Scaling of normal stresses in the turbulent boundary layer

Concentrating on the canonical zero pressure gradient (ZPG) turbulent boundary layer (TBL), different scalings of normal stresses, in particular of $$, have been proposed. In the range of Reynolds numbers where measurements are available, the best data collapse is obtained by scaling stream-wise fluctuations with the free stream velocity $U_{\infty}$. It is shown with the underlying RANS equations that this choice, together with the traditional Rotta outer scale $\delta^{\ast} U^+_{\infty}$ and the ``log law'' leads to a boundary layer thickness which decreases in the downstream direction. In other words, if one insists on both the traditional mean flow similarity and on scaling normal stresses with $U_{\infty}$, all (growing) TBLs seen so far are very far from their true asymptotic (shrinking) state. Alternative assumptions/scalings and their consequences will be discussed.