# Standard k-epsilon model

## Transport equations for standard $k$-$\epsilon$ model

For turbulent kinetic energy $k$

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

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 buoyancy

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

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

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