# 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

$\varepsilon = C_D {{k^{{3 \over 2}} } \over l}$
$C_D = 0.3$
$\sigma _k = 1$

where

$\tau _{ij} = 2\nu _T S_{ij} - {2 \over 3}k\delta _{ij}$

## References

1. Wilcox, D.C. (2004), Turbulence Modeling for CFD, ISBN 1-928729-10-X, 2nd Ed., DCW Industries, Inc..