Large-Reynolds-number asymptotics of the streamwise normal stress in zero-pressure-gradient turbulent boundary layers

A more poetic long title could be ‘A voyage from the shifting grounds of existing data on zero-pressure-gradient (abbreviated ZPG) turbulent boundary layers (abbreviated TBLs) to infinite Reynolds number’. Aided by the requirement of consistency with the Reynolds-averaged momentum equation, the ‘shifting grounds’ are sufficiently consolidated to allow some firm conclusions on the asymptotic expansion of the streamwise normal stress $\langle uu\rangle ^{+}$, where the $^{+}$ indicates normalization with the friction velocity $u_{{\it\tau}}$ squared. A detailed analysis of direct numerical simulation data very close to the wall reveals that its inner near-wall asymptotic expansion must be of the form $f_{0}(y^{+})-f_{1}(y^{+})/U_{\infty }^{+}+\mathit{O}(U_{\infty }^{+})^{-2}$, where $U_{\infty }^{+}=U_{\infty }/u_{{\it\tau}}$, $y^{+}=yu_{{\it\tau}}/{\it\nu}$ and $f_{0}$, $f_{1}$ are $\mathit{O}(1)$ functions fitted to data in this paper. This means, in particular, that the inner peak of $\langle uu\rangle ^{+}$ does not increase indefinitely as the logarithm of the Reynolds number but reaches a finite limit. The outer expansion of $\langle uu\rangle ^{+}$, on the other hand, is constructed by fitting a large number of data from various sources. This exercise, aided by estimates of turbulence production and dissipation, reveals that the overlap region between inner and outer expansions of $\langle uu\rangle ^{+}$ is its plateau or second maximum, extending to $y_{\mathit{break}}^{+}=\mathit{O}(U_{\infty }^{+})$, where the outer logarithmic decrease towards the boundary layer edge starts. The common part of the two expansions of $\langle uu\rangle ^{+}$, i.e. the height of the plateau or second maximum, is of the form $\,A_{\infty }-B_{\infty }/U_{\infty }^{+}+\cdots \,$with $A_{\infty }$ and $B_{\infty }$ constant. As a consequence, the logarithmic slope of the outer $\langle uu\rangle ^{+}$ cannot be independent of the Reynolds number as suggested by ‘attached eddy’ models but must slowly decrease as $(1/U_{\infty }^{+})$. A speculative explanation is proposed for the puzzling finding that the overlap region of $\langle uu\rangle ^{+}$ is centred near the lower edge of the mean velocity overlap, itself centred at $y^{+}=\mathit{O}(\mathit{Re}_{{\it\delta}_{\ast }}^{1/2})$ with $\mathit{Re}_{{\it\delta}_{\ast }}$ the Reynolds number based on free stream velocity and displacement thickness. Finally, similarities and differences between $\langle uu\rangle ^{+}$ in ZPG TBLs and in pipe flow are briefly discussed.