# Boussinesq eddy viscosity assumption

### From CFD-Wiki

(Boussinesq links Reynolds tensor to traceless deformation rate) |
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- | In 1877 Boussinesq postulated that the momentum transfer caused by turbulent eddies can be modeled with an eddy viscosity. This is in analogy with how the momentum transfer caused by the molecular motion in a gas can be described by a molecular viscosity. The Boussinesq assumption states that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way: | + | In 1877 Boussinesq postulated that the momentum transfer caused by turbulent eddies can be modeled with an eddy viscosity. This is in analogy with how the momentum transfer caused by the molecular motion in a gas can be described by a molecular viscosity. The Boussinesq assumption states that the [[Reynolds stress tensor]], <math>\tau_{ij}</math>, is proportional to the mean strain rate tensor, <math>S_{ij}</math>, and can be written in the following way, for an incompressible flow : |

- | :<math>\tau_{ij} = 2 \, \mu_t \, S_{ij}</math> | + | :<math>\tau_{ij} = 2 \, \mu_t \, S_{ij} </math> |

Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]]. The same equation can be written more explicitly as: | Where <math>\mu_t</math> is a scalar property called the [[Eddy viscosity|eddy viscosity]]. The same equation can be written more explicitly as: | ||

- | :<math> -\overline{ | + | :<math> -\rho \overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} \right) - \frac{2}{3} k \delta_{ij}</math> |

The Boussinesq eddy viscosity assumption is also often called the Boussinesq hypothesis or the Boussinesq approximation. | The Boussinesq eddy viscosity assumption is also often called the Boussinesq hypothesis or the Boussinesq approximation. |

## Revision as of 17:13, 17 February 2009

In 1877 Boussinesq postulated that the momentum transfer caused by turbulent eddies can be modeled with an eddy viscosity. This is in analogy with how the momentum transfer caused by the molecular motion in a gas can be described by a molecular viscosity. The Boussinesq assumption states that the Reynolds stress tensor, , is proportional to the mean strain rate tensor, , and can be written in the following way, for an incompressible flow :

Where is a scalar property called the eddy viscosity. The same equation can be written more explicitly as:

The Boussinesq eddy viscosity assumption is also often called the Boussinesq hypothesis or the Boussinesq approximation.

## References

**Boussinesq, J. (1877)**, "Théorie de l’Écoulement Tourbillant", Mem. Présentés par Divers Savants Acad. Sci. Inst. Fr., Vol. 23, pp. 46-50.