Laminar-Turbulent Patterns in Shear Flows : Evasion of Tipping, Saddle-Loop Bifurcation and Log scaling of the Turbulent Fraction

TL;DR

This paper models the transition to turbulence in shear flows using a reaction-advection-diffusion framework, revealing bifurcations, pattern formation, and a logarithmic scaling law consistent with experimental observations.

Contribution

It introduces a novel one-dimensional model capturing laminar-turbulent pattern dynamics and bifurcations, including saddle-loop scenarios and log scaling of turbulent fractions.

Findings

Identification of a saddle-loop bifurcation leading to solitary pulses.

Demonstration of logarithmic divergence of pattern wavelength.

Model reproduces experimental and numerical shear flow data.

Abstract

We analyze a one-dimensional two-scalar fields reaction advection diffusion model for the globally subcritical transition to turbulence. In this model, the homogeneous turbulent state is disconnected from the laminar one and disappears in a tipping catastrophe scenario. The model however exhibits a linear instability of the turbulent homogeneous state, mimicking the onset of the laminar-turbulent patterns observed in the transitional regime of wall shear flows. Numerically continuing the solutions obtained at large Reynolds numbers, we construct the Busse balloon associated with the multistability of the nonlinear solutions emerging from the instability. In the core of the balloon, the turbulent fluctuations, encoded into a multiplicative noise, select the pattern wavelength. On the lower Reynolds number side of the balloon, the pattern follows a cascade of destabilizations towards larger and larger, eventually infinite wavelengths. In that limit, the periodic limit cycle associated with the spatial pattern hits the laminar fixed point, resulting in a saddle-loop global bifurcation and the emergence of solitary pulse solutions. This saddle-loop scenario predicts a logarithmic divergence of the wavelength, which captures experimental and numerical data in two representative shear flows.