Topics in vortex motion

Six topics in incompressible, inviscid fluid flow involving vortex motion are presented. The stability of the unsteady flow field due to the vortex filament expanding under the influence of an axial compression is examined in the first chapter as a possible model of the vortex bursting observed in aircraft contrails. The filament with a stagnant core is found to be unstable to axisymmetric disturbances. For initial disturbances with the form of axisymmetric Kelvin waves, the filament with a uniformly rotating core is neutrally stable, but the compression causes the disturbance to undergo a rapid increase in amplitude. The time at which the increase occurs is, however, later than the observed bursting times, indicating the bursting phenomenon is not caused by this type of instability.

In the second and third chapters the stability of a steady vortex filament deformed by two-dimensional strain and shear flows, respectively, is examined. The steady deformations are in the plane of the vortex cross-section. Disturbances which deform the filament centerline into a wave which does not propagate along the filament are shown to be unstable and a method is described to calculate the wave number and corresponding growth rate of the amplified waves for a general distribution of vorticity in the vortex core.

In Chapter Four exact solutions are constructed for two-dimensional potential flow over a wing with a free ideal vortex standing over the wing. The loci of positions of the free vortex are found and the lift is calculated. It is found that the lift on the wing can be significantly increased by the free vortex.

The two-dimensional trajectories of an ideal vortex pair near an orifice are calculated in Chapter Five. Three geometries are examined, and the criteria for the vortices to travel away from the orifice are determined.

Finally, Chapter Six reproduces completely the paper, "Structure of a linear array of hollow vortices of finite cross-section," co-authored with G. R. Baker and P. G. Saffman. Free streamline theory is employed to construct an exact steady solution for a linear array of hollow, or stagnant cored vortices. If each vortex has area A and the separation is L, then there are two possible shapes if A^(1/2)/L is less than 0.38 and none if it is larger. The stability of the shapes to two-dimensional, periodic and symmetric disturbances is considered for hollow vortices. The more deformed of the two possible shapes is found to be unstable, while the less deformed shape is stable.

Studies in vortex motion

This thesis covers four different problems in the understanding of vortex sheets, and these are presented in four chapters.

In Chapter 1, free streamline theory is used to determine the steady solutions of an array of identical, hollow or stagnant core vortices in an inviscid, incompressible fluid. Assuming the array is symmetric to rotation through π radians about an axis through any vortex centre, there are two solutions or no solutions depending on whether A^(1/2)/L is less than or greater than 0.38 where A is the area of the vortex and L is the separation distance. Stability analysis shows that the more deformed shape is unstable to infinitesimal symmetric disturbances which leave the centres of the vortices undisplaced.

Chapter 2 is concerned with the roll-up of vortex sheets in homogeneous fluid. The flow over conventional and ring wings is used to test the method of Fink and Soh (1974). Despite modifications which improve the accuracy of the method, unphysical results occur. A possible explanation for this is that small scales are important and an alternate method based on "Cloud-in-Cell" techniques is introduced. The results show small scale growth and amalgamation into larger structures.

The motion of a buoyant pair of line vortices of opposite circulation is considered in Chapter 3. The density difference between the fluid carried by the vortices and the fluid outside is considered small, so that the Boussinesq approximation may be used. A macroscopic model is developed which shows the formation of a detrainment filament and this is included as a modification to the model. The results agree well with the numerical solution as developed by Hill (1975b) and show that after an initial slowdown, the vortices begin to accelerate downwards.

Chapter 4 reproduces completely a paper that has already been published (Baker, Barker, Bofah and Saffman (1974)) on the effect of "vortex wandering" on the measurement of velocity profiles of the trailing vortices behind a wing.

Ghosts of order on the frontier of chaos

What kinds of motion can occur in classical mechanics? We address this question by looking at the structures traced out by trajectories in phase space; the most orderly, completely integrable systems are characterized by phase trajectories confined to low-dimensional, invariant tori. The KAM theory examines what happens to the tori when an integrable system is subjected to a small perturbation and finds that, for small enough perturbations, most of them survive.

The KAM theory is mute about the disrupted tori, but, for two-dimensional systems, Aubry and Mather discovered an astonishing picture: the broken tori are replaced by "cantori," tattered, Cantor-set remnants of the original invariant curves. We seek to extend Aubry and Mather's picture to higher dimensional systems and report two kinds of studies; both concern perturbations of a completely integrable, four-dimensional symplectic map. In the first study we compute some numerical approximations to Birkhoff periodic orbits; sequences of such orbits should approximate any higher dimensional analogs of the cantori. In the second study we prove converse KAM theorems; that is, we use a combination of analytic arguments and rigorous, machine-assisted computations to find perturbations so large that no KAM tori survive. We are able to show that the last few of our Birkhoff orbits exist in a regime where there are no tori.

Universal quantum computation with trapped ions in thermal motion by adiabatic passage

We propose a universal quantum computation scheme for trapped ions in thermal motion via the technique of adiabatic passage, which incorporates the advantages of both the adiabatic passage and the model of trapped ions in thermal motion. Our scheme is immune from the decoherence due to spontaneous emission from excited states as the system in our scheme evolves along a dark state. In our scheme the vibrational degrees of freedom are not required to be cooled to their ground states because they are only virtually excited. It is shown that the fidelity of the resultant gate operation is still high even when the magnitude of the effective Rabi frequency moderately deviates from the desired value.

Design for all: designing for the motion-impaired user

Designing for all requires the adaptation and modification of current design best practices to encompass a broader range of user capabilities. This is particularly the case in the design of the human-product interface. Product interfaces exist everywhere and when designing them, there is a very strong temptation to jump to prescribing a solution with only a cursory attempt to understand the nature of the problem. This is particularly the case when attempting to adapt existing designs, optimised for able-bodied users, for use by disabled users. However, such approaches have led to numerous products that are neither usable nor commercially successful. In order to develop a successful design approach it is necessary consider the fundamental structure of the design process being applied. A three stage design process development strategy which includes problem definition, solution development and solution evaluation, should be adopted. This paper describes the development of a new design approach based on the application of usability heuristics to the design of interfaces. This is illustrated by reference to a particular case study of the re-design of a computer interface for controlling an assistive device.