This paper considers steady, slowly varying water waves propagating over a gently sloping bed on a steady current. The current varies linearly with depth, and so has constant vorticity ω. The analysis is two-dimensional and dissipation is neglected. Definitions, and expressions correct to second order in the amplitude, are given for the radiation stress, wave energy density E and total energy flux. An average La-grangian [Lscr ], obtained by heuristic arguments from Clebsch potentials, leads to the result that for this particular problem E equals the wave action [Lscr ]ω times the angular frequency ωrm relative to a frame of reference moving with the average-over-depth current velocity Um. This determines the variation of the amplitude with distance explicitly. An analytical expression for the height of the mean water surface is found by a heuristic argument which compares the conservation equations for total energy and wave action. All the results have been checked directly by substitution back into the basic equations. Graphs illustrate the effect of the vorticity ω on the wavelength, amplitude and set-down.

A model of the diffusive interface in double-diffusive convection at high Rayleigh number is proposed. The interface is assumed to have a double structure: two marginally stable boundary layers from which blobs or thermals arise on the outer edges of the interface, separated by a diffusive core across which all transport takes place by molecular diffusion. The model is time-independent and comparison is made with unsteady ‘run-down’ experiments on the assumption that the experiments run down through a sequence of equilibrium states each of which can be considered separately. The model predicts a constant ratio of the buoyancy fluxes of the two components at a value equal to the square root of the ratio of their molecular diffusivities, and individual fluxes in reasonable agreement with the available experimental data. Some time-dependent features of the model are also examined.

Two-dimensional time-periodic water waves on a gently sloping bottom are investigated under the classical long-wave assumptions that ε = h′0/λ′ and δ = H′/h′0 are small parameters (H′ being the wave height, h′0 a characteristic water depth and δ′ the horizontal scale for the oscillatory motion) and the assumption that δ/ε2 = O(1) as δ and ε tend to zero.

It is shown that for a bottom slope hx for which hx = o(ε3) the governing KdV equation with slowly varying coefficients (derived by Johnson 1973a) has a time-periodic solution which in the first approximation is a slowly varying cnoidal wave. The second-order approximation in an asymptotic expansion with respect to the bottom slope represents the deformation of this wave due to the sloping bottom. For hx [Gt ] ε5 this deformation is larger than the second-order contribution from the basic expansion with respect to wave amplitude which underlies the KdV equation itself; the calculation is then a consistent approximation to physical reality.

Numerical results for the deformation are given. Also, the wave profiles are compared with experiments on a plane of slope $h_x = \frac{1}{35}$ and show good agreement even for the large values of H’/h’ appropriate to waves rather close to breaking.

Convection in a porous cavity driven by heating in the horizontal is analysed by a number of different techniques which yield a fairly complete description of the two-dimensional solutions. The solutions are governed by two dimensionless parameters: the Darcy-Rayleigh number R and the cavity aspect ratio A. We first find solutions valid for shallow cavities, A → 0, by using matched asymptotic expansions. These solutions are given up to O(A6R4). For A fixed, we find regular expansions in R by semi-numerical techniques, up to O(R30) in some cases. Series-improvement techniques then enable us to cover the range 0 ≤ R ≤ ∞. A limited result regarding bifurcations is noted. Finally, for R → ∞ with A fixed, we propose a self-consistent boundary-layer theory which extends previous approximate work. The results obtained by these different methods of solution are in good agreement with each other and with experiments.

The supersonic flow field past a circular cone of semi-apex angle 7·5° at incidences up to 26° has been investigated experimentally at a free-stream Mach number of 2·94. The experiments were made using a five-hole conical-head directional probe. Since the flow past the cone was found to be conical, the flow phenomena may be described in a plane perpendicular to the cone axis. This paper gives the direction of the conical streamlines, the conical sonic line and the conical Mach number in such a plane, all of which may be deduced from the measurements. At incidences exceeding twice the cone semi-apex angle a separated flow regime was observed which consisted of two main vortices on either side of the leeward plane of symmetry, and probably also two secondary vortices close to the cone surface. From incidences of 17° onwards, an embedded region of conical supersonic flow terminated by a shock wave was revealed. At 22°, approximately, a second embedded shock wave was measured close to the cone surface and extended slightly across the leeward plane of symmetry. This shock wave may have been generated by the vortex system, which induced supersonic cross-flow components towards the cone.

Steady flow of an incompressible homogeneous fluid over shallow topography in a rotating annulus is considered. The flow is driven by a differentially rotating lid. The recycling nature of the system means that prescribed upstream conditions are not available to close the problem. Consideration of the balance of transport across streamlines for the geostrophic flow leads to a general circulation condition; namely $\Gamma(\psi) = \frac{1}{2}\Gamma_T(\psi)$, where ϕ is the stream function for the geostrophic flow, Λ is the circulation around a streamline and ΛT is the circulation around the same path calculated using the prescribed upper surface velocity. Using this condition, stream functions for the linear viscous and nonlinear quasi-inviscid flow can be found. Solutions for these two limits, and linearized perturbation solutions for the transition regime between them, are presented for flow over ridges in the annulus.

An International Advanced Course on the Thermomechanics of Magnetic Fluids was held from 3 to 7 October 1977 at the International Centre for Mechanical Sciences (CISM), Udine, Italy, under the direction of B. Berkovsky. The course was a joint CISM/UNESCO undertaking and its general objectives were to assess knowledge in the field, to identify specific problem areas and promising directions for future research and applications, and to assist in the establishment of active relations between national research groups. It was the first conference to bring together the diverse and dispersed community of researchers in the field; there were 34 participants from 16 countries and 19 papers were included in the programme.

An investigation based on the elongated-body theory of Lighthill (1960, 1970) and Wu (1971) is made of the carangiform swimming motions of fish, a mode of propulsion in which the amplitude of body undulation becomes significant only in the posterior half, or even third, of the length of the fish and the anterior part is relatively inflexible. For typical slender fish performing undulatory swimming motions, three hydro-dynamic features are taken into account: (a) the resistance to the lateral undulatory motions as well as longitudinal (tangential) frictional resistance, (b) the forces of interaction with the water associated with its inertial (virtual-mass) response to the lateral motions of the fish and (c) the reaction forces due to the vortex sheets shed from sharp trailing edges to the rear of the section of maximum span (depth), where a great variety of fins are generally found.

The active tail oscillations give rise to oscillatory side forces, to which the remainder of the body responds passively. These passive yawing motions are studied to find their amplitude, the yawing axis and any associated energy dissipation. The contribution of each of the above three forces is examined and the effects of the oscillation frequency, a slenderness parameter δ of the body, and the shape of the transverse cross-sections are considered.

The present theory is further applied to predict the turning movement when the fish changes direction, the way in which the above forces act during this process being investigated. A small perturbation analysis relative to a uniform rectilinear motion not only reveals whether or not the motion is dynamically stable, but also leads to a full description of the motion that follows a given initial perturbation. For finite perturbations a numerical method is adopted to find the time development of both the direction of motion relative to the direction of the initial uniform motion and the ratio of the kinetic energy of the body to the initial kinetic energy for different magnitudes I0 of the initial impulsive perturbation in the transverse direction. The final angle of turn is obtained in terms of I0 for different values of δ. The loss of kinetic energy during the turn divided by the initial kinetic energy is found to be very small for a small angle of turn.

Detailed analyses are made for rigid fish models in which the distribution of depth (or span) along the body length is quadratic with a maximum at the centre. Agreement of the present analyses with observations is fairly good at least qualitatively and some quantitative estimates are made of the side forces, both oscillatory and impulsive, exerted on the body by the tail. It is found that the vortex sheets shed from trailing edges increase the energy loss due to the yawing oscillations, in addition to the resistive dissipation, but contribute to the directional stability of rectilinear motion.

The Navier-Stokes equations for a two-layer flow are written in a curvilinear system of co-ordinates in which the height is measured from the interface. A technique for averaging the equations over an ensemble of wave surfaces which are not very different from each other is proposed. Moments which include deviations in the rate of surface displacement and in the slopes are dropped. It is assumed that the averaged equations describe the evolution of a large-scale velocity field. The moments of the turbulent velocity field are parameterized using the isotropic coefficient of turbulent viscosity, which, from dimensional considerations, is expressed in terms of the length scale growing linearly upwards and downwards from the interface and the turbulent kinetic energy. The equation defining the evolution of turbulent energy is derived without allowing for the curvilinearity of the system of co-ordinates. Laboratory experiments in a wind-water tunnel are simulated by integrating the equations numerically. The results are compared with measurements.

Free-stream turbulence is modelled by a low-intensity array of vortices where the vorticity is distributed continuously throughout the flow. This vorticity approaches and convects along a semi-infinite flat plate and the structure of the free-stream disturbances is altered by the impermeability condition at the plate. The analysis consists of tracking the vorticity as it convects with the uniform mean flow, determining the stream function induced by that vorticity field without imposing the impermeability condition, and finally superimposing an irrotational flow field which effects impermeability. The bisecting of a vortex as it encounters the plate yields a pair of vortices which rotate in the same direction. The combined heights of these vortices are less than the height of the original vortex. Small segments of a vortex which has been cut by the plate, but not through its centre, are completely absorbed by neighbouring vortices. Far from the leading edge in any direction, the disturbance pressure is O(ρq2), where q is the characteristic disturbance speed, and the streamline patterns convect with the mean flow. Near the leading edge, the fluctuating pressure is O(ρqU∞) because of unsteady vortex distortion and the velocity correlations reveal that Taylor's hypothesis is not valid. These correlations are based on averages over time and over all possible orientations of the vortex array. Vortex structures based on iso-vorticity contours are sometimes quite different from the structures based on disturbance streamlines.