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Prandtl's one-equation model

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{{Turbulence modeling}}
==Kinematic Eddy Viscosity==
==Kinematic Eddy Viscosity==
:<math>  
:<math>  
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:<math>
:<math>
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{{\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]
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{{\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]
</math>
</math>
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</math> <br>
</math> <br>
:<math>
:<math>
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     C_D  = 0.3 
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     C_D  = 0.08
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</math> <br>
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</math> <ref>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 <math>C_D</math>constant as 0.1643, but also uses another definition of the length scale and other constants (see [http://www.simuserve.com/phoenics/d_polis/d_lecs/pnalecs/lec7-22.htm#7 here]).</ref>
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<br>
:<math>
:<math>
   \sigma _k  = 1
   \sigma _k  = 1
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\tau _{ij}  = 2\nu _T S_{ij}  - {2 \over 3}k\delta _{ij}  
\tau _{ij}  = 2\nu _T S_{ij}  - {2 \over 3}k\delta _{ij}  
</math>
</math>
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:<math>l</math> is the turbulent length scale
== References ==
== References ==
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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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*{{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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*{{reference-paper|author=Emmons, H. W.|year=1954|title=Shear flow turbulence|rest=Proceedings of the 2nd U.S. Congress of Applied Mechanics, ASME}}
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*{{reference-paper|author=Glushko, G.|year=1965|title=Turbulent boundary layer on a flat plate in an incompressible fluid|rest=Izvestia Akademiya Nauk SSSR, Mekh, No 4, P 13}}
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==Footnotes==
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<div class="references-small">
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<references/>
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</div>
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----
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<i> Return to [[Turbulence modeling]] </i>
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[[Category:Turbulence models]]

Latest revision as of 19:43, 22 September 2010

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^{{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).

Return to Turbulence modeling

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