
Wall functions are a collection of semiempirical formulas and functions that in effect ``bridge'' or ``link'' the solution variables at the nearwall 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 lawofthewall for mean velocity yields
where
(10.82) 
and  =  von Kármán constant (= 0.42)  
=  empirical constant (= 9.793)  
=  mean velocity of the fluid at point  
=  turbulence kinetic energy at point  
=  distance from point to the wall  
=  dynamic viscosity of the fluid 
The logarithmic law for mean velocity is known to be valid for about 30 to 60. In FLUENT, the loglaw is employed when .
When the mesh is such that at the walladjacent cells, FLUENT applies the laminar stressstrain relationship that can be written as
It should be noted that, in FLUENT, the lawsofthewall for mean velocity and temperature are based on the wall unit, , rather than ( ). 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 lawofthewall for mean velocity, the lawofthewall 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 highPrandtlnumber 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 nearwall 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 lawofthewall implemented in FLUENT has the following composite form:
where is computed by using the formula given by Jayatilleke [ 117]:
(10.86) 
and
=  thermal conductivity of fluid  
=  density of fluid  
=  specific heat of fluid  
=  wall heat flux  
=  temperature at the cell adjacent to wall  
=  temperature at the wall  
=  molecular Prandtl number ( )  
=  turbulent Prandtl number (0.85 at the wall)  
=  26 (Van Driest constant)  
=  0.4187 (von Kármán constant)  
=  9.793 (wall function constant)  
=  mean velocity magnitude at 
Note that, for the segregated solver, the terms
will be included in Equation 10.85 only for compressible flow calculations.
The nondimensional thermal sublayer thickness, , in Equation 10.85 is computed as the 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 lawofthewall 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, , is computed from the intersection of the linear and logarithmic profiles, and stored.
During the iteration, depending on the value at the nearwall cell, either the linear or the logarithmic profile in Equation 10.85 is applied to compute the wall temperature or heat flux (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 10.85, the lawofthewall for species can be expressed for constant property flow with no viscous dissipation as
where is the local species mass fraction, and are molecular and turbulent Schmidt numbers, and is the diffusion flux of species at the wall. Note that and are calculated in a similar way as and , with the difference being that the Prandtl numbers are always replaced by the corresponding Schmidt numbers.
Turbulence
In the  models and in the RSM (if the option to obtain wall boundary conditions from the equation is enabled), the equation is solved in the whole domain including the walladjacent cells. The boundary condition for imposed at the wall is
where is the local coordinate normal to the wall.
The production of kinetic energy, , and its dissipation rate, , at the walladjacent cells, which are the source terms in the equation, are computed on the basis of the local equilibrium hypothesis. Under this assumption, the production of and its dissipation rate are assumed to be equal in the walladjacent control volume.
Thus, the production of is computed from
and is computed from
The equation is not solved at the walladjacent cells, but instead is computed using Equation 10.810.
Note that, as shown here, the wall boundary conditions for the solution variables, including mean velocity, temperature, species concentration, , and , 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 wallbounded 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 constantshear and local equilibrium hypotheses are the ones that most restrict the universality of the standard wall functions. Accordingly, when the nearwall flows are subjected to severe pressure gradients, and when the flows are in strong nonequilibrium, the quality of the predictions is likely to be compromised.
The nonequilibrium wall functions offered as an additional option can improve the results in such situations.
NonEquilibrium Wall Functions
In addition to the standard wall function described above (which is the default nearwall treatment) a twolayerbased, nonequilibrium wall function [ 128] is also available. The key elements in the nonequilibrium wall functions are as follows:
The lawofthewall for mean temperature or species mass fraction remains the same as in the standard wall function described above.
The loglaw for mean velocity sensitized to pressure gradients is
where
and is the physical viscous sublayer thickness, and is computed from
where .
The nonequilibrium wall function employs the twolayer concept in computing the budget of turbulence kinetic energy at the walladjacent cells, which is needed to solve the equation at the wallneighboring cells. The wallneighboring cells are assumed to consist of a viscous sublayer and a fully turbulent layer. The following profile assumptions for turbulence quantities are made:
where , and is the dimensional thickness of the viscous sublayer, defined in Equation 10.813.
Using these profiles, the cellaveraged production of , , and the cellaveraged dissipation rate, , can be computed from the volume average of and of the walladjacent cells. For quadrilateral and hexahedral cells for which the volume average can be approximated with a depthaverage,
and
where is the height of the cell ( ). For cells with other shapes (e.g., triangular and tetrahedral grids), the appropriate volume averages are used.
In Equations 10.815 and 10.816, the turbulence kinetic energy budget for the wallneighboring 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 nonequilibrium 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 wallneighboring cells. Thus, the nonequilibrium wall functions, in effect, partly account for nonequilibrium effects neglected in the standard wall function.
Standard Wall Functions vs. NonEquilibrium Wall Functions
Because of the capability to partly account for the effects of pressure gradients and departure from equilibrium, the nonequilibrium 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 (skinfriction 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 highReynoldsnumber, wallbounded flows. The nonequilibrium wall functions further extend the applicability of the wall function approach by including the effects of pressure gradient and strong nonequilibrium. 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 nearwall modeling approach combined with adequate mesh resolution in the nearwall region. FLUENT provides the enhanced wall treatment for such situations. This approach can be used with the three  models and the RSM.