Transition to disordered front propagation

Lead Research Organisation: University of Manchester
Department Name: Physics and Astronomy


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.

Planned Impact

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.
Description Most time-evolving systems are nonlinear. This means that even deterministic systems, in which randomness does not play a role, can exhibit complex, non-intuitive behaviour which reduces their predictability, for example through sensitivity to initial conditions commonly referred to as the Butterfly effect. The most alarming example of such a dynamical system is the Earth's climate. Nonlinear dynamical systems feature remarkable properties. They may undergo sudden changes in behaviour known as bifurcations upon variation of an external control parameter. They may also support multiple stable behaviours for fixed parameter values and transition between them upon application of a sufficiently large perturbation. Bubble propagation in a Hele-Shaw channel is a playground for understanding fundamental features of non-linear dynamics.

We study experimentally and numerically the propagation of an air bubble through a fluid-filled, geometrically-perturbed Hele-Shaw channel; a system which supports several stable modes of steady bubble propagation. During its transient evolution, a bubble may undergo several topological changes in the form of breakup and coalescence, depending on both initial conditions and control parameters. Long-term, either a single asymmetric or symmetric bubble is recovered or else multiple bubbles remain, whose
relative separation increases with time. We explore how the transient and long-term evolution is controlled by edge states available to the bubble - weakly unstable states with a stable manifold that determines the basin boundary separating initial conditions decaying to different long-term stable behaviours, which may change with each new topological configuration. Single-bubble edge states have also been identified and we have discovered sensitivity to initial conditions in a simple setting where the system can choose between only three long-term outcomes. We are currently focusing on the increasing sensitivity to initial conditions as the flow rate is increased and the associated transition to disorder.

This award has enabled the interdisciplinary training of three PDRAs, who have all secured further research positions as an outcome of this project.
Exploitation Route Our findings offer a framework as to how we interpret time-evolution in nonlinear dynamical systems. This framework is relevant to subcritical nonlinear processes where there are several stable states, so that finite-amplitude perturbations are required to transition from one state to another.
Sectors Energy,Environment,Manufacturing, including Industrial Biotechology

Description The key scientific impact from this grant is a framework in which to interpret complex dynamics in droplet flows. This has direct impact in other two-phase flows such as viscous fingering instabilities and also more broadly in complex systems where a key question concerns the ability to predict future dynamics. The project has enabled the development of a complex experimental facility which is currently used in a follow-up PhD project funded by an EPSRC DTP, from which two papers have been published so far, one further paper is under review and another in the final stages of preparation. WE have made significant headway in understanding of complexity in front propagation based on modern ideas from dynamical systems theory and we have by now met some of the objectives of the third work package. The PhD student is an outstanding experimentalist and scientist and we expect to exceed the objectives of the grant. The central scientific question we are seeking to resolve is whether disordered front propagation is a self-sustaining dynamical state of the system or a noise-driven phenomenon. The project also provides academic and non-academic impact in the availability of the numerical codes developed in the open-source, multi-physics, finite-element library oomph-lib. These developments in oomph-lib have recently enabled a collaboration with Moltex Energy, who are funding a post-doc for 6 months to determine the feasibility of liquid thermometers in cutting-edge nuclear technology.
First Year Of Impact 2020
Sector Energy
Impact Types Economic

Title Data to support "Bifurcations of drops and bubbles propagating in variable-depth Hele-Shaw channels" 
Description Text files containing data needed to reconstruct figures in the paper "Bifurcations of drops and bubbles propagating in variable-depth Hele-Shaw channels". 
Type Of Material Database/Collection of data 
Year Produced 2021 
Provided To Others? Yes  
Description Cambridge DAMTP 
Organisation University of Cambridge
Department Department of Applied Mathematics and Theoretical Physics (DAMTP)
Country United Kingdom 
Sector Academic/University 
PI Contribution The collaboration with DAMTP contributed to the development of the ideas that led to the funding of this project.
Collaborator Contribution We previously published three joint papers in the Journal of Fluid Mechanics.
Impact The collaboration is multi-disciplinary in the sense that it allies experimental physics and mathematics. The outcomes have been academic with the publication of three papers.
Start Year 2014
Description Keynote talk at APS/DFD annual meeting 2020 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Keynote invited presentation at major conference in the field: American Physical Society Annual meeting of the Division of Fluid Dynamics.
Year(s) Of Engagement Activity 2020
Description Keynote talk at EFMC14 in Patras, Greece 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Prestigious keynote lecture in leading European conference on fluid mechanics.
Year(s) Of Engagement Activity 2022
Description Leeds Institute of Fluid Dynamics colloquium 2019 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Online colloquium for LIFD with an audience of around 80.
Year(s) Of Engagement Activity 2019
Description Seminar KTH Stockholm 2020 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact Invitation to present online lecture in international seminar series.
Year(s) Of Engagement Activity 2020