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Wilcox's k-omega model

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{{Turbulence modeling}}
==Kinematic Eddy Viscosity ==
==Kinematic Eddy Viscosity ==
:<math>
:<math>
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==Closure Coefficients and Auxilary Relations==
==Closure Coefficients and Auxilary Relations==
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:<math>
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\alpha  = {{13} \over {25}}   
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\alpha  = {{5} \over {9}}   
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\beta  = \beta _0 f_\beta   
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</math>
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:<math>
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\beta ^*  = \beta _0^* f_{\beta ^* } 
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</math>
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:<math>
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\sigma  = {1 \over 2} 
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</math>
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\sigma ^*  = {1 \over 2} 
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</math>
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:<math>
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\beta _0  = {9 \over {125}}
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</math>
</math>
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f_\beta   = {{1 + 70\chi _\omega  } \over {1 + 80\chi _\omega  }}
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\beta = {{3} \over {40}}  
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</math>
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:<math>
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\chi _\omega  = \left| {{{\Omega _{ij} \Omega _{jk} S_{ki} } \over {\left( {\beta _0^* \omega } \right)^3 }}} \right|
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\beta^*  = {9 \over {100}}
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</math>
:<math>
:<math>
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\beta _0^* = {9 \over {100}}
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\sigma = {1 \over 2}
</math>
</math>
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f_{\beta ^* } = \left\{
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\sigma ^*  = {1 \over 2}   
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\begin{matrix}
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  {1,} & {\chi _k  \le 0}  \\
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  {{{1 + 680\chi _k^2 } \over {1 + 80\chi _k^2 }},} & {\chi _k > 0}  \\
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\end{matrix}
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  \right.
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</math>
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:<math>
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\chi _k  \equiv {1 \over {\omega ^3 }}{{\partial k} \over {\partial x_j }}{{\partial \omega } \over {\partial x_j }}
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</math>
</math>
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</math>
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== References ==
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l = {{k^{{1 \over 2}} } \over \omega }
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</math>
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#{{reference-paper|author=Wilcox, D.C. |year=1988|title=Re-assessment of the scale-determining equation for advanced turbulence models|rest=AIAA Journal, vol. 26, no. 11,  pp. 1299-1310}}
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== References ==
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[[Category:Turbulence models]]
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#{{reference-book|author=Wilcox, D.C. |year=2004|title=Turbulence Modeling for CFD|rest=ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc.}}
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Latest revision as of 16:52, 8 March 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

Contents

Kinematic Eddy Viscosity

\nu _T = {k \over \omega }

Turbulence Kinetic Energy

{{\partial k} \over {\partial t}} + U_j {{\partial k} \over {\partial x_j }} = \tau _{ij} {{\partial U_i } \over {\partial x_j }} - \beta ^* k\omega + {\partial \over {\partial x_j }}\left[ {\left( {\nu + \sigma ^* \nu _T } \right){{\partial k} \over {\partial x_j }}} \right]

Specific Dissipation Rate

{{\partial \omega } \over {\partial t}} + U_j {{\partial \omega } \over {\partial x_j }} = \alpha {\omega \over k}\tau _{ij} {{\partial U_i } \over {\partial x_j }} - \beta \omega ^2 + {\partial \over {\partial x_j }}\left[ {\left( {\nu + \sigma \nu _T } \right){{\partial \omega } \over {\partial x_j }}} \right]

Closure Coefficients and Auxilary Relations

\alpha = {{5} \over {9}}
\beta = {{3} \over {40}}
\beta^* = {9 \over {100}}
\sigma = {1 \over 2}
\sigma ^* = {1 \over 2}
\varepsilon = \beta ^* \omega k

References

  1. Wilcox, D.C. (1988), "Re-assessment of the scale-determining equation for advanced turbulence models", AIAA Journal, vol. 26, no. 11, pp. 1299-1310.
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