Liquid sheets are a generic occurrence in nature. They provide a key pathway by which liquid matter breaks up and is dispersed into smaller entities, as in the formation of sprays. However, the stability of liquid sheets is surprisingly subtle, and involves a range of physical effects, without which a quantitative understanding is impossible. As a result, even the linear stability of liquid sheets has remained open for over half a decade. Now, by a combination of experiment and thorough theoretical analysis by Tammisola et al. (J. Fluid Mech., this issue, vol. 672, 2011, pp. 5–32), we have finally gained a solid understanding of the key factors that control the initial stages of sheet breakup.

The stability of a plane liquid sheet is studied experimentally and theoretically, with an emphasis on the effect of the surrounding gas. Co-blowing with a gas velocity of the same order of magnitude as the liquid velocity is studied, in order to quantify its effect on the stability of the sheet. Experimental results are obtained for a water sheet in air at Reynolds number Rel = 3000 and Weber number We = 300, based on the half-thickness of the sheet at the inlet, water mean velocity at the inlet, the surface tension between water and air and water density and viscosity. The sheet is excited with different frequencies at the inlet and the growth of the waves in the streamwise direction is measured. The growth rate curves of the disturbances for all air flow velocities under study are found to be within 20% of the values obtained from a local spatial stability analysis, where water and air viscosities are taken into account, while previous results from literature assuming inviscid air overpredict the most unstable wavelength with a factor 3 and the growth rate with a factor 2. The effect of the air flow on the stability of the sheet is scrutinized numerically and it is concluded that the predicted disturbance growth scales with (i) the absolute velocity difference between water and air (inviscid effect) and (ii) the square root of the shear from air on the water surface (viscous effect).

We consider transient flow in a box containing an isolated buoyancy source, ventilated by a windward high-level opening and a leeward low-level opening, so that prevailing wind acts to oppose buoyancy-driven flow. Hunt & Linden (J. Fluid Mech., vol. 527, 2005, p. 27) demonstrated that two stable steady states can exist above a critical wind strength: buoyancy-driven displacement ventilation with a two-layer stratification and wind-driven mixing ventilation with the whole interior contaminated by buoyant fluid. We present two time-dependent models for this system: a nonlinear ordinary differential equation (ODE) model following Kaye & Hunt (J. Fluid Mech., vol. 520, 2004, p. 135), assuming ‘perfect’ vertical mixing of fluid within each layer, and a partial differential equation model assuming zero vertical mixing, following Germeles (J. Fluid Mech., vol. 71, 1975, p. 601).

We apply these models to an initial-value problem – the filling box with constant opposing wind. The interface between the upper hot plume fluid and the lower cool ambient air can dramatically overshoot its final level before relaxing to equilibrium; in some cases, a fully contaminated transient can occur before the buoyancy-driven two-layer steady state is reached. However, we find that for an initially completely uncontaminated box, the system converges to a stable wind-driven steady state whenever it exists. By analysing phase diagrams of the ODE model for the flow, we establish a general method of determining which final state is attained and also explain the hysteresis observed by Hunt & Linden (2005). We confirm these transient behaviours by conducting salt bath experiments in a recirculating flume tank and establish quantitative agreement between theory and experiment. Our ‘zero mixing’ model is more accurate than our ‘perfect mixing’ model for our experiments, as the upper layer remains stratified for a substantial time.

The stability of buoyancy-driven propagation of a fluid-filled crack through an elastic solid is studied using a combination of theory and experiments. For the theory, the lubrication approximation is introduced for fluid flow, and the surrounding solid is described by linear elasticity. Solutions are then constructed for a planar fluid front driven by either constant flux or constant volume propagating down a pre-cut conduit. As the thickness of the pre-cut conduit approaches zero, it is shown how these fronts converge to zero-toughness fracture solutions with a genuine crack tip. The linear stability of the planar solutions towards transverse, finger-like perturbations is then examined. Instabilities are detected that are analogous to those operating in the surface-tension-driven fingering of advancing fluid contact lines. Experiments are conducted using a block of gelatin for the solid and golden syrup for the fluid. Again, planar cracks initiated by emplacing the syrup above a shallow cut on the surface of the gelatin develop transverse, finger-like structures as they descend. Potential geological applications are discussed.

Whistling behaviour of two geometrically periodic systems, namely corrugated pipes and multiple side branch systems, is investigated both experimentally and numerically. Tests are performed on corrugated pipes with various lengths and cavity geometries. Experiments show that the peak-whistling Strouhal number, where the maximum amplitude in pressure fluctuations is registered, is independent of the pipe length. Experimentally, a decrease of the peak-whistling Strouhal number by a factor of two is observed with increasing confinement ratio, i.e. the ratio of pipe diameter to cavity width. A numerical methodology that combines incompressible flow simulations with vortex sound theory is proposed to estimate the acoustic source power in periodic systems. The methodology successfully predicts the Strouhal number ranges of acoustic energy production/absorption and the nonlinear saturation mechanism responsible for the stabilization of the limit cycle oscillation. The methodology predicts peak-whistling Strouhal numbers in agreement with experiments and explains the dependence of the peak-whistling Strouhal number on the confinement ratio. Combined with an energy balance, the proposed methodology is used to estimate the acoustic fluctuation amplitudes.

We study the dispersal of a plume of incompressible buoyant fluid injected into a confined sloping aquifer which has an outflow at a single fault which may be up-dip (up-slope) or down-dip (down-slope) from the point of injection. We develop a long-time asymptotic solution for the motion of the injected fluid. We show that for the case in which the outflow fault is up-dip from the point of injection, there is a critical injection rate above which the injected fluid floods the full depth of the aquifer, and we show that for the case in which the outflow fault is down-dip from the point of injection, there is a critical injection rate below which all injected fluid initially flows up-dip. Our analysis leads to expressions for the lateral extent of the injected fluid as a function of time, and we consider the implications of the model for the dispersal of supercritical carbon dioxide injected into deep saline aquifers. The work also indicates that the geometry of the system may have a significant effect on (i) the total volume of carbon dioxide which it is possible to sequester in a faulted aquifer and (ii) the interpretation of the dispersed position of any injected tracers.

The stability of the flow around a rotating cylinder in a fluid linearly stratified along the cylinder axis is studied numerically and experimentally for moderate Reynolds numbers. The flow is assumed potential and axisymmetric with an angular velocity profile Ω = 1/r2, where r is the radial coordinate. Neglecting density diffusion and non-Boussinesq effects, the properties of the linear normal modes are first provided. A comprehensive stability diagram is then obtained for Froude numbers between 0 and 3 and Reynolds numbers below 1000. The main result is that the potential flow, which is stable for a homogeneous fluid, becomes unstable for Froude number close to one and for Reynolds numbers larger than 360. The numerical results are then compared with experimental results obtained using shadowgraph and synthetic Schlieren techniques. Two symmetrical helical modes are found to be simultaneously unstable. We show that these modes exhibit an internal gravity wave structure extending far from the cylinder in agreement with the theory. Their wavelength and frequency are shown to be in good agreement with the numerical predictions for a large range of Froude and Reynolds numbers. These experimental results are the first indisputable evidence of the radiative instability.

The first evidence of a variety of nonlinear equilibrium states of travelling and stationary waves is provided in a two-dimensional granular plane Couette flow via nonlinear stability analysis. The relevant order-parameter equation, the Landau equation, has been derived for the most unstable two-dimensional perturbation of finite size. Along with the linear eigenvalue problem, the mean-flow distortion, the second harmonic, the distortion to the fundamental mode and the first Landau coefficient are calculated using a spectral-based numerical method. Two types of bifurcations, Hopf and pitchfork, that result from travelling and stationary instabilities, respectively, are analysed using the first Landau coefficient. The present bifurcation theory shows that the flow is subcritically unstable to stationary finite-amplitude perturbations of long wavelengths (kx ~ 0, where kx is the streamwise wavenumber) in the dilute limit that evolve from subcritical shear-banding modes (kx = 0), but at large enough Couette gaps there are stationary instabilities with kx = O(1) that lead to supercritical pitchfork bifurcations. At moderate-to-large densities, in addition to supercritical shear-banding modes, there are long-wave travelling instabilities that lead to Hopf bifurcations. It is shown that both supercritical and subcritical nonlinear states exist at moderate-to-large densities that originate from the dominant stationary and travelling instabilities for which kx = O(1). Nonlinear patterns of density, velocity and granular temperature for all types of instabilities are contrasted with their linear eigenfunctions. While the supercritical solutions appear to be modulated forms of the fundamental mode, the structural features of unstable subcritical solutions are found to be significantly different from their linear counterparts. It is shown that the granular plane Couette flow is prone to nonlinear resonances in both stable and unstable regimes, the signature of which is implicated as a discontinuity in the first Landau coefficient. Our analysis identified two types of modal resonances that appear at the quadratic order in perturbation amplitude: (i) a ‘mean-flow resonance’ which occurs due to the interaction between a streamwise-independent shear-banding mode (kx = 0) and a linear/fundamental mode kx ≠ 0, and (ii) an exact ‘1 : 2 resonance’ that results from the interaction between two waves with their wavenumber ratio being 1 : 2.

A model is derived for long peristaltic waves propagating steadily down a fluid-filled, axisymmetric tube. The waves are driven by imposing a radial force of prescribed form on the tube. The resulting deformation of the tube wall is modelled using linear elasticity and the internal flow using the lubrication approximation. Numerical solutions for periodic wave trains and solitary waves are presented, along with asymptotic solutions at both small and large forcing amplitudes. Large-amplitude periodic waves are characterized by narrow blisters adjoining long occluded sections of the tube, whereas a solitary wave of strong contraction produces a long inflated bow wave that propels a large quantity of fluid. A measure of pumping efficacy is given by the ratio of the net fluid displacement to the power input, and is highest for a large-amplitude solitary wave.

A mathematical model is developed for long peristaltic waves propelling a suspended rigid object down a fluid-filled axisymmetric tube. The fluid flow is described using lubrication theory and the deformation of the tube using linear elasticity. The object is taken to be either an infinitely long rod of constant radius or a parabolic-shaped lozenge of finite length. The system is driven by a radial force imposed on the tube wall that translates at constant speed down the tube axis and with a form chosen to generate a periodic wave train or a solitary wave. These waves exert a traction on the enclosed object, forcing it into motion. Periodic waves drive the infinite rod at a speed that attains a maximum at a moderate forcing amplitude and approaches approximately one quarter of the wave speed in the large-amplitude limit. The finite lozenge can be entrained and driven at the same speed as a solitary wave or periodic wave train if the forcing is sufficiently strong. For weaker forcing, the lozenge is either left behind the solitary wave or interacts repeatedly with the waves in the periodic train to generate stuttering forward progress. The threshold forcing amplitude for entrainment increases weakly with the radial span of the enclosed object, but strongly with the axial length, with entrainment becoming impossible if the object is too long.

In this paper, we perform direct numerical simulations (DNS) of turbulent boundary layers with nominal free-stream Mach number ranging from 0.3 to 12. The main objective is to assess the scalings with respect to the mean and turbulence behaviours as well as the possible breakdown of the weak compressibility hypothesis for turbulent boundary layers at high Mach numbers (M > 5). We find that many of the scaling relations, such as the van Driest transformation for mean velocity, Walz's relation, Morkovin's scaling and the strong Reynolds analogy, which are derived based on the weak compressibility hypothesis, remain valid for the range of free-stream Mach numbers considered. The explicit dilatation terms such as pressure dilatation and dilatational dissipation remain small for the present Mach number range, and the pressure–strain correlation and the anisotropy of the Reynolds stress tensor are insensitive to the free-stream Mach number. The possible effects of intrinsic compressibility are reflected by the increase in the fluctuations of thermodynamic quantities (p′rms/pw, ρ′rms/ρ, T′rms/T) and turbulence Mach numbers (Mt, M′rms), the existence of shocklets, the modification of turbulence structures (near-wall streaks and large-scale motions) and the variation in the onset of intermittency.

The wave pattern generated by a pressure source moving over the free surface of deep water at speeds, U, below the minimum phase speed for linear gravity–capillary waves, cmin, was investigated experimentally using a combination of photographic measurement techniques. In similar experiments, using a single pressure amplitude, Diorio et al. (Phys. Rev. Lett., vol. 103, 2009, 214502) pointed out that the resulting surface response pattern exhibits remarkable nonlinear features as U approaches cmin, and three distinct response states were identified. Here, we present a set of measurements for four surface-pressure amplitudes and provide a detailed quantitative examination of the various behaviours. At low speeds, the pattern resembles the stationary state (U = 0), essentially a circular dimple located directly under the pressure source (called a state I response). At a critical speed, but still below cmin, there is an abrupt transition to a wave-like state (state II) that features a marked increase in the response amplitude and the formation of a localized solitary depression downstream of the pressure source. This solitary depression is steady, elongated in the cross-stream relative to the streamwise direction, and resembles freely propagating gravity–capillary ‘lump’ solutions of potential flow theory on deep water. Detailed measurements of the shape of this depression are presented and compared with computed lump profiles from the literature. The amplitude of the solitary depression decreases with increasing U (another known feature of lumps) and is independent of the surface pressure magnitude. The speed at which the transition from states I to II occurs decreases with increasing surface pressure. For speeds very close to the transition point, time-dependent oscillations are observed and their dependence on speed and pressure magnitude are reported. As the speed approaches cmin, a second transition is observed. Here, the steady solitary depression gives way to an unsteady state (state III), characterized by periodic shedding of lump-like disturbances from the tails of a V-shaped pattern.

A theoretical model is presented for the generation of waves by a localized pressure distribution moving on the surface of deep water with speed near the minimum gravity–capillary phase speed, cmin. The model employs a simple forced–damped nonlinear dispersive equation. Even though it is not formally derived from the full governing equations, the proposed model equation combines the main effects controlling the response and captures the salient features of the experimental results reported in Diorio et al. (J. Fluid Mech., vol. 672, 2011, pp. 268–287 – Part 1 of this work). Specifically, as the speed of the pressure disturbance is increased towards cmin, three distinct responses arise: state I is confined beneath the applied pressure and corresponds to the linear subcritical steady solution; state II is steady, too, but features a steep gravity–capillary lump downstream of the pressure source; and state III is time-periodic, involving continuous shedding of lumps downstream. The transitions from states I to II and from states II to III, observed experimentally, are associated with certain limit points in the steady-state response diagram computed via numerical continuation. Moreover, within the speed range that state II is reached, the maximum response amplitude turns out to be virtually independent of the strength of the pressure disturbance, in agreement with the experiment. The proposed model equation, while ad hoc, brings out the delicate interplay between dispersive, nonlinear and viscous effects that takes place near cmin, and may also prove useful in other physical settings where a phase-speed minimum at non-zero wavenumber occurs.

Using an iterative algorithm, a family of stationary two-dimensional vortical dipoles is constructed, including translational (symmetric and asymmetric about the translation axis) and orbital (i.e. moving in circles) dipoles. The patches of uniform vorticity comprising a dipole possess symmetry about the axis passing through their centroids and are, generally, unequal in area and absolute value of vorticity. The solutions are discriminated by three parameters, the ratio of the areas of individual vortices, the ratio of their vorticities and the separation between the centroids of the patches. The dipole stability and evolution of unstable states are studied numerically with a contour dynamics method, where the perturbations allowed are, generally, asymmetric. The diagrams of convergence of the iterative algorithm (without any symmetry constraints) are built in three cross-sections of the parameter space: at opposite vorticity of the individual vortices, at equal areas of the vortices and at zero net circulation of the vortex pairs (when inequality of areas of the individual vortices is offset by inequality of the absolute values of vorticity). The convergence bound is shown to be close to the stability bound in the parameter space, and the larger is the separation, the stronger are the perturbations needed to move the dipole out of equilibrium. Typical scenarios of the evolution of unstable symmetric translational dipoles and weakly stable dipoles of other kinds are described, including the transition of a dipole into an oscillating tripole – the scenario that has not been discussed so far.

Reflection of an obliquely incident solitary wave at a vertical wall is studied experimentally in the laboratory wave tank. Precision measurements of water-surface variations are achieved with the aid of laser-induced fluorescent (LIF) technique and detailed features of the Mach reflection are captured. During the development stage of the reflection process, the stem wave is not in the form of a Korteweg–de Vries (KdV) soliton but a forced wave, trailing by a continuously broadening depression. Evolution of stem-wave amplification is in good agreement with the Kadomtsev–Petviashvili (KP) theory. The asymptotic characteristics and behaviours are also in agreement with the theory of Miles (J. Fluid Mech., vol. 79, 1977b, p. 171) except those in the neighbourhood of the transition between the Mach reflection and the regular reflection. The predicted maximum fourfold amplification of the stem wave is not realized in the laboratory environment. On the other hand, the laboratory observations are in excellent agreement with the previous numerical results of the higher-order model of Tanaka (J. Fluid Mech., vol. 248, 1993, p. 637). The present laboratory study is the first to sensibly analyse validation of the theory; note that substantial discrepancies exist from previous (both numerical and laboratory) experimental studies. Agreement between experiments and theory can be partially attributed to the large-distance measurements that the precision laboratory apparatus is capable of. More important, to compare the laboratory results with theory, the corrected interaction parameter is derived from proper interpretation of the theory in consideration of the finite incident wave angle. Our laboratory data indicate that the maximum stem wave can reach higher than the maximum solitary wave height. The wave breaking near the wall results in the substantial increase in wave height and slope away from the wall.

We investigate theoretically the statistics of the equilibria of two-dimensional droplets over random topographical substrates. The substrates are appropriately represented as families of certain stationary random functions parametrized by a characteristic amplitude and wavenumber. In the limit of shallow topographies and small contact angles, a linearization about the flat-substrate equilibrium reveals that the droplet footprint is adequately approximated by a zero-mean, normally distributed random variable. The theoretical analysis of the statistics of droplet shift along the substrate is highly non-trivial. However, for weakly asymmetric substrates it can be shown analytically that the droplet shift approaches a Cauchy random variable; for fully asymmetric substrates its probability density is obtained via Padé approximants. Generalization to arbitrary stationary random functions does not change qualitatively the behaviour of the statistics with respect to the characteristic amplitude and wavenumber of the substrate. Our theoretical results are verified by numerical experiments, which also suggest that on average a random substrate neither enhances nor reduces droplet wetting. To address the question of the influence of substrate roughness on wetting, a stability analysis of the equilibria must be performed so that we can distinguish between stable and unstable equilibria, which in turn requires modelling the dynamics. This is the subject of Part 2 of this study.

We examine the dynamics of a two-dimensional droplet spreading over a random topographical substrate. Our analysis is based on the formalism developed in Part 1 of this study, where a random substrate was modelled as band-limited white noise. The system of integrodifferential equations for the motion of the contact points over deterministic substrates derived by Savva and Kalliadasis (Phys. Fluids, vol. 21, 2009, 092102) is applicable to the case of random substrates as well. This system is linearized for small substrate amplitudes to obtain stochastic differential equations for the droplet shift and contact line fluctuations in the limit of shallow and slowly varying topographies. Our theoretical predictions for the time evolution of the statistical properties of these quantities are verified by numerical experiments. Considering the statistics of the dynamics allows us to fully address the influence of the substrate variations on wetting. For example, we demonstrate that the droplet wets the substrate less as the substrate roughness increases, illustrating also the possibility of a substrate-induced hysteresis effect. Finally, the analysis of the long-time limit of spreading dynamics for a substrate represented by a band-limited white noise is extended to arbitrary substrate representations. It is shown that the statistics of spreading is independent of the characteristic length scales that naturally arise from the statistical properties of a substrate representation.

The effect of a surface trip wire on the flow around a sphere is experimentally investigated at subcritical Reynolds numbers of Re = 0.5 × 105 – 2.8 × 105 based on the free-stream velocity U∞ and sphere diameter d. By varying the streamwise location (20° – 70° from the stagnation point) and diameter (0.33 × 10−2 < k/d < 1.33 × 10−2) of a trip wire, we measure the drag, surface pressure distribution and boundary layer velocity profiles above the sphere surface, and conduct flow visualization. Depending on the size and streamwise location of the trip wire, three different flow characteristics are observed above the sphere surface. For low Reynolds numbers, the disturbance induced by the trip wire decays downstream and main separation occurs at a streamwise location similar to that of a smooth sphere. As the Reynolds number is increased, laminar separation is delayed farther downstream by the disturbance from the trip wire and the transition to turbulence occurs along the separated shear layer, resulting in the flow reattachment to the sphere surface and thus forming a secondary separation bubble on the sphere surface. Then, the main separation is delayed due to high momentum near the surface and the drag is significantly reduced. When the trip wire produces even larger disturbances through the separation and reattachment right at the trip-wire location for higher Reynolds numbers, the boundary layer flow becomes turbulent soon after the trip-wire location and the main separation is delayed, resulting in drag reduction.

An analysis is made of the sound generated by the time-dependent throttling of a nominally steady stream of air through a small orifice into a flow-through resonant cavity. This is exemplified by the production of voiced speech, where air from the lungs enters the vocal tract through the glottis at a time-variable volume flow rate Q(t) controlled by oscillations of the glottis cross-section. Voicing theory has hitherto determined Q from a heuristic, reduced complexity ‘Fant’ differential equation. A new self-consistent, integro-differential form of this equation is derived in this paper using the theory of aerodynamic sound, with full account taken of the back-reaction of the resonant tract on the glottal flux Q. The theory involves an aeroacoustic Green's function (G) for flow–surface interactions in a time-dependent glottis, so making the problem non-self-adjoint. In complex problems of this type, it is not usually possible to obtain G in an explicit analytic form. The principal objective of this paper is to show how the Fant equation can still be derived in such cases from a consideration of the equation of aerodynamic sound and from the adjoint of the equation governing G in the neighbourhood of the ‘throttle’. The theory is illustrated by application to the canonical problem of throttled flow into a Helmholtz resonator.

Large-scale longitudinal vortices in high-speed turbulent separated flows caused by relatively small irregularities at the model leading edges or at the model surfaces are investigated in this paper. Oil-flow visualization and infrared thermography techniques were applied in the wind tunnel tests at Mach numbers 3 and 5 to investigate the nominally 2-D ramp flow at deflection angles of 20°, 25° and 30°. The surface contour anomalies have been artificially simulated by very thin strips (vortex generators) of different shapes and thicknesses attached to the model surface. It is shown that the introduced streamwise vortical disturbances survive over very large downstream distances of the order of 104 vortex-generator heights in turbulent supersonic flows without pressure gradients. It is demonstrated that each vortex pair induced in the reattachment region of the ramp is definitely a child of a vortex pair, which was generated originally, for instance, by the small roughness element near the leading edge. The dependence of the spacing and intensity of the observed longitudinal vortices on the introduced disturbances (thickness and spanwise size of vortex generators) and on the flow parameters (Reynolds numbers, boundary-layer thickness, compression corner angles, etc.) has been shown experimentally.