Asymptotic turbulent friction in 2D rough-walled flows

TL;DR

This study experimentally investigates the asymptotic turbulent friction in 2D rough-walled flows, revealing a different scaling law from 3D flows and linking friction to the turbulent spectrum's spectral exponent.

Contribution

It demonstrates that in 2D turbulent flows, the friction scales linearly with roughness, contrasting with the 3D case, and connects this to the turbulent spectral exponent.

Findings

In 2D flows, friction f scales linearly with roughness r.

The f -- r relation in 2D differs from the 3D case.

Friction is linked to the turbulent spectrum's spectral exponent.

Abstract

The friction f is the property of wall-bounded flows that sets the pumping cost of a pipeline, the draining capacity of a river, and other variables of practical relevance. For highly turbulent rough-walled pipe flows, f depends solely on the roughness length scale r, and the f -- r relation may be expressed by the Strickler empirical scaling f $\propto$ r$^{\frac{1}{3}}$. Here, we show experimentally that for soap film flows that are the two-dimensional (2D) equivalent of highly turbulent rough-walled pipe flows, f $\propto$ r and the f -- r relation is not the same in 2D as in 3D. Our findings are beyond the purview of the standard theory of friction but consistent with a competing theory in which f is linked to the turbulent spectrum via the spectral exponent $\alpha$: In 3D, $\alpha$ = 5/3 and the theory yields f $\propto$ r$^{\frac{1}{3}}$; in 2D, $\alpha$ = 3 and the theory yields f $\propto$ r.