Combustion Science and Technology A Simple Model

This article was downloaded by: [Lipatnikov, Andrei] On: 21 February 2011 Access details: Access Details: [subscription number 933743744] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 3741 Mortimer Street, London W1T 3JH, UK

Combustion Science and Technology

Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713456315

A Simple Model for Evaluating Conditioned Velocities in Premixed Turbulent Flames

V. A. Sabel'nikova; A. N. Lipatnikovb a ONERA/DEFA, Centre de Palaiseau, Palaiseau, France b Department of Applied Mathematics, Chalmers University of Technology, Gothenburg, Sweden Online publication date: 19 February 2011

To cite this Article Sabel'nikov, V. A. and Lipatnikov, A. N.(2011) 'A Simple Model for Evaluating Conditioned Velocities

in Premixed Turbulent Flames', Combustion Science and Technology, 183: 6, 588 — 613 To link to this Article: DOI: 10.1080/00102202.2010.528713 URL: http://dx.doi.org/10.1080/00102202.2010.528713

PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

Combust. Sci. and Tech., 183: 588–613, 2011 Copyright # Taylor & Francis Group, LLC ISSN: 0010-2202 print=1563-521X online DOI: 10.1080/00102202.2010.528713

A SIMPLE MODEL FOR EVALUATING CONDITIONED VELOCITIES IN PREMIXED TURBULENT FLAMES V. A. Sabel’nikov1 and A. N. Lipatnikov2 1

ONERA=DEFA, Centre de Palaiseau, Palaiseau, France Department of Applied Mathematics, Chalmers University of Technology, Gothenburg, Sweden

Downloaded By: [Lipatnikov, Andrei] At: 08:21 21 February 2011

2

A simple model is proposed to evaluate (a) the divergence of velocity vector conditioned on unburned mixture, and (b) the vector component normal to the mean flame brush in the flamelet regime of premixed turbulent combustion. The model involves a single constant and does not invoke an extra balance equation. To perform the first test of the model, six flames stabilized in impinging jets and experimentally investigated by 4 research groups were numerically simulated. In the computations, (a) approximations of the measured axial profiles of the mean combustion progress variable were invoked, (b) the well-known (Bray et al., 1998, and 2000) statistically steady and 1-dimensional Favre-averaged continuity and Euler equations were numerically integrated in order to approximate the measured axial profiles of the mean axial velocity, and, then, (c) the approximations were utilized in order to evaluate conditioned velocities and turbulent scalar flux using the proposed model supplemented with the BML approach and balance equation for the Favre-averaged combustion progress variable. The obtained agreement between the measured and computed axial profiles of the conditioned axial velocities or axial turbulent scalar flux was encouraging, thus, indicating that the proposed simple model is promising. Since the correlation between fluctuations of velocity and unity normal vectors, conditioned to flamelet surface, plays a key role in the model, the encouraging test results call for studying this correlation in future DNS. Moreover, further research into the difference in velocity conditioned on unburned mixture and velocity conditioned on the unburned side of flamelets is necessary for improving the model at the leading edge of a turbulent flame brush. Keywords: Conditioned velocities; Countergradient transport; Modeling; Premixed turbulent combustion; Stagnation flames

INTRODUCTION Turbulent scalar transport in premixed flames has challenged the combustion community for four decades since the phenomenon of the so-called countergradient diffusion was documented numerically by Libby and Bray (1981) and experimentally by Moss (1980) and by Yanagi and Mimura (1981). The phenomenon consists of the fact that turbulent transport of a scalar quantity (e.g., the combustion progress variable c introduced by Bray and Moss, 1977) may occur in the direction of an increase Received 19 April 2010; revised 29 June 2010; accepted 28 September 2010. Address correspondence to Andrei Lipatnikov, Department of Applied Mathematics, Chalmers University of Technology, Gothenburg 41296, Sweden. E-mail: [email protected] 588

EVALUATING CONDITIONED VELOCITIES

589

in the mean value of this quantity (i.e., qu00 c00 rc > 0). Here, u is the flow velocity vector, q is the density, overbars and overlines designate the Reynolds averages with c0 c c, and ~c qc= q is the Favre average with c00 c ~c. Countergradient scalar transport in premixed turbulent flames is commonly associated with a higher magnitude j ub j of velocity conditioned on burned mixture than the magnitude j uu j of velocity conditioned on unburned mixture. Because the probability of finding intermediate (between unburned and burned) states of a reacting mixture is substantially lower than unity in many weakly and moderately turbulent premixed flames, as reviewed elsewhere (Driscoll, 2008; Lipatnikov and Chomiak, 2010), the well-known BML model (Bray and Moss, 1977; Bray et al., 1985; Libby and Bray, 1981) is widely recognized to be a reasonable approximation. Within the framework of the model, the probability of finding burned mixture is equal to the Reynolds-averaged combustion progress variable c and


qu00 c00 ¼ q~cð1 ~cÞðub uu Þ

ð1Þ

(i.e., the direction of the flux qu00 c00 is controlled by the direction of the slip velocity vector Du ub uu ). For a statistically planar one-dimensional flame that propagates from right to left (or from left to right), @c=@x > 0 (or < 0) and the conditioned velocities uu and ub are positive (negative) i.e., qu00 c00 rc > 0 if jub j > juu j. Although various approaches to modeling the flux qu00 c00 have already been developed as reviewed elsewhere (Bray, 1995; Lipatnikov and Chomiak, 2010), most of them deal with balance equations either for qu00 c00 or for uu and ub . The use of such balance equations substantially complicates numerical simulations (especially if combustion in an engine is addressed) and requires invoking a number of assumptions and tuning constants in order to close the equations. The goal of the present paper is to propose much simpler equations for evaluating (a) the divergence of the conditioned velocity vector uu and (b) the normal (to the mean flame brush) component of that vector. In the next section, the aforementioned simple equations are obtained. In the third section, the approach is validated by computing axial turbulent scalar fluxes qu00 c00 and conditioned velocities uu and ub in six premixed turbulent flames stabilized in impinging jets and experimentally investigated by Cho et al. (1988), Cheng and Shepherd (1991), Li et al. (1994), and Stevens et al. (1998). In the fourth section, the model is compared with alternative approaches and a way of using it in LES is outlined. MODEL The present paper is restricted to the flamelet regime of premixed turbulent combustion (i.e., unburned c ¼ 0 and burned c ¼ 1 mixtures are considered to be separated by an infinitely thin self-propagating interface (flamelet) wrinkled and advected by turbulent eddies). Accordingly, r c ¼ 0 everywhere with the exception of the flamelet surface, where rc is the Dirac d function. We use a common assumption (valid for premixed combustion at low Mach numbers) that the velocity fields in unburned and burned mixtures are incompressible. The densities qu and qb of unburned and burned mixtures, respectively, are

590

V. A. SABEL’NIKOV AND A. N. LIPATNIKOV

constant. The principal idea of the model consists of using the assumption of incompressibility for determining the divergence of the velocity vector uu conditioned on unburned mixture. Due to the incompressibility of unburned mixture, r uu ¼ 0

ð2Þ

ð1 cÞr uu ¼ 0

ð3Þ

r ½ð1 cÞuu ¼ uu rc ¼ uu njrcj

ð4Þ

and the following equation

holds everywhere. Therefore,


where n is the unit vector normal to the flamelet and pointing to the unburned gas, that is jrcjn ¼ rc:

ð5Þ

Averaging the left-hand side (LHS) of Eq. (4), we obtain r ½ð1 cÞuu ¼ r ð1 cÞuu ¼ r ð1 cÞu ¼ r ½ð1 cÞuu :

ð6Þ

Here, we take into account that (i) the equality of ð1 cÞuu ¼ ð1 cÞu is valid for discontinuous functions such as the velocity u, because (1 c) ¼ 0 in burned mixture, and (ii) the velocity uu conditioned on unburned mixture is defined as follows (Libby, 1985): uu ð1cÞu 1c :

ð7Þ

r ½ð1 cÞ uu ¼ uu njrcj

ð8Þ

ð1 cÞr uu ¼ uu njrcj þ uu rc:

ð9Þ

Eqs. (4) and (6) yield

or

Equation (9) may be rewritten as follows: h i ð1 cÞr uu ¼ ðuu nÞf uu ðnÞf R

ð10Þ

using local instantaneous flame surface density (FSD) R jrcj

ð11Þ

ð qÞf qR R

ð12Þ

and the surface average


591

of any quantity q. Note that c ð nÞf ¼ njrcj ¼ rc ¼ r R R R

ð13Þ

by virtue of Eqs. (5), (11), and (12). Equation (11) looks different from the definition of FSD used for flamelets of a finite thickness (Trouve´ and Poinsot, 1994; Vervisch et al., 1995; Veynante and Vervisch, 2002), because infinitely thin flamelets are addressed in the present paper. Finally, introducing ðqÞ0f q ðqÞf

ð14Þ

we obtain


ðuu nÞf ¼ ½ð uu Þf þ ðuu Þ0f ½ð nÞf þ ðnÞ0f ¼ ðuu Þf ðnÞf þ ½ðuu Þ0f ðnÞ0f f :

ð15Þ

Consequently, Eq. (10) reads h i ð1 cÞr uu ¼ ½ðuu Þ0f ðnÞ0f f R þ uu ðuu Þf rc

ð16Þ

using Eq. (13). Eq. (16) is the straightforward consequence of the incompressibility assumption and is the principal result of the above analysis. It is not closed and, on the face of it, Eq. (16) appears to be difficult to use due to the unclosed correlation ½ðuu Þ0f ðn0 Þf f and the flow velocity ð uu Þf conditioned on the unburned side of flamelets. However, direct numerical simulations (DNS) have shown that (a) ðuu Þf uu in the largest part of a statistically planar, one-dimensional premixed turbulent flame brush with the exception of the leading edge (see Figure 4 reported by Im et al., 2004) and (b) the aforementioned correlation ‘‘remains nearly constant, small, and positive’’ (Im et al., 2004). Based on these DNS data, we propose to use the following simple equation ð1 cÞr uu ¼ bu0 R

ð17Þ

for evaluating the divergence of the velocity vector conditioned on unburned mixture in premixed turbulent flames. Here, b is a constant and u0 is the rms turbulent velocity at the leading edge of a turbulent flame brush. It is worth noting that a similar assumption of ðub Þf ub does not hold in premixed turbulent flames (e.g., see Figure 4 reported by Im et al., 2004). The difference in ð ub Þf and ub is associated with the following physical mechanism. When a fluid element moves within a flame brush, the element is accelerated by the pressure gradient induced by heat release, with the acceleration being inversely proportional to the fluid density. To simplify the discussion and to neglect the acceleration of the unburned mixture as compared with burned gas, let us assume that qb e ðnk Þ < w >c ðnk Þ

2

ð33Þ


k¼1

~ ðnÞ in (a) flames h4 and h6 investiFigure 4 Axial profiles of the normalized Favre-averaged axial velocity w gated by Li et al. (1994) and (b) sets 2 and 3 investigated by Stevens et al. (1998). Symbols show experimental data. Curves were computed. Thin and thick lines were obtained using Eqs. (32) and (33), respectively.



599

between the measured e (nk) and the computed c (nk), either the Reynolds (Cheng and Shepherd, 1991; Cho et al., 1988) or the averaged ¼ w ~ (Li et al., 1994; Stevens et al., 1998) axial velocities, evalFavre-averaged ¼ w uated within the flame brush only, has the minimum value. Moreover, to assess the influence of Q and g1 on the test results, an alternative criterion given by Eq. (32) was also used. Note that Eqs. (24), (25), (28), (32), and (33) involve neither normalized con u or w b nor normalized turbulent scalar flux qw00 c00 . Therefore, ditioned velocities w ~ can be approxia measured profile of the Favre-averaged normalized axial velocity w mated using the above method independently of the model to be tested. The approxi~ are shown in lines in Figure 4. Good agreement between thick mated profiles of w lines computed using Eq. (33) and the experimental data (symbols) obtained by Li et al. (1994) and by Stevens et al. (1998) within a flame brush makes subsequent tests of Eq. (20) solid and target-directed. Agreement between the experimental data and thin lines obtained using Eq. (32) is worse, especially for the data by Stevens et al. (1998). Thus, Figure 4 indicates that it is better to use Eq. (33) in the present test. Furthermore, for the experiments by Li et al. (1994), simulations were also performed using Eq. (33) in order to evaluate Q and g1 with variations in g1 being limited to the range of 0.055 g1 0.045 (flame h4) and 0.045 g1 0.055 (h6) based on the values of g1 ¼ 0.05 and 0.05, respectively, reported by Bray et al. (2000). In the latter flame, the restriction did not play any role and the minimum scatter N was obtained using g1 ¼ 0.049 (see Table 2) regardless of whether or not variations in g1 were restricted. The solid line in Figure 5 shows that the use of g1 ¼ 0.055 and Q ¼ 1.403 allows us to better approximate the mean axial velocity measured in the fresh mixture, in line with the results by Bray et al. (1998). However, within the flame brush, the agreement with the experimental data is slightly reduced. In the following, normalized scalar fluxes qw00 c00 computed using all three pairs of Q and g1 will be reported for the flame h4.

~ ðnÞ in flame h4 investigated by Li Figure 5 Axial profiles of the normalized Favre-averaged axial velocity w et al. (1994). Symbols show experimental data. Curves were computed using Q and g1 specified in legends.

600

V. A. SABEL’NIKOV AND A. N. LIPATNIKOV

As concerns the axial profiles of the Reynolds-averaged velocity u, reported by Cho et al. (1988) and by Cheng and Shepherd (1991), Eqs. (24–33) are not sufficient to approximate these profiles, because the flux qu00 c00 should be known in order to evaluate u using Eq. (23). For this reason, the approximated profiles of uðxÞ will be reported when testing Eq. (20). Finally, it is worth emphasizing that the test method discussed requires knowledge of the axial profiles of the mean combustion progress variable and mean axial velocity. Accordingly, the very interesting experiments by Kalt et al. (2002) are not utilized to assess Eq. (20) due to the lack of measured profiles of < c > (x) and < u > (x) in the cited paper. u of the Evaluation of conditioned velocities. The normal component w u was evaluated using Eq. (20), which reads normalized conditioned velocity vector w


u w

dc d ¼ ðSL þ bu0 ÞR dn U

ð34Þ

for the simulated flames. The same value, b ¼ 1.1 of a single model constant was set for all of the six simulated flames. The values of u0 invoked to close Eq. (34) are reported in Table 1. These values have been taken from the caption to Figure 2 in the paper by Cho et al. (1988), Figure 4b in the paper by Chen and Shepherd (1991), Figure 3 reported by Li et al. (1994) for flame h4, Figure 7 in the paper by Stevens et al. (1998) with u0 ¼ Uðqw002 = qÞ1=2 at n ¼ 0.4 and 0.3 for sets 2 and 3, respectively. For flame h6 investigated by Li et al. (1994), u0 was estimated using the data reported by Bray et al. ðn ¼ 1Þ (2000). According to the latter paper, (a) the turbulent kinetic energy k1 k was the same in flames h4 and h6 (p. 13); (b) Gww ¼ 1.7 and 2.4 at the leading edge of flames h4 and h6, respectively (cf. Figures 3b and 4c in the cited paper); where (c) Gww qw002 =ð qk1 Þ, see Eq. (5) therein. Therefore, the rms velocity u0 ¼ (k1Gww)1=2 at the leading edge of flame h6 is larger by a factor of 1.2 than at the leading edge of flame h4. b conditioned on b of the normalized velocity vector w The normal component w burned mixture was evaluated using either Eq. (23) for the flames investigated by Cho et al. (1988) and by Cheng and Shepherd (1991) or Eq. (22) for the flames investigated by Li et al. (1994) and by Stevens et al. (1998). Turbulent scalar flux was calculated using Eq. (1) for all the flames. The mean FSD invoked by Eq. (34) was evaluated using Eq. (26). Since the latter equation involves the flux qw00 c00 , iterations were required in order to numerically solve Eqs. (1), (22) or (23), (26), and (34). For the flames, investigated by Li et al. (1994) and by Stevens et al. (1998), first, Q and g1 were determined by solving Eqs. (24–25) with boundary conditions given u was calculated for the ~cðnÞ by Eq. (28) and, second, the conditioned velocity w ~ ðnÞ profiles shown in lines in Figures 3 and 4, respectively. Since Cho et al. and w (1988) and Cheng and Shepherd (1991) reported the profiles uðxÞ of the Reynoldsaveraged axial velocity, conditioned velocities are required in order to compare ~ ðnÞ obtained by numerically integrating the measured data with the profiles of w Eqs. (24–25). Accordingly, for flames 1 and 2 investigated by Cho et al. (1988) and by Cheng and Shepherd (1991), respectively, the problem could not be divided



601

Figure 6 Reynolds-averaged and conditioned axial velocities (a) in case 1 studied by Cho et al. (1988) and (b) in flame s9 investigated by Cheng and Shepherd (1991). Symbols show experimental data. Curves were computed. Thin and thick lines were obtained using Eqs. (32–33), respectively.

into two independent tasks, the determination of Q and g1 and the subsequent evaluation of uu , that is, Eqs. (24–25) were solved jointly with Eq. (34) for these two flames. In order to test the numerical method, simulations were also performed by replacing Eq. (34) with Eq. (17) and all other things being equal. In the latter case, (a) the radial slip velocity was neglected, that is, the normalized r uu was estimated u =dn þ 2~ to be equal to d w g, and (b) the following boundary condition uu ðc ¼ c0 Þ ¼ uðc ¼ c0 Þ was invoked, with weak sensitivity of computed results to variations in c0