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10.8.2 Wall Functions

Wall functions are a collection of semi-empirical formulas and functions that in effect ``bridge'' or ``link'' the solution variables at the near-wall cells and the corresponding quantities on the wall. The wall functions comprise

FLUENT offers two choices of wall function approaches:

Standard Wall Functions

The standard wall functions in FLUENT are based on the proposal of Launder and Spalding [ 141], and have been most widely used for industrial flows. They are provided as a default option in FLUENT.

Momentum

The law-of-the-wall for mean velocity yields


\begin{displaymath} U^* = \frac{1}{\kappa} \, \ln (E y^*) \end{displaymath} (10.8-1)

where


\begin{displaymath} U^* \equiv \frac{U_P C_{\mu}^{1/4} k_P^{1/2}}{\tau_w/\rho} \end{displaymath} (10.8-2)


\begin{displaymath} y^* \equiv \frac{\rho C_{\mu}^{1/4} k_P^{1/2} y_P}{\mu} \end{displaymath} (10.8-3)


and $\kappa$ = von Kármán constant (= 0.42)
  $E$ = empirical constant (= 9.793)
  $U_P$ = mean velocity of the fluid at point $P$
  $k_P$ = turbulence kinetic energy at point $P$
  $y_P$ = distance from point $P$ to the wall
  $\mu$ = dynamic viscosity of the fluid

The logarithmic law for mean velocity is known to be valid for $y^* >$ about 30 to 60. In FLUENT, the log-law is employed when $y^* > 11.225$.

When the mesh is such that $y^* < 11.225$ at the wall-adjacent cells, FLUENT applies the laminar stress-strain relationship that can be written as


\begin{displaymath} U^* = y^* \end{displaymath} (10.8-4)

It should be noted that, in FLUENT, the laws-of-the-wall for mean velocity and temperature are based on the wall unit, $y^*$, rather than $y^+$ ( $\equiv \rho u_{\tau} y/\mu$). These quantities are approximately equal in equilibrium turbulent boundary layers.

Energy

Reynolds' analogy between momentum and energy transport gives a similar logarithmic law for mean temperature. As in the law-of-the-wall for mean velocity, the law-of-the-wall for temperature employed in FLUENT comprises the following two different laws:

The thickness of the thermal conduction layer is, in general, different from the thickness of the (momentum) viscous sublayer, and changes from fluid to fluid. For example, the thickness of the thermal sublayer for a high-Prandtl-number fluid (e.g., oil) is much less than its momentum sublayer thickness. For fluids of low Prandtl numbers (e.g., liquid metal), on the contrary, it is much larger than the momentum sublayer thickness.

In highly compressible flows, the temperature distribution in the near-wall region can be significantly different from that of low subsonic flows, due to the heating by viscous dissipation. In FLUENT, the temperature wall functions include the contribution from the viscous heating [ 273].

The law-of-the-wall implemented in FLUENT has the following composite form:


\begin{displaymath} T^* \equiv \frac{\left(T_w - T_P\right) \rho c_p C_{\mu} ^{1... ..._c^2 \right\} & \; \; \; (y^* > y^*_T) \\ \end{array}\right . \end{displaymath} (10.8-5)

where $P$ is computed by using the formula given by Jayatilleke [ 117]:


\begin{displaymath} P = 9.24 \left[ \left(\frac{\rm Pr}{{\rm Pr}_t}\right)^{3/4}... ... \right] \left[1 + 0.28 e^{-0.007 {\rm Pr}/{\rm Pr}_t} \right] \end{displaymath} (10.8-6)

and


  $k_f$ = thermal conductivity of fluid
  $\rho$ = density of fluid
  $c_p$ = specific heat of fluid
  $\dot{q}$ = wall heat flux
  $T_P$ = temperature at the cell adjacent to wall
  $T_w$ = temperature at the wall
  ${\rm Pr}$ = molecular Prandtl number ( $\mu c_p/k_f$)
  ${\rm Pr}_t$ = turbulent Prandtl number (0.85 at the wall)
  $A$ = 26 (Van Driest constant)
  $\kappa$ = 0.4187 (von Kármán constant)
  $E$ = 9.793 (wall function constant)
  $U_c$ = mean velocity magnitude at $y^{*}=y_T^{*}$

Note that, for the segregated solver, the terms


\begin{displaymath}\frac{1}{2} \rho {\rm Pr} \frac{C_\mu^{1/4}k_P^{1/2}}{\dot{q}} U_P^2\end{displaymath}

and

\begin{displaymath}\frac{1}{2} \rho \frac{C_\mu^{1/4}k_P^{1/2}}{\dot{q}} \left\{{\rm Pr}_t U_P^2 + ({\rm Pr} - {\rm Pr}_t)U_c^2 \right\}\end{displaymath}

will be included in Equation  10.8-5 only for compressible flow calculations.

The non-dimensional thermal sublayer thickness, $y^*_T$, in Equation  10.8-5 is computed as the $y^*$ value at which the linear law and the logarithmic law intersect, given the molecular Prandtl number of the fluid being modeled.

The procedure of applying the law-of-the-wall for temperature is as follows. Once the physical properties of the fluid being modeled are specified, its molecular Prandtl number is computed. Then, given the molecular Prandtl number, the thermal sublayer thickness, $y^*_T$, is computed from the intersection of the linear and logarithmic profiles, and stored.

During the iteration, depending on the $y^*$ value at the near-wall cell, either the linear or the logarithmic profile in Equation  10.8-5 is applied to compute the wall temperature $T_w$ or heat flux $\dot{q}$ (depending on the type of the thermal boundary conditions).

Species

When using wall functions for species transport, FLUENT assumes that species transport behaves analogously to heat transfer. Similarly to Equation  10.8-5, the law-of-the-wall for species can be expressed for constant property flow with no viscous dissipation as


\begin{displaymath} Y^* \equiv \frac{\left(Y_{i,w} - Y_{i}\right) \rho C_{\mu}^{... ...^*) + P_c \right] & \; \; \; (y^* > y^*_c) \end{array}\right . \end{displaymath} (10.8-7)

where $Y_{i}$ is the local species mass fraction, ${\rm Sc}$ and ${\rm Sc}_t$ are molecular and turbulent Schmidt numbers, and $J_{i,w}$ is the diffusion flux of species $i$ at the wall. Note that $P_c$ and $y^*_c$ are calculated in a similar way as $P$ and $y^*_T$, with the difference being that the Prandtl numbers are always replaced by the corresponding Schmidt numbers.

Turbulence

In the $k$- $\epsilon$ models and in the RSM (if the option to obtain wall boundary conditions from the $k$ equation is enabled), the $k$ equation is solved in the whole domain including the wall-adjacent cells. The boundary condition for $k$ imposed at the wall is


\begin{displaymath} \frac{\partial k}{\partial n} = 0 \end{displaymath} (10.8-8)

where $n$ is the local coordinate normal to the wall.

The production of kinetic energy, $G_k$, and its dissipation rate, $\epsilon$, at the wall-adjacent cells, which are the source terms in the $k$ equation, are computed on the basis of the local equilibrium hypothesis. Under this assumption, the production of $k$ and its dissipation rate are assumed to be equal in the wall-adjacent control volume.

Thus, the production of $k$ is computed from


\begin{displaymath} G_k \approx \tau_w \frac{\partial U}{\partial y} = \tau_w \frac{\tau_w}{\kappa \rho C_{\mu}^{1/4} k_P^{1/2} y_P} \end{displaymath} (10.8-9)

and $\epsilon$ is computed from


\begin{displaymath} \epsilon_P = \frac{C_{\mu}^{3/4} k_P^{3/2}}{\kappa y_P} \end{displaymath} (10.8-10)

The $\epsilon$ equation is not solved at the wall-adjacent cells, but instead is computed using Equation  10.8-10.

Note that, as shown here, the wall boundary conditions for the solution variables, including mean velocity, temperature, species concentration, $k$, and $\epsilon$, are all taken care of by the wall functions. Therefore, you do not need to be concerned about the boundary conditions at the walls.

The standard wall functions described so far are provided as a default option in FLUENT. The standard wall functions work reasonably well for a broad range of wall-bounded flows. However, they tend to become less reliable when the flow situations depart too much from the ideal conditions that are assumed in their derivation. Among others, the constant-shear and local equilibrium hypotheses are the ones that most restrict the universality of the standard wall functions. Accordingly, when the near-wall flows are subjected to severe pressure gradients, and when the flows are in strong non-equilibrium, the quality of the predictions is likely to be compromised.

The non-equilibrium wall functions offered as an additional option can improve the results in such situations.

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 [ 128] 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 function described above.

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


\begin{displaymath} \frac{\tilde{U} C_{\mu}^{1/4} k^{1/2}}{\tau_w/\rho} = \frac{... ...\ln \left(E \, \frac{\rho C_{\mu}^{1/4} k^{1/2} y}{\mu}\right) \end{displaymath} (10.8-11)

where


\begin{displaymath} \tilde{U} = U - \frac{1}{2} \, \frac{d p}{d x} \left[\frac{y... ...rac{y - y_v}{\rho \kappa \sqrt{k}} + \frac{y_v^2}{\mu} \right] \end{displaymath} (10.8-12)

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


\begin{displaymath} y_v \equiv \frac{\mu y_v^*}{\rho C_{\mu}^{1/4} k_P^{1/2}} \end{displaymath} (10.8-13)

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:


\begin{displaymath} \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 . \end{displaymath} (10.8-14)

where $C_{\ell} = \kappa C_{\mu}^{-3/4}$, and $y_v$ is the dimensional thickness of the viscous sublayer, defined in Equation  10.8-13.

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,


\begin{displaymath} \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) \end{displaymath} (10.8-15)

and


\begin{displaymath} \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 \end{displaymath} (10.8-16)

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  10.8-15 and 10.8-16, the turbulence kinetic energy budget for the wall-neighboring cells is effectively sensitized to the proportions of the viscous sublayer and the fully turbulent layer, which varies widely from cell to cell in highly non-equilibrium flows. It effectively relaxes the local equilibrium assumption (production = dissipation) that is adopted by the standard wall function in computing the budget of the turbulence kinetic energy at wall-neighboring cells. Thus, the non-equilibrium wall functions, in effect, partly account for non-equilibrium effects neglected in the standard wall function.

Standard Wall Functions vs. Non-Equilibrium Wall Functions

Because of the capability to partly account for the effects of pressure gradients and departure from equilibrium, 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 severe pressure gradients and change rapidly. In such flows, improvements can be obtained, particularly in the prediction of wall shear (skin-friction coefficient) and heat transfer (Nusselt or Stanton number).

Limitations of the Wall Function Approach

The standard wall functions give reasonably accurate 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 and strong non-equilibrium. However, the wall function approach becomes 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 items listed above is a prevailing feature of the flow you are modeling, and if it is considered critically important to capture that feature for the success of your simulation, you must employ the near-wall modeling approach combined with adequate mesh resolution in the near-wall region. FLUENT provides the enhanced wall treatment for such situations. This approach can be used with the three $k$- $\epsilon$ models and the RSM.


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