Impact of fuel supply impedance and fuel staging on gas turbine ...

4.2 Theoretical <strong>and</strong> numerical combustion
4.2.2 Practical premixed combustion model
Similar to Eqn. (4.6), (4.7), (4.8) a Reynolds averaged equation <strong>of</strong> the mixture
fraction <strong>and</strong> the weighted reaction progress can be derived. For the weighted
reaction progress this relation yields:
∂( ¯ρ ˜F w )
+ ∂( ¯ρ ˜F w ũ i )
∂t ∂x i
(
2
= ∂ [(
ρ D+ µ ) ]
t ∂˜F w
+
∂x i Sc c,t ∂x i
ρ D+ µ t
Sc c,t
)( ∂ ˜Z M
∂x i
)
∂˜c
− ˜Z M ¯˙ω c . (4.21)
∂x i
The term − ¯ρ c ′′ u ′′
i
was replaced analogous to Eqn. (4.11) by µ t
Sc c,t
, where Sc c,t
denotes the turbulent Schmidt number, which is set by default to 0.9 [5] 4 . The
source term ¯˙ω c is defined as:
¯˙ω c = S¯
c − ∂ (
ρ D ∂˜c )
. (4.22)
∂x i ∂x i
To solve Eqn. (4.21), the turbulent flame speed closure (TFC) model is used for
the unknown averaged reaction source term S¯
c . The original idea <strong>of</strong> the TFCmodel
was presented by Zimont [138], whereas the complete formulation <strong>of</strong>
the model was described by Zimont <strong>and</strong> Lipatnikov [136]. Furthermore, the
TFC-model was improved <strong>and</strong> extended by Zimont et al. [137] <strong>and</strong> Polifke et
al. [97].
The approach is based on the idea to represent the source term as a spatial
progress <strong>of</strong> the flame front, which can be described as a function <strong>of</strong> the turbulent
flame speed s t :
¯ S c = ρ u s t |∇˜c|. (4.23)
|∇˜c| is the magnitude <strong>of</strong> the gradient <strong>of</strong> the progress variable <strong>and</strong> ρ u is the
density <strong>of</strong> the unburnt gas. This approach has the advantage that, for a given
4 Again, a unity Lewis number is assumed. In this case it follows that the Pr<strong>and</strong>tl number Pr = Sc.
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