Numerical Simulation of the Dynamics of Turbulent Swirling Flames

Turbulent Reacting Flows
2.4.2 Fundamental Transport Equations for LES Reacting Flows
The transport equations for LES are obtained by filtering the instantaneous
transport equations of conservation of mass, momentum, energy and species.
A detailed description of the instantaneous transport equations for reacting
flow is shown in [115, 152, 214, 219]. The filtered transport equations are [22,
152]:
• Conservation of mass
∂ρ
∂t + ∂ρũ i
∂x i
= 0, (2.32)
• Conservation of momentum
∂ρũ i
∂t
+ ∂ρũ i ũ j
∂x j
= − ∂pδ i j
∂x j
+ ∂τ i j
∂x j
+ ∂τ i j t
∂x j
, (2.33)
• Conservation of energy
∂ρẼ
∂t
+ ∂ρũ j Ẽ
∂x j
= − ∂[u i (pδ i j − τ i j ) + q j + q j t ]
∂x j
+ ˙ω T , (2.34)
• Species mass fraction
∂ρỸ k
∂t
+ ∂ρũ j Ỹk
∂x j
= ∂[J j,k + J j,k
t
]
∂x j
+ ˙ω k . (2.35)
where τ i j , q j , J j,k , ˙ω T and ˙ω k are the viscous stress tensor, the heat flux, the
diffusive species flux, the heat release and the reaction rate of the k th species.
The superscript t indicates subgrid turbulent terms which are modeled.
Additionally, the equation of state for an ideal gas mixture is defined by:
p = ρRT, (2.36)
R =
R ,
W mix
(2.37)
where R is the universal gas constant and W mix is the molecular weight of the
mixture.
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