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

Authors: V. A. Sabel'nikov; P. Bruel
Journal: Journal of Combustion
 (2011)

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## 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.


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## Body

## 1. Introduction

The determination of the burning velocity of turbulent premixed flamesST has been the subject of many experimental studies. As far as a given modelling of such reacting flows is concerned, the a priorivalue of ST  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 Dt, 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 (NB≤1) or counter-gradient turbulent flux (CGTF) (NB≥1).The Bray number was defined as NB=τSL/2αvu′ where τ designates the heat release parameter and αv 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 NB≥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 ofNB 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 variablec=(T-Tr)/(Tb-Tr), where subscripts r and b denote the states of the reactant mixture and the fully burnt products, respectively, and τ=(Tb-Tr)/Tr 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 (1)∂ρ¯∂t+∂ρ¯ũ∂x=0,∂ρ¯c̃∂t+∂ρ¯ũc̃∂x=-∂∂xρu′′c′′¯+w¯,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, ∂c̃/∂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 as(2)F=ρu′′c′′¯=FGTF+FCGTFFGTF<0 represents the gradient contribution while FCGTF>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 SL, the heat release parameter τ and the rms velocity u′, namely, ρu′′c′′¯=ρ¯c̃(1-c̃)(τSL-2αvu′). In this particular case, the counter-gradient contribution is equal to(3)FCGTF=ρ¯c̃(1-c̃)τSL,
and the gradient one reads as(4)FGTF=-2αvu′ρ¯c̃(1-c̃).The ratioFCGTF/|FGTF| is just equivalent to the Bray number NB. If we introduce Dt≈ltu′, where lt is the turbulence integral length scale and suppose that ∂c̃/∂x≈c̃(1-c̃)/lt, then the gradient contribution can be classically written as FGTF=-ρ¯Dt∂c̃/∂x. Here, the dependency of αv on the ratio between the flame thermal thickness and the turbulence integral length scale has been neglected. In the following analysis, we shall consider FGTF as given by (4) combined with a more general expression for FCGTF, namely,(5)FCGTF=ρ¯τSLf(c̃),
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(c̃) is a quadratic symmetric function, namely,(6)f(c̃)=c̃(1-c̃).If we now rewrite thec̃-equation combined with the continuity equation, one obtains(7)ρ¯∂c̃∂t+(ρ¯ũ+τSL∂∂c̃ρ¯f(c̃))∂c̃∂x=∂∂x(ρ¯Dt∂c̃∂x)+w¯.It is worth noting that the analysis could be done also in the case, not considered here, whereF 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 scaleû=u′,(2)
a time scalet̂=lt/u′,(3)
a length scalel̂ proportional to the integral turbulence scale, that is, l̂=lt,(4)
a reference turbulent diffusion coefficientDt0=ltu′ (lt is chosen in such a way that the proportionality coefficient between Dt0 and ltu′ is unity),(5)
a flame velocity scaleDt0/t̂=u′.Thus, one has(8)R∂c̃∂t*+(Ru*+NB2αv∂∂c̃(Rf))∂c̃∂x*=∂∂x*(RD∂c̃∂x*)+w¯*,
where x*=x/l̂, u*=ũ/û, t*=t/t̂, 𝒟=Dt/Dt0, R=ρ¯/ρu=1/1+τc̃, and w¯*=w¯(t̂/ρu). In the following and whenever unambiguous, we shall drop superscript *. So, with that convention, (8) reads as(9)R∂c̃∂t+(Ru+NB2αv∂∂c̃(Rf))∂c̃∂x=∂∂x(RD∂c̃∂x)+w¯,
and the nondimensional form of the continuity equation (1) is given by(10)∂R∂t+∂Ru∂x=0.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 ∂ρ¯f(c̃)/∂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 consumptionm. and the turbulent flame speed St such as m.=ρuSt, the preceding continuity and progress-variable nondimensional equations are written as(11)Ru=m.ρuu′=ρuStρuu′=ρbubρuu′=Λ,[Λ+NB2αvddc̃(Rf)]dc̃dx=ddx(RDdc̃dx)+w¯,
where Λ=St/u′.We can rewrite (11) as a first-order differential equation for P=(R𝒟/Λ)(dc̃/dx) on the c̃-domain [0,1], namely (12)[1+NB2αvΛddc̃(Rf)]P=PP′+w¯RDΛ2.
Introducing N̂B=NB2αv and Ω=w¯R𝒟, (12) is expressed as(13)PP′=[1+N̂BΛ(Rf)′]P-Ω1Λ2,
where ()′ denotes the derivation with respect to c̃.


## 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 variablec=(T-Tr)/(Tb-Tr), where subscripts r and b denote the states of the reactant mixture and the fully burnt products, respectively, and τ=(Tb-Tr)/Tr 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 (1)∂ρ¯∂t+∂ρ¯ũ∂x=0,∂ρ¯c̃∂t+∂ρ¯ũc̃∂x=-∂∂xρu′′c′′¯+w¯,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, ∂c̃/∂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 as(2)F=ρu′′c′′¯=FGTF+FCGTFFGTF<0 represents the gradient contribution while FCGTF>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 SL, the heat release parameter τ and the rms velocity u′, namely, ρu′′c′′¯=ρ¯c̃(1-c̃)(τSL-2αvu′). In this particular case, the counter-gradient contribution is equal to(3)FCGTF=ρ¯c̃(1-c̃)τSL,
and the gradient one reads as(4)FGTF=-2αvu′ρ¯c̃(1-c̃).The ratioFCGTF/|FGTF| is just equivalent to the Bray number NB. If we introduce Dt≈ltu′, where lt is the turbulence integral length scale and suppose that ∂c̃/∂x≈c̃(1-c̃)/lt, then the gradient contribution can be classically written as FGTF=-ρ¯Dt∂c̃/∂x. Here, the dependency of αv on the ratio between the flame thermal thickness and the turbulence integral length scale has been neglected. In the following analysis, we shall consider FGTF as given by (4) combined with a more general expression for FCGTF, namely,(5)FCGTF=ρ¯τSLf(c̃),
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(c̃) is a quadratic symmetric function, namely,(6)f(c̃)=c̃(1-c̃).If we now rewrite thec̃-equation combined with the continuity equation, one obtains(7)ρ¯∂c̃∂t+(ρ¯ũ+τSL∂∂c̃ρ¯f(c̃))∂c̃∂x=∂∂x(ρ¯Dt∂c̃∂x)+w¯.It is worth noting that the analysis could be done also in the case, not considered here, whereF 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 scaleû=u′,(2)
a time scalet̂=lt/u′,(3)
a length scalel̂ proportional to the integral turbulence scale, that is, l̂=lt,(4)
a reference turbulent diffusion coefficientDt0=ltu′ (lt is chosen in such a way that the proportionality coefficient between Dt0 and ltu′ is unity),(5)
a flame velocity scaleDt0/t̂=u′.Thus, one has(8)R∂c̃∂t*+(Ru*+NB2αv∂∂c̃(Rf))∂c̃∂x*=∂∂x*(RD∂c̃∂x*)+w¯*,
where x*=x/l̂, u*=ũ/û, t*=t/t̂, 𝒟=Dt/Dt0, R=ρ¯/ρu=1/1+τc̃, and w¯*=w¯(t̂/ρu). In the following and whenever unambiguous, we shall drop superscript *. So, with that convention, (8) reads as(9)R∂c̃∂t+(Ru+NB2αv∂∂c̃(Rf))∂c̃∂x=∂∂x(RD∂c̃∂x)+w¯,
and the nondimensional form of the continuity equation (1) is given by(10)∂R∂t+∂Ru∂x=0.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 ∂ρ¯f(c̃)/∂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 consumptionm. and the turbulent flame speed St such as m.=ρuSt, the preceding continuity and progress-variable nondimensional equations are written as(11)Ru=m.ρuu′=ρuStρuu′=ρbubρuu′=Λ,[Λ+NB2αvddc̃(Rf)]dc̃dx=ddx(RDdc̃dx)+w¯,
where Λ=St/u′.We can rewrite (11) as a first-order differential equation for P=(R𝒟/Λ)(dc̃/dx) on the c̃-domain [0,1], namely (12)[1+NB2αvΛddc̃(Rf)]P=PP′+w¯RDΛ2.
Introducing N̂B=NB2αv and Ω=w¯R𝒟, (12) is expressed as(13)PP′=[1+N̂BΛ(Rf)′]P-Ω1Λ2,
where ()′ denotes the derivation with respect to c̃.


## 3. Analysis of theP-Equation in the Vicinity of the Singular Points

Now, our primary objective is to determine the domain of existence ofΛ. In principle, we need for that to analyse the trajectories in the (P,c̃) 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 associatedΛ domain by examining the characteristic equations of (13) at both sides of the brush. The implication of the relative position of P and F=(N̂B/Λ)Rf for the P-trajectories when they are tangent to the characteristics lines is also analyzed. We denote F0′=dF(0)/dc̃=N̂B/Λ(Rf)′(0)=(N̂B/Λ)f′(0)=α and F1′=-dF(1)/dc̃=-N̂B/Λ(Rf)′(1)=-(1/(τ+1))(N̂B/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 by(14)s2-s(1+α)+1Λ2Ω′(0)=0.
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) FGTF+FCGTF<0 (the gradient contribution overcomes the counter-gradient one) and (2) the reverse situation prevails, that is, FGTF+FCGTF>0.(1)
Gradient transport prevails, that is,FGTF+FCGTF<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,FGTF+FCGTF>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(15)Λ2>f1Ω′(0).
Thus, for case 1 we have (16)f1Ω′(0)<Λ2<f2Ω′(0),α<1,
and for case 2(17)f1Ω′(0)<Λ2<f2Ω′(0),α>1,Λ2>f2Ω′(0),α>0.
Using the definitions of f1(α)=4/(1+α)2, f2(α)=1/α and α=(N̂B/Λ)f′(0), for case 1, one has (18)2Ω′(0)-N̂Bf′(0)<Λ<Ω′(0)N̂Bf′(0),Λ>N̂Bf′(0)
or in equivalent form (19)2Ω′(0)-N̂Bf′(0)<Λ<Ω′(0)N̂Bf′(0),N̂Bf′(0)<Ω′(0).
For case 2, we have (20)2Ω′(0)-N̂Bf′(0)<Λ<Ω′(0)N̂Bf′(0),N̂Bf′(0)>Ω′(0),Λ>Ω′(0)N̂Bf′(0),N̂B>0.
Introducing the variables (21)Λ̃=ΛΩ′(0),ÑB=N̂Bf′(0)Ω′(0),
we can rewrite for case 1 (22)2-ÑB<Λ̃<1ÑB,ÑB<1,
and for case 2 (23)2-ÑB<Λ̃<1ÑB,ÑB>1,Λ̃>1ÑB,ÑB>0.
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 (Λ̃,ÑB) 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-c̃) and F=β(1-c̃) (s>0 and β>0), the characteristic equation associated with (13) is given by(24)s2+s(1-β)+1Λ2Ωc̃′(1)=0.Figure 1
Domain of existence in the (Λ̃,ÑB) plane of gradient or counter-gradient total turbulent flux at the reactants side.The discriminantΔ1- 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,FGTF+FCGTF<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,FGTF+FCGTF>0⇔P<F. This implies that β>s3 or equivalently Λ2>(1/β)|Ω′(1)|.Using the relation(25)βα=11+τ|f′(1)|f′(0),
we transform the above inequalities into the following form:for case 1 (GTF)(26)Λ̃<1ÑBfB,for case 2 (CGTF)(27)Λ̃>1ÑBfB,
where (28)fB=(τ+1)f′(0)|f′(1)||Ω′(1)|Ω′(0).
Figures 2(a) and 2(b) present the domain of GTF and CGTF obtained from the analysis in the vicinity c̃→1 and for fB>1 and fB<1, respectively. In the latter case, two remarkable values ÑB1 and ÑB2of ÑB are introduced. They are defined by ÑB1=(1-1-fB)<1 and ÑB2=(1+1-fB)>1.Domain of existence in the (Λ̃,ÑB) plane of gradient or counter-gradient total turbulent flux at the products side (a) case fB>1 and (b) case fB<1.
(a)(b)There exists a maximum of four possible configurations, listed in Table1, 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 1ConfigurationG/GG/CGCG/GCG/CGReactants sideGGCGCGProducts sideGCGGCGLet us consider first the casefB>1. Analysis of inequalities (22)–(23) and (25)–(27) gives the following results.(i)
G/G configuration:(29)2-NB̃<Λ̃<1NB̃,NB̃<1.(ii)
G/CG configuration is absent.(iii)
CG/G configuration:(30){{2-NB̃<Λ̃<1NB̃,NB̃>1Λ̃>1NB̃,NB̃>0}sumoftwosetsΛ̃<1NB̃fB,NB̃>0}product.(iv)
CG/CG Configuration:(31){{2-NB̃<Λ̃<1NB̃,NB̃>1Λ̃>1NB̃,NB̃>0}sumoftwosetsΛ̃>1NB̃fB,NB̃>0}product.Domains of the above three possible configurations are given in Figure3. Now for the case fB<1, all four configurations are possible, namely,(i)
G/G configuration:(32)2-NB̃<Λ̃<fBNB̃for0<NB̃<NB̃1withNB̃1=1-1-fB<1.
Consequently, in this interval [0,NB̃1], we have Λ̃>1.(ii)
G/CG configuration:Figure 3
The domain of the three different possibilities for the turbulent flame brush structure in the plane (Λ̃,ÑB) for  fB>1.fB/NB̃<Λ̃<1/NB̃ for 0<NB̃<NB̃1 and 2-NB̃<Λ̃<1/NB̃ for NB̃1<NB̃<1 and again we have Λ̃>1.(iii)
CG/G configuration:max(0;2-NB̃)<Λ̃<fB/NB̃ for NB̃>NB̃2 with NB̃2=1+1-fB>1. In that case, we have Λ̃<1.(iv)
CG/CG configuration:(33)Λ̃>1NB̃for0<NB̃<1,Λ̃>2-NB̃for1≤NB̃<NB̃2,Λ̃>fBNB̃forNB̃>NB̃2.The different regions of the (Λ̃,ÑB) plane corresponding to these four configurations are presented in Figure 4. We note that the case fB<1 is realised for the classical eddy-break-up model where (34)w¯*=c̃(1-c̃)1+τc̃,
and if Dt=Dt0=const,   f(c̃)=c̃(1-c̃)-Veynante et al. case [6]. In that case, one has Ω(c̃)=c̃(1-c̃)/(1+τc̃)2 and consequently Ω′(0)=1 and Ω′(1)=-1/(1+τ)2. Thus, (35)fB=11+τ<1.
The case fB>1 is obtained if (36)w¯*=c̃(1-c̃)(1+ϰc̃),
where ϰ is some positive contant. So we have (37)Ω(c̃)=c̃(1-c̃)(1+ϰc̃)(1+τc̃),Ω′(0)=1,Ω′(1)=-1+ϰ1+τ.
For Veynante et al. [6], f(c̃)=c̃(1-c̃) and so (38)fB=(1+τ)1+ϰ1+τ=1+ϰ>1wheneverϰ>0.Figure 4
The domain of the three different possibilities for the turbulent flame brush structure in the plane (Λ̃,ÑB) 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 fB>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¯=c̃(1-c̃). In such a case, we have fB=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 NB=0. But we will consider here NB like a parameter not linked to τ. In such a situation, by denoting K=2N̂B and cw=Λ+N̂B, (11) reduced to that analyzed by Murray [16] to determine the wave speed cw, namely, (39)d2c̃dx2-(cw-Kc̃)dc̃dx+c̃(1-c̃)=0.
Murray [16] showed that travelling wave solutions exist for all cw≥cw(K) where (40)cw(K)={2ifK<2,k2+2kifK⩾2,
or equivalently with our notations(41)Λ={2-N̂BifN̂B<1,1N̂BifN̂B⩾1.
Thus, it appears that in that particular case, for N̂B⩾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 parameterfB (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.



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*Source: 821358-2011-07-09.xml*