# Linear eddy viscosity models

(Difference between revisions)
 Revision as of 14:41, 4 November 2009 (view source)m← Older edit Latest revision as of 18:38, 7 June 2011 (view source)Zhuding (Talk | contribs) m Line 15: Line 15: - :Note that that inclusion of $\frac{2}{3} \rho k \delta_{ij}$ in the linear constitutive relation is required by tensorial algebra purposes when solving for [[Two equation models|two-equation turbulence models]] (or any other turbulence model that solves a transport equations for $k$. + :Note that that inclusion of $\frac{2}{3} \rho k \delta_{ij}$ in the linear constitutive relation is required by tensorial algebra purposes when solving for [[Two equation models|two-equation turbulence models]] (or any other turbulence model that solves a transport equation for $k$.

## Latest revision as of 18:38, 7 June 2011

These are turbulence models in which the Reynolds stresses, as obtained from a Reynolds averaging of the Navier-Stokes equations, are modelled by a linear constitutive relationship with the mean flow straining field, as:

$- \rho \left\langle u_{i} u_{j} \right\rangle = 2 \mu_{t} S_{ij} - \frac{2}{3} \rho k \delta_{ij}$

where

• $\mu_{t}$ is the coefficient termed turbulence "viscosity" (also called the eddy viscosity)
• $k = \frac{1}{2} \left( \left\langle u_{1} u_{1} \right\rangle + \left\langle u_{2} u_{2} \right\rangle + \left\langle u_{3} u_{3} \right\rangle \right)$ is the mean turbulent kinetic energy
• $S_{ij}= \frac{1}{2} \left[ \frac{\partial U_{i}}{\partial x_{j}} + \frac{\partial U_{j}}{\partial x_{i}} \right] - \frac{1}{3} \frac{\partial U_{k}}{\partial x_{k}} \delta_{ij}$ is the mean strain rate

Note that that inclusion of $\frac{2}{3} \rho k \delta_{ij}$ in the linear constitutive relation is required by tensorial algebra purposes when solving for two-equation turbulence models (or any other turbulence model that solves a transport equation for $k$.

This linear relationship is also known as the Boussinesq hypothesis. For a deep discussion on this linear constitutive relationship, check section Introduction to turbulence/Reynolds averaged equations.

There are several subcategories for the linear eddy-viscosity models, depending on the number of (transport) equations solved for to compute the eddy viscosity coefficient.