Note: A useful distinction: the wall law is physics, the wall functions are numerics.

In turbulence studies, particularly for cases with complex geometries, resolving the physical phenomena occurring near the wall is generally costly in terms of computation time, because it requires a very fine mesh in those zones where the velocity and temperature gradients are very high. Yet it is essential to predict these phenomena correctly, because by simple conservation of the flow rate, a poor wall description of the velocity will lead to a poor description in the core of the flow. Rather than focusing the effort on a fine resolution, it is then common to focus the effort on a modelling of the wall velocity gradient, which makes it possible to keep a relatively coarse mesh at the wall. These approaches are known as "wall functions" or "wall treatment" and allow a notable decrease of the simulation times. They have long been integrated into most industrial computation codes.

Reichardt wall law

The laws available with the \(\overline{k}\)– \(\overline{\epsilon}\) model in TrioCFD are formulated so as to describe the entire boundary layer continuously. For the dimensionless velocity, the Reichardt wall law is used [143] :

\[U^{+}=\frac{1}{\kappa}\ln(1+\kappa y^{+})+A\left(1-e^{-y^{+}/11}-\frac{y^{+}}{11}e^{-y^{+}/3}\right) \]

with

\[A=\frac{1}{\kappa}\ln\left(\frac{E}{\kappa}\right) \]

The values of the constants are \(\kappa=0.415\) and \(E=9.11\). Asymptotically, the linear behaviour is recovered when \(y^{+}\) tends to zero, and the logarithmic behaviour when \(y^{+}\) becomes "large". Moreover, this law gives a reasonable description of the buffer zone. The quantities \(\overline{k}\) and \(\overline{\epsilon}\) are described, for all \(y^{+}\), by:

\[\begin{aligned} \overline{k}^{+} & = 0.07\,y^{+2}e^{-y^{+}/9}+\frac{1}{\sqrt{C_{\eta}}}\left(1-e^{-y^{+}/20}\right)^{2}\\ \overline{\epsilon}^{+} & = \frac{1}{\kappa(y^{+4}+15^{4})^{1/4}} \end{aligned} \]

These formulations respect the behaviour generally accepted at the wall, namely:

\[\overline{k}(y=0)=0\,;\qquad\frac{d\overline{k}}{dy}(y=0)=0\,;\qquad\frac{d\overline{\epsilon}}{dy}(y=0)=0 \]

Other wall treatments in TrioCFD

The law developed by Ciofalo and Collins [22] is also available, but only for the VDF discretization of TrioCFD (Cartesian meshes). It is also possible to modify the values of the constants of the logarithmic law, or to impose the friction velocity, but this seems of little use in industrial studies. Concerning LES, the law of Werner and Wengle is also implemented [182], as well as the TBLE (Thin Boundary Layer Equation) approach.