Revisit eddy viscosity in pressure-driven wall turbulence at high Reynolds number

Abstract

We investigate eddy-viscosity distributions in pressure-driven wall turbulence for three canonical configurations: plane closed-channel flow, open-channel flow with a free-slip surface, and pipe flow. Using direct numerical simulation (DNS) databases spanning friction Reynolds numbers $Re_{\tau}=$ 2000--12000, we infer the eddy viscosity from one-point statistics through the Boussinesq relation. The DNS-inferred eddy viscosity displays configuration-dependent behavior in the outer region, indicating that a single full-depth expression is not uniformly accurate for all three configurations. Building on the interpretation of eddy viscosity as the product of a velocity scale and a length scale, we extend the log-law scaling into the outer region. Specifically, we adopt a stress-based velocity scale and introduce an outer correction function to capture the remaining dependence on the outer coordinate. We then embed a compact parametric form of this correction into a Cess-type framework with van Driest near-wall damping, yielding a full-depth eddy-viscosity model. We assess the model using eddy-viscosity profiles, the log-law indicator function, and skin friction. The results show that the proposed model yields noticeable improvement for open-channel flow while remaining comparable to the classical Cess model for closed-channel flow and pipe flow. These findings underscore the role of outer boundary conditions in shaping the outer-region eddy viscosity and, consequently, mean-flow predictions.