# Nonlinear eddy viscosity models

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This is class of turbulence models for the RANS equations in which an eddy viscosity coefficient is used to relate the mean turbulence field to the mean velocity field, however in a nonlinear relationship

$- \rho \left\langle u_{i} u_{j} \right\rangle = 2 \, \mu_{t} \, F_{nl} \left( S_{ij}, \Omega_{ij}, \dots \right)$

where

• $F_{nl} \left( S_{ij}, \Omega_{ij}, \dots \right)$ is a nonlinear function possibly dependent on the mean strain and vorticity fields or even other turbulence variable
• $\mu_{t}$ is the coefficient termed turbulence "viscosity" (also called the eddy viscosity)
• $S_{ij}= \frac{1}{2} \left[ \frac{\partial U_{i}}{\partial x_{j}} + \frac{\partial U_{j}}{\partial x_{i}} \right] - \frac{1}{3} \frac{\partial U_{k}}{\partial x_{k}} \delta_{ij}$ is the mean strain rate
• $\Omega_{ij}= \frac{1}{2} \left[ \frac{\partial U_{i}}{\partial x_{j}} - \frac{\partial U_{j}}{\partial x_{i}} \right]$ is the mean vorticity