Low-dimensional models of the transition to turbulence
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Date
26/11/2019Author
Thomson, Stuart William
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Abstract
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.