Turbulence appearance and non-appearance in thin fluid layers

TL;DR

This paper investigates how turbulence in thin fluid layers depends on the driving mechanism, showing that wall-driven flows cannot sustain turbulence while pressure-driven flows can, leading to distinct flow structures and scaling laws.

Contribution

It demonstrates that flow driving method critically influences turbulence presence in thin layers, revealing new flow regimes and deriving a novel scaling law for the friction factor.

Findings

Wall-driven (Couette) flow cannot sustain turbulence at any viscosity or friction.

Pressure-driven (Poiseuille) flow can sustain turbulence and exhibits traveling wave structures.

A new scaling law for the friction factor as a function of Reynolds number is derived and validated.

Abstract

Flows in fluid layers are ubiquitous in industry, geophysics and astrophysics. Large-scale flows in thin layers can be considered two-dimensional (2d) with bottom friction added. Here we find that the properties of such flows depend dramatically on the way they are driven. We argue that wall-driven (Couette) flow cannot sustain turbulence at however small viscosity and friction. Direct numerical simulations (DNS) up to the Reynolds number $Re=10^6$ confirm that all perturbations die in a plane Couette flow. On the contrary, for sufficiently small viscosity and friction, we show that finite perturbations destroy the pressure-driven laminar (Poiseuille) flow. What appears instead is a traveling wave in the form of a jet slithering between wall vortices. For $10^4<Re<5\cdot10^4$, the mean flow has remarkably simple structure: the jet is sinusoidal with a parabolic velocity profile, vorticity is constant inside vortices, while the fluctuations are small. At higher $Re$ strong fluctuations appear, yet the mean traveling wave survives. Considering the momentum flux barrier in such a flow, we derive a new scaling law for the $Re$-dependence of the friction factor and confirm it by DNS.