The steady streaming generated in a pipe of slowly varying cross-section when a purely oscillatory pressure difference is maintained between its ends is considered. It is assumed that the perturbation of the pipe wall in the r, θ plane is small compared with the characteristic thickness of the Stokes layer associated with the oscillatory motion of the fluid. The first-order steady streaming is evaluated for the cases when this characteristic thickness is large and small compared with a typical radius of the pipe. In both these limits it is found that the geometry of the pipe is crucial in determining the nature of the induced steady streaming. If the ends of the pipe have the same mean radius it is found that the steady streaming consists of regions of recirculation between the nodes of the pipe. Otherwise the steady streaming is of a larger order of magnitude and has a component which represents a net flow towards the wider end of the pipe.

We consider the motion of the mass of fluid ejected through a sharp-edged orifice by the motion of a piston. The vorticity formed by viscous forces within the separated flow at the sharp edge rolls up to form a concentrated vortex which, after a development period, consists of a core of very fine scale turbulence surrounded by a co-moving bubble of much larger scale turbulence. This bubble entrains outer fluid, mixes with it, and deposits the majority into a wake together with some small fraction of the total vorticity of the ring. Enough fluid is retained to account for the slow growth of the whole fluid mass. A theory which takes account of both the growth process and the loss of vorticity is proposed. By comparison with experimental measurements we have determined that the entrainment coefficient for turbulent vortex rings has a value equal to 0.011 ± 0.001, while their effective drag coefficient is 0.09 ± 0.01. These results seem to be independent of Reynolds number to within experimental accuracy.

We discuss the theory of Kelvin wave propagation along an infinitely long coast-line which is straight except for small deviations which are treated as a stationary random function of distance along the coast. An operator expansion technique is used to derive the dispersion relation for the coherent Kelvin wave field. For the subinertial case σ = ω/f < 1 (ω = wave frequency, f = Coriolis parameter), it is shown that the wave speed is always decreased by the coastal irregularities. Moreover, while the coherent wave amplitude is unaltered, the energy flux along the coast is decreased by the irregularities. For the case σ > 1, however, we show that in the direction of propagation the wave is attenuated (with the energy being scattered into the random Poincaré and Kelvin wave modes) and that the wave speed is again decreased. Applications of the theory are made to the California coast and North Siberian coast to determine the decrease in phase velocity due to small coastal irregularities. For the California coast the percentage decrease is only about 1%. For the Siberian coast, however, the percentage decrease is about 25% for the K1 tide, and a minimum of 25% for the M2 tide. The attenuation of a Kelvin wave, however, appears to be due to very large scale irregularities. An estimate of the actual attenuation rate is not possible, though, because of the relatively short extent of coastal contours available for spectral analysis.

Although attention in this paper has been focused on Kelvin wave propagation, the method developed could readily be used to study the behaviour of other classes of waves trapped against a randomly perturbed boundary.

Fully developed turbulent flow through three concentric annuli was investigated experimentally for a Reynolds-number range Re = 2 × 104−2 × 105. Measurements were made of the pressure drop, the positions of zero shear stress and maximum velocity, and the velocity distribution in annuli of radius ratios α = 0.02, 0.04 and 0.1, respectively. The results for the key problem in the flow through annuli, the position of zero shear stress, showed that this position is not coincident with the position of maximum velocity. Furthermore, the investigation showed the strong influence of spacers on the velocity and shear-stress distributions. The numerous theoretical and experimental results in the literature which are based on the coincidence of the positions of zero shear stress and maximum velocity are not in agreement with reality.

The time development of non-axisymmetric disturbances on a shear layer in a uniformly rotating fluid is studied theoretically. It is assumed that the Rossby number, Ekman number, shear-layer length scale, and initial disturbance amplitude are small, and that disturbances grow if vorticity transfer from the shear flow exceeds Ekman-layer vorticity dissipation, a mechanism investigated by Busse (1968). Expressions for the ultimate instability amplitude are determined for specified relations among the small parameters, using the volume-integrated energy equation, the shape assumption, and approximations for the spatial dependence of disturbances. Disturbance growth is limited by modification of the initially-unstable shear-layer profile, and the eventual amplitude is shown to be fairly insensitive to the specific form of the profile. Using data from observations by Hide & Titman, the predicted maximum velocity amplitude of the non-axisymmetric motions for their experiments is approximately one-quarter of the velocity of the shear flow at the point of maximum gradient.

An investigation of the hydrodynamic stability of swirling flows having arbitrary Rossby numbers is described. A necessary condition for instability is derived for rigidly rotating flows and this condition is further refined in the specific case of a parabolic axial flow. Numerical results are presented for two azimuthal wave-numbers corresponding to the maximum growth rates of unstable perturbations as a function of Rossby number. It is found that the largest growth rates occur when the Rossby number is O(1) and that instability persists for surprisingly large values of this parameter. Previous explanations of the instability mechanism are discussed and it is concluded that these are only adequate in the limit of small Rossby number.

We expand the equations describing plane Poiseuille flow in Fourier series in the co-ordinates in the plane parallel to the bounding walls. There results an infinite system of equations for the amplitudes, which are functions of time and of the cross-stream co-ordinate. This system is drastically truncated and the resulting set of equations is solved accurately by a finite difference method. Three truncations are considered: (I) a single mode with dependence only on the downstream co-ordinate and time, (II) the mode of (I) plus its first harmonic, (III) a single three-dimensional mode. For all three cases, for a variety of initial conditions, the solutions evolve to a steady state as seen in a particular moving frame of reference. No runaways are encountered.

Measurements of the heat flux and temperature gradient in a layer of saline solution stably stratified by Soret diffusion and heated from below exhibit the onset of convection by the growth of very small oscillations which eventually trigger finite amplitude convection. The critical Rayleigh numbers and periods of oscillation are consistent with published theoretical calculations. Several non-linear modes are observed, even for the same heat flux, near the point of onset.

The approximations implicit in Bénard convection have been modified to include viscous dissipation. It is shown that both the influence of an adiabatic temperature gradient and of viscous dissipation are governed by the same dimensionless parameter Di = αgh/cp. Numerical calculations of finite amplitude convection are given for finite values of Di. It is found that increasing Di decreases flow velocities and finally stabilizes the flow.

This paper investigates the non-uniform motion of a thin plate of finite aspect ratio, with a rounded leading edge and sharp trailing edge, executing heaving and pitching oscillations at zero mean lift. Such vertical motions characterize the horizontal lunate tails with which cetacean mammals propel themselves, and the same motions, turned through 90° to become horizontal motions of sideslip and yaw, characterize the vertical lunate tails of certain fast-swimming fishes. An oscillating vortex sheet consisting of streamwise and spanwise components is shed to trail behind the body and it is this additional feature of the streamwise component resulting from the finiteness of the plate that makes this study a generalization of the two-dimensional treatment of lunate-tail propulsion by Lighthill (1970). The forward thrust, the power required, the energy imparted to the wake and the hydromechanical propulsive efficiency are determined for this general motion as functions of the physical parameters defining the problem: namely the aspect ratio, the reduced frequency, the feathering parameter and the position of the pitching axis. The dependence of the thrust coefficient and propulsive efficiency on these physical parameters, for the complete range of variation consistent with the assumptions of the problem, has been depicted graphically.

The interaction between an acoustic source, an unstable shear layer and a large inhomogeneous solid surface is studied, using an idealized model in which a vortex sheet is generated by uniform subsonic flow on one side of a semi-infinite plate, and subjected to line-source irradiation. Both the steady-state (time-harmonic) and initial-value (impulsive source) situations are examined. In particular, the time-harmonic field which can develop in a causal manner from a quiescent initial state is examined, and a specific criterion is given by which one may obtain the correct causal harmonic solution without explicit consideration of an initial-value problem. The satisfaction of this criterion demands not only the presence in the harmonic problem of the Helmholtz instability of an infinite vortex sheet (Jones & Morgan 1972), but additionally the presence of an edge-scatttered instability which in real time consists of a singular line plus a tail. The harmonic solution is discussed in some detail, and the consequences of omitting the unstable solutions and thereby violating causality are shown greatly to affect the diffracted field in some circumstances. The general features of the initial-value problem are also dealt with, the various waves being classified and their arrival times at any point being given in simple form. The paper ends with some speculations as to the applicability of these phenomena to the description of the process of ‘parametric amplification’, by which sound generated within a duct can be greatly amplified in the far field by triggering unstable modes on the shear layer shed from the duct.