Dr Ashley Willis

Position: Senior lecturer
Home page: http://apwillis.staff.shef.ac.uk
Telephone: (0114) 2223746
Office: H12 Hicks building
Photo of Ashley Willis


MAS280 Mechanics and Fluids Information  
MAS316 Mathematical modelling of natural systems Information Home page
MAS414 Mathematical Modelling of Natural Systems Information Home page


Interests: Fluid dynamics, Magnetohydrodynamics (MHD)
Research group: Fluid Dynamics
Publications: Preprint page, ArXiv, MathSciNet


Past grants, as Principal Investigator
Optimisation in fluid mechanics
Efficient capture of the dominant periodic orbits underlying turbulent fluid flow
Past grants, as Coinvestigator
Application to STFC for Consolidated Grant Support for the Solar Physics and Space Plasma Research Centre (SP2RC) STFC


Admissions Lead, Programme Leader: Study Abroad


2010/18-present: Lecturer/Senior Lecturer, School of Mathematics and Statistics, 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: Ph.D. Applied Mathematics, Newcastle University, U.K.

Research interests:

Google Scholar page for APW

Images/videos on my page

Founder of openpipeflow.org.

General interests: 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. Is there a nonlinear instability mechanism? If so, it is 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?

Fun stuff

Working arrangements:

Usually in my office, H12. To be sure, please email in advance.