The wall shear stress is needed as a boundary condition and calculated in the following way:

(4–389)

where denotes the density. Equation 4–389 is based on the friction velocity and the alternative velocity scale , which has been introduced to improve the behavior at separation. Both quantities are blended between the viscous sublayer and the logarithmic region.

The friction velocity is given by:

(4–390)

where

(4–391)

(4–392)

with =0.4187 and E=9.793. represents the mean velocity tangent to the wall at the wall-adjacent cell centroid. The non-dimensional distance from the wall is defined using :

(4–393)

denotes the distance from the wall at the wall-adjacent cell centroid and the molecular viscosity.

For the blending of the alternative velocity scale the following formulation is used:

(4–394)

where k denotes the turbulence kinetic energy at the wall-adjacent cell centroid.

For the transport equation of the turbulence kinetic energy, a zero flux boundary condition is specified not only in the logarithmic region, but also (artificially) for the integration down to the wall. The wall production term is blended between the viscous sublayer and logarithmic region. The production term can be written using and in the following way:

(4–395)

which gives the following relations for the viscous sublayer and logarithmic region:

(4–396)

(4–397)

Note that is not based on a gradient computation, but directly derived from the log-law relation in Equation 4–392.

The blended form of the wall production term finally reads:

(4–398)

In the transport equation of the specific dissipation rate, the value of ω at the wall is prescribed. The wall value is blended between the relations of the viscous sublayer and logarithmic region in the following way:

(4–399)

where and denote the density and the dynamic viscosity, respectively. u* represents the alternative velocity scale given in Equation 4–394.

Analytical solutions for can be given for both the laminar sublayer and the logarithmic region. This strategy can used to define a -insensitive near wall treatment by blending between these two analytical expressions depending on :

(4–400)

(4–401)

(4–402)

This formulation is the current default treatment in Ansys Fluent and is labelled further-on as "correlation". Incompressible plane Couette flow at Reynolds number 10⁶ has been used to calibrate the entire formulation for momentum and turbulence equations in order to minimize the variation of the wall shear stress over . This leads to a value of 1/3 for C_calib and 1.3 for C_exp. However, due to its analytical expression, this formulation offers only limited possibilities for fine tuning.

A further improvement was achieved by using a table look-up for for a wide range of values. Incompressible plane Couette flow has again been used to calibrate this formulation in order to minimize the variation of the wall shear stress over . For this purpose, a fine grid solution was created, and the resulting wall shear stress was used as a reference on coarser grids. A table was created by running the plane Couette flow for various meshes, storing the optimal and then transforming the obtained data sets into an efficient table look-up for .

Specific tables have been created for BSL, SST, and GEKO turbulence models and this method is further-on labelled as "tabulated". All other ω-based turbulence models are using the SST table look-up so that for example, the Transition SST, DES-SST, and SAS-SST (all implemented as separate models) are using a consistent table.

Both formulations agree well for high quality meshes with small values of y+ (y+ < 1) and moderate expansion ratios. Although these meshes are generally recommended, it might not always be possible to create such meshes due to the complexity of the applications. The table look-up offers a more accurate prediction of the wall shear stress, especially for poorer meshes having larger values of y+ and/or higher expansion ratios.

As an example of this improvement, the results of a zero pressure gradient flat plate boundary layer flow are presented below. The setup is based on the experiment of Wieghardt & Tillmann [783] and the Reynolds number based on inlet velocity and length of the flat plate is Re=10⁷. Figure 4.18: Comparison of the skin friction distribution on a fine (y+ ~ 0.1, left) and coarse (y+ ~ 22, right) mesh for the ‘correlation’ and ‘tabulated’ near wall treatments with SST. shows the skin friction distribution for a fine (y+ ~ 0.1) and coarse (y+ ~ 22) mesh computed with the SST turbulence model. The visual difference between the 2 formulations appears small and confirms that the correlation model is already reasonably close.

Figure 4.18: Comparison of the skin friction distribution on a fine (y+ ~ 0.1, left) and coarse (y+ ~ 22, right) mesh for the ‘correlation’ and ‘tabulated’ near wall treatments with SST.

However, looking at the overall drag reveals that the "tabulated" (new) model is superior for this case. The error in drag prediction is defined as

(4–403)

where the reference solution corresponds to the solution on a very fine mesh with y+ ~ 0.001. The results are summarized in Table 1 and it can be seen that the error is strongly reduced on the coarse mesh for the "tabulated" near wall treatment:

The near wall treatment of the energy equation follows a Kader blending method between the viscous sublayer and logarithmic law of the wall as outlined by Equation 4–379.