Searching for the log law in open channel flow

Abstract

We carry out direct numerical simulations of flow in a plane open channel at friction Reynolds number up to ${{Re}}_{\tau } \approx 6000$ . We find solid evidence for the presence of universal large-scale organization in the outer layer, with eddies that are larger and stronger than in the closed channel flow. As a result, velocity fluctuations are found to be stronger than in closed channels, throughout the depth. The inner-layer peak of the streamwise velocity variance is observed to grow logarithmically, as in Townsend's attached-eddy model (Townsend, The Structure of Turbulent Shear Flow, 2nd edn, Cambridge University Press, 1976), but saturation of the growth cannot be discarded based on the present data. Although we do not observe a clear outer peak of the streamwise velocity variance, we present substantial evidence that such a peak should emerge at a Reynolds number barely higher than achieved herein. The most striking feature of the flow is the presence of an extended logarithmic layer, with associated Kármán constant asymptoting to $k \approx 0.375$ , in line with observations made in shear-free Couette–Poiseuille flow (Coleman et al., Flow Turbul. Combust., vol. 99, issue 3, 2017, pp. 553–564). The virtual absence of a wake region and of corrective terms to the log law in the present flow leads us to conclude that deviations from the log law observed in internal flows are likely due to the effects of the opposing walls, rather than the presence of a driving pressure gradient.