On the difficulty of determining Karman 'constants' from DNS

The difficulty of determining the slope of the famed logarithmic law in the mean velocity profile in wall-bounded turbulent flows, the inverse of the Karman 'constant' κ, from direct numerical simulations (DNS) is discussed for channel flow. Unusual approaches, as well as the analysis of the standard log-indicator function are considered and analyzed, leading to the conclusion that a definitive determination of the channel flow κ from DNS with an uncertainty of, say, 2-3% will require the residue of the mean stream-wise momentum equation in the simulations to be reduced by at least an order of magnitude.