K-epsilon turbulence model

Definition

Consider an incompressible turbulent flow with kinematic viscosity $\nu$, mean velocity $\bar u_i$ and fluctuations $u_i'$ from the Reynolds decomposition. Let $k = \frac12 \overline{u_i' u_i'}$ be the turbulent kinetic energy and $\varepsilon = \nu \, \overline{(\partial u_i' / \partial x_j)(\partial u_i' / \partial x_j)}$ its dissipation rate. The standard $k$–$\varepsilon$ model closes the Reynolds-averaged Navier–Stokes equations as follows.

1. The Reynolds stress is given by the Boussinesq eddy viscosity hypothesis with eddy viscosity $\nu_t = C_\mu k^2 / \varepsilon$.
2. The field $k$ satisfies $$\frac{\partial k}{\partial t} + \bar u_j \frac{\partial k}{\partial x_j} = \frac{\partial}{\partial x_j} \left[ \left( \nu + \frac{\nu_t}{\sigma_k} \right) \frac{\partial k}{\partial x_j} \right] + P_k - \varepsilon.$$
3. The field $\varepsilon$ satisfies $$\frac{\partial \varepsilon}{\partial t} + \bar u_j \frac{\partial \varepsilon}{\partial x_j} = \frac{\partial}{\partial x_j} \left[ \left( \nu + \frac{\nu_t}{\sigma_\varepsilon} \right) \frac{\partial \varepsilon}{\partial x_j} \right] + C_{1\varepsilon} \frac{\varepsilon}{k} P_k - C_{2\varepsilon} \frac{\varepsilon^2}{k}.$$

Here $P_k = -\overline{u_i' u_j'} \, \partial \bar u_i / \partial x_j$ is the production of turbulent kinetic energy, and $C_\mu$, $\sigma_k$, $\sigma_\varepsilon$, $C_{1\varepsilon}$, $C_{2\varepsilon}$ are positive constants.