Burning Rate in Impinging Jet Flames

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.

1. Introduction

Although turbulent flame speed 𝑆_𝑡, 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 thickness Δ_𝑡 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 𝛿_𝐿/𝑅_𝑓 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 Δ_𝑡/𝑅_𝑓 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 of Δ_𝑡 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 velocity 𝑈_𝑡≡|∫^∞_−∞̄‌𝑊𝑑𝑥/{𝜌_𝑢[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 Δ_𝑡. 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 (𝑅_𝑓+Δ_𝑡)²/𝑅²_𝑓≈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 speed 𝑆_𝑡 and/or burning velocity 𝑈_𝑡 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−𝑅_𝑓(𝑡)]/Δ_𝑡(𝑡), rather than on two independent variables, time 𝑡 and radial distance 𝑟) was theoretically studied and the following two radii:𝑅_𝑓,𝑠=3^∞₀𝑐𝑟²𝑑𝑟2^∞₀𝑐𝑟𝑑𝑟⁻¹,(1)𝑅²_𝑓,𝑢=2^∞₀𝑐𝑑𝑟(2) 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 𝑈_𝑡≡∫^∞₀𝑊𝑟²𝑑𝑟/(𝜌_𝑢𝑅²_𝑓,𝑢), 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 𝑆_𝑡=(𝜌_𝑏/𝜌_𝑢)𝑑𝑅_𝑓,𝑠/𝑑𝑡, then 𝑆_𝑡=𝑈_𝑡, and (iii) in order for the difference in the observed flame speed 𝑑𝑅_𝑓,𝑠/𝑑𝑡 and a mean velocity 𝒰_𝑢 of the unburned mixture to be equal to 𝑆_𝑡 or 𝑈_𝑡, the velocity 𝒰_𝑢 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=𝑅_𝑓,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 𝑈_𝑡, 𝑆_𝑡, and 𝒰_𝑢 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:𝑐₀=𝑐̃𝜉=𝜌_𝑢𝜌_𝑏¹₀Ψ₂𝑑𝑐̃,(3) where𝜌_𝑢Ψ₂(𝜉)≡𝜌𝜉−^𝜉_−∞𝑑𝜌𝑑𝜁𝜁𝑑𝜁,(4) and 𝜉=(𝑥−𝑥_𝑓)/Δ_𝑡 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 Figure 1 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,𝑢_𝑡=¹₀Ω𝑑𝑧,(5) where the burning velocity 𝑢_𝑡=𝑈_𝑡/𝑈, burning rate Ω=𝑑𝑊/𝜌_𝑢𝑈, 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 𝜌_𝑢 of the unburned gas.

Table 1 

Experimental conditions.

Figure 1 

Premixed turbulent flame stabilized in an impinging jet.

(a)

(b)

(c)

Figure 2 

Axial profiles of (a) the Reynolds-averaged combustion progress variable 𝑐(𝑥), measured by Cho et al. [15] and by Cheng and Shepherd [16], and the Favre-averaged combustion progress variable 𝑐(̃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).

In the present work, the burning rate Ω is evaluated using the following well-known balance equation [19–24]:𝑑𝑑𝑧𝜌𝑤𝑐̃+2𝜌𝑔𝑐̃̃+𝑑𝑑𝑧𝜌𝑐̃(1−𝑐)̃𝑤_𝑏−𝑤_𝑢+2𝜌𝑐̃(1−𝑐)̃𝑔_𝑏−𝑔_𝑢=Ω,(6) 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 𝜌_𝑢=1 for simplicity, that is, symbol 𝜌 designates the density normalized using the density of unburned gas, 𝑤=𝑢/̃𝑈 is the normalized Favre-averaged axial velocity, which depend only on 𝑧 on the axis (𝜕𝑐/𝜕𝑟=𝜕𝜌/𝜕𝑟=𝜕𝑤/𝜕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]:𝜌𝐮^𝑐^=𝜌𝑐̃(1−𝑐)̃𝐮_𝑏−𝐮_𝑢(7) 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  𝑐(̃𝑧),𝜌(𝑧),𝑤(𝑧),𝑤_𝑢(𝑧),𝑤_𝑏(𝑧),𝑔_𝑢(𝑧),and𝑔_𝑏(𝑧) should be known. In the present work, the following well-known BML state equation [13]:𝜌_𝑏𝑐=𝜌𝑐̃=𝑐̃1+(𝜎−1)𝑐̃(8) 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=𝜌⁻¹_𝑏 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 function𝑐=12erfc√𝜋𝜉=1√𝜋^∞_𝜉√𝜋e^𝜁2𝑑𝜁(9) to calculate 𝑐(𝑧) and (ii) equation (8) to compute 𝑐[̃𝑐(𝑧)]. Here and in the following, 𝜉=(𝑧−𝑧_𝑓)/𝛿_𝑡, with the values of the normalized flame coordinate 𝑧_𝑓 and thickness 𝛿_𝑡=Δ_𝑡/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.

Second, 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:𝑑𝜌𝑤𝑑𝑧+2𝜌𝑔̃=0,(10) and radial impulse𝑑𝑑𝑧𝜌𝑤𝑔̃+3𝜌𝑔²̃+𝑑𝑑𝑧𝜌𝑐̃(1−𝑐)̃𝑤_𝑏−𝑤_𝑢𝑔_𝑏−𝑔_𝑢+3𝜌𝑐̃(1−𝑐)̃𝑔_𝑏−𝑔_𝑢²=𝑄,(11) equations supplemented with (i) the following boundary conditions:𝑤(1)=−1,𝑔̃(1)=𝑔₁,(12) (ii) the well-known BML relation [13]𝐮=𝐮̃‌+𝐮^=𝐮̃‌+(𝜎−1)𝜌𝐮^𝑐^,(13) 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.

(a)

(b)

Figure 3 

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)

Figure 4 

Axial profiles of the normalized Favre-averaged axial velocities 𝑤(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)

Figure 5 

Axial profiles of the normalized axial turbulent scalar fluxes 𝜌𝑤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.

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 𝑢_𝑡 obtained using this method are reported in the first row of Table 3.

Table 3 

Turbulent burning velocities.

These 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 profile 𝑤(𝑧) obtained in the unburned mixture is extrapolated by invoking the following analytical solution:𝑤_𝑒=2𝑔₁(1−𝑧)−𝑔²₁−𝑄(1−𝑧)²−1,𝑔_𝑒=𝑔₁−𝑔²₁−𝑄(1−𝑧),(14) 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 |𝑤(𝑧)|.

Second, the reference value 𝑐₀ 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=𝑧_𝑓 if (8) holds. Indeed, using (8) and (9), we have−𝜎𝜎−1𝑑𝜌𝑑𝜉=𝜎𝜌²𝑑𝑐̃𝑑𝜉=𝑑𝑐𝑑𝜉=−e^−𝜋𝜉2.(15) Subsequently, (4) and (15) readΨ₂=𝜌𝜉−𝜎−1𝜎^𝜉_−∞𝜁e^−𝜋𝜁2𝑑𝜁=𝜌𝜉+𝜎−12𝜋𝜎e^−𝜋𝜉2=𝜌𝜉−𝜎−12𝜋𝜎𝑑𝑐𝑑𝜉.(16) Therefore,¹₀Ψ₂𝑑𝑐̃=^−∞_∞𝜌𝜉−𝜎−12𝜋𝜎𝑑𝑐𝑑𝜉𝑑𝑐̃𝑑𝜉𝑑𝜉=^−∞_∞𝜉𝜎𝜌𝑑𝑐𝑑𝜉+12𝜋𝜎𝜌²𝑑𝜌𝑑𝜉𝑑𝑐𝑑𝜉𝑑𝜉=12𝜋𝜎^−∞_∞−1𝜌𝑑²𝑐𝑑𝜉²+1𝜌²𝑑𝜌𝑑𝜉𝑑𝑐𝑑𝜉𝑑𝜉=12𝜋𝜎^∞_−∞𝑑𝑑𝜉1𝜌𝑑𝑐𝑑𝜉𝑑𝜉=0(17) and 𝑧=𝑧_𝑓. In the present work, the normalized distance 𝑧=𝑧_𝑓 was taken from Table 2.

If the method is applied to process raw experimental data, then, the coordinate 𝑧₀ of the burning velocity surface may be calculated from the following equality:𝑐𝑧₀=0.5,(18) 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 to 𝑤_𝑒(𝑧_𝑓) calculated from (14); see filled circle in Figure 6.

Results of the test, shown in Table 3, indicate that the values of 𝑤_𝑒(𝑧_𝑓) (the second row) are very close to the normalized turbulent burning velocity 𝑢_𝑡 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 velocity 𝑈_𝑡 is controlled by processes within (or, maybe, at the leading edge of) turbulent flame brush, 𝑈_𝑡 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)𝑆_𝐿(𝑟/𝑅_𝑓)², and the burning velocity can easily be evaluated by analyzing the radial velocity profile measured ahead of the flame.

The approximate equality of 𝑢_𝑡 and |𝑤_𝑒(𝑧_𝑓)| 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:𝜕𝜕𝑡𝜌𝑐̃+𝜕𝜕𝑥_𝑗𝜌𝑢_𝑗̃𝑐̃=𝜕𝜕𝑥_𝑗𝜌𝐷_𝑡𝜕𝑐̃𝜕𝑥_𝑗+𝑈_𝑡||∇𝑐||̃,(19) for the Favre-averaged combustion progress variable, where 𝐷_𝑡 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 reads𝜌𝑤𝑑𝑐̃𝑑𝑧=𝑑_𝑡𝑑𝑑𝑧𝜌𝑑𝑐̃𝑑𝑧−𝑢_𝑡𝑑𝑐̃𝑑𝑧,(20) where 𝑑_𝑡=𝐷_𝑡/(𝑈𝑑) is the normalized turbulent diffusivity. Substitution of (8) and (9) into (20) yields𝜌𝑤𝑑𝑐̃𝑑𝑧=𝑑_𝑡𝛿_𝑡−2𝜋𝜉𝜌−𝜎−1𝜎e^−𝜋𝜉2𝑑𝑐̃𝑑𝑧−𝑢_𝑡𝑑𝑐̃𝑑𝑧.(21) Therefore, within the flame,𝜌𝑤=𝑑_𝑡𝛿_𝑡−2𝜋𝜉𝜌−𝜎−1𝜎e^−𝜋𝜉2−𝑢_𝑡.(22) Differentiating this equation and using (8), (10), and (15), we obtain2𝜌𝑔̃=−1𝛿_𝑡𝑑𝜌𝑤𝑑𝜉=2𝜋𝜌𝑑_𝑡𝛿²_𝑡.(23)

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−2𝜋𝑑_𝑡𝑧_∗−𝑧_𝑓𝛿²_𝑡−𝑢_𝑡=2𝑔₁1−𝑧_∗−𝑔²₁−𝑄1−𝑧_∗²−1,𝜋𝑑_𝑡𝛿²_𝑡=𝑔₁−𝑔²₁−𝑄1−𝑧_∗.(24) Simple algebraic manipulations with these two equations and (14) result in𝑢_𝑡=−𝑤_𝑒𝑧_𝑓+𝑔²₁−𝑄𝑧_𝑓−𝑧_∗².(25) 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 𝑢_𝑡≈−𝑤_𝑒(𝑧_𝑓) 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 𝑢_𝑡=−𝑤_𝑢(𝑧_𝑓).

Figure 7 

Term (𝑔21−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., 𝑢_𝑡=−𝑤_𝑒(𝑧_𝑓)) 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:𝑊=𝑈_𝑡||∇𝑐||̃(26) yields too strong dependence of 𝑊 on 𝑐̃ if the turbulent burning velocity 𝑈_𝑡 is evaluated using local Favre-averaged turbulence characteristics (turbulent kinetic energy 𝑘̃‌ and its dissipation rate 𝜀̃), 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:𝜕𝜕𝑡𝜌𝑐̃+𝜕𝜕𝑥_𝑘𝜌𝑢_𝑘̃𝑐̃=−𝜕𝜕𝑥_𝑘𝜌𝑢^_𝑘𝑐^+𝑊,(27) that is,−𝜕𝜕𝑥_𝑗𝜌𝑢^_𝑗𝑐^+𝑊=𝜕𝜕𝑥_𝑗𝜌𝐷_𝑡𝜕𝑐̃𝜕𝑥_𝑗+𝑈_𝑡||∇𝑐||̃,(28) within the framework of the extended Zimont model. If applied to a flame stabilized in an impinging jet, (28) reads−𝜕𝜕𝑧𝜌𝑐̃(1−𝑐)̃𝑤_𝑏−𝑤_𝑢_{1}−2𝜌𝑐̃(1−𝑐)̃𝑔_𝑏−𝑔_𝑢_{2}+Ω_3=𝑑_𝑡𝜕𝜕𝑧𝜌𝜕𝑐̃𝜕𝑧_{4}+𝑢_𝑡||∇𝑐||_5̃.(29)

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,𝑢_𝑡=¹₀Ω𝑑𝑧−¹₀2𝜌𝑐̃(1−𝑐)̃𝑔_𝑏−𝑔_𝑢𝑑𝑧,(30) 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 𝑢_𝑡 nor 𝑑_𝑡 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(𝑐)̃ into an expression for 𝑈_𝑡=𝑈_𝑡(𝑆²_𝐿/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).

Figure 8 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 𝑢_𝑡= const and 𝑑_𝑡= 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 Table 1.

Furthermore, curves 2 and 3 in Figure 7 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:−𝜕𝜕𝜉𝜌𝑢^_𝑛𝑐^+𝑊=𝜕𝜕𝜉𝜌𝐷_𝑡𝜕𝑐̃𝜕𝜉+𝑈_𝑡||∇𝑐||̃,(31) where the 𝜉-axis is locally normal to the mean flame brush and 𝑢_𝑛 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:−𝜕𝜕𝑧𝜌𝑐(̃1−𝑐)̃(𝑤_𝑏−𝑤_𝑢)_{I}+Ω_III=𝑑_𝑡𝜕𝜕𝑧𝜌𝜕𝑐̃𝜕𝑧_{IV}+𝑢_𝑡||∇𝑐||_Ṽ(32) 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 𝑑_𝑡 and 𝑢_𝑡 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:𝜌𝑤𝑑𝑐̃𝑑𝑧+2𝜌𝑐̃(1−𝑐)̃𝑔_𝑏−𝑔_𝑢=𝑑_𝑡𝛿_𝑡−2𝜋𝜉𝜌−𝜎−1𝜎e^−𝜋𝜉2𝑑𝑐̃𝑑𝑧−𝑢_𝑡𝑑𝑐̃𝑑𝑧,(33) 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 𝑢_𝑡 and 𝑑_𝑡 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.

Acknowledgments

The author would like to express his sincere thanks to Professors Jerzy Chomiak and Vladimir Sabel'nikov for valuable discussions. This work was supported by the Swedish Research Council (VR), Swedish Energy Agency, and by the Combustion Engine Research Center (CERC).