# Burning Rate in Impinging Jet Flames

Authors: A. N. Lipatnikov
Journal: Journal of Combustion
 (2011)

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## Abstract

A method for evaluating burning velocity in premixed turbulent flames stabilized in divergent
mean flows is quantitatively validated using numerical approximations of measured
axial profiles of the mean combustion progress variable, mean and conditioned axial velocities,
and axial turbulent scalar flux, obtained by four research groups from seven different
flames each stabilized in an impinging jet. The method is further substantiated by
analyzing the combustion progress variable balance equation that is yielded by the extended
Zimont model of premixed turbulent combustion. The consistency of the model
with the aforementioned experimental data is also demonstrated.


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## Body

## 1. Introduction

Although turbulent flame speedSt, by analogy with laminar burning, is often considered to be a basic characteristic of premixed combustion and was the main subject of numerous studies reviewed elsewhere [1–4], a precise definition of this quantity is difficult due to the following two key differences between laminar and turbulent premixed flames.First, while a typical laminar flame is thin as compared with both the flame curvature and scales of spatial nonuniformities of the incoming flow of fresh reactants, the mean thicknessΔt of a turbulent flame is comparable with length scales that characterize mean flow nonuniformities or the flame curvature. Due to the significant thickness of a turbulent flame brush, both the mean convective flux of the deficient reactant toward the flame and the mean flow velocity vary substantially across the flame brush, thus making a choice of a reference flux or a reference flow velocity (with respect to that flame speed should be defined) difficult [5, 6]. For instance, Figure 2 in a paper by Gouldin [6] indicates more than threefold decrease in the mean normal convective flux from the leading to the trailing edges of a V-shaped flame brush. Other experimental data that show the same effect were recently reviewed by Lipatnikov and Chomiak [7] (see Section 5.2in the cited paper). It is also worth remembering that the difference in the speeds of planar and curved laminar flames scales as a ratio δL/Rf of the flame thickness to the radius of its curvature [8], and this ratio is much less than unity in a typical laminar premixed flame. If a similar scaling is assumed to hold for a premixed turbulent flame, then the difference between the speeds of statistically planar and curved flames could be large [9], because a ratio of Δt/Rf is of unity order in a typical curved premixed turbulent flame.Second, while a typical laminar flame has a fully developed structure and speed, because a time scale of laminar flame development is very short in comparison with time scales characterizing the flow, a typical turbulent flame is a developing wave, with the growth ofΔt being the most striking manifestation of premixed turbulent flame development, as stressed elsewhere [3]. Due to the growth of a turbulent flame brush thickness, the observed speeds of different isotemperature (or isoconcentration) surfaces differ from one another.In the statistically planar one-dimensional case, the latter problem may be resolved by considering turbulent burning velocityUt≡|∫-∞∞¯W̅dx/{ρu[Y(∞)-Y(-∞)]}|, that is, the normalized total mass rate of consumption (or production) of the deficient reactant (or a main product) Y per unit mean flame area. However, in the case of a curved flame brush, the definition of the area is ambiguous due to a large thickness Δt. For instance, a ratio of the areas of surfaces associated with the leading and trailing edges of a statistically spherical turbulent flame brush may be as large as (Rf+Δt)2/Rf2≈4 if the mean flame radius and thickness are of the same order.Several methods have been proposed to resolve the above issues and to determine turbulent flame speedSt and/or burning velocity Ut in statistically stationary [5, 6] or expanding [10–12] flames. In particular, in [10], an expanding statistically spherical premixed turbulent flame with a self-similar mean structure (i.e., the mean density ρ¯(ξ) depends on a single normalized distance ξ=[r-Rf(t)]/Δt(t), rather than on two independent variables, time t and radial distance r) was theoretically studied and the following two radii:(1)Rf,s=(3∫0∞c¯r2dr)(2∫0∞c¯rdr)-1,(2)Rf,u2=2∫0∞c¯dr
were introduced. Here, c is the well-known combustion progress variable [13] and overbar designates Reynolds average. It was analytically shown that (i) if turbulent burning velocity is defined as follows Ut≡∫0∞W¯r2dr/(ρuRf,u2), then it is not affected straightforwardly by the rate of the growth of the mean flame brush thickness, (ii) if turbulent flame speed is defined as follows St=(ρb/ρu)dRf,s/dt, then St=Ut, and (iii) in order for the difference in the observed flame speed dRf,s/dt and a mean velocity 𝒰u of the unburned mixture to be equal to St or Ut, the velocity 𝒰u should be evaluated by extrapolating the mean flow velocity distribution u¯(r) in the unburned mixture ahead of the flame brush to the burning velocity surface characterized by r=Rf,u. Here, W¯ is the mean mass rate of product creation, that is, the source term in the well-known balance equation [13] for the Favre-averaged combustion progress variable.It is of interest to note that, as shown elsewhere [12], the above purely theoretical results are very close to the definition of the burning velocity of expanding statistically spherical premixed turbulent flames, proposed by Bradley et al. [11] based on different reasoning.Although the theoretical analysis in [10] was solely performed for expanding flames, either statistically planar or statistically spherical ones, it was also hypothesized that the proposed methods for determining Ut, St, and 𝒰u may be useful in other cases, for example, in statistically stationary premixed turbulent flames stabilized in divergent flows such as impinging jets. In particular, turbulent burning velocity in such a flame was hypothesized to be approximately equal to a velocity obtained by extrapolating the velocity distribution in the unburned gas ahead of the flame to a burning velocity surface characterized by the following reference value of the Favre-averaged combustion progress variable:(3)c0=c̃(ξ=ρuρb∫01Ψ2dc̃),
where(4)ρuΨ2(ξ)≡ρ¯ξ-∫-∞ξdρ¯dζζdζ,
and ξ=(x-xf)/Δt is normalized distance in the direction normal to the mean flame brush.In a subsequent paper [14], this hypothesis was indirectly supported by simulating a premixed turbulent flame stabilized in an impinging jet and by invoking two different RANS models of premixed turbulent combustion. However, in the cited paper, only numerical results were reported, and the discussed hypothesis has never been tested against experimental data, to the best of the author’s knowledge.The present work is aimed at filling this gap and testing the hypothesis against experimental data obtained by four research groups [15–18] from seven premixed turbulent flames each stabilized in an impinging jet.In the next section, the hypothesis is tested. Obtained results are discussed in the third section followed by conclusions.


## 2. Test

Because the studied flames (see Figure1 and Table 1) look statistically planar in the vicinity of the axis of an impinging jet, turbulent burning velocity may be evaluated by integrating the mean mass rate of product creation along the axis, that is,(5)ut=∫01Ωdz,
where the burning velocity ut=Ut/U, burning rate Ω=dW¯/ρuU, and axial distance z=x/d (z=0 at the wall) are normalized using the distance d between the burner exit and the wall, the mean flow velocity U at the burner exit, and the density ρu of the unburned gas.Table 1
Experimental conditions.
No.daUbFuelΦcSLσdu′eFlame(m)(m/s)(m/s)(m/s)10.0755CH41.00.365f7.5130.30Cho et al. [15], case 120.15gC2H61.00.76h8.0040.4Cheng and Shepherd [16], s930.033.6CH40.890.3077.0770.2Li et al. [17], h4i40.033.6CH40.890.3077.0770.2Li et al. [17], h650.0350.75CH40.60.0855.5440.06Stevens et al. [18], set 160.0353CH41.00.3657.5130.33Stevens et al. [18], set 270.0352.25CH41.30.2137.1120.18Stevens et al. [18], set 3ad is the distance between the jet exit and the wall (see Figure 1).bU is the mean axial flow velocity in the jet exit.cΦ is the equivalence ratio.dσ is the density ratio.eu′ is the rms turbulent velocity.fFor CH4-air mixtures, the values of the laminar flame speeds SL, reported in Table 1, were estimated using recent experimental data [26].gThis value was reported by Bray et al. [19] (see footnote on page 645 in the cited paper).hThis value was reported by Cheng and Shepherd [16].iThe main difference between the two flames studied by Li et al. [17] was the diameters (h= 4 and 6 mm) of holes in grids used to generate turbulence.Figure 1
Premixed turbulent flame stabilized in an impinging jet.Axial profiles of (a) the Reynolds-averaged combustion progress variablec¯(x), measured by Cho et al. [15] and by Cheng and Shepherd [16], and the Favre-averaged combustion progress variable c̃(z), measured (b) by Li et al. [17] and (c) by Stevens et al. [18]. Symbols show experimental data. Curves have been obtained using (9).
(a)(b)(c)In the present work, the burning rateΩ is evaluated using the following well-known balance equation [19–24]:(6)ddz(ρ̅w̃c̃)+2ρ̅g̃c̃+ddz[ρ̅c̃(1-c̃)(w¯b-w¯u)]+2ρ̅c̃(1-c̃)(g̅b-g̅u)=Ω,
which holds in the vicinity of the jet axis. Note that the radial turbulent scalar flux, that is, the last term on the left hand side (LHS) of (6), was taken into account in [23, 24], but was neglected in the other cited papers [19–22]. Here, we set ρu=1 for simplicity, that is, symbol ρ designates the density normalized using the density of unburned gas, w̃=ũ/U is the normalized Favre-averaged axial velocity, which depend only on z on the axis (∂c̅/∂r=∂ρ¯/∂r=∂w̃/∂r=0 at r=0 for symmetry reasons), and g=(d/U)(∂υ/∂r)r=0 is the normalized radial gradient of the radial velocity, calculated at the axis, with g̃ depending solely on x. The following well-known BML relation [25]:(7)ρu′′c′′¯=ρ¯c̃(1-c̃)(u¯b-u¯u)
was invoked to close the turbulent scalar flux ρu′′c′′¯ in the balance equation for c̃, with u¯u and u¯b designating velocity vectors conditioned to unburned and burned mixture, respectively.In order to evaluate the normalized burning rate by solving (6), the axial distributions of  c̃(z),ρ¯(z),w̃(z),w̅u(z),w̅b(z),g̅u(z),andg̅b(z) should be known. In the present work, the following well-known BML state equation [13]:(8)ρbc̅=ρ¯c̃=c̃1+(σ-1)c̃
and the experimental data reported by Cho et al. [15], Cheng and Shepherd [16], Li et al. [17], and Stevens et al. [18] were used for these purposes. Here, σ=ρb-1 is the density ratio. However, because the straightforward use of the experimental data could result in significant numerical errors when differentiating the data measured in a few points separated by substantial distances, smooth numerical approximations of these experimental data were obtained using the following method [24].First, following the widely accepted practice [19–23], the axial profiles of either the Reynolds-averaged combustion progress variable, reported by Cho et al. [15] and by Cheng and Shepherd [16], or the Favre-averaged one, measured by Li et al. [17] and by Stevens et al. [18], were approximated (see Figure 2) using (i) a complementary error function(9)c̅=12erfc⁡(πξ)=1π∫ξπ∞eζ2dζ
to calculate c̅(z) and (ii) equation (8) to compute c̃[c¯(z)]. Here and in the following, ξ=(z-zf)/δt, with the values of the normalized flame coordinate zf and thickness δt=Δt/d being reported in Table 2. As shown elsewhere [3, 27], this approximation works well in various premixed turbulent flames. The readers interested in a theoretical justification of (9) are referred to [28, 29].Table 2
Parameters of computations.
FlamezfδtQg110.2170.0801.2980.07420.5460.1410.6480.83230.1540.0621.733−0.18340.2320.1131.511−0.01650.3620.1120.8990.39860.2950.1571.2720.16970.2590.1161.2150.155Second, the profiles of the mean axial velocity and either axial conditioned velocities or axial turbulent scalar flux were computed by solving the following well-known [19–23] continuity:(10)dρ¯w̃dz+2ρ¯g̃=0,
and radial impulse(11)ddz(ρ¯w̃g̃)+3ρ¯g̃2+ddz[ρ¯c̃(1-c̃)(w¯b-w¯u)(g¯b-g¯u)]+3ρ¯c̃(1-c̃)(g¯b-g¯u)2=Q,
equations supplemented with (i) the following boundary conditions:(12)w̃(1)=-1,g̃(1)=g1, (ii) the well-known BML relation [13](13)u¯=ũ+u′′¯=ũ+(σ-1)ρu′′c′′¯,
and (iii) balance equations for g¯u and the difference in w¯b and w¯u, derived by the present author [30] and discussed in detail elsewhere [7, 31]. Here, Q=-d2/U2(∂2p¯/∂r2)r=0 and g1 are tuning parameters reported in Table 2.The readers interested in discussion and validation of this numerical model are referred to a recent paper [24]. For the goal of the present study, the validity of this model is a side issue, but the most important point consists of the fact that curves plotted in Figures 3, 4, and 5 well approximate experimental data shown in symbols. Therefore, these curves could be considered to be smooth approximations of the data, while further details of the approximation techniques are of minor importance as far as the evaluation of Ω by numerically integrating (6) is concerned.Axial profiles of the Reynolds-averaged and conditioned axial velocities in the flames investigated by Cho et al. [15] (a) and by Cheng and Shepherd [16] (b). Symbols show experimental data. Curves have been computed.
(a)(b)Axial profiles of the normalized Favre-averaged axial velocitiesw̃(z) in flames investigated by Li et al. [17] (a) and by Stevens et al. [18] (b). Symbols show experimental data. Curves have been computed.
(a)(b)Axial profiles of the normalized axial turbulent scalar fluxesρw′′c′′¯(z) in flames investigated by Li et al. [17] (a) and by Stevens et al. [18] (b). Symbols show experimental data. Curves have been computed.
(a)(b)Thus, in the present work, the reference method of evaluating turbulent burning velocity consists of numerically integrating (5) and (6) by invoking the smooth approximations of experimental data shown in lines in Figures 2, 3, 4, and 5. The values of ut obtained using this method are reported in the first row of Table 3.Table 3
Turbulent burning velocities.
Flame no.1234567ut0.0940.2550.0920.1310.1890.1440.116-w¯e(zf)0.0920.2540.0930.1340.1910.1440.116(g12-Q)[zf-z(c¯=0.1)]20.002−0.00020.0020.0050.0020.0080.004-∫012ρ¯c̃(1-c̃)(g¯b-g¯u)dz0.0060.0040.0040.0090.0040.0140.009[2ρ¯c̃(1-c̃)(g¯b-g¯u)/ρ¯w̃(dc̃/dz)]c¯=0.10.030.010.020.030.0090.030.025These reference values are used in order to test the method put forward in [10]. The latter method consists of the following.First, the normalized velocity profilew¯(z) obtained in the unburned mixture is extrapolated by invoking the following analytical solution:(14)w¯e=2g1(1-z)-(g12-Q)(1-z)2-1,g¯e=g1-(g12-Q)(1-z),
to (10) and (11) written in the constant density case (ρ=1) and supplemented with boundary conditions given by (12). In the present work, the parameters Q and g1 were taken from Table 2. If the method is applied to process raw experimental data, then these parameters may be tuned by comparing the profile of w¯(z) obtained from the oncoming flow of unburned mixture (e.g., see thin solid line in Figure 6) with profiles calculated using (14) with various Q and g1 (e.g., see dashed line in Figure 6).Figure 6
Tested method for determining turbulent burning velocity. Bold and thin solid lines show typical axial profiles of the Favre-averaged combustion progress variable and normalized axial velocity, respectively. Dashed curve calculated using (14) extrapolates the mean velocity profile from the oncoming flow to the flame brush. Filled circle shows turbulent burning velocity determined using the tested method. Filled triangle shows turbulent displacement speed associated commonly with the local minimum of |w̃(z)|.Second, the reference valuec0 associated with the burning velocity surface is calculated using (3), (4), and (9). It can easily be shown that the reference surface is located at z=zf if (8) holds. Indeed, using (8) and (9), we have(15)-σσ-1dρ¯dξ=σρ¯2dc̃dξ=dc¯dξ=-e-πξ2.
Subsequently, (4) and (15) read(16)Ψ2=ρ¯ξ-σ-1σ∫-∞ξζe-πζ2dζ=ρ¯ξ+σ-12πσe-πξ2=ρ¯ξ-σ-12πσdc¯dξ.
Therefore,(17)∫01Ψ2dc̃=∫∞-∞(ρ¯ξ-σ-12πσdc¯dξ)dc̃dξdξ=∫∞-∞(ξσρ¯dc¯dξ+12πσρ¯2dρ¯dξdc¯dξ)dξ=12πσ∫∞-∞(-1ρ¯d2c¯dξ2+1ρ¯2dρ¯dξdc¯dξ)dξ=12πσ∫-∞∞ddξ(1ρ¯dc¯dξ)dξ=0
and z=zf. In the present work, the normalized distance z=zf was taken from Table 2.If the method is applied to process raw experimental data, then, the coordinatez0 of the burning velocity surface may be calculated from the following equality:(18)c¯(z0)=0.5,
using the measured profile of the Reynolds-averaged combustion progress variable. If the Favre-averaged combustion progress variable is measured, then the proper reference value may be calculated substituting c¯=0.5 into (8). Under typical conditions, the reference value of c̃≈0.1 could be used (e.g., see bold solid line in Figure 6).Third, turbulent burning velocity is considered to be equal tow¯e(zf) calculated from (14); see filled circle in Figure 6.Results of the test, shown in Table3, indicate that the values of w¯e(zf) (the second row) are very close to the normalized turbulent burning velocity ut calculated using (5) in all the studied flames, thus validating the considered method.


## 3. Discussion

From the physical standpoint, the above results could be interpreted as follows. Although the turbulent burning velocityUt is controlled by processes within (or, maybe, at the leading edge of) turbulent flame brush, Ut can be evaluated by properly analyzing the velocity field in front of the flame brush, because the changes in this field, induced by the flame, are controlled by the burning velocity and the density ratio. For instance, in the simplest hypothetical case of an expanding spherical infinitely thin laminar flame, the velocity in unburned mixture is equal to (σ-1)SL(r/Rf)2, and the burning velocity can easily be evaluated by analyzing the radial velocity profile measured ahead of the flame.The approximate equality ofut and |w¯e(zf)| may further be supported by invoking the so-called Zimont [32–34] model of premixed turbulent combustion, extended, and validated by the present author [35–37]. As reviewed elsewhere [3, 27], the model is successfully used by a number of different groups to simulate various turbulent flames.The model deals with the following balance equation:(19)∂∂t(ρ¯c̃)+∂∂xj(ρ¯ũjc̃)=∂∂xj(ρ¯Dt∂c̃∂xj)+Ut|∇c̃|,
for the Favre-averaged combustion progress variable, where Dt is turbulent diffusivity. Note that a basically similar balance equation was first introduced into the combustion literature by Prudnikov [38]. Along the axis of a statistically stationary flame stabilized in an impinging jet, this equation reads(20)ρ¯w̃dc̃dz=dtddz(ρ¯dc̃dz)-utdc̃dz,
where dt=Dt/(Ud) is the normalized turbulent diffusivity. Substitution of (8) and (9) into (20) yields(21)ρ¯w̃dc̃dz=dtδt(-2πξρ¯-σ-1σe-πξ2)dc̃dz-utdc̃dz.
Therefore, within the flame,(22)ρ¯w̃=dtδt(-2πξρ¯-σ-1σe-πξ2)-ut.
Differentiating this equation and using (8), (10), and (15), we obtain(23)2ρ¯g̃=-1δtdρ¯w̃dξ=2πρ¯dtδt2.Let us consider the leading edge of a turbulent flame brush and match (22) and (23), which are valid within the flame, with (14), which is valid in the unburned mixture. Because ρ¯→1,  c¯→0, and ξ≫1 at the leading edge z→z*, we obtain(24)-2πdtz*-zfδt2-ut=2g1(1-z*)-(g12-Q)(1-z*)2-1,πdtδt2=g1-(g12-Q)(1-z*).
Simple algebraic manipulations with these two equations and (14) result in(25)ut=-we(zf)+(g12-Q)(zf-z*)2.
According to the present simulations (see the third row in Table 3 and curves 1 in Figure 7), the last term in (25) is small at the leading edges of all the seven flames addressed in the present paper. Therefore, the model predicts that ut≈-we(zf) in a premixed turbulent flame stabilized in a circular impinging jet. Note that a similar analysis of a flame stabilized in a planar impinging jet yields exactly ut=-wu(zf).Figure 7
Term(g12-Q)(zf-z)2 in (25) (1) and mean axial velocities calculated using (2) equation (14) and (3) equation (22) versus the Reynolds-averaged combustion progress variable. Results shown in bold and thin lines have been obtained for flames h4 and h6, respectively, investigated by Li et al. [17].Certainly, the above substantiation of the studied method (i.e.,ut=-w¯e(zf)) by invoking the extended Zimont model may be put into question by referring to a paper by Bray et al. [21] who claimed that the model failed in predicting the burning rate within flames stabilized in impinging jets, particularly, in flames nos. 2–5 studied in the present paper. More specifically, Bray et al. [21] have found that the following simple expression:(26)W¯=Ut|∇c̃|
yields too strong dependence of W¯ on c̃ if the turbulent burning velocity Ut is evaluated using local Favre-averaged turbulence characteristics (turbulent kinetic energy k̃ and its dissipation rate ɛ̃), which depend substantially on c̃.As pointed out elsewhere [39], the model tested by Bray et al. [21] differed substantially from the model developed and validated by Zimont and Lipatnikov [35–37]. In particular, the latter model provides a joint closure of the sum of the transport and reaction terms in the following well-known [13] balance equation:(27)∂∂t(ρ¯c̃)+∂∂xk(ρ¯ũkc̃)=-∂∂xk(ρuk′′c′′¯)+W¯,
that is,(28)-∂∂xjρu′′jc′′¯+W¯=∂∂xj(ρ¯Dt∂c̃∂xj)+Ut|∇c̃|,
within the framework of the extended Zimont model. If applied to a flame stabilized in an impinging jet, (28) reads(29)-∂∂z[ρ¯c̃(1-c̃)(w¯b-w¯u)]︸1-2ρ¯c̃(1-c̃)(g¯b-g¯u)︸2+Ω︸3=dt∂∂z(ρ¯∂c̃∂z)︸4+ut|∇c̃|︸5.Different terms in (29) are shown in Figure 8 for the flame h4 investigated by Li et al. [17], and similar results were obtained for the other six flames. In these calculations, the burning velocity used by the extended Zimont model was evaluated by integrating (29), that is,(30)ut=∫01Ωdz-∫012ρ¯c̃(1-c̃)(g¯b-g¯u)dz,
and the normalized turbulent diffusivity was set in order to satisfy (29) at a point characterized by the maximum magnitude of the diffusion term 4. Thus, neither ut nor dt depend on z. As discussed earlier [39], the use of a constant turbulent burning velocity within such flames appears to be more consistent with the underlying physics than substitution of k̃(c̃) into an expression for Ut=Ut(SL2/k̃,…), because the Favre-averaged k̃ strongly overestimates the turbulent kinetic energy due to the substantial contribution of the unburned-burned intermittency into k̃, for example, see Figure 1 in [39].Figure 8
Different terms in (29), normalized using d and U, versus the Favre-averaged combustion progress variable. Curves have been computed in the flame h4 investigated by Li et al. [17]. Term numbers are specified in legends. Curve 6 shows the difference between the LHS and RHS of (29).Figure8 shows that although the source terms on the LHS and RHS of (29) depend differently on the mean combustion progress variable (compare with curves 3 and 5), (29) is sufficiently accurate in all the studied flames (see curve 6 which shows the difference between the LHS and RHS).Moreover, (23), which results from the extended Zimont model, yields a constant radial gradient g̃ of the Favre-averaged radial velocity across a premixed turbulent flame stabilized in an impinging jet provided that ut= const and dt= const. Numerical results plotted in Figure 9 show that g̃ is approximately constant in the range c¯<0.8 of turbulent flame brush, thus further validating the model in the largest part of turbulent flame brush.Figure 9
Normalized radial gradient of the Favre-averaged radial velocity versus the Reynolds-averaged combustion progress variable. Numbers in legends correspond to flame numbers in Table1.Furthermore, curves 2 and 3 in Figure7 show velocities calculated using (14) and (22), respectively, at the leading edges of the two flames investigated by Li et al. [17]. The latter curves have been calculated by neglecting term proportional to e-πξ2, as it vanishes at the leading edge. The fact that curves 2 and 3 are very close to one another further supports the extended Zimont model. Similar results were computed for other studied flames.Thus, the extended Zimont model is consistent with the experimental data obtained from premixed turbulent flames stabilized in impinging jets and, therefore, may be used to substantiate the tested method for evaluating turbulent burning velocity in such flames, as done above (Although the model was also put into question by Peters [40], his arguments were rebutted in earlier papers [28, 29] .).Finally, it is worth raising the following issue. To the the best of the present author's knowledge, the key equation of the extended Zimont model, that is, (19), has yet been substantiated by basic arguments [3, 28, 29, 32, 34, 38] only in the case of a statistically planar one-dimensional premixed turbulent flame. Therefore, strictly speaking, the extended Zimont model yields the following joint closure:(31)-∂∂ξρun′′c′′¯+W¯=∂∂ξ(ρ¯Dt∂c̃∂ξ)+Ut|∇c̃|,
where the ξ-axis is locally normal to the mean flame brush and un is the flow velocity in this direction. Accordingly, the model may be considered to avoid closing the turbulent scalar flux in the tangential direction, and the following balance equation:(32)-∂∂z[ρ¯c̃(1-c̃)(w¯b-w¯u)]︸I+Ω︸III=dt∂∂z(ρ¯∂c̃∂z)︸IV+ut|∇c̃|︸V
may be applied to a flame stabilized in an impinging jet by referring to the same extended Zimont model. While (29) invokes this model and a gradient closure of the transverse turbulent scalar flux (the closure terms vanishes along the jet axis for symmetry reasons), (32) invokes the same model and the BML (7) in order to close the transverse flux (the corresponding terms are not shown on the LHS and RHS of (32), because they cancel one another).Terms I and III on the LHS of (32) are equal to terms 1 and 3, respectively, on the LHS of (29), but the former equation does not involve a counterpart to term 2, which is much smaller than terms 1 and 5 (see Figure 8). Our simulations show that terms IV and V on the RHS of (32) are close to terms 4 and 5, respectively, on the RHS of (29), with the difference between the two sets of terms resulting from different dt and ut substituted into the two equations. Indeed, integration of (32) leads us to (5), with the difference between it and (30) consisting of the second term on the RHS of the latter equation. Comparison of the first and fourth rows in Table 3 shows that the integrated transverse scalar flux is much less than the integrated heat release rate in all the seven flames addressed in the present paper, with the largest difference (10%) being computed in flame no. 6 (set 2 investigated by Stevens et al. [18]). Thus, Figure 8 qualitatively describes the behavior of both terms 1–5 in (29) and terms I–V in (32) in all the flames studied here, but term II vanishes in the latter equation.If (29) is substituted with (32), then (21) should also be substituted with the following equation:(33)ρ¯w̃dc̃dz+2ρ¯c̃(1-c̃)(g¯b-g¯u)=dtδt(-2πξρ¯-σ-1σe-πξ2)dc̃dz-utdc̃dz,
that is, the second term is introduced into the LHS. Accordingly, (22)–(25), which result from (21), do not result from (33). However, because the introduced term is much less than the mean convection term at the leading edges of all the seven flames simulated here (see the fifth row in Table 3), (25) appears to be sufficiently accurate even if (32) and (33) are invoked. At the same time, the variations in g̃ within the studied flames (see Figure 9) may in part be caused by the transverse flux introduced into (33), as well as by the eventual dependencies of ut and dt on z due to variations in turbulence characteristics within the flames.


## 4. Conclusions

A method for evaluating burning velocity in premixed turbulent flames stabilized in divergent mean flows, proposed earlier by Lipatnikov and Chomiak [10], was quantitatively validated using numerical approximations of measured axial profiles of the mean combustion progress variable, mean and conditioned axial velocities, and axial turbulent scalar flux, obtained by Cho et al. [15], Cheng and Shepherd [16], Li et al. [17], and Stevens et al. [18] from seven different flames each stabilized in an impinging jet.The method is further substantiated by analyzing the combustion progress variable balance equation that is yielded by the extended Zimont model of premixed turbulent combustion. The compatibility of the model with the aforementioned experimental data is also shown.



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