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Linear eddy viscosity models

Linear eddy viscosity turbulence model selections include:

# Background

Under the Boussinesq hypothesis [6], the deviatoric anisotropic stress is considered proportional to the traceless mean rate of strain:

$- \rho \tensor{R}_\mathit{dev} = - \rho \av{\u' \otimes \u'} + \frac{2}{3} \rho k \tensor{I} = \mu_t \left[ 2\, \tensor{S} - \left( \frac{2}{3} \div \u \right) \tensor{I} \right]$

where $$\tensor{S}$$ is the symmetric tensor

$\tensor{S} = \frac{1}{2}\left( \grad \av{\u} + \grad \left( \av{\u} \right)^T \right)$

$- \rho \tensor{R}_\mathit{dev} = \mu_t \left( \grad \av{\u} + \grad \left( \av{\u} \right)^T \right) + \mu_t \left( \frac{2}{3} \div \u \right)\tensor{I}$

where $$\mu_t$$ is the dynamic eddy viscosity. The momentum equation therefore becomes:

$\ddt{\rho \av{\u}} + \div \left( \rho \av{\u} \otimes \av{\u} \right) = \vec{g} - \grad \av{p}' + \div \left( \mu_\mathit{eff} \grad \av{\u} \right) + \div \left[ \mu_\mathit{eff} \, \mathrm{dev2}\left(\left(\grad \av{\u}\right)^T \right) \right]$

where $$\mu_\mathit{eff}$$ is the effective dynamic eddy viscosity:

$\mu_\mathit{eff} = \mu + \mu_t$

i.e. the sum of the laminar and turbulent contributions.

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