The present study describes, first, an efficient algorithm for computing solutions in terms of capillary–gravity solitary waves of the irrotational Euler equations with a free surface and, second, provides numerical evidences of the existence of an infinite number of generalised solitary waves (solitary waves with undamped oscillatory wings). Using conformal mapping, the unknown fluid domain, which is to be determined, is mapped into a uniform strip of the complex plane. In the transformed domain, a Babenko-like equation is then derived and solved numerically.

The disturbance wave pattern produced from a harmonic point source in a compressible flat-plate boundary layer is computed using linear stability theory and the direct numerical integration approach. Receptivity coefficients are computed for a wide band of least-stable spanwise modes generated at the source, which are followed in the streamwise direction in order to study the wave-interference pattern. The effect of boundary-layer growth on the development of linear waves is determined by using the method of multiple scales. Results are presented for Mach numbers of 0, 2 and 7. It is found that disturbances spread in wedge-shaped regions behind the source and the wedge angle decreases with Mach number. The lateral spreading angle for the instability waves turns out to be quite close to the angle found experimentally for lateral contamination of turbulence. It is found that, owing to wave cancellation, the computed maximum disturbance amplitude is significantly lower than that obtained by following the most-amplified single normal mode.

In this paper we revisit the Landau and Levich analysis of a coating flow in the case where the flow in the thin liquid film is supported by a Rayleigh surface acoustic wave (SAW), propagating in the solid substrate. Our theoretical analysis reveals that the geometry of the film evolves under the action of the propagating SAW in a manner that is similar to the evolution of films that are being deposited using the dip coating technique. We show that in a steady state the thin-film evolution equation reduces to a generalized Landau–Levich equation with the dragging velocity, imposed by the SAW, depending on the local film thickness. We demonstrate that the generalized Landau–Levich equation has a branch of stable steady state solutions and a branch of unstable solutions. The branches meet at a saddle-node bifurcation point corresponding to the threshold value of the SAW intensity. Below the threshold value no steady states were found and our numerical computations suggest a gradual thinning of the liquid film from its initial geometry.

This work aims to study the dynamics of and noise generated by large-scale structures in a Mach 0.9 turbulent jet of Reynolds number $6.2\times 10^{5}$ using plasma-based excitation of shear layer instabilities. The excitation frequency is varied to produce individual or periodic coherent ring vortices in the shear layer. First, two-point cross-correlations are used between the acoustic near field and far field in order to identify the dominant noise source region. The large-scale structure interactions are then investigated by stochastically estimating time-resolved velocity fields using time-resolved near-field pressure traces and non-time-resolved planar velocity snapshots (obtained by particle image velocimetry) by means of an artificial neural network. The estimated time-resolved velocity fields show multiple mergings of large-scale structures in the shear layer, and indicate that disintegration of coherent ring vortices is the dominant aeroacoustic source mechanism for the jet studied here. However, the merging of vortices in the initial shear layer is also identified as a non-trivial noise source mechanism.

In this paper, we analyse by numerical simulations the nonlinear dynamics of the elliptic instability in the configurations of a single strained vortex and a system of two counter-rotating vortices. We show that although a weakly nonlinear regime associated with a limit cycle is possible, the nonlinear evolution far from the instability threshold is, in general, much more catastrophic for the vortex. In both configurations, we put forward some evidence of a universal nonlinear transition involving shear layer formation and vortex loop ejection, leading to a strong alteration and attenuation of the vortex, and a rapid growth of the vortex core size.

Collapsing cylindrical metallic shells have been used to compress magnetic fluxes and generate megagauss magnetic fields. Such shells experience large, rapidly growing accelerations and their symmetry can be completely destroyed by Rayleigh-Taylor instabilities. This paper presents a theoretical study of the Rayleigh-Taylor instability for radially accelerated incompressible cylindrical shells submitted to the pressure of much lighter media. Low-amplitude flute perturbations are considered and Fourier-analyzed in the azimuthal angle. A fourth-order linear differential system with time-dependent coefficients is derived, which determines the two interface-displacements. Stability criteria are discussed. When the perturbation wavelength is much greater or much smaller than the shell thickness, the differential system splits into two independent differential equations and results are greatly simplified; analytical solutions are available for some cases.

The case of axial field compression (A.F.C.) is discussed as an application. Numerical solutions give the time behaviour of all possible initially given disturbances. The initial perturbations, which are able to reach the axis during their development and which are consequently dangerous in magnetic field compression experiments, have been calculated. Results are consistent with the few experimental data available. They show that the degree of symmetry of cylindrical devices has to be extremely good in order to get successful compressions.

Finally, non-linear and compressibility effects have been taken into account for some A.F.C. cases, solving the full non-linear fluid equations numerically.

In recent years, the independent measurement of wall shear stress with oil-film or oil-drop interferometry has become a cornerstone of turbulent-boundary-layer research as many arguments depend critically on a precise knowledge of the skin friction ${\it\tau}_{w}^{\ast }$. To our knowledge, all practitioners of oil-drop interferometry have so far used the leading-order similarity solution for asymptotically thin, wedge-shaped, two-dimensional oil films established by Tanner & Blows (J. Phys. E: Sci. Instrum., vol. 9, 1976, pp. 194–202) to relate the evolution of drop thickness to ${\it\tau}_{w}^{\ast }$. It is generally believed that this procedure, if carefully implemented, yields the true time-averaged ${\it\tau}_{w}^{\ast }$ within $\pm 1\,\%$ or possibly better, but the systematic errors due to the finite thickness of the oil film have never been determined. They are analysed here for oil films with a thickness of the order of a viscous unit in a zero-pressure-gradient turbulent boundary layer. Neglecting spanwise surface curvature and surface tension effects, corrections due to the secondary air boundary layer above the oil film are derived with a linearised triple-layer approach that accounts for the turbulent shear-stress perturbation by means of modified van-Driest-type closure models. In addition, the correction due to processing oil drops with a slight streamwise surface curvature as if they were exact wedges is quantified. Both corrections are evaluated for oil-drop interferograms acquired in a zero-pressure-gradient turbulent boundary layer at a Reynolds number of around 3500, based on displacement thickness, and are shown to produce a reduction of the friction velocity relative to the basic Tanner and Blows theory of between $-0.1\,\%$ and $-1.5\,\%$, depending on the mixing-length model. Despite the uncertainty about the true correction, the analysis allows the formulation of some guidelines on where and when to analyse interference fringes in order to minimise the error on the measured wall shear stress.

We consider the effect of evaporation on the aggregation of a number of elastic objects due to a liquid’s surface tension. In particular, we consider an array of spring–block elements in which the gaps between blocks are filled by thin liquid films that evaporate during the course of an experiment. Using lubrication theory to account for the fluid flow within the gaps, we study the dynamics of aggregation. We find that a non-zero evaporation rate causes the elements to aggregate more quickly and, indeed, to contact within finite time. However, we also show that the final number of elements within each cluster decreases as the evaporation rate increases. We explain these results quantitatively by comparison with the corresponding two-body problem and discuss their relevance for controlling pattern formation in elastocapillary systems.

Mean-velocity and turbulence measurements have been made in the turbulent near wake of a flat plate at various Reynolds numbers in order to investigate the low-Reynolds-number effects in this region. The results indicate that the low-Reynolds-number effects are significant enough to partially explain the discrepancies in the existing mean-velocity data. It has been found that, while the Reynolds-number-independent, inner-law similarity of the boundary layers continues to exist, the width of the inner wake that develops within the inner-law region scales with the outer variable. Therefore, the mean velocity near the wake centreline depends on the Reynolds number. It is conjectured that this is due to the influence of the large eddies of the outer layer on the spreading of the inner wake.

Measured turbulence quantities indicate that sudden changes occurring just downstream of the trailing edge are independent of the Reynolds number, but the subsequent development of the turbulent stress profiles depends on the Reynolds number. The Reynolds shear stress and the mean-velocity profiles within the inner wake show approximate similarity.

Two-dimensional vortex shedding behind an inclined flat plate in uniform shear flow is numerically investigated by means of an inviscid discrete-vortex approximation. The points of appearance of the vortices are fixed in the plane of the plate at a short distance downstream of the edges of the plate. The strengths of the vortices are determined from the Kutta condition. On the assumption that the steadily periodic flow patterns are independent of initial conditions, the numerical calculations are performed for an inclined flat plate immersed in an incompressible fluid which is set in motion impulsively from rest with the velocity profile of uniform shear flow. The results of analysis show that the Strouhal number of vortex shedding and the time-averaged values of other hydrodynamic characteristics of the flow such as the outer-edge velocity of the separated shear layers, the convective velocity of the shear layers and the drag force exerted on the plate vary closely linearly with the shear parameter of the approaching shear flow. A linear relation between the Strouhal number and the shear parameter is confirmed by an air-tunnel experiment. The effects of the shear parameter on the calculated vortex patterns in the wake are also presented.

Local and global three-dimensionality of cavity interfaces near detachment was examined. Cavities forming on hydrophilic test objects at higher Reynolds numbers (Re>105) exhibited a local flow structure in the cavity interface called ‘divots’. Divots resulted from a local breakdown of the two-dimensional laminar boundary separation. Divots did not form on hydrophobic test objects. Instead, at higher Reynolds numbers (Re>105), the cavity at detachment was composed of a series of wedge shaped structures. Flows with strong adverse pressure gradients upstream of cavity detachment exhibited only local three-dimensionality near one cavity detachment. Flows with weak adverse pressure gradients upstream of cavity detachment were more susceptible to breakdown into global three-dimensionality. This was the case for cavitation on the hydrofoils. Holographic particle imaging velocimetry (HPIV) was used to examine the spanwise and streamwise variation of the flow upstream of the cavity detachment. Three-dimensionality of the cavity detachment was associated with strong variations of the flow upstream of the cavity in the direction perpendicular to the mean flow direction.

Hydrodynamic modulation of wind waves by long surface waves in a wave tank is investigated, at wind speeds ranging from 1.5 to 10 m s−1. The results are compared with the linear, non-dissipative, theory of Longuet-Higgins & Stewart (1960), which describes the modulation of a group of short gravity waves due to straining of the surface by currents produced by the orbital motions of the long wave, and work done against the radiation stresses of the short waves. In most cases the theory is in good agreement with the experimental results when the short waves are not too steep, and the rate of growth due to the wind is relatively small. At the higher wind speeds, the effects of wind-wave growth, dissipation and wave-wave interactions are dominant.

Synthetic jets are zero-net-mass-flux actuators that can be used in a range of flow control applications. For some applications, the scaling of the trajectory of the jet with actuation and cross-flow parameters is important. This scaling is investigated for changes in the friction Reynolds number, changes in the velocity ratio (defined as the ratio between the mean jet blowing velocity and the free-stream velocity) and changes in the actuation frequency of the jet. A distinctive aspect of this study is the high-Reynolds-number turbulent boundary layers (up to $Re_{\unicode[STIX]{x1D70F}}=12\,800$) of the cross-flow. To our knowledge, this is the first study to investigate the effect of the friction Reynolds number of the cross-flow on the trajectory of an (unsteady) jet, as well as the first study to systematically investigate the scaling of the trajectory with actuation frequency. A broad range of parameters is varied (rather than an in-depth investigation of a single parameter) and the results of this study are meant to indicate the relative importance of each parameter rather than the exact influence on the trajectory. Within the range of parameters explored, the critical ones are found to be the velocity ratio as well as a non-dimensional frequency based on the jet actuation frequency, the cross-flow velocity and the jet dimensions. The Reynolds number of the boundary layer is shown to have only a small effect on the trajectory. An expression for the trajectory of the jet is derived from the data, which (in the limit) is consistent with known expressions for the trajectory of a steady jet in a cross-flow.

We have shown that high-viscosity drops in two-dimensional linear creeping flows with a non-zero vorticity component may have two stable stationary states. One state corresponds to a nearly spherical, compact drop stabilized primarily by rotation, and the other to an elongated drop stabilized primarily by capillary forces. Here we explore consequences of the drop bistability for the dynamics of highly viscous drops. Using both boundary-integral simulations and small-deformation theory we show that a quasi-static change of the flow vorticity gives rise to a hysteretic response of the drop shape, with rapid changes between the compact and elongated solutions at critical values of the vorticity. In flows with sinusoidal temporal variation of the vorticity we find chaotic drop dynamics in response to the periodic forcing. A cascade of period-doubling bifurcations is found to be directly responsible for the transition to chaos. In random flows we obtain a bimodal drop-length distribution. Some analogies with the dynamics of macromolecules and vesicles are pointed out.

The properties of a dilute bidisperse particle–gas suspension under low Reynolds number, high Stokes number conditions are studied in the limit τcτv using a perturbation analysis in the small parameter v, which is proportional to the ratio of timescales τc/τv. Here, τc is the time between successive collisions of a particle, and tv is the viscous relaxation time. The leading-order distribution functions for the two species are isotropic Gaussian distributions, and are identical to the molecular velocity distributions in a two-component gas at equilibrium. Balance equations are written for the mean and mean-square velocities, using a distribution function that is a small perturbation from the isotropic Gaussian. The collisional terms are calculated by performing an ensemble average over the relative configurations of the colliding particles, and the mean velocity and velocity variances are calculated correct to O(v2) by solving the balance equations. The difference in the mean velocities of the two species is O(v) smaller than the mean velocity of the suspension, and the fluctuating velocity is O(v½) smaller than the mean velocity.

Using a time-periodic perturbation of a two-dimensional steady separation bubble on a plane no-slip boundary to generate chaotic particle trajectories in a localized region of an unbounded boundary layer flow, we study the impact of various geometrical structures that arise naturally in chaotic advection fields on the transport of a passive scalar from a local ‘hot spot’ on the no-slip boundary. We consider here the full advection-diffusion problem, though attention is restricted to the case of small scalar diffusion, or large Péclet number. In this regime, a certain one-dimensional unstable manifold is shown to be the dominant organizing structure in the distribution of the passive scalar. In general, it is found that the chaotic structures in the flow strongly influence the scalar distribution while, in contrast, the flux of passive scalar from the localized active no-slip surface is, to dominant order, independent of the overlying chaotic advection. Increasing the intensity of the chaotic advection by perturbing the velocity field further away from integrability results in more non-uniform scalar distributions, unlike the case in bounded flows where the chaotic advection leads to rapid homogenization of diffusive tracer. In the region of chaotic particle motion the scalar distribution attains an asymptotic state which is time-periodic, with the period the same as that of the time-dependent advection field. Some of these results are understood by using the shadowing property from dynamical systems theory. The shadowing property allows us to relate the advection-diffusion solution at large Péclet numbers to a fictitious zero-diffusivity or frozen-field solution, corresponding to infinitely large Péclet number. The zero-diffusivity solution is an unphysical quantity, but is found to be a powerful heuristic tool in understanding the role of small scalar diffusion. A novel feature in this problem is that the chaotic advection field is adjacent to a no-slip boundary. The interaction between the necessarily non-hyperbolic particle dynamics in a thin near-wall region and the strongly hyperbolic dynamics in the overlying chaotic advection field is found to have important consequences on the scalar distribution; that this is indeed the case is shown using shadowing. Comparisons are made throughout with the flux and the distributions of the passive scalar for the advection-diffusion problem corresponding to the steady, unperturbed, integrable advection field.

We consider the behaviour of solutions to the nonlinear shallow-water equations which describe wave runup on a plane beach, concentrating on the behaviour at and just behind the moving shoreline. We develop regular series expansions for the hydrodynamic variables behind the shoreline, which are valid for any smooth initial condition for the waveform. We then develop asymptotic descriptions of the shoreline motion under localized initial conditions, in particular a localized Gaussian waveform: we obtain estimates for the maximum runup and drawdown of the wave, for its maximum velocities and the forces it is able to exert on objects in its path, and for the conditions under which such a wave breaks down. We show how these results may be extended to include initial velocity conditions and initial waveforms which may be approximated as the sum of several Gaussians. Finally, we relate these results tentatively to the observed behaviour of a tsunami.

The instability of an anticyclonic solid-body rotating eddy embedded on a quiescent environment is studied, for all possible values of the parameters of the unperturbed state, i.e. the vortex's relative thickness and rotation rate. The Coriolis force is fundamental for the existence of the eddy (because the pressure force has a centrifugal direction) and therefore this analysis pertains to the study of mesoscale vortices in the ocean or the atmosphere, as well as those in other planets.

These eddies are known to be stable when the ‘second’ layer is assumed imperturbable (infinitely deep); however, here these vortices are found to be unstable in the more realistic case of an active environment layer, which may be arbitrarily thick.

Three basic types of instability are found, classified according to the dynamic structure of the growing perturbation field, in both layers: baroclinic instability (Rossby-like in both layers), Sakai instability (Poincaré-like in the vortex layer and Rossby-like in the environment), and Kelvin–Helmholtz instability (Poincaré-like in both layers). In addition, there is a hybrid instability, which goes continuously from the baroclinic to the Sakai types, as the rotation rate is increased.

The problem is constrained by the conservation of pseudoenergy and angular pseudomomentum, which are quadratic (to lowest order) in the perturbation. The requirement that both integrals of motion vanish for a growing disturbance, determines the structure of the latter in both layers. Furthermore, that constraint restricts the region, in parameter space, where each type of instability is present.

Experiments have been made to investigate the motion generated by a body moving along the axis of a uniformly rotating fluid.

Part of the investigation concerns the motion generated in a cylinder whose radial dimensions are much greater than those of the body. Measurements have been made of the velocities of particles on the axis of rotation both ahead of and behind the body, and the results indicate that there is a significant axial motion generated by the body over a wide range of Rossby numbers. A measurement of the instantaneous velocity profile ahead of the body, determined as a function of the radius, agrees fairly well with a low Rossby number calculation of the flow due to a circular disk (Morgan 1951). In addition, the forward influence of the body has been measured as a function of the Rossby number and from these results it is suggested that the body has a finite influence far upstream at all Rossby numbers and that the blocking phenomenon first reported by Taylor (1922) probably occurs for all values of the Rossby number (UΩa) less than a critical value which is about 0·7.

Experiments have also been made in a long cylindrical tube which acts as a wave guide. At large distances from the body the separate effects of the various modes can be observed and hence it is possible to measure the flow corresponding to an individual wave-number: these measurements show that, as a result of the Doppler effect, the motion a large distance ahead of the body is different from that far behind (see Lighthill 1967). Moreover, the experiments indicate that no disturbances propagate ahead of the body when its velocity exceeds the maximum group velocity in the fluid, but that disturbances trail behind the body when its velocity is far in excess of the maximum group velocity. Measurements of the maximum group velocity are in good agreement with the theoretical value.

The redistribution of mean momentum and vorticity, along with the mechanisms underlying these redistribution processes, is explored for post-laminar flow in fully developed, pressure driven, channel flow. These flows, generically referred to as transitional, include an instability stage and a nonlinear development stage. The central focus is on the nonlinear development stage. The present analyses use existing direct numerical simulation data sets, as well as recently reported high-resolution molecular tagging velocimetry measurements. Primary considerations stem from the emergence of the effects of turbulent inertia as represented by the Reynolds stress gradient in the mean differential statement of dynamics. The results describe the flow evolution following the formation of a non-zero Reynolds stress peak that is known to first arise near the critical layer of the most unstable disturbance. The positive and negative peaks in the Reynolds stress gradient profile are observed to undergo a relative movement toward both the wall and centreline for subsequent increases in Reynolds number. The Reynolds stress profiles are shown to almost immediately exhibit the same sequence of curvatures that exists in the fully turbulent regime. In the transitional regime, the outer inflection point in this profile physically indicates a localized zone within which the mean dynamics are dominated by inertia. These observations connect to recent theoretical findings for the fully turbulent regime, e.g. as described by Fife, Klewicki & Wei (J. Discrete Continuous Dyn. Syst., vol. 24, 2009, p. 781) and Klewicki, Fife & Wei (J. Fluid Mech., vol. 638, 2009, p. 73). In accord with momentum equation analyses at higher Reynolds number, the present observations provide evidence that a logarithmic mean velocity profile is most rapidly approximated on a sub-domain located between the zero in the Reynolds stress gradient (maximum in the Reynolds stress) and the outer region location of the maximal Reynolds stress gradient (inflection point in the Reynolds stress profile). Overall, the present findings provide evidence that the dynamical processes during the post-laminar regime and those operative in the high Reynolds number regime are connected and describable within a single theoretical framework.