Streamwise energy-transfer mechanisms in zero- and adverse-pressure-gradient turbulent boundary layers

The present study investigates streamwise ($\overline{u^2}$) energy-transfer mechanisms in the inner and outer regions of turbulent boundary layers (TBLs). Particular focus is placed on the $\overline{u^2}$-production, its inter-component and wall-normal transport as well as dissipation, all of which become statistically significant in the outer region with increasing friction Reynolds number ($Re_{\tau}$). These properties are analyzed using published data sets of zero, weak and moderately strong adverse-pressure-gradient (APG) TBLs across a decade of $Re_{\tau}$, revealing similarity in energy-transfer pathways for all these TBLs. It is found that both the inner and outer peaks of $\overline{u^2}$ are always associated with local maxima in the $\overline{u^2}$-production and its inter-component transport, and the regions below/above each of these peaks are always dominated by wall-ward/away-from-wall transport of $\overline{u^2}$, thereby classifying the $\overline{u^2}$-profiles into four distinct regimes. This classification reveals existence of phenomenologically similar energy-transfer mechanisms in the `inner' and `outer' regions of moderately strong APG TBLs, which meet at an intermediate location coinciding with the minimum in $\overline{u^2}$ profiles. Given that the wall-ward/away-from-wall transport of $\overline{u^2}$ is governed by the $\rm Q_4$(sweeps)/$\rm Q_2$(ejections) quadrants of the Reynolds shear stress, it is argued that the emergence of the $\overline{u^2}$ outer peak corresponds with the statistical dominance of $\rm Q_4$ events in the outer region. Besides unravelling the dynamical significance of $\rm Q_2$ and $\rm Q_4$ events in the outer region of turbulent boundary layers, the present analysis also proposes new phenomenological arguments for testing on canonical wall-turbulence data at very high $Re_{\tau}$.