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Prandtl's one-equation model

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Turbulence modeling
Turbulence
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  1. Linear eddy viscosity models
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      1. Prandtl's one-equation model
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      3. Spalart-Allmaras model
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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.

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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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