This paper considers the propagation and scattering of waves in dispersive and non-dispersive media containing random inhomogeneities. A detailed discussion of the propagation of the coherent component of the wave field is presented, and a rather general method for obtaining the absorption coefficient and scattering cross-section per unit volume of medium is described. A major objective is to derive macroscopic equations for the propagation of the coherent field in the sense that the damping due to scattering appears in the wave equation as pseudo-viscosity. The paper is concluded by a detailed application of the theory to the problem of the propagation of surface gravity waves over a stretch of shallow water with a ‘rough’ bed. By including finite amplitude effects a balance is achieved between the dispersion of the pseudo-viscous term and the non-linear convective terms, which enables the steady profile of a weak bore to be calculated.

A small-disturbance solution is obtained for the steady two-dimensional flow over a sinusoidal wall of an inviscid gas in vibrational or chemical non-equilibrium. The results are based on a single, linear, third-order partial differential equation, which plays the same role here as does the Prandtl–Glauert equation in equilibrium flow. The solution is valid throughout the range from subsonic to supersonic speeds and for all values of the rate parameter from equilibrium to frozen flow (in both of which limits it reduces to Ackert's classical solution of the Prandtl–Glauert equation). The results illustrate in simple fashion some of the properties of non-equilibrium flow, such as the occurrence of pressure drag at subsonic speeds and the absence of the discontinuous phenomena that characterize the Prandtl–Glauert theory when the flow changes from subsonic to supersonic.

This paper describes the results of experiments conducted on a circular jet simultaneously excited by two different acoustic tones. By varying the phase between two signals at harmonically related frequencies, control can be exercised on the process of harmonic generation – sometimes the process being virtually destroyed. This is shown to be so for both harmonic and subharmonic generation, but the latter is more difficult to control.

The influence of leading-edge separation vortices on the Weis-Fogh (1973) lift- generation mechanism for insect hovering is investigated. The analysis employs a vortex-shedding model (Edwards 1954; Cheng 1954) and represents an extension of Lighthill's (1973, 1975) analysis of an inviscid model without separated vortices. Results of the study compare reasonably well with observations on a laboratory model at high Reynolds number (Maxworthy 1979), confirming that vortex separation significantly enhances the initial circulation on each of the wings. Unlike the un- separated-model solution, this circulation was found to depend on the history of the wing motion and to increase with a large opening angle.

A submerged sphere is considered to be absorbing power from an incident wave through an integrated mooring and power take-off system. It is shown that the power absorbed (as characterized by the absorption length) depends on the hydrodynamic properties of the sphere; in particular on the added-mass and damping coefficients. These coefficients are determined and the results used to study the power absorption properties of the sphere. Curves are given showing the variation of the absorption length with wavenumber, for differing depths of submergence.

A sphere exhibits a prolonged residence time when settling through a stable stratification of miscible fluids due to the deformation of the fluid-density field. Using a Green's function formulation, a first-principles numerically assisted theoretical model for the sphere–fluid coupled dynamics at low Reynolds number is derived. Predictions of the model, which uses no adjustable parameters, are compared with data from an experimental investigation with spheres of varying sizes and densities settling in stratified corn syrup. The velocity of the sphere as well as the deformation of the density field are tracked using time-lapse images, then compared with the theoretical predictions. A settling rate comparison with spheres in dense homogeneous fluid additionally quantifies the effect of the enhanced residence time. Analysis of our theory identifies parametric trends, which are also partially explored in the experiments, further confirming the predictive capability of the theoretical model. The limit of infinite fluid domain is considered, showing evidence that the Stokes paradox of infinite fluid volume dragged by a moving sphere can be regularized by density stratifications. Comparisons with other possible models under a hierarchy of additional simplifying assumptions are also presented.

A simple frictional wind-driven model of the Antarctic circumpolar current is examined. The geometry includes what are thought to be the main features effecting the current, namely a gap corresponding to Drake Passage and a partial barrier corresponding to the South American peninsula. Solutions obtained by both numerical and analytical methods are presented. The analytic solution, valid for small values of a friction parameter, enables the total rate of transport of water by the current to be calculated as a function of the friction parameter and the wind stress distribution. The width which controls the rate of transport by the current tends to be the narrowest encountered rather than the average width as assumed in earlier zonal models. However, the current spreads out to several times this width due to frictional effects. The values of the eddy viscosity required to give a rate of transport of the observed order are about 103 cm2/s for a bottom-friction model and of the order of 108 cm2/s for a lateral friction model.

A droplet may ‘walk’ across the surface of a vertically vibrating bath of the same fluid, due to the propulsive interaction with its wave field. This hydrodynamic pilot-wave system exhibits many dynamics previously believed to exist only in the quantum realm. Starting from first principles, we derive a discrete-time fluid model, whereby the bath–droplet interactions are modelled as instantaneous. By analysing the stability of the fixed points of the system, we explain the dynamics of a walking droplet and capture the quantisations for multiple-droplet interactions. Circular orbits in a harmonic potential are studied, and a double quantisation of chaotic trajectories is obtained through systematic statistical analysis.

The evolution of a turbulent spot in an accelerating laminar boundary-layer flow was investigated. The type of boundary layer chosen for this experiment resembles in every respect the flow in the vicinity of a stagnation point theoretically described by Falkner and Skan. The rate of growth of the spot was significantly inhibited by the favourable pressure gradient in all three directions. It became much shorter and narrower in comparison with a similar spot generated in a Blasius boundary layer at comparable distances from its origin and comparable Reynolds numbers. The celerities of its boundaries did not scale with the local free-stream velocity as they do in the absence of a pressure gradient. Dimensional analysis was used to identify and correlate the independent variables determining the size, the convection speed, and the relative rate of growth of this spot

The familiar arrowhead shape of the spot gave way to a rounded triangular shape with the trailing interface being straight and perpendicular to the direction of streaming. The familiar Tollmien-Schlichting wave packet was not observed in this pressure gradient because the surrounding boundary layer was very stable at the Re considered. Since the arrowhead shape of the spot is associated with the breakdown of the waves within the packet it cannot occur below the critical Re. The relative size of the ‘calmed region’ following the spot also diminished; however, one could only speculate as to the origin of this region.

A near-wall subgrid-scale (SGS) model is used to perform large-eddy simulation (LES) of the developing, smooth-wall, zero-pressure-gradient flat-plate turbulent boundary layer. In this model, the stretched-vortex, SGS closure is utilized in conjunction with a tailored, near-wall model designed to incorporate anisotropic vorticity scales in the presence of the wall. Large-eddy simulations of the turbulent boundary layer are reported at Reynolds numbers based on the free-stream velocity and the momentum thickness in the range . Results include the inverse square-root skin-friction coefficient, , velocity profiles, the shape factor , the von Kármán ‘constant’ and the Coles wake factor as functions of . Comparisons with some direct numerical simulation (DNS) and experiment are made including turbulent intensity data from atmospheric-layer measurements at . At extremely large , the empirical Coles–Fernholz relation for skin-friction coefficient provides a reasonable representation of the LES predictions. While the present LES methodology cannot probe the structure of the near-wall region, the present results show turbulence intensities that scale on the wall-friction velocity and on the Clauser length scale over almost all of the outer boundary layer. It is argued that LES is suggestive of the asymptotic, infinite Reynolds number limit for the smooth-wall turbulent boundary layer and different ways in which this limit can be approached are discussed. The maximum of the present simulations appears to be limited by machine precision and it is speculated, but not demonstrated, that even larger could be achieved with quad- or higher-precision arithmetic.

In this paper direct numerical evidence of flow-induced incipient cavitation is presented through lattice Boltzmann simulations of multiphase flows with a non-ideal thermodynamic equation of state. Cavitation emerges spontaneously as a result of the underlying non-ideal interactions, with no need for any modelling criteria based on the fluid variables, such as pressure or stress tensor. The onset of cavitation is well captured by Joseph’s minimum tension criteria, (Joseph, J. Fluid Mech., vol. 366, 1998, pp. 367–378; Dabiri, Sirignano & Joseph, Phys. Fluids, vol. 19, 2007, 072112), complemented with surface tension corrections, as proposed by Brennen (Cavitation and Bubble Dynamics, Oxford University Press, 1995). The simulations also show that the cavitation number (${C}_{N} $) proves to be a poor predictor of the onset of cavitation. Finally, strong dependence of the bubble morphology on the surface tension is also highlighted.

We present experimental results of the generation of internal gravity waves by a turbulent buoyant plume impinging upon the interface between a uniform density layer of fluid and a linearly stratified layer. The wave field is observed and its properties are measured non-intrusively using axisymmetric Schlieren. In particular, we determine the fraction of the energy flux associated with the plume at the neutral buoyancy level that is extracted by the waves. On average, this was found to be approximately 4%. Within the limits of the experimental parameters, the maximum vertical displacement amplitude of waves were found to depend linearly upon the maximum penetration height of the plume beyond the neutral level. The frequency of the waves was found to lie in a narrow range relative to the buoyancy frequency. The results are used to interpret the generation of waves in the atmosphere by convective storms impinging upon the tropopause via the mechanical oscillator effect.

This paper presents a comprehensive study of the near-bed hydrodynamics at non-moving streambeds based on laboratory experiments in open-channel flows. Pressure and velocity measurements were made with an array of up to 15 miniaturized piezo-resistive pressure sensors within the bed and slightly above it, and a two-dimensional particle-image-velocimetry (PIV) system measuring in streamwise vertical or horizontal planes. Three different types of bed materials were studied covering typical natural streambed conditions. The range of the global Reynolds number covered in the experiments was from 20000 to 200000. This study provides new insights into the flow structure over gravel beds based on the PIV measurements in both streamwise vertical and horizontal planes. In a streamwise vertical plane, large-scale wedge-like flow structures were observed where a zone of faster fluid over-rolled a zone with slower fluid. The resulting shear layer was inclined along the flow at an angle of 10°–25° to the bed, and was populated with clockwise rotating eddies. This mechanism occurred with sufficient frequency and shape to leave an ‘imprint’ in the velocity statistics. Typically, the described flow pattern is formed near the bed and is approximately scaled with the height of the logarithmic layer, although the biggest structures extended over the whole flow depth. In a horizontal near-bed plane, turbulent structures formed a patched ‘chessboard’ pattern with regions of lower and higher velocities that were elongated in the streamwise direction. Their lateral extension was typically two to four times the equivalent sand roughness with lengths up to several water depths. The dimensions of the regions were increasing linearly with the distance from the bed. These findings are consistent with conceptual models originally developed for smooth-wall flows. They also support observations made in rough-bed flume experiments, numerical simulations and natural rivers. Spatial fields of bed-pressure fluctuations were reconstructed by applying Taylor's frozen turbulence hypothesis on time data obtained with an array of pressure sensors. Based on the conditional sampling of velocity patterns associated with pressure-drop events a distinct bed-destabilizing flow-pressure pattern was identified. If a high-speed fluid in the wake of a large-scale wedge-like flow structure reaches the vicinity of the bed, a phenomenon akin to a Bernoulli effect leads to a distinctive low-pressure pattern. The resulting force may exceed the particles' submerged weight and is assumed to be able to give an initial lift to the particle. As a result, the exposed area of a particle is amplified and its angle of repose is reduced, increasing the probability for entrainment.

In the thermal convection problem with free boundaries, the interaction of two ‘roll’ disturbances is considered. The problem is reduced to a pair of non-linear ordinary differential equations, which should also provide a model for the interaction of two disturbances in more general situations than that for which these equations have been derived. The equations contain several parameters which necessitates a discussion of various possible types of solution. Some representative results are: (1) under certain circumstances, an equilibrium state may be composed of a mixture of a linearly stable disturbance and a linearly unstable disturbance; (2) for the thermal convection problem, when the Rayleigh number is slightly above the minimum critical value, the equilibrium state will contain only one of two linearly unstable disturbances. These and other results are compared with experimental observations.

In this paper experiments are reported on a condensing steam jet. Superheated steam is injected at the bottom centre of a cylindrical water vessel, resulting in a turbulent jet with Reynolds numbers varying between $7.9{\times}10^4$ and $18.1{\times}10.4$, depending on the bulk temperature of the water. Near the steam inlet, the flow is two-phase with rapidly condensing steam. Downstream of a development region the jet is essentially single-phase. Using particle image velocimetry in a vertical plane through the central axis, instantaneous velocity fields of the single-phase region have been measured at a rate of 15Hz. The velocity field in this region is found to be self-similar, i.e. the width of the jet, $r_{1/2}$, increases linearly with increasing distance to the virtual origin, and a Gaussian profile prevails if velocities and distances are properly scaled. The spreading rate is equal to the one usually found in single-phase jets, and temperature independent. The virtual origin of the jet is positioned at a temperature-dependent distance (3–7 nozzle diameters) upstream of the steam inlet, and this distance is shown to correlate with the length of the condensation region. The turbulence intensity is found to be similar to the intensities usually reported for single-phase jets, although full isotropy is only reached at a distance of 15 nozzle diameters from the nozzle. The jet exhibits a slow wobbling motion, which can be attributed to instability of the backflow resulting from the confinement. When the measurements are compensated for this wobble, a slightly smaller spreading rate is obtained, which indicates that unconditional averaging may conceal significant flow structuring.

The behaviour of dipolar vortices in a rotating fluid with a sloping bottom (simulating the variation of the Coriolis parameter on the Earth, with the direction of steepest bottom slope corresponding with the northern direction) has been investigated in the laboratory. Dipoles were generated by moving a vertical cylinder through the fluid. Dye photographs provided qualitative information, whereas quantitative information about the evolving flow field was obtained by streak photography. Dipoles initially directed under a certain angle relative to the west–east axis showed meandering or cycloid-like trajectories. Soeme symmetries between east-travelling dipoles (ETD's) and west-travelling dipoles (WTD's) were observed. ETD's are stable in the trajectory sense: a small deviation from zonal motion results in small oscillations around the equilibrium latitude. WTD's are unstable: small initial deviations produce large displacements in northern or southern directions. This asymmetry arises because the vorticity of a dipole moving westward is anticorrelated with the ambient vorticity, while the vorticities are correlated when the dipole moves eastward. ETD's increase in size and eventually split into two independent monopoles, the rate of growth depending on the gradient of planetary vorticity. WTD's are initially more compact but owing to the large displacements in the meridional direction strong asymmetries in the circulation of the two halves are produced, resulting in a large deformation of the weaker part. The experimental observations show good qualitative agreement with analytical and numerical results obtained using a modulated point-vortex model.

Possible solutions of the Navier-Stokes equations are given representing certain finite disturbances in plane Poiseuille flow which vary with distance parallel to the bounding walls. These solutions are based on infinitesimal disturbances which vary exponentially with distance (upstream or downstream) instead of with time, and they are more closely related to the disturbances investigated experimentally than the corresponding ‘time-dependent’ solutions.

This paper provides a unique, detailed evaluation of basic aeroacoustic theory applied to low-Mach-number (M = 0.08) cold jets. In contrast to most prior studies comparing theoretical predictions of jet noise with experimental results, our comparison uses a relatively complete knowledge of the flow field and employs vortex sound theory — an acoustic analogy which is shown to be insensitive to those aspects of the flow field about which our knowledge is incomplete. The primary result is that the measured sound field directivity of vortex ring pairing in circular jets is very similar to that predicted by theory: a stationary, axisymmetric, lateral quadrupole. This directivity is very unlike the monotonic polar dependence found in time-average measures of jet noise fields and unlike the directivity found in similar excited jet experiments. Although not perfect, the agreement between experiment and theory here is satisfyingly close in comparison to the discrepancies found by Huerre & Oighton (1983). Our result also proves that pairing of purely axisymmetric coherent structures is not the dominant sound source in low-Mach-number jets and that vortex asymmetry must be an essential aspect of the vortex motions which produce noise in such jets.

Profiles of gasdynamic parameters in self-similar blast waves, taking into account the influence of conduction and radiation fluxes due to high temperatures attained at the centre, are determined. In the blast-wave equations these fluxes are expressed in terms of the Fourier law for heat conduction and a differential expression for radiative transport in a semi-grey gas model. Various boundary conditions are considered in order to account for different ways in which blast waves are initiated and driven. Similarity requirements are implemented in the solution by compatible functional forms of gas conductivity and absorptivity, as well as the opacity of the shock front. This formulation yields a two-point boundary-value problem, which is then transformed into an initial-value problem in order to facilitate the integration. As a particular example, a detailed solution for the constant-energy case is obtained, covering the whole range of relative heat-transfer effects expressed in terms of radiative to gasdynamic energy fluxes, from the adiabatic flow field, on one extreme, to the isothermal, on the other.