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Near-wall treatment for k-omega models

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(the formula for kp is wrong)
 
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{{Turbulence modeling}}
{{Turbulence modeling}}
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==Standard wall function approach==
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As described in [[Two equation turbulence models]] low and high reynolds number treatments are possible.
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There are two possible ways of implementing wall functions in a finite volume code:
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==Standard wall functions==
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* Additional source term in the momentum equations.
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Main page: [[Two equation models#Near-wall treatments| Two equation near-wall treatments]]
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* Modification of turbulent viscosity in cells adjacent to solid walls.
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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.
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For <math>k</math> the boundary conditions imposed at the solid boundary are:
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Using the compact version of log-law
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<table width="70%"><tr><td>
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<math>
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\frac{U}{u_\tau} = \frac{1}{\kappa} \ln E y^{+}
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</math></td><td width="5%">(1)</td></tr></table>
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where <math>E=9.8</math> is equivalent to additive constants in [[Law of the wall]], and and using <math>\tau_w = \rho u_\tau^2</math> we obtain:
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:<math>
:<math>
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  \tau_w = \frac{\rho u_\tau \kappa U}{\ln Ey^{+}},
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\begin{matrix}
 +
\frac{\partial k}{\partial n} = 0 & & \frac{\partial \omega}{\partial n} = 0
 +
\end{matrix}
</math>
</math>
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On the other hand, the linear interpolation for shear stress, rembembering that <math>U|_{y=0}=0</math>, is:
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where <math>n</math> is the normal to the boundary.
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:<math>
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  \tau_w = (\nu_t + \nu)\frac{U_p}{y_p}.
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</math>
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Comparing the above equations we obtain an expression for turbulent viscosity can be obtained:
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<table width="70%"><tr><td>
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<math>
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  \nu_t = \nu\left( \frac{y^{+}\kappa}{\ln Ey^{+}} - 1\right).
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</math> </td><td width="5%">(2)</td></tr></table>
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Note that <math>u_\tau</math> has been been incorporated in <math>y^+</math>. 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 <math>y^{+}</math> has to be employed.  
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After multiplying log law (1) by <math>y_p/\nu</math> and after reorganising some terms we get:
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Moreover the centroid values in cells adjacent to solid wall are specified as
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<table width="70%"><tr><td>
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<math>
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  \frac{\kappa U_p y_p}{\nu} = y^{+}\ln{Ey^{+}}.
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</math></td><td width="5%">(3)</td></tr></table>
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This equation can be solved numerically with respect to <math>y^+</math> for example via root searching algorithms e.g. Newton method for specified <math>U_p</math>, <math>y_p</math> and <math>\nu</math>. One iteration in a Newton method for (3) is
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:<math>
:<math>
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   y^{+}_{n+1} = \frac{\frac{\kappa U_p y_p}{\nu} + y^{+}_{n}}{1 + \ln E y_n^{+}}.
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\begin{matrix}
 +
   k_p = \frac{u^2_\tau}{\sqrt{C_\mu}},
 +
&&
 +
  \omega_p = \frac{u_\tau}{\sqrt{C_\mu}\kappa y_p} = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p}.
 +
\end{matrix}
</math>
</math>
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Thus obtained <math>y^{+}</math> is then substituted to (2) Eventually the estimated <math>u_\tau</math> serves also to define the values of turbulent quantities in the cell adjacent to the wall:
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In the alternative approach <math>k</math> 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 <math>\omega</math> equations is known for both viscous and log layer
:<math>
:<math>
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   k_p = \frac{u_\tau^2}{\sqrt{C_\mu}}
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\begin{matrix}
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   \omega_\text{vis} = \frac{6\nu}{\beta y^2} &  \omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}
 +
\end{matrix}
</math>
</math>
 +
where <math>y</math> is the cell centroid distance from the wall. Using this a blending can take the following form:
:<math>
:<math>
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   \omega_p = \frac{\sqrt{k_p}}{{C_\mu^{1/4}}\kappa y_p},
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   \omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},
</math>
</math>
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which are the values for <math>k</math> and <math>\omega</math> according to Wilcox(1993) asymptotic analysis of log layer. These wall functions for <math>k</math> and <math>\omega</math> are the results of the solution of model equation for the logarithmic layer.
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Note that for low <math>y</math> values the <math>1/y^2</math> will dominate and therefore viscous value of <math>\omega</math> will be reproduced. Conversely, for larger values of <math>y</math>, <math>1/y</math> will be dominant and logarithmic value will be recovered.
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The above methodology is known to produce spurious results in separated flows, where, by definition, <math>u_\tau = 0</math> at the separation and reattachment point. Many extension of this approach has been proposed.
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==Automatic wall treatments==
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Menter suggested a mechanism that switches automatically between HRN and LRN treatments.
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''The full description to appear soon. The idea is based on blending:''
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 +
Subsequently Menter proposes also blending for friction velocity. Friction velocity for viscous and logarithmic region are:
:<math>
:<math>
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  \omega_\text{vis} = \frac{6\nu}{\beta y^2}
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\begin{matrix}
 +
u^\text{vis}_\tau = \frac{U}{y^{+}}  & &  u_\tau^\text{log} = \frac{U}{\log E y^{+}}
 +
\end{matrix}
</math>
</math>
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And the blending suggested:
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:<math>
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  \omega_\text{log} = \frac{u_\tau}{C_\mu^{1/4} \kappa y}
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</math>
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:<math>
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  \omega_p = \sqrt{\omega_{\text{vis}}^2 + \omega_{\text{log}}^2},
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</math>
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:<math>
:<math>
   u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},
   u_\tau = \sqrt[4]{(u_\tau^{\text{vis}})^4 + (u_\tau^{\text{log}})^4},
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== References ==
== References ==
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* {{reference-paper|author=Wilcox, D.|year=1993|title=Turbulence Modeling for CFD|rest='DCV Industries, Inc. La Canada, California'}}
 
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* {{reference-paper|author=Bredberg, J.|year=2000|title=On the Wall Boundary Condition for Turbulence Models|rest='Internal Report, Department of Thermo and Fluid Dynamics, Chalmers University of Tecyhnology Gotebord, Sweden'}}
 
* {{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=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=''}}
* {{reference-paper|author=ANSYS|year=2006|title=FLUENT Documentation|rest=''}}

Latest revision as of 10:40, 31 March 2021

Turbulence modeling
Turbulence
RANS-based turbulence models
  1. Linear eddy viscosity models
    1. Algebraic models
      1. Cebeci-Smith model
      2. Baldwin-Lomax model
      3. Johnson-King model
      4. A roughness-dependent model
    2. One equation models
      1. Prandtl's one-equation model
      2. Baldwin-Barth model
      3. Spalart-Allmaras model
    3. Two equation models
      1. k-epsilon models
        1. Standard k-epsilon model
        2. Realisable k-epsilon model
        3. RNG k-epsilon model
        4. Near-wall treatment
      2. k-omega models
        1. Wilcox's k-omega model
        2. Wilcox's modified k-omega model
        3. SST k-omega model
        4. Near-wall treatment
      3. Realisability issues
        1. Kato-Launder modification
        2. Durbin's realizability constraint
        3. Yap correction
        4. Realisability and Schwarz' inequality
  2. Nonlinear eddy viscosity models
    1. Explicit nonlinear constitutive relation
      1. Cubic k-epsilon
      2. EARSM
    2. v2-f models
      1. \overline{\upsilon^2}-f model
      2. \zeta-f model
  3. Reynolds stress model (RSM)
Large eddy simulation (LES)
  1. Smagorinsky-Lilly model
  2. Dynamic subgrid-scale model
  3. RNG-LES model
  4. Wall-adapting local eddy-viscosity (WALE) model
  5. Kinetic energy subgrid-scale model
  6. Near-wall treatment for LES models
Detached eddy simulation (DES)
Direct numerical simulation (DNS)
Turbulence near-wall modeling
Turbulence free-stream boundary conditions
  1. Turbulence intensity
  2. Turbulence length scale

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

Contents

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}}, && \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", .
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