The acoustic radiation pressure exerted by a plane — progressive or standing — sound wave on a compressible sphere suspended freely in a viscous fluid is calculated. In deriving the general expression for the radiation pressure, it is supposed that the radius of the sphere is arbitrary. Two limiting cases of interest are then considered. In the first of these, it is assumed that the sound wavelength is much larger than the radius of the sphere which is, in turn, much larger than the viscous wavelength, it being supposed that this condition is satisfied both outside and inside the sphere. In the second case, the situation is investigated when the radius of the sphere is small compared with the viscous wavelength which is, in turn, much smaller than the sound wavelength, it being supposed that this condition is satisfied, as before, both outside and inside the sphere. It is shown that in both cases the expressions for the radiation pressure are drastically different from the well-known expressions for the radiation pressure in a perfect fluid: the calculation of the radiation pressure from the formulae obtained for a perfect fluid in the cases when the effect of viscosity is not negligible gives both quantitatively and qualitatively wrong results.

In a two-dimensional incompressible fluid, the barotropic instability of isolated circular vortices can lead to multipole formation. The multipoles we study here are composed of a core vortex surrounded by two or more identical satellite vortices, of opposite-sign vorticity to the core, and the total circulation is zero. First, we present the generation of multipoles from unstable piecewise-constant monopoles perturbed on a monochromatic azimuthal mode. The stationary multipoles formed by this nonlinear evolution retain the same energy, circulation and angular momentum as the original monopoles, but possess a lower enstrophy. These multipolar steady states are then compared to multipolar equilibria of the Euler equation, obtained either analytically by a perturbation expansion or numerically via a relaxation algorithm. Finally the stability of these equilibria is studied. Quadrupoles (one core vortex bound to three satellites) prove relatively robust, whether initially perturbed or not, and resist severe permanent deformations (mode-2 shears or strains of amplitude up to 0.1ζ(max). Amplification of the mode-3 deformation proves more destructive. More complex multipoles degenerate in less than a turnover period into end-products of a lesser complexity, via vortex splitting, pairing or merging. We use the conservation of integral properties to classify the large variety of instability mechanisms along physical guidelines. To conclude, we synthetize the connections between these various vortex forms.

The mechanisms of coherent tripole formation from unstable shielded circular vortices are analysed in the context of two-dimensional incompressible flows. Three stages are identified during the transformation process: the linear growth of the initial normal mode perturbation, its nonlinear amplification and the finite-amplitude saturation under the tripolar form. We give a geometrical discussion of the mutual influence of the core vortex and of the satellites generated from the shield. The role of the angular momentum in determining the finite amplitude saturation is demonstrated using a simple elliptical model of the core vortex associated with two point-vortex satellites. The long-time asymmetric breaking of the tripole into a dipole and a monopole is shown to be driven by the erosion of the core vortex by stripping and diffusion. Finally the influence of bottom topography on tripole formation is considered, providing a rich phenomenology when the height of the topography is varied.

The leading optical approximation to a slowly varying solitary crest on constant depth is the plane soliton solution with the local values of amplitude and orientation substituted. This leads to two nonlinear hyperbolic equations for the local amplitude and inclination of the crest that have been reported by several authors and predict the formation of progressive wave jumps, or shocks, from any initial perturbation of the crest. In comparison to numerical solutions of the Boussinesq equations we find that this optical approximation fails to reproduce essential properties of the crest dynamics, in particular that the crest modulations are damped and that well-defined wave jumps do not necessarily evolve. One purpose of the present work is to include such features in an amended optical approximation.

We obtain the leading correction to the ‘local soliton’ solution by a multiple scale technique. In addition to a modification to the wave profile the perturbation expansion also yields a diffracted wave system and a celerity that depends on the curvature of the crest. The principle of energy conservation then leads us to a second-order optical approximation consisting of transport equations of mixed hyperbolic/parabolic nature. Under additional assumptions the transport equations can be reduced to the well-known Burgers equation.

Numerical simulations of the Boussinesq equations are performed for modulations on otherwise straight crests and radially converging solitons. The improved optical, or ray, theory reproduces all essential features and agrees closely with the numerical solution in both cases. Contrary to purely hyperbolic optical descriptions the present theory also predicts wave jumps of finite width that are consistent with the triad solution of Miles (1977).

The present work indicates that while sinusoidal waves often are appropriately described by the lowest-order physical optics, higher-order corrections must be expected to be important for single crested waves.

Measurements in turbulent channel flow with forced oscillations covering a wide range of frequencies (ω+ = 0.03–0.0005) and amplitudes (10–70% of centreline velocity) are presented and discussed. Phase averages of the velocity <u> across the flow, and of the wall shear stress <τ>, as well as the turbulent fluctuations <u′u′> and <t′t′> are obtained with LDA and hot-film techniques. The time-mean quantities, except u’2, are only slightly affected by the imposed oscillations whatever their frequency and amplitude. It is shown that the appropriate similarity parameter for the oscillating quantities ũ and ĩ is the non-dimensional Stokes length l+s (or the frequency ω+ = 2/l+2s). In the regime of high-frequency forcing (l+s < 10) the oscillating flow ũ and ĩ are governed by purely viscous shear forces although the time-mean flow is fully turbulent. This behaviour may be explained by the physical significance of l+s. At lower frequency l+s 10, the oscillating flow is influenced by the turbulence, in particular the amplitude of ĩ increases with respect to the Stokes amplitude and becomes proportional to l+s. The relative amplitude of <u′u′> and <t′t′> decreases sharply with increasing forcing frequency once l+s < 25. This decay of the turbulence response is faster for the wall shear stress. For forcing frequencies such that l+s > 12, <u′u′> and <t′t′> lag behind <u> and <τ> by respectively about 75 and 130 viscous time units. These lags decrease by a factor 2 at higher forcing frequencies. It is shown that in the log layer, the turbulence modulation diffuses away from the wall with a diffusivity equal to that of the time-mean turbulence. The imposed oscillations are felt down to the small scales of the turbulence as may be evidenced from the cyclic modulation of the Taylor microscale, the skewness and the flatness factors of δu′/δt. The modulations of the skewness and the flatness go through a maximum around l+s = 12.

The derivation of a new vectorial bedload formulation for the transport of coarse sediment by fluid flow is presented in the first part of the paper. This relation has been developed for slopes up to the angle of repose both in the streamwise and transverse directions. The pressure distribution is assumed to be hydrostatic. The bed shear stress for the onset of particle motion and mean particle velocity are obtained from the mean force balance on a particle. A new generalized Bagnold hypothesis is introduced to calculate the sediment content of the bedload layer. The new formulation possesses two innovative features. It is fully nonlinear and vectorial in nature, in addition, it behaves smoothly up to the angle of repose.

A mathematical model of the time evolution of straight river channels is presented in the second half of the paper. This study focuses on the evolution process due to bank erosion in the presence of bedload only. The bed and bank material is taken to be coarse, non-cohesive and uniform in size. The sediment continuity and the fluid momentum conservation equations describe the time evolution of the bed topography and flow field. These equations are coupled through the fluid shear stress acting on the bed. This bed shear stress distribution is predicted with the aid of a simple algebraic turbulent closure model. As regards the computation of the sediment flux, the new fully nonlinear vectorial formulation is found to perform well and renders the evolution model fully mechanistic.

The formation of an erosional front in the time development of straight river channels has been so far obscured in physical experiments. Herein, with the help of the new bedload formulation, the existence and migration speed of the front of erosion are inferred from the analysis of the sediment continuity equation.

The model successfully describes the time relaxation of an initially trapezoidal channel toward an equilibrium cross-sectional shape, as evidenced by comparison with experimental data. This equilibrium is characterized by a constant width, vanishing sediment transport in the transverse direction, and a small but non-vanishing streamwise transport rate of bed sediment.

Averaged equations governing the motion of equal rigid spheres suspended in a potential flow are derived from the equation for the probability distribution. A distinctive feature of this work is the derivation of the disperse-phase momentum equation by averaging the particle equation of motion directly, rather than the microscopic equation for the particle material. This approach is more flexible than the usual one and leads to a simpler and more fundamental description of the particle phase. The model is closed in a systematic way (i.e. with no ad hoc assumptions) in the dilute limit and in the linear limit. One of the closure quantities is related to the difference between the gradient of the average pressure and the average pressure gradient, a well-known problem in the widely used two-fluid engineering models. The present result for this quantity leads to the introduction of a modified added mass coefficient (related to Wallis's ‘exertia’) that remains very nearly constant with changes in the volume fraction and densities of the phases. Statistics of this coefficient are provided and exhibit a rather strong variability of up to 20% among different numerical simulations. A detailed comparison of the present results with those of other investigators is given in § 10.

As a further illustration of the flexibility of the techniques developed in the paper, in Appendix C they are applied to the calculation of the so-called ‘particle stress’ tensor. This derivation is considerably simpler than others available in the literature.

A train of uniform two-dimensional gravity waves in deep water is known to be unstable to certain sideband disturbances. If the time of propagation is sufficiently long for the fourth-order terms to be important, the sidebands may grow at unequal rates, resulting in a downward shift of peak frequency. But this shift is only a temporary phase of a recurrent evolution process. Recent work by us (Hara & Mei 1991) has shown that wind and dissipation can help maintain this downshift at large time. In this paper we examine a similar two-dimensional problem for capillary–gravity waves. The basic flow in air and water is assumed to be steady, horizontally uniform and turbulent; the wave-induced flow in both media is assumed to be laminar. Evolution equations are deduced with wind and dissipation included in such a way that their influence is comparable to the asymmetric spectral evolution. After finding the initial growth rates of unstable sidebands, the nonlinear development of modulational instability is examined by integrating the evolution equations numerically. Computed results show that persistent downshift of frequency can happen for relatively long waves, but upshift occurs for very short waves.

This is the second part of a three-part study of the stability of vertically oriented double-diffusive interfaces having an imposed vertical stable temperature gradient. In this study, flow is forced within a fluid of infinite extent by a prescribed excess of compositionally buoyant material between two parallel interfaces. Compositional diffusivity is ignored while thermal diffusivity and viscosity are finite. The stability of the interfaces is analysed first in the limit that they are close together (compared with the salt-finger lengthscale), then for general spacing. Attention is focused on whether the preferred mode of instability is varicose or sinuous and whether its wavevector is vertical or oblique.

The interfaces are found to be unstable for some wavenumber for all values of the Prandtl number and interface spacing. The preferred mode of instability for closely spaced interfaces is varicose and vertical for Prandtl number less than about 9, sinuous oblique for Prandtl number between 9 and 15 and sinuous vertical for larger Prandtl number. For general spacing each of the four possible modes of instability is preferred for some range of Prandtl number and interface separation, with no clear pattern of preference, except that the sinuous oblique mode is preferred for widely separated interfaces. The growth rate of the preferred mode is largest for interfaces having separations of from 1 to 3 salt-finger lengths.

A series of laboratory experiments was carried out to examine the interaction between stratification and turbulence at an inversion layer, with the objective of gaining insight into certain wave–turbulence encounters in the atmosphere. A three-layer stratified fluid system, consisting of a (thick) strongly stratified inversion layer, sandwiched between an upper turbulent layer and a lower non-turbulent weakly stratified layer, was employed. Oscillating-grid-induced shear-free turbulence was used in the upper layer. During the experiments, a fourth (interfacial) layer developed in the region between the inversion and the turbulent layer, and most of the wave–turbulence interactions were confined to this layer. Detailed measurements of the vertical velocity structure, internal-wave parameters and mixing characteristics were made in the stratified layers and, whenever possible, the results were compared to available theoretical predictions.

A shock front in the concentration between two miscible fluids flowing through a porous medium becomes dispersed owing to the heterogeneous structure of the porous medium. If the fluids have equal viscosity and density and the heterogeneity of the porous medium is statistically homogeneous, the length of the dispersion zone between the fluids is known to increase as (βX)½, where β is the dispersivity and X is the average displacement distance. At present the dispersivity is considered to be a property of only the porous medium. For the case where the fluids differ in density and/or in viscosity we have investigated the effect of the dynamics of the fluid flow on the magnitude of the dispersivity β and on the validity of the X½ dependence of the dispersion zone's length. First, we measured the dispersivity in a 1.8 m long sandstone core with brine displacing water and with gas displacing oil. The measurements demonstrate that the dispersivity does indeed depend on the displacement velocity. Second, we monitored the expansion of the dispersion zone using detailed numerical simulations of the flow in a porous medium with statistically homogeneous heterogeneity. We found that the dispersion zone does grow as X½ in the presence of a density contrast and a viscosity contrast, in spite of the nonlinearity of the governing equations. Third, we quantified the magnitude of the dispersivity by means of a random-walk model and tested the model against the experiments and the numerical simulations. Experiments, simulations and the model show that the dispersivity is strongly dependent on the displacement velocity in the conditionally stable flow regime. They also show that a nearly non-dispersive development of the shock front between the fluids occurs when gravity segregation dominates the dispersive effect of the porous medium. Even a very small difference in density, such as that between water and brine, can suppress the dispersivity significantly.

Magnetohydrodynamic flows in a U-bend and in a right-angle bend are considered with reference to the toroidal concepts of self-cooled liquid-metal blankets. The ducts composing the bends have rectangular cross-sections. The applied magnetic field is aligned with the toroidal duct and perpendicular to ducts supplying liquid metal. For high Hartmann numbers the flow region is divided into cores and boundary layers of different types. The magnetohydrodynamic equations are reduced to a system of partial differential equations governing wall electric potentials and the core pressure. The system is solved numerically. The results show that the flow is very sensitive to variations of certain parameters, such as the wall conductance ratio and the aspect ratio of the toroidal duct cross-section. Depending on these parameters, the flow exhibits a variety of qualitatively different flow patterns. In particular, structures of helical and vortex type are obtained. A high-velocity jet occurs at the plasma-facing first wall and there is mixing of the fluid in the toroidal duct. These factors lead to desirable heat-transfer conditions.

The turbulent shear layer and the associated recirculation region on the sidewall formed in flow separation from the forward corner of a square cylinder have been studied with one-component laser-Doppler velocimetry. Because of vortex shedding, the flow is approximately periodic, and is treated as a separated flow undergoing largeamplitude forcing at the shedding frequency. Phase (ensemble)-averaged velocities and turbulence intensities were obtained, and a close relationship in phase and amplitude between phase-averaged turbulence intensities and gradients of phase-averaged velocity is found in much of the flow region. The similarity behaviour of the phase-averaged profiles in the shear layer as well as the streamwise growth of the shear layer are investigated. While phase-averaged velocity profiles collapse well in similarity coordinates, normalized turbulence intensities exhibit systematic deviations from similarity. Shear-layer growth also departs markedly from the linear growth law of unforced plane mixing layers. The effect of the recirculation is suggested as a possible explanation for some of these deviations. Similarities to and differences from steady and forced mixing layers, steady separated flows with recirculation, and unsteady boundary layers are discussed.