# Linear eddy viscosity models

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 Revision as of 17:26, 30 October 2009 (view source)← Older edit Revision as of 22:28, 30 October 2009 (view source)mNewer edit → Line 1: Line 1: {{Turbulence modeling}} {{Turbulence modeling}} - 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: + 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: :[itex] :[itex]

## Revision as of 22:28, 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:

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

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.

There are several subcategories for the linear eddy-viscosity models, depending on the number of (transport) equations are solved for to compute the eddy viscosity coefficient.