Heat release modeling

FPVA-based model

V. Terrapon and H. Pitsch

1Stanford PSAAP Center - Working draft

Why is turbulent combustion challenging?

QUALITATIVE

A large range of scales is involved!

Typical engineering application

Turbulent flow

Chemistry

10-9 10-6 10-3 100 103

Integral length scale

Reaction zone thickness

Kolmogorov length scale

Reaction time scale

[m] [s]

➔ Equations are very stiff➔ DNS is usually not possible

(LES/RANS)

2Stanford PSAAP Center - Working draft

Closure is requiredSource term is highly non-linear

Decaying isotropic turbulence

€

∂ρ ̃ Y i∂t

+∇ ρuYi( ) =∇ ρDi∇Yi( ) + ˙ m i

Average species transport equation:

€

∂ρ ̃ Y i∂t

= ˙ m i

The source term:

Average species transport equation simplifies:

Diffusive and convective terms can usually be treated like for the NS equations

€

˙ m i = W iν ij K fj X k[ ]ν kj

'

− Krj Xk[ ]ν kj

''

k =1

Nsp

∏k =1

N sp

∏ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

j =1

Nreac

∑

with

€

K fj = A fjTβ j exp −

E j

RT ⎛ ⎝ ⎜

⎞ ⎠ ⎟

Species mass fraction

DNS

From M. Ihme

Flamelets

From mean quantities

3Stanford PSAAP Center - Working draft

Tabulated chemistry as a potential solution

€

0 =ρχ2

∂2φ∂Z 2 + ˙ m φ

Transformation and asymptotic approximation leads to equations for the flame micro-structure similar to the flamelet equations

Flame strain is measured by the scalar dissipation rate

Mixture fraction

€

χ=2D∂Z∂x ⎛ ⎝ ⎜

⎞ ⎠ ⎟2

Steady diffusion flame solution for H2/air flame at χst =1

Scale separation allows us to pre-compute and tabulate chemistry as function of Z and χ

Zst

4Stanford PSAAP Center - Working draft

Diffusion flame regime

€

∂φ∂τ

= ˙ m φ

Auto-ignition regime

€

∂φ∂τ

=ρχ2

∂ 2φ∂Z 2 + ˙ m φ

Combustion model for high-speed flows based on tabulated chemistry

Assumptions Current combustion model•Species mass fractions independent of compressibility effects•Flame micro-structure

•Based on Flamelet Progress Variable Approach (FPVA)*

•Tabulated chemistry•Temperature computed from species mass fractions and energy (not from chemistry table)•3 additional transport equations for the mean and variance of the mixture fraction and for the progress variable

•Complex chemistry mechanism•Only few additional scalars to solve•Turbulence – chemistry interaction

Pros•Compressibility effects on chemistry not accounted for•Accuracy of ignition dynamics

Cons

+ -

* Pierce & Moin, 2004; Ihme et al., 20055

Stanford PSAAP Center - Working draft

Model algorithmTransported

variables

NSE

Turbulence

Combustion

Tabulated chemistry• Table lookup• Pre-computed

chemistry

Temperature• Newton

iteration• Not looked up

from table

All other quantities

• EOS • Mixing

rules

Chemistry does not account for compressibility

effects and viscous heating

6Stanford PSAAP Center - Working draft

Source term of progress variable C has a strong pressure dependence

•Dynamics of flame strongly dependent on pressure•Quadratic dependence (mostly 2 body interactions)•Rescaling of source term for progress variableSc(P) = Sc(Pref) P2/P2

ref

7Stanford PSAAP Center - Working draft

Rescaled source term as function of the progress variable

•Dynamics of flame strongly dependent on pressure•Quadratic dependence (mostly 2 body interactions)•Rescaling of source term for progress variableSc(P) = Sc(Pref) P2/P2

ref

8Stanford PSAAP Center - Working draft

Z = Zst = 0.028

P