# Two equation turbulence models

Two-equation turbulence models are one of the most common type of turbulence models. Models like the $k-\epsilon$ model and the $k-\omega$ model have become industry standard models and are commonly used for most type of engineering problems. Two-equation turbulence models are also very much still an active area of research and new refined two-equation models are still being developed.

By definition, two-equation models have two extra transport equations to represent the turbulent properties of the flow. Most often one of the transported variables is the turbulent energy, $k$. The second transported variable can vary depending on what type of two-equation model it is. Common choices are the turbulent dissipation, $\epsilon$ or the specific dissipation, $\omega$. In any case the second variable should be thought of as the variable that transports and determines the scale of the turbulence (length-scale or time-scale), whereas the first variable, $k$, determines the strength or energy in the turbulence.

The basis for all two-equation models is the Boussinesq eddy viscosity assumption, which postulates that the Reynolds stress tensor is proportional to the strain rate tensor:

$\tau_{ij} = 2 \, \mu_t \, S_{ij}$

Or the same equation written out more explicitly:

$-\overline{\rho u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right)$