Transition to disordered front propagation

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The transition to turbulence in pressure-driven pipe flow has remained the greatest unsolved problem in fluid mechanics since Reynolds' pioneering experiments in the late nineteenth century. Although Poiseuille flow in a cylindrical pipe is linearly stable for all values of the Reynolds number - the ratio of inertial to viscous forces - turbulence can appear for Re > 2000 in the form of localised puffs advected down the pipe, if perturbations exceed a finite-amplitude threshold. In the last twenty years, significant progress in the understanding of the transition to turbulence in pipe flow, and more generally shear flows, has been achieved by focusing on the nonlinear dynamics of these flows. The central question underlying this proposal is whether the complex transition scenario uncovered for shear flows may arise in other fluid mechanical systems.We focus on a canonical flow, Saffman-Taylor fingering in a confined channel - parallel plates separated by a narrow gap so that the width to depth aspect ratio is very large - which is an archetype for front propagation and pattern formation. The displacement of a more viscous fluid (oil) by a less viscous fluid (air) under constant volume-flux (or pressure head) yields patterns ranging from to the steady propagation of a single air finger to unsteady front propagation - highly branched patterns which arise through repeated tip-splitting events and finger competition. This transition exhibits striking similarities with shear flow transition in that (a) the single propagating finger solution of a depth-averaged model is known to be linearly stable up to very large values of the driving parameter, and (b) the threshold value of the driving parameter for transition was found experimentally to be very sensitive to the level of perturbations in the system.In shear flow turbulence, a key theoretical concept is the interpretation of localised turbulent puffs as edge states - weakly unstable states with a stable manifold that determines the basin boundary separating initial conditions decaying to laminar flow from those growing to turbulence. The fundamental hypothesis to be investigated in the proposed research is that unstable solutions of the Saffman-Taylor flow are edge states that underlie both the transition from the steadily propagating Saffman-Taylor finger to the experimentally observed complex patterns, and the dynamics of the patterns themselves. This hypothesis stems from preliminary experimental observations and time-dependent numerical simulations of a depth averaged model, which indicates destabilisation of a bubble through the transient exploration of weakly unstable solutions of the Saffman-Taylor problem, when a large value of parameter is applied from rest.The shear flow transition also exhibits excitable dynamics, in that below threshold a turbulent puff excited by a localised perturbation is a transient excursion from laminar flow, which eventually decays on long time scales. Beyond threshold, a turbulent fixed point appears that enables localised patches of turbulence to grow. We will investigate whether excitable dynamics underlie transition in the Saffman-Taylor problem. We will apply a range of localised or spatially-distributed topographical perturbations of known amplitude in order to probe the dynamical response of the interface and establish the transition threshold as a function of perturbation and driving parameter.Finally, a yet unproven hypothesis of shear flow transition is that turbulence can be characterised by a chaotic meandering between unstable solutions. The Saffman-Taylor fingering problem exhibits a much simpler spatial structure partly because nonlinearities only occur within interfacial conditions. Hence, we will attempt to to characterise disordered front propagation and assess the above hypothesis for the Saffman-Taylor transition scenario.