# Linear eddy viscosity models

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 Revision as of 22:28, 30 October 2009 (view source)m← Older edit Latest revision as of 18:38, 7 June 2011 (view source)Zhuding (Talk | contribs) m (3 intermediate revisions not shown) Line 1: Line 1: {{Turbulence modeling}} {{Turbulence modeling}} - These are turbulence models in which the [[Introduction to turbulence/Reynolds averaged equations|Reynolds stresses]], as obtained from a [[Introduction to turbulence/Reynolds averaged equations|Reynolds averaging of the Navier-Stokes equations]], are modelled by a ''linear constitutive relationship'' with the ''mean'' flow straining field, such as: + These are turbulence models in which the [[Introduction to turbulence/Reynolds averaged equations|Reynolds stresses]], as obtained from a [[Introduction to turbulence/Reynolds averaged equations|Reynolds averaging of the Navier-Stokes equations]], are modelled by a ''linear constitutive relationship'' with the ''mean'' flow straining field, as: :$:[itex] - - \rho \left\langle u_{i} u_{j} \right\rangle = \mu_{t} \left[ S_{ij} - \frac{1}{3} S_{kk} \delta_{ij} \right] + - \rho \left\langle u_{i} u_{j} \right\rangle = 2 \mu_{t} S_{ij} - \frac{2}{3} \rho k \delta_{ij}$ [/itex] - where $\mu_{t}$ is the coefficient termed turbulence "viscosity" (also called the eddy viscosity), and $S_{ij}$ is the ''mean'' strain rate defined by: + 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 models|two-equation turbulence models]] (or any other turbulence model that solves a transport equation for $k$. - :$- S_{ij}= \frac{1}{2} \left[ \frac{\partial U_{i}}{\partial x_{j}} + \frac{\partial U_{j}}{\partial x_{i}} \right] -$ 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]]. 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 are solved for to compute the eddy viscosity coefficient. + 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. # [[Algebraic turbulence models|Algebraic models]] # [[Algebraic turbulence models|Algebraic models]] - ##[[Cebeci-Smith model]] - ##[[Baldwin-Lomax model]] - ## [[Johnson-King model]] - ## [[A roughness-dependent model]] # [[One equation turbulence models|One equation models]] # [[One equation turbulence models|One equation models]] - ## [[Prandtl's one-equation model]] - ## [[Baldwin-Barth model]] - ## [[Spalart-Allmaras model]] # [[Two equation models]] # [[Two equation models]] - ## [[k-epsilon models]] - ### [[Standard k-epsilon model]] - ### [[Realisable k-epsilon model]] - ### [[RNG k-epsilon model]] - ### [[Near-wall treatment for k-epsilon models]] - ## [[k-omega models]] - ### [[Wilcox's k-omega model]] - ### [[Wilcox's modified k-omega model]] - ### [[SST k-omega model]] - ### [[Near-wall treatment for k-omega models]] - ## [[Two equation turbulence model constraints and limiters]] - ### [[Kato-Launder modification]] - ### [[Durbin's realizability constraint]] - ### [[Yap correction]] - ### [[Realisability and Schwarz' inequality]] [[Category:Turbulence models]] [[Category:Turbulence models]]

## 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.