The friction factor of two-dimensional rough-boundary turbulent soap film flows

TL;DR

This paper predicts the friction factor in two-dimensional turbulent soap-film flows with rough boundaries, revealing different scaling laws depending on the dominant cascade and Reynolds number, supported by numerical simulations.

Contribution

It introduces a theoretical framework for friction factor scaling in 2D turbulent soap-film flows considering both energy and enstrophy cascades, validated by numerical simulations.

Findings

Friction factor scales as Re^{-1/2} in enstrophy-dominated flows at intermediate Re.

At large Re, the friction factor is proportional to roughness r in enstrophy-dominated flows.

Numerical simulations agree with theoretical predictions and show data collapse scaling similar to 3D pipe flow phenomena.

Abstract

We use momentum transfer arguments to predict the friction factor $f$ in two-dimensional turbulent soap-film flows with rough boundaries (an analogue of three-dimensional pipe flow) as a function of Reynolds number Re and roughness $r$, considering separately the inverse energy cascade and the forward enstrophy cascade. At intermediate Re, we predict a Blasius-like friction factor scaling of $f\propto\textrm{Re}^{-1/2}$ in flows dominated by the enstrophy cascade, distinct from the energy cascade scaling of $\textrm{Re}^{-1/4}$. For large Re, $f \sim r$ in the enstrophy-dominated case. We use conformal map techniques to perform direct numerical simulations that are in satisfactory agreement with theory, and exhibit data collapse scaling of roughness-induced criticality, previously shown to arise in the 3D pipe data of Nikuradse.