The recent article by Schoof (J. Fluid Mech., vol. 712, 2012, pp. 552–578) provides a technically demanding solution to the problem of determining ice-stream margin evolution. It is important in opening the way to the future theoretical description of how the ice sheets will melt and sea level will rise as the climate warms. But the sophistication of the mathematics should not operate as a mask to an examination of the credibility of the model.

Vortex formation and force generation of clapping plates with various aspect ratios ($AR$) and stroke angles were investigated. Experiments were performed with a pair of hinged rectangular plates that were rotated symmetrically in a static fluid, and defocusing digital particle image velocimetry was employed to measure the three-dimensional flow field. Single-plate cases were also studied to compare with clapping plate cases. As $AR$ decreases, both circulation of the tip vortex and area enclosed by the vortex loop increase inversely. An empirical power-law relationship with a negative exponent is found between total impulse and $AR$ for a given stroke angle. The sensitivity of the force generated by the plates to the change of $AR$ is larger at the smaller stroke angle because of faster acceleration and deceleration. The increase in impulse per plate from the single-plate case to the clapping double-plate case is larger for lower $AR$. These results reveal that low $AR$ wings are more efficient in propulsive force generation in some specific modes of unsteady flapping flight. The evolution of the wake structures is found to depend on $AR$ and stroke angle.

The details of the sound sources within two initially laminar subsonic jets corresponding to different inflow conditions, one fully laminar and the other with nozzle boundary layer forcing, are studied using the Goldstein acoustic analogy. The statistics of the acoustic analogy equivalent sources are generated from large eddy simulations (LES) and issues associated with the range of validity of these are explored. The predicted comparative importance of various source terms confirms the results of previous studies, with the exception of one extra term that involves the longitudinal and the lateral source components. Agreement within 1 dB over the frequency range from Strouhal number $\mathit{St}= 0. 8$ to $\mathit{St}= 6$, which was shown to best correspond to the validity range of the acoustic source model for the LES data available, is found between noise predictions using the acoustic analogy model and those of a previously conducted LES–ILEE (isotropic linearized Euler equation) control surface method for the nominally laminar inflow jet case. The acoustic analogy is used to determine the source length scales which contribute to noise at different radiated angles to the jet, how they are distributed and how they change with change of inflow conditions. For the vortex pairing frequency, using a new acoustic source decomposition technique based on the correlation length scale, two types of noise sources are found. One source, which corresponds to the peak source amplitude, is located at the upstream end of the jet and is associated with the vortex pairing. The other one, which is the dominant source, is associated with jet mixing at the end of the jet potential core. It is also shown that boundary layer forcing leads to a reduction in the contribution to the noise of the large source length scales in comparison with the fully laminar case.

We study two-phase stratified flow where the bottom layer is a thin laminar liquid and the upper layer is a fully developed gas flow. The gas flow can be laminar or turbulent. To determine the boundary between convective and absolute instability, we use Orr–Sommerfeld stability theory, and a combination of linear modal analysis and ray analysis. For turbulent gas flow, and for the density ratio $r= 1000$, we find large regions of parameter space that produce absolute instability. These parameter regimes involve viscosity ratios of direct relevance to oil and gas flows. If, instead, the gas layer is laminar, absolute instability persists for the density ratio $r= 1000$, although the convective/absolute stability boundary occurs at a viscosity ratio that is an order of magnitude smaller than in the turbulent case. Two further unstable temporal modes exist in both the laminar and the turbulent cases, one of which can exclude absolute instability. We compare our results with an experimentally determined flow-regime map, and discuss the potential application of the present method to nonlinear analyses.

The Cahn–Hilliard model is increasingly often being used in combination with the incompressible Navier–Stokes equation to describe unsteady binary fluids in a variety of applications ranging from turbulent two-phase flows to microfluidics. The thickness of the interface between the two bulk fluids and the mobility are the main parameters of the model. For real fluids they are usually too small to be directly used in numerical simulations. Several authors proposed criteria for the proper choice of interface thickness and mobility in order to reach the so-called ‘sharp-interface limit’. In this paper the problem is approached by a formal asymptotic expansion of the governing equations. It is shown that the mobility is an effective parameter to be chosen proportional to the square of the interface thickness. The theoretical results are confirmed by numerical simulations for two prototypal flows, namely capillary waves riding the interface and droplets coalescence. The numerical analysis of two different physical problems confirms the theoretical findings and establishes an optimal relationship between the effective parameters of the model.

Solute transport within solidifying binary alloys occurs predominantly by convection from narrow liquid chimneys within a porous mushy layer. We develop a simple model that elucidates the dominant structure and driving forces of the flow, which could be applied to modelling brine fluxes from sea ice, where a cheaply implementable approach is essential. A horizontal density gradient within the mushy layer in the vicinity of the chimneys leads to baroclinic torque which sustains the convective flow. In the bulk of the mushy layer, the isotherms are essentially horizontal. In this region, we impose a vertically linear temperature field and immediately find that the flow field is a simple corner flow. We determine the strength of this flow by finding a similarity solution to the governing mushy-layer equations in an active region near the chimney. We also determine the corresponding shape of the chimney, the vertical structure of the solid fraction and the interstitial flow field. We apply this model first to a periodic, planar array of chimneys and show analytically that the solute flux through the chimneys is proportional to a mush Rayleigh number. Secondly we extend the model to three dimensions and find that an array of chimneys can be characterized by the average drainage area alone. Therefore we solve the model in an axisymmetric geometry and find new, sometimes nonlinear, relationships between the solute flux, the Rayleigh number and the other dimensionless parameters of the system.

This paper presents time-averaged data for high-Reynolds-number hypersonic shock-wave/boundary-layer interactions, using a body of revolution to achieve high standards of two-dimensionality. The data are collected at nominal Mach 8.9, but a calibration is included that permits weak flow gradients in the test section to be incorporated as part of the data interpretation or flow modelling. The axisymmetric turbulent test boundary layer is developed on a hollow cylinder, aligned axially with the flow. The shock-wave interaction with this boundary layer is then generated by two separate configurations. Firstly, an impinging shock-wave case, that uses a concentric cowl to radiate an axisymmetric shock system onto the test boundary layer: for this case both an attached flow and a separated flow interaction are formed. Secondly, use of a conical-flare afterbody to produce a separated flow interaction. Quantitative data are presented for surface pressures and heat transfer, supported by some schlieren visualization and surface oil flows. A restricted CFD programme is included to assist the interpretation of the experiments.

We study the influence of low-frequency vibration on Marangoni instability in a layer of a binary mixture with the Soret effect. A linear stability analysis is performed numerically for perturbations of a finite wavelength (short-wave perturbations). Competition between long-wave and short-wave modes is found: the former ones are critical at smaller absolute values of the Soret number $\chi $, whereas the latter ones lead to instability at higher $\vert \chi \vert $. In both cases the vibration destabilizes the layer. Two variants of calculations are performed: via Floquet theory (linear asymptotic stability) and taking noise into consideration (empirical criterion). It is found that fluctuations substantially reduce the domains of stability. Further, while studying a limiting case within the empirical criterion, we have found a short-wave instability mode overlooked in former investigations of coupled Rayleigh–Marangoni convection in a layer of pure liquid.

We investigate the motion of a suspended non-Brownian sphere past a fixed cylindrical or spherical obstacle in the limit of zero Reynolds number for arbitrary particle–obstacle aspect ratios. We consider both a suspended sphere moving in a quiescent fluid under the action of a uniform force as well as a uniform ambient velocity field driving a freely suspended particle. We determine the distribution of particles around a single obstacle and solve for the individual particle trajectories to comment on the transport of dilute suspensions past an array of fixed obstacles. First, we obtain an expression for the probability density function governing the distribution of a dilute suspension of particles around an isolated obstacle, and we show that it is isotropic. We then present an analytical expression – derived using both Eulerian and Lagrangian approaches – for the minimum particle–obstacle separation attained during the motion, as a function of the incoming impact parameter, i.e. the initial offset between the line of motion far from the obstacle and a parallel line that goes through its centre. Further, we derive the asymptotic behaviour for small initial offsets and show that the minimum separation decays exponentially. Finally we use this analytical expression to define an effective hydrodynamic surface roughness based on the net lateral displacement experienced by a suspended sphere moving past an obstacle.

This study concerns the rectilinear and periodic paths of an axisymmetric solid body (short-length cylinder and disk of diameter $d$ and thickness $h$) falling in a vertical tube of diameter $D$. We investigated experimentally the influence of the confinement ratio ($S= d/ D\lt 0. 8$) on the motion of the body, for different aspect ratios ($\chi = d/ h= 3$, $6$ and $10$), Reynolds numbers ($80\lt Re\lt 320$) and a density ratio between the fluid and the body close to unity. For a given body, the Reynolds number based on its mean vertical velocity is observed to decrease when $S$ increases. The critical Reynolds number for the onset of the periodic motion decreases with $S$ in the case of thin bodies ($\chi = 10$), whereas it appears unaffected by $S$ for thicker bodies ($\chi = 3$ and $6$). The characteristics of the periodic motion are also strongly modified by the confinement ratio. A thick body ($\chi = 3$) tends to go back to a rectilinear path when $S$ increases, while a thin body ($\chi = 10$) displays oscillations of growing amplitude with $S$ until it touches the tube (at about $S= 0. 5$). For a given aspect ratio, however, the amplitudes of the oscillations follow a unique curve for all $S$, which depends only on the relative distance of the Reynolds number to the threshold of path instability. In parallel, numerical simulations of the wake of a body held fixed in a uniform confined flow were carried out. The simulations allowed us to determine in this configuration the effect of the confinement ratio on the thresholds for wake instability (loss of axial symmetry at $R{e}_{c 1} $ and loss of stationarity at $R{e}_{c 2} $) and on the maximal velocity ${V}_{w} $ in the recirculating region of the stationary axisymmetric wake. The evolution with $\chi $ and $S$ of ${V}_{w} $ at $R{e}_{c 1} $ was used to define a Reynolds number $R{e}^{\ast} $. Remarkably, for a freely moving body, $R{e}^{\ast} $ remains almost constant when $S$ varies, regardless of the nature of the path.

The response of a two-dimensional liquid foam to a localized air injection is investigated experimentally and theoretically. The experiments show a rich phenomenology, with two essentially distinct behaviours, depending on the injection conditions. At low flux, the injected air forms a central bubble that grows inside the foam and induces plastic rearrangements, without film rupture. This ‘pure swelling’ regime is reminiscent of ductile fracture. In this regime, the central bubble shows fingering patterns beyond a certain velocity. The dependence among the swelling rate, the injection overpressure and the other control parameters, namely cell gap, bubble size and foam area, is captured by a simple balance between the pressure drop and bubble/wall friction under a radial assumption. Fingering is successfully modelled by the linear stability analysis of an azimuthal perturbation of the radial model; yield stress becomes an important parameter to determine the finger width. At high injection rate, films are broken and narrow cracks form rapidly through the foam, reminiscent of brittle fracture. Criteria for the transition between ductile and brittle behaviours are investigated, both at the local and global scales.

In this paper we solve two initial value problems for two-dimensional internal gravity waves. The waves are contained in a uniformly stratified, square-shaped domain whose sidewalls are tilted with respect to the direction of gravity. We consider several disturbances of the initial stream function field and solve both for its free evolution and for its evolution under parametric excitation. We do this by developing a structure-preserving numerical method for internal gravity waves in a two-dimensional stratified fluid domain. We recall the linearized, inviscid Euler–Boussinesq model, identify its Hamiltonian structure, and derive a staggered finite difference scheme that preserves this structure. For the discretized model, the initial condition can be projected onto normal modes whose dynamics is described by independent harmonic oscillators. This fact is used to explain the persistence of various classes of wave attractors in a freely evolving (i.e. unforced) flow. Under parametric forcing, the discrete dynamics can likewise be decoupled into Mathieu equations. The most unstable resonant modes dominate the solution, forming wave attractors.

A spherical drop is constrained by a solid support arranged as a latitudinal belt. This belt support splits the drop into two deformable spherical caps. The edges of the support are given by lower and upper latitudes yielding a ‘spherical belt’ of prescribed extent and position: a two-parameter family of constraints. This is a belt-constrained Rayleigh drop. In this paper we study the linear oscillations of the two coupled spherical-cap surfaces in the inviscid case, and the viscous case is studied in Part 2 (Bostwick & Steen, J. Fluid Mech., vol. 714, 2013, pp. 336–360), restricting to deformations symmetric about the axis of constraint symmetry. The integro-differential boundary-value problem governing the interface deformation is formulated as a functional eigenvalue problem on linear operators and reduced to a truncated set of algebraic equations using a Rayleigh–Ritz procedure on a constrained function space. This formalism allows mode shapes with different contact angles at the edges of the solid support, as observed in experiment, and readily generalizes to accommodate viscous motions (Part 2). Eigenvalues are mapped in the plane of constraints to reveal where near-multiplicities occur. The full problem is then approximated as two coupled harmonic oscillators by introducing a volume-exchange constraint. The approximation yields eigenvalue crossings and allows post-identification of mass and spring constants for the oscillators.

A spherical drop is constrained by a solid support arranged as a latitudinal belt. The spherical belt splits the drop into two deformable spherical caps. The edges of the belt support are given by lower and upper latitudes, yielding a support of prescribed extent and position: a two-parameter family of geometrical constraints. In this paper we study the linear oscillations of the two coupled surfaces in the viscous case, the inviscid case having been dealt with in Part 1 (Bostwick & Steen, J. Fluid Mech., vol. 714, 2013, pp. 312–335), restricting to axisymmetric disturbances. For the viscous case, limiting geometries are the spherical-bowl constraint of Strani & Sabetta (J. Fluid Mech., vol. 189, 1988, pp. 397–421) and free viscous drop of Prosperetti (J. Méc., vol. 19, 1980b, pp. 149–182). In this paper, a boundary-integral approach leads to an integro-differential boundary-value problem governing the interface disturbances, where the constraint is incorporated into the function space. Viscous effects arise due to relative internal motions and to the no-slip boundary condition on the support surface. No-slip is incorporated using a modified set of shear boundary conditions. The eigenvalue problem is then reduced to a truncated set of algebraic equations using a spectral method in the standard way. Limiting cases recover literature results to validate the proposed modification. Complex frequencies, as they depend upon the viscosity parameter and the support geometry, are reported for both the drop and bubble cases. Finally, for the drop, an approximate boundary between over- and under-damped motions is mapped over the constraint parameter plane.

A planar Savart water sheet uniformly seeded with small air bubbles in a large surface concentration is studied as a model experiment of the so-called ‘effervescent’ atomization process. This two-dimensional setup allows for a quantitative observation of all the steps of the sheet’s disintegration into a collection of disjointed droplets. The bubbles are heterogeneous nucleation sites which puncture the sheet with holes. The dynamics of the opening of holes competes with the simultaneous nucleation rate of new holes in a statistically stationary fashion. The liquid constituting the sheet is then transiently concentrated in a web of ligaments of various lengths and diameters, at the junction between adjacent holes. Their breakup produces the final spray. We provide a complete description of the ligament web statistics when nucleation is synchronous, and we show that the drop size dispersion from the breakup of a single ligament is responsible for the shape of the overall spray drop size distribution.

The present experimental and numerical study focuses on the vortical structures encountered in steady and pulsed low-blowing-ratio transverse jets ($0. 150\leq \mathit{BR}\leq 4. 2$), a configuration hardly discussed in the literature. Under unforced conditions at low blowing ratio, a stable leading-edge shear-layer rollup is identified inside the jet pipe. As the blowing ratio is increased, the destabilization and evolution of this structure sheds light on the formation mechanisms of the well-known transverse jet vortical system. A discussion on the nature of the counter-rotating vortex pair in low-blowing-ratio transverse jets is also provided. Under forced conditions, the experimental observations support and extend numerical results of previous fully modulated jet studies. Large-eddy simulation results provide scaling parameters for the classification of starting vortices for partly modulated jets, as well as information on their three-dimensional dynamics. The counter-rotating vortex pair initiation is observed and detailed in both Mie scattering visualizations and simulations. The observations support a mechanism based on stretching of the starting vortical structures because of inviscid induction and partial leapfrogging. Two modes of cross-flow ingestion inside the jet pipe are described as the pulsed jet cycles from high to low values of blowing ratio.

Gravity-driven geophysical mass flows often consist of a heterogeneous fluid–solid mixture. The complex interplay between the components leads to phenomena such as lateral levee formation in avalanches, or a granular front and an excess fluid pore pressure in debris flows. These effects are very important for predicting runout and the forces on structures, yet they are only partially represented in simplified shallow flow theories, since rearrangement of the mixture composition perpendicular to the main flow direction is neglected. In realistic flows, however, rheological properties and effective basal drag may depend strongly on the relative concentration of the components. We address this problem and present a depth-averaged model for shallow mixtures that explicitly allows for rearrangement in this direction. In particular we consider a fluid–solid mixture that experiences bulk horizontal motion, as well as internal sedimentation and resuspension of the particles, and therefore resembles the case of a debris flow. Starting from general mixture theory we derive bulk balance laws and an evolution equation for the particle concentration. Depth-integration yields a shallow mixture flow model in terms of bulk mass, depth-averaged particle concentration, the particle vertical centre of mass and the depth-averaged velocity. This new equation in this model for the particle vertical centre of mass is derived by taking the first moment, with respect to the vertical coordinate, of the particle mass conservation equation. Our approach does not make the Boussinesq approximation and results in additional terms coupling the momentum flux to the vertical centre of mass. The system is hyperbolic and reduces to the shallow-water equations in the homogeneous limit of a pure fluid or perfect mixing. We highlight the effects of sedimentation on resuspension and finally present a simple friction feedback which qualitatively resembles a large-scale experimental debris flow data set acquired at the Illgraben, Switzerland.

We derive an equation relating the pressure at the flat bed and the profile of an irrotational steady water wave, valid for all classical solutions of the governing equations for water waves. This permits the recovery of the surface wave from pressure measurements at the bed. Although we focus on periodic waves, the extension to solitary waves is straightforward. We illustrate the usefulness of the equation beyond the realm of linear theory by investigating the regime of shallow-water waves of small amplitude and by presenting a numerical example.

An exact expression ${\mathscr{E}}_{a} $ for available potential energy density in Boussinesq fluid flows (Roullet & Klein, J. Fluid Mech., vol. 624, 2009, pp. 45–55; Holliday & McIntyre, J. Fluid Mech., vol. 107, 1981, pp. 221–225) is shown explicitly to integrate to the available potential energy ${E}_{a} $ of Winters et al. (J. Fluid Mech., vol. 289, 1995, pp. 115–128). ${\mathscr{E}}_{a} $ is a positive definite function of position and time consisting of two terms. The first, which is simply the indefinitely signed integrand in the Winters et al. definition of ${E}_{a} $, quantifies the expenditure or release of potential energy in the relocation of individual fluid parcels to their equilibrium height. When integrated over all parcels, this term yields the total available potential energy ${E}_{a} $. The second term describes the energetic consequences of the compensatory displacements necessary under the Boussinesq approximation to conserve vertical volume flux with each parcel relocation. On a pointwise basis, this term adds to the first in such a way that a positive definite contribution to ${E}_{a} $ is guaranteed. Globally, however, the second term vanishes when integrated over all fluid parcels and therefore contributes nothing to ${E}_{a} $. In effect, it filters the components of the first term that cancel upon integration, isolating the positive definite residuals. ${\mathscr{E}}_{a} $ can be used to construct spatial maps of local contributions to ${E}_{a} $ for direct numerical simulations of density stratified flows. Because ${\mathscr{E}}_{a} $ integrates to ${E}_{a} $, these maps are explicitly connected to known, exact, temporal evolution equations for kinetic, available and background potential energies.

Self-sustained oscillations resulting from fluid–solid instabilities, such as the flutter of a flexible flag in axial flow, can be used to harvest energy if one is able to convert the solid energy into electricity. Here, this is achieved using piezoelectric patches attached to the surface of the flag, which convert the solid deformation into an electric current powering purely resistive output circuits. Nonlinear numerical simulations in the slender-body limit, based on an explicit description of the coupling between the fluid–solid and electric systems, are used to determine the harvesting efficiency of the system, namely the fraction of the flow kinetic energy flux effectively used to power the output circuit, and its evolution with the system’s parameters. The role of the tuning between the characteristic frequencies of the fluid–solid and electric systems is emphasized, as well as the critical impact of the piezoelectric coupling intensity. High fluid loading, classically associated with destabilization by damping, leads to greater energy harvesting, but with a weaker robustness to flow velocity fluctuations due to the sensitivity of the flapping mode selection. This suggests that a control of this mode selection by a careful design of the output circuit could provide some opportunities to improve the efficiency and robustness of the energy harvesting process.