Flamelet Generated Manifold

The Flamelet Generated Manifold (FGM) model is different from the Chemical Equilibrium (CE) and Steady Laminar Flamelet (SLF) models since a progress variable which represents the reaction progress is included to parameterize the thermo-chemistry. The FGM model is appropriate for modeling premixed and partially-premixed flames.

In the current implementation, either a 0D constant pressure reactor or a 1D premixed strained or freely propagating flamelet can be used to generate the manifolds. The thermo-chemistry is parameterized by mixture fraction, heat loss ratio (same definition as in CE), and progress variable.

Unnormalized Progress Variable

The FGM model solves a transport equation for an unnormalized progress variable, $y$ .

The unnormalized progress variable is defined as:

Figure 1. EQUATION_DISPLAY

y = Σ (W_{k} Y_{k})

(3532)

where $W_{k}$ is the $k$ th species weight and $Y_{k}$ is the mass fraction of the $k$ th species.

The $y$ Favre averaged transport equation can be written:

Figure 2. EQUATION_DISPLAY

\frac{\partial ρ y}{\partial t} + \nabla \cdot (ρ u y) - \nabla \cdot (Γ_{y} \nabla y) = {\dot{ω}}_{y}

(3533)

Note	Overbars (for RANS averaging) and overtildes (for Favre averaging) are excluded for clarity.

where

{\dot{ω}}_{y}

is the unnormalized progress variable source term and

Γ_{y}

is the diffusivity which is computed from the material properties:

Figure 3. EQUATION_DISPLAY

Γ_{y} = ρ D_{l a m} + \frac{μ_{t}}{σ_{t, y}}

(3534)

$D_{l a m}$ is the laminar diffusivity.
$μ_{t}$ is the turbulent viscosity.
$σ_{t, y}$ is the turbulent Schmidt number for the unnormalized progress variable.

The progress variable is then defined as:

Figure 4. EQUATION_DISPLAY

c = \frac{y - y_{u}}{y_{b} - y_{u}}

(3535)

Where:

$y$ is the unnormalized progress variable of the mixture Eqn. (3532).
$y_{u}$ is the unnormalized progress variable at the initial unburnt state.
$y_{b}$ is the unnormalized progress variable at the burnt (equilibrium) state.

Unnormalized Progress Variable Variance

There are two options available to compute the Unnormalized Progress Variable Variance:

Algebraic Relationship
This option computes the unnormalized progress variable variance using:
Figure 5. EQUATION_DISPLAY

$y_{v a r} = c_{v} Δ^{2} (\nabla {y_{m e a n})}^{2}$

(3536)

where,

$c_{v}$ = model constant

$Δ$ = mesh size

The algebraic relationship option is appropriate for Large Eddy Simulations (LES).

Transport Equation
This option computes the unnormalized progress variable variance using a transport equation:

Figure 6. EQUATION_DISPLAY

$\frac{\partial}{\partial t} (ρ y_{v a r}) + \nabla • (ρ U y_{v a r} - Γ_{y} \nabla y_{v a r}) = \frac{2 μ_{t}}{σ_{t, y_{var}}} t r {(\nabla y)}^{2} - C_{d} ρ \frac{ε}{k} y_{v a r}$

(3537)

where
Figure 7. EQUATION_DISPLAY

$y_{v a r} = \bar{{(y - y_{m e a n})}^{2}}$

(3538)

$σ_{t, y_{var}}$ = turbulent Schmidt number for the progress variable variance

$C_{d}$ = dissipation constant

Progress Variable Source

Different source terms are used for the term

{\dot{ω}}_{y}

in Eqn. (3533), depending on the option that is selected for the progress variable source:

FGM Kinetic Rate
The source term, ${\dot{ω}}_{y}$ in Eqn. (3533) is calculated from the chemical kinetic reaction rate, which is interpolated from the FGM table.
Coherent Flame Model (CFM)
The source term, ${\dot{ω}}_{y, c f m}$ in Eqn. (3533) is calculated based on the CFM flame propagation method Eqn. (3555).
Turbulent Flame Closure (TFC)
The source term, ${\dot{ω}}_{y, t f c}$ in Eqn. (3533) is calculated based on the TFC flame propagation method Eqn. (3556).

Tabulation for FGM

The independent variables are:

Mixture Fraction $Z$
Heat Loss Ratio $γ$
Progress Variable $c$

Governing Equations for 0D Ignition Reactor Type

During the tabulation process, the following equations are solved for each

Z

and

γ

Figure 8. EQUATION_DISPLAY

{\begin{matrix} \frac{\partial Y_{i}}{\partial t} = R_{i, k i n} \\ \frac{\partial T}{\partial t} = - Σ_{i = 1}^{N} \frac{h_{i} R_{i, k i n}}{C_{p}} \end{matrix}

(3539)

Each point at physical time instant

t

is converted to the corresponding progress variable

c

from Eqn. (3535).

Governing Equations for 1D Premixed Strained Reactor Type

For 1D premixed flamelets, the following opposed-flow premixed equations are solved in progress variable space at a fixed mixture fraction, see Eqn. (3535).

The equations for an opposed-flow strained premixed flame in reaction progress space, assuming unity Lewis number, are:

Figure 10. EQUATION_DISPLAY

\frac{\partial Y_{i}}{\partial τ} = χ_{c} \frac{\partial^{2} Y_{i}}{\partial c^{2}} + \frac{{\dot{ω}}_{i}}{ρ} - \frac{\partial Y_{i}}{\partial c} \frac{{\dot{ω}}_{c}}{ρ}

(3541)

Figure 11. EQUATION_DISPLAY

\frac{\partial T}{\partial τ} = \frac{χ_{c}}{C_{p}} (\frac{\partial C_{p}}{\partial c} + Σ_{i = 1}^{N} C_{p_{i}} \frac{\partial Y_{i}}{\partial c}) \frac{\partial T}{\partial c} + χ_{c} \frac{\partial^{2} T}{\partial c^{2}} - \frac{1}{ρ C_{p}} Σ_{i = 1}^{N} h_{i} {\dot{ω}}_{i} - \frac{\partial T}{\partial c} \frac{{\dot{ω}}_{c}}{ρ}

(3542)

where

τ

is pseudo-time,

T

is temperature,

Y_{i}

is the mass fraction of the

i t h

species,

C_{p}

is the heat capacity, and

ρ

is the density.

h_{i}

and

{\dot{ω}}_{i}

are the enthalpy and the reaction rate of the

i t h

species, respectively.

{\dot{ω}}_{c}

is the progress variable reaction rate, calculated from Eqn. (3532) and Eqn. (3535). The equations are integrated in pseudo-time until the unsteady terms on the left hand side are small. The residual is defined as the sum of the normalized absolute values of the unsteady terms over all points, for temperature and post-processing species. The equations are considered converged when this residual is less than the specified Flamelet Convergence Tolerance parameter.

The progress variable dissipation rate—that is calculated at any

Z

and

c

in the manifold for 1D premixed FGM tables—is defined as:

Figure 12. EQUATION_DISPLAY

χ_{c} (Z, c) = χ_{Z} \exp [- 2 {({e r f c}^{- 1} (2 c))}^{2}]

(3543)

where

{e r f c}^{- 1}

is the inverse of the complementary error function,

χ_{Z}

is the scalar dissipation function in the

Z

direction, which is given as:

Figure 13. EQUATION_DISPLAY

χ_{Z} = χ_{Z, s t o} \exp [- 2 {({e r f c}^{- 1} (\frac{Z}{Z_{s t o}}))}^{2}]

(3544)

where

χ_{Z, s t o}

is the specified maximum dissipation rate of the progress variable at stoichiometric conditions.

Governing Equations for 1D Premixed Freely Propagating Reactor Type

The equations for a freely propagating premixed flame in progress variable space, assuming mixture averaged diffusion are [765]:

Figure 14. EQUATION_DISPLAY

\frac{\partial Y_{i}}{\partial τ} = g_{c}^{2} D_{i} \frac{\partial^{2} Y_{i}}{\partial c^{2}} + \frac{g_{c}}{ρ} \frac{\partial}{\partial_{c}} (ρ g_{c} (D_{i} + Y_{c} V_{c})) \frac{\partial Y_{i}}{\partial c} - \frac{g_{c}}{ρ} \frac{\partial}{\partial_{c}} (ρ g_{c} Y_{i} V^{*}) + \frac{{\dot{ω}}_{i}}{ρ} - \frac{\partial Y_{i}}{\partial c} \frac{{\dot{ω}}_{c}}{ρ}

(3545)

Figure 15. EQUATION_DISPLAY

\frac{\partial T}{\partial τ} = \frac{g_{c}}{ρ} (\frac{1}{C_{p}} \frac{\partial}{\partial c} (g_{c} λ) + \frac{\partial}{\partial c} (ρ g_{c} Y_{c} V_{c}) - \frac{ρ g_{c}}{C_{p}} \underset{i}{Σ} C_{p, i} (- D_{i} \frac{\partial Y_{i}}{\partial c} + Y_{i} V^{*})) \frac{\partial T}{\partial c} + \frac{g_{c}^{2} λ}{ρ C_{p}} \frac{\partial^{2} T}{\partial c^{2}} - \frac{1}{ρ C_{p}} \underset{i}{Σ} h_{i} {\dot{ω}}_{i} - \frac{\partial T}{\partial c} \frac{{\dot{ω}}_{c}}{ρ}

(3546)

Figure 16. EQUATION_DISPLAY

\frac{\partial g_{c}}{\partial τ} = - \frac{g_{c}^{2}}{ρ} \frac{\partial^{2}}{\partial c^{2}} (ρ g_{c} Y_{c} V_{c}) + \frac{g_{c}}{ρ} \frac{\partial {\dot{ω}}_{c}}{\partial_{c}} - \frac{{\dot{ω}}_{c}}{ρ} \frac{\partial g_{c}}{\partial c}

(3547)

The laminar flame speed is determined from the solution by:

Figure 17. EQUATION_DISPLAY

S_{u} = \frac{1}{ρ_{0}} (- \frac{\partial (g_{c} ρ Y_{c} V_{c})}{\partial c} + \frac{{\dot{ω}}_{c}}{g_{c}})

(3548)

where

g_{c}

is the gradient of progress in physical space,

D_{i}

is the diffusion coefficient species

i

λ

is the thermal conductivity, and

ρ_{0}

is the unburnt gas density.

Y_{c} {\tilde{V}}_{c}

is the diffusion flux for progress variable:

Figure 18. EQUATION_DISPLAY

Y_{c} V_{c} = Σ_{i} W_{i} (- D_{i} \frac{\partial Y_{i}}{\partial c} + Y_{i} V^{*})

(3549)

W_{i}

represents the species weight used to calculate the progress variable, and

V^{*}

is the correction term for species diffusion:

Figure 19. EQUATION_DISPLAY

V^{*} = Σ_{i} D_{i} \frac{\partial Y_{i}}{\partial c}

(3550)

Optimal Progress Variable Weights

During tabulation when using optimal species weights, a generic optimization algorithm is used to minimize gradients in reaction progress variable space. The progress variable is defined to be a linear combination of CO₂, CO, H₂O, and H₂ mass fractions. The species weights are optimized for the cost function and constraints described below. The cost function is defined as the sum of the maximum gradient of each dependent variable as function of progress variable, for all table points in mixture fraction, mixture fraction variance, and heat loss ratio space:

Figure 20. EQUATION_DISPLAY

C = \underset{t p}{Σ} \underset{k}{Σ} C_{t p, k}

(3551)

where

t p

denotes all table points in mixture fraction, mixture fraction variance, and heat loss ratio space, and

k

denotes a dependent variable. The dependent variables include temperature, progress variable source term, and all tabulated species.

C_{t p, k}

is the cost for a given dependent variable and table point, defined as:

Figure 21. EQUATION_DISPLAY

C_{t p, k} = \frac{\max_{y, t p} | \frac{d ϕ_{k, t p} (y)}{d y} |}{ϕ_{k}^{\max} - ϕ_{k}^{\min}}

(3552)

where

ϕ_{k}

is the dependant variable,

y

is the unnormalized progress variable.

ϕ_{k}^{\max}

and

ϕ_{k}^{\min}

are the maximum and minimum values of

ϕ_{k}

in the full table. This global optimization yields lower derivatives in progress variable space, and thus lower sensitivity to errors in progress variable in the CFD simulations.

Integration for FGM

Turbulent fluctuations are modeled with statistically independent assumed shape Beta PDFs for and

c

, and a delta function PDF for enthalpy

γ

(heat loss ratio).

After integration, the FGM table has five independent variables:

Mixture Fraction $Z_{m e a n}$
Heat Loss Ratio $γ$
Progress Variable $c_{m e a n}$
Mixture Fraction Variance $Z_{var}$
Progress Variable Variance $c_{var}$

For each point,

Z_{m e a n}

γ

c_{m e a n}

Z_{var}

, and

c_{var}

, the averaged quantity

Q

(from density weighted averages) is determined by:

Figure 22. EQUATION_DISPLAY

Q (Z_{m e a n}, c_{m e a n}, γ, Z_{var}, c_{var}) = \int_{0}^{1} \int_{0}^{1} \int_{- 1}^{1} Q (Z, c, γ) P (Z, c, γ) d Z d c d γ = \int_{0}^{1} \int_{0}^{1} Q (Z, c, γ) P (Z, c) d Z d c = \int_{0}^{1} \int_{0}^{1} Q (Z, c, γ) P_{c} (c | Z) P_{Z} (Z) d Z d c

(3553)

here, a delta function is used for the PDF of the heat loss ratio.

P_{Z} (Z)

is a beta PDF that is parameterized by

Z_{m e a n}

and

Z_{var}

(see Beta Probability Density Function (PDF)).

P_{c} (c | Z)

is a beta PDF of

c

, conditioned on mixture fraction, that is parameterized by

c_{m e a n}

and

c_{var}