EPSRC Reference: 
EP/P026044/1 
Title: 
Transition to disordered front propagation 
Principal Investigator: 
Juel, Professor A 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Physics and Astronomy 
Organisation: 
University of Manchester, The 
Scheme: 
Standard Research 
Starts: 
01 July 2017 
Ends: 
31 December 2019 
Value (£): 
479,041

EPSRC Research Topic Classifications: 
Continuum Mechanics 
Fluid Dynamics 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
The transition to turbulence in pressuredriven 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 finiteamplitude 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, SaffmanTaylor 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 volumeflux (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 tipsplitting events and finger competition. This transition exhibits striking similarities with shear flow transition in that (a) the single propagating finger solution of a depthaveraged 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 SaffmanTaylor flow are edge states that underlie both the transition from the steadily propagating SaffmanTaylor finger to the experimentally observed complex patterns, and the dynamics of the patterns themselves. This hypothesis stems from preliminary experimental observations and timedependent numerical simulations of a depth averaged model, which indicates destabilisation of a bubble through the transient exploration of weakly unstable solutions of the SaffmanTaylor 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 SaffmanTaylor problem. We will apply a range of localised or spatiallydistributed 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 SaffmanTaylor 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 SaffmanTaylor transition scenario.

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