# Standard k-epsilon model

(Difference between revisions)
Jump to: navigation, search
 Revision as of 00:14, 14 September 2005 (view source)Zxaar (Talk | contribs)← Older edit Revision as of 00:21, 14 September 2005 (view source)Zxaar (Talk | contribs) Newer edit → Line 19: Line 19: [/itex] [/itex]

+ + + + + == 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 :
+ $+ S \equiv \sqrt{2S_{ij} S_{ij}} +$

## 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

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