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4.12.3 Non-Equilibrium Wall Functions

In addition to the standard wall function described above (which is the default near-wall treatment) a two-layer-based, non-equilibrium wall function [ 166] is also available. The key elements in the non-equilibrium wall functions are as follows:

The law-of-the-wall for mean temperature or species mass fraction remains the same as in the standard wall functions described above.

The log-law for mean velocity sensitized to the pressure gradients is

 \frac{\widetilde{U} C_{\mu}^{1/4} k^{1/2}}{\tau_w/\rho} = \f... ...\ln \left(E \, \frac{\rho C_{\mu}^{1/4} k^{1/2} y}{\mu}\right) (4.12-14)


 \widetilde{U} = U - \frac{1}{2} \, \frac{d p}{d x} \left[\fr... ...rac{y - y_v}{\rho \kappa \sqrt{k}} + \frac{y_v^2}{\mu} \right] (4.12-15)

and $y_v$ is the physical viscous sublayer thickness, and is computed from

 y_v \equiv \frac{\mu y_v^*}{\rho C_{\mu}^{1/4} k_P^{1/2}} (4.12-16)

where $y_v^* = 11.225$.

The non-equilibrium wall function employs the two-layer concept in computing the budget of turbulence kinetic energy at the wall-adjacent cells, which is needed to solve the $k$ equation at the wall-neighboring cells. The wall-neighboring cells are assumed to consist of a viscous sublayer and a fully turbulent layer. The following profile assumptions for turbulence quantities are made:

 \tau_t = \left\{ \begin{array}{ll} 0, & y < y_v \\ \tau_w, ... ...\frac{k^{3/2}}{{C_{\ell}}^* y}, & y > y_v \end{array} \right . (4.12-17)

where ${C_{\ell}}^* = \kappa C_{\mu}^{-3/4}$, and $y_v$ is the dimensional thickness of the viscous sublayer, defined in Equation  4.12-16.

Using these profiles, the cell-averaged production of $k$, $\overline{G_k}$, and the cell-averaged dissipation rate, $\overline{\epsilon}$, can be computed from the volume average of $G_k$ and $\epsilon$ of the wall-adjacent cells. For quadrilateral and hexahedral cells for which the volume average can be approximated with a depth-average,

 \overline{G_k} \equiv \frac{1}{y_n} \, \int_0^{y_n} \tau_t \... ...o C_{\mu}^{1/4} k_P^{1/2}} \, \ln \left(\frac{y_n}{y_v}\right) (4.12-18)


 \overline{\epsilon} \equiv \frac{1}{y_n} \, \int_0^{y_n} \ep... ...C_{\ell}}^*} \, \ln \left(\frac{y_n}{y_v}\right)\right] \, k_P (4.12-19)

where $y_n$ is the height of the cell ( $y_n = 2 y_P$). For cells with other shapes (e.g., triangular and tetrahedral grids), the appropriate volume averages are used.

In Equations  4.12-18 and 4.12-19, the turbulence kinetic energy budget for the wall-neighboring cells is effectively depends on the proportions of the viscous sublayer and the fully turbulent layer, which varies widely from cell to cell in highly non-equilibrium flows. The nonequilibrium wall functions account for the effect of pressure gradients on the distortion of the velocity profiles. In such cases the assumption of local equilibrium, when the production of the turbulent kinetic energy is equal to the rate of its distruction, is no longer valid. Therefore, the non-equilibrium wall functions, in effect, partly account for the non-equilibrium effects that are neglected in the standard wall functions.

Standard Wall Functions vs. Non-Equilibrium Wall Functions

Because of the capability to partly account for the effects of pressure gradients, the non-equilibrium wall functions are recommended for use in complex flows involving separation, reattachment, and impingement where the mean flow and turbulence are subjected to pressure gradients and rapid changes. In such flows, improvements can be obtained, particularly in the prediction of wall shear (skin-friction coefficient) and heat transfer (Nusselt or Stanton number).


Non-equilibrium wall functions are available with the following turbulence closures:
  • k- $\epsilon$ models

  • Reynolds Stress Transport models

Limitations of the Wall Function Approach

The standard wall functions give reasonable predictions for the majority of high-Reynolds-number wall-bounded flows. The non-equilibrium wall functions further extend the applicability of the wall function approach by including the effects of pressure gradient; however, the above wall functions become less reliable when the flow conditions depart too much from the ideal conditions underlying the wall functions. Examples are as follows:

If any of the above listed features prevail in the flow you are modeling, and if it is considered critically important for the success of your simulation, you must employ the near-wall modeling approach combined with the adequate mesh resolution in the near-wall region. ANSYS FLUENT provides the enhanced wall treatment for such situations. This approach can be used with the $k$- $\epsilon$ and the RSM models.

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