In this paper, a concept of a floating elastic wave energy converter consisting of a disk-shaped elastic plate is proposed. The floating plate is moored to the seabed through a series of power take-off (PTO) units. A theoretical model based on the linear potential flow theory and eigenfunction matching method is developed to study the hydroelastic characteristics and evaluate wave power absorption of the device. The PTO system is simulated as a discrete PTO, and moreover, it is also modelled as a continuum PTO to represent the case when the PTO system is composed of a large number of PTO units. The continuum PTO approximation is tested against the discrete PTO simulation for accuracy. Two methods are proposed to predict the wave power absorption of the device. After running convergence analysis and model validation, the present model is employed to do a multiparameter impact analysis. The device adopting a continuum PTO system is found to capture wave power efficiently in an extensive range of wave frequencies. For the continuum PTO system, it is theoretically possible to adopt optimised PTO damper and stiffness/mass to guarantee the absorption of 100 % of the energy flux available in one circular component of the plane incident wave.

In this paper we study the generation of Tollmien–Schlichting waves initiated by vibrations of a wall where the wall is coated with a thin liquid film in a transonic flow regime. Motion of fluids are described by the two-dimensional Navier–Stokes equations assuming the Reynolds number is large. To find asymptotic solutions of the transonic boundary layer, we conduct an inspection analysis on the affine transformations of the triple-deck model for a subsonic flow and the unsteady full potential equations, with the intention of obtaining the order quantity of the free-stream Mach number in the transonic flow. We construct a modified triple-deck model for the transonic flow by considering the scalings of the perturbations that lead to the viscous–inviscid interaction problem for the flow in a subsonic regime. In particular, we are interested in the region where the subsonic scalings become invalid as the flow approaches transonic regime. We assume the wall oscillates in the vertical direction to the oncoming flow and these vibrations are periodic in time. We outline the process where the flow in the boundary layer converts the wall vibration perturbations into the instability modes which are measured by the receptivity coefficient. The viscous–inviscid interaction problem describes the stability of the boundary layer on the lower branch of the neutral curve. We show that the governing equations for the air viscous sublayer and the film flow are quasi-steady. The equation describing the inviscid layer of the airflow is unsteady and its referred to as the unsteady Kàrmàn–Guderley equation. The influence of the film surface tension is expressed through normal shear stress condition at the interface. We present an analytic formula for the amplitude of the Tollmien–Schlichting waves that are formed in the boundary layer. We analysed our model with different values of surface tension, initial film thickness and Kàrmàn–Guderley parameter. Depending on the value of these parameters, the initial amplitude of the instability waves may grow or decay.

Inkjet printing, originally invented for text and pattern printing, is now central to many industrial applications, such as printed electronics, flexible electronics, 3-D printing of mechanical and even biological devices. However, constrained by the droplets’ ejection mechanism, the accuracy of traditional inkjet printing is limited by the size of its orifice, and it is difficult to achieve a volume of droplets at the femtolitre scale, which hinders its further application in the above fields. To this end, we propose the confined interface vibration inkjet printing (CIVIJP) technique, which is capable of printing patterns in a liquid environment with droplet size much smaller than the orifice from which they are dispensed. Here, further systematic study of the mechanism of printing in a liquid environment was carried out with the assistance of a high-speed imaging technique. It is found for the first time that a single pulse stimulation applied on the piezo-ceramic of the inkjet nozzle can trigger damping oscillations of the oil/water interface confined by the orifice, which can last more than 500 $\mathrm {\mu }$s. By adjusting the intensity of single pulse stimulation, the size and quantity of the dispensed droplets can be controlled in a wide range, which is obviously different from traditional droplet ejection in a gaseous environment. This work reveals the underlying physics between the pulse stimulation and the interface behaviours, as well as the physics between the interface behaviours and the size and number of dispensed droplets, enriching the fundamental theory of the inkjet printing in liquid phase.

Chemically active, or autophoretic, particles that isotropically emit or absorb solute molecules undergo spontaneous self-propulsion when their activity is increased beyond a critical Péclet number ($Pe$). Here, we conduct numerical computations, using a spectral element based method, of a rigid, spherical autophoretic particle in unsteady rectilinear translation. The particle can be freely suspended (or ‘unforced’) or subject to an external force field (or ‘forced’). The motion of an unforced particle progresses through four regimes as $Pe$ is increased: quiescent, steady, stirring and chaos. The particle is stationary in the quiescent regime, and the solute profile is isotropic about the particle. At $Pe=4$ the fore–aft symmetry in the solute profile is broken, resulting in its steady self-propulsion. Our computations indicate that the self-propulsion speed scales linearly with $Pe-4$ near the onset of self-propulsion, as has been predicted in previous studies. A further increase in $Pe$ gives rise to the stirring regime at $Pe\approx 27$, where the fluid undergoes recirculation, while the particle remains essentially stationary. As $Pe$ is increased even further, the particle dynamics is marked by chaotic oscillations at $Pe\approx 55$ and higher, which we characterize in terms of the mean square displacement and velocity autocorrelation of the particle. Our results for an autophoretic particle under a weak external force are in good agreement with recent asymptotic predictions (Saha, Yariv & Schnitzer, J. Fluid Mech., vol. 916, 2021, p. A47). Additionally, we demonstrate that the strength and temporal scheduling of the external force may be tuned to modulate the chaotic dynamics at large $Pe$.

In this article, we study numerically the dispersion of colloids in a two-dimensional cellular flow in the presence of an imposed mean salt gradient. Owing to the additional scalar, the colloids do not follow exactly the Eulerian flow field, but have a (small) extra-velocity proportional to the salt gradient, $\boldsymbol {v}_{dp}=\alpha \boldsymbol {\nabla } S$, where $\alpha$ is the phoretic constant and $S$ the salt concentration. We study the demixing of an homogenous distribution of colloids and how their long-term mean velocity $\boldsymbol {V_m}$ and effective diffusivity $D_{eff}$ are influenced by the phoretic drift. We observe two regimes of colloids dynamics depending on a blockage criterion $R=\alpha G L/\sqrt {4 D_cD_s}$, where $G$ is the mean salt gradient amplitude, $L$ the length scale of the flow and $D_c$ and $D_s$ the molecular diffusivities of colloids and salt. When $R<1$, the mean velocity is strongly enhanced with $V_m \propto \alpha G \sqrt {Pe_s}$, ${Pe}_s$ being the salt Péclet number. When $R > 1$, the compressibility effect due to the phoretic drift is so strong that a depletion of colloids occurs along the separatrices inhibiting cell-to-cell transport.

The wind energy industry relies on computationally efficient engineering-type models to design wind farms. Typically these models do not account for the effect of atmospheric stratification in either the boundary layer or the free atmosphere. This study proposes a new analytical model for fully developed wind-turbine arrays in conventionally neutral atmospheric boundary layers frequently encountered in nature. The model captures the effect of the free-atmosphere stratification, Coriolis force, wind farm layout and turbine operating condition on the wind farm performance. The model is developed based on the physical insight derived from large-eddy simulations. We demonstrate that the geostrophic drag law (GDL) for flow over flat terrain can be extended to flow over fully developed wind farm arrays. The presence of a vast wind farm significantly increases the wind farm friction velocity compared with flow over flat terrain, which is modelled by updated coefficients in the GDL. The developed model reliably captures the vertical wind speed profile inside the wind farm. Furthermore, the power production trends observed in simulations are reliably reproduced. The wind farm performance, normalized by the geostrophic wind speed, decreases as the free-atmosphere thermal stability increases or the Coriolis force decreases. In addition, we find that the optimal wind farm performance is obtained at a lower thrust coefficient than the Betz limit, which indicates that optimal operating conditions for turbines in a wind farm are different than for a single turbine.

This work focuses on the dynamics of a train of solid particles, separated by a distance $L$, flowing near a deformable interface formed by two co-flowing immiscible fluids in a microchannel of height $h$. Our study includes a systematic analysis of the influence of the governing parameters (fluids viscosity ratio, interface and particle positions, Reynolds $Re$ and capillary $Ca$ numbers and the inter-particle distance $L$) on the hydrodynamic force $f$ exerted on the particle. In the pure inertial regime with non-deformable interfaces $Ca=0$, the particle is driven towards the wall (interface) when the particle is close to the interface (wall). Up to three neutral equilibrium positions $f=0$, two of them stable, are found in this limit. The contrary is obtained in the pure capillary regime $Re=0$. In this limit, we also carried out an asymptotic analysis in the distinguished limits of very large and very small surface tension. In the latter case, the amplitude of the interface deformation induced by the particle is large, comparable to its diameter, but its influence is limited to a small region upstream and downstream of the particle. In the limit of very large surface tension, the amplitude of the interface deformation is small but the presence of the particle modifies the shape of the interface in a region of length $2\lambda$, much larger than the particle diameter $d$. The parameter $\lambda$, introduces an additional characteristic length that determines the asymptotic behaviour of the flow properties in the limit of large surface tension.

This work reveals that the dynamic response of a spherical cap bubble in contact with a rigid wall depends on the effective contact angle at the instant prior to collapse. This parameter allows us to discriminate between two regimes in which the mechanisms of interaction between the collapsing bubble and its surrounding medium differ significantly: when the contact angle is smaller than $90^{\circ }$, a classical jet directed towards the wall is observed, whereas if the initial contact angle is larger than $90^{\circ }$, an annular re-entrant jet parallel to the wall appears. We show that this change of behaviour can be explained using the impulse potential flow theory for small times, which shows the presence of a singularity on the initial acceleration of the contact line when the contact angle is larger than $90^{\circ }$. Direct numerical simulations show that although viscosity regularises the solution at $t > 0$, the solution remains singular at $t=0$. In these circumstances, numerical and experimental results show that the collapse of flat bubbles can eventually lead to the formation of a vortex ring that unexpectedly induces long-range effects. The role of the bubble geometry at the instant of maximum expansion on the overall collapse process is shown to be well captured by the impulse potential flow theory, which can be generalised easily to other bubble shapes. These results may find direct application in the interpretation of geophysical flows as well as the control and design of biomedical, naval, manufacturing and sonochemistry applications.

We study the two-dimensional Rayleigh–Bénard instability subject to the combined effects of a solid–liquid phase boundary and shear using linear stability theory and energy analysis. We consider two thermal states of the solid (isothermal and conducting), and two types of shear that can arise in different contexts. When the melting temperature is equal to that held at the top boundary, three instability modes can arise with the increase of Reynolds number Re that characterizes the shear intensity: a boundary mode, a mixed boundary–bulk mode and a bulk flow mode. When the melting temperature lies between the top and the bottom boundaries, the introduction of Couette flow, independent of its intensity, always leads to the mixed mode, whereas the instability with Poiseuille flow is dominated by the bulk flow mode once Re exceeds a critical value, below which the mixed mode dominates. The energy analysis suggests that there exist two mechanisms by which the shear flow affects the system: one is by inhibiting the upward heat flux and another is by absorbing energy from the perturbed hydrodynamic field. These two mechanisms can play totally different roles in different cases. Results in the high-Re regime indicate that, when Re exceeds its classical threshold, i.e. Re = 5772.2 for Poiseuille flow, the Tollmien–Schlichting instability will be dominant in the present system.

An ‘inverted flag’ – a flexible plate clamped at its trailing edge – undergoes large-amplitude flow-induced flapping when immersed in a uniform and steady flow. Here, we report direct numerical simulations of a related single degree-of-freedom mechanical system: a rigid plate attached at its trailing edge to a torsional spring. This system is termed a ‘rigid inverted flag’ and exhibits the dynamical states reported for the (flexible) inverted flag, with additional behaviour. This finding shows that the flapping dynamics of inverted flags is not reliant on their continuous flexibility, i.e. many degrees of freedom. The rigid inverted flag exhibits additional, novel states including a heteroclinic-type orbit that results in small-amplitude flapping, and a number of chaotic large-amplitude flapping regimes. We show that the various routes to chaos are driven by a series of periodic states, including at least two which are subharmonic. The instability and competition between these periodic states lead to chaos via type-I intermittency, mode competition and mode locking. The rigid inverted flag allows these periodic states and their subsequent interaction to be explained simply: they arise from an interaction between a preferred vortex shedding frequency and a single natural frequency of the structure. The dynamics of rigid inverted flags is yet to be studied experimentally, and this numerical study provides impetus for such future work.

The processing of dense suspensions is a crucial step in many industries including mining; the production of ceramics; and the manufacture of pharmaceuticals. It is widely reported that these suspensions exhibit nonlinear behaviours such as shear thinning and thickening, with particle surface contacts recently being accepted as a primary culprit in the latter. In light of this, the modelling community have started to explore the role of particle surface tribology, predominantly by incorporating Coulombic friction models borrowed from the field of dry granular matter. Full details of the interactions between particle surfaces remain unclear, however, and it is suggested that physical interlocking of particle asperities may be key. Here, we use particle-based simulations to explore explicitly the effect of interlocking on the rheology of dense suspensions of micron-sized solids in a Newtonian fluid. Our simplified model recovers shear thinning, thickening and jamming phenomena commonly seen in experiments.

Dewetting of a substrate, i.e. the liquid film retraction under partial wetting conditions, has been studied extensively over the past decades. This theoretical work deals with the dewetting phenomenon in the presence of liquid evaporation into the pure vapour atmosphere driven by substrate superheating with respect to the saturation temperature corresponding to the vapour pressure. The dynamic profile of the vapour–liquid interface is analysed numerically by using the generalised lubrication approximation that can be applied to an interface slope larger than its conventional counterpart. This approximation uses a phenomenological argument to increase the precision of calculations. For small slopes, it reduces to the conventional lubrication approach. Several nanoscale effects are included in the theory to obtain more precise results: the Kelvin effect, the hydrodynamic slip, the Marangoni effect, the vapour recoil and the interfacial thermal resistance. Their relative importance is discussed. The results include the film shape evolution, the apparent contact angle and the contact line speed, all defined by the substrate superheating and its wetting properties. These dependencies agree with existing experimental results. It is found that the dewetting is accelerated by evaporation. The numerical results are compared to those of the multiscale theory that uses two main parameters, the Voinov angle (the apparent angle obtained within the microregion problem) and the Voinov length. The latter is studied for several fluids as a function of superheating, and an approximate expression for it is proposed. A good agreement between the numerics and the multiscale theory shows the validity of the latter. This means that the dewetting speed in the presence of evaporation can be calculated with an analytical expression that requires only the Voinov angle as a numerical input. Such results are important for many applications, for instance, bubble growth in boiling or film dynamics in heat pipes.

We propose an efficient model for the description of the three-dimensional (3-D) evolution of breaking water waves in the nearshore region. A fundamental property of the model is its intrinsic ability to account for the 3-D dynamics of vorticity and the energy dissipation induced by wave breaking. In particular, the vorticity evolution is achieved through the use of mollified operators, an approach similar in spirit to that adopted in smoothed particle hydrodynamics. Further, since the model is based on depth-semi-averaged equations with a core structure similar to that of nonlinear shallow-water equations, it takes advantage of well-known numerical methods for hyperbolic equations, while permitting computation of local flows. Finally, the model relies on a limited number of tunable parameters and a very simple breaking criterion. All the above aspects allow for a simple and reliable representation of the main features of wave breaking at the time and spatial scales typical of the nearshore wave dynamics. A number of benchmarks are used to explore the properties of the model, which is tuned only once for all cases. Wave height decay rates are well described for both sloshing (thin) and shoaling (thick) spillers, and a good description is also provided of the vorticity field. A final run of an impulsive wave over a submerged breakwater is used to illustrate the representation of the 3-D vorticity dynamics.

Shock tubes are a crucial source of experimental data for the aerothermodynamic modelling of atmospheric entry vehicles. Notably, many chemical-kinetic and radiative models are validated directly against optical measurements from these facilities. Typically, the incident shock speed at the location of the experimental measurement is taken to be representative of the test slug; however, the shock velocity can vary substantially upstream of this location. These variations have been long posited as a source of disagreement with computational predictions, although a definitive link has proved elusive. This work describes a series of experiments which aim to isolate and confirm the importance of the shock deceleration effect. This is achieved by generating different shock trajectories and comparing the post-shock trends in atomic oxygen emission and electron density. These trends are shown to be directly linked to the upstream shock speed variations using a recently developed numerical tool (LASTA). The close agreement of the comparisons confirms the importance of shock speed variation for shock tube experiments; these findings have direct and potentially critical relevance for all such studies, both past and present.

We explore air invasion in a dead-end compliant water-filled microchannel containing a constriction. The phenomenon is driven by the pervaporation of the liquid present in the channel through the surrounding medium. The penetration is intermittent, jerky and characterised by a stop-and-go dynamics as the bubble escapes the constriction. We demonstrate that this sequence of arrest and jump of the bubble is due to an elasto-capillary coupling between the air–liquid interface and the elastic medium. When the interface enters the constriction, its curvature increases strongly, leading to a depressurisation within the liquid-filled channel that drives a compression of the channel. As the interface is forced to leave the constriction at a given threshold pressure, due to the ongoing loss of liquid content by pervaporation, the pressure is suddenly released, which gives rise to a rapid propagation of the air bubble away from the constriction, and a restoration of the rest shape of the channel. Combining macroscopic observations and confocal imaging, we present a comprehensive experimental study of this phenomenon. In particular, the effect of the channel geometry on the time of arrest in the constriction and the jump length is investigated. Our novel microfluidic design succeeds in mimicking the role of inter-vessel pits in plants, which transiently stop the propagation of air embolisms during long and severe droughts. It is expected to serve as a building block for further biomimetic studies in more complex leaf-like architectures, in order to recover this universal phenomenon of intermittent propagation reported in real leaves.

During solid surface impact, a falling drop's energy is transformed into oscillations of its liquid/gas interface. We consider drop deposition during oblique impact in the capillary-ballistic regime characterized by high Reynolds number and moderate Weber number. We treat this as an inverse problem showing that post-impact observations of the frequency spectrum and modal partition of energy allow one to determine a drop's pre-impact characteristics and wetting properties. Our analysis is useful for quantifying contact-line dissipation during inertial spreading and can be used as a diagnostic technique for determining substrate wetting properties.

An axisymmetric squirmer in a Bingham viscoplastic fluid is studied numerically to determine the effect of a yield stress environment on locomotion. The nonlinearity of the governing equations necessitates numerical methods, which are accomplished by solving a variable-viscosity Stokes equation with a finite element approach. The effects of stroke modes, both pure and combined, are investigated, and it is found that for the treadmill or ‘neutral’ mode, the swimmer in a yield stress fluid has a lower swimming velocity and uses more power. However, the efficiency of swimming reaches its maximum at a finite yield limit. In addition, for higher yield limits, higher stroke modes can increase the swimming velocity and hydrodynamic efficiency of the treadmill swimmer. The higher-order odd-numbered squirming modes, particularly the third stroke mode, can generate propulsion by themselves that increases in strength as the viscoplastic nonlinearity increases to a specific limit. These results are closely correlated with the confinement effects induced by the viscoplastic rigid surface surrounding the swimming body, showing that swimmers in viscoplastic environments, both biological and artificial, could potentially employ other non-standard swimming strategies to optimize their locomotion.

Our understanding of homogeneous vapour bubble growth is currently restricted to asymptotic descriptions of their limiting behaviour. While attempts have been made to incorporate both the inertial and thermal limits into bubble growth models, the early stages of bubble growth have not been captured. By accounting for both the changing inertial driving force and the thermal restriction to growth, we present an inertio-thermal model of homogeneous vapour bubble growth, capable of accurately capturing the evolution of a bubble from the nano- to the macro-scale. We compare our model predictions with: (a) published experimental and numerical data, and (b) our own molecular simulations, showing significant improvement over previous models. This has potential application in improving the performance of engineering devices, such as ultrasonic cleaning and microprocessor cooling, as well as in understanding of natural phenomena involving vapour bubble growth.

We investigate the effect of inertial particles dispersed in a circular patch of finite radius on the stability of a two-dimensional Rankine vortex in semi-dilute dusty flows. Unlike the particle-free case where no unstable modes exist, we show that the feedback force from the particles triggers a novel instability. The mechanisms driving the instability are characterized using linear stability analysis for weakly inertial particles and further validated against Eulerian–Lagrangian simulations. We show that the particle-laden vortex is always unstable if the mass loading $M>0$. Surprisingly, even non-inertial particles destabilize the vortex by a mechanism analogous to the centrifugal Rayleigh–Taylor instability in radially stratified vortex with density jump. We identify a critical mass loading above which an eigenmode $m$ becomes unstable. This critical mass loading drops to zero as $m$ increases. When particles are inertial, modes that fall below the critical mass loading become unstable, whereas modes above it remain unstable but with lower growth rates compared with the non-inertial case. Comparison with Eulerian–Lagrangian simulations shows that growth rates computed from simulations match well the theoretical predictions. Past the linear stage, we observe the emergence of high-wavenumber modes that turn into spiralling arms of concentrated particles emanating out of the core, while regions of particle-free flow are sucked inward. The vorticity field displays a similar pattern which leads to the breakdown of the initial Rankine structure. This novel instability for a dusty vortex highlights how the feedback force from the disperse phase can induce the breakdown of an otherwise resilient vortical structure.

A significant challenge in the experimental or computational characterisation of porous bodies and wind turbines is the correction of the obtained flow quantities for wall interference effects. Conventional corrective models are based on the Rankine–Froude theory, which is valid when the body solidity, or turbine induction factor, is sufficiently low. To resolve this issue, this work presents a new corrective model that builds on an extension of the Rankine–Froude theory, valid at arbitrary solidities, coupled with the method of mirror images to account for the existence of channel walls. The predictions of the new model are validated using laboratory and numerical experiments of porous plates and wind turbines. The results show that the new model performs equally as well as conventional ones when the solidity is low, but becomes increasingly more accurate as the latter grows.