# Near-wall treatment for k-omega models

(Difference between revisions)
 Revision as of 12:23, 31 October 2011 (view source)← Older edit Latest revision as of 22:24, 3 November 2011 (view source) (→Automatic wall treatments) (16 intermediate revisions not shown) Line 1: Line 1: - ==Standard wall function approach== + {{Turbulence modeling}} - There are two possible ways of implementing wall functions in a finite volume code: + As described in [[Two equation turbulence models]] low and high reynolds number treatments are possible. - * Additional source term in the momentum equations. + ==Standard wall functions== - * Modification of turbulent viscosity in cells adjacent to solid walls. + Main page: [[Two equation models#Near-wall treatments| Two equation near-wall treatments]] - 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. + 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. - Using the compact version of log-law + Moreover the centroid values in cells adjacent to solid wall are specified as -
+ - $+ - \frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+} + -$(1)
+ - where $E=9.8$ is equivalent to additive constants, and and using $\tau_w = \rho u_\tau^2$ we obtain: + :$:[itex] - \tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}}, + \begin{matrix} + k_p = \frac{u^2_\tau}{\sqrt{C_\mu}y_p}, + && + \omega_p = \frac{u_\tau}{\sqrt{C_\mu}\kappa y_p} = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p}. + \end{matrix}$ [/itex] - On the other hand, the linear interpolation for shear stress, rembembering that $U|_{y=0}=0$, is: + 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 :$:[itex] - \tau_w = (\nu_t + \nu)\frac{U_p}{y_p}. + \begin{matrix} + \omega_\text{vis} = \frac{6\nu}{\beta y^2} & \omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y} + \end{matrix}$ [/itex] - + where $y$ is the cell centroid distance from the wall. Using this a blending can take the following form: - 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 + :$:[itex] - y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}. + \omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},$ [/itex] - 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: + 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: :$:[itex] - k_p = \frac{u_\tau^2}{\sqrt{C_\mu}} + \begin{matrix} + u^\text{vis}_\tau = \frac{U}{y^{+}} & & u_\tau^\text{log} = \frac{U}{\log E y^{+}} + \end{matrix}$ [/itex] + And the blending suggested: :$:[itex] - \omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p}, + u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},$ [/itex] - 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== ==FLUENT== Line 68: Line 64:

- Reference: + == References == - FLUENT 6.2 Documentation, 2006
+ + * {{reference-paper|author=Menter, F., Esch, T.|year=2001|title=Elements of industrial heat transfer predictions|rest='COBEM 2001, 16th Brazilian Congress of Mechanical Engineering.'}} + * {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}

## Latest revision as of 22:24, 3 November 2011

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", .