# Prandtl's one-equation model

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## 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 $C_D$constant as 0.1643, but also uses another definition of the length scale and other constants (see here).