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Linear eddy viscosity models

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These are turbulence models in which the Reynolds stresses as obtained from a Reynolds averaging of the Navier-Stokes equations) are modelled by a linear constitutive relationship with the mean flow straining field, such as:

- \rho \left\langle  u_{i} u_{j} \right\rangle = \mu_{t} \left[ S_{ij} - \frac{1}{3} S_{kk} \delta_{ij} \right]

where \mu_{t} is the coefficient termed turbulence "viscosity" (also called the eddy viscosity), and S_{ij} is the mean strain rate defined by:

S_{ij}= \frac{1}{2} \left[ \frac{\partial U_{i}}{\partial x_{j}} + \frac{\partial U_{j}}{\partial x_{i}} \right]

This linear relationship is also known as the Boussinesq hypothesis. For a deep discussion on this linear constitutive relationship, check section Introduction to turbulence/Reynolds averaged equations.

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