Optimization in Fluid Mechanics

Lead Research Organisation: University of Bristol
Department Name: Mathematics


This project aims to realise the full potential of optimisation as a theoretical tool to study fluid mechanics motivated by our need to better understand and control flows around us. As an exemplar, the drag experienced by vehicles as they move through either air or water is a huge consumer of energy and source of carbon emissions which the UK urgently needs to reduce. In the past, optimisation has generally only been used with simplified constraints such as the linearised Navier-Stokes equations to keep problems tractable. Recently, however, two breakthroughs now strongly suggest that the solutions to more sophisticated optimisation problems can be successfully computed and a recent experiment highlights what may be achieved using clever geometry design.

This project will seek to exploit these exciting advances by developing new optimisation-based approaches to treat three key problems in fluid mechanics: 1) how to systematically search for new nonlinear flow solutions to the governing Navier-Stokes equations; 2) how to manipulate nonlinear stability via boundary geometry to design more energy-efficient fluid flows in pipelines; and 3) how to calculate the best rigorous upper estimates of energy consumption (or drag) in fully turbulent shear and convective flows.

Planned Impact

The proposed research, which seeks to develop the full potential of optimisation techniques in fluid mechanics, will have an impact on knowledge, society, people and the economy in the following ways.

The impact on knowledge will be through the mathematical techniques developed with beneficiaries being applied mathematicians, physicists and engineers. The key hope is to show how optimisation techniques can be used to answer new questions as well as making existing techniques more accessible and appealing (easier to apply). Our research results will be disseminated by journal publications, seminars, conference presentations and by posting on open access arXiv servers. Computer codes developed during the work will be made freely available on the web.

The impact on society will be the raised profile of the two University schools hosting the research (Bristol and Sheffield) and the international reputation of the UK in the research area of fluid mechanics. This, in turn, will attract people seeking training into the UK and to these Schools increasing their income streams. Another tangible impact will be the eagerness of non-UK researchers to actively seek out and collaborate with UK researchers.

The impact on people will most immediately be through the RAs who will receive training in, and experience using, the modern optimisation techniques which form the subject of the grant. Also, presuming a successful outcome of the grant, the PI and CI will continue to develop the optimisation approach further and consequently will train future postgraduates in this research area.

One of the objectives of the grant is (via optimisation) to develop a more energy-efficient pipeline in which the flow has a greater tolerance of ambient noise before becoming turbulent. There is therefore the clear potential of reducing the energy consumption associated with pumping fluids over long distances and as a result carbon emissions. Contact with BP and Schlumberger will ensure that professional engineers will be made aware of progress and possible benefits to the economy.


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Kerswell R.R. (2018) Nonlinear nonmodal stability theory in Annual reviews of fluid mechanics

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Olvera D (2017) Exact coherent structures in stably stratified plane Couette flow in Journal of Fluid Mechanics

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Wilson A (2018) Can libration maintain Enceladus's ocean? in Earth and Planetary Science Letters

Related Projects

Project Reference Relationship Related To Start End Award Value
EP/P001130/1 01/12/2016 31/08/2017 £322,600
EP/P001130/2 Transfer EP/P001130/1 01/09/2017 31/03/2020 £261,160
Description One of the key objectives of the grant was to explore whether adding new constraints to a variational problem would improve the upper bound estimate of the heat flux through a fluid convecting turbulently. Interestingly, for the remaining constraints available from the governing Boussinesq equation, this was not found to be the case. The implication is that the mathematical technique used (the background method) has been exhausted and is of no further use. This is an important albeit negative result which terminates investigation of the method. This result is documented in an article currently in press at the Journal of Fluid Mechanics (accepted 15th Dec 2019).

The time-stepping approach was developed further and a limitation found (for the some cases no proof was available that it would converge). This is also documented in the Journal of Fluid Mechanics article (in press).

Results have also been obtained optimally designing baffles to relaminarise turbulent pipe flows (two papers are currently under review). Interestingly the baffle is more effective than anticipated in relaminarising the flow but the cost in terms of enhanced pressure resistance is also more costly tha envisaged. This was another unanticipated finding.

The grant is still ongoing and 3 papers (2 under review and one in press) are still to be added.
Exploitation Route The time-stepping approach for solving the variational problems considered can be taken up by other applied mathematicians and engineers. The approach to designing a baffle to relaminarise turbulent pipe flows is also ripe for further development. In fact, the postdoc working on it intends to continue working on this problem when he sets up his own group at the Harbin Institute of Technology, China where he has been appointed as an assistant professor. The PI anticipates a future collaboration between Cambridge and Harbin.
Sectors Energy