Large-eddy Simulation of Realistic Gas Turbine Combustors

2 Mathematical Formulation
We solve the variable density, low-Mach number, Navier-Stokes equations for the gas-phase.
The formulation is based on flamelet progress-variable (FPV) approach developed by Pierce
& Moin [5] for LES of non-premixed, turbulent combustion.
The liquid phase is treated in
the Lagrangian framework with efficient particle tracking scheme on unstructured grids, which
allows simulation of millions of independent droplet trajectories. A summary of the filtered
Eulerian/Lagrangian equations and subgrid models for unclosed terms and droplet dynamics is
given below.
2.1 Filtered LES Equations for Gas-Phase
The gas phase continuity, scalar, and momentum equations are,
∂ ( ρ g ũ j
)
∂x j
= − ∂ρ g
∂t + Ṡm (1)
( )
∂ ρ g ˜Z
∂t
( )
∂ ρ g ˜C
∂t
∂ ( ρ g ũ i
)
∂t
+
+
)
(
∂
(ρ g ˜Zũj
= ∂ ∂
ρ
∂x j ∂x g ˜α ˜Z
)
Z − ∂q Zj
+
j ∂x j ∂x ṠZ (2)
j
)
(
∂
(ρ g ˜Cũj
= ∂ ∂
ρ
∂x j ∂x g ˜α ˜C
)
C − ∂q Cj
+ ˙ω C (3)
j ∂x j ∂x j
+ ∂ ( ρ g ũ i ũ j
)
∂x j
= − ∂p + ∂(2µ ˜S ij )
− ∂q ij
+
∂x i ∂x j ∂x Ṡi (4)
j
where
˜S ij = 1 ( ∂ũi
+ ∂ũ )
j
2 ∂u j ∂u i
− 1 3 δ ∂ũ k
ij . (5)
∂x k
Here ρ g is the gas-phase density, u j the velocity vector, p the pressure, µ the dynamic
viscosity, δ ij the Kronecker symbol, Z the mixture fraction, C the progress variable, α Z and α C
the scalar diffusivities, and ˙ω C the source term due to chemical reactions. The additional term
4