Linear eddy viscosity models

**Turbulence modeling**
	Turbulence
	RANS-based turbulence models Linear eddy viscosity models Algebraic models Cebeci-Smith model ; Baldwin-Lomax model ; Johnson-King model ; A roughness-dependent model ; ; One equation models Prandtl's one-equation model ; Baldwin-Barth model ; Spalart-Allmaras model ; ; Two equation models k-epsilon models Standard k-epsilon model ; Realisable k-epsilon model ; RNG k-epsilon model ; Near-wall treatment ; ; k-omega models Wilcox's k-omega model ; Wilcox's modified k-omega model ; SST k-omega model ; Near-wall treatment ; ; Realisability issues Kato-Launder modification ; Durbin's realizability constraint ; Yap correction ; Realisability and Schwarz' inequality ; ; ; ; Nonlinear eddy viscosity models Explicit nonlinear constitutive relation Cubic k-epsilon ; EARSM ; ; v2-f models model ; model ; ; ; Reynolds stress model (RSM) ;
	Large eddy simulation (LES) Smagorinsky-Lilly model ; Dynamic subgrid-scale model ; RNG-LES model ; Wall-adapting local eddy-viscosity (WALE) model ; Kinetic energy subgrid-scale model ; Near-wall treatment for LES models ;
	Detached eddy simulation (DES)
	Direct numerical simulation (DNS)
	Turbulence near-wall modeling
	Turbulence free-stream boundary conditions Turbulence intensity ; Turbulence length scale ;

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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.

@@ Line 1: / Line 1: @@
-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:
+{{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, as:
 :<math>
-- \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}
 </math>
-where <math>\mu_{t} </math> is the coefficient termed turbulence "viscosity" (also called the eddy viscosity), and <math>S_{ij} </math> is the ''mean'' strain rate defined by:
+where
+:*<math>\mu_{t} </math> is the coefficient termed turbulence "viscosity" (also called the eddy viscosity)
+:*<math>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)</math> is the mean turbulent kinetic energy
+:*<math>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}
+</math> is the ''mean'' strain rate
+:Note that that inclusion of <math>\frac{2}{3} \rho k \delta_{ij}</math> 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 <math>k</math>.
-:<math>
-S_{ij}= \frac{1}{2} \left[ \frac{\partial U_{i}}{\partial x_{j}} + \frac{\partial U_{j}}{\partial x_{i}} \right]
-</math>
 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.
+# [[Algebraic turbulence models|Algebraic models]]
+# [[One equation turbulence models|One equation models]]
+# [[Two equation models]]
 [[Category:Turbulence models]]

Linear eddy viscosity models

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Latest revision as of 18:38, 7 June 2011

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