Dr. Ashley P. Willis
Senior Lecturer of Applied Mathematics
Room H12,
Hicks Building
School of Mathematics and Statistics
University of Sheffield
S3 7RH, U.K.
+44 (0)114 2223746
a.p.willis/a/sheffield.ac.uk
Office hours:
Usually in my office, H12.
To be sure, please email in advance.
2010/18- : Lecturer/Senior Lecturer, School of Mathematics & Statistics, University of Sheffield, U.K.
2008-10 : EU Marie Curie Fellow (PI), Laboratoire d'Hydrodynamique (LadHyX), Ecole Polytechnique, Paris, France
2005-08 : EPSRC Research Associate, School of Mathematics, University of Bristol, U.K.
2002-05 : NERC Research Fellow (RA), School of Earth Sciences, University of Leeds, U.K.
2002 : PhD Applied Mathematics, Newcastle University, U.K.
Research
Pattern formation in fluid flows (transition to turbulence, nonlinear dynamics, stability, chaos). Astrophysical fluid dynamics and magnetohydrodynamics (flows in planetary interiors, accretion discs, magnetic field generation).
The modelling of fluid flows is perhaps the traditional test-bed for the development of new methods, and there are many applications in the area of clean energy, such as thermal exchangers in solar and geothermal systems, and in wind and wave energy capture. Turbulent flow is characterised by chaotic whirls or eddies, which effectively enhance diffusion and dissipate kinetic energy. This might aid mixing or heat transfer, but dramatically increases the cost of pumping fluids. Therefore, understanding how and when turbulence appears is typically crucial in applications.
On a larger scale, astrophysical applications include the dynamics of flows inside planets and proto-planetary flows about a central mass (accretion discs). Magnetic fields typically play an important role in determining the nature of these astrophysical flows.
PhD Projects
I am looking for motivated students who would like to apply and broaden their mathematical skills. I expect you to be familiar with differential equations and vector calculus; you do not need to have taken advanced courses on fluids.
Project examples:
Transition to chaotic or turbulent flow can linked to solutions of the Navier-Stokes equations, but which solutions are important, and how can we improve methods to calculate them? The solutions we have found so far typically only capture eddies of a single length scale.. how can we find solutions that capture the multi-scale nature of turbulence? Single-scale examples for the flow through a pipe can be seen here.
Can we use optimisation to reduce energy expenditure? For example, experiments have shown that, surprisingly, introducing a partial blockage in a pipe can eliminate turbulence, rather than just stirring the flow up to produce more turbulence. This makes pumping the fluid a lot easier downstream of the blockage, but pumping fluid past the blockage itself it is more expensive. Can we create this effect by less expensive means? Can we find an 'optimal' strategy?
Some planets, like Earth, have a magnetic field, others do not. Typically the magnetic field is produced through the motion of electrically conducting fluid, like liquid iron in Earth's core. But what properties of the flow are essential for magnetic field growth? In this article I found the optimal flow for magnetic field growth (and minimal magnetic Reynolds number), but what is it about this flow that is so special? What happens under more realistic constraints on the flow, such as the geometry of the boundaries, e.g. within in a sphere or spherical shell?
Observations of astrophysical flows around a central body (accretion discs) suggest that these flows are turbulent, while mathematically the flows should be linearly stable. So is there a nonlinear instability mechanism? If so, it occurs at too large flow rate to compute directly in simulation; astrophysical flow rates are huge. Can we develop a reduced model that captures the essential physics to investigate this nonlinear dynamics?
For further information please contact me. Please note that all PhD applications must be made through the university system: School info on PhD places.
Some of my work:
I am the founder and owner of openpipeflow.org, a free and fast simulation code for the study of pipe flow, particularly as a dynamical system. Non-problem-specific codes for dynamical systems include...
A Jacobian-free Newton-Krylov (JFNK) code (MATLAB / FORTRAN90) for finding solutions of high-dimensional nonlinear equations (or low-dimensional if you like!).
A Double pendulum android app: with two double-pendula side-by-side, it is specifically designed to show how initially similar states diverge - the chaotic system exhibits sensitivity to initial conditions (SIC). It also shows how the chaotic paths follow particular repeating patterns, periodic orbits.
A description of the mathematics behind the following videos can be found here.
Collaborator's pages
Elena Marensi , EPSRC Research Assocate, Sheffield, Optimization in Fluid Mechanics ;
Marc Avila Bremen; Predrag Cvitanović Georgia Tech ( chaosbook.org ); Yohann Duguet LIMSI, Paris; Dave Gubbins Leeds; Yongyun Hwang Imperial; Rich Kerswell DAMTP, Cambridge; Jeorge Peixinho PIMM, Paris.
Sheffield Fluid Dynamics Group
SoMaS Applied Mathematics Seminars