Fluid dynamics and nonlinear waves
The application of differential geometry to the equations of fluid mechanics has a long and distinguished history. Contemporary geometric descriptions of nonlinear partial differential equations appear to play a key role in studying vortex tubes and sheets, the fundamental building blocks of turbulence.
Research projects
Research leads
Professor Tom Bridges
Water waves
Dr Matthew Turner
Water waves, vortex dynamics and hydrodynamic stability
Water waves
Shallow water sloshing and its implications for wave energy harvesting. Current work being led by Tom Bridges and Matthew Turner is focussing on the coupling between vessel motion and fluid motion in 3D. The vessel motion is modelled using the geometric mechanics framework for rigid body motion, and the fluid motion is modelled using a hierarchy of models from the shallow water equations, to irrotational 2D and 3D fluid motion, and up to the full inviscid Euler equations with exact representation of vorticity.
Also of interest is using ideas from complex analysis, in particular conformal mappings, to develop effecting numerical schemes to solve coupled sloshing problems. Utilizing the Schottky-Klein Prime function we can solve problems which involve multiply connected domains, such as vessels containing baffles.
Vortex dynamics
Understanding the dynamics of vortices is important when considering things such as thermal and sedimentary transport in the oceans. Current research in this area led by Matthew Turner focuses on investigating how coherent vortices react to nonlinear perturbations, in particular looking at thresholds in parameter space for where these nonlinear perturbations persist for large times.
Also of interest currently is trying to predict when a pair of point vortices on the plane with some solid boundary propagate in a leap-frogging motion. Recent work has determined a numerical criteria, based upon vortex strengths and initial separation, for two vortices external to a circular cylinder. This work is now being extended to other doubly- and multiply connected domains.
Hydrodynamic stability
Current research in this area led Matthew Turner is looking at the absolute instability properties of spatially periodic flows, in particular a mixing layer with a spatially periodic bounding wall. The approach to determining the stability properties using Floquet theory in the spatial direction in order to form modal solutions and derive a dispersion relation for wave-like solutions. The global stability properties are then sought by looking for saddle points of the complex dispersion relation.
