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.
Small Inertia Effect on Depressurization Behavior due to Translational Motion of Spherical Drop. 1st Report. Theoretical Analysis of Outer Expansion Effect and Perturbed Component of Reynolds Number.
