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Two equation turbulence models

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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)

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