# Research in the Waves Group

A wave is *a means of transferring energy from one place to another
without bulk motion*. Waves appear all over physics, whether
light waves, sound waves, elastic waves or other less obvious
varieties. Wave motion is ubiquitous, and an understanding of wave
motion is of fundamental importance.

The Waves Group studies problems of wave motion, usually, but not exclusively, drawn from fluid dynamics. These include problems as diverse as turbomachinery noise, theoretical hydrodynamic stability, fluid-structure interactions and development of computational methods. This work has a great variety of applications and much of our research is funded by industry.

Below is a broad brush description of some of the current themes of our work. We are interested in other problems as well.

## Acoustics and turbomachinery noise

Research has been conducted on a wide of problems arising in the generation and propagation of sound. Most interest has focused on turbomachinery noise, and much of the work has been sponsored by Rolls Royce and Hitachi. High levels of acoustic radiation are produced when turbulence is ingested into a fan and by the hydrodynamic interaction between the various blade rows, and we have studied a range of problems concerned with the sound produced when unsteady vortical flow interacts with a solid object such as a fan blade. Other work has been concerned with predicting the effects of aeroengine intake scarfing, whereby the the shape of the engine intake is changed so as to produce some improvement in the noise exposure on the ground.

Swirling flows with relevance to the flow behind a rotating fan, and the effects of swirl on the stability and acoustics of diverging jets are also under investigation, as described below.

## Hydrodynamic stability and fluid-structure interaction

Although, for certain problems, it is relatively simple to write down
solutions of the governing equations of fluid dynamics, is it is less
simple to show that these solutions can observed in the real world.
To do this, one has to show that the solution is *stable*, and
the stability of flows is an old problem of hydrodynamics.

The extension of fluid stability theory to handle interactions between the fluid and any elastic or other structures present is an interesting topic of research. A variety of counter-intuitive and complicated behaviours can occur; for example, damping can become a destabilizing effect. The picture to the left shows the response of a finite baffle in mean flow to (single period) forcing. Although the forcing is periodic, frequencies generated in the startup persist and change the nature of the solution at large times, meaning a classical frequency analysis does not tell the whole story.

## Theoretical Acoustics

We are interested not only in applications of waves, but also in the fundamental mathematical methods used to model and investigate waves and related behaviour. For example, one powerful but complex method is the Wiener-Hopf technique, which is used for a broad collection of PDEs which arise in acoustic, finance, hydrodynamic, elasticity, potential and electromagnetic theories. It is an elegant method based on the exploitation of the analyiticity properties of the functions. For the scalar Wiener-Hopf the solution can be expressed in terms of a Cauchy type integral, although in more complicated scalar Wiener-Hopf equations (as often occur in practice) the exact solution is difficult or slow to compute. We are currently working on developing approximate methods for solving such problems which are easily implementable, reliable and have explicit error bounds. Moreover, the general theory for solving matrix Wiener-Hopf problems is not yet known, with only a few matrix Wiener-Hopf problems having ever been solved. This is an area of active research.

## Swirling flows and transition to turbulence

We are also interested in a range of issues associated with the acoustics and stability of swirling flow. The behaviour of small-amplitude wave motions in irrotational steady flow is well-understood, but interesting effects arise in the presence of mean vorticity. Duct acoustics in strong swirl is relevant to the flow behind a rotating fan, and the effects of swirl on the stability and acoustics of diverging jets are also under investigation.

High rotation can produce novel nonlinear solutions which give insight into the complex transition processes observed in shear flows. As such, we are working on analysing the effects of swirl and rotation on the sub-critical and super-critical stability properties of cylindrical and Cartesian shear flows.

## Biological acoustics

There are many instances of waves in biology. For example, the flight of most owl species is effectively silent, a feat which allows the owl to acoustically target and sneak up on its prey in total darkness; this noise reduction is achieved through specialised plumage (as shown in the picture on the left). We are developing theoretical models incorporating these unique attributes, with the dual aims of helping explain the mystery of silent owl flight and guiding the development of quieter airframes and fluid-loaded blades, such as those found on helicopter rotors and wind turbines.