On the universality of inertial energy in the log layer of turbulent boundary layer and pipe flows

Recent experiments in high Reynolds number pipe flow have shown the apparent obfuscation of the k(x)(-1) behaviour in spectra of streamwise velocity fluctuations (Rosenberg et al. in J Fluid Mech 731: 46-63, 2013). These data are further analysed here from the perspective of the log r behaviour in second-order structure functions, which have been suggested as a more robust diagnostic to assess scaling behaviour. A detailed comparison between pipe flows and boundary layers at friction Reynolds numbers of Re-tau approximate to 5000-20,000 reveals subtle differences. In particular, the log r slope of the pipe flow structure function decreases with increasing wall distance, departing from the expected 2A(1) slope in a manner that is different to boundary layers. Here, A(1) approximate to 1.25, the slope of the log law in the streamwise turbulence intensity profile at high Reynolds numbers. Nevertheless, the structure functions for both flows recover the 2A(1) slope in the log layer sufficiently close to the wall, provided the Reynolds number is also high enough to remain in the log layer. This universality is further confirmed in very high Reynolds number data from measurements in the neutrally stratified atmospheric surface layer. A simple model that accounts for the 'crowding' effect near the pipe axis is proposed in order to interpret the aforementioned differences.