# Boussinesq eddy viscosity assumption

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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 trace-less mean strain rate tensor, <math>S_{ij}^*</math>, and can be written in the following way: |

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

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+ | Note that for incompressible flow: | ||

+ | |||

+ | :<math>\frac{\partial u_k}{\partial x_k} = 0</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 19:06, 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 trace-less mean strain rate tensor, , and can be written in the following way:

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

Note that for incompressible flow:

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.