Refinement of the logarithmic law of the wall

Abstract

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 η = y/δ with a slope proportional to the von Kármán constant, κ = 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 + = U /u τ = (1/κ) ln(y * /a), where y * = yU S /ν with U S = yS(y) being the local shear velocity and the two flow-independent constants κ = 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 η = 0.5 for internal flows, and η = 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.

The Law of the Wall and von Kármán Constant: An Ongoing Controversial Debate

“…The setting a = 0.334 (which is close to a = 0.36 applied by Laadhari [68]) recovers U + = κ −1 ln y + + 5.03 implied by the PVM, i.e., Laadhari's model recovers the implications of the PVM.…”

“…The difference is that Luchini does not make an attempt to introduce different von Kármán constants and other constants for different flows considered. Another model that supports the validity and universality of the log-law is the model of Laadhari [68]. The model reads U + = κ −1 ln(y * /a), where y * = y +2 S + and κ = 0.40.…”

“…Recent analyses in favor of universality were presented, e.g., in Refs. [56,[66][67][68][69][70]. In particular, a comprehensive analysis of available numerical and experimental data up to very high Re was presented recently based on observational physics criteria [9,10].…”