@article {Dawes3066,
author = {Dawes, J. H. P. and Giles, W. J.},
title = {Turbulent transition in a truncated one-dimensional model for shear flow},
volume = {467},
number = {2135},
pages = {3066--3087},
year = {2011},
doi = {10.1098/rspa.2011.0225},
publisher = {The Royal Society},
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 localized in the spanwise direction with those resulting from sinusoidal perturbations. For spanwise-localized initial conditions, the subcritical transition to a {\textquoteleft}turbulent{\textquoteright} state (i) takes place more abruptly, with a boundary between laminar and turbulent flows that appears to be much less {\textquoteleft}structured{\textquoteright} 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 E0 required for a spanwise-localized initial perturbation to excite a chaotic transient has a power-law scaling with the Reynolds number E0\~{}Rep with p≈-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≈-2. The distributions of lifetimes of chaotic transients at the fixed Reynolds number are found to be consistent with exponential distributions.},
issn = {1364-5021},
URL = {http://rspa.royalsocietypublishing.org/content/467/2135/3066},
eprint = {http://rspa.royalsocietypublishing.org/content/467/2135/3066.full.pdf},
journal = {Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences}
}