Flame Propagation

The flame surface density (FSD)

Σ

transport equation for RANS simulations is written as:

图 1. EQUATION_DISPLAY

\begin{matrix} \frac{\partial Σ}{\partial t} + \nabla\cdot (v Σ) - \nabla\cdot [(D + \frac{μ_{t}}{S c}) \nabla (\frac{Σ}{ρ})] = & Σ [\min (C_{d i v u} \nabla\cdot v, 0) + F_{Σ_{p r o d}} (\max (C_{d i v u} \nabla\cdot v, 0) + C α Γ \frac{ε}{k} + C F_{l s} (\frac{2}{3} \frac{ρ_{u}}{ρ_{b}} U_{l} Σ \frac{1 - c}{c})) - β U_{l} Σ \frac{1}{1 - \bar{c}} - F_{l s} (\frac{2}{3} \frac{1}{(γ p)} \frac{\partial p}{\partial t})] + S_{c o n v} \end{matrix}

(3909)

where:

$D$ is the molecular diffusivity
$C_{d i v u}$ is an empirical parameter
$C$ is a correction factor that takes into account the flame chemical timescale and the flame interaction with the walls
$F_{l s}$ is the laminar strain factor
$Γ$ is the ITNFS (Net Flame Stretch) function
$ρ_{u}$ and $ρ_{b}$ are the densities of the unburnt and burnt gases, respectively
$U_{l}$ is the effective laminar flame speed calculated by:

图 2. EQUATION_DISPLAY

$U_{l} = S_{l} Q_{l}$

(3910)

where $S_{l}$ is the adiabatic laminar flame speed and $Q_{l}$ is a heat-loss effect correction factor
$γ$ is the isentropic coefficient calculated by:

图 3. EQUATION_DISPLAY

$γ = \frac{c_{p}}{c_{v}}$

(3911)
$S_{c o n v}$ is an additional contribution to the FSD from convection at the spark plug
$μ_{t}$ is the turbulent viscosity
$p$ is the thermodynamic pressure
$\bar{c}$ is the Reynolds-averaged progress variable:
图 4. EQUATION_DISPLAY

$\bar{c} = 1 - \frac{ρ Y_{f}}{ρ_{u} Y_{u, f}}$

(3912)
$α$ and $β$ are empirical coefficients for the production and destruction terms, respectively.
$Y_{u, f}$ is the mean fuel tracer mass fraction.

The correction factor,

C

in Eqn. (3909), is calculated by:

图 5. EQUATION_DISPLAY

C = \frac{q_{w}}{{1 + q_{w} [\frac{α Γ ε}{k} + \frac{2}{3} \frac{ρ_{u}}{ρ_{b}} U_{l} Σ \frac{(1 - c)}{c}] τ_{c}}}

(3913)

where

τ_{c}

is the chemical timescale:

图 6. EQUATION_DISPLAY

τ_{c} = \frac{δ_{l}}{U_{l} Z e}

(3914)

in which

δ_{l}

is the laminar flame thickness:

图 7. EQUATION_DISPLAY

δ_{l} = \frac{2 μ_{b}}{ρ_{u} U_{l} \Pr}

(3915)

and

Z e

is the Zeldovich number.

q_{w}

is the wall flame quenching factor:

图 8. EQUATION_DISPLAY

q_{w} = {\begin{matrix} 1 & y^{+} / y_{c}^{+} > D_{q u e r a t} \\ 0 & y^{+} / y_{c}^{+} < D_{q u e r a t} \end{matrix}

(3916)

where

y_{c}^{+} = 11.5

and

D_{q u e r a t}

is a parameter.

Laminar Flame Speed

The adiabatic laminar flame speed in Eqn. (3910) is given by:

Extended Metghalchi-Keck Correlation

图 9. EQUATION_DISPLAY

S_{l} (Φ) = S_{l, 0} {(\frac{T_{u}}{T_{0}})}^{a} {(\frac{p}{p_{0}})}^{b_{\mod}} \max [1 - U_{l a m 1} X_{r e s}, \exp (U_{l a m 2} X_{r e s})]

(3917)

where

S_{l, 0}

,

a

, and

b

are functions of the equivalence ratio

Φ

and fuel, in the Metghalchi and Keck [820] correlation, or the Gulder correlation [769].

$T_{0}$ and $p_{0}$ represent the reference temperature ( $300 K$ ) and pressure $101325 P a$ , respectively.
$X_{r e s}$ is the mole fraction of the residual gas.
$U_{l a m 1}$ and
$U_{l a m 2}$ are the first and second coefficients for EGR respectively, with default values of 2.1 and -3.
$b$ is also corrected as:
- When $p / p_{r e f} > p_{t r a n s l}$ :
  图 10. EQUATION_DISPLAY
  
  $b_{\mod} = b - (0.08 (\frac{(\frac{p}{p_{0}}) - p_{t r a n s l}}{U_{l a m 3}} + \frac{p_{t r a n s l}}{U_{l a m 3 b e l o w}}))$
  
  (3918)
- Otherwise:
  图 11. EQUATION_DISPLAY
  
  $b_{\mod} = b - (0.08 (\frac{\frac{p}{p_{0}}}{U_{l a m 3 b e l o w}}))$
  
  (3919)
$p_{t r a n s l}$ is the normalized transitional pressure value for pressure scaling, $U_{l a m 3}$ is the coefficient for pressure scaling with (default value of 40), and $U_{l a m 3 b e l o w}$ is the coefficient for pressure scaling below the transitional pressure with (default value of 60).

The laminar flame speed can be changed by adjusting the Flame Speed Multiplier scaling factor.

Universal Laminar Flame Speed Correlations: Simcenter STAR-CCM+ identifies the most appropriate laminar flame speed correlation for each individual fuel in a mixture of fuels and then uses the Hirasawa method [821] to calculate the combined laminar flame speed for the blended mixture of fuels that are specified Eqn. (3571).

Turbulence Flame Quenching

Turbulence usually enhances the flame speed and fuel consumption rates in turbulent premixed flames. However, turbulence levels that are too high can break up the flame altogether, leading to extinction.

The guiding principle is based on the Karlovitz number ${K a}_{δ}$ , the ratio between the chemical and smallest eddies (Kolmogorov) timescales. If ${K a}_{δ}$ is greater than a certain threshold, then the flame transitions into the so-called broken regime and breaks up [822].

In ECFM combustion, these effects are modeled by introducing the flame quenching factor:

q = \frac{1}{2} (1 + \tanh (100 (1 - {K a}_{δ})))

(3921)

with:

{K a}_{δ} = δ^{2} \frac{{υ_{η}}^{2}}{{S_{l}}^{2}}

(3922)

where:

$δ$ is the flame inner-layer relative thickness (approximately 0.1).
$S_{l}$ is the laminar flame speed.

The flame quenching factor $q$ ( $0 < q < 1$ ) multiplies the production source terms in the flame surface density equation for ECFM combustion (for both RANS and LES).

For RANS, $υ_{η}$ is the characteristic Kolmogorov velocity scale:

υ_{η} = {(\frac{μ ϵ}{ρ})}^{\frac{1}{4}}

(3923)

For LES, the characteristic Kolmogorov velocity scale is [823]:

υ_{η} = \frac{μ}{ρ η}

(3924)

where:

η = \frac{Δ x}{L_{η}}

(3925)

with:

L_{η} = a {(\frac{2 {Δ x}^{6} {| S |}^{3}}{{(\frac{μ}{ρ})}^{3}})}^{b}

(3926)

$μ$ is the dynamic viscosity.
$ρ$ is the density.
$Δ x$ is the filter length (equal to the cell characteristic size).
$S$ is the strain rate.
$ϵ$ is the turbulence dissipation rate.
$a = 0.3328$
$b = 0.2651$

Premixed Reaction Mechanism

: For the premixed stage of combustion, there are a blend of three reactions for auto-ignition and flame propagation (extended to oxygenated fuels):; 图 13. EQUATION_DISPLAY

$α (1 - r_{C O}) [C_{n} H_{m} O_{p} + (n + \frac{m}{4} - \frac{p}{2}) O_{2} \to n {C O}_{2} + \frac{m}{2} H_{2} O]$

(3927); 图 14. EQUATION_DISPLAY

$(α r_{C O}) [C_{n} H_{m} O_{p} + (\frac{n}{2} + \frac{m}{4} - \frac{p}{2}) O_{2} \to n C O + \frac{m}{2} H_{2} O]$

(3928); 图 15. EQUATION_DISPLAY

$(1 - α) [C_{n} H_{m} O_{p} + (\frac{n}{2} - \frac{p}{2}) O_{2} \to n C O + \frac{m}{2} H_{2}]$

(3929)
The parameter $r_{C O}$ determines the bias towards a CO-rich reaction—even in lean combustion. $α$ is given by:; 图 16. EQUATION_DISPLAY

$α = \frac{4 (n + \frac{m}{4} - \frac{p}{2}) / Φ - 2 (n - p)}{2 n + m}$

(3930); in which the equivalence ratio is based on Eqn. (3333).

Premixed Fuel Burning Rate

: The fuel burning rate ${\dot{ω}}_{f u}$ due to flame propagation in the unburnt mixed gas region is given by:
图 17. EQUATION_DISPLAY

${\dot{ω}}_{f u} = - ρ_{u} U_{l} Y_{u, f}^{m} Σ$

(3931)