Numerical Simulation of the Dynamics of Turbulent Swirling Flames

Turbulent Reacting Flows
2.4.2.1 Modeling of Subgrid Terms
The subgrid turbulent terms in the transport equations need to be modeled.
Most subgrid models are based on the eddy-viscosity assumption
(Boussinesq’s hypothesis). The subgrid stress tensor τ i j t is modeled by:
τ i j t = −ρ ( u i u j − ũ i ũ j
)
, (2.47)
= 2ρν t
(
˜S i j − 1 3 ˜S kk δ i j
). (2.48)
where ν t is the SGS turbulent viscosity. Some models proposed for ν t are
shown in the next section.
The other terms as the subgrid diffusive and heat flux are modeled by:
• The Subgrid scale diffusive species flux:
where,
t ( )
J j,k = ρ u i Y k − ũ i Ỹ k , (2.49)
(
)
= −ρ D t ∂Ỹ k
c,t
k
− Ỹ k Ṽ i . (2.50)
∂x j
D t k = ν t
S t , (2.51)
c,k
Ṽ i
c,t
=
N∑
µ t
k=1 ρS t c,k
∂Ỹ k
∂x j
. (2.52)
S t c,k
is the turbulent Schmid number equal to 0.6 for all species [22].
• The Subgrid heat flux:
q t j = ρ ( ũ i E − ũ i Ẽ ) , (2.53)
∂ ˜T N∑
= −λ t + J t j,k
˜hs,k .
∂x j
(2.54)
k=1
where,
λ t = µ tC p
P t r
(2.55)
The turbulent Prandtl number P t r
is equal to 0.6 [22].
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