Turbulent transition in a truncated one-dimensional model for shear flow

TL;DR

This paper introduces a simplified one-dimensional shear flow model that captures complex spatio-temporal dynamics of turbulence transition, highlighting differences between localized and sinusoidal perturbations and analyzing transient behaviors.

Contribution

The model allows detailed numerical study of shear flow transition, revealing new insights into the effects of localized perturbations and transient chaos with a novel scaling law for initial energy.

Findings

Spanwise-localised perturbations cause more abrupt transition.

Chaotic transient lifetimes follow exponential distributions.

Minimum initial energy scales as Re^-4.3, differing from previous models.

Abstract

We present a reduced model for the transition to turbulence in shear flow that is simple enough to admit a thorough numerical investigation while allowing spatio-temporal dynamics that are substantially more complex than those allowed in previous modal truncations. Our model allows a comparison of the dynamics resulting from initial perturbations that are localised in the spanwise direction with those resulting from sinusoidal perturbations. For spanwise-localised initial conditions the subcritical transition to a `turbulent' state (i) takes place more abruptly, with a boundary between laminar and `turbulent' flow that is appears to be much less `structured' and (ii) results in a spatiotemporally chaotic regime within which the lifetimes of spatiotemporally complicated transients are longer, and are even more sensitive to initial conditions. The minimum initial energy $E_0$ required for a spanwise-localised initial perturbation to excite a chaotic transient has a power-law scaling with Reynolds number $E_0 \sim Re^p$ with $p \approx -4.3$. The exponent $p$ depends only weakly on the width of the localised perturbation and is lower than that commonly observed in previous low-dimensional models where typically $p \approx -2$. The distributions of lifetimes of chaotic transients at fixed Reynolds number are found to be consistent with exponential distributions.