Uncertainty analysis of the von Kármán constant

In 1930, von Kármán presented an expression for the mean velocity distribution in channel and pipe flows that can be transformed into the today well-known logarithmic velocity distribution. At the same time, he also formulated the logarithmic skin friction law and obtained a value of 0.38 for the constant named after him through pipe flow pressure drop measurements. Different approaches to determine the von Kármán constant from mean velocity measurements have been proposed over the last decades, sometimes giving different results even when employed on the same data, partly because the range over which the logarithmic law should be fitted is also under debate. Up to today, the research community has not been able to converge toward a single value and the favored values range between 0.36 and 0.44 for different research groups and canonical flow cases. The present paper discusses some pitfalls and error sources of commonly employed estimation methods and shows, through the use of boundary layer data from Österlund (1999) that von Kármán’s original suggestion of 0.38 seems still to be valid for zero pressure gradient turbulent boundary layer flows. More importantly, it is shown that the uncertainty in the determination of the von Kármán constant can never be less than the uncertainty in the friction velocity, thereby yielding a realistic uncertainty for the most debated constant in wall turbulence.