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Linear eddy viscosity models

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{{Turbulence modeling}}
{{Turbulence modeling}}
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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:
:<math>  
:<math>  
-
- \rho \left\langle  u_{i} u_{j} \right\rangle = \mu_{t} \left[ S_{ij} - \frac{1}{3} S_{kk} \delta_{ij} \right]
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- \rho \left\langle  u_{i} u_{j} \right\rangle = 2 \mu_{t} S_{ij} - \frac{2}{3} \rho k \delta_{ij}
</math>
</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>.
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:<math>
 
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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]].
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]]
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##[[Cebeci-Smith model]]
 
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##[[Baldwin-Lomax model]]
 
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## [[Johnson-King model]]
 
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## [[A roughness-dependent model]]
 
# [[One equation turbulence models|One equation models]]
# [[One equation turbulence models|One equation models]]
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## [[Prandtl's one-equation model]]
 
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## [[Baldwin-Barth model]]
 
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## [[Spalart-Allmaras model]]
 
# [[Two equation models]]
# [[Two equation models]]
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## [[k-epsilon models]]
 
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### [[Standard k-epsilon model]]
 
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### [[Realisable k-epsilon model]]
 
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### [[RNG k-epsilon model]]
 
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### [[Near-wall treatment for k-epsilon models]]
 
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## [[k-omega models]]
 
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### [[Wilcox's k-omega model]]
 
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### [[Wilcox's modified k-omega model]]
 
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### [[SST k-omega model]]
 
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### [[Near-wall treatment for k-omega models]]
 
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## [[Two equation turbulence model constraints and limiters]]
 
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### [[Kato-Launder modification]]
 
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### [[Durbin's realizability constraint]]
 
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### [[Yap correction]]
 
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### [[Realisability and Schwarz' inequality]]
 
[[Category:Turbulence models]]
[[Category:Turbulence models]]

Latest revision as of 18:38, 7 June 2011

Turbulence modeling
Turbulence
RANS-based turbulence models
  1. Linear eddy viscosity models
    1. Algebraic models
      1. Cebeci-Smith model
      2. Baldwin-Lomax model
      3. Johnson-King model
      4. A roughness-dependent model
    2. One equation models
      1. Prandtl's one-equation model
      2. Baldwin-Barth model
      3. Spalart-Allmaras model
    3. Two equation models
      1. k-epsilon models
        1. Standard k-epsilon model
        2. Realisable k-epsilon model
        3. RNG k-epsilon model
        4. Near-wall treatment
      2. k-omega models
        1. Wilcox's k-omega model
        2. Wilcox's modified k-omega model
        3. SST k-omega model
        4. Near-wall treatment
      3. Realisability issues
        1. Kato-Launder modification
        2. Durbin's realizability constraint
        3. Yap correction
        4. Realisability and Schwarz' inequality
  2. Nonlinear eddy viscosity models
    1. Explicit nonlinear constitutive relation
      1. Cubic k-epsilon
      2. EARSM
    2. v2-f models
      1. \overline{\upsilon^2}-f model
      2. \zeta-f model
  3. Reynolds stress model (RSM)
Large eddy simulation (LES)
  1. Smagorinsky-Lilly model
  2. Dynamic subgrid-scale model
  3. RNG-LES model
  4. Wall-adapting local eddy-viscosity (WALE) model
  5. Kinetic energy subgrid-scale model
  6. Near-wall treatment for LES models
Detached eddy simulation (DES)
Direct numerical simulation (DNS)
Turbulence near-wall modeling
Turbulence free-stream boundary conditions
  1. Turbulence intensity
  2. Turbulence length scale

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.

  1. Algebraic models
  2. One equation models
  3. Two equation models
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