# Boussinesq eddy viscosity assumption

(Difference between revisions)
 Revision as of 12:35, 7 April 2011 (view source)← Older edit Latest revision as of 09:15, 3 January 2012 (view source)Peter (Talk | contribs) m (Reverted edits by Reverse22 (talk) to last revision by Peter) (11 intermediate revisions not shown) Line 1: Line 1: - 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]], $\tau_{ij}$, is proportional to the trace-less mean strain rate tensor, $S_{ij}^*$, and can be written in the following way: + In 1877 Boussinesq postulated{{Citation + |  last =Boussinesq | first = J. + |  journal = Mémoires présentés par divers savants à l'Académie des Sciences + |  number = 1 + |  pages = 1-680 + |  title = Essai sur la théorie des eaux courantes + |  volume = 23 + |  address =  Paris + |  year = 19877 + }} + {{Citation + |  last =Schmitt | first = F.G. + |  journal = Comptes Rendus Mécanique + |  number = (9-10) + |  pages = 617-627 + |  title = About Boussinesq’s turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity + |  volume = 335 + |  url = http://hal.archives-ouvertes.fr/hal-00264386/fr/ + |  year = 2007 + }} + 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]], $\tau_{ij}$, is proportional to the trace-less mean strain rate tensor, $S_{ij}^*$, and can be written in the following way: :$\tau_{ij} = 2 \, \mu_t \, S_{ij}^* - \frac{2}{3} \rho k \delta_{ij}$ :$\tau_{ij} = 2 \, \mu_t \, S_{ij}^* - \frac{2}{3} \rho k \delta_{ij}$ - Where $\mu_t$ is a scalar property called the [[Eddy viscosity|eddy viscosity]]. The same [http://www.merchantos.com pos system] equation can be written more explicitly as: + Where $\mu_t$ is a scalar property called the [[Eddy viscosity|eddy viscosity]]. The same equation can be written more explicitly as: :$-\rho \overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} - \frac{2}{3} \frac{\partial U_k}{\partial x_k} \delta_{ij} \right) - \frac{2}{3} \rho k \delta_{ij}$ :$-\rho \overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} - \frac{2}{3} \frac{\partial U_k}{\partial x_k} \delta_{ij} \right) - \frac{2}{3} \rho k \delta_{ij}$ Line 12: Line 32: 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. - + == See also == + *[[Two equation models]] + *[[Introduction to turbulence/Reynolds averaged equations#Turbulence closure problem and eddy viscosity|Turbulence closure problem and eddy viscosity]] ==References== ==References== - + {{reflist}} - {{reference-paper|author=Boussinesq, J.|year=1877|title=Essai sur la théorie des eaux courantes|rest=Mémoires présentés par divers savants à l'Académie des Sciences XXIII, 1, pp. 1-680}} + - + - {{reference-paper|author=Schmitt, F.G.|year=2007|title=About Boussinesq’s turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity|rest=Comptes Rendus Mécanique, vol. 335 (9-10), pp. 617-627; doi:10.1016/j.crme.2007.08.004}} + {{stub}} {{stub}}

## Latest revision as of 09:15, 3 January 2012

In 1877 Boussinesq postulated[1][2] 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, $\tau_{ij}$, is proportional to the trace-less mean strain rate tensor, $S_{ij}^*$, and can be written in the following way:

$\tau_{ij} = 2 \, \mu_t \, S_{ij}^* - \frac{2}{3} \rho k \delta_{ij}$

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

$-\rho \overline{u'_i u'_j} = \mu_t \, \left( \frac{\partial U_i}{\partial x_j} + \frac{\partial U_j}{\partial x_i} - \frac{2}{3} \frac{\partial U_k}{\partial x_k} \delta_{ij} \right) - \frac{2}{3} \rho k \delta_{ij}$

Note that for incompressible flow:

$\frac{\partial U_k}{\partial x_k} = 0$

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