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Standard k-epsilon model

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== Effect of Bouyancy ==
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P_b = \beta g_i \frac{\mu_t}{{\rm Pr}_t} \frac{\partial T}{\partial x_i}
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where Pr<sub>t</sub>  is the turbulent Prandtl number for energy and g<sub>i</sub>  is the component of the gravitational vector in the ith direction. For the standard and realizable -  models, the default value of Pr<sub>t</sub>  is 0.85.
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The coefficient of thermal expansion, <math> \beta </math> , is defined as <br>
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\beta = - \frac{1}{\rho} \left(\frac{\partial \rho}{\partial T}\right)_p
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== Model Constants ==
== Model Constants ==

Revision as of 00:32, 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}}

Effect of Bouyancy

P_b = \beta g_i \frac{\mu_t}{{\rm Pr}_t} \frac{\partial T}{\partial x_i}



where Prt is the turbulent Prandtl number for energy and gi is the component of the gravitational vector in the ith direction. For the standard and realizable - models, the default value of Prt is 0.85.

The coefficient of thermal expansion, \beta , is defined as
\beta = - \frac{1}{\rho} \left(\frac{\partial \rho}{\partial T}\right)_p

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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