# Standard k-epsilon model

(Difference between revisions)
 Revision as of 22:43, 13 September 2005 (view source)Zxaar (Talk | contribs)← Older edit Revision as of 00:14, 14 September 2005 (view source)Zxaar (Talk | contribs) Newer edit → Line 1: Line 1: + == Transport Equation for standard k-epsilon model == + + For k
$\frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_i} (\rho k u_i) = \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j}\right] + P_k + P_b - \rho \epsilon - Y_M + S_k$ $\frac{\partial}{\partial t} (\rho k) + \frac{\partial}{\partial x_i} (\rho k u_i) = \frac{\partial}{\partial x_j} \left[ \left(\mu + \frac{\mu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j}\right] + P_k + P_b - \rho \epsilon - Y_M + S_k$ - +
+ For dissipation $\epsilon$

Line 9: Line 13: [/itex] [/itex] + == Modeling turbulent viscosity == + Turbulent viscosity is modelled as:
$[itex] \mu_t = \rho C_{\mu} \frac{k^2}{\epsilon} \mu_t = \rho C_{\mu} \frac{k^2}{\epsilon}$ [/itex] +
+ + == Model Constants == $[itex] C_{1 \epsilon} = 1.44, \;\; C_{2 \epsilon} = 1.92, \;\; C_{\mu} = 0.09, \;\; \sigma_k = 1.0, \;\; \sigma_{\epsilon} = 1.3 C_{1 \epsilon} = 1.44, \;\; C_{2 \epsilon} = 1.92, \;\; C_{\mu} = 0.09, \;\; \sigma_k = 1.0, \;\; \sigma_{\epsilon} = 1.3$ [/itex]

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

## Model Constants

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