A spherical or cylindrical cavity containing quiescent gas begins to contract at high constant radial speed, driving an axisymmetric shock wave inward to collapse at the centre. We analyse the flow field by expanding the solution in powers of time, and calculate 40 terms by delegating the arithmetic to a computer. Analysis of the series for the radius of the shock wave confirms Guderley's local self-similar solution for the focusing, including recent refined values for his similarity exponent, and yields higher terms in his local expansion. In the range of adiabatic exponent where the Guderley solution has been shown not to be unique we find, in accord with a conjecture of Gel'fand, that the smallest admissible similarity exponent is realized.

The influence of leading-edge separation vortices on the Weis-Fogh (1973) lift- generation mechanism for insect hovering is investigated. The analysis employs a vortex-shedding model (Edwards 1954; Cheng 1954) and represents an extension of Lighthill's (1973, 1975) analysis of an inviscid model without separated vortices. Results of the study compare reasonably well with observations on a laboratory model at high Reynolds number (Maxworthy 1979), confirming that vortex separation significantly enhances the initial circulation on each of the wings. Unlike the un- separated-model solution, this circulation was found to depend on the history of the wing motion and to increase with a large opening angle.

By producing thermal fluctuations with a mandohe and helium fluctuations with chimneys attached to the grid bars, the mixing of temperature and helium fluctuations as well as the decay of temperature and helium variance and their flux is investigated in decaying grid-generated turbulence. The helium, temperature and velocity fluctua- tions were measured with a modified ‘Way-Libby’ interference probe (Way & Libby 1970, 1971). It is shown that, as for temperature variance, the helium-variance decay rate is a function the ratio of the helium length scale to the velocity length scale. It is also shown that the decay of the cross-correlation between temperature and helium fluctuations is slow if both scalars are introduced close to the grid, but rapid if each scalar is introduced at a different distance from the grid, and hence at different scales. The results corroborate those of the inference method of Warhaft (1981), which is extended here to examine other cases. A particularly unexpected finding is that under certain circumstances the two-scalar cross-correlation may actually increase with distance from the grid, although the scalar covariance decreases. The return to isotropy of helium flux and temperature flux is also investigated and is shown to be slow if the scalar flux is produced near the grid bars, but faster if the flux is produced further downstream. For all the measurements helium and temperature were passive additives.

The properties of reflection and transmission of internal gravity waves incident upon a shear layer containing a critical level are investigated. The shear layer is modelled by a hyperbolic tangent profile. In the Boussinesq approximation, the differential equation governing the propagation of these waves can then be transformed into Heun's equation. For large Richardson numbers this equation can be approximated by an equation that has solutions in terms of hypergeometric functions. For these values of the Richardson number the reflection coefficient proves to be strongly dependent on the place of the critical level in the shear flow. If the Doppler-shifted frequency is an odd function of the height difference with respect to the critical level, the reflection and transmission coefficients can be evaluated in closed form.

Over-reflection is possible for sufficiently small wavenumbers and Richardson numbers. It is pointed out that over-reflection and over-transmission cannot occur in a stable flow and that resonant over-reflection is not possible in our model.

When a shock moves in a non-uniform medium, its motion is influenced both by the non-uniformity ahead of the shock and also by a wave which overtakes the shock. In this paper the overtaking wave is examined in an analytic manner by considering its formation and subsequent propagation. This analysis is then combined with the known ‘freely propagating’ description of the shock motion, in which the overtaking wave is ignored, and results in a description of the shock motion in integral form. For the particular problem of a strong shock propagating in a medium with a power-law density variation, the freely propagating shock law has previously been compared with the available similarity solution. The estimate of the effect of the overtaking wave presented in this paper is shown to provide a significant improvement to the description of the shock motion in this instance.