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Standard k-epsilon model

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== Production of k ==
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<math>
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P_k = - \rho \overline{u'_i u'_j} \frac{\partial u_j}{\partial x_i} 
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<math> P_k = \mu_t S^2 </math>
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Where <math> S </math> is the modulus of the mean rate-of-strain tensor, defined as : <br>
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<math>
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S \equiv \sqrt{2S_{ij} S_{ij}}
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</math>

Revision as of 00:21, 14 September 2005

Contents

Transport Equation for standard k-epsilon model

For k
\frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_i} (\rho k u_i) = \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


For dissipation \epsilon

\frac{\partial}{\partial t} (\rho \epsilon) + \frac{\partial}{\partial x_i} (\rho \epsilon u_i) = \frac{\partial}{\partial x_j} \left[\left(\mu + \frac{\mu_t}{\sigma_{\epsilon}} \right) \frac{\partial \epsilon}{\partial x_j} \right] + C_{1 \epsilon}\frac{\epsilon}{k} \left( P_k + C_{3 \epsilon} P_b \right) - C_{2 \epsilon} \rho \frac{\epsilon^2}{k} + S_{\epsilon}

Modeling turbulent viscosity

Turbulent viscosity is modelled as:
\mu_t = \rho C_{\mu} \frac{k^2}{\epsilon}



Production of k

P_k = - \rho \overline{u'_i u'_j} \frac{\partial u_j}{\partial x_i}
P_k = \mu_t S^2

Where S is the modulus of the mean rate-of-strain tensor, defined as :
S \equiv \sqrt{2S_{ij} S_{ij}}


Model Constants

C_{1 \epsilon} = 1.44, \;\; C_{2 \epsilon} = 1.92, \;\; C_{\mu} = 0.09, \;\; \sigma_k = 1.0, \;\; \sigma_{\epsilon} = 1.3

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