Turbulent-laminar patterns in shear flows without walls

TL;DR

This paper introduces a simplified, low-dimensional model of shear flows without walls that accurately captures turbulent-laminar patterns and transition phenomena, reducing computational costs and applicable to various flow geometries.

Contribution

The authors develop a minimal Fourier-mode model for stress-free shear flows that reproduces key turbulent-laminar transition features and extends to different flow configurations.

Findings

Model captures spot growth, turbulent bands, and uniform turbulence.

Quantitative agreement with turbulence in interior plane Couette flow.

Stress-free flow models for plane Poiseuille and pipe flows successfully replicate transition structures.

Abstract

Turbulent-laminar intermittency, typically in the form of bands and spots, is a ubiquitous feature of the route to turbulence in wall-bounded shear flows. Here we study the idealised shear between stress-free boundaries driven by a sinusoidal body force and demonstrate quantitative agreement between turbulence in this flow and that found in the interior of plane Couette flow -- the region excluding the boundary layers. Exploiting the absence of boundary layers, we construct a model flow that uses only four Fourier modes in the shear direction and yet robustly captures the range of spatiotemporal phenomena observed in transition, from spot growth to turbulent bands and uniform turbulence. The model substantially reduces the cost of simulating intermittent turbulent structures while maintaining the essential physics and a direct connection to the Navier-Stokes equations. We demonstrate the generic nature of this process by introducing stress-free equivalent flows for plane Poiseuille and pipe flows which again capture the turbulent-laminar structures seen in transition.