PhD projects

PhD opportunity!

Applications to study may be received at any time (for the following, or anything else in my research areas).  Applications for School-funded scholarships open October 2026, must be submitted by early January 2027, and are for an October 2027 start.

Modelling of flow transition and heat transfer for advanced geothermal systems  

This PhD project is an exciting opportunity to work at the intersection of advanced applied mathematics and industrial energy innovation.  In collaboration with SLB (slb.com)—a global leader in energy technology—this project aims to understand the fundamental mechanisms of turbulence and flow transition in the context of next-generation geothermal systems.

Project Overview: An unexplored nonlinear landscape

Advanced Geothermal Systems (AGSs) are a promising frontier for sustainable energy - they aim to overcome geographical limitations by transferring heat through deep closed-loop pipe networks, and have the advantage over wind-energy of being continuous and reliable.  More widely, fluid flow through pipes is a central mechanism for energy transfer across countless applications, from domestic heating systems to cutting-edge technologies like solar collectors and heat pumps. 

However, the efficiency of these systems depends critically on the nature of the flow — whether it is smooth and orderly or chaotic and turbulent. Counterintuitively, increased heating often suppresses turbulence, thereby reducing the effectiveness of heat transfer itself! Under such conditions, it is difficult to predict the performance of an application in advance, and experimental data can exhibit considerable scatter. The horizontal orientation of AGSs’s closed-loop pipe networks is especially interesting, as the buoyancy introduces an upward flow component that disrupts traditional turbulence mechanisms, creating a rich and largely unexplored space of nonlinear flow behaviour.

New Approach: Dynamical systems and Coherent structures.

The project will build on recent breakthroughs in dynamical-systems approaches to turbulence in isothermal flows, which have enabled substantial developments in our understanding of the appearance of turbulence.  Several newly-established methods are now in place to extract fundamental ‘coherent structures’, such as travelling wave and periodic orbit solutions, that distil the fundamental physical processes of the system.  New optimisation techniques now make it possible to navigate the high-dimensional space of turbulent systems, allowing researchers to identify critical perturbations that can trigger or suppress turbulence. These tools will be applied to design flow modifications that can directly enhance heat transfer in AGS applications.

Research Environment & Training

The successful applicant will be jointly supervised by leading academics from the School of Mathematical and Physical Sciences (MPS) and the School of Mechanical, Aerospace and Civil Engineering (MAC) at the University of Sheffield, with support from SLB. Regular meetings with the industrial partner and the possibility of a research placement will ensure a strong link between theoretical development and practical application.

You’ll join the interdisciplinary Sheffield Fluid Mechanics Group (Group Website) — a large community of researchers spanning mathematics and engineering sciences.  You will be part of a multidisciplinary network of students within MPS and students on related projects in MAC, including an ongoing PhD project focused on improving computational models of AGS flows.

You will develop high-value transferrable skills with applications across academia and industry. In addition, fluid dynamics itself is a very employable area: “The total UK turnover of firms engaged in fluid dynamics exceeds £200 billion and together they employ over 500,000 people, illustrating how fluid dynamics activity is often embedded in larger organisations.” [1].  This project includes training in: Nonlinear dynamical systems modelling, Large-scale simulations and high-performance computing, Parallel programming and optimisation techniques.    [1]  https://eprints.whiterose.ac.uk/id/eprint/178990/1/Our%20Fluid%20Nation%20-%20The%20Impact%20of%20Fluid%20Dynamics%20in%20the%20UK.pdf

Why Sheffield?

Sheffield offers a comprehensive graduate development programme, including training in research planning, academic writing, and leadership. You’ll have opportunities to supervise undergraduate research projects, helping you build valuable mentoring and teaching experience.

If you're excited by the idea of using advanced mathematics and computation to help shape the future of clean energy, this project offers a rare opportunity.

The University of Sheffield is one of the leading Russell Group universities in the UK. We carry out cutting-edge research with strong links to industry. When you enrol to do a PhD with us, you will be working with world-leading academics and have access to top of the range facilities. As a PhD student you will have the opportunity to gain skills not only to conduct research, but also to take your career to the next level, whether you want to stay in academia, go into industry or the public sector, or set up your own company. You will have access to a range of training and support services to help you excel in your studies and beyond.

Sheffield is also a great place to live!  Check it out...

Supervised by 

Dr Ashley Willis (School of Mathematical & Physical Sciences), second supervisor  Prof Shuisheng He (School of Mechanical Civil & Aerospace Engineering) and Dr Ashley Willis (School of Mathematical & Physical Sciences), and in Collaboration with SLB.

How to apply

Please see this page PhD Study.

Please email a.p.willis@sheffield.ac.uk to discuss your interest in and suitability for the project prior to submitting your application. 

Funding Notes

Applications to study may be received at any time.  Applications for School-funded scholarships open October 2026, must be submitted by early January 2027, and are for an October 2027 start.