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Boundary Layer Drag Reduction via Periodic Control of Chaotic Systems


Project Description

Turbulent fluid flows are ubiquitous in the engineering sciences. Particularly prominent examples include pipeline transport of liquids, air flow over aircraft wings and around land vehicles, and water flow around ship hulls. Much of the energy put into these systems is used to overcome the friction or viscous turbulent drag force on the boundaries of the flow, such as the pipe wall or the aircraft wing. Recent experiments and simulations indicate that a saving of up to 25% of the turbulent drag force is achievable by introducing weak oscillations of the flow boundaries perpendicular to the flow direction. There is an intuitive explanation for this reduction, but as yet there is no mathematical foundation or method to predict or to control the extent to which the drag force may be reduced.

This project will develop a mathematical framework and methodology for reducing fluid drag by drawing on recent advances in the dynamical systems approach to fluid flow. The dynamical systems approach describes turbulence in terms of a walk between individual flow structures such as "exact coherent structures" or "unstable periodic orbits", rather than the traditional approach which uses statistics to provide averages over all the structures. Such invariant solutions, that are embedded in chaotic dynamics, have been studied in various contexts, most often in low-dimensional systems like the Lorenz equations or in non-dissipative systems that arise in quantum chaos. Recently, invariant solutions have been identified in many fluid flows. If we can weakly periodically manipulate the flow such that lower-drag structures are visited more frequently during the turbulent ’walk’ than they otherwise would be, then we can reduce the long-time turbulent drag in a targeted, predictable manner.

To start with, the project will examine the effects of weak periodic forcing on low-dimensional chaotic systems such as the Lorenz equations (used to model atmospheric convection) and the Kuramoto-Sivashinski equation (used to model chemical reactions and instabilities of flame fronts). This will develop the foundations of a mathematical description of optimal weak periodic control of chaotic systems. Later in the project, the methodology will be applied to the Navier-Stokes equations in the setting of plane Couette flow or channel flow in order to examine the effect of periodic forcing on coherent structures embedded in realistic flows.

Key skills developed through the project include the mathematics of dynamical systems theory, direct numerical simulation of fluid flows, and advanced techniques at the interface between these two areas.

Interested applicant should send an email to Dr Eaves with a CV and cover letter.

Funding Notes

This is a 36 month fully funded PhD position covering tuition fees and an annual stipend set at UKRI rates (2020/21 Stipend: £15,285). The candidate must have no restrictions on how long they can stay in the UK and have been ordinarily resident in the UK for at least 3 years prior to the start of the studentship (with some further constraint regarding residence for education, further guidance can be found on the EPSRC website). Applicants from EU countries other than the UK who do not comply with the residency criteria are only eligible for a fees-only PhD studentship award.

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