Werner and Wengle [63] use a piecewise function for the velocity in wall coordinates:

\begin{equation} u^+(y^+) = \begin{cases} y^+ & y^+ \leq A^{\frac{1}{1-B}} \\ A (y^+)^B & \text{otherwise} \end{cases} \end{equation}

with

\begin{equation} A = 8.3 \quad \text{and} \quad B = \frac 1 7. \end{equation}

The relations

\begin{equation} u^+ = \frac{U_p}{u_\tau} \quad \text{and} \quad y^+ \frac{y_p u_\tau}{\nu_w} \end{equation}

are defined for the average velocity within the first cell. For \(y^+ \leq A^{\frac{1}{1-B}}\), the friction velocity can be obtained from the average cell velocity magnitude \(U_p\) by solving \(u^+ = y^+\) for the friction velocity. For greater values of \(y^+\) the average velocity in wall coordinates in the first cell needs to be determined by averaging the piecewise function. This yields a piecewise definition for \(u_\tau(U_p)\). Finally, the intersection between the two pieces in terms of \(U_p\) needs to be computed. At the intersection, the linear part of the function extends over the complete cell height \(\Delta y^+ = A^{\frac{1}{1-B}}\). Solving the definitions of the wall coordinates gives

\begin{equation} U_p = \frac{(y^+)^2 \nu_w}{y_p} = \frac{(\Delta y^+)^2 \nu_w}{4 y_p} \leq \frac{\nu_w}{4 y_p} A^{\frac{2}{1-B}} \end{equation}

for the linear part of \(u^+(y^+)\).

Combining all the above yields the friction velocity as a function of the wall parallel velocity

\begin{equation} u_\tau(U_p) = \begin{cases} \sqrt{\frac{U_p \nu_w}{y_p}} & U_p \leq \frac{\nu_w}{4 y_p} A^{\frac{2}{1-B}} \\ \left[ \frac{1+B}{A} \left( \frac{\nu_w}{2 y_p} \right)^B U_p + \frac{1-B}{2} A^\frac{1+B}{1-B} \left( \frac{\nu_w}{2 y_p} \right)^{1+B} \right]^{\frac{1}{1+B}} & \text{otherwise} \end{cases}. \end{equation}

Activate with control file command STOKES_BC.