We consider a head-on collision between two solitary waves on the surface of an inviscid homogeneous fluid. A perturbation method which in principle can generate an asymptotic series of all orders, is used to calculate the effects of the collision. We find that the waves emerging from (i.e. long after) the collision preserve their original identities to the third order of accuracy we have calculated. However a collision does leave imprints on the colliding waves with phase shifts and shedding of secondary waves. Each secondary wave group trails behind its primary, a solitary wave. The amplitude of the wave group diminishes in time because of dispersion. We have also calculated the maximum run-up amplitude of two colliding waves. The result checks with existing experiments.

Sound waves can be generated in a colloid by the application of an alternating electric field. In this paper we describe the method for calculating this and the related electro-acoustic phenomenon of electric fields generated by sound waves. As an illustration of the procedure, we obtain formulae for these two effects for a suspension of spherical particles with thin double layers, in a parallel plate geometry.

We propose a diagnostic technique to detect instability waves in a subsonic round jet using a phased microphone array. The detection algorithm is analogous to the beam-forming technique, which is typically used with a far-field microphone array to localize noise sources. By replacing the reference solutions used in the conventional beam-forming with eigenfunctions from linear stability analysis, the amplitudes of instability waves in the axisymmetric and first two azimuthal modes are inferred. Experimental measurements with particle image velocimetry and a database from direct numerical simulation are incorporated to design a conical array that is placed just outside the mixing layer near the nozzle exit. The proposed diagnostic technique is tested in experiments by checking for consistency of the radial decay, streamwise evolution and phase correlation of hydrodynamic pressure. The results demonstrate that in a statistical sense, the pressure field is consistent with instability waves evolving in the turbulent mean flow from the nozzle exit to the end of the potential core, particularly near the most amplified frequency of each azimuthal mode. We apply this technique to study the effects of jet Mach number and temperature ratio on the azimuthal mode balance and evolution of instability waves. We also compare the results from the beam-forming algorithm with the proper orthogonal decomposition and discuss some implications for jet noise.

The mechanism of wake-induced vibrations (WIV) of a pair of cylinders in a tandem arrangement is investigated by experiments. A typical WIV response is characterized by a build-up of amplitude persisting to high reduced velocities; this is different from a typical vortex-induced vibration (VIV) response, which occurs in a limited resonance range. We suggest that WIV of the downstream cylinder is excited by the unsteady vortex–structure interactions between the body and the upstream wake. Coherent vortices interfering with the downstream cylinder induce fluctuations in the fluid force that are not synchronized with the motion. A favourable phase lag between the displacement and the fluid force guarantees that a positive energy transfer from the flow to the structure sustains the oscillations. If the unsteady vortices are removed from the wake of the upstream body then WIV will not be excited. An experiment performed in a steady shear flow turned out to be central to the understanding of the origin of the fluid forces acting on the downstream cylinder.

In this paper we intend to predict the magnitude of the contribution to the Reynolds stress of bursting events: ‘ejections’, ‘sweeps’, ‘inward interactions’ and ‘outward interactions’. We shall do this by making use of the conditional probability distribution of the Reynolds stress − uv, which can be derived by applying the cumulant-discard method to the Gram-Charlier probability distribution of the two variables u and v. The Reynolds-stress fluctuations in openchannel flows over smooth and rough beds are measured by dual-sensor hot-film anemometers, whose signals are conditionally sampled and sorted into the four quadrants of the u, v plane by using a high-speed digital data processing system.

We shall verify that even the third-order conditional probability distribution of the Reynolds stress shows fairly good agreement with the experimental results and that the sequence of events in the bursting process, i.e. ejections, sweeps and interactions, is directly related to the turbulent energy budget in the form of turbulent diffusion. Also, we shall show that the roughness effect is marked in the area from the wall to the middle of the equilibrium region, and that sweeps appear to be more important than ejections as the roughness increases and as the distance from the wall decreases.

Direct numerical simulation of the incompressible Navier–Stokes equations is used to study flows where laminar boundary-layer separation is followed by turbulent reattachment forming a closed region known as a laminar separation bubble. In the simulations a laminar boundary layer is forced to separate by the action of a suction profile applied as the upper boundary condition. The separated shear layer undergoes transition via oblique modes and Λ-vortex-induced breakdown and reattaches as turbulent flow, slowly recovering to an equilibrium turbulent boundary layer. Compared with classical experiments the computed bubbles may be classified as ‘short’, as the external potential flow is only affected in the immediate vicinity of the bubble. Near reattachment budgets of turbulence kinetic energy are dominated by turbulence events away from the wall. Characteristics of near-wall turbulence only develop several bubble lengths downstream of reattachment. Comparisons are made with two-dimensional simulations which fail to capture many of the detailed features of the full three-dimensional simulations. Stability characteristics of mean flow profiles are computed in the separated flow region for a family of velocity profiles generated using simulation data. Absolute instability is shown to require reverse flows of the order of 15–20%. The three-dimensional bubbles with turbulent reattachment have maximum reverse flows of less than 8% and it is concluded that for these bubbles the basic instability is convective in nature.

A drop of fluid, initially held spherical by surface tension, will deform when an electric or magnetic field is applied. The deformation will depend on the electric/magnetic properties (permittivity/permeability and conductivity) of the drop and of the surrounding fluid. The full time-dependent low-Reynolds-number problem for the drop deformation is studied by means of a numerical boundary-integral technique. Fluids with arbitrary electrical properties are considered, but the viscosities of the drop and of the surrounding fluid are assumed to be equal.

Two modes of breakup have been observed experimentally: (i) tip-streaming from drops with pointed ends, and (ii) division of the drop into two blobs connected by a thin thread. Pointed ends are predicted by the numerical scheme when the permittivity of the drop is high compared with that of the surrounding fluid. Division into blobs is predicted when the conductivity of the drop is higher than that of the surrounding fluid. Some experiments have been reported in which the drop deformation exhibits hysteresis. This behaviour has not in general been reproduced in the numerical simulations, suggesting that the viscosity ratio of the two fluids can play an important role.

Detailed understanding of flowing granular materials is severely hampered by the deficiencies of present experimental methods. To help increase the information base, a computer simulation has been developed to describe two-dimensional unidirectional flows of inelastic fully rough particles. This paper presents the results of a Couette shear-flow simulation. The results include distributions of velocity, density and granular temperature (a measure of the kinetic energy contained in the random particle motions). The effects of density and shear rate on the granular temperature are explored. Shear and normal forces on the solid walls are compared with experimental and theoretical results. The behaviour of the particles in the simulated flow is examined and assessments are made of the collision angle and velocity distributions. The development of a distinct, ‘layered’ microstructure is observed in high-density granular flows.

Measurements of skin friction and mean-velocity profiles have been made in fully developed flows in pipes and channels in the Reynolds number range 1000 < Re < 10000. These measurements, and observations of hot-wire signals, indicate rather remarkable differences between two-dimensional and axially symmetric flows and also make it difficult to give a precise definition of the term ‘fully developed turbulent flow’.

Direct simulations of the turbulent shear layer are performed for subsonic to supersonic Mach numbers. Fully developed turbulence is achieved with profiles of mean velocity and turbulence intensities that agree well with laboratory experiments. The thickness growth rate of the shear layer exhibits a large reduction with increasing values of the convective Mach number, Mc. In agreement with previous investigations, it is found that the normalized pressure–strain term decreases with increasing Mc, which leads to inhibited energy transfer from the streamwise to cross-stream fluctuations, to the reduced turbulence production observed in DNS, and, finally, to reduced turbulence levels as well as reduced growth rate of the shear layer. An analysis, based on the wave equation for pressure, with supporting DNS is performed with the result that the pressure–strain term decreases monotonically with increasing Mach number. The gradient Mach number, which is the ratio of the acoustic time scale to the flow distortion time scale, is shown explicitly by the analysis to be the key quantity that determines the reduction of the pressure–strain term in compressible shear flows. The physical explanation is that the finite speed of sound in compressible flow introduces a finite time delay in the transmission of pressure signals from one point to an adjacent point and the resultant increase in decorrelation leads to a reduction in the pressure–strain correlation.

The dependence of turbulence intensities on the convective Mach number is investigated. It is found that all components decrease with increasing Mc and so does the shear stress.

DNS is also used to study the effect of different free-stream densities parameterized by the density ratio, s = ρ2/ρ1, in the high-speed case. It is found that changes in the temporal growth rate of the vorticity thickness are smaller than the changes observed in momentum thickness growth rate. The momentum thickness growth rate decreases substantially with increasing departure from the reference case, s = 1. The peak value of the shear stress, uv, shows only small changes as a function of s. The dividing streamline of the shear layer is observed to move into the low-density stream. An analysis is performed to explain this shift and the consequent reduction in momentum thickness growth rate.

The structure and growth of the internal boundary layer which forms downstream of a sudden change from a smooth to a rough surface under zero pressure gradient conditions has been studied experimentally. To keep pressure disturbances due to the roughness change small, the level of the rough surface was depressed, so that the crest of the roughness was aligned with the level of the smooth surface. It has been found that, in the region near the change, the structure of the internal layer is largely independent of that in the almost undisturbed outer layer, whilst both the zero time delay and the moving axis integral length scales in the internal layer are significantly reduced below those on the smooth wall. The growth-rate of the internal layer is similar to that of the zero pressure gradient boundary layer, whilst the level of turbulence inside the internal layer is high because of the large turbulent energy production near the rough wall. From the mixing length results, and an analysis of the turbulent energy equation, it is deduced that the internal layer flow near the wall is not in energy equilibrium, and hence the concept of inner layer similarity breaks down. From an initially self-preserving state on the smooth wall, the turbulent boundary layer approaches a second self-preserving state on the rough wall well downstream of the roughness step.

We study the impact and subsequent retraction of liquid droplets upon high-speed impact on hydrophobic surfaces. Extensive experiments show that the drop retraction rate is a material constant and does not depend on the impact velocity. We show that on increasing the Ohnesorge number, $\Oh\,{=}\,\eta/\sqrt{\rho R_{\rm I} \gamma}$, the retraction, i.e. dewetting, dynamics crosses from a capillary-inertial regime to a capillary-viscous regime. We rationalize the experimental observations by a simple but robust semi-quantitative model for the solid-liquid contact line dynamics inspired by the standard theories for thin-film dewetting.

It is well known that subgrid models such as Smagorinsky's cannot be used for the spatially growing simulation of the transition to turbulence of flat-plate boundary layers, unless large-amplitude perturbations are introduced at the upstream boundary: they are over-dissipative, and the flow simulated remains laminar. This is also the case for the structure-function model (SF) of Métais & Lesieur (1992). In the present paper we present a sequel to this model, the filtered-structure-function (FSF) model. It consists of removing the large-scale fluctuations of the field before computing its second-order structure function. Analytical arguments confirm the superiority of the FSF model over the SF model for large-eddy simulations of weakly unstable transitional flows. The FSF model is therefore used for the simulation of a quasi-incompressible (M∞ = 0.5) boundary layer developing spatially over an adiabatic flat plate, with a low level of upstream forcing. With the minimal resolution 650 × 32 × 20 grid points covering a range of streamwise Reynolds numbers Rex1 ε [3.4 × 105, 1.1 × 106], transition is obtained for 80 hours of time-processing on a CRAY 2 (whereas DNS of the whole transition takes about ten times longer). Statistics of the LES are found to be in acceptable agreement with experiments and empirical laws, in the laminar, transitional and turbulent parts of the domain. The dynamics of low-pressure and high-vorticity distributions is examined during transition, with particular emphasis on the neighbourhood of the critical layer (defined here as the height of the fluid travelling at a speed equal to the phase speed of the incoming Tollmien–Schlichting waves). Evidence is given that a subharmonic-type secondary instability grows, followed by a purely spanwise (i.e. time-independent) mode which yields peak-and-valley splitting and transition to turbulence. In the turbulent region, flow visualizations and local instantaneous profiles are provided. They confirm the presence of low- and high-speed streaks at the wall, weak hairpins stretched by the flow and bursting events. It is found that most of the vorticity is produced in the spanwise direction, at the wall, below the high-speed streaks. Isosurfaces of eddy viscosity confirm that the FSF model does not perturb transition much, and acts mostly in the vicinity of the hairpins.

The sound generated by vortex pairing in a two-dimensional compressible mixing layer is investigated. Direct numerical simulations (DNS) of the Navier–Stokes equations are used to compute both the near-field region and a portion of the acoustic field. The acoustic analogy due to Lilley (1974) is also solved with acoustic sources determined from the near-field data of the DNS. It is shown that several commonly made simplifications to the acoustic sources can lead to erroneous predictions for the acoustic field. Predictions based on the quadrupole form of the source terms derived by Goldstein (1976a, 1984) are in excellent agreement with the acoustic field from the DNS. However, despite the low Mach number of the flow, the acoustic far field generated by the vortex pairings cannot be described by considering compact quadrupole sources. The acoustic sources have the form of modulated wave packets and the acoustic far field is described by a superdirective model (Crighton & Huerre 1990). The presence of flow–acoustic interactions in the computed source terms causes the acoustic field predicted by the acoustic analogy to be very sensitive to small changes in the description of the source.

Results of an experimental investigation of the evolution of a nonlinear wave train on deep water are reported. The initial stage of evolution is found to be characterized by exponential growth of a modulational instability, as was first discovered by Benjamin ' Feir. At later stages of evolution it is found that the instability does not lead to wave-train disintegration or loss of coherence. Instead, the modulation periodically increases and decreases, and the wave train exhibits the Fermi–Pasta–Ulam recurrence phenomenon. Results of an earlier study of nonlinear wave packets by Yuen ' Lake, in which solutions of the nonlinear Schrödinger equation were shown to provide quantitatively correct descriptions of the properties of nonlinear wave packets, are applied to describe the experimentally observed wave-train phenomena. A comparison between the laboratory data and numerical solutions of the nonlinear Schrödinger equation for the long-time evolution of nonlinear wave trains is given.

From a series of experiments using simplified mechanical models we suggest certain minor modifications to the Weis-Fogh (1973)–Lighthill (1973) explanation of the so-called ‘clap and fling’ mechanism for the generation of large lift coefficients by insects in hovering flight. Of particular importance is the production and motion of a leading edge, separation vortex that accounts for virtually all of the circulation generated during the initial phase of the ‘fling’ process. The magnitude of this circulation is substantially larger than that calculated using inviscid theory. During the motion that subsequently separates the wings, the vorticity over each of them is convected and combined to become a tip vortex of uniform circulation spanning the space between them. This combined vortex moves downwards as a part of a ring, of large impulse, that is then continuously fed from quasi-steady separation bubbles that move with the wings as they continue to open at a large angle of attack. Such effects are able to account for the large lift forces generated by the insect.

Results of laboratory experiments are presented in which a finite volume of homogeneous fluid was released instantaneously into another fluid of slightly lower density. The experiments were performed in a channel of rectangular cross-section, and the two fluids used were salt water and fresh water. As previously reported, the resulting gravity current, if viscous effects are negligible, passes through two distinct phases: an initial adjustment phase, during which the initial conditions are important, and an eventual self-similar phase, in which the front speed decreases as t−1/3 (where t is the time measured from release). The experiments reported herein were designed to emphasize the inviscid motion. From our observations we argue that the current front moves steadily in the first phase, and that the transition to the inviscid self-similar phase occurs when a disturbance generated at the endwall (or plane of symmetry) overtakes the front. If the initial depth of the heavy fluid is equal to or slightly less than the total depth of the fluid in the channel, the disturbance has the appearance of an internal hydraulic drop. Otherwise, the disturbance is a long wave of depression. Measurements of the duration of the initial phase and of the speed and depth of the front during this phase are presented as functions of the ratio of the initial heavy fluid depth to the total fluid depth. These measurements are compared with numerical solutions of the shallow-water equations for a two-layer fluid.

The effect of high levels of free-stream turbulence on the transition in a Blasius boundary layer is studied by means of direct numerical simulations, where a synthetic turbulent inflow is obtained as superposition of modes of the continuous spectrum of the Orr–Sommerfeld and Squire operators. In the present bypass scenario the flow in the boundary layer develops streamwise elongated regions of high and low streamwise velocity and it is suggested that the breakdown into turbulent spots is related to local instabilities of the strong shear layers associated with these streaks. Flow structures typical of the spot precursors are presented and these show important similarities with the flow structures observed in previous studies on the secondary instability and breakdown of steady symmetric streaks.

Numerical experiments are performed by varying the energy spectrum of the incoming perturbation. It is shown that the transition location moves to lower Reynolds numbers by increasing the integral length scale of the free-stream turbulence. The receptivity to free-stream turbulence is also analysed and it is found that two distinct physical mechanisms are active depending on the energy content of the external disturbance. If low-frequency modes diffuse into the boundary layer, presumably at the leading edge, the streaks are induced by streamwise vorticity through the linear lift-up effect. If, conversely, the free-stream perturbations are mainly located above the boundary layer a nonlinear process is needed to create streamwise vortices inside the shear layer. The relevance of the two mechanisms is discussed.

Experiments on fluidization with water of spherical particles falling against gravity in columns of rectangular cross-section are described. All of them are dominated by inertial effects associated with wakes. Two local mechanisms are involved: drafting and kissing and tumbling into stable cross-stream arrays. Drafting, kissing and tumbling are rearrangement mechanisms in which one sphere is captured in the wake of the other. The kissing spheres are aligned with the stream. The streamwise alignment is massively unstable and the kissing spheres tumble into more stable cross-stream pairs of doublets which can aggregate into larger relatively stable horizontal arrays. Cross-stream arrays in beds of spheres constrained to move in two dimensions are remarkable. These arrays may even coalesce into aggregations of close-packed spheres separated by regions of clear water. A somewhat weaker form of cooperative motion of cross-stream arrays of rising spheres is found in beds of square cross-section where the spheres may move freely in three dimensions. Horizontal arrays rise where drafting spheres fall because of greater drag. Aggregation of spheres seems to be associated with relatively stable cooperative motions of horizontal arrays of spheres rising in their own wakes.

The effect of viscous dissipation in natural convection is appreciable when the induced kinetic energy becomes appreciable compared to the amount of heat transferred. This occurs when either the equivalent body force is large or when the convection region is extensive. Viscous dissipation is considered here for vertical surfaces subject to both isothermal and uniform-flux surface conditions. A perturbation method is used and the first temperature perturbation function is calculated for Prandtl numbers from 10−2 to 104. The magnitude of the effect depends upon a dissipation number, which is not expressible in terms of the Grashof or the Prandtl number.