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Realisable k-epsilon model

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Revision as of 23:06, 18 September 2005 by Zxaar (Talk | contribs)
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Transport Equations

 \frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_j} (\rho k u_j) = \frac{\partial}{\partial x_j} \left [ \left(\mu + \frac{\mu_t}{\sigma_k}\right) \frac{\partial k} {\partial x_j} \right ] + P_k + P_b - \rho \epsilon - Y_M + S_k

 \frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_j} (\rho \epsilon u_j) = \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_{\epsilon}}\right) \frac{\partial \epsilon}{\partial x_j} \right ] + \rho \, C_1 S \epsilon - \rho \, C_2 \frac{{\epsilon}^2} {k + \sqrt{\nu \epsilon}} + C_{1 \epsilon}\frac{\epsilon}{k} C_{3 \epsilon} P_b + S_{\epsilon}


 C_1  =  \max\left[0.43, \frac{\eta}{\eta + 5}\right] , \;\;\;\;\; \eta  =  S \frac{k}{\epsilon}, \;\;\;\;\; S =\sqrt{2 S_{ij} S_{ij}}

In these equations,  P_k represents the generation of turbulence kinetic energy due to the mean velocity gradients, calculated in same manner as standard k-epsilon model.  P_b is the generation of turbulence kinetic energy due to buoyancy, calculated in same way as standard k-epsilon model.

Modelling Turbulent Viscosity

 \mu_t = \rho C_{\mu} \frac{k^2}{\epsilon}

 C_{\mu} = \frac{1}{A_0 + A_s \frac{k U^*}{\epsilon}}
  U^* \equiv \sqrt{S_{ij} S_{ij} + \tilde{\Omega}_{ij} \tilde{\Omega}_{ij}}   ;
 \tilde{\Omega}_{ij}  =  \Omega_{ij} - 2 \epsilon_{ijk} \omega_k      ;
  \Omega_{ij}  =  \overline{\Omega_{ij}} - \epsilon_{ijk} \omega_k

where  \overline{\Omega_{ij}} is the mean rate-of-rotation tensor viewed in a rotating reference frame with the angular velocity  \omega_k  . The model constants  A_0  and  A_s  are given by:
 A_0 = 4.04, \; \;  A_s = \sqrt{6}  \cos \phi

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