# Near-wall treatment for k-omega models

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## Standard wall function approach

There are two possible ways of implementing wall functions in a finite volume code:

• Additional source term in the momentum equations.
• Modification of turbulent viscosity in cells adjacent to solid walls.

The source term in the first approach is simply the difference between logarithmic and linear interpolation of velocity gradient multiplied by viscosity (the difference between shear stresses). The second approach does not attempt to reproduce the correct velocity gradient. Instead, turbulent viscosity is modified in such a way as to guarantee the correct shear stress.

Using the compact version of log-law

 $\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}$ (1)

where $E=9.8$ is equivalent to additive constants, and and using $\tau_w = \rho u_\tau^2$ we obtain:

$\tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},$

On the other hand, the linear interpolation for shear stress, rembembering that $U|_{y=0}=0$, is:

$\tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.$

Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:

 $\nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).$ (2)

Note that $u_\tau$ has been been incorporated in $y^+$. The latter remains the only unknown in the equation and has to be estimated for the current velocity field. In the standard approach this cannot be done explicitly and instead an implicit way of obtaining $y^{+}$ has to be employed.

After multiplying log law (1) by $y_p/\nu$ and after reorganising some terms we get:

 $\frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.$ (3)

This equation can be solved numerically with respect to $y^+$ for example via root searching algorithms e.g. Newton method for specified $U_p$, $y_p$ and $\nu$. One iteration in a Newton method for (3) is

$y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.$

Thus obtained $y^{+}$ is then substituted to (2) Eventually the estimated $u_\tau$ serves also to define the values of turbulent quantities in the cell adjacent to the wall:

$k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}$
$\omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},$

which are the values for $k$ and $\omega$ according to Wilcox2006 asymptotic analysis of log layer. These wall functions for $k$ and $\omega$ are the results of the solution of model equation for the logarithmic layer.

The above methodology is known to produce spurious results in separated flows, where, by definition, $u_\tau = 0$ at the separation and reattachment point. Many extension of this approach has been proposed.

## 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

• Bredberg, J. (2000), "On the Wall Boundary Condition for Turbulence Models", 'Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'.
• ANSYS (2006), "FLUENT Documentation", .