The energy budget at the outer peak of $\overline{u^2}$ in turbulent pipe flow.

We use the NSTAP data from the Princeton superpipe (Vallikivi PhD thesis 2014) to examine the spatial \& spectral energy fluxes close to the outer peak in $\overline {u^2}$ in the range of Reynolds numbers, $2.1 \times 10^6 \le Re_D \le 6.0 \times 10^6$, for which the ratio of hot-wire length to Kolmogorov length scale is $ l/\eta \approx 10$. Previous results (Hultmark {\it et al.} PRL, {\bf 108}, 2012) suggest that the outer peak emerges at $Re_D \approx 1.1 \times 10^6$, its position exhibiting a locus $y_m^+=0.23(Re^+)^{0.67}$. We note that this is close to the position of the well-known ``mesolayer'', which we have also described as the intermediate layer with scaling ($y_m^+ \propto \sqrt{Re_\tau}, u_m$), where $u_m$ is the rms velocity at $y=y_m$ (Diwan \& Morrison, TSFP11 -- see also the presentation in the session, ``Turbulent Boundary Layers''). It is straightforward to show that the locus of $u_m$ is close to that for the production, $P_m(\overline{u^2})$, where the local-equilibrium approximation approximately holds. Therefore, spectral dynamics are most Kolmogorov-like because spatial transport is minimal. We examine the inertial scaling of the axial velocity spectra and low-order structure functions to explain the importance of intermediate scaling.