Making a normal-mode assumption, we shall investigate stability with respect to non-axisymmetric perturbations of an inhomogeneous incompressible fluid rotating between two perfectly conducting, infinite, coaxial cylinders in the presence of an axial and a toroidal magnetic field. We shall establish sufficient conditions for stability, discuss the westward-drift nature of unstable modes and estimate upper bounds on the azimuthal phase speeds and growth rates. There will also be a discussion of the validity of the sufficient conditions for stability when the normal-mode assumption is not made, so that the stability is based on an initial-value problem.

The motion of a continuously stratified fluid in a rotating annulus is investigated experimentally. The flow is driven by the differential rotation of a lid on the annulus in contact with the fluid. For small enough Rossby numbers the basic flow is found to be axisymmetric, but in contrast to the unstratified case the zonal flow is not independent of the depth, and there is significant vertical shear in the interior. As the Rossby number of the flow is increased the axisymmetric flow becomes unstable to non-axisymmetric baroclinic disturbances. Some features of these disturbances as well as the time response to the motion of the lid are examined.

This is a review of an IUTAM symposium held in Canberra on 20–23 July 1976. The subject matter ranged widely: from surf on beaches to waves of oceanic scale, and from elegant analysis to observations from tide gauges.

The slope and curvature distributions of wind waves along two principal axes (upwind-downwind and cross-wind) have been measured in a laboratory tank under various wind velocities. In both directions, the slope distributions are very closely Gaussian, and the components of the mean-square water-surface slope vary loga,rithmically with the friction velocity of the wind. As the windvelocityincreases, the ratio of the upwind-downwind and cross-wind components increases and lies between 0.5 and 0.6 at high wind velocities in the gravity-governed regime of wind-wave interaction. The radius of .water-surface curvature, along either direction of measurement, is generally found to be greater at a steeper viewing angle from the normal to the mean water surface. The average radius of curvature of the disturbed surface varies inversely with the friction velocity of the wind. The ratio of the upwind-downwind and cross-wind components of the average radius of curvature is unity at all wind velocities, indicating that the wind-disturbed water surface is isotropic on the smallest scale. Other results show that both the slope and the curvature distributions are asymmetric along the upwind-downwind direction, either because of the presence of parasitic capillaries or because of the occurrence of wave breaking. The results also indicate that even the high frequency portion of the spectrum is saturated locally but the spectrum is not universal, and that the long waves suppress the growth of the nearly saturated ripples.

The wave-mechanical condition (Landahl 1972) for breakdown of an unsteady laminar flow into strong small-scale secondary instabilities is applied to some simple stratified inviscid shear flows. The cases considered have one or two discrete density interfaces and simple discontinuous or continuous velocity profiles. A primary wavelike disturbance to such a flow produces a perturbation velocity that is discontinuous at a density interface. The resulting instantaneous system, defined as the sum of the mean flow and the primary oscillation, develops a local secondary shear-flow instability that has a group velocity equal to the arithmetic mean of the instantaneous velocities on the two sides of the interface. According to the breakdown criterion, the disturbed flow will become critical whenever this velocity reaches a value equal to the phase velocity of the primary wave. The calculations show that for a single density interface breakdown may occur for low to moderate wave amplitudes in a fairly narrow range of Richardson numbers on the stable side of the stability boundary. On the other hand, in the unstable regime maximum wave slopes of order unity may be reached before breakdown occurs; this conclusion is in qualitative agreement with experiments. When the system includes two density interfaces, it is found that there exists a range of high Richardson numbers far into the stable regime for which breakdown may take place even for very small and zero wave interface deflexions.

The results of the linear theory for the flow of a supersonic relaxing gas past a slender body of revolution are analysed in regions where its predictions of wavelet position begin to break down. In this way new variable systems can be found which make it possible to discuss the correct nonlinear wave behaviour far from the body. The situation depends upon three especially important parameters, namely the thickness ratio ε of the body, the ratio δ of relaxing-mode energy to thermal energy and the ratio λ of a relaxation length to a typical body length. After establishing general results from the linear theory, the conical body is treated in some detail. This makes it possible to demote λ as an important parameter, although its restoration does prove useful at one point in the analysis, and results are derived for shock-wave behaviour when ord 1 [ges ] δ > ord ε4, δ = ord ε4 and δ < ord ε4. In the first range of δ fully dispersed waves are essential, although they are fully established only at great distances from the cone; in the second range of δ partly dispersed waves seem to be the most likely to appear, and in the third range relaxation effects are second-order modifications of a basically frozen-flow field. Practical situations may well fall into the first of these categories.

The linearized stability of plane Couette flow is investigated here, without using the Orr–Sommerfeld equation. Rather, an unusual symmetry of the problem is exploited to obtain a complete set of modes for perturbations of the unbounded (no walls) flow. An explicit Green's function is constructed from these modes. The unbounded flow is shown to be rigorously stable. The bounded case (with walls) is investigated by using a ‘method of images’ with the unbounded Green's function; the stability problem in this form reduces to an algebraic characteristic equation (not a differential-equation eigenvalue problem), involving transcendental functions defined by integral representations.

The stability of circular Couette flow when the outer cylinder is at rest and the inner is modulated both with and without a mean shear is examined in the narrow-gap limit. Disturbances are assumed to be axisymmetric. Two criteria are used to determine conditions for stability; the first requires that the motion be strongly stable, the second only that disturbances of arbitrary initial energy decay from cycle to cycle. The behaviour of critical parameters as a function of frequency is similar for the linear and the energy analysis. The range of Reynolds numbers bounded above by certain instability and below by conditional nonlinear stability is enlarged by modulation.

Measurements of a separating two-dimensional incompressible boundary layer with an airfoil-type pressure distribution are reported. Unique mean and fluctuation velocity measurements and the distribution of the fraction of the time γp during which the flow moves downstream were obtained in the separated region using a directionally sensitive laser anemometer. Linearized hot-film anemometer measurements of mean velocities, turbulent shearing stress and intensities, eddy speeds, spectra and dissipation were made for γp > 0·8. The wall shearing stress, bursting frequencies, wall speed and spanwise structure were obtained using flush-surface hot-film sensors. The turbulent/non-turbulent interfacial intermittency γ and the frequency of passage of turbulent bulges were determined using smoke as a turbulence marker and the laser anemometer system for illumination and signal detection.

Upstream of separation the velocity profile correlations of Perry & Schofield (1973) are supported within the uncertainty of the data. Normal-stress effects are very important, influencing $-\overline{uv}/\overline{q^2} $ and the dissipation length correlations, and directly providing sizable terms in the momentum and turbulence energy equations. The criteria of Sandborn for turbulent separation and fully developed separation are found to hold. Downstream of separation there is apparent similarity of $\overline{u^2}$, U and γp throughout the shear flow. The passive low velocity backflow near the wall apparently just serves to satisfy continuity requirements after the energetic outer-region flow has deflected away from the wall upon separation.

The wall bursting frequency nA scales on outer velocity and length scales, with U∞/δnA ≈ 10, or about twice the value observed for zero-pressure-gradient flows. The non-dimensional spanwise spacing of wall eddies is given approximately by the relation λzUM/v ≈ 100 upstream of separation, where $U_M = (- \overline{uv}_{\max})^{\frac{1}{2}}$. The speed of wall eddies is found to be about 14Uτ.

An investigation of the existence of purely thermal intermittency has been undertaken for the flow produced by a step change in temperature in a decaying homogeneous turbulent field. The results show that a thermal interface can be defined and that it is a random function of space and time, having gross characteristics similar to the turbulent interface in a free shear flow. A technique for identifying the hot and cold regions of the fluid in a systematic way was developed and this permitted conditional averages of the mean and fluctuating temperature to be formed.

Steady two-dimensional viscous motion within a circular cylinder generated by (a) the rotation of part of the cylinder wall and (b) fluid entering and leaving through slots in the wall is considered. Studied in particular are moving-surface problems with continuous and discontinuous surface speeds, simple inflow–outflow problems and the impinging-jet problem within a circle. The analytical solutions at zero Reynolds number are given for the last two types of problem. Numerical results are obtained at various Reynolds numbers from the integral representation of the solution. At zero Reynolds number this approach involves a quadrature around the circumference of the cylinder. At other Reynolds numbers it involves an iterative–integral technique based on the use of the ‘clamped-plate’ biharmonic Green's function.