A model for simulating the process of growth, collapse and rebound of a cavitation bubble travelling along the flow through a convergent–divergent nozzle producing a cavitating water jet is established. The model is based on the Rayleigh–Plesset bubble dynamics equation using as inputs ambient pressure and velocity profiles calculated with the aid of computational fluid dynamics (CFD) flow modelling. A variable time-step technique is applied to solve the highly nonlinear second-order differential equation. This technique successfully solves the Rayleigh–Plesset equation for wide ranges of pressure variation and bubble original size and saves considerable computing time. Inputs for this model are the pressure and velocity data from CFD calculation. To simulate accurately the process of bubble growth, collapse and rebound, a heat transfer model, which includes the effects of conduction plus radiation, is developed to describe the thermodynamics of the incondensable gas inside the bubble. This heat transfer model matches previously published experimental data well. Assuming that single bubble behaviour also applies to bubble clouds, the calculated distance from the nozzle exit travelled by the bubble to the point where the bubble size becomes invisible is taken to be equal to the bubble cloud length observed. The predictions are compared with experiments carried out in a cavitation cell and show good agreement for different nozzles operating at different pressure conditions.

Marine ice sheets are continental ice masses resting on bedrock below sea level. Their dynamics are similar to those of land-based ice sheets except that they must couple with the surrounding floating ice shelves at the grounding line, where the ice reaches a critical flotation thickness. In order to predict the evolution of the grounding line as a free boundary, two boundary conditions are required for the diffusion equation describing the evolution of the grounded-ice thickness. By analogy with Stefan problems, one of these conditions imposes a prescribed ice thickness at the grounding line and arises from the fact that the ice becomes afloat. The other condition must be determined by coupling the ice sheet to the surrounding ice shelves. Here we employ matched asymptotic expansions to study the transition from ice-sheet to ice-shelf flow for the case of rapidly sliding ice sheets. Our principal results are that the ice flux at the grounding line in a two-dimensional ice sheet is an increasing function of the depth of the sea floor there, and that ice thicknesses at the grounding line must be small compared with ice thicknesses inland. These results indicate that marine ice sheets have a discrete set of steady surface profiles (if they have any at all) and that the stability of these steady profiles depends on the slope of the sea floor at the grounding line.

The instability modes arising within simple non-reversing pulsatile flows in a circular tube with a smooth axisymmetric constriction are examined using global Floquet stability analysis and direct numerical simulation. The sectionally averaged pulsatile flow is represented with one harmonic component superimposed on a time-mean flow. We have previously identified a period-doubling global instability mechanism associated with alternating tilting of the vortex rings that are ejected out of the stenosis throat with each pulse. Here we show that while alternating tilting of vortex rings is the primary instability mode for comparatively larger reduced velocities associated with long pulse periods (or low Womersley numbers), for lower reduced velocities that are associated with shorter pulse periods the primary instability typically manifests as azimuthal waves (Widnall instability modes) of low wavenumber that grow on each vortex ring. Convective shear-layer instabilities are also supported by the types of flow considered. To provide an insight into the comparative role of these types of instability, which have still shorter temporal periods, we also introduce high-frequency low-amplitude perturbations to the base flows of the above global instabilities. For the range of parameters considered, we observe that the dominant features of the primary Floquet instability persist, but that the additional presence of the convective instability can have a destabilizing effect, especially for long pulse periods.

Numerical results on thermally driven nonlinear magnetoconvection in a rapidly rotating fluid spherical shell are reported. A uniform magnetic field that is parallel to the rotation axis is imposed externally. The Ekman number is 2 × 10−6, representing a state of negligible viscosity, as in the Earth's core. The convection pattern is characterized by a few large-scale vortex columns superimposed on a fast westward (retrograde) zonal flow. In the equatorial region, an anticyclonic vortex is intensified, in which an induced axial magnetic field is stored. Interaction between the magnetized vortex and the zonal flow leads to a thin jet at the western side of the vortex. The jet is also characterized by a thin electric current sheet caused by a steep gradient of the axial magnetic field. Because of this structure, the jet region can be designated as a magnetic front by analogy with fronts in mid-latitude atmospheric cyclones. It can be estimated from an order-of-magnitude analysis that the jet width decreases in inverse proportion to the zonal flow speed, and that the jet speed and the sheet-like electric current are proportional to the square of the zonal flow speed.

A theory is presented for the mass transport induced by a small-amplitude progressive wave propagating in water over a thin layer of viscoelastic mud modelled as a Voigt medium. Based on a sharp contrast in length scales near the bed, the boundary-layer approximation is applied to the Navier–Stokes equations in Lagrangian form, which are then solved for the first-order oscillatory motions in the mud and the near-bed water layers. On extending the analysis to second order for the mass transport, it is pointed out that it is inappropriate, as was done in previous studies, to apply the complex viscoelastic parameter to a higher-order analysis, and also to suppose that a Voigt body can undergo continuous steady motion. In fact, the time-mean motion of a Voigt body is only transient, and will stop after a time scale given by the ratio of the viscosity to the shear modulus. Once the mud has attained its steady deformation, the mass transport in the overlying water column can be found as if it were a single-layer system. It is found that the near-bed mass transport has non-trivial dependence on the mud depth and elasticity, which control the occurrence of resonance. Even when the resonance is considerably damped by viscosity, the mass transport in water over a viscoelastic layer can be dramatically different, in terms of magnitude and direction, from that over a rigid bed.

In this paper we present mean velocity distributions measured in several different wave flumes. The flows shown involve different types of mechanical wavemakers, channels of differing sizes, and two different end conditions. In all cases, when surface waves, nominally deep-water Stokes waves, are generated, counterflowing Eulerian flows appear that act to cancel locally, i.e. not in an integral sense, the mass transport associated with the Stokes drift. No existing theory of wave–current interactions explains this behaviour, although it is symptomatic of Gerstner waves, rotational waves that are exact solutions to the Euler equations. In shallow water (kH ≈ 1), this cancellation of the Stokes drift does not hold, suggesting that interactions between wave motions and the bottom boundary layer may also come into play.

We present a numerical model of the hydrodynamic interactions between two capsules freely suspended in a simple shear flow. The capsules are identical and each consists of a liquid droplet enclosed by a thin hyperelastic membrane, devoid of bending resistance and obeying a neo-Hookean constitutive law. The two capsules are slightly prestressed with a given inflation ratio in order to avoid the small deformation instability due to compression observed for a single capsule in simple shear flow. The viscosity ratio between the interior and exterior fluids of the capsule is taken to be unity and creeping flow conditions are assumed to prevail. The boundary-element method is used with bi-cubic B-splines as basis functions on a structured mesh in order to discretize the capsule surface. A new method using two grids with initially orthogonal pole axes is developed to eliminate polar singularities in the load calculation and to allow for long computation times. Two capsules suspended in simple shear flow usually have different velocities and thus eventually pass each other. We study this crossing process as a function of flow strength and initial particle separation. We find that hydrodynamic interactions during crossing lead to large shape alterations, elevated elastic tensions in the membrane and result in an irreversible trajectory shift of the capsules. Furthermore, a tendency towards buckling is observed, particularly during the separation phase where large pressure differences occur. Our results are in qualitative agreement with those obtained for a pair of interacting liquid droplets but show the specific role played by the membrane of capsules.

Vortex shedding behind a cylinder can be controlled by placing another small cylinder behind it, at low Reynolds numbers. This has been demonstrated experimentally by Strykowski & Sreenivasan (J. Fluid Mech. vol. 218, 1990, p. 74). These authors also provided preliminary numerical results, modelling the control cylinder by the innovative application of boundary conditions on some selective nodes. There are no other computational and theoretical studies that have explored the physical mechanism. In the present work, using an over-set grid method, we report and verify numerically the experimental results for flow past a pair of cylinders. Apart from providing an accurate solution of the Navier–Stokes equation, we also employ an energy-based receptivity analysis method to discuss some aspects of the physical mechanism behind vortex shedding and its control. These results are compared with the flow picture developed using a dynamical system approach based on the proper orthogonal decomposition (POD) technique.

Effects of wall roughness/topography on flows in strongly confined (micro-)channels are studied by means of lattice Boltzmann simulations. Whereas wall roughness in macroscopic channels is considered to be relevant only for high-Reynolds-number turbulent flows (where the flow is turbulent even for smooth walls), it is shown in this paper that, in micro-channels, surface roughness may even modify qualitative features of the flow. In particular, a transition from laminar to unsteady flow is observed. It is found that this roughness-induced transition is strongly enhanced as the channel width is decreased. The reliability of our results is checked by computing the viscous shear stress and the Reynolds stress across the channel, their sum following the theoretical prediction for the stress balance perfectly. Furthermore, the solutions obtained obey the transformation rules of the Navier–Stokes equation: When expressed in reduced (dimensionless) units, results for various channel dimensions, forcing term or dynamic viscosity are identical provided that the channel shape and the Reynolds number are unchanged. The time evolution of the velocity fluctuations at the initial stages of the transition to flow instability is monitored. It is found that fluctuations first occur in the vicinity of the rough wall, supporting the interpretation of wall roughness as a source of fluctuations and thus flow instability. In addition to their physical significance, our results provide further evidence for the reliability of the lattice Boltzmann method in dealing with complex unsteady flows.

This paper presents a study of the global response of a fluid to impulsive and localized forcing; it has been motivated by the recent laboratory experiments on the locomotion of water-walking insects reported in Hu, Chan & Bush (Nature, vol. 424, 2003, p. 663). These insects create both waves and vortices by their rapid leg strokes and it has been a matter of some debate whether either form of motion predominates in the momentum budget. The main result of this paper is to argue that generically both waves and vortices are significant, and that in linear theory they take up the horizontal momentum with share 1/3 and 2/3, respectively.

This generic result, which depends only on the impulsive and localized nature of the forcing, is established using the classical linear impulse theory, with adaptations to weakly compressible flows and flows with a free surface. Additional general comments on experimental techniques for momentum measurement and on the wave emission are given and then the theory is applied in detail to water-walking insects.

Owing to its generality, this kind of result and the methods used to derive it should be applicable to a wider range of wave–vortex problems in the biolocomotion of water-walking animals and elsewhere.

It has recently been shown that laminar, pressure-driven flow of a conducting fluid in a helical pipe can generate a dynamo. Geometrical constraints have hitherto required a relatively small Reynolds number, and a much larger magnetic Reynolds number, Rm. Here, a configuration with two interwoven helical pipes is considered which is shown to drive a dynamo at a Reynolds number of a few hundred and Rm > 30. Various computer animations of the dynamo are available with the online version of the paper. It is found that hydrodynamic instabilities may inhibit the dynamo, but may also be regularized by it. It is also shown that a dynamo pump is possible, with flow down one pipe generating a field which drives flow in the second. Movies are available with the online version of the paper.

An obliquely inclined circular water jet, impinging on a flat horizontal surface, confers a series of hydraulic jump profiles, pertaining to different jet inclinations and jet velocities. These jump profiles are non-circular, and can be broadly grouped into two categories, based on the angle of jet inclination, φ, made with horizontal. Jumps corrosponding to the range (25° < φ≤ 90°) are observed to be bounded by smooth curves, whereas those corresponding to φ≤ 25° are characterized by distinct corners. The present work attempts to find a geometric and hydrodynamic characterization of the spatial patterns formed as a consequence of such non-circular hydraulic jump profiles. Flow-visualization experiments are conducted to depict the shape of demarcating boundaries between supercritical and subcritical flows, and the corresponding radial jump locations are obtained. Theoretical calculations are also executed to obtain the radial locations of the jumps with geometrically smooth profiles. Comparisons are subsequently made between the theoretical predictions and the experimental observations, and a good agreement between these two can be observed. Jumps with corners, however, turn out to be comprised of strikingly contrasting profiles, which can be attributed to the ‘jump–jet’ interaction and the ‘jump-jump’ interaction mechanisms. A phenomenological explanation is also provided, by drawing an analogy from the theory of shock-wave interactions.

A proper orthogonal decomposition (POD) analysis and low-dimensional modelling of thermally driven two-dimensional flow of air in a horizontal rotating cylinder, subject to the Boussinesq approximation, is considered. The problem is unsteady due to the harmonic nature of the gravitational buoyancy force with respect to the rotating observer and is characterized by four dimensionless numbers: gravitational Rayleigh number (Rag), the rotational Rayleigh number (RaΩ), the Taylor number (Ta) and Prandtl number (Pr). The data for the POD analysis are obtained by numerical integration of the governing equations of mass, momentum and energy. The POD is applied to the computational data for RaΩ varying in the range 102–106 while Rag and Pr are fixed at 105 and 0.71 respectively. The ratio of Ta to RaΩ is fixed at 100 so that the results apply to physically realistic situations. A new criterion, in the form of appropriately defined error norms, for assessing the truncation error of the POD expansion is proposed. It is shown that these error norms reflect the accuracy of the POD-based reconstructions of a given data ensemble better than the widely employed average energy criterion. The translational symmetry in both space and time of the pair of modes having degenerate (equal) eigenvalues confirms the presence of travelling waves in the flow for several different RaΩ values. The shifts in space and time of the structure of the degenerate modes are utilized to estimate the wave speeds in a given direction. The governing equations for the fluctuations are derived and low-dimensional models are constructed by employing a Galerkin procedure. For each of the five values of RaΩ, the low-dimensional models yield accurate qualitative as well as quantitative behaviour of the system. Sufficient modes are included in the low-dimensional models so that the modelling of the unresolved scales of motion is not needed to stabilize their solution. Not more than 20 modes are required in the low-dimensional models to accurately model the system dynamics. The ability of low-dimensional models to accurately predict the system behaviour for a set of parameters different from those from which they were constructed is also examined.

Experiments and theory on the rupture of a free plane viscous film are reported. The relatively thick film, with a typical thickness of the order of 0.1–0.6 mm, rests between two long parallel needles. When the film is punctured, a hole is formed with the rim on the front. The hole expands, reaches the needles, and propagates along them with a constant velocity of the order of 2–50 cm s−1. The Reynolds numbers for the present experiments are relatively small, 0.002 ≤ Re ≤ 0.34. A crude theory for propagation velocity of the rim is proposed; the theory compares well with the experimental data. The rupture profile is visually similar to a U-shaped curve. Crude equations for the rupture profile are derived, and their solutions are consistent with the experimental observations. A theory for propagation velocity and profile of the rupture, applicable to all Reynolds numbers, is proposed.

A three-dimensional numerical model is developed to simulate the process of collision between an evaporative droplet and a high-temperature particle. This phenomenon is of direct relevance to many engineering process operations, such as fluid catalytic cracking (FCC), polyethylene synthesis, and electronic materials coating. In this study, the level-set method and the immersed-boundary method are combined to describe the droplet–particle contact dynamics in a fixed Eulerian grid. The droplet deformation is captured by one level-set function while the solid–fluid boundary condition is imposed on the particle surface through the immersed-boundary method involving another level-set function. A two-dimensional vapour-layer model is developed to simulate the vapour flow dynamics. Equations for the heat transfer characteristics are formulated for each of the solid, liquid and gas phases. The incompressible flow-governing equations are solved using the finite-volume method with the ALE (arbitrary Lagrangian Eulerian) technique. The simulation results are validated through comparisons with experimental data obtained from the new experimental set-up designed in this study. An important feature of the droplet impacting on a particle with film boiling is that the droplet undergoes a spreading, recoiling and rebounding process, which is reproduced by the numerical simulation based on the model. Details of the collision such as spread factor, contact time and temperature distribution are provided. Simulations are also conducted to examine the effects of the particle size and the collision velocity. Although the value for the maximum spread factor is larger for a higher impact velocity and for a smaller particle, the contact time is independent of the impact velocity and particle size. Both the normal collision and the oblique collision are considered in this study.

Laboratory experiments and numerical simulations of oscillating spin-up in a square tank have been conducted to investigate the production of small-scale vorticity near the no-slip sidewalls of the container and the formation and subsequent decay of wall-generated quasi-two-dimensional vortices. The flow is made quasi-two-dimensional by a steady background rotation, and a small sinusoidal perturbation to the background rotation leads to the periodic formation of eddies in the corners of the tank by the roll-up of vorticity generated along the sidewalls. When the oscillation period is greater than the time scale required to advect a full-grown corner vortex to approximately halfway along the sidewall, dipole structures are observed to form. These dipoles migrate away from the walls, and the interior of the tank is continually filled with new vortices. The average size of these vortices appears to be largely controlled by the initial formation mechanism. Their vorticity decays from interactions with other stronger vortices that strip off filaments of vorticity, and by Ekman pumping at the bottom of the tank. Subsequent interactions between the weaker ‘old’ vortices and the ‘young’ vortices result in the straining, and finally the destruction, of older vortices. This inhibits the formation of large-scale vortices with diameters comparable to the size of the container.

The laboratory experiments revealed a k−5/3 power law of the energy spectrum for small-to-intermediate wavenumbers. Measurements of the intensity spectrum of a passive scalar were consistent with the Batchelor prediction of a k−1 power law at large wavenumbers. Two-dimensional numerical simulations, under similar conditions to those in the experiments (with weak Ekman decay), were also performed and the simultaneous presence of a k−5/3 and k−3−ζ (with 0 < ζ « 1) power spectrum is observed, with the transition occurring at the wavenumber at which vorticity is injected from the viscous boundary layer into the interior. For higher Ekman decay rates, steeper spectra are obtained for the large wavenumber range, with ζ = O(1) and proportional to the Ekman decay rate. Movies are available with the online version of the paper.

The statistical properties of fully developed planar turbulent Couette–Poiseuille flow result from the simultaneous imposition of a mean wall shear force together with a mean pressure force. Despite the fact that pure Poiseuille flow and pure Couette flow are the two extremes of Couette–Poiseuille flow, the statistical properties of the latter have proved resistant to scaling approaches that coherently extend traditional wall flow theory. For this reason, Couette–Poiseuille flow constitutes an interesting test case by which to explore the efficacy of alternative theoretical approaches, along with their physical/mathematical ramifications. Within this context, the present effort extends the recently developed scaling framework of Wei et al. (2005a) and associated multiscaling ideas of Fife et al. (2005a, b) to fully developed planar turbulent Couette–Poiseuille flow. Like Poiseuille flow, and contrary to the structure hypothesized by the traditional inner/outer/overlap-based framework, with increasing distance from the wall, the present flow is shown in some cases to undergo a balance breaking and balance exchange process as the mean dynamics transition from a layer characterized by a balance between the Reynolds stress gradient and viscous stress gradient, to a layer characterized by a balance between the Reynolds stress gradient (more precisely, the sum of Reynolds and viscous stress gradients) and mean pressure gradient. Multiscale analyses of the mean momentum equation are used to predict (in order of magnitude) the wall-normal positions of the maxima of the Reynolds shear stress, as well as to provide an explicit mesoscaling for the profiles near those positions. The analysis reveals a close relationship between the mean flow structure of Couette–Poiseuille flow and two internal scale hierarchies admitted by the mean flow equations. The averaged profiles of interest have, at essentially each point in the channel, a characteristic length that increases as a well-defined ‘outer region’ is approached from either the bottom or the top of the channel. The continuous deformation of this scaling structure as the relevant parameter varies from the Poiseuille case to the Couette case is studied and clarified.

The aim of the present work, final of a three-part series, is to analyse in detail flow motions within the swash zone and define suitable shoreline boundary conditions for the longshore flow for wave-averaged circulation models. The analyses of Parts 1 and 2 are extended to cover horizontally two-dimensional flows. An analytical solution for the longshore motion representing the drift velocity of the whole swash zone water mass is found. This is seen to be approximated well by the ratio between the time integral of the longshore momentum flux crossing the swash lower boundary and the swash zone net water volume. Further, a complete set of shoreline boundary conditions, taking into account wave–wave interactions, is obtained on the basis of fully numerical solutions of the nonlinear shallow-water equations. The main focus of the work is to clarify the structure of the shoreline boundary conditions for the longshore flow, but attention has also been paid to their derivation and assessment from the numerical solutions. The latter have been obtained on the basis of a fairly broad range of input wave conditions which, though biased towards those typical of reflective beaches, are believed to cover conditions also typical of moderate dissipative beaches. Two main terms are found to contribute to the longshore drift velocity: (i) a term, proportional to the shallow-water velocity, accounting for short-wave interactions, frictional swash zone forces and continuous forcing due to non-breaking wave nonlinearities and (ii) a drift-type term representing the momentum transfer due to wave breaking.

An experimental study, investigating the mean flow and turbulence in the wind drift layer formed beneath short wind waves was conducted. The degree to which these flows resemble the flows that occur in boundary layers adjacent to solid walls (i.e. wall-layers) was examined. Simultaneous DPIV (digital particle image velocimetry) and infrared imagery were used to investigate these near-surface flows at a fetch of 5.5 m and wind speeds from 4.5 to 11 m s−1. These conditions produced short steep waves with dominant wavelengths from 6 cm to 18 cm. The mean velocity profiles in the wind drift layer were found to be logarithmic and the flow was hydrodynamically smooth at all wind speeds. The rate of dissipation of turbulent kinetic energy was determined to be significantly greater in magnitude than would occur in a comparable wall-layer. Microscale breaking waves were detected using the DPIV data and the characteristics of breaking and non-breaking waves were compared. The percentage of microscale breaking waves increased abruptly from 11% to 80% as the wind speed increased from 4.5 to 7.4 m s− and then gradually increased to 90% as the wind speed increased to 11 m s−. At a depth of 1 mm, the rate of dissipation was 1.7 to 3.2 times greater beneath microscale breaking waves compared to non-breaking waves. In the crest–trough region beneath microscale breaking waves, 40% to 50% of the dissipation was associated with wave breaking. These results demonstrated that the enhanced near-surface turbulence in the wind drift layer was the result of microscale wave breaking. It was determined that the rate of dissipation of turbulent kinetic energy due to wave breaking is a function of depth, friction velocity, wave height and phase speed as proposed by Terray et al. (1996). Vertical profiles of the rate of dissipation showed that beneath microscale breaking waves there were two distinct layers. Immediately beneath the surface, the dissipation decayed as ζ−0.7 and below this in the second layer it decayed as ζ−2. The enhanced turbulence associated with microscale wave breaking was found to extend to a depth of approximately one significant wave height. The only similarity between the flows in these wind drift layers and wall-layers is that in both cases the mean velocity profiles are logarithmic. The fact that microscale breaking waves were responsible for 40%–50% of the near-surface turbulence supports the premise that microscale breaking waves play a significant role in enhancing the transfer of gas and heat across the air–sea interface.

A computational study has been performed to examine the amplification of three-dimensional flow features as a vortex with small-amplitude helical perturbations impinges on a circular cylinder whose axis is parallel to the nominal vortex axis. For sufficiently weak vortices with sufficiently small core radius in an inviscid flow, three-dimensional perturbations on the vortex core are indefinitely amplified as the vortex wraps around the cylinder front surface. The paper focuses on the effect of viscosity in regulating amplification of three-dimensional disturbances and on assessing the ability of two-dimensional computations to accurately model parallel vortex–cylinder interaction problems. The computations are performed using a multi-block structured finite-volume method for an incompressible flow, with periodic boundary conditions along the cylinder axis. Growth of three-dimensional flow features is examined using a proper-orthogonal decomposition of the Fourier-transformed vorticity field in the azimuthal and axial directions. The interaction is examined for different axial wavelengths and amplitudes of the initial helical vortex waves and for three different Reynolds numbers.