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 localized in the spanwise direction with those resulting from sinusoidal perturbations. For spanwise-localized initial conditions, the subcritical transition to a 'turbulent' state (i) takes place more abruptly, with a boundary between laminar and turbulent flows that appears to be much less 'structured' and (ii) results in a spatio-temporally chaotic regime within which the lifetimes of spatio-temporally complicated transients are longer, and are even more sensitive to initial conditions. The minimum initial energy E(0) required for a spanwise-localized initial perturbation to excite a chaotic transient has a power-law scaling with the Reynolds number E(0) similar to Re(p) with p approximate to -4.3. The exponent p depends only weakly on the width of the localized perturbation and is lower than that commonly observed in previous low-dimensional models where typically p approximate to -2. The distributions of lifetimes of chaotic transients at the fixed Reynolds number are found to be consistent with exponential distributions.