Turbulent inertia and the onset of log region in pipe flows

The wallnormal ($y$)-location where the log-region begins in wall-turbulence is the same location where the turbulent inertia or TI (${\rm d}\left<-uv\right>/{\rm d}y $) and the pressure gradient terms from the mean-momentum equation start balancing each other. This location is closely followed by the location, $y_m^+$ where TI vanishes (before becoming negative in the log-region). Dynamics of TI is elucidated using DNS data of pipe flow at $\delta^+ \approx$ 500, 1000 and 2000. We decompose TI as (i) velocity-vorticity correlations ($\left< v \omega_z \right> + \left<- w \omega_y \right>$) and their co-spectra, and (ii) wall-normal gradient of the Reynolds shear stress co-spectra (${\partial \Phi_{-uv}}/{\partial y}$). One interesting result is that the onset of the log-region moves outward with increasing Reynolds number as $ \sim \sqrt{\delta^+}$ because the eddies located close to $y_m^+$ are influenced by large scale accelerating motions of the type $\left<- w \omega_y \right>$ related to vorticity stretching.