Flame Propagation

There are two flame propagation models in Simcenter STAR-CCM+ that are used to calculate the flame movement in space, for premixed and partially-premixed systems, through calculation of the turbulent flame speed.

Coherent Flame Model (CFM)
The CFM model assumes that combustion occurs in the flamelet region. The mean turbulent reaction rate can be expressed as the product of the flame surface density, the laminar flame speed, and the unburnt density. A transport equation is solved for the flame surface density.
Turbulent Flame Speed Closure (TFC)
The TFC combustion model assumes that in a premixed combustion system, the reaction takes place in a thin layer which separates reactants and products. The mean reaction rate is closed using the turbulent flame speed.

These flame position models solve for the flame position by transporting a reaction progress variable to obtain the reaction progress:

Figure 1. EQUATION_DISPLAY

\frac{\partial ρ y_{m e a n}}{\partial t} + \nabla \cdot (ρ u y_{m e a n}) - \nabla \cdot (Γ_{y} \nabla y_{m e a n}) = {\dot{ω}}_{y, t f c} (o r {\dot{ω}}_{y, c f m})

(3554)

When the TFC/CFM model is used alongside the Chemical Equilibrium (CE) model or Steady Laminar Flamelet (SLF) model, $y$ is taken as the fuel mass fraction. However, when the TFC/CFM model is selected with the Flamelet Generated Manifold (FGM) model, $y$ is the unnormalized progress variable which is defined by the FGM model in Eqn. (3532).

CFM

The source term is calculated as:

Figure 2. EQUATION_DISPLAY

{\dot{ω}}_{y, c f m} = - (ρ_{u} S_{l} Σ)

(3555)

where:

$ρ_{u}$ is the unburnt density
$S_{l}$ is the laminar flame speed
$Σ$ is the flame area density, which is defined as the flame area per unit volume

There are options for the laminar flame speed:

TFC

The Turbulent Flame Speed Closure (TFC) model propagates premixed flame fronts at specified flame speeds. The source term for the unnormalized progress variable is calculated using one of the following methods:

Turbulent flame speed based source:

Figure 3. EQUATION_DISPLAY

${\dot{ω}}_{y, t f c} = A ρ_{u} S_{t} | \nabla y_{m e a n} |$

(3556)
There are options for the turbulent flame speed, $S_{t}$ :
- Zimont
- Peters
User defined source:
The unnormalized progress variable source term is computed from a user-specified source for the normalized progress variable.

In a partially premixed flame, the averaged value for any quantity $Q$ at location $\underset{̲}{x}$ and time $t$ , $\tilde{Q} (\underset{̲}{x}, t)$ , can be calculated with the following form:

Figure 4. EQUATION_DISPLAY

\tilde{Q} (\underset{̲}{x}, t) = \int_{0}^{1} \int_{0}^{1} Q (Z, c) P (Z, c; \underset{̲}{x}, t) d Z d c

(3557)

Here $P (Z, c; \underset{̲}{x}, t)$ is the joint pdf of the mixture fraction $Z$ and progress variable $c$ .

The joint pdf can be decomposed to a conditional pdf $P_{z} (Z | c; \underset{̲}{x}, t)$ and a marginal pdf $P_{c} (c; \underset{̲}{x}, t)$ :

Figure 5. EQUATION_DISPLAY

P (Z, c; \underset{̲}{x}, t) = P_{z} (Z | c; \underset{̲}{x}, t) P_{c} (c; \underset{̲}{x}, t)

(3558)

Using the classical flamelet approximation of a bimodal form:

Figure 6. EQUATION_DISPLAY

P_{c} (c; \underset{̲}{x}, t) = c_{m e a n} δ (1 - c) + (1 - c_{m e a n}) δ (c)

(3559)

where $δ$ is a $δ$ function, we can get:

Figure 7. EQUATION_DISPLAY

\tilde{Q} (\underset{̲}{x}, t) = (1 - c_{m e a n}) {\tilde{Q}}^{u} (\underset{̲}{x}, t) + c_{m e a n} {\tilde{Q}}^{b} (\underset{̲}{x}, t)

(3560)

where ${\tilde{Q}}^{u} (\underset{̲}{x}, t)$ is the unburnt state and ${\tilde{Q}}^{b} (\underset{̲}{x}, t)$ is the burned state. In this model, the burned state is from the Chemical Equilibrium or Steady Laminar Flamelet chemistry model.

Flame Area Density

The equation for $σ$ is:

Figure 8. EQUATION_DISPLAY

\frac{\partial (ρ σ)}{\partial t} + \nabla \cdot (ρ u σ) - \nabla \cdot (ρ Γ_{σ} \nabla σ) = S_{Σ} + \nabla \cdot (Γ_{σ} σ \nabla ρ)

(3561)

The source term $S_{Σ}$ includes the flame area production by stretch and destruction by fuel consumption and is calculated using the following equation:

Figure 9. EQUATION_DISPLAY

S_{Σ} = α K_{t} Σ - β \frac{ρ_{u} Y_{f t} S_{l} (1 + a \sqrt{k} / S_{l})}{ρ Y_{f}} Σ^{2}

(3562)

α

β

and

a

are model parameters.

Σ

is the flame area density, which is defined as the flame area per unit volume.

σ

is the flame area per unit mass.

Figure 10. EQUATION_DISPLAY

Σ = ρ σ

(3563)

where

ρ

is the gas density.

The method for $K_{t}$ calculation is described in the Net Flame Stretch K_t Calculation section.

Net Flame Stretch K_t Calculation

The methods that are developed in [753] are used. $K_{t}$ is tabulated with two parameters:

Figure 11. EQUATION_DISPLAY

Γ_{K} = \frac{K_{t}}{ε / k} = f (\frac{u'}{S_{l}}, \frac{I_{l}}{δ_{l}})

(3564)

$u'$ is the turbulence intensity that is obtained from the following equation:

Figure 12. EQUATION_DISPLAY

u' = \sqrt{2 / 3 k}

(3565)

$I_{l}$ is the integral length scale that is defined as:

Figure 13. EQUATION_DISPLAY

I_{l} = \frac{C_{μ}^{3 / 4}}{κ} \cdot \frac{k^{3 / 2}}{ε}

(3566)

where $C_{μ}$ and $κ$ are the turbulent viscosity coefficient and von Karman constant of the K-Epsilon model, respectively.

$δ_{l}$ is the thermal laminar flame thickness that is calculated as follows:

Figure 14. EQUATION_DISPLAY

δ_{l} = 2 \frac{μ_{b}}{{P r ρ}_{u} S_{l}}

(3567)

In this expression, $P r$ is the laminar Prandtl number of the burnt gas and assumed to be 0.9. $μ_{b}$ is the molecular viscosity of the burnt gas that is calculated from Sutherland’s law:

Figure 15. EQUATION_DISPLAY

μ_{b} = \frac{μ_{1} T_{b}^{1.5}}{μ_{2} + T_{b}}

(3568)

Here, $μ_{1}$ and $μ_{2}$ are constants with the values of 1.457e-6 and 110, respectively. $T_{b}$ is the burnt gas temperature. $Γ_{k}$ has the following form:

Figure 16. EQUATION_DISPLAY

Γ_{k} = Γ_{p} - B \cdot Γ_{q}

(3569)

$Γ_{p}$ and $Γ_{q}$ are the flame production and quench due to the stretch; they follow the empirical correlation in [753]. $B$ is a model parameter.

Flame Propagation

Flame Area Density

Net Flame Stretch Kt Calculation

Net Flame Stretch K_t Calculation