Premixed turbulent combustion
2.9 Equations for the Mean and the Variance of G
To obtain a formulation that is consistent with the well-established use of Favre averages in turbulent combustion, we split G and the velocity vectorv into
2.9 Equations for the mean and the variance of G 115 Favre means and fluctuations:
G= ˜G + G, v = ˜v + v. (2.138) Here ˜G and ˜v are at first viewed as unconditional averages. Since in a turbulent flame G was interpreted as the scalar distance between the instantaneous and the mean flame front, evaluated at G(x, t) = G0, the Favre mean ˜G= ρG/ ¯ρ represents the Favre average of that distance. If G(x, t) = G0 is defined to lie in the unburnt mixture immediately ahead of the thin flame structure, as is often assumed for the corrugated flamelets regime, the density at G(x, t) = G0
is constant and equal toρu. Similarly, if it is an iso-temperature surface, as is assumed for the thin reaction zones regime, changes of the density along that surface are expected to be small. In both cases the Favre average ˜G is approximately equal to the conventional mean value ¯G. Using Favre averages rather than conventional averages, which might have appeared more appropri-ate for a nonconserved quantity like the scalar G, therefore has no practical consequences.
The Favre mean velocity ˜v is an unconditional average, but in the end of the analysis of the analysis its conditional counterpart is needed. The conditional mean velocity can be measured by taking averages over the entire flame brush, conditioned at the location of instantaneous flame front. It will be modeled in a similar way as the conditional variance by assuming that it is equal to the unconditional value at the mean flame front position.
Introducing (2.138) into (2.121) leads to the following equation for the Favre mean value of G:
ρ¯∂ ˜G
∂t + ¯ρ ˜v · ∇ ˜G + ∇ · ( ¯ρ
vG)= ρs0Lσ − (ρ D) κσ.¯ (2.139)
The last term in this equation is proportional to the molecular diffusivity D. In the large Reynolds number limit it is expected to be small compared to the other terms in the equation. Since turbulent modeling is consistent in that limit only, we will neglect this term in the following. We will find its turbulent equivalent, however, below.
An equation for the variance G2 may be derived by subtracting (2.139) from (2.121) to obtain an equation for G. After multiplying this by 2Gand averaging one obtains
ρ¯∂ G2
∂t + ¯ρ ˜v·∇ G2+∇ ·( ¯ρ
vG2)= −2 ¯ρvG·∇ ˜G − ¯ρ ˜ω− ¯ρ ˜χ −(ρ D) σ.(2.140) Details of the derivation may be found in Peters (1999). As noted before, the condition ˜G(x, t) = G0now defines the location of the mean flame front, while the Favre variance G2represents the turbulent flame brush thickness.
Let us at first consider the sink terms in (2.140). The Favre kinematic restora-tion ˜ω corresponds to ¯ω defined before and is important in the corrugated flamelets regime only. It is defined as
ω = −2˜ ρsL0
Gσ / ¯ρ (2.141)
and is to be modeled similarly to (2.135). The Favre scalar dissipation is defined as
χ = 2(ρ D) ˜ (∇G)2/ ¯ρ (2.142) and is to be modeled in the thin reaction zone regime similarly to (2.137).
The last term (ρ D) σ in (2.140) represents a curvature term (cf. Peters, 1999). Since it is proportional to the molecular diffusivity, it is small compared to the other terms in (2.140) in the limit of large Reynolds numbers and will not be considered further.
Since closure of the sink terms ˜ω and ˜χ is different in the two regimes, a combined expression must be sought if (2.140) is to be used for a general model of premixed turbulent combustion. The order of magnitude analysis performed on (2.118) shows that the dominant term in the corrugated flamelets regime is kinematic restoration whereas in the thin reaction zones regime it is scalar dissipation. In Peters (1999) the sum of both sink terms ˜ω and ˜χ, valid for both regimes, was modeled as
ω + ˜χ = c˜ s
ε˜
˜kG2 , (2.143)
where csis a modeling constant. Peters (1999) predicted 2.0 for this constant, which was confirmed by Wenzel (2000) on the basis of DNS calculations. If gas expansion effects are taken into account, a smaller value appears to be appropriate, as will be discussed in Section 2.12.
Let us now consider the modeling of the correlation
vG appearing in both (2.139) and (2.140). The last term∇ · ( ¯ρvG) on the l.h.s of (2.139) is a turbulent transport term. A classical gradient transport approximation cannot be used for this term, because this would lead to an elliptic equation for ˜G, which is inconsistent with the mathematical character of the G-equation. To obtain for (2.139) the same mathematical form as that of (2.121) the transport term must be modeled as a curvature term. In fact, a transformation similar to (2.111) shows that the second-order elliptic operator that would result from a gradient flux approximation can be split into a normal diffusion and a curvature term:−∇ · ( ¯ρv
G)= ∇ · ( ¯ρ Dt∇ ˜G) = ˜n · ∇( ¯ρ Dt˜n· ∇ ˜G) − ¯ρ Dtκ|∇ ˜G|.˜ (2.144)2.9 Equations for the mean and the variance of G 117 Here Dtis the turbulent diffusivity and ˜n and ˜κ are defined as in (2.54) and (2.64), respectively, but with ˜G instead of G. Since diffusion normal to G-isolines is already contained in the burning velocity and therefore does not appear in the instantaneous G-equation, it cannot appear in the equation for ˜G for the same reason. We therefore must remove the normal diffusion term in (2.144) to obtain
−∇ · ( ¯ρ
vG)= − ¯ρ Dtκ|∇ ˜G|.˜ (2.145) This curvature term avoids the formation of cusps of the mean flame front.It is worth noting that by imposing|∇ ˜G| = 1 by reinitialization of the ˜G-field outside ˜G(x, t) = G0, normal diffusion automatically vanishes and the formal removal of the normal diffusion term would not have been necessary.
For the turbulent production term in (2.140) classical gradient transport modeling is appropriate since second-order derivatives are not involved, and we have
−v
G· ∇ ˜G = Dt(∇ ˜G)2. (2.146) For the same reason as above the turbulent transport term in the variance Equation (2.140) must be modeled in a way to avoid turbulent diffusion normal to the mean flame front. It has been noted in Section 2.8 that the conditional vari-ance should not depend on the coordinate normal to the flame front. However, since transport in the tangential direction is permitted, we replace the turbulent transport term in (2.140) by a gradient transport approximation in tangential direction only:−∇( ¯ρ
vG2)= ∇||· ( ¯ρ Dt∇||G2). (2.147) The tangential diffusion operator may be calculated by subtracting the normal diffusion ˜n· ∇( ¯ρ Dt˜n· ∇ G2) from the diffusive operator of G2.The equation for ˜G, (2.139), models the propagation of the mean turbulent premixed flame front. As in laminar combustion, where the mass flow rate through a steady one-dimensional flame determines the laminar burning veloc-ity s0L, as defined in (2.4), we will consider the case of a steady planar turbulent flame, in order to determine the turbulent burning velocity s0T, assuming that it is a quantity that depends on local mean quantities only. Whether this assumption is justified will be discussed at the end of Section 2.12 in detail. If it is, the mass flow rate through the flame is constant and equal to ( ¯ρs0T). One then obtains from (2.139) the equation
ρs¯ 0T
∂ ˜G
∂x = ρsL0
σ .¯ (2.148)
Here, as before, x is the coordinate normal to the mean turbulent flame surface pointing toward the burnt gas. Similarly to (2.60) one has for the mean turbulent flame
d x= d ˜G
|∇ ˜G|. (2.149)
The gradient d ˜G/dx in (2.148) may therefore be replaced by |∇ ˜G|, which leads to
ρs¯ T0
|∇ ˜G| = ρs0L
σ.¯ (2.150)
This equation relates the turbulent burning velocity s0Tto the mean gradient ¯σ.
Both are conditional quantities to be evaluated at the mean flame front. Using this and the closure assumptions derived above, Equations (2.139) and (2.140) for ˜G and G2become
ρ¯∂ ˜G
∂t + ¯ρ ˜v · ∇ ˜G = ρs¯ T0
|∇ ˜G| − ¯ρ Dtκ|∇ ˜G|,˜ (2.151)
ρ¯∂ G2
∂t + ¯ρ ˜v · ∇ G2= ∇||· ( ¯ρ Dt∇||G2)+ 2 ¯ρ Dt(∇ ˜G)2− csρ¯ε˜
˜kG2. (2.152) It is easily seen that (2.151) has the same form as (2.121) and therefore shares its mathematical properties. It also is valid at ˜G(x, t) = G0 only, while the solution outside of that surface depends on the ansatz for ˜G(x, t) that is intro-duced. The same argument holds for (2.152) since the conditional variance is a property defined at the flame front. The solution of that equation will provide the conditional value (G2)0at the mean flame surface ˜G(x, t) = G0. Following (2.128) and (2.132), we see that its square root is a measure of the flame brush thicknessF,t, which for an arbitrary value of|∇ ˜G| at the front, will be defined as
F,t = (G2(x, t))1/2
|∇ ˜G|
˜
G=G0
. (2.153)
To solve (2.151), a model for the turbulent burning velocity sT0 must be provided. A first step would be to use empirical correlations from the literature.
Alternatively, since sT0 is related to ¯σ by (2.150), a modeled balance equation for the mean gradient ¯σ will be derived. It will turn out that ¯σ represents the flame surface area ratio, which is a conditional mean quantity to be calculated from its balance equation at ˜G(x, t) = G0.