An experimental investigation of a confined rectangular jet in crossflow was performed. The rectangular jet is highly confined in that it spans almost 80% of the crossflow duct, rather than issuing into a semi-infinite crossflow. Furthermore, the jet is confined in the cross-stream direction because it issues into a relatively narrow duct. In addition, the flow rate of the secondary jet is large (up to 50% of the crossflow flow rate) which also influences the jet–crossflow interaction. Configurations of this type are found in a variety of different industrial manufacturing processes used to mix product streams.

A systematic variation of three pertinent parameters, i.e. momentum ratio, injection angle and development length, was performed. A full factorial experiment was run using three velocity ratios ($Vr\,{=}\,0.5, 1.0, 1.5$), three downstream distances ($x/D_{h}\,{=}\,6, 10, 19$) and six injection angles ($\alpha\,{=}\,18^\circ, 24^\circ$, 30^\circ, 48^\circ, 60^\circ, 90^\circ$). A planar Mie scattering mixing diagnostic system was used to evaluate the relative mixing effectiveness at various conditions within the parameter space studied. Three regimes for the jet–crossflow interaction and the resulting scalar concentration field were revealed: ‘wall jet’, ‘fully lifted jet’ and ‘reattached jet’. To understand the flow physics in these regimes, a more detailed exploration of the secondary flow and coherent structures was required. This was accomplished by acquiring velocity field data at measurement locations and conditions that demarcate the different regimes ($\alpha\,{=}\,30^{\circ}$ and $48^{\circ}, Vr\,{=}\,1.0$ and 1.5, $x/D_{h}\,{=}\, 3, 6, 10$, 15 and 19) using a laser-Doppler velocimetry (LDV) system. The combined scalar concentration and velocity field data provided an understanding of the large-scale mixing and the role of coherent structures and their evolution. The investigation revealed that the flow does not necessarily develop symmetrically and also highlighted some of the effects of confinement.

Droplet impact on a dry solid surface is investigated experimentally. A small obstacle made by layers of adhesive tape is located on the solid surface at some distance from the impact centre. The splashing of the drop starts at the tape, as a sheet of liquid shoots upwards. Angle, speed and dynamics of this liquid sheet are investigated as a function of the distance from the impact centre to the obstacle and its height. Reynolds and Weber numbers are kept constant. Volume-of-fluid simulations reproduce the experiments qualitatively. Measured sheet angles are compared with the predictions of a simple theory.

Stereoscopic particle image velocimetry (PIV) measurements are made in streamwise–spanwise and inclined cross-stream planes (inclined at $45^\circ$ and $135^\circ$ to the principal flow direction) of a turbulent boundary layer at moderate Reynolds number ($\hbox{\it Re}_\tau\,{\sim} 1100$). Two-point spatial velocity correlations computed using the PIV data reveal results that are consistent with an earlier study in which packets of hairpin vortices were identified by a feature-detection algorithm in the log region, but not in the outer wake region. Both streamwise–streamwise ($R_{\hbox{\scriptsize\it uu}}$) and streamwise–wall-normal ($R_{uw}$) correlations are significant for streamwise displacements of more than 1500 wall units. Zero crossing data for the streamwise fluctuating component $u$ reveal that streamwise strips between zero crossings of 1500 wall units or longer occur more frequently for negative $u$ than positive $u$, suggesting that long streamwise correlations in $R_{\hbox{\scriptsize\it uu}}$ are dominated by slower streamwise structures. Additional analysis of $R_{ww}$ correlations suggests that the long streamwise slow-moving regions contain discrete zones of strong upwash over extended streamwise distances, as might occur within packets of angled hairpin vortices. At a wall-normal location outside of the log region ($z/\delta \,{=}\, 0.5$), the correlations are shorter in the streamwise direction and broader in the spanwise direction. Correlations in the inclined cross-stream plane data reveal good agreement with the streamwise–spanwise plane. $R_{\hbox{\scriptsize\it uu}}$ in the $45^\circ$ plane is more elongated along the in-plane wall-normal direction than in the $135^\circ$ plane, which is consistent with the presence of hairpin packets with a low-speed region lifting away from the wall.

The classical edge-wave problem is addressed by scaling the governing equations, for small slope ($\varepsilon$) at the shore, according to the exact edge-wave solution (for uniform slope) which is based on the Gerstner solution of the water-wave problem. The bottom is allowed to be any suitable profile which varies on the scale of this small parameter; a multiple-scale method is then employed to construct the solution. The leading-order equations – which are a version of the shallow-water equations – are fully nonlinear, but an appropriate exact travelling-wave solution exists; the next term in the asymptotic expansion, valid for $\varepsilon\,{\to}\,0$, is also found and, from this, uniformity conditions are deduced. The results are used to describe the run-up pattern produced by edge waves at the shoreline, based on any mode other than the first; this pattern corresponds closely with what is observed, and also with the exact solution for uniform slope everywhere. The surface wave, from the shoreline, seawards, is described for various depth profiles (such as a constant depth at infinity or with a sand bar close inshore). The problem for the first mode, which corresponds to a non-uniformity in the expansion, is briefly discussed; in this case, it is not possible to find an exact closed-form solution.

The corresponding analysis in the case when a longshore current (varying on the same scale as the depth) is flowing, in addition to a general depth profile, is also presented, and the notion of an effective depth profile is confirmed. Finally, a brief mention is made of model equations for edge waves (which have single-mode exact solutions); these may provide the basis for further investigations into the interaction of modes.

A long-wavelength weakly nonlinear analysis is used to investigate the possibility for resonant energy exchange between low-mode internal waves and counter-rotating roll vortices known as Langmuir circulation. The analysis is based on a two-layer ocean model in which the Langmuir circulation is confined to the upper layer and counter-propagating internal waves travel along the sharp thermocline normal to the axes of the vortices. An asymptotically consistent description of the slow-time behaviour is obtained by making a WKBJ approximation to treat the comparatively high-frequency internal-wave reflections identified in Part 1. When the vortices and waves are modelled as linearly neutral modes, the resulting dynamics take the form of nonlinear oscillations. The theory suggests that Langmuir cells may transiently lose stability to standing internal-wave disturbances whose nodes are aligned with the cell downwelling zones. An exact solution of the Langmuir-circulation–standing-wave interaction is used to gain insight into the nonlinear instability mechanism. As in Part 1, the modification of the linear internal-wave dynamics by the Craik–Leibovich ‘vortex force’ is found to be crucial to the interaction.

The breakup of a free thin liquid film subjected to an impulsive acceleration is investigated. A soap film is stretched on a frame at the exit of a shock tube. As the shock impacts the film, the film accelerates within a very short time and detaches from the frame at a constant velocity function of the shock strength. The liquid thickness modulations amplify and eventually the film is perforated with a number of holes, subsequently growing in radius and connecting to each other. The initially connex film is left in the form of a web of liquid ligaments which break into droplets. Both the hole density and formation time depend on the film velocity. We analyse these observations with an impulsive Rayleigh–Taylor instability incorporating liquid surface tension. It is shown to account for both the mode selection and its associated time of growth, providing a criterion for the film bursting time and hole density.

Novel experimental results are presented of the transient motion of a large spherical-cap bubble and of the displaced liquid motion in a closed cylindrical vertical tube of finite length. This is a fundamental fluid mechanics problem and has direct application to Space flight where liquids in fuel tanks are exposed to large changes in acceleration. The initial spherical cap shape is produced by a thin membrane which can be considered equivalent to a large surface tension. The bubble is released by puncturing the membrane which, subsequently, retracts in a time of order 1 ms. Apart from the generation of shear instabilities of very short wavelength, the membrane withdrawal has a negligible effect on the bubble and liquid motions and can thus be considered instantaneous. The displacements of the bubble, the annular liquid sheet and the geyser front were determined from high-speed video images from which velocities and accelerations were calculated. The results are compared with a theoretical model of the unsteady bubble and sheet velocities.

The unsteady interaction of a moving shock wave with nearly homogeneous and isotropic decaying compressible turbulence has been studied experimentally in a large-scale shock tube facility. Rectangular grids of various mesh sizes were used to generate turbulence with Reynolds numbers based on Taylor's microscale ranging from 260 to 1300. The interaction has been investigated by measuring the three-dimensional velocity and vorticity vectors, the full velocity gradient and rate-of-strain tensors with instrumentation of high temporal and spatial resolution. This allowed estimates of dilatation, compressible dissipation and dilatational stretching to be obtained. The time-dependent signals of enstrophy, vortex stretching/tilting vector and dilatational stretching vector were found to exhibit a rather strong intermittent behaviour which is characterized by high-amplitude bursts with values up to 8 times their r.m.s. within periods of less violent and longer lived events. Several of these bursts are evident in all the signals, suggesting the existence of a dynamical flow phenomenon as a common cause. Fluctuations of all velocity gradients in the longitudinal direction are amplified significantly downstream of the interaction. Fluctuations of the velocity gradients in the lateral directions show no change or a minor reduction through the interaction. Root mean square values of the lateral vorticity components indicate a 25% amplification on average, which appears to be very weakly dependent on the shock strength. The transmission of the longitudinal vorticity fluctuations through the shock appears to be less affected by the interaction than the fluctuations of the lateral components. Non-dissipative vortex tubes and irrotational dissipative motions are more intense in the region downstream of the shock. There is also a significant increase in the number of events with intense rotational and dissipative motions. Integral length scales and Taylor's microscales were reduced after the interaction with the shock in all investigated flow cases. The integral length scales in the lateral direction increase at low Mach numbers and decrease during strong interactions. It appears that in the weakest of the present interactions, turbulent eddies are compressed drastically in the longitudinal direction while their extent in the normal direction remains relatively the same. As the shock strength increases the lateral integral length scales increase while the longitudinal ones decrease. At the strongest interaction of the present flow cases turbulent eddies are compressed in both directions. However, even at the highest Mach number the issue is more complicated since amplification of the lateral scales has been observed in flows with fine grids. Thus the outcome of the interaction strongly depends on the initial conditions.

Numerical solutions to the steady two-dimensional compressible Euler equations corresponding to a compressible analogue of the Mallier & Maslowe (Phys. Fluids, vol. A 5, 1993, p. 1074) vortex are presented. The steady compressible Euler equations are derived for homentropic flow and solved using a spectral method. A solution branch is parameterized by the inverse of the sound speed at infinity, $c_{\infty}^{-1}$, and the mass flow rate between adjacent vortex cores of the corresponding incompressible solution, $\epsilon$. For certain values of the mass flux, the solution branches followed numerically were found to terminate at a finite value of $c_{\infty}^{-1}$. Along these branches numerical evidence for the existence of extensive regions of smooth steady transonic flow, with local Mach numbers as large as 1.276, is presented.

The scattering of waves in a two-layer fluid of varying mean depth is examined in a three-dimensional context using linear theory. A variational technique is used to construct a particular type of approximation which has the effect of removing the vertical coordinate and reducing the problem to two coupled partial differential equations in two independent variables. A transformation of this differential equation system leads to a particularly simple approximate representation of the scattering process. The theory is applied to two-dimensional scattering, for which a set of symmetry relations is derived. A selection of numerical results is presented to illustrate the principle interest in the problem, namely the energy transfer between surface and interfacial waves induced by bed undulations.

Previous studies of the organized motion mostly focused on large-scale structures; investigations of other scales have been relatively scarce because of the difficulty of extracting the structures of these scales using conventional vortex-detection techniques. A wavelet multi-resolution technique based on an orthogonal wavelet transform has been applied to analysing the velocity data simultaneously obtained in two orthogonal planes in the turbulent near-wake of a circular cylinder. Using this technique, the flow is decomposed into a number of wavelet components based on their characteristic or central frequencies. The flow structure of each wavelet component is examined in terms of sectional streamlines and vorticity contours. The wavelet component at a central frequency of $f_{0}$, the same as the vortex-shedding frequency, exhibits the characteristics of the Kármán vortices, thus providing a validation of the analysis technique. The spanwise vorticity contours of the wavelet component at $f_{0}$ display a secondary spanwise structure near the saddle point, whose vorticity is opposite-signed to that of the Kármán vortices. This structure is observed for the first time and its occurrence is consistent with the streamwise decay in the vorticity strength of the spanwise structures. Two-point velocity correlations of wavelet components in the lateral and spanwise directions and the wavelet auto-correlation function indicate that the wavelet components of $f_{0}$ and 2$f_{0}$ are relatively large-scale and organized, displaying considerable two-dimensionality. The component of 4$f_{0}$ is significantly less organized and highly three-dimensional, as indicated by much smaller spanwise correlation coefficients. The components at frequencies of 8$f_{0}$ and higher have virtually zero correlation coefficients.

Direct numerical simulation of a turbulent channel flow in a periodic domain of relatively wide spanwise extent, but minimal streamwise length, is carried out at Reynolds numbers $\Rey_\tau\,{=}\,137$ and 349. The large-scale structures previously observed in studies of turbulent channel flow using huge computational domains are also shown to exist even in the streamwise-minimal channels of the present study. Moreover, the limitation of the streamwise length of the domain enforces the interaction between large-scale structures and near-wall structures, which consequently makes it tractable to extract a simple cycle of processes sustaining the structures in the present channel flow. It is shown that the large-scale structures are generated by the collective behaviour of near-wall structures and that the generation of the latter is in turn enhanced by the large-scale structures. Hence, near-wall and large-scale structures interact in a co-supporting cycle.

Electrokinetic flow instabilities occur under high electric fields in the presence of electrical conductivity gradients. Such instabilities are a key factor limiting the robust performance of complex electrokinetic bio-analytical systems, but can also be exploited for rapid mixing and flow control for microscale devices. This paper reports a representative flow instability phenomenon studied using a microfluidic T-junction with a cross-section of 11 $\umu$m by 155 $\umu$m. In this system, aqueous electrolytes of 10:1 conductivity ratio were electrokinetically driven into a common mixing channel by a steady electric field. Convectively unstable waves were observed with a nominal threshold field of $0.5\,\hbox{kV}\,\hbox{cm}^{-1}$, and upstream propagating waves were observed at $1.5\,\hbox{kV}\,\hbox{cm}^{-1}$. A physical model has been developed for this instability which captures the coupling between electric and flow fields. A linear stability analysis was performed on the governing equations in the thin-layer limit, and Briggs–Bers criteria were applied to select physically unstable modes and determine the nature of instability. The model predicts both qualitative trends and quantitative features that agree very well with experimental data, and shows that conductivity gradients and their associated bulk charge accumulation are crucial for such instabilities. Comparison between theory and experiments suggests the convective role of electro-osmotic flow. Scaling analysis and numerical results show that the instability is governed by two key controlling parameters: the ratio of dynamic to dissipative forces which governs the onset of instability, and the ratio of electroviscous to electro-osmotic velocities which governs the convective versus absolute nature of instability.

The elastic effects of an insoluble surfactant on the formation and evolution of two-dimensional Faraday waves is investigated numerically. We analyse the influence of the elasticity of the surface-active agent on the amplitude of the vertical vibration needed to excite two-dimensional standing waves on the free surface. The numerical solutions show that the interface is always subharmonically excited at the onset and that the presence of the surfactant requires a higher external force to induce standing waves. They also show that the magnitude of the external amplitude is related to the temporal phase shift that exists between the evolution of the surfactant concentration and the free-surface shape. A detailed description of the time-varying velocity fields and interfacial distribution of surfactants helps to provide insight into the mechanisms ruling the phenomenon.

The two-dimensional problem of the free sloshing of an inviscid fluid in a vertically walled tank with an arbitrary bed shape is solved at both first and second order in the Stokes expansion of the velocity potential. The approach employed at both orders uses Green's functions for a flat bed in conjunction with the Cauchy–Riemann equations to derive integral equations for the tangential flux along the varying bed. The first- and second-order potentials everywhere in the fluid may then be related to these fluxes. Significant analytic progress is made with the calculation of various contributions to the integral equations at second order. The equations at first and second order are ultimately solved using a variational principle equivalent to the Galerkin method, giving efficient and accurate results. In particular, the work involved in determining the second-order solution is no more intensive than in solving the first-order problem. The first-order solution is shown to reproduce known results for specific bed shapes. The method is applied to a range of bed shapes and the second-order correction to the free-surface elevation is illustrated.

The effect of pressure gradients on the stability of creeping flows of Newtonian fluids in channels lined with an incompressible and impermeable neo-Hookean material is examined in this work. Three different configurations are considered: (i) pressure-driven flow between a rigid wall and a wall lined with a neo-Hookean material; (ii) pressure-driven flow between neo-Hookean-lined walls; and (iii) combined Couette–Poiseuille flow between a rigid wall and a neo-Hookean-lined wall. In each case, a first normal stress difference whose magnitude depends on depth arises in the base state for the solid, and linear stability analysis reveals that this leads to a short-wave instability which is removed by the presence of interfacial tension. For sufficiently thick solids, low-wavenumber modes become unstable first as the applied strain increases above a critical value, whereas for sufficiently thin solids, high-wavenumber modes becomes unstable first. Comparison of the dimensionless critical strains shows that configurations (i) and (ii) are more difficult to destabilize than Couette flow past a neo-Hookean solid. For configuration (iii), the nonlinear elasticity of the solid leads to two physically distinct critical conditions, in contrast to what happens when a linear elastic material is used. The mechanisms underlying the behaviour of the critical strains are explained through an analysis of the interfacial boundary conditions.