Analysis of mixture fraction fluctuations generated by spray vaporisation

Chapter 1 A MODEL DESCRIPTION OF THE EFFECTS OF VARIABLE FUEL-AIR MIXTURE COMPOSITION ON TURBULENT FLAME PROPAGATION Jer^ome Helie Institut Francais du Petrole Division Techniques d'Applications Energetiques Rueil-Malmaison, France Arnaud Trouve University of Maryland Department of Fire Protection Engineering College Park, MD, USA atrouve@eng.umd.edu Abstract Strati ed direct-injection spark-ignition engines feature a fuel-air mix- ing eld characterized by large spatio-temporal variations of the mean mixture composition, e( i ), as well as regions with signi cant statistical uctuations around e. Modi cations to the Coherent Flame Model (CFM) are proposed in the present study to account for the e ects of variable mixture strength on the primary premixed ame, as well as for the downstream formation of a secondary non-premixed reaction zone. Compared to previous work, the domain of validity of the present modi cations is extended to the case of combustion with occurences of premixed ame extinction. The modeling strategy is based on previous results from direct numerical simulations as well as a theoretical analysis of a simpli ed problem due to Kolmogorov, Petrovskii and Piskunov (KPP). The KPP problem corresponds to a one-dimensional, turbulent ame propagating steadily into frozen turbulence and frozen fuel-air distribution, and provides a convenient framework to test the modi ed CFM model. In this simpli ed but somewhat generic con guration, the following trends are predicted: for small variations in mixture strength, in the absence of premixed ame extinction, partial fuel-air premixZ x ;t Z i:e: 1 2 ing has a net negative (positive) impact on the turbulent ame speed, T , for mixtures close to stoichiometry (close to the ammability limits); for large variations in mixture strength, in the presence of premixed ame extinction, partial fuel-air premixing has a net negative impact on T and results in increased levels of secondary burning. s i:e: s Keywords: Turbulent ame propagation, modeling, partially-premixed combustion, Internal Combustion engines, spark-ignition, direct-injection 1. Introduction Direct-injection spark-ignition (DI-SI) engines represent one of the most promising technology to achieve further reductions in fuel consumption in automotive transport (Zhao et al., 1997) (Takagi, 1998). Under light- or part-load operation, these engines use fuel strati cation to provide additional control on combustion performance and establish overall fuel lean operating conditions. In strati ed DI-SI engines, the reactive mixture features temporal and spatial variations in the statisticallyaveraged fuel-air composition, with stoichiometric to slightly rich conditions (Ze  Z ) achieved at the spark plug at the time of ignition, and very lean conditions (Ze  Z ) sustained close to the cylinder walls. In addition, the reactive mixture also features regions with signi cant statistical uctuations in the charge distribution, (Z 0 =Ze ) = O(1). Statistical uctuations are produced by turbulent mixing in regions with strong fuel strati cation as well as by the late vaporization of over-sized liquid fuel droplets. These spatio-temporal and statistical Z -variations correspond to inhomogeneities in mixture strength into which the sparkignited turbulent ame propagates. The propagation of turbulent ames into mixtures with variable fuelair composition has been described recently using direct numerical simulations (DNS) in a series of related studies (Poinsot et al., 1996) (Helie & Trouve, 1998) (Haworth et al., 2000). The numerical con gurations considered in the DNS studies correspond to propane-air ames propagating into isotropic turbulent ow and into non-homogeneous reactants characterized by di erent levels of unmixedness. They also correspond to a regime of ame- ow conditions where the ow to ame velocity scale ratio is large, (u0 =s ) > 1, and the variations in mixture strength remain small, (Z 0 =Ze )  1. The latter restriction is used as an intermediate step in the DNS studies, where the initial focus is on in ammable mixtures, i:e: on mixture compositions within the propane-air ammability limits and without the additional complication of premixed ame extinction. st st L 3 Turbulent Flame Propagation in Inhomogeneous Mixtures Unburnt gas Partially burnt gas inhomogeneous mixture composition 1st stage : Premixed Flame 2nd stage : Diffusion Flames Figure 1.1. A representation of turbulent ame 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 ame front followed by a secondary stage corresponding to multiple non-premixed ames. The primary premixed stage produces partially burnt gas, 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. i:e The DNS results show that : (1) under lean-rich conditions, the reaction zone can be described as a staged premixed/non-premixed combustion system (Fig. 1) ; (2) the rst premixed stage can be conveniently described using classical laminar amelet concepts; (3) for (small) variations in mixture strength around mean stoichiometric conditions, Ze  Z , unmixedness tends to have a net negative impact on the overall mean premixed reaction rate (i:e: on the turbulent ame speed); (4) this e ect is related to changes in the mean amelet structure, i:e: to a decrease of the mean amelet mass burning rate per unit ame surface area with increasing values of Z 0 ; (5) for (u0 =s ) > 1, the e ect of unmixedness on the turbulent premixed ame surface area remains negligible. The general objective of the present study is to examine how the DNS results above may be incorporated into current model descriptions of turbulent ame propagation in SI engines. A number of previous modeling e orts aimed at introducing the e ects of fuel-air unmixedness into several classical turbulent combustion models may be found in the literature. Previous e orts include extensions of widely used amelet models, such as the Bray-Moss-Libby (BML) model (Bray, 1990) (Bray & Libby, 1994), the Coherent Flame Model (CFM) (Candel et al., 1990) (Fichot et al., 1993), and the G-equation model (Peters, 1992) (Wirth & Peters, 1992). For instance, an early extension of CFM to the case of combustion of mixtures with partially premixed reactants may be found in Veynante et al. (1991). This extension includes a description of the e ects of mean fuel strati cation as well as a description of secondary non-premixed burning. The e ects of statistical variations in mixture st L 4 strength are, however, neglected. Note that this modi ed version of CFM has been used recently to simulate strati ed charge combustion in DI-SI engines (Baritaud et al., 1996) (Duclos & Zolver, 1998). Furthermore, a recent extension of BML to the case of partially premixed systems may also be found in Lahjaily et al. (1998). This extension includes a description of fuel-air mixing e ects associated with both a spatially variable mean and the presence of turbulent uctuations, but remains limited to the case of lean combustion without secondary burning. Finally, an extension of the G-equation model may be found in Muller et al. (1994). While this extension is originally proposed to describe ame propagation in the stabilization region of turbulent lifted jet di usion ames, it may also be applied to the problem of ame propagation in strati ed SI engines. The present paper is a continuation of earlier work in Veynante et al., Baritaud et al. and Duclos & Zolver aimed at adapting CFM to treat turbulent ame propagation in mixtures with variable fuel-air composition. It is also an extension of previous work in Helie & Trouve (2000) where a preliminary version of a modi ed CFM, was proposed using DNS for basic guidance and accounting for uctuations in mixture strength as well as secondary burning. This preliminary version, however, was restricted to the case of in ammable fuel-air mixtures (the small-Z 0 regime). The case of mixtures characterized by large values of Z 0 with occurences of premixed ame extinction is now considered in the following. 2. The modi ed Coherent Flame Model 2.1. Formulation of CFM-Z Following the modeling strategy proposed in Veynante et al. (1991) and Haworth et al. (2000), we adopt a conditional two- uid approach to describe the two combustion stages (Fig. 1). The conditional formulation allows to distinguish between fresh fuel in the unburnt gas, that is consumed by the primary premixed ame, and fuel fragments in the partially burnt gas, that are consumed by the secondary nonpremixed ames. We accordingly use the following decomposition : Yf = Yc 0 + Yc 1 , where Yc 0  (Y I=), Yc 1  (Y (1 , I )=), with Y the fuel mass fraction and I a marker for the unburnt gas (see Helie & Trouve, 2000, for more details). Yc 0 and Yc 1 are used as principal variables and are obtained as solutions of modeled balance equations: F F F F F F F F F F @Yc 0 + @ (ue Yc 0) = @ (  @ Yc 0 ) , !_ @t @x @x  @x t F i i F F F; i F i 0!1 , !_ 0!2 F; (1.1) 5 Turbulent Flame Propagation in Inhomogeneous Mixtures @Yc 1 + @ (ue Yc 1) = @ (  @ Yc 1 ) + !_ 0!1 , !_ 1!2 (1.2) @t @x @x  @x where !_ 0!2 and !_ 1!2 are the mean fuel mass reaction rates associated with the rst and second combustion stages, and where !_ 0!1 is F F t i F F; i i F; F F; i F; F; the mean fuel mass leakage term that represents transformation of fuel in the unburnt gas to excess fuel in the partially burnt gas. Closure models for these terms will be presented in the next section. Note that to achieve our speci c objective of capturing the e ects of variations in mixture strength, the closure models will require a description of the mixing eld, i:e: a description of the probability density function (Pdf) for mixture fraction. We adopt here a standard presumed 002 . -Pdf for Z , where pe(Z ) is parametrized in terms of Ze and Zg 0 1 002 , the premixed ame Finally, in addition to Yc , Yc , Ze and Zg surface density  is treated as a principal variable and is obtained from the following balance equation: @  + @ ue  = @ (  @  ) +   , D (1.3) F F i t  @t @x @x  @x where  and D are the modi ed turbulent ame stretch and -dissipation t i i i t term, and are described below. 2.2. Closure models We now brie y summarize the closure models used to describe the source and sink terms present in Eqs. 1-3 and refer the reader to Helie & Trouve (2000) for a more detailed presentation. A new important feature in the present version of CFM-Z is the use of Z -conditioning to distinguish between in ammable (Z < Z < Z with Z and Z the lean and rich fuel-air mixture ammability limits) and non-in ammable (0  Z  Z or Z  Z  1) mixture compositions, that are respectively associated with burning and extinguished premixed amelets. For instance, primary burning is associated with ammable fuel-air conditions and is described using a modi ed amelet expression: !_ 0!2 = hm_ i  (1.4) R R where hm_ i = ( LR m_ (Z )pe(Z )dZ )=( LR pe(Z )dZ ) is the re-normalized mean fuel mass burning rate per unit premixed ame surface area. Fuel mass transfer across the primary ame has contributions from both burning and extinguished premixed amelets and is described using the following decomposition: !_ 0!1 = !_ 0!1 + !_ 0!1 (1.5) L L R R F; L b Z L b Z Z L F; Z F; ;b F; ;e L R 6 We write: Y 0 , Y 0 !_ 0!2 ; !_ 0!1 = 0! 2 Y 0 ,Y 1 Y 0 ,Y 1 (1.6) R 1 R e where Y 0 = Y Z , Y = L Y (Z )pe(Z )dZ , n = 0 or 1, in which Y 0(Z ) and Y 1 (Z ) respectively describe the mixture composition upstream and downstream of the primary ame and are estimated by the classical pure mixing and Burke-Schumann solutions. Secondary burning is simply described using a classical, mixing controlled model proposed by Magnussen & Hjertager (1976): !_ F; Y 0!1 = ;b 1 b !_ F; b b F; b F; F; F; b F; F; F; Z F;n F F; b F; b F; ;e F; F;n Z F; 0 1 1 1 Yd2 A  c @ !_ 1!2 = A  k Min Y ; r (1.7) O F F; s where classical coupling relations are used to relate the mean oxidizer mass fraction Yd2 1 to Yc 0 , Yc 1 , and Ze. Finally, the CFM-Z modi cations to the -equation are based on the following Z -sensitive expressions of D and  : O F F t D = Y F; Y F;  = Z t 0 b R Z hm_ i 2 Yc 0 0 ,Y (1.8) L b F; 1 b F  (Z )pe(Z )dZ (1.9) t L Z In Eq. 9, the ame stretch  (Z ) is estimated using the closure expression proposed by Meneveau & Poinsot (1991) where  is related to the Kolmogorov time scale:  = (= )1 2 , with a function of the relative ow to ame velocity and length scale ratios, (u0 =s ) and R (l =l ) = (l =(D =s )). We obtain:  = (= )1 2 LR (Z )pe(Z )dZ . While this last expression may be used directly, the following approximation is adopted in order to avoid a costly integration of over the in ammable Z -domain: t t t t = k k L t F t th L t t = Z Z k k  u0  Z R l ; (1.10) hs i D =hs i ( L pe(Z )dZ ) R R where hs i = ( LR s (Z )pe(Z )dZ )=( LR pe(Z )dZ ), and where  depends R on fuel-air mixture composition via hs i and ( LR pe(Z )dZ ).  = (  )1 2 t k t L Z L b Z Z t = b th L b Z Z L t Z Z L b The present model follows closely the modeling strategy proposed by Helie & Trouve (2000) and accounts for premixed ame extinction due to Z 7 Turbulent Flame Propagation in Inhomogeneous Mixtures 1.00 turbulent flame speed Z/Zst=1 Z/Zst=0.75 0.75 0.50 Z/Zst=1.5 Z/Zst=1.75 0.25 0.0 0.5 1.0 1.5 rms mixture fraction Figure 1.2. Variation of the turbulent ame speed, sT , with the degree of unmixed- ness, ( e), for di erent mean mixture compositions, 0 75  ( e st )  1 75. The turbulent ow eld is the same in all cases : L ( st ) = 10 and t F ( st ) = 100. The variations T ( e ) are based on both a KPP analysis (lines) and a numerical evaluation (circles) of CFM-Z. T ( e ) is made non-dimensional with T ( st 0). 0 Z =Z : 0 u =s s Z; Z Z =Z Z : l =l Z 0 s Z; Z 0 s Z ; non-in ammable fuel-air compositions via the following two ingredients: an additional fuel mass leakage term across the primary burning zone, !_ 0!1 ; and a modi ed turbulent ame stretch  . Note that the last term on the right-hand side of Eq. 11 is dominant for large values of Z 0 and is responsible in CFM-Z for the reduced values of stretch, premixed ame surface area and turbulent ame speed obtained with large levels of fuel-air unmixedness. F; 3. ;e t Numerical evaluation of CFM-Z Let us now consider the ideal problem due to Kolmogorov, Petrovskii and Piskunov (1937), referred to hereafter as the KPP problem, and corresponding to a statistically one-dimensional, plane, turbulent ame that propagates steadily into frozen turbulence and frozen fuel-air distribution. This KPP con guration is considered in several studies in the literature, because its simplicity allows the derivation of exact analytical expressions (Hakberg & Gosman, 1984) (Duclos et al., 1993) (Fichot et al., 1993). For instance, the analytical methods presented in Fichot et al. (1993) and Duclos et al. (1993) show that the turbulent ame speed predicted by CFM-Z is equal to : s = 2 (  T t   )1 2 (1.11) = t Therefore, the e ects of fuel-air unmixedness on s depend exclusively on their in uence in the model description of  . T t 8 The present KPP formulation uses Eqs. 1-3 with constant values of k, , Ze and Zg002 , coupled with the usual conservation equations for mass, momentum and energy. The CFM-Z closure models of x2 are combined with the -Pdf description of pe(Z ) and a description of m_ (Z ) from a amelet library. Mixture fraction is de ned using Z  0:06, r  3:64, Y 1 = 1, Y 12  0:233. The model constants are  =  = 1, = 2:1, = 1, A = 2. In the energy equation, we use  (Z ) = 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 nite di erence solver similar to the solver used in previous DNS studies (Helie & Trouve, 1998). The solutions provide information on both the turbulent ame speed and the turbulent ame structure. For instance, gure 2 presents the estimated variations of s with Z 0 , for di erent values of Ze. Consistent with previous results (Helie & Trouve, 2000), in the small-Z 0 regime, i:e: for (Z 0 =Ze ) < 0:2, s is predicted to decrease with increasing values of Z 0 when the mixture compositions are close to stoichiometry, whereas an opposite trend is predicted when the mixture compositions are close to the ammability limits. In the largeZ 0 regime, i:e: for (Z 0 =Ze)  0:2, s is a ected primarily by occurences of premixed ame extinction and predicted to decrease with increasing values of Z 0 . In Fig. 2, the numerical results are also compared to the analytical expression in Eq. 11. The excellent agreement supports the extension of the KPP analysis to the case of partially premixed systems. L st F s F O st T T mean conditional mass fraction T 0.2 YO2 0 0.1 YO2 1 0 YF 1 YF 0.0 −20 −10 0 10 20 30 distance across the flame Figure 1.3. Spatial variations of the conditional mean fuel and oxidizer mass fractions across the ame. ( e st ) = 0 75, ( e) = 1 , L ( st ) = 10 and t F ( st ) = 100. The distance 1 is made non-dimensional with the turbulent integral length scale. The unburnt (burnt) gas side corresponds to 1 0 ( 1 0). Z =Z : 0 Z =Z : 0 u =s Z x x < x > l =l Z 9 REFERENCES We now examine a typical CFM-Z ame structure. Figure 3 is of particular interest since it provides an illustration of the conditional approach perspective. In this perspective, the reactants present in the unburnt gas (x1 ! ,1) are consumed by primary burning or leaked to the partially burnt gas, and Yc 0 and Yd2 0 must vanish when crossing the ame. In addition, the excess reactants in the partially burnt gas are produced by leaking across the primary ame and consumed by secondary burning, and Yc 1 and Yd2 1 increase or descrease, depending on the relative weights of these two processes. Secondary burning lasts until one of the reactant is depleted and accordingly in Fig. 3, Yc 1 ! 0 when x1 ! +1. F F O O F 4. Conclusion The Coherent Flame Model (CFM) is modi ed in the present study to treat turbulent ame propagation in mixtures with variable fuel-air composition. The modi cations include: a description of the reaction zone as a staged premixed/non-premixed combustion system using a conditional two- uid approach; a description of primary burning in the premixed stage using simple extensions of classical amelet expressions; a description of secondary burning in the non-premixed stage using a mixing controlled model; a description of statistical uctuations in mixture strength using a standard presumed -Pdf approach for mixture fraction; a description of premixed ame extinction associated with nonin ammable fuel-air compositions via an additional fuel mass leakage term across the primary burning zone and a modi ed expression for the turbulent ame stretch. The modi ed CFM model is tested numerically in a simpli ed con guration due to Kolmogorov, Petrovskii and Piskunov (KPP). The KPP problem corresponds to a plane, turbulent ame propagating steadily into frozen turbulence and frozen fuel-air distribution. The following trends are predicted from the KPP analysis: in the absence of signi cant premixed ame extinction, fuel-air unmixedness has a net negative impact on the turbulent ame speed for variations in mixture strength around mean stoichiometric conditions, and a net positive impact for variations close to the ammability limits; in the presence of signi cant premixed ame extinction, fuel-air unmixedness has a net negative impact on s and results in increased levels of secondary burning. T 10 References Baritaud, T. A., Duclos, J. M. and Fusco, A. (1996). Proc. Combust. Inst. 26:2627{2635. Bray, K. N. C. (1990). Proc. R. Soc. Lond. A 431:315{335. Bray, K. N. C. and Libby, P. A. (1994). in Turbulent Reacting Flows (P. A. Libby and F. A. Williams, Eds.), Academic Press, pp. 115{151. Candel, S. 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