Turbulent Flame Speed Closure

When using the Complex Chemistry model, the turbulent flame speed closure (TFC) model uses the concept of turbulent flame speed to model the turbulent-chemistry interactions.

The TFC model identifies cells in which premixed combustion takes place (based on the Takeno index), and in those cells, removes the diffusion term and scales the reaction rates by $F_{t f c}$ :

图 1. EQUATION_DISPLAY

\frac{\partial ρ Y_{k}}{\partial t} + \nabla\cdot (ρ v Y_{k}) - \nabla\cdot (ρ D {\nabla Y}_{k}) = F_{t f c} ρ {\dot{ω}}_{k}

(3434)

where:

$ρ$ is the density.
$Y_{k}$ is the $k$ th species mass fraction.
$D$ is the diffusion term.
$F_{t f c}$ is the reaction rate multiplier.
${\dot{ω}}_{k}$ is the reaction source term for the $k$ th species.

F_{t f c}

is the ratio between the source term for the tracking species

Y_{c} ≔ Y_{C O 2} + Y_{H 2 O}

from the TFC model transport equation and the reaction source term for

Y_{c}

from the complex chemistry model transport equation.

图 2. EQUATION_DISPLAY

F_{t f c} ≔ \frac{{\dot{ω}}_{Y_{c}}^{t f c}}{{\dot{ω}}_{Y_{c}}^{c c}}

(3435)

The TFC reaction rate that is used in clustering is the target source from the TFC model for $Y_{c}$ , that is ${\dot{ω}}_{Y_{c}}^{t f c}$ .

${\dot{ω}}_{Y_{c}}^{t f c}$ is a function of $S_{t}$ , the desired turbulent flame velocity vector, given by Zimont or Peters, and directed towards the fresh gases, where:

图 3. EQUATION_DISPLAY

S_{t} ≔ S_{t} n_{f}

(3436)

and:

图 4. EQUATION_DISPLAY

n_{f} ≔ - \frac{\nabla Y_{c}}{| \nabla Y_{c} |}

(3437)

Then:

图 5. EQUATION_DISPLAY

{\dot{ω}}_{Y_{c}}^{t f c} = - ρ_{u} S_{t} \cdot \nabla Y_{c} + (1 - c) ρ_{u} S_{t} \cdot \nabla Y_{c}^{u} + c ρ_{u} S_{t} \cdot \nabla Y_{c}^{e q}

(3438)

where:

$ρ_{u}$ is the density of the unburnt gases.
$c$ is the progress variable.
$Y_{c}^{u}$ is the unburnt conditional mass fraction of $Y_{c}$ . A full transport equation is solved for this variable.
$Y_{c}^{e q}$ is an approximate value of the equilibrium mass fraction of $Y_{c}$ .

The first term on the right-hand side of Eqn. (3438) accounts for the actual flame propagation and is the most dominant, the second for stratifications in the residual (unburnt) gases, and the third for stratifications in the equilibrium (burnt) gases.

A wall quenching model proposed by Ranasinghe and Malalasekera [804] is implemented where a quenching non-dimensional distance factor $q_{w}$ , based on the Peclet number $P e$ , is applied to the source term ${\dot{ω}}_{Y_{c}}^{t f c}$ :

图 6. EQUATION_DISPLAY

\begin{matrix} q_{w} = \frac{P e - {P e}_{q}}{{P e}_{\max} - {P e}_{q}}; & 0 < q_{w} < 1 \end{matrix}

(3439)

with:

P e = d / δ_{l}

where:

${P e}_{q} = 4$
${P e}_{\max} = 40$
$d$ is the distance from the closest wall
$δ_{l}$ is the laminar flame thickness

The diffusivity is also knocked down in the pre-mixed active region (flame brush), which is identified based on the Takeno index number.

Metghalchi Laminar Flame Speed

The correlation that is proposed by Metghalchi and Keck [768] is calculated as follows:

图 7. EQUATION_DISPLAY

S_{l} = S_{l 0} {(\frac{T_{u}}{T_{0}})}^{α} {(\frac{p}{p_{0}})}^{β} (1 - 2.1 Y_{E G R})

(3440)

where

p

is the pressure,

T

is the temperature, the subscripts

0

and

u

denote reference and unburnt gas properties, respectively,

S_{l}

is the laminar flame speed, and

Y_{E G R}

is the mass fraction of any exhaust gas recirculation (EGR) that is present. The default value for the reference temperature

T_{0}

298 K

and for the reference pressure

P_{0}

101325 P a

. The reference laminar flame speed

S_{l 0}

and the exponents

α

and

β

depend on the equivalence ratio

ϕ

of the fuel. The exponents are defined as:

图 8. EQUATION DISPLAY

α = 2.18 - 0.8 (ϕ - 1)

(3441)

图 9. EQUATION DISPLAY

β = - 0.16 + 0.22 (ϕ - 1)

(3442)

The reference laminar speed

S_{l 0}

is a weak function of fuel type and is fit by a second-order polynomial of the form:

图 10. EQUATION DISPLAY

S_{l 0} = B_{m} + B_{2} {(ϕ - ϕ_{m})}^{2}

(3443)

where coefficients

ϕ_{m}

B_{m}

and

B_{2}

are specified in the following table:


Fuel	$ϕ_{m}$	$B_{m}$	$B_{2}$
Methanol	1.11	36.92	-140.51
Propane	1.08	34.22	-138.65
Isooctane	1.13	26.32	-84.72

0.4 \leq p \leq 50

atm,

300 \leq T \leq 700

K, and

0.8 \leq ϕ \leq 1.5

, the authors claim that the laminar flame speed is within 10% of the measured data.

ϕ \geq 0.65

is recommended because:

None of the combustion models can accurately predict burning in lean mixtures.
The laminar flame speed correlation function progressively diverges from the experimentally observed data in lean mixtures.

Gulder Laminar Flame Speed

The laminar flame speed correlation, which Gülder proposed [769] is calculated as follows:

图 11. EQUATION_DISPLAY

S_{l} = Z W ϕ^{η} \exp [- ξ {(ϕ - 1.075)}^{2}] {(\frac{T_{u}}{T_{0}})}^{α} {(\frac{P}{P_{0}})}^{β} (1 - 2.1 Y_{E G R})

(3444)

where

Y_{E G R}

is the mass fraction of any exhaust gas recirculation (EGR) that is present, and

Z

W

η

ξ

α

and

β

are fuel-dependent constants that are defined in the table below:


Fuel	$Z$	$W$	$η$	$ξ$	$α$	$β$
Fuel	$Z$	$W$	$η$	$ξ$	$α$	$ϕ < 1$	$ϕ > 1$
Methane	1	0.422	0.15	5.18	2.00	–0.5	–0.5
Propane	1	0.446	0.12	4.95	1.77	–0.2	–0.2
Methanol	1	0.492	0.25	5.11	1.75	–0.2/ $\sqrt{ϕ}$	-0.2 $ϕ$
Ethanol	1	0.465	0.25	6.34	1.75	–0.17/ $\sqrt{ϕ}$	-0.17 $\sqrt{ϕ}$
Iso-octane	1	0.4658	–0.326	4.48	1.56	–0.22	–0.22

Zimont Turbulent Flame Speed

Zimont used the following correlation for the turbulent flame speed [805]:

图 12. EQUATION_DISPLAY

S_{t} = 0.5 G {(u′)}^{3 / 4} S_{l}^{1 / 2} α_{u}^{- 1 / 4} I_{l}^{1 / 4}

(3445)

Here,

u′

is the turbulent velocity,

S_{l}

is the laminar flame speed,

α_{u}

is the unburnt thermal diffusivity of the unburnt mixture, and

I_{l}

is the integral turbulent length scale.

The stretch factor

G

takes the stretch effect into account by representing the probability of unquenched flamelets which is obtained by integrating the log-normal distribution of the turbulent dissipation rate:

图 13. EQUATION_DISPLAY

G = \frac{1}{2} e r f c [- \frac{1}{\sqrt{2 σ}} (ln \frac{ε_{c r}}{ε} + \frac{σ}{2})]

(3446)

where

e r f c

is a complementary error function and

σ

is the standard deviation of the distribution of

ε

computed with the following equation:

图 14. EQUATION_DISPLAY

σ = μ_{s t r} ln (I_{l} / η)

(3447)

where

I_{l}

is the integral turbulent length scale,

η

is the Kolmogorov micro-scale, and

μ_{s t r}

is an empirical model coefficient with a default value of 0.28.

ε_{c r}

is the turbulent dissipation rate at the critical strain rate

g_{c r}

图 15. EQUATION_DISPLAY

ε_{c r} = 15 ν g_{c r}^{2}

(3448)

where $ν$ is the kinematic viscosity of the fluid.

A high value for

g_{c r}

suggests no occurrence of the flame stretch. One method to compute

g_{c r}

is to assume that it is proportional to the chemical time scale:

图 16. EQUATION_DISPLAY

g_{c r} = \frac{B S_{L}^{2}}{α_{u}}

(3449)

where the value of constant $B$ is user-defined.

Peters Turbulent Flame Speed

The Peters correlation [806] for turbulent flame speed has the following form:

图 17. EQUATION_DISPLAY

S_{t} = S_{l} (1 + σ_{t})

(3450)

where:

图 18. EQUATION_DISPLAY

\begin{matrix} σ_{t} = - A B + \sqrt{{(A B)}^{2} + C \frac{u′ I_{l}}{S_{l} δ_{l}^{0}}} \\ A = \frac{A_{4} B_{3}^{2}}{2 B_{1}} \\ \begin{matrix} B = \frac{I_{l}}{δ_{l}^{0}} c_{e w} \\ C = A_{4} B_{3}^{2} \end{matrix} \end{matrix}

(3451)

where

δ_{l}^{0}

is the laminar flame thickness, and

c_{e w}

is Ewald’s corrector which has a default value of 1.0.

$A_{1}$ , $A_{4}$ , $B_{1}$ , and $B_{3}$ are model constants with default settings of 0.37, 0.78, 2.0, and 1.0, respectively.

Exhaust Gas Recirculation

In the unburnt gases, the mass fraction from exhaust gas recirculation (EGR)

Y_{E G R}

is calculated as:

图 19. EUQATION_DISPLAY

Y_{E G R} = \frac{Y_{c}^{u}}{Y_{c}^{s t}}

(3452)

where

Y_{c}^{u}

is an approximation of

Y_{c} = Y_{C O 2} + Y_{H 2 O}

in the unburnt gases, and

Y_{c}^{s t}

is a fully burnt mixture at stoichiometric conditions.

In the flame brush and behind the flame brush, a region average is used for the EGR mass fraction.

Coupling with the Relax to Equilibrium Model

If NH₃ is the only fuel, it is recommended to use the Relax to Chemical Equilibrium option.

When using the Relax to Chemical Equilibrium option, the chemistry is reduced to a binary model:

图 20. EQUATION_DISPLAY

{\dot{ω}}_{k} = {\begin{matrix} 0 & i f c < ϵ \\ F_{t f c} \frac{Y_{k}^{e q} - Y_{k}}{Δ t} & o t h e r w i s e \end{matrix}

(3453)

where $Y_{k}^{e q}$ is the $k$ th species mass fraction at equilibrium, $c$ is the normalized progress variable, and $ϵ = 10^{- 3}$ .

Reactions that occur ahead of the flame front are frozen. Behind the flame, the mixture is forced to immediate equilibrium and the chemistry becomes time-scale free. This feature is useful for in-cylinder applications and premixed or partially-premixed mixtures in which you are not interested in knock or emission modeling.

Flame Speed Multiplier

The flame speed multiplier is a scale factor applied to $S_{l}$ obtained from any of the Laminar Flame Speed methods listed above.