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 trace-less 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}^* - \frac{2}{3} \rho k \delta_{ij}</math> | :<math>\tau_{ij} = 2 \, \mu_t \, S_{ij}^* - \frac{2}{3} \rho k \delta_{ij}</math> | ||
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:<math>\frac{\partial U_k}{\partial x_k} = 0</math> | :<math>\frac{\partial U_k}{\partial x_k} = 0</math> | ||
| - | The Boussinesq eddy viscosity assumption is also often called the | + | The Boussinesq eddy viscosity assumption is also often called the Boussinesq hypothesis or the Boussinesq approximation. |
==References== | ==References== | ||
Revision as of 23:09, 1 August 2011
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), "Essai sur la théorie des eaux courantes", Mémoires présentés par divers savants à l'Académie des Sciences XXIII, 1, pp. 1-680.
Schmitt, F.G. (2007), "About Boussinesq’s turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity", Comptes Rendus Mécanique, vol. 335 (9-10), pp. 617-627; doi:10.1016/j.crme.2007.08.004.
