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Title: Is the von Karman constant universal across wall-bounded flows?
Plain: 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: 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.
Settled by: 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.
Kind: well-posed
Answer type: value
References: (empty)
Literature status: contested
Status note: 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.
Posed since: 1930