Determination of the Burning Velocity Domain of a Statistically Stationary Turbulent Premixed Flame in Presence of Counter-Gradient Transport

Abstract

The present study aims at providing a complete picture of the various propagation scenarios that a statistically stationary turbulent premixed flame may possibly undergo. By explicitly splitting the scalar turbulent flux between its gradient and counter-gradient contributions, the scalar governing equation is rewritten as an ordinary differential equation in the phase space. Then, an analysis of the characteristic equations in the vicinity of the reactants and products side is carried out. The domain of existence of the propagation velocity is then determined and positioned over the relevant Bray number range. It is shown in particular that when a counter-gradient transport at the cold leading edge of the flame is dominant, there still exists a possibility of observing a steady regime of propagation. This conclusion is compatible with recent experimental data and observations based on the analysis of direct numerical simulations.

1. Introduction

The determination of the burning velocity of turbulent premixed flames 𝑆_𝑇 has been the subject of many experimental studies. As far as a given modelling of such reacting flows is concerned, the a priori value of 𝑆_𝑇 for statistically stationary one-dimensional turbulent flame is often estimated through the direct application of the results of the Kolmogorov-Petrovskii-Piskounov (KPP) theory [1] which showed that, for a one-dimensional reaction zone without heat release described through the evolution of a single progress variable 𝑐 and with a constant diffusion coefficient 𝐷_𝑡, the steady propagation of the reaction zone is controlled by the behavior of the reaction rate at the reactants side and by the value of the diffusion coefficient. In many situations, experiments (Moss [2], Cheng and Shepherd [3], Troiani et al. [4], Zimmer et al. [5]) as well as direct numerical simulations (DNS) (Veynante et al. [6], Nishiki [7], Hauguel [8], Lee and Huh [9]) show that an expression of the turbulent transports via a gradient expression is not appropriate as some mechanisms may promote a transport with a sign identical to that of the mean scalar gradient. An extensive review of relevant experimental and DNS data can be found in the paper of Lipatnikov and Chomiak [10]. In such situations, the introduction of the Bray number proposed by Veynante et al. [6] proved to be helpful to discriminate between flow configurations featuring gradient turbulent flux (GTF) only (𝑁_𝐵≤1) or counter-gradient turbulent flux (CGTF) (𝑁_𝐵≥1).The Bray number was defined as 𝑁_𝐵=𝜏𝑆_𝐿/2𝛼_𝑣𝑢^′ where 𝜏 designates the heat release parameter and 𝛼_𝑣 some order unity efficiency function which depends mainly on (and is an increasing function of) the ratio between the turbulence integral length scale and the flame thermal thickness. For 𝑁_𝐵≥1, one of the most prominent mechanism leading to CGTF is directly related to the differential effect of the pressure gradient on the pockets of reactants (heavy) and products (lighter). But it should, nevertheless, be stressed, that even when such a mechanism is predominant, there still exists a gradient turbulent flux acting at scales much smaller than those of the pockets and for which the turbulent transport coefficient cannot be estimated through a turbulence model which integrates the whole turbulence scales. Accordingly, and following in that respect Veynante et al. [6], Zimont and Biagioli [11] or Lipatnikov and Chomiak [12], this suggests to consider the turbulent scalar flux as resulting from the competition between gradient and counter-gradient mechanisms.

Along these lines, we propose to study the propagation properties of a turbulent premixed flame by considering such a flux decomposition. First, an extended KPP analysis is developed. It combines all the possible scenarios with the change of 𝑁_𝐵 and provides a general analysis whose results are not qualitatively dependent anymore on the modelling closures chosen but permit a clear and comprehensive discrimination between all the various propagation regimes.

We note that, in the framework of an eddy-break-up model, an analytical attempt aimed at explaining the influence of the presence of CGTF on the turbulent flame propagation properties has been done by Corvellec et al. [13] who considered an idealized one-dimensional premixed flame that propagates through high-Reynolds-number frozen turbulence. These authors dealt only with a flame brush formed by a GTF zone followed by a CGTF one. The scalar flux in each zone was expressed by employing a classical gradient formulation with the peculiarity of using a strictly negative diffusion coefficient for the CGTF zone. Thanks to this mathematical “trick”, it was then possible to employ the KPP technique to study the characteristics of the governing equation at both sides of the flame brush and at the intermediate point at which the flux is zero. This approach did not put into evidence a qualitative change in the nature of the burning velocity domain which appears to be still a semi-infinite interval. The lower bound though appeared to be controlled by both sides in such a situation. It drew also the attention on the fact that the KPP results cannot be extrapolated directly in such situations (in accordance to some extent with the numerical simulations of Bradley et al. [14]). The results of Corvellec et al. [13] cannot be considered as being of general implication since they are contingent to the recourse to an effective diffusion coefficient and are limited to the analysis of only one flame structure, that is, a GTF zone followed by a CGTF one. The Bray number is absent in the theory developed in [13] whereas DNS [6, 9] and recent experiments on turbulent premixed flames [4, 5] show that a different mean flame structure may be encountered that is, a CGTF zone followed by a GTF one. Thus, the present approach is more general than those followed previously since it permits to cover all the two-zone mean flame structures that can be thought of. It shows in particular that as far as the burning velocity domain is concerned, qualitative differences with the KPP and Corvellec et al. [13] results are obtained.

2. Mathematical Formulation

2.1. Governing Equations

An unsteady 1-D freely developing turbulent isenthalpic premixed flame is considered here and is described through the evolution of a single progress variable 𝑐=(𝑇−𝑇_𝑟)/(𝑇_𝑏−𝑇_𝑟), where subscripts 𝑟 and 𝑏 denote the states of the reactant mixture and the fully burnt products, respectively, and 𝜏=(𝑇_𝑏−𝑇_𝑟)/𝑇_𝑟 defines the heat release parameter. Thus, in a fixed coordinate system (𝑂,𝑥) the Favre averaged equation for 𝑐 and the continuity equation to be used later on, are given by 𝜕𝜌+𝜕𝜕𝑡𝜌𝑢𝜕𝜕𝑥=0,𝜌𝑐̃+𝜕𝜕𝑡𝜌𝑢𝑐̃𝜕𝜕𝑥=−𝜕𝑥𝜌𝑢^′′𝑐^′′+𝑤, (1)𝜌𝑢^′′𝑐^′′ is the turbulent flux, and 𝑤 is the mean reaction rate whose exact expression is not required at this stage. The flame is propagating from right to left, that is, 𝜕𝑐/̃𝜕𝑥⩾0.

We consider here the case for which counter-gradient mechanism is largely present in the flame as it is the case when the turbulence intensity is sufficiently low and/or the heat release parameter sufficiently large (see [6] for further details). Accordingly, the turbulent flux is expressed as𝐹=𝜌𝑢^′′𝑐^′′=𝐹_GTF+𝐹_CGTF (2)𝐹_GTF<0 represents the gradient contribution while 𝐹_CGTF>0 corresponds to the counter-gradient part. Such a decomposition is compatible with the findings of Veynante et al. [6] who derived an algebraic expression for the flux as a function of the laminar flame velocity 𝑆_𝐿, the heat release parameter 𝜏 and the rms velocity 𝑢^′, namely, 𝜌𝑢^′′𝑐^′′=𝜌𝑐(̃1−𝑐)̃(𝜏𝑆_𝐿−2𝛼_𝑣𝑢^′). In this particular case, the counter-gradient contribution is equal to𝐹_CGTF=𝜌𝑐̃(1−𝑐)̃𝜏𝑆_𝐿, (3) and the gradient one reads as𝐹_GTF=−2𝛼_𝑣𝑢^′𝜌𝑐̃(1−𝑐)̃. (4)

The ratio 𝐹_CGTF/|𝐹_GTF| is just equivalent to the Bray number 𝑁_𝐵. If we introduce 𝐷_𝑡≈𝑙_𝑡𝑢^′, where 𝑙_𝑡 is the turbulence integral length scale and suppose that 𝜕𝑐/̃𝜕𝑥≈𝑐(̃1−𝑐)̃/𝑙_𝑡, then the gradient contribution can be classically written as 𝐹_GTF=−𝜌𝐷_𝑡𝜕𝑐/̃𝜕𝑥. Here, the dependency of 𝛼_𝑣 on the ratio between the flame thermal thickness and the turbulence integral length scale has been neglected. In the following analysis, we shall consider 𝐹_GTF as given by (4) combined with a more general expression for 𝐹_CGTF, namely,𝐹_CGTF=𝜌𝜏𝑆_𝐿𝑓(𝑐)̃, (5) where 𝑓 is a positive continuous function of 𝑐̃ such that 𝑓(0)=𝑓(1)=0. In the case of Veynante et al. [6], 𝑓(𝑐)̃ is a quadratic symmetric function, namely,𝑓(𝑐)̃=𝑐̃(1−𝑐)̃. (6)

If we now rewrite the 𝑐̃-equation combined with the continuity equation, one obtains𝜌𝜕𝑐̃+𝜕𝑡𝜌𝑢+𝜏𝑆_𝐿𝜕𝜕𝑐̃(𝑐)̃𝜕𝑐̃𝜌𝑓=𝜕𝜕𝑥𝜕𝑥𝜌𝐷_𝑡𝜕𝑐̃+𝜕𝑥𝑤. (7)

It is worth noting that the analysis could be done also in the case, not considered here, where 𝐹 would be expressed without the introduction of a turbulent diffusion coefficient (e.g., using directly the decomposition of Veynante et al. [6]).

We rewrite (7) in a nondimensional form by defining the following reference parameters:(1)a velocity scale 𝑢=𝑢^′,(2)a time scale 𝑡̂=𝑙_𝑡/𝑢^′,(3)a length scale 𝑙̂ proportional to the integral turbulence scale, that is, 𝑙̂=𝑙_𝑡,(4)a reference turbulent diffusion coefficient 𝐷_𝑡0=𝑙_𝑡𝑢^′ (𝑙_𝑡 is chosen in such a way that the proportionality coefficient between 𝐷_𝑡0 and 𝑙_𝑡𝑢^′ is unity),(5)a flame velocity scale 𝐷_𝑡0/𝑡̂=𝑢^′.

Thus, one has𝑅𝜕𝑐̃𝜕𝑡^∗+𝑅𝑢^∗+𝑁_𝐵2𝛼_𝑣𝜕𝜕𝑐̃𝜕𝑐̃𝑅𝑓𝜕𝑥^∗=𝜕𝜕𝑥^∗𝜕𝑐̃𝑅𝒟𝜕𝑥^∗+𝑤^∗, (8) where 𝑥^∗=𝑥/𝑙̂, 𝑢^∗=𝑢/𝑢, 𝑡^∗=𝑡/𝑡̂, 𝒟=𝐷_𝑡/𝐷_𝑡0, 𝑅=𝜌/𝜌_𝑢=1/1+𝜏𝑐̃, and 𝑤^∗=𝑤(𝑡/̂𝜌_𝑢). In the following and whenever unambiguous, we shall drop superscript *. So, with that convention, (8) reads as𝑅𝜕𝑐̃+𝜕𝑡𝑅𝑢+𝑁_𝐵2𝛼_𝑣𝜕𝜕𝑐̃𝜕𝑐̃𝑅𝑓=𝜕𝜕𝑥𝜕𝑐̃𝜕𝑥𝑅𝒟+𝜕𝑥𝑤, (9) and the nondimensional form of the continuity equation (1) is given by𝜕𝑅+𝜕𝑡𝜕𝑅𝑢𝜕𝑥=0. (10)

At this stage, it is important to underline that the form of (7) allows us to give a physical interpretation of the influence of CGTF on the flame structure (following mainly the reasoning adopted in [6]). The term of CGTF is a nonlinear convection term which tends to move both sides toward each other because of the change of sign of 𝜕𝜌𝑓(𝑐)̃/𝜕𝑐̃ (positive at the fresh reactants side and then negative at the burnt products side). Such a “thinning” effect has to be counter-balanced by the gradient transport term in order to obtain a steady state flame propagation regime.

2.2. Steady-State Regime of Flame Propagation

Let us consider a steady state regime of flame propagation. Accordingly, in a coordinate system attached to the mean flame brush and introducing the flame mass consumption 𝑚̇ and the turbulent flame speed 𝑆_𝑡 such as 𝑚̇=𝜌_𝑢𝑆_𝑡, the preceding continuity and progress-variable nondimensional equations are written as𝑚̇𝑅𝑢=𝜌_𝑢𝑢^′=𝜌_𝑢𝑆_𝑡𝜌_𝑢𝑢^′=𝜌_𝑏𝑢_𝑏𝜌_𝑢𝑢^′=Λ,Λ+𝑁_𝐵2𝛼_𝑣𝑑𝑑𝑐̃𝑑𝑐̃𝑅𝑓=𝑑𝑑𝑥𝑑𝑐̃𝑑𝑥𝑅𝒟+𝑑𝑥𝑤, (11) where Λ=𝑆_𝑡/𝑢^′.

We can rewrite (11) as a first-order differential equation for 𝑃=(𝑅𝒟/Λ)(𝑑𝑐/̃𝑑𝑥) on the 𝑐̃-domain [0,1], namely 𝑁1+^𝐵2𝛼_𝑣Λ𝑑𝑑𝑐̃𝑅𝑓𝑃=𝑃𝑃^′+𝑤𝑅𝒟Λ². (12) Introducing 𝑁_𝐵=𝑁_𝐵2𝛼_𝑣 and Ω=𝑤𝑅𝒟, (12) is expressed as𝑃𝑃^′=𝑁1+^𝐵Λ𝑅𝑓^′1𝑃−ΩΛ², (13) where ()^′ denotes the derivation with respect to 𝑐̃.

3. Analysis of the P-Equation in the Vicinity of the Singular Points

Now, our primary objective is to determine the domain of existence of Λ. In principle, we need for that to analyse the trajectories in the (𝑃,𝑐)̃ phase plane in the KPP-like manner. It is clear that a priori, due to the presence of the CGTF derivative 𝑓^′, it is difficult to perform the KPP analysis in such a case.

Thus, leaving aside the question of determining the trajectories themselves, we focus on the determination of the associated Λ domain by examining the characteristic equations of (13) at both sides of the brush. The implication of the relative position of 𝑃 and 𝐹=(𝑁_𝐵/Λ)𝑅𝑓 for the 𝑃-trajectories when they are tangent to the characteristics lines is also analyzed. We denote 𝐹^′₀=𝑑𝐹(0)/𝑑𝑐̃=𝑁_𝐵/Λ(𝑅𝑓)^′(0)=(𝑁_𝐵/Λ)𝑓^′(0)=𝛼 and 𝐹^′₁=−𝑑𝐹(1)/𝑑𝑐̃=−𝑁_𝐵/Λ(𝑅𝑓)^′(1)=−(1/(𝜏+1))(𝑁_𝐵/Λ)𝑓^′(1)=𝛽=(1/(𝜏+1))𝛼|𝑓^′(1)|/𝑓^′(0) and introduce the two functions 𝑓₁(𝛼)=4/(1+𝛼)² and 𝑓₂(𝛼)=1/𝛼, such that 𝑓₂(𝛼)>𝑓₁(𝛼) for all 𝛼 but 𝛼=1 for which 𝑓₂(1)=𝑓₁(1).(a)Reactants Side. In the limit 𝑐̃→0⁺ and assuming that 𝑃=𝑠𝑐̃ and 𝐹=𝛼𝑐̃ (𝑠>0 and 𝛼>0), the characteristic equation associated with (13) is given by𝑠²()+1−𝑠1+𝛼Λ²Ω^′(0)=0. (14) The required positiveness of the discriminant Δ₀⁺=(1+𝛼)²−(4/Λ²)Ω^′(0) is assured as soon as Λ²>𝑓₁(𝛼)Ω^′(0). The corresponding roots 𝑠₁=(1+𝛼)/2+Δ₀⁺/4 and 𝑠₂=(1+𝛼)/2−Δ₀⁺/4 are both positive with 𝑠₁>𝑠₂. Accordingly, the reactants side is an improper unstable node and all the trajectories but one (the characteristic line 𝑠₁𝑐̃ itself) are tangent to the 𝑠₂𝑐̃ line. Two cases are to be considered in the vicinity of 𝑐̃=0: (1) 𝐹_GTF+𝐹_CGTF<0 (the gradient contribution overcomes the counter-gradient one) and (2) the reverse situation prevails, that is, 𝐹_GTF+𝐹_CGTF>0.(1) Gradient transport prevails, that is, 𝐹_GTF+𝐹_CGTF<0⇔𝑃>𝐹. This implies that 𝑠₂>𝛼. Solving this inequality yields 𝛼<1 and Λ²<𝑓₂Ω^′(0). It is worth noting that when 𝛼→0, one recovers the KPP domain.(2)Counter-gradient transport prevails, that is, 𝐹_GTF+𝐹_CGTF>0⇔𝑃<𝐹. Two scenarios meet these inequalities, namely.(i)𝛼>𝑠₁. Solving this inequality yields 𝛼>1 and Λ²<𝑓₂Ω^′(0).(ii)𝑠₂<𝛼<𝑠₁. This implies that Λ²>𝑓₂Ω^′(0) for 𝛼>0.

In both cases (1, 2), we must satisfy also Λ²>𝑓₁Ω^′(0). (15) Thus, for case 1 we have 𝑓₁Ω^′(0)<Λ²<𝑓₂Ω^′(0),𝛼<1, (16) and for case 2𝑓₁Ω^′(0)<Λ²<𝑓₂Ω^′(0)Λ,𝛼>1,₂>𝑓₂Ω^′(0),𝛼>0. (17) Using the definitions of 𝑓₁(𝛼)=4/(1+𝛼)², 𝑓₂(𝛼)=1/𝛼 and 𝛼=(𝑁_𝐵/Λ)𝑓^′(0), for case 1, one has 2Ω^′(0)−𝑁_𝐵𝑓^′(0)Ω<Λ<^′(0)𝑁_𝐵𝑓^′(0),Λ>𝑁_𝐵𝑓^′(0) (18) or in equivalent form 2Ω^′(0)−𝑁_𝐵𝑓^′(0)Ω<Λ<^′(0)𝑁_𝐵𝑓^′(0),𝑁_𝐵𝑓^′(0)<Ω^′(0). (19) For case 2, we have 2Ω^′(0)−𝑁_𝐵𝑓^′(0)Ω<Λ<^′(0)𝑁_𝐵𝑓^′(0),𝑁_𝐵𝑓^′(0)>Ω^′(0),ΩΛ>^′(0)𝑁_𝐵𝑓^′(0),𝑁_𝐵>0. (20) Introducing the variables ΛΛ=Ω^′(0),𝑁_𝐵=𝑁_𝐵𝑓^′(0)Ω^′(0), (21) we can rewrite for case 1 2−𝑁_𝐵<1Λ<𝑁_𝐵,𝑁_𝐵<1, (22) and for case 2 2−𝑁_𝐵<1Λ<𝑁_𝐵,𝑁_𝐵1>1,Λ>𝑁_𝐵,𝑁_𝐵>0. (23) It is important to note that relations in (23) are the sum of two sets, but not the product. Figure 1 illustrates the domain of existence of GTF and CGTF in the plane (Λ,𝑁_𝐵) that followed from the analysis of (13) (for P) in the vicinity 𝑐̃→0.(b)Products Side. In the limit 𝑐̃→1⁻ and assuming that 𝑃=𝑠(1−𝑐)̃ and 𝐹=𝛽(1−𝑐)̃ (𝑠>0 and 𝛽>0), the characteristic equation associated with (13) is given by 𝑠²+1+𝑠1−𝛽Λ²Ω^′_̃𝑐(1)=0. (24)

Figure 1 

Domain of existence in the (Λ,𝑁𝐵) plane of gradient or counter-gradient total turbulent flux at the reactants side.

The discriminant Δ₁⁻ is always positive and there exists only one positive root 𝑠₃=(𝛽−1)/2+Δ₁⁻/4. Accordingly, the products side is a saddle and all the trajectories are tangent to the 𝑠₃𝑐̃ line. Again, two situations are to be considered:(1)GTF prevails, that is, 𝐹_GTF+𝐹_CGTF<0⇔𝑃>𝐹. This implies that 𝛽<𝑠₃ or equivalently Λ²<(1/𝛽)|Ω^′(1)|. In the limit 𝛽→0, the original KPP behavior is recovered, that is, the absence of constraint.(2)CGTF prevails, that is, 𝐹_GTF+𝐹_CGTF>0⇔𝑃<𝐹. This implies that 𝛽>𝑠₃ or equivalently Λ²>(1/𝛽)|Ω^′(1)|.

Using the relation 𝛽𝛼=1𝑓1+𝜏^′(1)𝑓^′(0), (25) we transform the above inequalities into the following form:

for case 1 (GTF) 1Λ<𝑁_𝐵𝑓_𝐵, (26)

for case 2 (CGTF) 1Λ>𝑁_𝐵𝑓_𝐵, (27) where 𝑓_𝐵=()𝑓𝜏+1^′(0)𝑓^′(1Ω)^′(1)Ω^′(0). (28) Figures 2(a) and 2(b) present the domain of GTF and CGTF obtained from the analysis in the vicinity 𝑐̃→1 and for 𝑓_𝐵>1 and 𝑓_𝐵<1, respectively. In the latter case, two remarkable values 𝑁_𝐵1 and 𝑁_𝐵2of 𝑁_𝐵 are introduced. They are defined by 𝑁_𝐵1=(1−1−𝑓_𝐵)<1 and 𝑁_𝐵2=(1+1−𝑓_𝐵)>1.

(a)

(b)

Figure 2 

Domain of existence in the (Λ,𝑁𝐵) plane of gradient or counter-gradient total turbulent flux at the products side (a) case 𝑓𝐵>1 and (b) case 𝑓𝐵<1.

There exists a maximum of four possible configurations, listed in Table 1, which correspond to the total turbulent flux behavior (gradient (G) or counter-gradient (CG)) in the vicinity of the singular points 𝑐̃=0 and 𝑐̃=1.

Table 1 

Let us consider first the case 𝑓_𝐵>1. Analysis of inequalities (22)–(23) and (25)–(27) gives the following results.(i) G/G configuration: 2−𝑁_𝐵<1Λ<𝑁_𝐵,𝑁_𝐵<1. (29)(ii)G/CG configuration is absent.(iii)CG/G configuration: 2−𝑁_𝐵<1Λ<𝑁_𝐵,𝑁_𝐵1>1Λ>𝑁_𝐵,𝑁_𝐵1>0sumoftwosetsΛ<𝑁_𝐵𝑓_𝐵,𝑁_𝐵>0product. (30)(iv)CG/CG Configuration: 2−𝑁_𝐵<1Λ<𝑁_𝐵,𝑁_𝐵1>1Λ>𝑁_𝐵,𝑁_𝐵1>0sumoftwosetsΛ>𝑁_𝐵𝑓_𝐵,𝑁_𝐵>0product. (31)

Domains of the above three possible configurations are given in Figure 3. Now for the case 𝑓_𝐵<1, all four configurations are possible, namely,(i) G/G configuration:2−𝑁_𝐵<𝑓Λ<^𝐵𝑁_𝐵for0<𝑁_𝐵<𝑁_𝐵1with𝑁_𝐵1=1−1−𝑓_𝐵<1. (32) Consequently, in this interval [0,𝑁_𝐵1], we have Λ>1.(ii)G/CG configuration:

Figure 3 

The domain of the three different possibilities for the turbulent flame brush structure in the plane (Λ,𝑁𝐵) for 𝑓𝐵>1.

𝑓_𝐵/𝑁_𝐵<Λ<1/𝑁_𝐵 for 0<𝑁_𝐵<𝑁_𝐵1 and 2−𝑁_𝐵<Λ<1/𝑁_𝐵 for 𝑁_𝐵1<𝑁_𝐵<1 and again we have Λ>1.(iii)CG/G configuration:

max(0;2−𝑁_𝐵)<Λ<𝑓_𝐵/𝑁_𝐵 for 𝑁_𝐵>𝑁_𝐵2 with 𝑁_𝐵2=1+1−𝑓_𝐵>1. In that case, we have Λ<1.(iv) CG/CG configuration:1Λ>𝑁_𝐵for0<𝑁_𝐵𝑁<1,Λ>2−_𝐵for1≤𝑁_𝐵<𝑁_𝐵2,𝑓Λ>^𝐵𝑁_𝐵for𝑁_𝐵>𝑁_𝐵2. (33)

The different regions of the (Λ,𝑁_𝐵) plane corresponding to these four configurations are presented in Figure 4. We note that the case 𝑓_𝐵<1 is realised for the classical eddy-break-up model where 𝑤^∗=𝑐̃(1−𝑐)̃1+𝜏𝑐̃, (34) and if 𝐷_𝑡=𝐷_𝑡0=const,   𝑓(𝑐)̃=𝑐(̃1−𝑐)̃-Veynante et al. case [6]. In that case, one has Ω(𝑐)̃=𝑐(̃1−𝑐)̃/(1+𝜏𝑐)̃² and consequently Ω^′(0)=1 and Ω^′(1)=−1/(1+𝜏)². Thus, 𝑓_𝐵=11+𝜏<1. (35) The case 𝑓_𝐵>1 is obtained if 𝑤^∗=𝑐̃(1−𝑐)̃(1+𝜘𝑐)̃, (36) where 𝜘 is some positive contant. So we have Ω(𝑐)̃=𝑐̃(1−𝑐)̃(1+𝜘𝑐)̃(𝑐)̃,Ω1+𝜏_′(0)=1,Ω^′(1)=−1+𝜘.1+𝜏 (37) For Veynante et al. [6], 𝑓(𝑐)̃=𝑐(̃1−𝑐)̃ and so 𝑓_𝐵=(1+𝜏)1+𝜘1+𝜏=1+𝜘>1whenever𝜘>0. (38)

Figure 4 

The domain of the three different possibilities for the turbulent flame brush structure in the plane (Λ,𝑁𝐵) for 𝑓𝐵>1.

It should be noted that experimental data [3, 15] suggest that 𝑤^∗ is shifted towards the burnt gas side. Such a situation can be modelled by relation (36) with 𝜘>0, and, in this case, we can have 𝑓_𝐵>1. As was shown previously, in such a situation, the configuration G/CG is absent. Such a result is compatible with DNS results for 1-D freely propagating turbulent premixed flame [6, 9]. It is also in correspondance with the recent experimental data of Troiani et al. [4] who investigated bluff-body stabilized turbulent premixed flames. These authors showed that the radial component of the scalar flux (which is the counterpart of 𝐹) could exhibit a transition between a counter-gradient to a gradient behavior when moving along the normal to the mean front from reactants toward products.

4. A Particular Configuration: The Murray's Equation

The Murray's equation [16] is a good example of the convection-diffusion equation, of the type considered here, to demonstrate that the common viewpoint that at the leading edge it is the diffusion term which always drives the flame propagation is not universal. Such an equation is obtained as a particular case of (9) corresponding to 𝑅=1, 𝑢=0, 𝒟=1, and 𝑤=𝑐(̃1−𝑐)̃. In such a case, we have 𝑓_𝐵=1,   𝑓^′(0)=|𝑓^′(1)|=1,Ω^′(0)=|Ω^′(1)|=1, and 𝛽=𝛼. We note that formally though, 𝑅=1 corresponds to 𝜏=0 leading to 𝑁_𝐵=0. But we will consider here 𝑁_𝐵 like a parameter not linked to 𝜏. In such a situation, by denoting 𝐾=2𝑁_𝐵 and 𝑐_𝑤=Λ+𝑁_𝐵, (11) reduced to that analyzed by Murray [16] to determine the wave speed 𝑐_𝑤, namely, 𝑑²𝑐̃𝑑𝑥²−𝑐_𝑤𝑐̃𝑑𝑐̃−𝐾+𝑐̃(𝑐)̃𝑑𝑥1−=0. (39) Murray [16] showed that travelling wave solutions exist for all 𝑐_𝑤≥𝑐_𝑤(𝐾) where 𝑐_𝑤(𝐾)=𝑘2if𝐾<2,2+2𝑘if𝐾⩾2, (40) or equivalently with our notations𝑁Λ=2−_𝐵if𝑁_𝐵1<1,𝑁_𝐵if𝑁_𝐵⩾1. (41) Thus, it appears that in that particular case, for 𝑁_𝐵⩾1 and in the zone 𝑐̃→0, it is the counter-gradient transport that prevails (see the domain CG/G in Figures 3 and 4).

5. Concluding Remark

A methodology that permits to study the domain of the tubulent flame velocities for a steady regime of a turbulent premixed flame propagation has been presented. It has been shown how the introduction of the parameter 𝑓_𝐵 (defined by 32) that combines both the counter-gradient flux and the mean reaction rate behavior at both edges of the flame can be used to determine the domain where a steady regime of propagation may exist for any combination of scalar fluxes that dominate at the flame edges. It is worth noticing though that the present analysis of the sole velocity domain does not prove the existence/unicity of the steady solutions. Consequently, future work will concentrate on the use of numerical simulations to study these questions.

Acknowledgments

The support of the first author by ONERA as well as the support of the second author by CNRS are greatly acknowledged.