When linear acoustic theory is applied to the thickness noise problem of a supersonic propeller, it can give rise to a surface on which the pressure is discontinuous or singular. A method is described for obtaining the equation of this surface (when it exists), and the pressure field nearby; jumps, logarithms and inverse square roots occur, and their coefficients may be calculated exactly. The special case of a blade with a straight radial edge gives a cusped cone, whose sheets, each with a different type of discontinuity or singularity in pressure, are separated by lines of cusps; the coefficients in formulae for the pressure near the surface tend to infinity as a cusp line is approached, in proportion to the inverse quarter power of distance from the line. These results determine regions of space where nonlinear effects are important, and they suggest a strong analogy with sonic boom.

The effects of diffraction by a long barrier on second-order long waves forced by sinusoidally modulated short incident waves are examined for a two-layered model ocean. When the group velocity of the short waves lies between the phase velocities of the longest baroclinic and barotropic modes, long internal waves of the frequency equal to twice the modulational frequency of the short waves are found to radiate away from the edge ray which divides the geometrical shadow and the illuminated region. In particular the baroclinic wave can penetrate the shadow. This penetration occurs when the internal long wave is not resonated by short surface waves.

For a bathymetry consisting of parallel bottom contours, wide-angle parabolic models are developed to describe the diffraction of linear water waves. The first model, developed by operator correspondence, extends the validity of conventional forms of the parabolic model for wave angles up to 70° from the assumed wave direction. Through the use of Fourier decomposition, wave models valid to 90° are developed for three different lateral boundary conditions. By application, it is shown that the diffraction of waves through gaps or around structures is governed by the initial wave condition at the structure, which can be expanded into progressive and evanescent wave modes. Away from the structure, the wave field consists of only the progressive wave modes, which disperse according to their direction of propagation, the water depth and Snell's Law. Examples are shown for oblique waves through a gap, directional seas past a breakwater, a plane wave with varying crest amplitude, and finally for the diffraction of waves into a channel.

We extend the recent work on Bragg scattering of water waves by one-dimensional parallel bars of sinusoidal profile to two-dimensional, doubly sinusoidal bed waves. The resonance condition governing the phase matching between the incident, scattered and bed waves is now more complicated and a much richer variety of resonant reflection can occur. In particular, for one normally incident wave there can be two reflected waves forming a standing wave in the transverse direction. Solutions for a wide strip of bed waves are discussed for incident water waves satisfying approximately the Bragg resonance condition. Modifications for a two-dimensional array of hemispheroids are also given. Possible application to the design of submerged breakwaters is suggested.

Similarity solutions have been found for steady two-dimensional laminar flow in which dense fluid is emitted upwards from a horizontal plane into a laminar shear flow or into a uniform flow. The solutions also apply to a light fluid released at an upper horizontal surface. The Navier-Stokes equations and the diffusion-advection equation are simplified by making the Boussinesq approximation and the boundary-layer approximation, which here also implies that pressure is hydrostatic.

For an oncoming linear shear flow representing flow near a solid surface, a similarity solution is obtained with depth proportional to $x^{\frac{1}{3}}$ where x is the horizontal coordinate. Horizontal velocity and concentration of dense fluid both increase as $x^{\frac{1}{3}}$, so that the solution represents fluid propagating upstream along the surface, and diffusing vertically to be swept downstream again. Numerical solutions for vertical profiles of velocity and concentration are presented for a Schmidt or Prandtl number σ between 0.71 and infinity. Two alternative sets of boundary conditions are possible. In one set, the pressure above the boundary layer is unchanged but the velocity profile is displaced upwards. In the second, this displacement is forced to be zero with the result that a pressure gradient is generated in the outer flow. These two boundary conditions are known to apply to disturbances in a laminar boundary-layer on horizontal lengthscales respectively greater or smaller than the triple-deck scale.

With a uniform velocity upstream and a stress-free boundary, representing flow at a free surface, similarity solutions exist only for a plume growing downstream from the source of a buoyancy flux B, with depth increasing as x½ and concentration decreasing as x−½. When gravity has negligible effects, so that B = 0, the solution is a Gaussian plume. With finite B, there is an adverse gradient of hydrostatic pressure and the plume is decelerated so that it is deeper than in neutral flow. Numerical solutions for σ = 0.71 reveal that there is a maximum buoyancy flux Bcrit above which no similarity solution exists. This occurs with a non-zero value of the surface velocity. For B < Bcrit it is found that there are in fact two possible solutions. One has surface velocity greater than at the critical flux and tends to the passive Gaussian plume as B → 0. In the other, surface velocity decreases from Bcrit, reaching zero at a non-zero value of B. Similar behaviour is found in an asymptotic solution for very large σ.

We present calculations of the change in phase speed of one train of water waves in the presence of another. We use a general method, based on Zakharov's (1968) integral equation. It is shown that the change in phase speed of each wavetrain is directly proportional to the square of the amplitude of the other. This generalizes the work of Longuet-Higgins & Phillips (1962) who considered gravity waves only.

In the important case of gravity-capillary waves, we present the correct form of the Zakharov kernel. This is used to find the expressions for the changes in phase speed. These results are then checked using a perturbation method based on that of Longuet-Higgins & Phillips (1962). Agreement to 6 significant digits has been obtained between the calculations based on these two distinct methods. Full numerical results in the form of polar diagrams over a wide range of wavelengths, away from conditions of triad resonance, are provided.

We have modelled the wall region of a turbulent boundary layer by expanding the instantaneous field in so-called empirical eigenfunctions, as permitted by the proper orthogonal decomposition theorem (Lumley 1967, 1981). We truncate the representation to obtain low-dimensional sets of ordinary differential equations, from the Navier–Stokes equations, via Galerkin projection. The experimentally determined eigenfunctions of Herzog (1986) are used; these are in the form of streamwise rolls. Our model equations represent the dynamical behaviour of these rolls. We show that these equations exhibit intermittency, which we analyse using the methods of dynamical systems theory, as well as a chaotic regime. We argue that this behaviour captures major aspects of the ejection and bursting events associated with streamwise vortex pairs which have been observed in experimental work (Kline et al. 1967). We show that although this bursting behaviour is produced autonomously in the wall region, and the structure and duration of the bursts is determined there, the pressure signal from the outer part of the boundary layer triggers the bursts, and determines their average frequency. The analysis and conclusions drawn in this paper appear to be among the first to provide a reasonably coherent link between low-dimensional chaotic dynamics and a realistic turbulent open flow system.

We study the evolution of unsteady two-dimensional vorticity structures surrounded by fluid at rest. The flow is initiated by a short fluid impulse in a horizontal layer of mercury and is constrained to be two-dimensional by a vertical uniform magnetic field. The impulse is generated by an electric pulse between two electrodes, and a flow circulation can be produced by diverting part of the current through the external frame. The velocity field is measured from the streaks of small particles floating on the free upper surface, and the vorticity is then obtained by means of an analytical interpolation and differentiation. The flow always evolves toward a set of independent steady structures with symmetry which are either circular vortices (monopoles) or couples (dipoles). The latter have a linear or circular steady motion depending on the flow circulation around them. The region of non-zero vorticity is always close to a circle. The steadiness is confirmed by plotting the vorticity versus the stream function in the frame of reference moving with the couple. We obtain a curve, as appropriate for a steady solution of the Euler equation. The slope of this curve is either constant or has no maximum. We suggest that this result could correspond to a general stability condition. The interaction between two symmetric couples at various angles of incidence yields two new couples by exchange of their vortices. Oscillations of the resulting couples are often damped by releasing a circular vortex.

Motion pictures taken at Duck Creek, a clear stream 6.5 m wide and 35 cm deep near Pinedale, Wyoming, provide detailed, quantitative information on both the modes of motion of individual bedload particles and the collective motions of large numbers of them. Bed shear stress was approximately 6 Pa (60 dynes cm−2), which was about twice the threshold for movement of the 4 mm median diameter fine gravel bed material; and transport was almost entirely as bedload. The displacements of individual particles occurred mainly by rolling of the majority of the particles and saltation of the smallest ones, and rarely by brief sliding of large, angular ones. Entrainment was principally by rollover of the larger particles and liftoff of the smaller ones, and infrequently by ejection caused by impacts, whereas distrainment was primarily by diminution of fluid forces in the case of rolling particles and by collisions with larger bed particles in the case of saltating ones. The displacement times averaged about 0.2−0.4 s and generally were much shorter than the intervening repose times. The collective motions of the particles were characterized by frequent, brief, localized, random sweep-transport events of very high rates of entrainment and transport, which in the aggregate transported approximately 70% of the total load moved. These events occurred 9% of the time at any particular point of the bed, lasted 1–2 s, affected areas typically 20–50 cm long by 10–20 cm wide, and involved bedload concentrations approximately 10 times greater than background. The distances travelled during displacements averaged about 15 times the particle diameter. Despite the differences in their dominant modes of movement, the 8–16 mm particles typically travelled only about 30% slower during displacement than the 2–4 mm ones, whose speeds averaged 21 cm s−1. Particles starting from the same point not only moved intermittently downstream but also dispersed both longitudinally and transversely, with diffusivities of 4.6 and 0.26 cm2 s−1, respectively. The bedload transport rates measured from the films were consistent with those determined conventionally with a bedload sampler. The 2–4 mm particles were entrained 6 times faster on finer areas of the bed, where 8–16 mm particles covered 6% of the surface area, than on coarser ones, where they covered 12%, even though 2–4 and 4–8 mm particles covered practically the same percentage areas in both cases. The 4–8 and 8–16 mm particles, in contrast, were entrained at the same rates in both cases. To within the statistical uncertainty, the rates of distrainment balanced the rates of entrainment for all three sizes, and were approximately proportional to the corresponding concentrations of bedload.

Experiments on surface waves were made using a cylindrical container oscillated horizontally with the period T close to that associated with the two known degenerate modes. Outside a certain region in the (T,x0)-plane, where x0 is the amplitude of the forcing displacement, surface waves exhibit either of the two kinds of regular motions whose amplitudes are constant. Within this region, however, the wave amplitude slowly changes, expressing the irregular or periodic motion of surface waves. In order to analyse these motions in detail, the slow evolution of four variables associated with the amplitudes and phases of the two modes is computed from the free-surface displacement at two measuring points. It is shown that the most common attractor corresponding to the irregular wave motion is the strange attractor with a positive maximum Liapunov exponent and a correlation dimension of 2.1–2.4. Furthermore, another kind of chaotic attractor and a few periodic orbits are found in a small parametric region. The route to chaos associated with period-doubling bifurcation is also observed. The above experimental results are compared with the solutions to weakly nonlinear evolution equations derived by Miles. We find that the equations can explain well many of the experimental results on regular and irregular wave motions. In particular, the most common chaotic attractors both in the experiments and in the theory have similar shapes in a phase space, and also yield similar values for maximum Liapunov exponents and correlation dimensions.

The extension of the classic Rayleigh–Bénard problem of a horizontal layer heated from below to the three-dimensional convection in rectangular boxes is dealt with in this paper both numerically and experimentally. Also discussed is the influence of shear flows in tilted boxes and the transition to time-dependent oscillatory convection. Three-dimensional numerical simulations allow the calculation of stationary solutions and the direct simulation of oscillatory instabilities. We limited ourselves to laminar and transcritical flows. For studying the particular characteristics of three-dimensional convection in horizontal containers, we carefully selected two container geometries with aspect ratios of 10:4:1 and 4:2:1. The onset of steady cellular convection in tilted boxes is calculated by an iterative application of a combined finite-difference method and a Galerkin method. The appearance of longitudinal and transverse convection rolls is determined by means of inter-ferometrical measuring techniques and is compared with the results of the linear stability theory. The spatial flow structure and the transition to oscillatory convection is calculated for selected examples in the range of supercritical Rayleigh numbers. Experimental investigations concerning the stability behaviour of the steady solutions with regard to time-dependent disturbances show a distinct influence of the Prandtl number and confirm the importance of nonlinear effects.

The effect of dissipation on the flow of a stratified fluid over topography is considered in the weakly nonlinear, long-wave limit for the case when the flow is near resonance, i.e. the basic flow speed is close to a linear long-wave speed for one of the long-wave modes. The two types of dissipation considered are the dissipation due to viscosity acting in boundary layers and/or interfaces and the dissipation due to viscosity acting in the fluid as a whole. The effect of changing bottom topography on the flow produced by a force moving at a resonant velocity is also considered. In this case, the resonant condition is that the force velocity is close to a linear long-wave velocity for one of the long-wave modes. It is found that in most cases, these extra effects result in the formation of a steady state, in contrast to the flow without these effects, which remains unsteady for all time. The flow resulting under the action of boundary-layer dissipation is compared with recent experimental results.

We consider a plane, steady, homogeneous flow of circular disks. The disks are identical, smooth, and inelastic. We adopt the assumption of molecular chaos and introduce an anisotropic Maxwellian velocity distribution function based on the full second moment of the velocity fluctuations. In the limits of dilute and dense flows, we determine approximate analytic solutions of the balance law for the second moment that result in stresses whose qualitative behaviour and magnitudes are in good agreement with numerical simulations.

A fully nonlinear model is developed for the unidirectional propagation of periodic gravity wave groups in deep water, in which the shape of the group envelopes changes cyclically. It is intended to describe the slow-time evolution of wave groups on the open ocean surface, and to generalize the cyclic recurrence that can occur during the sideband modulation of Stokes waves and Schrödinger wave groups. The weak nonlinear interactions are shown to concentrate the wave energy at the centre of each group at regular intervals, causing the waves there to be of greater height locally in space and time. This is suggested as one mechanism for the local wave breaking that is observed on the open ocean surface. The cyclically recurring wave groups may be interpreted as the limit-cycle stage in a progression from uniform wave groups to chaos on the forced, damped, ocean surface.

The minimum wind speed for wind wave generation has been investigated in a laboratory wind-wave flume using a sensitive slope gauge to measure the initial wavelets about 10 μm high. The growth at very low wind speeds was higher than predicted by the viscous shear-flow instability theory. Assuming that the growth is exponential, the inception wind speed at which the growth rate becomes positive can be defined. It occurred at (friction velocity) u* ≈ 2 cm/s, somewhat lower than the u* ≈ 4–5 cm/s predicted by shear-flow instability theory. However, the observed growth rates were close to the theory at higher wind speeds when the waves were higher than 1 mm. The effect of temperature on the wind speed at which the waves become readily visible is shown to be appreciable and in keeping with the temperature dependent viscous damping. Other sources of growth are discussed. Our estimates show that the Phillips resonance mechanism might be sufficiently effective to generate the observed growth at very low wind speeds.

A classical transport enhancement problem is concerned with increasing the heat transfer in a system while minimizing penalties associated with shear stress, pressure drop, and viscous dissipation. It is shown by Reynolds' analogy that viscous dissipation in a wide class of flows scales linearly with the Nusselt number and quadratically with the Reynolds number. It thus follows that transport enhancement optimization is equivalent to a problem in hydrodynamic stability theory; a more unstable flow will achieve the same Nusselt number at a lower Reynolds number, and therefore at a fraction of the dissipative cost. This transport-stability theory is illustrated in a numerical study of supercritical (unsteady) two-dimensional flow in an eddy-promoter channel comprising a plane channel with an infinite periodic array of cylindrical obstructions.

It is shown that the addition of small cylinders to a plane channel results in stability modes that are little changed in form or frequency from plane-channel Tollmien-Schlichting waves. However, eddy-promoter flows are dramatically less stable than their plane-channel counterparts owing to cylinder-induced shear-layer instability (with critical Reynolds numbers on the order of hundreds rather than thousands), and thus these flows yield heat transfer rates commensurate with those of a plane-channel turbulent flow but at much lower Reynolds number. Small-cylinder supercritical eddy-promoter flows are shown to roughly preserve the convective-diffusive Reynolds analogy, and it thus follows from the transport-stability theory that eddy-promoter flows achieve the same heat transfer rates as plane-channel turbulent flows while incurring significantly less dissipation.

A numerical study is described of the Boussinesq flow past a sphere of a viscous, incompressible and non-diffusive stratified fluid. The approaching flow has uniform velocity and linear stratification. The Reynolds number Re (= 2ρ0Ua/μ) based on the sphere diameter is 200 and the internal Froude number F(= U/Na) is varied from 0.25 to 200. Here U is the velocity, N the Brunt-Väisälä frequency, a the radius of the sphere, μ the viscosity and ρ0 the mean density. The numerical results show changes in the flow pattern with Froude number that are in good agreement with earlier theoretical and experimental results. For F < 1, the calculations show the flow passing round rather than going over the obstacle, and confirm Sheppard's simple formula for the dividing-streamline height. When the Froude number is further reduced (F < 0.4), the flow becomes approximately two-dimensional and qualitative agreement with Drazin's three-dimensional low-Froude-number theory is obtained. The relation between the wavelength of the internal gravity wave and the position of laminar separation on the sphere is also investigated to obtain the suppression and induction of separation by the wave. It is also found that the lee waves are confined in the spanwise direction to a rather narrow strip just behind the obstacle as linear theory predicts. The calculated drag coefficient CD of the sphere shows an interesting Froude-number dependence, which is quite similar to the results given by experiments. In this study not only CD but also the pressure distribution which contributes to the change of CD are obtained and the mechanism of the change is closely examined.

In this paper the mathematical formulation associated with waves trapped by submerged cylinders is recast as a standard eigenvalue problem. In this way the existence is proven of trapped waves for every frequency Ω, and for arbitrary geometry of the submerged cylinder. At the same time a simple expression for the first eigenvalue and eigenmode, correct in the limits Ω → 0 or Ω → ∞, is derived. The expression can be a useful approximation for a structure relatively transparent to the wave action such as, for instance, a semisubmersible platform.

In a companion paper the existence of trapped waves over submerged cylinders has been analysed, and a necessary condition for their excitation was derived. In the present paper, this study is extended to obtain physically more important results. First we consider a more realistic geometry, namely a finite, although slender, cylinder. Second we derive the necessary and sufficient conditions for the excitation of trapped modes; and lastly, the induced resonant response is studied with the multiple-scales technique. It is shown then that the wave amplitude satisfies an equation similar to the resonant nonlinear oscillator.

Two plates placed in tandem in a duct flow shed vortices, which can excite and sustain an acoustic resonance associated with the duct. The sound can in turn ‘feed back’ and ‘lock’ the vortex shedding rate to the sound frequency. The experimental conditions under which loud resonant sound is sustained are described in this paper. The acoustic sources are predicted by combining a vortex model of the flow field with a finite-element solution of the sound field, and then using Howe's theory of aerodynamic sound to calculate the energy exchange between the flow and the sound field. Only in certain regions near the plates is substantial net energy exchange possible; the direction of energy transfer depends on the spacing of the plates. The region around the trailing edge of the upstream plate is found to be always a net acoustic source during resonance, while the region around the downstream plate is a net source or sink depending on the phase of the acoustic cycle at which vortices arrive there, which in turn depends on plate spacing and flow velocity. The net source region around the downstream plate is suppressed over a wide range of plate spacings by splitting this plate at midspan and rejoining it so that one half is offset in the flow direction by the distance a vortex travels in half a sound cycle.