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, $\tau_{ij}$ , 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.

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.

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Boussinesq eddy viscosity assumption

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