The scaling properties of the mean wall-normal velocity $V(x,y)$ in zero-pressure-gradient laminar and turbulent boundary-layer flows are investigated using numerical simulation data, physical experiment data, and integral analyses of the governing equations. The maximum mean wall-normal velocity $V_{\infty }$ and the boundary-layer thickness $\delta$ are evidenced to be the proper scaling for $V$ over most if not all of the boundary layer. This is different from the behavior of the mean streamwise velocity $U$ or the turbulent shear stress $T=-\rho \langle uv\rangle$, which depend on different characteristic length scales in the regions near and away from the surface, respectively. The reason for this apparent difference in scaling behaviors is described physically relative to the downstream development of the $U$ velocity profile and the mechanisms of boundary-layer growth. Insights pertaining to this are further surmised from an analytical relationship for the ratio of the displacement to momentum thickness, i.e., shape factor $H$. Integral analyses using the continuity and mean momentum equation show that $U_{\infty }{V}_{\infty }/u_{\tau }^2=H$, where $u_{\tau }$ is the friction velocity. Both the laminar similarity solution and direct numerical simulation data in post-transitional flows convincingly support this relation. Over the transitional regime, data of sufficiently high quality are lacking to check if this relation remains valid.

• Received 26 August 2016

DOI:https://doi.org/10.1103/PhysRevFluids.1.082401