Intermediate scaling and logarithmic invariance in turbulent pipe flow

A three-layer asymptotic structure for turbulent pipe flow is proposed revealing, in terms of intermediate variables, the existence of a Reynolds-number-invariant logarithmic region for the streamwise mean velocity and variance. The formulation proposes a local velocity scale (which is not the friction velocity) for the intermediate layer and results in two overlap layers. We find that the near-wall overlap layer is governed by a power law for the pipe for all Reynolds numbers, whereas the log law emerges in the second overlap layer (the inertial sublayer) for sufficiently high Reynolds numbers (Re-tau). This provides a theoretical basis for explaining the presence of a power law for the mean velocity at low Re-tau and the coexistence of power and log laws at higher Re-tau. The classical von Karman (.) and Townsend-Perry (A(1)) constants are determined from the intermediate-scaled log-law constants;. shows a weak trend at sufficiently high Re-tau but falls within the commonly accepted uncertainty band, whereas A1 exhibits a systematic Reynolds-number dependence until the largest available Re-tau. The key insight emerging from the analysis is that the scale separation between two adjacent layers in the pipe is proportional to root Re-tau (rather than Re-tau) and therefore the approach to an asymptotically invariant state can be expected to be slow.