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

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.