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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
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      2. Baldwin-Lomax model
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      4. A roughness-dependent model
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      1. Prandtl's one-equation model
      2. Baldwin-Barth model
      3. Spalart-Allmaras model
    3. Two equation models
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        1. Standard k-epsilon model
        2. Realisable k-epsilon model
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        4. Near-wall treatment
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        2. Wilcox's modified k-omega model
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      3. Realisability issues
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        2. Durbin's realizability constraint
        3. Yap correction
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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}

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. However, CHAM lists the constant as 0.1643 (see here)

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