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Prandtl's one-equation 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
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        2. Realisable k-epsilon model
        3. RNG k-epsilon model
        4. Near-wall treatment
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      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
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      1. \overline{\upsilon^2}-f model
      2. \zeta-f model
  3. Reynolds stress model (RSM)
Large eddy simulation (LES)
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  2. Dynamic subgrid-scale model
  3. RNG-LES model
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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^{{1 \over 2}} l = C_D {{k^2 } \over \varepsilon }

Model


{{\partial k} \over {\partial t}} + U_j {{\partial k} \over {\partial x_j }} = \tau _{ij} {{\partial U_i } \over {\partial x_j }} - C_D {{k^{{3 \over 2}} } \over l} + {\partial  \over {\partial x_j }}\left[ {\left( {\nu  + {{\nu _T } \over {\sigma _k }}} \right){{\partial k} \over {\partial x_j }}} \right]


Closure Coefficients and Auxilary Relations


 \varepsilon  = C_D {{k^{{3 \over 2}} } \over l}

    C_D  = 0.08
[1]



   \sigma _k  = 1


where


\tau _{ij}  = 2\nu _T S_{ij}  - {2 \over 3}k\delta _{ij}
l is the turbulent length scale

References

  • Wilcox, D.C. (2004), Turbulence Modeling for CFD, ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc..
  • Emmons, H. W. (1954), "Shear flow turbulence", Proceedings of the 2nd U.S. Congress of Applied Mechanics, ASME.
  • Glushko, G. (1965), "Turbulent boundary layer on a flat plate in an incompressible fluid", Izvestia Akademiya Nauk SSSR, Mekh, No 4, P 13.

Footnotes

  1. The exact constant used by Prandtl is currently unknown by the author. Wilcox mentions that other researchers (Emmons 1954 and Glushko 1965) have used a value ranging from 0.07 to 0.09. Prandtl's one equation model can be written in a slightly different way with different constants. For example, CHAM lists the C_Dconstant as 0.1643, but also uses another definition of the length scale and other constants (see here).

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