# Near-wall treatment for k-omega models

As described in Two equation turbulence models low and high reynolds number treatments are possible.

## Standard wall functions

Main page: Two equation near-wall treatments

For $k$ the boundary conditions imposed at the solid boundary are:

$\begin{matrix} \frac{\partial k}{\partial n} = 0 & & \frac{\partial \omega}{\partial n} = 0 \end{matrix}$

where $n$ is the normal to the boundary.

Moreover the centroid values in cells adjacent to solid wall are specified as

$\begin{matrix} k_p = \frac{u^2_\tau}{\sqrt{C_\mu}y_p}, && \omega_p = \frac{u_\tau}{\sqrt{C_\mu}\kappa y_p} = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p}. \end{matrix}$

In the alternative approach $k$ production terms is modified.

## Automatic wall treatments

The purpose of automatic wall treatments is to make results insensitive with respect to wall mesh refinement. Many blending approaches have been proposed. The one by Menter takes advantage of the fact that the solution to $\omega$ equations is known for both viscous and log layer

$\begin{matrix} \omega_\text{vis} = \frac{6\nu}{\beta y^2} & \omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y} \end{matrix}$

where $y$ is the cell centroid distance from the wall. Using this a blending can take the following form:

$\omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},$

Note that for low $y$ values the $1/y^2$ will dominate and therefore viscous value of $\omega$ will be reproduced. Conversely, for larger values of $y$, $1/y$ will be dominant and logarithmic value will be recovered.

Subsequently Menter proposes also blending for friction velocity. Friction velocity for viscous and logarithmic region are:

$\begin{matrix} u^\text{vis}_\tau = \frac{U}{y^{+}} & & u_\tau^\text{log} = \frac{U}{\log E y^{+}} \end{matrix}$

And the blending suggested:

$u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},$

## FLUENT

Both k- omega models (std and sst) are available as low-Reynolds-number models as well as high-Reynolds-number models.

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, that means:

If the Transitional Flows option is enabled in the Viscous Model panel, low-Reynolds-number variants will be used, and, in that case, mesh guidelines should be the same as for the enhanced wall treatment
(y+ at the wall-adjacent cell should be on the order of y+ = 1. However, a higher y+ is acceptable as long as it is well inside the viscous sublayer (y+ < 4 to 5).)

If Transitional Flows option is not active, then the mesh guidelines should be the same as for the wall functions.
(For [...] wall functions, each wall-adjacent cell's centroid should be located within the log-law layer, 30 < y+ < 300. A y+ value close to the lower bound y+ = 30 is most desirable.)

## References

• Menter, F., Esch, T. (2001), "Elements of industrial heat transfer predictions", 'COBEM 2001, 16th Brazilian Congress of Mechanical Engineering.'.
• ANSYS (2006), "FLUENT Documentation", .