# Mathematical Modelling of Water Wave Impacts (COOKERU16SF)

This project is no longer listed in the FindAPhD
database and may not be available.

for PhD studentship opportunities
• Dr M Cooker
• No more applications being accepted
• Self-Funded PhD Students Only

## Project Description

When a sea wave travels from the deep ocean into the shallow water near the coast it tends to steepen and overturn as a breaking wave. The impact of such waves exerts erosive forces on beaches, cliffs and potentially damaging forces on engineered structures. A breaking wave can make a force that is impulsive; that is say one that is very large but very short-lived. The theory of Cooker and Peregrine (1995) has been used to understand better the impulsive pressure distribution in the water, and the overall wave-force on the structure. That paper treats the most violent flows, which are well approximated by solving mixed two-dimensional boundary-value problems. The challenge of this project is to extend the work to understand the influence of the third dimension on the distribution of peak pressure. This is important because when we want to model the impact of a plane, breaking wave onto a sea wall when the wall’s shape is not just a plane. For example re-entrant corners in a seawall are known to allow an increase the local impulsive pressure, and damage can start at such corners. Another example of the influence of the third dimension comes from Cox and Cooker (2001). They modelled the confined spaces of a crack in a seawall, and showed from mathematically that the distribution of maximum pressure can act to widen and or deepen the crack.

The student will learn how to model the violent flow in the context of inviscid fluid mechanics, by posing appropriate mixed boundary conditions for Laplace’s equation. The student will learn a variety of techniques for solving the equations and satisfying the boundary conditions: exactly using analysis, approximately with asymptotics and numerically by efficient computations.

The interpretation of the results has important consequences for the sudden change in the wave-water’s velocity field, as well as modelling the distribution of peak pressure at the time of impact. Cooker (2013) reviews violent wave impact and shows that adding the third dimension to the geometry of this type of fluid-dynamical problems remains largely unexplored terrain.

## Funding Notes

This PhD project is offered on a self-funding basis. It is open to applicants with funding or those applying to funding sources. Details of tuition fees can be found at http://bit.ly/1Jf7KCr

A bench fee is also payable on top of the tuition fee to cover specialist equipment or laboratory costs required for the research. The amount charged annually will vary considerably depending on the nature of the project and applicants should contact the primary supervisor for further information about the fee associated with the project.

## References

i) M.J. Cooker and D.H. Peregrine (1995) Pressure impulse theory for liquid impact problems. Journal of Fluid Mechanics, 297, 193—214.
ii) S.J. Cox and M.J. Cooker (2001) The pressure impulse in a fluid saturated crack in a sea wall. Coastal Engineering, 42, 241--256.
iii) M.J. Cooker (2013) A theory for the impact of a wave breaking onto a permeable barrier with jet generation. Journal of Engineering Mathematics, 79, 1—12.