An oscillating body floating at the water surface produces a field of self-generated waves. When the oscillation induces a difference in fore–aft wave amplitude squared, these self-generated waves can be used as a mechanism to propel the body horizontally across the surface (Longuet-Higgins 1977 Proc. R. Soc. Lond. A, vol. 352, no. 1671, pp. 463–480). The optimisation of this wave-driven propulsion is the interest of this work. To study the conditions necessary to produce optimal thrust we will consider a shallow water set-up where a periodically oscillating pressure source acts as the body. In this framework, an expression for the thrust is derived by relation to the difference in fore–aft amplitude squared. The conditions on the source for maximal thrust are explored both analytically and numerically in two optimal control problems. The first case is where a bound is imposed on the norm of the control function to regularise it. Secondly, a more physically motivated case is studied where the power injected by the source is bounded. The body is permitted to have a drift velocity $U$. When scaled with the wave speed $c$, the dimensionless velocity $v=U/c$ divides the study into subcritical, critical and supercritical regimes and the optimal conditions are presented for each. The result in the bounded power case is then used to demonstrate how the modulation of power injected can slowly change the cruising velocity from rest to supercritical velocities.

The interaction of an object with an unsteady flow is non-trivial and is still far from being fully understood. When an aerofoil or hydrofoil, for example, undergoes time-dependent motion, nonlinear flow phenomena such as dynamic stall can emerge. The present work experimentally investigates the interaction between a hydrofoil and surface gravity waves. The waves impose periodic fluctuations of the velocity magnitude and orientation, causing a steadily translating hydrofoil to be susceptible to dynamic stall at large wave forcing amplitudes. Simultaneous measurement of both the forces acting on the hydrofoil and the flow around it by means of particle image velocimetry (PIV) are performed, to properly characterise the hydrofoil–wave interaction. In an attempt at alleviating the impact of the flow unsteadiness via passive flow control, a bio-inspired tubercle geometry is applied along the hydrofoil leading edge. This geometry is known to delay stall in steady cases but has scarcely been studied in unsteady flow conditions. The vortex structures associated with dynamic stall are identified, and their trajectories, dimension and strength characterised. This analysis is performed for both straight- and tubercled-leading-edge geometries, with tubercles found to qualitatively modify the flow behaviour during dynamic stall. In contrast to previous studies, direct measurements of lift do not evidence any strong modification by tubercles. Drag-driven horizontal force fluctuations, however, which have not previously been measured in this context, are found to be strongly attenuated. This decrease is quantified and a physical model based on the flow observations is finally proposed.

This article presents experimental measurements of breaking wave impact loads on a vertical cylinder. The focus is on the influence of some of the breaking wave properties on the measured force. These properties are the distance to breaking, $\delta$, defined as the distance between the breaking location and the front face of the cylinder, and the breaking strength, characterised here by the $\varGamma$ parameter proposed by Derakhti et al. (J. Fluid Mech., 2018, vol. 848, p. R2). The wave characteristics are obtained through numerical simulations of the breaking waves using a fully nonlinear potential flow solver. Seven breaking waves with different breaking strengths have been considered. For each wave, the distance to breaking has been systematically varied and the resulting impact force time-history was measured. It is found that except for the two less intense breaking cases, corresponding to values of $\varGamma$ lower than one, there is a value of $\delta$ for which the magnitude of the impact force is maximum. Small variations of the distance to breaking $\delta$ strongly influence the impact force time-history and its maximum. A linear relationship is observed between the maximum force and the breaking strength $\varGamma$. For the wave cases with values of $\varGamma$ higher than one, the maximum impact force is observed when the distance to breaking $\delta$ is close to 5 % of the wavelength.

A time-domain model of an ice shelf interacting with ocean water in a finite domain is developed, which combines Kirchhoff–Love plate theory with the shallow-water wave equations. In particular, the domain is divided into an open-water region and a region in which the ocean is covered by an ice shelf. Boundary conditions, together with continuity conditions at the ice–water interface, lead to a nonlinear matrix eigenvalue problem, which is solved numerically to obtain the natural modes and frequencies of the system. These form the basis for reconstructing the transient response to wave forcing using a spectral method. Simulations show how wave packets excite multiple modes and generate interference patterns through boundary reflections. Since the method solves the initial value problem in a geometry containing both an open-ocean region and an ice-shelf-covered region, it provides a foundation for simulating sequential break-up of ice shelves due to wave-induced mechanical stresses, and contributes to broader efforts to model ice shelf disintegration under ocean forcing.

The reconfiguration of flexible aquatic vegetation and the associated forces have been extensively studied under two-dimensional flow conditions – such as unidirectional currents, pure waves and co-directional wave–current flows. However, behaviour under more complex, orthogonal wave–current flows remains largely unexplored. In coastal environments, such orthogonal flows arise when waves propagate perpendicular to a longshore current. To improve understanding of how aquatic vegetation helps protect coastlines and attenuates waves, we extended existing effective-length scaling laws that were validated in pure currents, pure waves, and co-directional waves and currents to orthogonal wave–current conditions by introducing new definitions of the Cauchy number. Experiments were conducted in a wave–current basin, where cylindrical rubber stems were mounted on force transducers to measure hydrodynamic forces. Stem velocities were extracted from video recordings to compute the relative velocity between the flow and the stems. Incorporating the phase shift between flow and stem velocities into the force models significantly improved predictions. Comparison of predicted and measured forces showed good agreement for both pure wave and wave–current scenarios, underscoring the importance of phase shifts and velocity reduction for force estimation. Our hypothesised effective-length scaling parameters under wave–current conditions were validated, but with a higher scaling coefficient due to inertial effects from the larger material aspect ratio. These findings offer new insights into the hydrodynamics of flexible structures under complex coastal flow conditions.

We develop a weakly nonlinear theory to revisit the water hammer phenomenon resulting from slow valve manoeuvres. The hydraulic head at the valve is known to be nonlinearly coupled with the flow velocity via a relation derived from Bernoulli’s principle, so that solutions have been so far obtained only via numerical models. The governing equations and boundary conditions indeed yield a nonlinear boundary-value problem, which is here solved using a perturbation approach, Laplace transform and complex analysis. We obtain space- and time-dependent analytical solutions in all of the pipe and validate our results by comparison with standard numerical methods (i.e. Allievi’s method) for determining the exact behaviour of the hydraulic head at the valve. Additionally, we derive algebraic practically relevant closed form expressions for predicting the maximum and minimum hydraulic head values during both valve closure and opening manoeuvres.

This research examines in detail the complex nonlinear forces generated when steep waves interact with vertical cylindrical structures, such as those typically used as offshore wind turbine foundations. These interactions, particularly the nonlinear wave forces associated with the secondary load cycle, present unanswered questions about how they are triggered. Our experimental campaigns underscore the occurrence of the secondary load cycle. We also investigate how the vertical distributions of the scattering force, pressure field and wave field affect the nonlinear wave forces associated with the secondary load cycle phenomena. A phase-based harmonic separation method isolates harmonic components of the scattering force’s vertical distribution, pressure field and wave field. This approach facilitates the clear separation of individual harmonics by controlling the phase of incident waves, which offers new insights into the mechanisms of the secondary load cycle. Our findings highlight the importance of complex nonlinear wave–structure interactions in this context. In certain wave regimes, nonlinear forces are locally larger than the linear forces, highlighting the need to consider the secondary load cycle in structural design. In addition, a novel discovery emerges from our comparative analysis, whereby very high-frequency (over the fifth in harmonic and order) oscillations, strongly correlated to wave steepness, have the potential to play a role in structural fatigue. This new in-depth analysis provides a unique insight regarding the complex interplay between severe waves and typical cylindrical offshore structures, adding to our understanding of the secondary load cycle for applications related to offshore wind turbine foundations.

We numerically investigate the hydrodynamics of an actively heaving flexible foil flapping under a wave surface. The coupled level set and volume-of-fluid method is used to capture the air–water interface, and the immersed-boundary method is used to capture the fluid–structure interaction. A sinusoidal heaving motion is imposed at the foil’s leading edge, and its posterior parts oscillate passively according to its flexible characteristics, allowing dynamic interactions with the wave-induced flow. The propulsive performance of the foil is examined for the influence of three main factors: the ratio of the heaving frequency ($f_{\!f}$) to the wave frequency ($f_w$), the phase difference between the heaving motion and the incident wave ($\mathit \varPhi$) and the submergence depth of the foil ($D$). At $\mathit \varPhi = 0$, the results reveal that the propulsion of the flexible foil benefits from flapping near the wave surface when $f_{\!f}/f_w = 0.5$, and the propulsive efficiency is optimised at $D/L = 1$, where $L$ is the foil’s length. However, when $f_{\!f}/f_w$ = 1.0 and 2.0, the propulsion of the flexible foil is hindered near the wave surface. This hydrodynamic hindrance is closely related to vortex splitting and roll-up phenomena, which induce the formation of a drag wake. By adjusting the phase difference $\mathit \varPhi$, the hindrance in the flexible foil propulsion can be mitigated to enhance propulsive performance. To further understand the relationship between the flapping kinematics and propulsive dynamics, we perform a scaling analysis based on lift force and added mass force, offering good quantification of propulsive performance.

Research on water wave metamaterials based on local resonance has advanced rapidly. However, their application to floating structures for controlling surface gravity waves remains underexplored. In this work, we introduce the floating metaplate, a periodic array of resonators on a floating plate that leverages locally resonant bandgaps to effectively manipulate surface gravity waves. We employ the eigenfunction matching method combined with Bloch’s theorem to solve the wave–structure interaction problem and obtain the band structure of the floating metaplate. An effective model based on averaging is developed, which agrees well with the results of numerical simulation, elucidating the mechanism of bandgap formation. Both frequency- and time-domain simulations demonstrate the floating metaplate’s strong wave attenuation capabilities. Furthermore, by incorporating a gradient in the resonant frequencies of the resonators, we achieve the rainbow trapping effect, where waves of different frequencies are reflected at distinct locations. This enables the design of a broadband wave reflector with a tuneable operation frequency range. Our findings may lead to promising applications in coastal protection, wave energy harvesting and the design of resilient offshore renewable energy systems.

When a water wave group encounters a floating body, it forces the body into motion; this motion radiates waves that modify the wave group. This study considers a floating body in the form of a two-dimensional (2-D) rectangular block constrained to heaving motion. The focus is on how the 2-D block modifies infragravity (IG) waves, a type of nonlinear low-frequency wave in the wave group. The IG waves transmitted beyond the block comprise two types: (i) bound IG waves generated by nonlinear interactions of first-order carrier waves, and (ii) free IG waves released due to discontinuities in flow potential created by the block. A systematic parameter sweep reveals that, when heaving motion is allowed, the transmitted IG waves differ significantly from those of stationary blocks. In some cases, heaving motion enables attenuation of the total transmitted IG waves, while stationary blocks cannot achieve similar effects. Only small-sized blocks are considered; they are ‘small’ compared with the IG wavelengths. The findings are relevant to dual-purpose wave energy converters designed for energy generation and coastal protection, floating breakwaters and other small-sized floating structures such as ships and some icebergs: the heaving motion of these objects may modify IG waves, thereby influencing harbour resonance, near-shore currents, beach erosion, wave forcing on ice shelves and coastal inundation.

We developed a numerical method to investigate the effects of flow properties and phase transition between a gas and a liquid on sloshing-induced impact pressures acting on the walls of a partially filled tank. The conservation equations of mass, momentum and energy, as well as a transport equation for the volume fraction, were solved by considering flow compressibility, surface tension and phase transition. We modelled the phase transition by employing a mass transfer model, and validated our numerical method against experimental data. We investigated the effects of flow compressibility and density ratio between gas and liquid, representing a range similar to that of natural gas and hydrogen. We examined the effects of phase transition on sloshing-induced impact loads caused by a single-impact wave with gas pockets. Compressibility, density ratio and phase transition significantly affected the flow of the liquid–gas interface in the tank and, consequently, the impact pressure. The gas compressibility, caused by a single-wave impact with gas pockets, reduced the impact pressures significantly. Although the influence of density ratio on impact pressures is often emphasised, we demonstrated that, for impacts with gas pockets, the gas density was decisive and not the density ratio. With increasing gas density, the shape of the liquid–gas interface changed, and the pressure peak decreased. For the cases investigated, the viscosity of the liquid phase hardly influenced the impact pressures. Furthermore, the phase change during condensation considerably reduced the impact pressure peak. The pressure fluctuations after the first impact were strongly damped due to the vaporisation process.

The interaction between the flow in a channel with multiple obstructions on the bottom and an elastic ice sheet covering the liquid is studied for the case of steady flow. The mathematical model employs velocity potential theory and fully accounts for the nonlinear boundary conditions at the ice/liquid interface and on the channel bottom. The integral hodograph method is used to derive the complex velocity potential of the flow, explicitly containing the velocity magnitude at the interface. This allows the boundary-value problem to be reduced to a system of nonlinear equations for the unknown velocity magnitude at the ice/liquid interface, which is solved using the collocation method. Case studies are carried out for a widened rectangular obstruction, whose width exceeds the wavelength of the interface, and for arrays of triangular ripples forming the undulating bottom shape. The influence of the bottom shape on the interface is investigated for three flow regimes: the subcritical regime, $F \lt F_{{cr}}$, for which the depth-based Froude number is less than the critical Froude number, and the interface perturbation decays upstream and downstream of the obstruction; the ice-supercritical and channel-subcritical regime, $F_{cr} \lt F \lt 1$, for which two waves of different wavelengths extend upstream and downstream to infinity; and the channel-supercritical regime, $F \gt 1$, for which the hydroelastic wave extends downstream to infinity. The results revealed a trapped interface wave above the rectangular obstruction and the ripple patch. The resonance behaviour of the interface over the undulating bottom occurs when the period of ripples approaches the wavelength of the ice/liquid interface.

This study explores interfacial waves in a three-layer fluid system, focusing on the coupling effects between the two interfaces. These effects include resonance induced by inertial coupling and damping caused by viscous coupling. A linear theoretical framework is developed to describe the coupled wave motion and evaluate the impact of interfacial coupling under viscous damping. Additionally, a semi-analytical model is introduced to accurately capture resonance frequency shifts and phase differences due to viscosity. The spiral structure of interfacial waves predicted by the models is confirmed experimentally using the background oriented Schlieren (BOS) method. Further, the model is validated by excellent agreement between theoretical predictions and ultrasonic measurements of wave amplitudes and phase differences. Finally, the study examines mechanical coupling and energy transfer between interfaces under external forcing, elucidating the formation of spiral waves. The accurate treatment of viscous boundary conditions by the semi-analytical model also enables its extension to multilayer fluid systems.

The problem of a uniform current interacting with bodies submerged beneath a homogeneous ice sheet is considered, based on linearised velocity potential theory for fluid and elastic thin plate theory for ice sheet. This problem is commonly solved by the boundary element method (BEM) with the Green function, which is highly effective except when the Green function becomes singular, and the direct solution of the BEM is no longer possible. However, flow behaviour, body force and ice sheet deflection near the critical Froude numbers are of major practical interest, such as in ice breaking. The present work successfully resolves this challenge. A modified boundary integral equation (BIE) is derived, which converts the singular Green function term to a far-field one and removes the singularity. The BIE is then imposed at infinity for additional unknowns in the far field. It is proved that the solution is finite and continuous at the critical Froude number $F = F_c$, where the body starts generating travelling waves, and finite but discontinuous at depth-based Froude number $F = 1^\pm$. Case studies are conducted for single and double circular cylinders and an elliptical cylinder with various angles of attack. A comprehensive analysis is made on the hydrodynamic forces and the generated flexural gravity wave profiles, and their physical implications are discussed. It is also concluded that the method developed in this paper is not confined to the present case but is also applicable to a variety of related problems when the BEM fails at the critical points.

We report on the experimental and theoretical characterisation of shallow water wave guiding along a curved wave guide. A curved beam of fixed height and width positioned at the bottom of a wave tank generates an effective step-like perturbation which can guide surface water waves. We construct a linear wave theory for this wave propagation and characterise the parameter region where wave guiding can develop, as well as the possible guided modes, their profile and propagation constant. The theoretical analysis is supported by experimental surface wave data. A good agreement is found between experimental data and theoretical predictions, which gives insight into the possible harnessing of wave-guiding phenomena for energy harvesting.

Linear hydroelastic waves propagating in a frozen channel are investigated. The channel has a rectangular cross-section, finite depth and infinite length. The liquid in the channel is an inviscid, incompressible liquid and covered with ice. The ice is modelled as a thin elastic plate of variable thickness clamped to the channel walls. The thickness is constant along the channel length and varying across it. The flow induced by ice deflections is potential. The problem reduced to a problem of wave profiles across the channel and was solved using a piecewise linear approximation of a shape of the thickness. Normal modes are calculated to ensure continuous deflections, slopes, bending stresses and shear forces in the ice plate. Two thickness distributions are studied: in Case I, the thickness is constant at a middle segment and linearly increases at edge segments over the channel’s width; in Case II, the thickness linearly decreases at the edge segments. In Case I, there is one segment with the thin part of the ice cover where, as expected, the oscillations of the ice plate will be concentrated. In Case II, there are two such areas, separated by the middle segment with the thick part of the ice cover. Dispersion relations, phase and group velocities, wave profiles and strain distributions in the ice plate are studied. Results show that the properties of periodic hydroelastic waves are significantly influenced by the ice thickness distribution across the channel.

Breaking wave impacts on rigid structures have been extensively studied, yet the role of structural elasticity in shaping the impact and response remains insufficiently understood. In this study, we experimentally investigate the hydroelastic behaviour of a vertical cantilever plate subjected to multimodal solitary breaking wave impacts. The plate is mounted near the still water level on a 1 : 10 sloping beach, and the wave height-to-depth ratio ($H/h$) is varied from 0.15 to 0.40 to systematically control the impact type from non-breaking to highly aerated wave impacts. We show that aeration significantly affects hydroelastic impacts. The spatio-temporal extent of the impact pressure on the elastic plate increases with air entrapment, while the peak pressure becomes highly sensitive as the wave approaches the flip-through regime. Pressure oscillations associated with bubble formation induce high-frequency structural vibrations, particularly under low-aeration conditions. Furthermore, we find that the elasticity has a limited effect on the peak pressure, impact duration and impulse, but increases the maximum quasi-hydrostatic force on the plate for the scenarios investigated. Following the impact, two distinct free-top deflections are identified, i.e. a deflection $\Delta x_{\textit{imp}}$ with high acceleration induced by the impact pressure and a deflection $\Delta x_{{hp}}$ with high magnitude caused by the maximum quasi-hydrostatic pressure. These deflections scale with the Cauchy number as $\Delta x_{\textit{imp}}/l \sim Ca_{\textit{imp}}/6$ and $\Delta x_{{hp}}/l \sim Ca_{{hp}}/12$ (where l is the plate length), exhibiting parabolic and linear trends with $H/h$, respectively. This work presents a benchmark dataset and introduces a predictive law for structural deflection, providing practical insights into hydroelastic effects across various impact regimes.

Waves propagating over oscillating periodic structures can be reflected and attenuated either by Bragg scattering or by local resonance. In this work, we focus on the interplay between surface gravity waves and submerged resonators, investigating the effect of the local resonance on wave propagation. The study is performed using a state of the art numerical simulation of the Navier–Stokes equation in two-dimensional form with free boundary and moving bodies. A volume of fluid interface technique is employed for tracking the free surface, and an immersed boundary method for the fluid–structure interaction. A wave maker is placed at one end of the flume and an absorbing beach at the other. The evolution in space of a monochromatic wave interacting with up to four resonators coupled only fluid mechanically is presented. We evaluate the efficiency of the system in terms of wave amplitude attenuation and energy transfers between the fluid and the solid phase. The results indicate that, near resonance conditions, both wave reflection and energy dissipation increase significantly. Conversely, far from resonance, waves can propagate through the system with minimal dissipation, even in the presence of numerous resonators. Moreover, when the time scale associated with the resonator’s restoring force is longer than the wave period, the resonators tend to follow the wave motion, oscillating with an amplitude comparable to that of the wave. In contrast, when the two time scales are similar, the resonator motion becomes amplified, resulting in stronger velocity gradients and enhanced viscous dissipation.

The nonlinear disturbance caused by either a localised pressure distribution moving at constant speed on the free surface of a liquid of finite depth or a flow over a topographic obstacle, is investigated using (i) the weakly nonlinear forced Kadomtsev–Petviashvili equation which is valid for depth-based Froude numbers near unity and (ii) the fully nonlinear free-surface Euler system. The presence of a steady v-shaped Kelvin wave pattern downstream of the forcing is established for this model equation, and the wedge angle is characterised as a function of the depth-based Froude number. Inspired by this analysis, it is shown that the wake can be eliminated via a careful choice of the forcing distribution and that, significantly, the corresponding nonlinear wave-free solution is stable so that it could potentially be seen in a physical experiment. The stability is demonstrated via the numerical solution of an initial value problem for both the model equation and the fully nonlinear Euler system in which the steady wave-free state is attained in the long-time limit.

A small sphere fixed at various drafts was subjected to unidirectional broad-banded surface gravity wave groups to investigate nonlinear exciting forces. Testing several incident wave phases and amplitudes permitted the separation of nonlinear terms using phase-based harmonic separation methods and amplitude scaling arguments, which identified third-order forces within the wave frequency range, i.e. third-order first-harmonic forces. A small-body approximation with instantaneous volumetric corrections reproduced the third-order first-harmonic heave forces very well in long waves, and at every tested draft. Further analysis of the numerical model shows these effects are primarily due to instantaneous buoyancy changes, which for a spherical geometry possess a cubic relationship with the wave elevation. These third-order effects may be important for applications such as heaving point absorber wave energy converters, where they reduce the first-harmonic exciting force by ${\sim} 10\, \%$ in energetic operational conditions, an important consideration for power capture.