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12.5.3 Wall Boundary Conditions

The wall boundary conditions for the $k$ equation in the $k$- $\omega$ models are treated in the same way as the $k$ equation is treated when enhanced wall treatments are used with the $k$- $\epsilon$ models. This means that all boundary conditions for wall-function meshes will correspond to the wall function approach, while for the fine meshes, the appropriate low-Reynolds-number boundary conditions will be applied.

In FLUENT the value of $\omega$ at the wall is specified as

 \omega_w = \frac{\rho \, (u^*)^2}{\mu} \omega^+ (12.5-53)

The asymptotic value of $\omega^+$ in the laminar sublayer is given by
 \omega^+ = \min \left(\omega_w^+, \frac{6}{\beta_i (y^+)^2}\right) (12.5-54)

where
 \omega_w^+ = \left\{ \begin{array}{ll} \left(\frac{50}{k_s... ...\\ \frac{100}{k_s^+} & k_s^+ \ge 25 \end{array} \right. (12.5-55)

where
 k_s^+ = \max \left(1.0, \frac{\rho k_s u^*}{\mu}\right) (12.5-56)

and $k_s$ is the roughness height.

In the logarithmic (or turbulent) region, the value of $\omega^+$ is
 \omega^+ = \frac{1}{\sqrt{\beta_\infty^*}} \frac{du_{\rm turb}^+}{dy^+} (12.5-57)

which leads to the value of $\omega$ in the wall cell as
 \omega = \frac{u^*}{\sqrt{\beta_\infty^*} \kappa y} (12.5-58)

Note that in the case of a wall cell being placed in the buffer region, FLUENT will blend $\omega^+$ between the logarithmic and laminar sublayer values.


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