Large-$Re$ asymptotics of the stream-wise normal stress in the ZPG turbulent boundary layer

Models for the stream-wise normal stress $\langle uu \rangle^+$ in wall-bounded turbulent flows have been proposed that lead to a log-law in the classical overlap layer (and part of the outer layer). Matching to the wall-layer immediately leads to $\langle uu \rangle^+_{\mathrm{inner}}\sim\ln(Re)$, i.e. to a mixed scaling in the inner layer. While this appears compatible with the observed $Re-\,$dependence of the inner peak, it is shown, in the case of the ZPG TBL, to be incompatible with DNS data and the Reynolds-averaged momentum equation. Matching inner and outer expansions of $\langle uu \rangle^+$ in terms of $1/U^+_{\infty}$ will be presented which are consistent with experimental data and DNS, and allow extrapolation to infinite Reynolds number.