A modified coherent flame model to describe turbulent flame propagation in mixtures with variable composition

Proceedings of the Combustion Institute, Volume 28, 2000/pp. 193–201

A MODIFIED COHERENT FLAME MODEL TO DESCRIBE TURBULENT FLAME PROPAGATION IN MIXTURES WITH VARIABLE COMPOSITION J. HE´LIE1 and A. TROUVE´2 1

Centre de Recherche en Combustion Turbulente Institut Franc¸ais du Pe´trole 92852 Rueil-Malmaison, France 2 Department of Mechanical Engineering George Washington University Washington, DC 20052, USA

Stratified direct-injection spark-ignition engines feature both large- and small-scale spatial variations in unburned mixture composition. Modifications of the coherent flame model (CFM) are proposed in the present study to account for the effects of variable mixture strength on the primary premixed flame, as well as for the formation of a secondary non-premixed reaction zone downstream of the premixed flame. The domain of validity of the present modifications is restricted to the case of small variations in mixture strength, without the additional complication of premixed flame extinction. The modeling strategy is based on previous results from direct numerical simulations as well as a theoretical analysis of a simplified problem by Kolmogorov, Petrovskii, and Piskunov (KPP). The KPP problem corresponds to a one-dimensional, turbulent flame propagating steadily into frozen turbulence and frozen fuel-air distribution, and it provides a convenient framework to test the modified CFM model. In this simplified but somewhat generic configuration, two radically different situations are predicted: for variations in mixture strength around mean stoichiometric conditions, unmixedness tends to have a net negative impact on the turbulent flame speed; in contrast, for variations in mixture strength close to the flammability limits, unmixedness tends to have a net positive impact on the turbulent flame speed. While featuring a restricted domain of validity, the proposed modifications to the CFM set the basis for future developments and are well suited in particular for an extension of the model to the case of combustion with occurrences of premixed flame extinction.

Introduction Direct-injection spark-ignition (DI-SI) engines represent one of the most promising technologies to achieve further reductions in fuel consumption in automotive transport [1,2]. Under light- or part-load operation, these engines use fuel stratification to provide additional control of combustion performance and establish overall fuel-lean conditions. In stratified DI-SI engines, the reactive mixture features both large- and small-scale variations in fuel/ air composition [1,2]. These variations correspond to inhomogeneities in mixture strength into which the spark-ignited turbulent flame propagates. The propagation of turbulent flames into mixtures with variable fuel/air composition has been described recently using direct numerical simulation (DNS) [3–5]. The DNS studies correspond to propane/air flames propagating into isotropic turbulent flow and non-homogeneous reactants. They also correspond to a combustion regime where the flow to flame velocity scale ratio is large, (u⬘/sL) ⬎ 1, and the variations in mixture strength remain small, (Z⬘/ Z˜) K 1. The latter restriction is used as an intermediate step in the DNS studies, where the initial

focus is on flammable mixtures, that is, mixture compositions within the propane/air flammability limits and without premixed flame extinction. The DNS results show that (1) under lean-rich conditions, the reaction zone can be described as a staged, premixed/non-premixed combustion system [4,5] (Fig. 1); (2) the first premixed stage can be described using classical laminar flamelet concepts [3–5]; (3) for (small) variations in mixture strength around mean stoichiometric conditions, Z˜ ⬇ Zst, unmixedness tends to have a net negative impact on the overall mean premixed reaction rate [4]; (4) this effect is related to changes in the mean flameletstructure, that is, to a decrease of the mean flamelet mass burning rate per unit flame surface area [4]; and (5) for (u⬘/sL) ⬎ 1, the effect of unmixedness on the turbulent premixed flame surface area remains negligible [3–5]. The general objective of the present study is to examine how the above DNS results may be incorporated into current model descriptions of turbulent flame propagation in SI engines. Previous similar modeling efforts may be found in the literature [6– 10]. For instance, an early extension of the coherent flame model (CFM) [11,12] to the case of partially

193

194

TURBULENT COMBUSTION—Premixed Flame Modeling

Fig. 1. A representation of turbulent flame propagation into non-homogeneous reactants. The reaction zone can be described as a staged combustion system with a primary stage corresponding to a propagating premixed flame front followed by a secondary stage corresponding to multiple non-premixed flames. The primary premixed stage produces partially burned gas, that is, a mixture composed of hot combustion products mixed with excess fuel fragments or with excess air. The excess reactants subsequently mix and burn in the secondary stage.

premixed combustion may be found in Ref. [6]. This extension includes a description of the effects of large-scale (grid-resolved) fuel stratification as well as a description of secondary non-premixed burning. The effects of small-scale (sub-grid scale) variations in mixture strength are, however, neglected. Extensions of the Bray-Moss-Libby (BML) model [13,14] and the G-equation model [15,16] may be found in Refs. [9] and [10]. The present paper is a continuation of earlier work in Refs. [6–8] aimed at adapting CFM to treat turbulent flame propagation in mixtures with variable fuel/air composition. The proposed modifications to CFM use the DNS results of Refs. [3–5] for basic guidance and account for small-scale variations in mixture strength as well as secondary burning. While setting the basis for future developments, the modified CFM model is here limited to the flame-flow regime that is described in the DNS studies, that is, to (u⬘/sL) ⬎ 1 and (Z⬘/Z˜) K 1. Coherent Flame Model We adopt CFM [11,12] as our starting model for premixed turbulent combustion. In its most basic form, CFM uses two coupled balance equations for YF and flame surface denmean fuel mass fraction ⬃ sity R ⳵q¯ ⬃ YF ⳵t

Ⳮ

⳵ (q¯ u˜i⬃ YF ) ⳱ ⳵xi

⬃F ⳵ m ⳵Y q¯ t ⳮ m ˙ LR ⳵xi rF ⳵xi

冢

冣

(1)

⳵R ⳵u˜ R Ⳮ i ⳱ ⳵t ⳵xi ⳵ mt ⳵R Ⳮ ␣jiR ⳮ b ⳵xi rR ⳵xi

冢

冣

m ˙ LR2

⬃F ⳮ YF,1) q¯ (Y

(2)

The last term on the right-hand side of equation 1 describes fuel consumption due to combustion. The last two terms on the right-hand side of equation 2 correspond to production of R by aerodynamic straining and dissipation of R by flame propagation effects. In the R-production term, the turbulent flame stretch, jt, is related to the Kolmogorov time scale 1/2

jt ⳱

冢me 冣

ckˆ

(3)

t

where ck is a function of the relative flow to flame velocity and length scale ratios u⬘/sL and lt/lF [17]. For simplicity, we assume here irreversible singlestep chemistry and unity Lewis numbers F Ⳮ rsO2 → (1 Ⳮ rs)P The mixture fraction Z is then defined as [18] ⬁ ⬁ Z ⬅ (YF ⳮ YO2/rs Ⳮ YO /rs)/(YF⬁ Ⳮ YO /rs) 2 2

Under perfectly premixed conditions, Z is a known parameter and the mean oxidizer mass fraction is simply obtained from ⬁ ⬁ ⬃ Y⬃ O2 ⳱ YO2(1 ⳮ Z) ⳮ rs(YFZ ⳮ YF )

(4)

In equations 1–4, CFM gives a description of the effects of the turbulent flow, via mt, e, u⬘, and lt, on premixed flame propagation. mt, e, u⬘, and lt are typically obtained from the k–e turbulence closure model [11,12]. Furthermore, while being restricted to the case of homogeneous reactants, CFM also provides a description of the influence of the overall unburned gas mixture composition, as characterized by Z. In equations 1–4, the Z variations of m ˙ L, YF,1, and ck allow CFM to capture changes in the turbulent flame speed and flame structure in response to changes in the overall fuel/air ratio [19]. Note that by solving an additional balance equation for Z,

TURBULENT FLAME PROPAGATION IN INHOMOGENEOUS MIXTURES

large-scale variations in fuel/air composition can easily be incorporated into CFM while keeping the same formulation. This formulation, however, remains insufficient to account for small-scale variations in mixture strength as well as secondary burning. We now introduce modifications to CFM that address this problem.

195

the closure models will require a description of the mixing field, that is, a description of the probability density function (pdf) for mixture fraction. We adopt a standard presumed b-pdf for Z, where p˜(Z) is parametrized in terms of [20]

⬃

Z˜ and Z⬙2

Furthermore, using equations 4 and A5 Modified Coherent Flame Model

0 ˆ1 Yˆ O2 and YO2

Formulation

are related to

Following the modeling strategy proposed in Refs. [5,6], we adopt a conditional two-fluid approach to describe the two combustion stages (Fig. 1). The conditional formulation makes it possible to distinguish between fresh fuel in the unburned gas that is consumed by the primary premixed flame, and fuel fragments in the partially burned gas that are consumed by the secondary non-premixed flames. We accordingly use the following decomposition

⬃ Yn ⳱ Yˆ0n Ⳮ Yˆ1n

Yˆ0F, Yˆ1F, and Z˜ 0 Yˆ O2 ⳱

⬅

0 ⳮ Yˆ O2

˜ and ⬃ Yˆ0F, Yˆ1F, Z, Z⬙2

with I a marker for the unburned gas and J a marker for the partially/fully burned gas ( J ⳱ 1 ⳮ I) (see appendix A). Yˆ0F and Yˆ1F are used as principal variables and are obtained as solutions of modeled balance equations: ⳵q¯ Yˆ0F ⳵ ⳵ m ⳵Yˆ0F Ⳮ (q¯ u˜iYˆ0F) ⳱ q¯ t ⳵t ⳵xi ⳵xi rF ⳵ xi

冣

ⳮ x ˙ F,0→1 ⳮ x ˙ F,0→2

⳵R ⳵u˜ R ⳵ mt ⳵R Ⳮ i ⳱ Ⳮ ␣j*R ⳮ DR t ⳵t ⳵xi ⳵xi rR ⳵xi

冢

冣

(9)

where j*t and DR are the modified flame stretch and R-dissipation term and are described below. Note that compared to the original CFM, the present conditional formulation corresponds to an increase in the number of principal variables from 2 to 5. This formulation is referred to hereafter as CFM-Z. Closure Models

冣

Ⳮ x ˙ F,0→1 ⳮ x ˙ F,1→2

R is treated as a principal variable and is obtained from

(5)

⳵q¯ YˆF1 ⳵ ⳵ m ⳵YˆF1 Ⳮ (q¯ u˜iYˆ1F) ⳱ q¯ t ⳵t ⳵xi ⳵xi rF ⳵ xi

冢

(8)

Finally, in addition to

¯ ), Yˆn1 ⬅ (qYnJ/q¯ ), n ⳱ F or O2 (qYnI/q

冢

(7)

⬁ ⬁˜ 1 ˆ0 ˆ1 ˜ Yˆ O2 ⳱ YO2(1 ⳮ Z) ⳮ rs(YFZ ⳮ YF ⳮ YF)

where Yˆ0n

˜ Y⬁O2(1 ⳮ Z) YˆF0 ⬁˜ YFZ

(6)

where x ˙ F,0→2 and x ˙ F,1→2 are the mean fuel mass reaction rates associated with the first and second combustion stages, and where x ˙ F,0→1 is the mean fuel mass leakage term that represents transformation of fuel in the unburned gas to excess fuel in the partially burned gas. Closure models for these terms will be presented later. Note that to achieve our specific objective of capturing the effects of variations in mixture strength,

We now turn to the closure models used to describe the source and sink terms present in equations 5, 6, and 9. The mixture composition upstream and downstream of the primary flame is estimated by the expressions in equations A1 and A2. In addition, implicit in the modeling is the assumption of statistical independence between the mixture fraction Z and the premixed reaction progress variable c. This assumption allows us to write (with q0 constant) p(Z|c˙ ⳱ 0) ⬇ p˜(Z). We also assume that p(Z|c ⳱ 1) ⬇ p˜(Z). While questionable, this choice is believed to be of secondary importance to the overall CFM-Z performance and is proposed as a way of avoiding unnecessary complications. x ˙ F,0→2 is described in CFM-Z using a standard flamelet expression x ˙ F,0→2 ⳱ 具m ˙ L典 R

(10)

196

TURBULENT COMBUSTION—Premixed Flame Modeling

now compare the value of 具m ˙ L典 obtained under arbitrary fuel/air mixing conditions, as characterized by Z˜ and

⬃

Z⬘ ⳱ (Z⬙2 )1/2 to that obtained for the same value of Z˜, but under perfectly premixed conditions, Z⬘ ⳱ 0. For simplicity, we further assume that the unmixed case corresponds to a binary mixture with a double-delta Zpdf: p˜(Z) ⳱ w1d(Z ⳮ Z1) Ⳮ w2d(Z ⳮ Z2) We then have Z˜ ⳱ w1Z1 Ⳮ w2Z2, 具m ˙ L典 ⳱ w1m ˙ L(Z1) Ⳮ w2m ˙ L(Z2)

Fig. 2. Variation of the fuel mass burning rate per unit flame surface area, m ˙ L, with the mixture fraction, Z. The solid curve gives an estimate of m ˙ L(Z), in the case of a C3H8/air, laminar premixed flame, and is obtained numerically using a detailed chemical-kinetic mechanism [21]. The symbols correspond to reactive mixtures characterized by different mean mixture compositions (top, (Z¯/Zst) ⳱ 1; bottom, (Z¯/Zst) ⬇ 1.5) and different levels of unmixedness (squares, perfectly premixed case; circles, unmixed case with a double-delta p˜(Z) distribution at Z1 and Z2). m ˙ L and Z are made non-dimensional with their values obtained at stoichiometry, m ˙ L(Zst) ⬇ 0.59 kg/m2/s, Zst ⬇ 0.06.

and, in the (x, y) plots of Fig. 2, the point of coordinates (Z˜, 具m ˙ L典) must therefore lie on the straight ˙ L (Z1)] to [Z2, m ˙L line segment that connects [Z1, m (Z2)]. The location of this point is easily compared in the plots to the location of the point [Z˜, m ˙ L (Z˜)] that represents perfect premixing. Fig. 2 shows that depending on the concavity of sL(Z) in the range Z1 ⱕ Z ⱕ Z2, 具m ˙ L典 will be larger (upward concavity) or smaller (downward concavity) than m ˙ L(Z˜). Figure 2 is fully consistent with the analysis presented in Ref. [4]. However, while the DNS study in Ref. [4] is limited to the case Z˜ ⳱ Zst, a more complete picture is proposed in Fig. 2 where two radically different situations are predicted. For variations in mixture strength around mean stoichiometric conditions (close to the flammability limits), sL(Z) exhibits downward (upward) concavity, and increasing levels of Z⬘ will result in decreasing (increasing) values of 具m ˙ L典. We now turn to x ˙ F,0→1. Both x ˙ F,0→1 and x ˙ F,0→2 describe two aspects of the same burning flamelet physics, and the corresponding models are therefore closely related. Appendix B shows that the modeling of x ˙ F,0→1 is also constrained by the realizability requirement in equation B3. We propose to write x ˙ F,0→1 ⳱

where 具m ˙ L典 ⳱

0

x ˙ F,0→2

(11)

where

1

冮

YF,1 YF,0 ⳮ YF,1

m ˙ L(Z)p˜(Z)dZ

This straightforward extension of the CFM closure is consistent with previous DNS-based findings [3,4]. Let us consider for instance a propane/air, laminar premixed flame. The variations of m ˙ L with Z are presented in Fig. 2. These variations are estimated based on an expression of m ˙ L in terms of sL, m ˙ L ⳱ q0sL(YF,0 ⳮ YF,1), with YF,0 and YF,1 given by equations A1 and A2, and variations of sL with Z obtained numerically using a detailed chemical–kinetic mechanism proposed by Peters [21,22]. Let us

YF,0 ⳱ YF⬁Z˜ and YF,1 ⳱

1

冮Y 0

˜ (Z)dZ F,1(Z)p

Because of the coupling relation between fuel mass and oxidizer mass during combustion [18], we have (YF,0 ⳮ YF,1) ⳱ (YO2,0 ⳮ YO2,1)/rs Therefore YF,1/(YF,0 ⳮ YF,1) ⳱

␾YO2,0/(YO2,0 ⳮ YO2,1), ⳮ 1 ⱖ ␾ ⳮ 1

TURBULENT FLAME PROPAGATION IN INHOMOGENEOUS MIXTURES

and using equation 11, the inequality in equation B3 is found to be always satisfied. We choose to describe secondary burning using a classical, mixing controlled model proposed by Magnussen and Hjertager [23]: x ˙ F,1→2 ⳱ A q¯

1 e Yˆ min YˆF1 , O2 k rs

冢

冣

(12)

The most critical choices in CFM-Z are made below, when considering modifications to the R equation. We start with the CFM expression of DR given in equation 2. An equivalent expression is in terms of sL and c¯ [11,12]

using the DNS results of Refs. [3–5]. As mentioned in the introduction, for (u⬘/sL) ⬎ 1, the total premixed flame surface area SV is approximately independent of Z⬘. We now examine the implications of this DNS result to the modeling of j*t . We continue to consider the KPP model problem and further assume a statistically uniform fuel/air distribution. By space-averaging equation 5, the following global balance equation for fresh fuel mass is obtained: q0sTYF,0 ⳱

DR ⳱ b s*R /(q¯ YˆF0 /q0YF,0) L

具m ˙ L典R2

YF,0 ⳮ YF,1

q¯ Yˆ0F

(13)

We now consider the CFM-Z modifications to j*t . Let us consider the ideal problem due to Kolmogorov, Petrovskii, and Piskunov [24], referred to hereafter as the KPP problem, and corresponding to a statistically one-dimensional, plane, turbulent flame that propagates steadily into frozen turbulence and frozen fuel/air distribution. This KPP configuration is considered in several studies in the literature because its simplicity allows the derivation of exact analytical expressions [12,25,26]. For instance, the analytical methods presented in Refs. [12] and [26] show that the turbulent flame speed predicted by CFM-Z is equal to sT ⳱ 2

冢rm

t

R

冣

Ⳮ⬁

冮

ⳮ⬁

Rdx1

we obtain SV sT ⳱ ˜ S0 R sL(Z)

(15)

˜ 具m ˙ L典 YF,0 ⳮ YF,1(Z) (16) ˜ m ˙ L(Z) YF,0 ⳮ YF,1 and where Q(Z˜) designates the value of Q obtained under perfectly premixed conditions. Equation 15 is another version of the classical flamelet result that gives sT as the product of SV times a mean flamelet speed. This flamelet speed is here expressed as the product of sL(Z˜) times the flamelet factor R. Note that R ⳱ 1 for Z⬘ ⳱ 0. In addition, for small Z⬘, the second term on the right-hand side of equation 16 remains close to unity, and R is therefore approximately equal to 具m ˙ L典/m ˙ L(Z˜). The variations of this ratio with Z⬘ were discussed earlier in this section and are illustrated in Fig. 2. Combining equations 14 and 15, we see that SV varies with Z⬘ like (j*t )1/2/R. Therefore, the requirement that SV is independent of Z⬘ implies that j*t is proportional to R2: ˜ j*t ⳱ R2 jt(Z) R⬅

1/2

⳱ R2

冢me 冣 t

1/2

␣j*t

(x ˙ F,0→1 Ⳮ x ˙ F,0→2)dx1

where

where s*L remains to be specified. The discussion in appendix B indicates that the choice of s*L is constrained by the realizability requirement that R → 0 ˆ0F → 0 (c¯ → 1). We adopt here the expression when Y given in equation B5: YF,0

ⳮ⬁

SV ⳱ S0

2

DR ⳱ b

Ⳮ⬁

冮

where integration is along the flame normal x1 direction. Using equations 10 and 11 and

DR ⳱ b sLR2/(1 ⳮ c¯) Using equation A5, a straightforward extension to the case of non-homogeneous reactants is readily obtained

197

(14)

Therefore, the effects of fuel/air unmixedness on turbulent flame speed depend exclusively on their influence in the model descriptions of j*t . For instance, if j*t is estimated by its CFM expression in equation 3 and is thereby assumed to be independent of Z⬘, sT will also be independent of Z⬘. Conversely, any desired effect of Z⬘ on sT must be incorporated into CFM via a modified closure expression for j*t . The choice of a closure model for j*t can be made

ck

l 冢s u⬘(Z)˜ , l (Z) ˜ 冣 t

L

(17)

F

where jt(Z˜) is the standard CFM expression for flame stretch. With this closure model, non-zero values of Z⬘ induce modifications of the mean flamelet speed, via deviations of R from unity; the turbulent flame speed is modified in the exact same proportions, sT ⬃ R; and the total premixed flame surface area remains unaffected, SV ⳱ SV(Z˜). Numerical Evaluation of CFM-Z in the KPP Problem While CFM-Z is intended for numerical simulations of combustion in practical engineering systems,

198

TURBULENT COMBUSTION—Premixed Flame Modeling

Fig. 3. Variation of the turbulent flame speed, sT, with the degree of unmixedness, (Z⬘/Z˜), as obtained for different mean mixture compositions, 0.75 ⱕ (Z˜/Zst) ⱕ 1.75. The turbulent flowfield is the same in all cases: u⬘/sL(Zst) ⳱ 10 and lt/lF(Zst) ⳱ 100. The variations sT(Z˜, Z⬘) are based on both a KPP analysis (lines) and a numerical evaluation (circles) of CFM-Z. sT(Z˜, Z⬘) is made non-dimensional with its value obtained under perfectly premixed conditions, sT(Z˜).

Fig. 4. Spatial variations of the conditional mean fuel and oxidizer mass fractions, Y0F, Y1F, Y0O2, and Y1O2, across the flame. (Z˜/Zst) ⳱ 1.5, (Z⬘/Z˜) ⳱ 0.20, u⬘/sL(Zst) ⳱ 10, and lt/lF(Zst) ⳱ 100. The distance x1 is made non-dimensional with the turbulent integral length scale. Y1O2 , is multiplied by a factor of 25 to facilitate the graphical display. The unburned (burned) gas side corresponds to x1 ⬍ 0 (x1 ⬎ 0).

ˆ ˆ ˆ

ˆ

ˆ

we consider here an intermediate validation step where the model is tested in the ideal KPP configuration. The mathematical formulation uses equations 5, 6, and 9 with constant values of k, e, Z˜, and ⬃ Z⬙2, and coupled with the usual conservation equations for mass, momentum, and energy. The CFMZ closure models are combined with the b-pdf description of p˜(Z), m ˙ L(Z) in Fig. 2, and YF,0(Z) and

YF,1(Z) in equations A1 and A2. Mixture fraction is ⬁ defined using Zst ⬇ 0.06, rs ⬇ 3.64, Y⬁F ⳱ 1, YO ⬇ 2 0.233. The model constants are rF ⳱ rR ⳱ 1, ␣ ⳱ 2.1, b ⳱ 1, A ⳱ 2. In the energy equation, we use s(Zst) ⳱ 5 and assume for simplicity that the heat of reaction per unit mass of fuel is independent of mixture strength. The numerical solutions of the KPP problem are obtained using a high-order finite difference solver similar to the solver used in previous DNS studies [4]. The solutions provide information on both the turbulent flame speed and the turbulent flame structure. For instance, Fig. 3 presents the estimated variations of sT with Z⬘, for different values of Z˜. Note that for any given choice of Z˜, the restriction of CFM-Z to flammable mixtures introduces an upper limit in the range of admissible Z⬘ values. Typically, (Z⬘/Z˜) ⬍ 0.2. In this small Z⬘ regime, sT is predicted to decrease with increasing values of Z⬘ when the mixture compositions are close to stoichiometry, whereas an opposite trend is predicted when the mixture compositions are close to the flammability limits. The effects of fuel/air unmixedness are particularly pronounced for very rich mixtures, with predicted increases in sT up to as much as 90%. Finally, in Fig. 3, the numerical results are also compared to the analytical expression in equation 14. The excellent agreement supports the extension of the KPP analysis to the case of partially premixed systems. We now examine a typical CFM-Z flame structure. Fig. 4 is of particular interest since it provides an illustration of the conditional approach perspective. In this perspective, the reactants present in the unburned gas (x1 → ⳮ⬁) are consumed by primary burning or leaked to the partially burned gas, and ˆ0F and YˆO0 2 must vanish when crossing the flame. In Y addition, the excess reactants in the partially burned gas are produced by leaking across the primary flame ˆ1F and and consumed by secondary burning, and Y 1 ˆ YO2 increase or decrease, depending on the relative weights of these two processes. Secondary burning lasts until one of the reactants is depleted, and accordingly in Fig. 4 Yˆ1O2 → 0 when x1 → Ⳮ⬁. Conclusion The CFM was modified in the present study to treat turbulent flame propagation in mixtures with variable fuel/air composition. The modifications include a description of the reaction zone as a staged, premixed/non-premixed combustion system using a conditional two-fluid approach; a description of primary burning in the premixed stage using simple extensions of classical flamelet expressions; a description of secondary burning in the non-premixed stage using a mixing controlled model; a description

TURBULENT FLAME PROPAGATION IN INHOMOGENEOUS MIXTURES

of sub-grid scale fluctuations in mixture strength using a standard presumed b-pdf approach for mixture fraction; and a correction factor introduced in the model description of turbulent flame stretch. The proposed modifications to CFM are based on previous results from DNS [3–5]. The DNS studies describe simplified flame-flow configurations corresponding to large turbulent intensities and flammable mixtures, and the scope of the present paper is limited accordingly to a regime where (u⬘/sL) ⬎ 1 and (Z⬘/Z˜) ⬍ 0.2. The latter restriction is used as an intermediate step in the present modeling effort as it removes the additional complication of premixed flame extinction. The correction factor introduced in the description of turbulent flame stretch is also based on a theoretical analysis of a simplified problem due to Kolmogorov, Petrovskii, and Piskunov [24]. The ideal KPP problem corresponds to a plane, turbulent flame propagating steadily into frozen turbulence and frozen fuel/air distribution. The modified CFM model was tested numerically in the KPP configuration, and the following trends were predicted: for variations in mixture strength around mean stoichiometric conditions, unmixedness tends to have a net negative impact on the turbulent flame speed; in contrast, for variations close to the flammability limits, unmixedness tends to have a net positive impact on sT. Consistent with previous DNS results [3–5], these effects are related to changes in the mean flamelet structure of the primary premixed flame, whereas the total premixed flame surface area remains independent of Z⬘.

In this appendix, we first introduce a generalized premixed reaction progress variable, c, to distinguish between pre- and post-primary flame gases. We then use c to define the fluid markers I and J and the associated conditional means. The mixture composition in the unburned gas is simply given by the classical pure mixing solution: ⬁ YF,0 ⳱ Y⬁FZ and YO2,0 ⳱ YO (1 ⳮ Z) 2

(A1)

Furthermore, if the chemistry is sufficiently fast, the mixture composition in the partially burned gas may be estimated using the classical Burke–Schumann solution [18]: YF,1

Z ⳮ Zst ⳱ YF max 0, and 1 ⳮ Zst ⬁

冢

冣

⬁

YO2,1 ⳱ YO2

冢

冣

Z max 0,1 ⳮ Zst

time scales, one can use the following definition for c [3,4]: c ⬅ (YF,0 ⳮ YF)/(YF,0 ⳮ YF,1) In the present (Z, c)-formulation, statistical averages may be expressed as double integrals over Z and c: ¯ ⳱ Q

(A2)

Assuming that the transition from unburned state to partially burned state is fast compared to mixing

1

1

冮 冮 0

Q(Z, c)p(Z, c)dZdc

0

With p(Z, c) ⳱ p(Z|c)p(c), and using the classical flamelet approximation p(c) ⬇ prob(c ⳱ 0)d(c) Ⳮ prob(c ⳱ 1)d(1 ⳮ c) [13,14], we obtain ¯ ⳱ prob(c ⳱ 0)Q0 Ⳮ prob(c ⳱ 1)Q1 Q

(A3)

Equation A3 is a central relation in the BML framework [13,14] that gives, for instance, c¯ ⳱ prob(c ⳱ 1) and (1 ⳮ c¯) ⳱ prob(c ⳱ 0). We now turn to the definitions of I and J. I(c) ⬅

cf

冮

0

d(c ⳮ c⬘)dc⬘

and J(c) ⳱ 1 ⳮ I(c), where cf defines the premixed flame front location (cf ⬇ 0.8). Using the flamelet assumption of a bimodal p(c), we find I¯ ⳱ prob(c ⱕ cf) ⬇ prob(c ⳱ 0) ⳱ (1 ⳮ c¯) and J¯ ⳱ 1 ⳮ I¯ ⬇ c¯

(A4)

Using equations A3 and A4, the conditional mean fresh reactants mass fractions are YˆF0 ⬅

Appendix A: Definitions for the Conditional Approach

199

qYFI qY ⳱ 0 F,0 (1 ⳮ c¯) and q¯ q¯ 0 Yˆ O2 ⳱

YO2,0 YF,0

YˆF0

(A5)

where q0 is assumed to be constant. Assuming statistical independence between Z and c, we may write ⬁ ˜ YF,0 ⬇ Y⬁FZ˜ and YO2,0 ⬇ YO (1 ⳮ Z) 2

With these approximations, equation A5 is used in CFM-Z to relate, respectively, c¯ and Yˆ0O2 to the prinˆ0F and Z˜. cipal variables Y

Appendix B: Realizability Requirements Realizability Requirement for the Source Terms Associated with the First Combustion Stage The CFM-Z relations in equations 7 and 8 may also be cast in alternative forms involving the mean rates of change during primary burning:

200

TURBULENT COMBUSTION—Premixed Flame Modeling

x ˙ O2,0→2 ⳱ rs x ˙ F,0→2 rs (x ˙ Ⳮx ˙ F,0→2) ␾ F,0→1

x ˙ O2,0→1 Ⳮ x ˙ O2,0→2 ⳱

(B1) (B2)

where ␾ is the local grid-cell equivalence ratio

␾ ⬅ (rsYF,0/YO2,0) Combining these equations together, we obtain x ˙ O2,0→2/(x ˙ O2,0→1 Ⳮ x ˙ O2,0→2) ⳱

␾x˙ F,0→2/(x˙ F,0→1 Ⳮ x˙ F,0→2) Now, given the present sign convention, a basic requirement in CFM-Z is that the mean rates of change in equations B1 and B2 are all positive. x ˙ F,0→1, x ˙ F,0→2 and x ˙ O2,0→2 are given by equations 10, 11, and B1 and are therefore positive. The requirement that x ˙ O2,0→1 ⱖ 0 is not, however, unconditionally satisfied in the model. Using equations B1 and B2, this requirement is satisfied when x ˙ F,0→1 ⱖ (␾ ⳮ 1) max(0, x ˙ F,0→2)

(B3)

Realizability Requirement for the Flame Surface Dissipation Term In Ref. [26], the modeling of DR is shown to be constrained by the realizability requirement that R → 0 when the premixed reaction is close to completion, that is, when

⬃ Y →Y

(c¯ → 1) When applied to equations 1 and 2, this requirement is shown to limit the range of admissible values of b according to b ⱖ (rF/rR). A strictly identical analysis applied to CFM-Z leads to the following new constraint: F

F,1

rF 具m ˙ L典 (B4) rR q0(YF,0 ⳮ YF,1)s*L where 具m ˙ L典, YF,0, and YF,1 are evaluated for c¯ → 1. In general, it is difficult to guarantee that the requirement in equation B4 is always satisfied. Such difficulties are conveniently avoided by making a judicious choice for s*L. For instance, the CFM and CFM-Z constraints are identical when the following model for s*L is adopted: b ⱖ

s*L ⳱

具m ˙ L典

(B5) q0(YF,0 ⳮ YF,1) In CFM-Z, we use equation B5, rF ⳱ rR and b ⳱ 1, and the inequality in equation B4 is always satisfied. Nomenclature Symbols A c

model constant in equation 12 premixed reaction progress variable

cf

value of c used to define the premixed flame location DR R-dissipation term in equation 9 I marker for the unburned gas (I ⳱ 1 upstream of the premixed flame, I ⳱ 0 downstream) J marker for the partially/fully burned gas ( J ⳱ 1 ⳮ I) k mean (Favre-averaged) turbulent kinetic energy lF laminar flame thickness lt integral length scale of the turbulent flow m ˙L laminar fuel mass burning rate per unit premixed flame surface area p(c) probability density function (pdf) of c p˜(Z) density-weighted pdf of Z p(Z, c) joint pdf of Z and c p(Z|c ⳱ c*) pdf of Z conditioned on c ⳱ c* prob(c ⳱ c*) statistical probability of c ⳱ c* R flamelet factor (equation 16) rs stoichiometric oxidizer/fuel mass ratio sL laminar flame speed turbulent flame speed sT total premixed flame surface area, SV SV ⬅ 兰V RdV projected area of the premixed flame S0 surface on (x2 ⳮ x3) planes t time T fluid temperature u⬘ turbulent rms velocity, u⬘ ⬅ (2k/3)1/2 ui xi-component of the flow velocity vector xi Cartesian i-coordinate component, x1 is the direction of mean flame propagation Yn mass fraction of species n Z mixture fraction Z⬘ rms fluctuation in mixture fraction, Z⬘ ⳱ ( Z⬙2 )1/2␣ Z⬙ fluctuation in mixture fraction in a Favre decomposition stoichiometric value of mixture fracZst tion ␣ model constant in equation 2 b model constant in equation 2 model function for the turbulent ck flame stretch, see equation 3 and Ref. [17] e mean (Favre-averaged) rate of dissipation of turbulent kinetic energy jt turbulent flame stretch turbulent kinematic viscosity mt x ˙ n,0→1 mass conversion rate of n in the unburned gas to excess n in the partially burned gas x ˙ n,0→2 mass reaction rate for species n associated with primary burning

⬃

TURBULENT FLAME PROPAGATION IN INHOMOGENEOUS MIXTURES

x ˙ n,1→2

␾ q rF rR R s Subscripts F, O2, P 0 1

mass reaction rate for species n associated with secondary burning local grid-cell equivalence ratio, ␾ ⬅ (rsYF,0/YO2,0) mass density model constant in equation 1 model constant in equation 2 flame surface density heat release factor, s ⬅ (T1(Z) ⳮ T0)/ T0 fuel, oxidizer, products value in the unburned gas (upstream of the premixed flame, c ⳱ 0) value in the partially/fully burned gas (downstream of the premixed flame, c ⳱ 1)

Superscripts * CFM-Z modified value ⬁ value in the feeding streams Averaging symbols ¯ Q ˜ Q 具Q典 ˆ0 Q ˆ1 Q Q0 Q1

standard ensemble-average ˜ density-weighted (Favre-) average, Q ⬅ (qQ/q) ¯ CFM-Z Z-average, 具Q典 ⬅ 兰10 Q(Z)p˜(Z)dZ CFM-Z average conditioned on being ˆ0 ⬅ in the unburned gas, Q (qQI/q) ¯ CFM-Z average conditioned on being ˆ1 ⬅ in the partially burned gas, Q (qQJ/q) ¯ Z-average conditioned on being in the ¯ 0 ⬅ 兰10 Q0(Z)p(Z|c unburned gas, Q ⳱ 0)dZ Z-average conditioned on being in the ¯ 1 ⬅ 兰10 partially burned gas, Q Q1(Z)p(Z|c ⳱ 1)dZ REFERENCES

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