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Wilcox's modified k-omega model

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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 = {{13} \over {25}}
\beta = \beta _0 f_\beta
\beta ^* = \beta _0^* f_{\beta ^* }
\sigma = {1 \over 2}
\sigma ^* = {1 \over 2}
\beta _0 = {9 \over {125}}
f_\beta = {{1 + 70\chi _\omega } \over {1 + 80\chi _\omega }}
\chi _\omega = \left| {{{\Omega _{ij} \Omega _{jk} S_{ki} } \over {\left( {\beta _0^* \omega } \right)^3 }}} \right|
\beta _0^* = {9 \over {100}}
f_{\beta ^* } = \left\{ \begin{matrix} {1,} & {\chi _k \le 0} \\ {{{1 + 680\chi _k^2 } \over {1 + 80\chi _k^2 }},} & {\chi _k > 0} \\ \end{matrix} \right.
\chi _k \equiv {1 \over {\omega ^3 }}{{\partial k} \over {\partial x_j }}{{\partial \omega } \over {\partial x_j }}
\varepsilon = \beta ^* \omega k
l = {{k^{{1 \over 2}} } \over \omega }


References

  1. Wilcox, D.C. (2004), Turbulence Modeling for CFD, ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc..
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