Boussinesq eddy viscosity assumption

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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.

References

↑ Boussinesq, J. (19877), "Essai sur la théorie des eaux courantes", Mémoires présentés par divers savants à l'Académie des Sciences 23 (1): 1-680
↑ Schmitt, F.G. (2007), "About Boussinesq’s turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity", Comptes Rendus Mécanique 335 ((9-10)): 617-627

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[0] Boussinesq, J. (19877), "Essai sur la théorie des eaux courantes", Mémoires présentés par divers savants à l'Académie des Sciences 23 (1): 1-680

[1] Schmitt, F.G. (2007), "About Boussinesq’s turbulent viscosity hypothesis: historical remarks and a direct evaluation of its validity", Comptes Rendus Mécanique 335 ((9-10)): 617-627

[1]

[2]

Boussinesq eddy viscosity assumption

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