Linear eddy viscosity models
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| - | These are turbulence models in which the [[Introduction to turbulence/Reynolds averaged equations|Reynolds stresses]] are modelled by a ''linear constitutive relationship'' with the ''mean'' flow straining field, such as: | + | These are turbulence models in which the [[Introduction to turbulence/Reynolds averaged equations|Reynolds stresses]] as obtained from a [[Introduction to turbulence/Reynolds averaged equations|Reynolds averaging of the Navier-Stokes equations]]) are modelled by a ''linear constitutive relationship'' with the ''mean'' flow straining field, such as: |
:<math> | :<math> | ||
Revision as of 17:14, 30 October 2009
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]](/W/images/math/5/f/8/5f8568434101e54fd4cb409ab338744a.png)
where 
is the coefficient termed turbulence "viscosity" (also called the eddy viscosity), and 
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]](/W/images/math/1/b/a/1baa28f3f8103315b8311820780a8205.png)
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.
