Low-dimensional models of the transition to turbulence

The transition to turbulence in shear flows such as pressure driven pipe flow or plane Couette flow presents an interesting theoretical problem: how do we understand the existence of chaos when the laminar flow is stable to infinitesimal perturbations? A number of approaches to the problem have been used in recent years and a great deal of progress has been made towards understanding the transition, often utilising low-dimensional models to generate hypotheses. In the first part of this thesis I study the behaviour of a system of partial differential equations based on the damped Kuramoto-Sivashinsky equation which exhibit a subcritical transition to turbulence as a control parameter is varied. Typical lifetimes of the system are measured and align with the scenario for shear flows; they have an exponential distribution for a given value of the control parameter, and the typical lifetime scale superexponentially with that parameter. Coherent structures are found numerically and the linear stability measured to create a bifurcation diagram which is reminiscent of the ones found in shear flows. In the second part of the thesis a link is drawn between the apparent dynamical role of the lower branch states of the extended KS equation and the current understanding of transitional turbulence as belonging to the universality class of Directed Percolation (DP). A novel DP model is introduced which has a third state which represents the behaviour of the lower branches and the critical exponents of the system are measured and found to agree with the expected exponents for 1+1 dimensional DP. A non-universal parameter is found which varies with the strength of the bouncing behaviour, although it is unclear if it is possible to measure this parameter in a meaningful way for a real flow. Finally, in the third part of the thesis the extended KS equation is studied in an extended spatial domain, to confirm the hypothesis that this system also belongs to the DP universality class. Critical exponents are measured and found to agree with 1+1 DP. This confirms that the system has a transition which reproduces many of the important features subcritical fluid flows like pressure driven pipe flow, or plane Couette flow