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

(4–338)

where

(4–339)

is the dimensionless velocity.

(4–340)

is the dimensionless distance from the wall.

The range of values for which wall functions are suitable depend on the overall Reynolds number of the flow. The lower limit always lies in the order of ~15. Below this limit, wall functions will typically deteriorate and the accuracy of the solutions cannot be maintained (for exceptions, see Scalable Wall Functions). The upper limit depends strongly on the Reynolds number. For very high Reynolds numbers (for example, ships, airplanes), the logarithmic layer can extend to values as high as several thousand, whereas for low Reynolds number flows (for example, turbine blades, and so on.) the upper limit can be as small as 100. For these low Reynolds number flows, the entire boundary layer is frequently only of the order of a few hundred units. The application of wall functions for such flows should therefore be avoided as they limit the overall number of nodes one can sensibly place in the boundary layer. In general, it is more important to ensure that the boundary layer is covered with a sufficient number of (structured) cells than to ensure a certain value.

In Ansys Fluent, the log-law is employed when . When the mesh is such that at the wall-adjacent cells, Ansys Fluent applies the laminar stress-strain relationship that can be written as

(4–341)

It should be noted that, in Ansys Fluent, the law-of-the-wall for mean velocity and temperature are based on the wall unit, , rather than (). These quantities are approximately equal in equilibrium turbulent boundary layers.

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 Ansys Fluent is composed of 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 (for example, oil) is much less than its momentum sublayer thickness. For fluids of low Prandtl numbers (for example, liquid metal), on the contrary, it is much larger than the momentum sublayer thickness.

(4–342)

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 Ansys Fluent, the temperature wall functions include the contribution from the viscous heating  [673].

The law-of-the-wall is implemented in Ansys Fluent for the non-dimensional temperature using the wall scaling:

(4–343)

Here the convective-conductive part and viscous heating part are modeled using the following composite forms:

(4–344)

(4–345)

where is computed by using the formula given by Jayatilleke  [281]:

(4–346)

Note that, for the pressure-based solver, the terms

and

will be included in Equation 4–343 only for compressible flow calculations.

The non-dimensional thermal sublayer thickness, , in Equation 4–343 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 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, , is computed from the intersection of the linear and logarithmic profiles, and stored.

During the iteration, depending on the value at the near-wall cell, either the linear or the logarithmic profile in Equation 4–343 is applied to compute the wall temperature or heat flux (depending on the type of the thermal boundary conditions).

The function for given by Equation 4–346 is relevant for the smooth walls. For the rough walls, however, this function is modified as follows:

(4–347)

where is the wall function constant modified for the rough walls, defined by . To find a description of the roughness function , you may refer to Equation 4–413 in the Fluent Theory Guide.

When using wall functions for species transport, Ansys Fluent assumes that species transport behaves analogously to heat transfer. Similarly to Equation 4–343, the law-of-the-wall for species can be expressed for constant property flow with no viscous dissipation as

(4–348)

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.

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 wall-adjacent cells. The boundary condition for imposed at the wall is

(4–349)

where is the local coordinate normal to the wall.

The production of kinetic energy, , and its dissipation rate, , at the wall-adjacent 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 wall-adjacent control volume.

Thus, the production of is based on the logarithmic law and is computed from

(4–350)

and is computed from

(4–351)

The equation is not solved at the wall-adjacent cells, but instead is computed using Equation 4–351. and Reynolds stress equations are solved as detailed in y+-Insensitive Near-Wall Treatment for ω-based Turbulence Models and Wall Boundary Conditions, respectively.

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 Ansys 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 from the ideal conditions that are assumed in their derivation. Among others, the constant-shear and local equilibrium assumptions 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 are offered as an additional option, which can potentially improve the results in such situations.

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).

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. For such situations, Ansys Fluent provides the enhanced wall treatment (available for the - and the RSM models), as well as the Menter-Lechner near-wall treatment (available for the - model).

In Ansys Fluent’s near-wall model, the viscosity-affected near-wall region is completely resolved all the way to the viscous sublayer. The two-layer approach is an integral part of the enhanced wall treatment and is used to specify both and the turbulent viscosity in the near-wall cells. In this approach, the whole domain is subdivided into a viscosity-affected region and a fully-turbulent region. The demarcation of the two regions is determined by a wall-distance-based, turbulent Reynolds number, , defined as

(4–359)

where is the wall-normal distance calculated at the cell centers. In Ansys Fluent, is interpreted as the distance to the nearest wall:

(4–360)

where is the position vector at the field point, and is the position vector of the wall boundary. is the union of all the wall boundaries involved. This interpretation allows to be uniquely defined in flow domains of complex shape involving multiple walls. Furthermore, defined in this way is independent of the mesh topology.

In the fully turbulent region (; ), the - models or the RSM (described in Standard, RNG, and Realizable k-ε Models and Reynolds Stress Model (RSM)) are employed.

In the viscosity-affected near-wall region (), the one-equation model of Wolfstein  [716] is employed. In the one-equation model, the momentum equations and the equation are retained as described in Standard, RNG, and Realizable k-ε Models and Reynolds Stress Model (RSM). However, the turbulent viscosity, , is computed from

(4–361)

where the length scale that appears in Equation 4–361 is computed from  [103]

(4–362)

The two-layer formulation for turbulent viscosity described above is used as a part of the enhanced wall treatment, in which the two-layer definition is smoothly blended with the high-Reynolds number definition from the outer region, as proposed by Jongen  [284]:

(4–363)

where is the high-Reynolds number definition as described in Standard, RNG, and Realizable k-ε Models or Reynolds Stress Model (RSM) for the - models or the RSM. A blending function, , is defined in such a way that it is equal to unity away from walls and is zero in the vicinity of the walls. The blending function has the following form:

(4–364)

The constant determines the width of the blending function. By defining a width such that the value of will be within 1% of its far-field value given a variation of , the result is

(4–365)

Typically, would be assigned a value that is between 5% and 20% of . The main purpose of the blending function is to prevent solution convergence from being impeded when the value of obtained in the outer layer does not match with the value of returned by the Wolfstein model at the edge of the viscosity-affected region.

The field in the viscosity-affected region is computed from

(4–366)

The length scales that appear in Equation 4–366 are computed from Chen and Patel [103]:

(4–367)

If the whole flow domain is inside the viscosity-affected region (), is not obtained by solving the transport equation; it is instead obtained algebraically from Equation 4–366. Ansys Fluent uses a procedure for the blending of that is similar to the -blending in order to ensure a smooth transition between the algebraically-specified in the inner region and the obtained from solution of the transport equation in the outer region.

The constants in Equation 4–362 and Equation 4–367, are taken from  [103] and are as follows:

(4–368)

To have a method that can extend its applicability throughout the near-wall region (that is, viscous sublayer, buffer region, and fully-turbulent outer region) it is necessary to formulate the law-of-the wall as a single wall law for the entire wall region. Ansys Fluent achieves this by blending the linear (laminar) and logarithmic (turbulent) law-of-the-wall using a function suggested by Kader [286]:

(4–369)

where the blending function is given by:

(4–370)

where and .

Similarly, the general equation for the derivative is

(4–371)

This approach allows the fully turbulent law to be easily modified and extended to take into account other effects such as pressure gradients or variable properties. This formula also guarantees the correct asymptotic behavior for large and small values of and reasonable representation of velocity profiles in the cases where falls inside the wall buffer region ().

The enhanced wall functions were developed by smoothly blending the logarithmic layer formulation with the laminar formulation. The enhanced turbulent law-of-the-wall for compressible flow with heat transfer and pressure gradients has been derived by combining the approaches of White and Cristoph  [706] and Huang et al.  [260]:

(4–372)

where

(4–373)

and

(4–374)

(4–375)

(4–376)

where is the location at which the log-law slope is fixed. By default, . The coefficient in Equation 4–372 represents the influences of pressure gradients while the coefficients and represent the thermal effects. Equation 4–372 is an ordinary differential equation and Ansys Fluent will provide an appropriate analytical solution. If , , and all equal 0, an analytical solution would lead to the classical turbulent logarithmic law-of-the-wall.

The laminar law-of-the-wall is determined from the following expression:

(4–377)

Note that the above expression only includes effects of pressure gradients through , while the effects of variable properties due to heat transfer and compressibility on the laminar wall law are neglected. These effects are neglected because they are thought to be of minor importance when they occur close to the wall. Integration of   Equation 4–377 results in

(4–378)

Enhanced thermal wall functions follow the same approach developed for the profile of . The unified wall thermal formulation blends the laminar and logarithmic profiles according to the method of Kader  [286]:

(4–379)

where the notation for and is the same as for standard thermal wall functions (see Equation 4–343). Furthermore, the blending factor is defined as

(4–380)

where is the molecular Prandtl number, and the coefficients and are defined as in Equation 4–370.

Apart from the formulation for in Equation 4–379, the enhanced thermal wall functions follow the same logic as for standard thermal wall functions (see Energy), resulting in the following definition for turbulent and laminar thermal wall functions:

(4–381)

(4–382)

where the quantity is the value of at the fictitious "crossover" between the laminar and turbulent region. The function is defined in the same way as for the standard wall functions.

A similar procedure is also used for species wall functions when the enhanced wall treatment is used. In this case, the Prandtl numbers in Equation 4–381 and Equation 4–382 are replaced by adequate Schmidt numbers. See Species for details about the species wall functions.

The boundary conditions for the turbulence kinetic energy are similar to the ones used with the standard wall functions (Equation 4–349). However, the production of turbulence kinetic energy, , is computed using the velocity gradients that are consistent with the enhanced law-of-the-wall (Equation 4–369 and Equation 4–371), ensuring a formulation that is valid throughout the near-wall region.

The wall shear stress is needed as a boundary condition and is calculated as:

(4–383)

where denotes the density. Both of the friction velocities ( and ) are blended between the viscous sublayer and the logarithmic region. The following formulation is used for the blending of :

(4–384)

For the friction velocity , the following formulation is used:

(4–385)

The main idea of the Menter-Lechner near-wall treatment is to add a source term to the transport equation of the turbulence kinetic energy that accounts for near-wall effects. The standard - model is modified as shown in the following equations (for the sake of simplicity, buoyancy effects are not included):

(4–386)

(4–387)

(4–388)

where , , , , and .

The additional source term is active only in the viscous sublayer and accounts for low-Reynolds number effects. It automatically becomes zero in the logarithmic region. The exact formulation of this source term is at this point proprietary and is therefore not provided here.

In combination with the Menter-Lechner near-wall treatment, the iterative treatment and linearization of the - two-equation model has been improved. This modification is activated by default when this near-wall treatment is used.

The Menter-Lechner near-wall treatment can be used together with the standard, realizable, and RNG - turbulence models.