The method of apparent masses is utilized to compute the initial lift and drag of an airfoil that starts impulsively from rest. Analytical solutions are obtained for in viscid incompressible flow past a slightly cambered airfoil at a small angle of attack. For a Joukowski airfoil with a cusped trailing edge, it is found that increasing camber or angle of attack will cause increases in both initial lift and drag, whereas increasing thickness will result in an opposite effect. Effects of trailing-edge angle are examined by considering the symmetric Kármán-Trefftz airfoil. The result shows that both lift and drag vanish at the initial instant if the airfoil has a finite trailing-edge angle.

A numerical method is developed for solution of the full nonlinear equations governing irrotational flow with a free surface and variable bed topography. It is applied to the unsteady motion of non-breaking water waves of arbitrary magnitude over a horizontal bed. All horizontal variation is approximated by truncated Fourier series. This and finite-difference representation of the time variation are the only necessary approximations. Although the method loses accuracy if the waves become sharp-crested at any stage, when applied to non-breaking waves the method is capable of high accuracy.

The interaction of one solitary wave overtaking another was studied using the Fourier method. Results support experimental evidence for the applicability of the Korteweg-de Vries equation to this problem since the waves during interaction are long and low. However, some deviations from the theoretical predictions were observed - the overtaking high wave grew significantly at the expense of the low wave, and the predicted phase shift was found to be only roughly described by theory. A mechanism is suggested for all such solitary-wave interactions during which the high and fast rear wave passes fluid forward to the front wave, exchanging identities while the two waves have only partly coalesced; this explains the observed forward phase shift of the high wave.

For solitary waves travelling in opposite directions, the interaction is quite different in that the amplitude of motion during interaction is large. A number of such interactions were studied using the Fourier method, and the waves after interaction were also found to be significantly modified - they were not steady waves of translation. There was a change of wave height and propagation speed, shown by the present results to be proportional to the cube of the initial wave height but not contained in third-order theoretical results. When the interaction is interpreted as a solitary wave being reflected by a wall, third-order theory is shown to provide excellent results for the maximum run-up at the wall, but to be in error in the phase change of the wave after reflection. In fact, it is shown that the spatial phase change depends strongly on the place at which it is measured because the reflected wave travels with a different speed. In view of this, it is suggested that the apparent time phase shift at the wall is the least-ambiguous measure of the change.

The pulsed ultrasonic Doppler velocimeter has been used extensively in transcutaneous measurement of the velocity of blood in the human body. It would be useful to evaluate turbulent flow with this device in both medical and non-medical applications. However, the complex behaviour and limitations of the pulsed Doppler velocimeter when applied to random flow have not yet been fully investigated.

In this study a three-dimensional stochastic model of the pulsed ultrasonic Doppler velocimeter for the case of a highly focused and damped transducer and isotropic turbulence is presented. The analysis predicts the correlation and spectral functions of the Doppler signal and the detected velocity signal. The analysis addresses specifically the considerations and limitations of measuring turbulent intensities and one-dimensional velocity spectra.

Results show that the turbulent intensity can be deduced from the broadening of the spectrum of the Doppler signal and a mathematical description of the effective sample-volume directivity.

In the measurement of one-dimensional velocity spectra at least two major complicacations are identified and quantified. First, the presence of a time-varying, broad-band random process (the Doppler ambiguity process) obscures the spectrum of the random velocity. This phenomenon is similar to that occurring in laser anemometry, but the ratio of the level of the ambiguity spectrum to the largest detected velocity spectral component can be typically two to three orders of magnitude greater for ultrasonic technique owing to the much greater wavelength.

Secondly, the spatial averaging of the velocity field in the sample volume causes attenuation in the measured velocity spectrum. For the ultrasonic velocimeter, this effect is very significant.

The influence of the Doppler ambiguity process can be reduced by the use of two sample volumes on the same acoustic beam. The signals from the two sample volumes are cross-correlated, removing the Doppler ambiguity process, while retaining the random velocity. The effects of this technique on the detected velocity spectrum are quantified explicitly in the analysis for the case of a three-dimensional Gaussianshaped sample-volume directivity.

An extensive experimental programme in both laminar and turbulent flow was undertaken to examine the validity of all of the major implications of the model of the pulsed ultrasonic Doppler velocimeter for turbulent flow developed in part 1 of this investigation. The turbulence measurements were made in fully developed flow at the centre of a 6·28 cm diameter pipe. The Reynolds number of the flow ranged from 6000 to 40000. The carrier frequency of the ultrasonic velocimeter was 4·7 MHz.

Measurements of the turbulence intensity and of the one-dimensional velocity spectra made with the ultrasonic velocimeter are compared with the analysis and with the actual quantities as measured by a hot-film anemometer. The experimental results are in agreement with theoretical predictions.

Measurements of one-dimensional turbulence spectra with reduced ambiguity spectra made by the two sample volume methods described in part 1 are presented. The results verify the analysis and indicate that an improvement in the useful dynamic range of the velocity power spectrum of nearly three orders of magnitude can realistically be achieved.

A description of MHD turbulence at low magnetic Reynolds number and large interaction parameter is proposed, in which attention is focussed on the role of insulating walls perpendicular to a uniform applied magnetic field. The flow is divided in two regions: the thin Hartmann layers near the walls, and the bulk of the flow. In the latter region, a kind of electromagnetic diffusion along the magnetic field lines (a degenerate form of Alfvén waves) is displayed, which elongates the turbulent eddies in the field direction, but is not sufficient to generate a two-dimensional dynamics. However the normal derivative of velocity must be zero (to leading order) at the boundaries of the bulk region (as at a free surface), so that when the length scale l⊥ perpendicular to the magnetic field is large enough, the corresponding eddies are necessarily two-dimensional. Furthermore, if l⊥ is not larger than a second limit, the Hartmann braking effect is negligible and the dynamics of these eddies is described by the ordinary Navier-Stokes equations without electromagnetic forces. MHD then appears to offer a means of achieving experiments on two-dimensional turbulence, and of deducing velocity and vorticity from measurements of electric field.

Passive scalar convection by a prescribed random velocity field is represented in terms of integral equations. Primitive perturbation expansions are constructed by iterating these integral equation representations as in Kraichnan (1977). First and second iterations of elemental functions within these expansions are assumed quadratically integrable with respect to space and time. That is, they are assumed to belong to the space L2. Line-renormalized perturbation expansions are constructed, corresponding to these primitive perturbation expansions, which converge almost everywhere. The direct-interaction approximation and the Lagrangian-history direct-interaction approximation are the simplest truncations of the appropriate line-renormalized perturbation expansions.