. These filtered conditional means are displayed in Fig. 22 and compared with measurements. The temperature and CO responses are reproduced at the flame base. The CO mass fraction is overestimated at X = 70D. Among the various possible causes associated with chemistry description, this could be attributed to diffusion flame burning, confirming the need for accounting for diffusion flame behavior downstream of the zone where the flame is stabilized.

wise plane of a jet flame [14,49], 

 (Yi |Z ∗ ) (X)  ∞  (Yi |Z ∗ )(X, r, t)P˜ (Z ∗ ; X, r, t)2π r dr = 0 , ∞ ˜ ∗ 0 P (Z ; X, r, t)2π r dr (34)

5. Conclusion The impact of burnt gases on flame stabilization has been studied using autoignition and premixedflamelet chemistry tabulations, under the operating conditions of a laboratory lifted jet flame in a vitiated

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ies have shown that diffusion can limit or propagates ignition through mixing layers [17,50–53]. It would therefore be of interest in future works to tabulate diffusion with an additional diffusion terms, for instance using unsteady diffusion flamelets [35].

Acknowledgments

(a)

Part of this work was performed during the 2006 Summer Program of the Center for Turbulence Research (CTR) at Stanford University. Computing resources were provided by IDRIS-CNRS and CRIHAN. Financial support by the European Commission through the project “Toward Innovative Methods for Combustion Prediction in Aeronautic Engines” (TIMECOP-AE, Project N: AST5-CT-2006-030828) is also gratefully acknowledged.

Appendix A

(b) Fig. 22. Conditional mean versus mixture fraction. Symbol: measurements (instantaneous quantity). Solid line: LES (filtered quantity). (a) Temperature. (b) CO mass fraction. From bottom to top, streamwise position X/D = 40, 50, and 70.

coflow investigated by Cabra et al. [26]. The differences and links between these tabulations have been discussed from the balance equations of species mass fractions expressed in terms of mixture fraction and a measure of the progress of reaction, which are here the two control parameters of the chemical look-up tables. Large-eddy simulation using a combination of these chemistry tabulations was found to reproduce most of the experimentally observed statistical properties of the jet flame in the vitiated coflow. A novel subgrid scale closure for the scalar dissipation rate of a reactive species has also been discussed and tested from the simulations. Downstream of the turbulent flame base, the diffusion flame part of the problem was not carefully addressed in the present work; further developments are needed to improve the chemistry tabulation strategy in this zone of the reacting flow. Similarly, the autoignition table was built without accounting for molecular diffusion. In Eq. (20), effects of diffusion are included in the premixed flame table only. This is a serious hypothesis, DNS stud-

The SGS structure function model is a direct transposition in physical space of the spectral turbulent viscosity. A detailed description of the SGS model in its preliminary form may be found in Métais and Lesieur [54]; the filtered version given below is due to Ducros et al. [55]. It was successfully applied to a large variety of configurations, including transitional flows [39]. The SGS viscosity is cast in

1/2 −3/2 , νt (x, Δ, t) = 0.0014CK Δ F˜2n=3 (x, Δ, t) (A.1) where CK = 1.4 is the Kolmogorov constant and Δ the filter size. F˜2n=3 is obtained from the structure function approximated over six neighboring points:  n=3 (Δ, t) = 1  u˜ n=3 − u˜ n=3 2 F˜2i,j,k i,j,k i+1,j,k 6 2  n=3  + u˜ n=3 i,j,k − u˜ i−1,j,k   n=3 2 + u˜ n=3 i,j +1,k − u˜ i,j,k 2  n=3  + u˜ n=3 i,j,k − u˜ i,j −1,k   n=3 2 + u˜ i,j,k+1 − u˜ n=3 i,j,k   n=3 2 . + u˜ n=3 i,j,k − u˜ i,j,k−1

(A.2)

The n = 3 exponent in the expression for the SGS viscosity (Eq. (A.1)) indicates that the structure function is applied to a velocity field that has been filtered three times using a Laplace type filter: n=m n=m = u˜ n=m u˜ n=m+1 i+1,j,k − 2u˜ i,j,k + u˜ i−1,j,k i,j,k n=m n=m + u˜ n=m i,j +1,k − 2u˜ i,j,k + u˜ i,j −1,k n=m n=m + u˜ n=m i,j,k+1 − 2u˜ i,j,k + u˜ i,j,k−1 .

(A.3)

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References [1] C. Schneider, A. Dreizler, J. Janicka, Flow Turbulence Combust. 74 (1) (2005) 103–127. [2] G. Subramanian, R. Bounaceur, A.P. Cruz, L. Vervisch, Combust. Sci. Technol. 179 (9) (2007) 1937–1962. [3] A. Cavaliere, M. de Joannon, Prog. Energy Combust. Sci. 30 (4) (2004) 329–366. [4] D. Tabacco, C. Innarella, C. Bruno, Combust. Sci. Technol. 174 (2002). [5] B.B. Dally, E. Riesmeier, N. Peters, Combust. Flame 137 (2005). [6] C.D. Pierce, P. Moin, J. Fluid Mech. 504 (2004) 73–97. [7] L. Selle, G. Lartigue, T. Poinsot, R. Koch, K.-U. Schildmacher, W. Krebs, B. Prade, P. Kaufmann, D. Veynante, Combust. Flame 137 (4) (2004) 489–505. [8] W.-W. Kim, S. Menon, H.C. Mongia, Combust. Sci. Technol. 143 (1999) 25–62. [9] H. Pitsch, H. Steiner, Phys. Fluids 12 (10) (2000) 2541– 2554. [10] O. Stein, A. Kempf, J. Janicka, Combust. Sci. Technol. 179 (2007) 173–189. [11] P. Domingo, L. Vervisch, S. Payet, R. Hauguel, Combust. Flame 143 (4) (2005) 566–586. [12] A. Gourara, F. Roger, H.-Y. Wang, J.-M. Most, Combust. Flame 144 (3) (2006) 592–604. [13] R. Cabra, J.Y. Chen, R.W. Dibble, A.N. Karpetis, R.S. Barlow, Combust. Flame 143 (4) (2005) 491–506. [14] L. Vervisch, R. Hauguel, P. Domingo, M. Rullaud, J. Turbulence 5 (4) (2004) 1–36. [15] R.R. Cao, S.B. Pope, A.R. Masri, Combust. Flame 142 (4) (2005) 438–453. [16] A.R. Masri, R. Cao, S.B. Pope, G.M. Goldin, Combust. Theory Model. 8 (1) (2003). [17] K. Gkagkas, R.P. Lindstedt, Proc. Combust. Inst. 31 (1) (2007) 1559–1566. [18] R. Cabra, T. Myhrvold, J.-Y. Chen, R.W. Dibble, A.N. Karpetis, R.S. Barlow, Proc. Combust. Inst. 29 (2002) 1881–1888. [19] Z. Wu, A.R. Masri, R.W. Bilger, Flow Turbulence Combust. 76 (1) (2006) 61–81. [20] W.P. Jones, S. Navarro-Martinez, O. Röhl, Proc. Combust. Inst. 31 (2) (2007) 1765–1771. [21] W.P. Jones, S. Navarro-Martinez, Combust. Flame 150 (3) (2007) 170–187. [22] L. Muñiz, M.G. Mungal, Combust. Flame 111 (1/2) (1997) 16–31. [23] S.K. Marley, K.M. Lyons, K.A. Watson, Flow Turbulence Combust. 72 (1) (2004) 29–47. [24] Y. Sommerer, D. Galley, T. Poinsot, S. Ducruix, S. Veynante, J. Turbulence 5 (1) (2004). [25] P. Domingo, L. Vervisch, Proc. Combust. Inst. 31 (2007) 1657, 1664. [26] R. Cabra, J.Y. Chen, R.W. Dibble, A.N. Karpetis, R.S. Barlow, http://www.me.berkeley.edu/cal/VCB/Data/ VCMAData.html. [27] M. Frenklach, T. Bowman, G. Smith, B. Gardiner, http://www.me.berkeley.edu/gri-mech. [28] R.J. Kee, J.F. Grcar, M.D. Smooke, J.A. Miller, A Fortran program for modeling steady laminar one-

[29] [30] [31] [32] [33]

[34] [35] [36] [37] [38] [39]

[40] [41] [42]

[43] [44] [45] [46] [47]

[48] [49] [50] [51] [52] [53] [54] [55]

dimensional premixed flames, SAND85-8240, Technical report, Sandia National Laboratories, 1985. T. Poinsot, D. Veynante, Theoretical and Numerical Combustion, R.T. Edwards, Inc., 2005. O. Gicquel, N. Darabiha, D. Thevenin, Proc. Combust. Inst. 28 (2000) 1901–1908. G.R. Ruetsch, L. Vervisch, A. Liñán, Phys. Fluids 6 (7) (1995) 1447–1454. S.D. Terry, K.M. Lyons, Combust. Sci. Technol. 177 (11) (2005) 2091–2112. B. Fiorina, O. Gicquel, L. Vervisch, N. Darabiha, S. Carpentier, Combust. Flame 140 (3) (2005) 147– 160. J.A. van Oijen, F.A. Lammers, L.P.H. de Goey, Combust. Flame 127 (3) (2001) 2124–2134. N. Peters, Turbulent Combustion, Cambridge Univ. Press, 2000. P. Domingo, L. Vervisch, K.N.C. Bray, Combust. Theory Model. 6 (4) (2002) 529–551. K. Bray, P. Domingo, L. Vervisch, Combust. Flame 141 (2005) 431–437. D. Veynante, L. Vervisch, Prog. Energy Combust. Sci. 28 (2002) 193–266. M. Lesieur, O. Métais, P. Comte, Large-Eddy Simulations of Turbulence, Cambridge Univ. Press, Cambridge, UK, 2005. N. Chakraborty, N. Swaminathan, Phys. Fluids 19 (2007) 045104. K.N.C. Bray, Proc. Combust. Inst. 26 (1996) 1–26. C. Dopazo, Non-isothermal turbulent reactive flows: Stochastic approaches, Ph.D. thesis, University of New York, Stony Brook, 1973. A. Mura, V. Robin, M. Champion, Combust. Flame 149 (1–2) (2007) 217–224. F. Ducros, F. Laporte, T. Soulères, V. Guinot, P. Moinat, B. Caruelle, J. Comput. Phys. 161 (2000) 114–139. T. Poinsot, S.K. Lele, J. Comput. Phys. 1 (101) (1992) 104–129. M. Klein, A. Sadiki, J. Janicka, J. Comput. Phys. 186 (2) (2002) 652–665. J.C.R. Hunt, A.A. Wray, P. Moin, Eddies, stream, and convergence zones in turbulent flows. In: Annual Research Briefs, Center for Turbulence Research, Stanford, 1988. D. Veynante, R. Knikker, J. Turbulence 7 (2006), doi:10.1088/1468-5248/5/1/037. T. Landenfeld, A. Sadiki, J. Janicka, Flow Turbulence Combust. 68 (2) (2002) 111–135. E. Mastorakos, T.A. Baritaud, T.J. Poinsot, Combust. Flame 109 (1–2) (1997) 198–223. E. van Kalmthout, D. Veynante, S. Candel, Proc. Combust. Inst. 26 (1996) 35. R. Hilbert, D. Thévenin, Combust. Flame 128 (1–2) (2002) 22–37. X.L. Zheng, J. Yuan, C.K. Law, Proc. Combust. Inst. 30 (1) (2004) 415–421. O. Metais, M. Lesieur, J. Fluid Mech. 239 (1992) 157– 194. F. Ducros, P. Comte, M. Lesieur, J. Fluid Mech. 326 (1996) 1–36.