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RNG k-epsilon model

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RNG k-epsilon model

Transport equations for k and \epsilon are


 \frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_i} (\rho k u_i) = \frac{\partial}{\partial x_j} \left(\alpha_k \mu_{\rm eff} \frac{\partial k}{\partial x_j}\right) + P_k + P_b - \rho \epsilon

 \frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_i} (\rho \epsilon u_i) = \frac{\partial}{\partial x_j} \left(\alpha_{\epsilon} \mu_{\rm eff} \frac{\partial \epsilon}{\partial x_j}\right) + C_{1 \epsilon}\frac{\epsilon}{k} \left( G_k + C_{3 \epsilon} G_b \right) - C_{2\epsilon} \rho \frac{\epsilon^2}{k} - R_{\epsilon}

Turbulent viscosity is modelled as


d \left(\frac{\rho^2 k}{\sqrt{\epsilon \mu}} \right) = 1.72 \frac{\hat{\nu}}{\sqrt{{\hat{\nu}}^3-1+C_\nu}} d{\hat{\nu}}



\hat{\nu} = \mu_{\rm eff}/\mu

and


C_\nu  \approx 100

Term  R_{\epsilon}

The term  R_{\epsilon} is modelled as


R_{\epsilon} = \frac{C_\mu \rho \eta^3 (1-\eta/\eta_0)}{1+\beta\eta^3} \frac{\epsilon^2}{k}

\eta \equiv Sk/\epsilon

\eta_0 = 4.38

\beta = 0.012

The transport equation for \epsilon can be re-written as:


\frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_i} (\rho \epsilon u_i) = \frac{\partial}{\partial x_j} \left(\alpha_{\epsilon} \mu_{\rm eff} \frac{\partial \epsilon}{\partial x_j}\right) + C_{1 \epsilon}\frac{\epsilon}{k} \left( G_k + C_{3 \epsilon} G_b \right) - C_{2\epsilon}^* \rho \frac{\epsilon^2}{k}

C_{2\epsilon}^* \equiv C_{2\epsilon} + {C_\mu \eta^3 (1-\eta/\eta_0)\over 1+\beta\eta^3}



C_{1\epsilon} = 1.42, \; \; C_{2\epsilon} = 1.68
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