This page lists potential PhD projects in fluids and solids, offered by faculty members of the School of Mathematics. For further details of the School's research in this area, please see the Applied Mathematics Research Group page. For further information about an individual project, please contact the listed supervisor. For information about submitting an application, please see our Research Degrees page.

Further related projects can be found in separate lists for Biological and Industrial Mathematics and Mathematical Physics.

## Modelling the Breaking-Wave Flows and Forces On and Inside a Porous Structure Modelling the Breaking-Wave Flows and Forces On and Inside a Porous Structure

Supervisor: Dr Mark Cooker.

When a sea wave overturns and breaks onto a shingle beach some of the water is forced into the spaces between the pebbles. Shingle is a porous medium, and water injected at the surface causes the water that already saturates the beach to move under the applied pressure gradients. The novelty of researching this flow is the extreme unsteadiness of the forcing, due to the sudden wave impact.

The project will model the flow field by adapting the theory of Cooker and Peregrine (1995). In addition to the flow, the large spatial pressure differences in the fluid phase cause forces on the individual rocks of the beach. The nett force may have a large enough magnitude to move the shingle, especially those relatively free rocks on the beach surface. Shingle rocks up to the size of large boulders are known to be displaced the width of the beach by a storm. An understanding of the coupling between the flow and the shingle or boulder movement, will help us to model coastal structures.

Breakwaters are often protected by (or composed of) shaped concrete blocks, with large void-spaces between the blocks. The void-spaces allow wave water to flow in and out easily, and they lessen the energy of regular periodic waves. But the mechanics of the flow and forces acting on and inside such structures is poorly understood when the waves are breaking. Rubble-mound breakwaters can be damaged dramatically quickly under storm-wave conditions.  (Y. Goda (2000) reviews the engineering aspects of waves and breakwaters.)

The project will adapt and extend existing theories of violent flow inside a porous structure, with a view to describing the displacements of the solid elements that make up the structure. The applicant must be familiar with theoretical fluid mechanics, water waves, and boundary-value problems for Laplace's equation.

References:

## Fluid-Structure Interaction Fluid-Structure Interaction

Supervisor: Prof Alexander Korobkin.

The problems of fluid-structure interaction are important in biology, medicine, offshore and polar engineering, naval fields, as well as in many industrial applications. The project is concerned with the deformations of an elastic body responding to hydrodynamic excitations and simultaneously the modification of these excitations owing to the body deformation. Such problems are coupled, which implies that the elastic deformations of the body depend on the hydrodynamic forces and vice versa. The hydrodynamic loads cannot be treated simply as external pre-calculated loads. Such problems are difficult to study both theoretically and numerically.

In the problem of elastic body impact onto liquid free surface, for example, the region of contact between the liquid and the body surface is unknown and has to be determined as part of the solution. Both experiments and numerical calculations show that the hydrodynamic loads during a fluid-structure interaction are difficult to predict and measure. On the other hand, the stresses in the body induced by the interaction are rather stable and can be well predicted even by simplified mathematical models. There is still no explanation to this phenomenon.

Elasticity plays an important role in cavitation and separation of the liquid surface from the surface of vibrating body. An elastic body vibrating at high frequency in a liquid may create a layer of bubbles around it and, in this way, may significantly reduce its interaction with the surrounding media. There are many other unusual phenomena associated with the fluid-structure interaction, which could not be observed in fluid dynamics or structural dynamics separately but only in the processes where the fluid and structure are put in contact and interact strongly one with another.

References:

## Filtering in Large Eddy Simulation Filtering in Large Eddy Simulation

Supervisor: Dr Jennifer Ryan

This project will combine tools from theoretical and computational applied mathematics and numerical analysis to address fundamental issues in the development of explicit filtering techniques critical for Large Eddy Simulation (LES) for turbulence modelling of compressible flow.  The aim of this project is to create a mathematically rigorous and computationally efficient explicit filter that allows for grid independent LES by exploiting superconvergence and useful superconvergence extraction through Smoothness-Increasing Accuracy-Conserving (SIAC) filtering.

SIAC filtering allows for the full resolution of the approximation and its derivatives and has proven successful in areas such as error reduction for linear acoustics and streamline visualization. SIAC filtering will be further developed, modified and adapted to resolve the under-resolved small scale numerical solutions needed for LES.  These approaches will translate to such areas as wind and tidal energy, a critical subject in exploration of Energy
Engineering and has the potential to affect areas such as ocean modelling, acoustics, combustion, etc.

References:

## Boundary Layer Flows Past Multiple Aerofoils and/or Ground Effects Boundary Layer Flows Past Multiple Aerofoils and/or Ground Effects

Supervisor: Dr Richard Purvis

This project will examine high Reynolds number flow past multiple aerofoils, focusing on three-dimensional arrangements (eg rotor blades) and/or ground effects. High-speed flow past a single aerofoil is much studied mathematically and is well understood. A thin boundary-layer exists near the blade within which all the viscous effects are contained, and this matches to an outer inviscid flow. The viscous boundary layer flow can be solved first and this drives the inviscid region. This changes if another blade is added behind the first (or the aerofoils are located near the ground or a water surface — so called wing-in-ground effect) as the outer inviscid problem is then coupled to the inner viscous problem. This viscous–inviscid interaction means that both regions must be solved hand-in-hand.

Previous work has looked at two-dimensional arrays of blades, as a model for a rotor blade, as well as single and sequences of aerofoils near to a solid surface. The current project would look to extend these models. Depending on interest it could focus on three-dimensional problems, specifically planar flow past aligned sequences of blades, or for the flow generated by an alignment more akin to a rotor blade. The inner problem can be solved if the blades are symmetric and it would be interesting to examine the viscous–inviscid coupled problem that arises with non-symmetry. An alternative focus would be to extend previous work on ground effects (for one or several aerofoils) to include high speed flow past aerofoils near to a water interface, such as for wing-in-ground effect vehicles operating over water.

References

1. S P Kirby, R Purvis (2013), Interactive planar multi-blade flows with a global angle of attack, E. J. Mech. B/Fluids 41,150-162.
2. S P Kirby, R Purvis (2013), High Reynolds number flow past many blades in extreme ground effect, Q. J. Mech Appl. Math. 66, 165-184.
3. R. Purvis, F. Smith (2004). Planar flow past two or more blades in ground effectQ. J. Mech Appl. Math. 57(1), 137-160.

## Air Cushioning in Water Impact Air Cushioning in Water Impact

Supervisor: Dr Richard Purvis

During the final moments before the impact of a water droplet onto a wall or into a water layer, air cushioning can delay touchdown and cause an air-bubble to be formed near to where the impact occurs. Recent experiments using high speed photography have captured pictures this air-bubble during a solid sphere impact into a water layer (similar experiments have shown the same behaviour for droplet impacts onto both solid surfaces and into water layers).

Mathematical models of this process have been shown to have surprisingly good agreement to the measured bubble size and are seen to capture much of the behaviour as impact is approached. However, there is still much that is not understood about this air cushioning phenomena. This project will look at some of these open problems, perhaps focusing on stability as touchdown is approached, the influence of surface tension or considering how the local geometry of an impacting solid influences the air cushioning. It would also be interesting to investigate how air influences the splashing behaviour after impact has occurred, as the development of a splash after impact has been shown experimentally to be very different if carried out in a vacuum.

References:

## Oscillations in Flow Through an Elastic-Walled Tube Oscillations in Flow Through an Elastic-Walled Tube

Supervisor: Dr Robert Whittaker

Flow-induced oscillations of fluid-conveying elastic vessels arise in many engineering and biomechanical systems. Examples include pipe flutter, wheezing during forced expiration from the pulmonary airways, and the development of Korotkoff sounds during blood pressure measurement by sphygmomanometry.

Experimental studies of flow in collapsible tubes are typically performed with a Starling resistor. A finite-length elastic tube is mounted between two rigid tubes and flow is driven through the system. The collapsible segment is contained inside a pressure chamber which allows the external pressure acting on the elastic tube to be controlled. If the pressure outside the tube becomes sufficiently large, it will buckle non-axisymmetrically. Once buckled, the tube is very flexible leading to strong fluid-structure interaction. Experiments show that in this buckled state, the elastic tube segment has a propensity to develop large-amplitude self-excited oscillations of great complexity when the flow rate is increased beyond a certain value.

This project aims to further our understanding of some of the mechanisms that can lead to this instability of flow through an elastic-walled tube. The fluid flow will be described by the Navier–Stokes equations, and an appropriate elastic model will be used for the tube wall. Whittaker et al (2010)  developed a relatively simple model for small amplitude, long-wavelength, high-frequency oscillations. Work is currently underway on extensions to include shear effects, wall inertia and axial bending. This project will start by working to relax some of the remaining assumptions in the previous model, e.g. adding nonlinear effects and allowing for different cross-sectional shapes. The project will likely focus on developing reduced analytic models (which may need to be solved analytically or numerically) though there is also scope for conducting full-scale numerical simulations.

References:

## Breathers in Weakly Nonlinear and Fully Nonlinear Wave Models Breathers in Weakly Nonlinear and Fully Nonlinear Wave Models

Supervisors: Dr Emilian Parau and Dr Davide Proment

The Nonlinear Schrodinger (NLS) equation is a very commonly used in physics to model various types of weakly nonlinear systems.  For instance, it has been derived in optics, water gravity waves, internal waves, water gravity-capillary waves, and flexural-gravity water waves. The one-dimensional NLS is integrable and different types of solutions like solitons and breathers (e.g. Peregrine, Ma, Akhmediev) have been found. Sometimes those particular solutions manifest themselves also in the fully-nonlinear models whose NLS is represent the weakly nonlinear approximation.  For example, numerical simulations for the fully-nonlinear deep water gravity wave equation have shown the existence of localised breathers.

In the proposed research, the existence of breathers and multi-breathers will be investigated in other fully-nonlinear
equations for water waves such as the cases of internal waves, capillary waves, gravity-capillary waves and flexural-gravity waves. Moreover, a stability analysis of breather solutions will be attempted for the easier weakly nonlinear models such as NLS or Dysthe equation, and for the fully-nonlinear equations.

References:

• F Dias, C Kharif (1999). Nonlinear gravity and capillary-gravity waves, Ann. Rev. Fluid Mech. 31, 301-346.
• A.I. Dyachenko, V.E. Zakharov (2008). On the Formation of Freak Waves on the Surface of Deep Water, Pis'ma v ZhETF, 88 (5), 356-359. [JETP Lett., 88 (5), 307-311].
• K.B. Dysthe (1979). Note on a modification to the nonlinear Schrodinger equation for application to deep water waves. Proc. R. Soc. Lond. A 369, 105-114.
• P. Guyenne, E.I. Parau (2012). Computations of fully-nonlinear hydroelastic solitary waves on deep water, J. Fluid Mech. 713, 307-329.
• A. Slunyaev, E. Pelinovsky, A. Sergeeva, A. Chabchoub, N. Hoffmann, M. Onorato, and N. Akhmediev (2013). Super-rogue waves in simulations based on weakly nonlinear and fully nonlinear hydrodynamic equations, Phys. Rev. E 88, 012909.
• V.E. Zakharov, A.I. Dyachenko (2010). About shape of giant breather, Eur. J. Mech. B / Fluids, 29, 127-131.