We derive the dynamics of small waves riding on larger waves using a canonical, Hamiltonian formulation. The small waves are treated linearly and their energy is derived to all orders in the scale separation between the waves. Our results are similar to those of Longuet-Higgins (1987), but we have extended his calculations to include gravity-capillary waves and to allow for a more general, two-dimensional, large-wave field. Our result for the small-wave Hamiltonian is expressed in both Eulerian (horizontal) coordinate system and a non-inertial system determined by the large wave's surface. On further assuming scale separation between the small and large waves the averaged Lagrangian equations and the action density are derived. Action conservation is explicitly demonstrated.

In a recent paper by Lawrence & Weinbaum (1986) an unexpected new behaviour was discovered for a nearly spherical body executing harmonic oscillations in unsteady Stokes flow. The force was not a simple quadratic function in half-integer powers of the frequency parameter λ2 = −ia2ω/ν, as in the classical solution of Stokes (1851) for a sphere, and the force for an arbitrary velocity U(t) contained a new memory integral whose kernel differed from the classical t−½ behaviour derived by Basset (1888) for a sphere. A more general analysis of the unsteady Stokes equations is presented herein for the axisymmetric flow past a spheroidal body to elucidate the behaviour of the force at arbitrary aspect ratio. Perturbation solutions in the frequency parameter λ are first obtained for a spheroid in the limit of low- and high-frequency oscillations. These solutions show that in contrast to a sphere the first order corrections for the component of the drag force that is proportional to the first power of λ exhibit a different behaviour in the extreme cases of the steady Stokes flow and inviscid limits. Exact solutions are presented for the middle frequency range in terms of spheroidal wave functions and these results are interpreted in terms of the analytic solutions for the asymptotic behaviour. It is shown that the force on a body can be represented in terms of four contributions; the classical Stokes and virtual mass forces; a newly defined generalized Basset force proportional to λ whose coefficient is a function of body geometry derived from the perturbation solution for high frequency; and a fourth term which combines frequency and geometry in a more general way. In view of the complexity of this fourth term, a relatively simple correlation is proposed which provides good accuracy for all aspect ratios in the range 0.1 < b/a < 10 where exact solutions were calculated and for all values of λ. Furthermore, the correlation has a simple inverse Laplace transform, so that the force may be found for an arbitrary velocity U(t) of the spheroid. The new fourth term transforms to a memory integral whose kernel is either bounded or has a weaker singularity than the t−½ behaviour of the Basset memory integral. These results are used to propose an approximate functional form for the force on an arbitrary body in unsteady motion at low Reynolds number.

Experiments by Binnie showed that unsteady waves were produced by flow through a channel with symmetric, wavy sidewalls, with waves propagating both upstream and downstream. However, the first-order solution to this problem that was obtained by Yih is a set of steady waves. The steady solution is shown to be unstable to a pair of infinitesimal disturbance waves which satisfy the resonance conditions of Phillips. For the Froude-number range used by Binnie, a pair of disturbances has been found such that one wave propagates upstream, one propagates downstream, and the amplitudes have an exponential growth. The Froude numbers outside the range of Binnie are also shown to be unstable. The steady waves produced by flow through an antisymmetric channel are shown to be unstable in the same manner.

This paper begins by reviewing Bethe's (1942) work on the subject. He considered the propagation of a normal shock wave in a medium with an arbitrary equation of state. Difficulties arise if one attempts to extend his theory to systems containing plane oblique shocks or the reflection or refraction of such shocks. The object of the present paper is to resolve these difficulties. General conditions for the local thermodynamic equilibrium and thermodynamie stability, of a non-equilibrium system in steady-state, adiabatic, flow are summarized by the principle of maximum entropy production, which gives \[ \Delta s\geqslant 0;\quad {\rm d}(\Delta s)= 0;\quad {\rm d}^2(\Delta s) < 0, \] for ht, constant, where s is the specific entropy and ht is the specific total enthalpy; it is deduced from the second law. Conversely the consequences of Δs < 0, d(Δs) ≠ 0, d2(Δs) = 0, are discussed and may lead to either an impossibility or to some form of instability such as unsteadiness, or a change in the structure of the system (a catastrophe).

Countercurrent flow is a generalized plane Couette flow in which a shear-induced drift current is opposed by a pressure-driven return flow such that the resulting mass flux is zero. This type of flow is encountered in both environmental fluid mechanics and tribology. Measurements are described that were undertaken in steady turbulent countercurrent flow, generated in a novel type of apparatus between smooth walls, at Reynolds numbers, expressed in terms of surface velocity and depth of the flow, from 200 to 20000; the critical Reynolds number of laminar to turbulent transition as determined herein is approximately 1750. The laboratory facility is explained, and experimental data are presented on mean velocities, pressure gradient, turbulence intensities, Reynolds stresses, and energy spectra. It is found that the velocity distributions follow the universal law of the wall in the drift current, but that the flow is undeveloped in the return portion of the flow. The non-dimensional longitudinal pressure gradient increases with Reynolds number, and a semi-empirical law of resistance is proposed and experimentally verified. Turbulence intensities in the drift current increase toward the shearing surface but are essentially constant in the opposing pressure-driven flow. The distribution of the Reynolds stress is found to be consistent with previous measurements obtained in this type of flow using different apparatus, but only approximately follows the theoretically linear distribution. Energy spectra in the shear-induced portion of the flow involve higher frequencies closer to the shearing surface than in the return current. There, the spectral energy is essentially the same throughout the flow, reflecting the constancy of turbulent intensity in this region.

A steady isolated vortex is produced in a horizontal layer of mercury (of thickness a), subjected to a uniform vertical magnetic field. The vortex is forced by an electric current going from an electrode in the lower plane to the circular outer frame. The flow is investigated by streak photographs of small particles following the free upper surface, and by electric potential measurements. The agreement with the asymptotic theory for high values of the Hartmann number M is excellent for moderate electric currents. In particular all the current stays in the thin Hartmann layer of thickness a/M, except in the vortex core of horizontal extension a/M½. For higher currents, the size of the core becomes larger and depends only on the local interaction parameters. When the current is switched off, we measure the decay due to the Hartmann friction. A similar study is carried out for a vortex created by an initial electric pulse, and for a pair of vortices of opposite sign. For all these examples, the dynamics can be described by the two-dimensional Navier-Stokes equations with Hartmann friction, except in the vortex cores. Finally a vortex is produced near a lateral wall and a detachment of the boundary layer parallel to the magnetic field occurs, by which a second vortex of opposite sign is generated.

We present an experimental and theoretical study of the effects of localized mixing on a stratified fluid contained in a reservoir. In the experiments, mixing is accomplished by means of a vertically oscillating, horizontal grid located near the water surface at one end of the reservoir. Once the grid is set in motion, a mixed layer forms immediately beneath it. As this layer deepens, a horizontal pressure gradient builds up which drives an outflow of mixed fluid into the unmixed interior of the reservoir. This outflow slows, and eventually brings to a halt, mixed-layer deepening under the grid. At this equilibrium depth, the vertical velocity of the entrainment interface induced by the outflow exactly equals the velocity at which the entrainment interface would move downwards because of mixing. This equilibrium state persists until the outflow intrusion is blocked by the far wall of the reservoir, at which time deepening resumes under the grid. Asymptotically, the fluid reaches a state in which mixed-layer deepening is independent of position.