Transition to disordered front propagation

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.

1) The UK fluid dynamics community will benefit from the training of two PDRAs with a unique inter-disciplinary outlook. The multi-disciplinary programme of research outlined in this proposal will provide the PDRAs with scientific training of outstanding quality both in breadth and depth, which meets the recommendations of the IRMS 2010 for fluid dynamics, in combining computational fluid dynamics with experiments. This will help to develop the UK's expertise and the output will be highly-skilled individuals suitable for employment in industry or academia.

2) The project will also be of a more general benefit to the applied scientific community, both in academia and industry, because oomph-lib, the scientific computing library within which the numerical methods will be implemented, is available as "open source" software.The library is already widely used (the oomph-lib mailing list currently has more than 350 subscribers) and is employed for research projects in academia and industry (e.g. by Thales Underwater Ltd. who are involved in an EPSRC-funded KTA project that explores its use in sonar applications). Following the approach adopted in many previous oomph-lib-based research projects, the newly-developed functionality (particularly the routines for computation of stable (and unstable) manifolds) will be augmented by detailed tutorials and then incoporated into the library, allowing its re-use in many other applications.

3) The subject lends itself to the type of public engagement activities at science fairs that AH (co-I) undertakes on a regular basis and for which funds are requested as part of this proposal. The complex dendritic patterns generated by repeated tip-splitting are fascinating to a general audience, and the analogy with shear flow transition is striking; comparisons between experimental and theoretical/numerical results can demonstrate the power of mathematical modelling in a real-world problem.