The rough wall and smooth wall boundary conditions are formulated continually [67] (in the second edition). The transition value from smooth to rough wall formulation is dependent on the distance of the first cell center from the wall \(y_1\) in wall coordinates \(y^+\).

\begin{eqnarray} \left. k \right|_w &=& 0 \\ \left. \omega \right|_w &=& \begin{cases} {10 \frac{6\nu}{\beta y_1^2}} & k_s^+ \leq \sqrt{\frac{2500\beta}{60}} y^+ \\ {\frac{2500\nu}{k_s^2}} & \sqrt{\frac{2500\beta}{60}} y^+ < k_s^+ \leq 25 \\ \frac{100}{k_s}\sqrt{\frac{\nu U_1}{y_1}} & k_s^+ > 25 \end{cases} \end{eqnarray}

Here \(U_1\) denotes the mean velocity at the first cell center from the wall. The equivalent sand grain roughness is denoted by \(k_s\) with

\begin{equation} k_s^+ = \frac{u_\tau k_s}{\nu}. \end{equation}

The model constant \(\beta\) of the \(\omega\)-equation is defined as in Wilcox k-w and Menter SST, respectively.