Refinement of the logarithmic law of the wall

Available direct numerical simulation of turbulent channel flow at moderately high Reynolds numbers data show that the logarithmic diagnostic function is a linearly decreasing function of the outer-normalized wall distance eta = y/delta with a slope proportional to the von Karman constant, kappa = 0.4. The validity of this result for turbulent pipe and boundary layer flows is assessed by comparison with the mean velocity profile from experimental data. The results suggest the existence of a flow-independent logarithmic law (U) over bar (+) = (U) over bar /u(tau) = (1/kappa) ln(y*/a), where y* = yUs/nu with U-S = yS(y) being the local shear velocity and the two flow-independent constants kappa = 0.4 and a = 0.36. The range of its validity extends from the inner-normalized wall distance y(+) = 300 up to half the outer-length scale eta = 0.5 for internal flows, and eta = 0.2 for zero-pressure-gradient turbulent boundary layers. Likewise, and within the same range, the mean velocity deficit follows a flow-dependent logarithmic law as a function of a local mean-shear-based coordinate. Furthermore, it is illustrated how the classical friction laws for smooth pipe and zero-pressure-gradient turbulent boundary layer are recovered from this scaling.