Distance-from-the-wall scaling in turbulent boundary layers

An assessment of self-similarity in the inertial sublayer of turbulent boundary layers (TBL) is presented by simultaneously considering the streamwise and wall-normal velocities. Here, we utilise carefully conducted subminiature $\times$-probe experiments at high Reynolds number. Moreover, the turbulent stresses are compared against results from a synthetic flow where the distance-from-the wall ($z$-) scaling is strictly enforced, following the Attached Eddy Hypothesis. We show that not all stresses approach the asymptotic solution at an equal rate as the friction Reynolds number ($Re_\tau$) is increased. Specifically, the motions which contribute to the wall-normal variance and Reynolds shear stress are found to follow the asymptotic solution at a relatively lower $Re_\tau$ even when the streamwise variance does not, and this trend is attributed to the contribution from attached eddies. Based on these findings, the Reynolds shear stress cospectra, through its $z$-scaling, are used to assess the wall-normal limits where self-similarity applies within the TBL. The limits are found to be consistent with the recent observations that the self-similar region starts and ends at viscous scaled wall-distances of $\mathcal{O}$$(\sqrt{Re_\tau})$ and $\mathcal{O}$$(Re_\tau)$ respectively.