In the present paper we formulate a predictive theory of the formation of sand ripples under sea waves. The theory is based on a linear stability analysis of a flat sandy bottom subject to a viscous oscillatory flow. The conditions for decay or amplification of a bottom perturbation are determined along with the wavelength of the most unstable component as a function of the Reynolds number of the flow and of the Froude and Reynolds numbers of the sediments. A comparison between theoretical findings and experimental data supports the validity of the present theory. An analytical solution for viscous oscillatory flow over a small-amplitude wavy bottom is determined for arbitrary values of the ratio r between the amplitude of fluid displacement and the wavelength of bottom waviness. Previous works by Lyne (1971) and Sleath (1976), who considered small or large values of r, are thus extended.

In the present paper we formulate a theory to predict the time development of sand ripples characterized by small but finite amplitude under the action of surface gravity waves. The theory is based on a weakly nonlinear stability analysis of a flat sandy bottom subject to viscous oscillatory flow. The parameters of the problem (namely the Reynolds number of the flow and the Reynolds and Froude numbers of sediments) are assumed to fall within a neighbourhood of the critical conditions determined in Blondeaux (1990). The analysis can predict the actual ripple height, wavelength and profile when flow separation is absent, i.e. for the case of rolling-grain ripples. Assuming Sleath's (1984) criterion for separation, the values of the relevant parameters at which transition from rolling-grain ripples to vortex ripples occurs are predicted. A comparison between theoretical findings and experimental data supports the validity of the present theory.

The dynamics and breakup of a bubble in an axisymmetric, time-periodic straining flow has been investigated via analysis of an approximate dynamic model and also by time-dependent numerical solutions of the full fluid mechanics problem. The analyses reveal that in the neighbourhood of a stable steady solution, an $O(\epsilon^{\frac{1}{3}})$ time-dependent change of bubble shape can be obtained from an O(ε) resonant forcing. Furthermore, the probability of bubble breakup at subcritical Weber numbers can be maximized by choosing an optimal forcing frequency for a fixed forcing amplitude.

Vortex ‘shedding’ behind circular cylinders can be altered and suppressed altogether (or ‘controlled’) over a limited range of Reynolds numbers, by a proper placement of a second, much smaller, cylinder in the near wake of the main cylinder. This new and dramatic suppression of vortex ‘shedding’ is the subject of this paper. Details of the phenomenon are documented through parallel experimental and numerical investigations, including flow visualization. Temporal growth rate measurements of the velocity fluctuations reveal that the presence of the smaller cylinder reduces the growth rate of the disturbances leading to vortex ‘shedding’, and that its suppression, accompanied by the disappearance of sharp spectral peaks, coincides with negative temporal growth rates. It is argued that the presence of the secondary cylinder has the effect of altering the local stability of the flow by smearing and diffusing concentrated vorticity in the shear layers behind the body; a related effect is that the secondary cylinder diverts a small amount of fluid into the wake of the main cylinder. A unified explanation of the formation and suppression of the vortex street is attempted, and it is suggested that the vortex ‘shedding’ is associated with temporally unstable eigenmodes which are heavily weighted by the near field. It is also shown that absolute instability is relevant, up to a point, in explaining vortex shedding, whose suppression can similarly be associated with altering the instability in the near wake region from absolute to convective.

This work is an experimental investigation of the turbulent concentration field formed when the nozzle gas from a round, momentum-driven, free turbulent jet mixes with gas entrained from a quiescent reservoir. The measurements, which were made with a non-intrusive laser-Rayleigh scattering diagnostic at Reynolds numbers of 5000, 16000, and 40000, cover the axial range from 20 to 90 jet exit diameters and resolve the full range of temporal and spatial concentration scales. Reynolds-number-independent and Reynolds-number-dependent similarities are investigated. The mean and r.m.s. values of the concentration are found to be consistent with jet similarity laws. Concentration fluctuation power spectra are found to be self-similar along rays emanating from the virtual origin of the jet. The probability density function for the concentration is also found to be self-similar along rays. Near the centreline of the jet, the scaled probability density function of jet fluid concentration is found to be nearly independent of the Reynolds number.

In the first part of the paper it is shown that the impulse and virial theorems of inviscid incompressible fluid mechanics are special cases of a more general theorem from which an infinity of relations can be obtained. Depending on the problem, only a finite number of these relations may be independent. An application of these results is in the approximate study of the hydrodynamic interaction of bodies. As an example, in the second part of the paper, the case of two freely translating, nonlinearly pulsating bubbles is considered. It is found that in certain parameter ranges the force between the bubbles has a sign opposite to what would be expected on the basis of the linear theory of Bjerknes forces.

This note presents numerical results on the stability of large-amplitude gravity waves on deep water. The results are then used to predict new two-dimensional superharmonic instabilities. They are due to collisions of eigenvalues of opposite signatures, confirming the recent condition for instability of MacKay & Saffman (1986).

A fairly simple theoretical model of an anisotropic compliant wall has been developed. It has been used to undertake a comprehensive numerical study of boundary-layer stability over such walls. The study is based on linearized theory, makes the usual quasi-parallel-flow approximation, uses the Blasius profile as the basic undisturbed flow and assumes two-dimensional disturbances. An investigation is carried out of the effects of anisotropic wall compliance on the Tollmien–Schlichting waves and the two previously identified wall modes, namely travelling-wave flutter and divergence. In addition global convergence techniques are used to search for other possible instabilities.

An asymptotic theory, valid for high Reynolds numbers, is also presented. This can provide accurate estimates of the eigenvalues. It is applicable to a much wider class of compliant walls than the relatively simple model used for the numerical study. An important use of the asymptotic theory is to help identify and elucidate the various energy-exchange mechanisms responsible for stabilization or destabilization of the instabilities. A reduction in the production of disturbance energy by the Reynolds shear stress is the main reason for the favourable effect of anisotropic wall compliance on instability growth. Other energy-exchange mechanisms, which have been found to make a significant contribution, include energy transfer from the disturbance to the mean flow due to the interaction of the fluctuating shear stress and the displaced mean flow, and the work done by the perturbations in wall pressure and shear stress.

It is found that anisotropic wall compliance confers very considerable advantage with respect to reduction in instability growth rate and transition delay. Using a fairly conservative criterion an almost ten-fold rise in transitional Reynolds number is predicted for anisotropic walls having the appropriate properties. Anisotropic wall compliance makes travelling-wave flutter much more sensitive to viscous effects and has a considerable stabilizing influence. The application of global convergence methods has led to the discovery of an anomalous spatially growing eigenmode which, according to conventional interpretation, would represent an instability. Further study of an appropriate initial-value problem has revealed that the new eigenmode is probably not an instability and that, for compliant walls, complex wavenumbers with positive real and negative imaginary parts do not necessarily correspond to an instability.

One-dimensional detonation wave propagation in channels of varying cross-section is reconsidered and studied in detail. Different analytical solutions are given for the case of an accelerated detonation wave in a converging channel and for a decelerated detonation wave in a diverging channel. Separation of the leading shock and the reaction zone in the second case is taken into account. Two- and three-dimensional problems of geometrical detonation wave dynamics can be solved by adapting the well-known approach of Whitham, but Whitham's method is based on a suitable one-dimensional analytical model.

The effect of polymer additive on the growth and collapse of a spherical vapour bubble is investigated theoretically, under conditions appropriate for cavitation (negligible influence of heat transfer, Newtonian viscosity, etc.). The polymer-induced stress is calculated using the yo-yo model of the polymer dynamics in transient extensional flows (Ryskin 1987a). The resulting equation of bubble dynamics is solved numerically; an approximate analytical solution is also obtained. It is found that the growth of a bubble is not affected by the polymer, but the final stage of the collapse is. After following closely the classical inviscid-fluid solution, the collapse is abruptly arrested, and the bubble wall velocity is reduced to nearly zero. The peak acoustic pressure of the radiated sound is also reduced, and the high-frequency part of the acoustic spectrum is sharply curtailed.

The transition of strictly two-dimensional Poiseuille flow from laminar to chaotic behaviour is studied through full numerical simulation of spatially periodic channels with fairly large longitudinal aspect ratios. The successive bifurcations are studied in detail and their physical mechanism is elucidated. The Liapunov exponents of the flow are measured and shown to be positive at large Reynolds numbers. Isolated, permanent patches of unsteady behaviour, resembling the turbulent ‘puffs’ observed in circular pipes, are found at low Reynolds numbers and shown to be important for the transition to chaos. The flow exhibits several other phenomena present in natural three-dimensional flows, including wall sweeps, ejections, and intermittency.

The efficiency of a perforated screen as a sound absorber can be greatly increased when a rigid surface is placed behind the screen, essentially because the sound can then interact many times with the perforations. We consider a practical application for a backed perforated screen with a bias flow through the perforations: the ‘screech liner’. This is a perforated lining which is inserted in the afterburner section of jet engines to suppress the acoustically driven combustion instability commonly known as screech. A pressure drop across the screen ensures that a bias flow of cool air is produced; this flow protects the liner from the intense heat in the afterburner.

Our analysis was developed in answer to a clear need for a theory which can predict the optimal geometry and bias flow to produce a highly absorptive liner. We show that it is theoretically possible to absorb all the sound at a particular frequency. Experimental results are presented which show encouraging agreement with the theoretical predictions.

Screech is thought to be the excitation of a transverse resonant oscillation in the jet pipe, but the insertion of a liner inevitably changes the frequency of such resonances because the boundary condition at the wall is altered. We examine the effect of a liner on the resonances which occur in a cylinder and show that a well-designed liner may suppress resonances over a range of frequencies.

The effect of the hot axial jet flow on the performance of a liner has not previously received attention. A simple model to account for this flow is included in our analysis.

This is the third of a series of papers which investigates the evolution of a binary alloy that is cooled from above and releases buoyant residual fluid as one component of the alloy is preferentially incorporated within the solid. This paper focuses on the compositional zonation that is produced when the melt is completely solidified. Parts 1 and 2 considered the temperature of the cooled boundary to be greater than the eutectic temperature of the alloy so that only partial solidification of the alloy could occur. Here we extend the study by investigating the effects that arise when the cooling temperature is less than the eutectic temperature. The formation of a completely solid layer results, which extends from the cooling plate down to a mushy zone of dendritic crystals and interstitial melt. The melt below this mushy layer is convectively unstable because it is cooled from above. This generates vigorous thermal convection. Eventually the melt becomes completely solidified and a compositionally zoned solid is formed. The solid below the cooling plate is shown to be of fixed bulk composition until it reaches the depth that the interface between the mushy layer and melt occupied when the melt first became saturated. Below this level, the composition of the solid decreases with depth until it merges into the solid growing from the floor, whose composition is nearly equal to that of the heavier component of the alloy. Results of laboratory experiments with aqueous solutions of sodium sulphate are in good agreement with our quantitative predictions. The model is used to give an interpretation of profiles of the composition of magnesium oxide found within solidified komatiite lavas.

Experiments on two-dimensional cavitating hydrofoils show important differences in global behaviour of flows according to the population of air nuclei conveyed by the liquid. By means of visualization techniques and flow modelling, the major features of attached-cavity flows and transient-bubble flows are revealed. The main topics of the paper are: cavitation inception in either regime, hydrofoil saturation and the sweeping away of a cavity by bubbles. The main conditions for the validity of the λ−3 similitude rule are delineated. Special attention is given to the mechanism of interaction between the exploding bubbles, the attached cavity and the boundary layer. Estimates of the critical number of active nuclei for saturation and cavity suppression which agree with experimental results are given.

Two-component laser velocimeter measurements in a fully developed turbulent water channel flow with polymer injection were used to examine the effect of polymer injection on the Reynolds stresses and the production terms in the Reynolds stress transport equations. These measurements show that while the root-mean-square (r.m.s.) fluctuation level of the streamwise velocity was increased, the r.m.s. of the wall-normal velocity and the Reynolds shear stress were reduced. The decrease in the Reynolds shear stress resulted from altered contributions from the quadrants of the (u,v)-plane. Although the Reynolds shear stress decreased, the magnitude of the velocity fluctuation products which most contributed to that stress increased. Production of the streamwise Reynolds normal stress was decreased but production of the Reynolds shear stress was unchanged. This shows that the processes represented by pressure–strain correlation terms in the Reynolds stress transport equations may be directly affected by the polymer.

Turbulent boundary layer experiments have been conducted at various Reynolds numbers on smooth walls and also on ‘k-type’ and ‘d-type’ rough walls. Both the spectral results and the broadband turbulence intensity results strongly support the Townsend (1976) attached eddy hypothesis and the Perry & Chong (1982) model. The spectral results obtained using the ‘flying’ hot-wire technique show the errors involved when using Taylor's (1938) hypothesis for converting the spectra from the frequency domain to the wavenumber domain. If the viscous dissipation spectral region is taken into account, the broadband turbulence intensity results agree well with the attached eddy hypothesis. The inconsistency of the various constants given in Perry, Lim & Henbest (1987) for the smooth and rough walls has been explained and removed. Lack of spatial resolution of the hot wires explains to some extent the scatter in the turbulence intensity of the component normal to the wall. This spatial resolution effect is most pronounced in the near-wall region at high Reynolds number and has been corrected by using the method of Wyngaard (1968).

Using an array of ×-probes aligned in the plane of mean shear in the turbulent far wake of a circular cylinder, instantaneous velocity vector patterns are obtained from which stream-function approximations and sectional streamlines are derived. Conditional patterns obtained using different methods for detecting the organized motion are essentially independent of the particular method used. The spatial arrangement of the organized motion about the flow centreline varies in a continuous manner between opposing and alternating modes, the latter being nearly twice as common as the former. Results presented include conditional patterns for the opposing and alternating modes and the relative contributions made by each mode to the Reynolds stresses. A modified Rankine vortex kinematic model, based as much as possible on experimental data and incorporating both modes, yields mean velocity and Reynolds stress distributions which agree well with experiment. A quasi-three-dimensional version of the model implies that large spanwise vortices and shear-aligned double rollers represent the same three-dimensional organized motion from two different viewpoints.

Simultaneous measurements have been made of large-scale approximations of the three components of the vorticity vector in the far wake of a slightly heated cylinder. Analysis of the results is directed towards the properties of large-scale spanwise vortices and shear-aligned double-roller vortices and the relationship between them. The relationship between vorticity and temperature is also studied. The probability density function of the inclination of the vorticity vector in the (x, y)-plane exhibits a maximum at about 45° in the central part of the wake. The p.d.f. of the inclination of the vorticity vector in the (x, z)-plane supports the existence of a concentration of spanwise vorticity near the outer part of the wake. In this region, there is significant correlation between vorticity and temperature. Conditional averages of velocity and temperature fluctuations and vorticity components are presented with respect to two types of detection, the first focusing on peaks in the ωx and ωy signals, and the second focusing on aspects of the large spanwise structures. Results from the two procedures are consistent, although different in emphasis, and are interpreted in terms of the downstream development of the wake. The upstream ends of large structures stretch and shear-align smaller, probably older, vortices while transporting heat and momentum defect outwards. Near the centreplane, the downstream ends of large structures transport colder, high-speed, spanwise-vortical fluid towards the centreplane, but ωx and ωy are not correlated with either temperature or the large structure. In (x, z)-planes, cuts through the shear-aligned vortices would appear to show double rollers.

A class of steady conically similar axisymmetrical flows of viscous incompressible fluid is studied. The motion is driven by a vortex half-line or conical vortex in the presence of a rigid conical wall or in free space. The dependence of the solutions on parameters (say, the vortex circulation) is analysed. At some finite parameter values the solutions lose existence, i.e. a flow collapse takes place. This is correlated with the appearance of a sink singularity at the symmetry axis. Analytical estimates of the critical parameter values are performed, together with numerical calculations of sounds on the solution existence region in parameter space.

Asymptotic analysis of the near-critical regimes shows that a strong axial jet develops. The jet momentum becomes infinite at the critical parameter value and a singularity occurs. These paradoxical features seem to be typical of conically similar viscous flows. Reasons for the paradox and ways of overcoming it are discussed. Solution non-uniqueness and a hysteresis phenomenon are found in the Serrin problem. Possible applications of the results to model some geophysical and asrophysical phenomena are outlined.

The force F(x) and torque G(x) acting on a conducting body or particle suspended at position x in a magnetic field ${\rm Re} (\hat{\boldmath B}_0({\boldmath x}){\bf e}^{-{\rm i}\omega t})$ are determined to leading order in the ratio of the scale a of the particle to the scale L of the field. When the particle is spherical, F decomposes naturally into an irrotational ‘lift’ force FL and a solenoidal ‘drag’ force FD related to G by \[ F^{\rm D} + {\textstyle\frac{1}{2}}\nabla \wedge G = 0. \] This relationship is important when the action of the field on a suspension of such spheres is considered, because it implies that the net effective force per unit volume acting to generate bulk flow is zero in any region where the concentration c is uniform. However, non-uniformity is generated by the force ingredient FL and bulk flow is then generated through interaction of G and ∇c. These effects are demonstrated for two examples involving rotating and travelling fields. Interactions of a finite number N of spheres are also considered, and in particular it is shown that when the field is a uniform rotating one, the governing dynamical system is at leading order Hamiltonian with four independent integral invariants. When N [ges ] 4, the system in general exhibits chaos.