9.3.1 Closure for ECFM Source Terms

$P_1$ represents the production of flame area density by turbulent flame stretching, and is modeled as

where $K_t$ is a turbulent time scale and $\alpha_1$ is a constant with a default value of 1.6. The constant $\alpha_0$ (default of 1) is a user-specified linear blending between the Intermediate Turbulent Net Flame Stretch (ITNFS) term, $\Gamma_K$, for low turbulence levels at $\alpha_0 = 1$, and a straightforward turbulent time scale source when $\alpha_0 = 0$ for high turbulence levels.

The ITNFS term, $\Gamma_K$, can be specified either as a constant or calculated as a function of the two parameters $u'/U_l$ and $l_t/\delta_L^0$, where $u'$ is the turbulent velocity fluctuation, $U_l$ is the laminar flame speed, $l_t$ is the integral turbulent length scale and $\delta_L^0$ is the laminar flame thickness.

The expression for $\Gamma_K$ is given by:

where $s$ is defined as

and $\sigma_1$ is

The ITNFS term, $\Gamma_K$, is sensitive to the laminar flame thickness. ANSYS FLUENT provides several options for the calculation of this quantity:

• Constant (user-specified) value

• Meneveau Flame Thickness [ 223]

  The laminar flame thickness is calculated as

  
  

  
  

  where $\alpha$ is the local unburnt thermal diffusivity.

• Poinsot Flame Thickness [ 274]

  The flame thickness is evaluated as in Equation  9.3-6 but an additional term, $\Gamma_p$ is added to $\Gamma_K$. $\Gamma_p$ is calculated as

  
  

  
  

  where

  
  

  
  
  

• Blint Correction Flame Thickness [ 30]

  This includes a correction due to rapid expansion of the gas:

  
  

  
  

  where $T_u$ is the unburned temperature, and $T_b$ denotes the burned temperature.

The term $P_2$ in Equation  9.3-1 models the influence of dilatation on the production of flame area density. The term is given by

where the constant $\alpha_2$ has a default of 1.

The term $P_3$ models the effect of thermal expansion of the burned gas on the flame area density, and is given by

where $\alpha_3$ is 1 by default.

The flame area destruction term $D$ is modeled as

where $\beta$ is a constant with a default value of 1.

As formulated, the model can become singular for $c=0$ and $c=1$, which is handled by limiting $c$. Further, the production terms $P_1$ and $P_2$ can be non-zero in regions where the mixture is outside the flammability limits, which is unphysical. Accordingly, ANSYS FLUENT sets the production terms to zero when the laminar flame speed is less than a very small value. The stability of the solution is enhanced by ensuring that the laminar flame speed in the destruction term is always greater than a small, finite value. Inspection of the function for $\Gamma_K$ shows that a singularity exists in Equation  9.3-3 for $s$ = -0.4 which can occur when the turbulent integral length scale is small compared to the laminar flame thickness. To prevent the singularity, the quantity $(s + 0.4)$ is limited to a small positive number. This results in a small net turbulent flame stretch term in laminar zones. These numerical limiting constants can be adjusted in the TUI.