Scaling of mean skin friction in turbulent boundary layers, and fully-developed pipe and channel flows

TL;DR

This paper presents a new theoretical derivation and a semi-empirical model that unify the scaling of skin friction across zero-pressure-gradient turbulent boundary layers, pipes, and channels, demonstrating near-universal behavior when accounting for boundary condition effects.

Contribution

The work introduces a dynamically consistent scaling theory for skin friction applicable to multiple flow types and proposes a universal finite-Re model with an empirical correction for boundary conditions.

Findings

Finite-Re model fits individual flow data well

Universal scaling improves data collapse across flows

Outer boundary condition effects can be empirically corrected

Abstract

An asymptotic $-1/2$ power-law scaling and a semi-empirical finite-$Re$ model were recently presented by Dixit et al. (2020) for skin friction in zero-pressure-gradient (ZPG) turbulent boundary layers (TBLs). In this work, a new derivation is presented which shows that these relations (i) fundamentally represent a dynamically-consistent scaling of skin friction for nominally two-dimensional ZPG TBLs and fully-developed pipes and channels, and (ii) apply individually to each of these flows. The new theoretical arguments are based on transfer of kinetic energy from mean flow to large eddies of turbulence and depend neither on flow geometry nor outer boundary condition, both of which distinguish one type of flow from the other. Using skin friction data from the literature, it is demonstrated that the finite-$Re$ model describes, as predicted by the theory, data from individual flows remarkably well; these data cover the complete range of laboratory/simulation Reynolds numbers to date. It is, however, observed that performance of the model degrades while attempting to describe data from all flows in a universal fashion. Differences in outer boundary condition and large-scale structures amongst different types of flows appear to be responsible for this degradation. An empirical correction based on Clauser's shape factor, is proposed to absorb the outer boundary condition effects into the scaling of skin friction. This correction leads to a new universal scaling and a robust, semi-empirical, universal finite-$Re$ model for skin friction in ZPG TBLs, pipes and channels. Remarkable collapse of data from all flows in the new scaling underscores the importance of a dynamically-consistent approach towards revealing universality of skin friction in wall turbulence.