# Linear eddy viscosity models

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{{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, | + | 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, as: |

:<math> | :<math> | ||

- | - \rho \left\langle u_{i} u_{j} \right\rangle = \mu_{t} | + | - \rho \left\langle u_{i} u_{j} \right\rangle = 2 \mu_{t} S_{ij} - \frac{2}{3} \rho k \delta_{ij} |

</math> | </math> | ||

- | where <math>\mu_{t} </math> is the coefficient termed turbulence "viscosity" (also called the eddy viscosity) | + | where |

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+ | :*<math>\mu_{t} </math> is the coefficient termed turbulence "viscosity" (also called the eddy viscosity) | ||

+ | :*<math>k = \frac{1}{2} \left( \left\langle u_{1} u_{1} \right\rangle + \left\langle u_{2} u_{2} \right\rangle + \left\langle u_{3} u_{3} \right\rangle \right)</math> is the mean turbulent kinetic energy | ||

+ | :*<math>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} | ||

+ | </math> is the ''mean'' strain rate | ||

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+ | :Note that that inclusion of <math>\frac{2}{3} \rho k \delta_{ij}</math> in the linear constitutive relation is required by tensorial algebra purposes when solving for [[Two equation models|two-equation turbulence models]] (or any other turbulence model that solves a transport equations for <math>k</math>. | ||

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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]]. | 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]]. |

## Revision as of 20:15, 3 November 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, as:

where

- is the coefficient termed turbulence "viscosity" (also called the eddy viscosity)
- is the mean turbulent kinetic energy
- is the
*mean*strain rate

- Note that that inclusion of in the linear constitutive relation is required by tensorial algebra purposes when solving for two-equation turbulence models (or any other turbulence model that solves a transport equations for .

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 solved for to compute the eddy viscosity coefficient.