Is the von Karman constant universal across wall-bounded flows?
In plain words
Close to a smooth wall, the mean speed of a turbulent flow grows with the logarithm of distance from the wall, with a slope set by the von Karman constant. Pipes, channels and boundary layers give slightly different values, and it is unclear whether one number holds for all.
Precise statement
In the overlap layer of smooth-wall turbulence, $U+ = (1/\kappa) \ln y+ + B$, with $U+ = U/u_{\tau}, y+ = y u_{\tau}/\nu$ and $u_{\tau}$ the friction velocity. Determine the $\mathrm{Re}_{\tau} \to \infty$ limit of $\kappa$ for pipe, channel and zero-pressure-gradient boundary layer, and whether the three coincide; reported values range from about 0.37 to 0.42. Answer: $\kappa$ for each flow with uncertainty below 0.005.
What would settle it
High-accuracy wall-shear and profile measurements in all three geometries at $\mathrm{Re}_{\tau} > 10^{4}$ with matched instrumentation, plus DNS at $\mathrm{Re}_{\tau} \sim 10^{4}$ or above.
Status in the literature
Flow-dependent values were proposed by Nagib and Chauhan (2008); pipe facility data and channel DNS up to $\mathrm{Re}_{\tau} \sim 10^{4}$ have not closed the question.