Flamelet presumed PDF models

Cook 1997: A laminar flamelet approach to subgrid-scale chemistry in turbulent flows

Reaction model

χ22yiξ2=m˙ρ

χ=χ0F(ξ)

F(ξ)=exp(2[erf1(2ξ1)]2)

We solve the flamelet equations for yi. Enthalpy is given by simple mixing:

h=hξ=0(1ξ)+hξ=1(ξ).

All other thermochemical quantites can then be computed from yi, h, and P. Denote these quantities, and yi, h generically as ϕ. We then have

ϕ=ϕ(ξ,χ0).

This can be considered the flamelet reaction model.

Mixing model

Average quantities are computed by convolving over the joint pdf:

ϕ¯=ϕ(ξ,χ0)P(ξ,χ0)dχ0dξ.

The joint pdf is modeled assuming independence between ξ and χ0:

P(ξ,χ0)P(ξ)P(χ0).

P(ξ) is modeled as a β-PDF: P(ξ)=Pβ(ξ;ξ¯,ξ2), which is parameterized by the mean and variance of ξ.

P(χ0) is modeled as a delta function: P(χ0)=δ(χ0χ0).

χ0 is computed by averaging χ=χ0F(ξ) above:

χ=χ0F(ξ)
χ¯=χ0F(ξ)P(ξ,χ0)dχ0dξ,
X=χ0F(ξ)Pβ(ξ)δ(χ0χ0)dχ0dξ,
X=χ0F(ξ)Pβ(ξ)dξ.

This gives

χ0=χ¯Pβ(ξ)dξ.

RANS

χ¯ is commonly modeled as

χ¯=cχϵ¯k¯ξ2,

where cχ=2. See Pitsch 1998

LES

χ¯ may be modeled as

χ¯=ρ¯(D+Dt)|ξ¯|2

The turbulent diffusivity Dt may be computed using the Smagorinsky model.

Dissipation pdf

  • The scalar dissipation rate pdf is sometimes modeled as a lognormal distribution:
  • In this case, the distribution depends on the mean and variance.
  • See Heyl and Bockhorn 2001
    • They used a constant variance.

Flamelet Progress Variable (FPV) models

  • Instead of parameterizing the flamelets in terms of ξ and χ0, the FPV model parameterize in terms of ξ and a progress variable.
  • Selected references: