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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^{+}

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).

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^{+}}.

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}}


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.

Automatic wall treatments










This means that all boundary conditions for

In Fluent, that means:




  • ANSYS (2006), "FLUENT Documentation", .
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