3.2.12. Turbulent Reacting Mixing Models

In a reacting flow it is sometimes necessary to limit the finite-rate chemistry reaction when it occurs in regions of high turbulence. In this case, turbulent mixing of the reactants limits the finite-rate (laminar) chemical reaction, the time scale of which can be calculated as

(3.421) $t_{\rm{c}} = N_{\rm{R}} \frac{\sum_{i=1}^{N_{\rm{S}}} {Y_{i}}} {\sum_{i=1}^{N_{\rm{S}}} {\frac{\rm{d}Y_i}{\rm{d}t}}}$

where $N_{\rm{R}}$ is the number of reactions, ${N_{\rm{S}}}$ is the number of species in the reaction, and $Y_{i}$ represents the concentration of species $i$ . The denominator in (3.421) is the production rate of species $i$ .

3.2.12.1. Modified EDC Model (PARENTE)

One such model, named here as the PARENTE model, is adopted from [61] which derives explicit dependencies between the EDC model coefficients and the turbulent Reynolds ( ${\rm{Re_{t}}}$ ) and Damkohler ( ${\rm{Da}}$ ) dimensionless numbers. This model is based on EDC and is used to modify the chemistry reactions prescribed in the gas phase. The PARENTE turbulent reacting mixing model can can be used in addition to or in lieu of the built-in EDC turbulent combustion model found in EDC Turbulent Combustion Model. The current model is described as

(3.422) ${\rm{Da}} = \max\left(\min\left(\frac{1}{t_{\rm{c}}}\sqrt{\frac{\nu}{\epsilon}}, 10 \right),1\times 10^{-10}\right)$

(3.423) ${\rm{Re_{t}}} = \frac{k^2}{\nu\epsilon}$

(3.424) $C_{\rm{\tau}} = \min\left(\frac{C_1}{\sqrt{\rm{Da}(\rm{Re_{t}}+1)}}, 2.1377\right)$

(3.425) $C_{\rm{\gamma}} = \max\left(\min\left(C_2\sqrt{\rm{Da}(\rm{Re_{t}}+1)}, 5.0\right), 0.4082\right)$

(3.426) $\gamma_{\rm{L}} = C_{\rm{\gamma}}\left(\frac{\nu\epsilon}{k^2}\right)^{\frac{1}{4}}$

(3.427) $\tau^* = C_{\rm{\tau}}\sqrt{\frac{\nu}{\epsilon}}$

where $k$ represents the turbulent kinetic energy, $\epsilon$ is the turbulent dissipation and $\nu$ is the kinematic viscosity of the gas. The constants $C_1$ and $C_2$ can be specified by the user, but default to $C_1=0.05774$ and $C_2=0.5$ . The modified reaction time scale, $\tau^*$ is used to solve the reaction equation from time $t^n$ to $t^{n+\tau^*}$ . Following this calculation, the reaction source terms for species ( $S^*_Y{_i}$ ) and enthalpy ( $S^*_H$ ) can be computed. These computed sources are further scaled via the multiplication of a coefficient $\kappa$ which is calculated as

(3.428) $\kappa &= 1 & \text{if } \gamma_{\rm{L}}\geq 1 \kappa &= \max\left(\min\left(\frac{\gamma_{\rm{L}}^{e_1}} {1-\gamma_{\rm{L}}^{e_2}}, 1\right) ,0\right), &\text{otherwise}$

where $e_1 = e_2 = 3.0$ in our implementation of the model. Following the evaluation of $\kappa$ , the final source terms for species and enthalpy can be calculated as

(3.429) $S_{Y_i} = \kappa S^*_{Y_i}$

(3.430) $S_H = \kappa S^*_H$

respectively.