Six consecutive solitary waves with identical wave height and separation time are generated to study the flow structures during the uprush–downwash interactions in the swash zone. Using particle image velocimetry, the cross-shore velocity fields are captured. Two different wave conditions are examined with different wave-height-to-water-depth ratios, i.e. $H_o/h=0.11$ and 0.22. The uprush–downwash interaction reaches quasi-steady state from the third solitary wave for both cases. For the former case, a weak non-stationary hydraulic jump appears during the downwash flow for all the six consecutive waves. The weak hydraulic jump evolves into a momentarily ‘stationary’ broken bore when the next wave arrives. For the latter case, the larger wave height generates stronger wave breaking. No non-stationary hydraulic jump is observed as the duration of downwash flow is relatively short. The flow reverses to the onshore direction before the downwash Froude number reaches the hydraulic jump condition. The temporal and spatial evolution of turbulence structure at the quasi-steady state is quantified using the spatial spectral analysis, the integral length scale and turbulence eddy viscosity. The results suggest that the large-scale energy generated during the uprush–downwash interaction modified the slope of the turbulence energy spatial spectrum in the inertial subrange from $-$5/3 to $-$1 in the larger length scale region, indicating the energy cascade depends not only on the dissipation rate, but also on the turbulent kinetic energy from the large-scale turbulence structure because of the large-scale energy injection in the inertial subrange.

Undulating and breaking bores are generated in the laboratory using a programmable long-stroke wavemaker. By changing the stroke length and the speed of the wavemaker, both non-decaying and decaying bores are generated and studied. Bore strength, height and duration are measured and compared with the solutions derived by using the method of characteristics, with excellent agreement. The measurements for inundation depth, runup height and flood duration are checked with the formulas presented in Barranco & Liu (J. Fluid Mech., vol. 915, 2021). The comparisons show that the formulas are also accurate for the non-decaying bores generated by the wavemaker. The maximum inundation depth predicted by the formula for zero bore length at the beach toe agrees with the laboratory observations for decaying bores. Using a high-speed particle image velocimetry system, the ensemble-averaged velocities and fluctuating velocities under undulating bores and breaking bores are measured in constant water depth and in the vicinity of the still water shoreline. Detailed analyses of the velocity fields are presented and discussed. For the undulating bore a long quiescent flood duration is observed, while for the breaking bore the up-rush flow changes into down-rush flow almost linearly.

Both one-dimensional in the horizontal direction (1DH, dispersive and non-dispersive) and two-dimensional in the horizontal direction (2DH) axisymmetric (approximate, non-dispersive) analytical solutions are derived for water waves generated by moving atmospheric pressures. For 1DH, three wave components can be identified: the locked wave propagating with the speed of the atmospheric pressure, $C_p$, and two free wave components propagating in opposite directions with the respective wave celerity, according to the linear frequency dispersion relationship. Under the supercritical condition ($C_p > C$, which is the fastest celerity of the water wave), the leading water wave is the locked wave and has the same sign (i.e. phase) as the atmospheric pressure, while the trailing free wave has the opposite sign. Under the subcritical condition ($C >C_p$) the fastest moving free wave component leads, and its free surface elevation has the same sign as the atmospheric pressure. For a long atmospheric pressure disturbance, the induced free surface profile mimics that of the atmospheric pressure. The 2DH problem involves an axisymmetric atmospheric pressure decaying in the radial direction as $O(r^{-1/2})$. Due to symmetry, only two wave components, locked and free, appear. The tsunami DART data captured during Tonga's volcanic eruption event are analysed. Corrections are necessary to isolate the free surface elevation data. Comparisons between the corrected DART data and the analytical solutions, including the arrival times of the leading locked waves and the trailing free waves, and the amplitude ratios, are in agreement in order of magnitude.

An analytical solution is developed for studying transient water wave-induced responses inside an unsaturated poroelastic seabed of finite thickness. The soil skeleton and the pore fluid are compressible and the constitutive relationship of the soil skeleton is described by Hooke's law. Assuming that the horizontal length scale of wave motion is much larger than the seabed thickness, the leading-order analytical solutions for the seabed responses, including pore fluid pressure and soil skeleton motion, are obtained. The present solutions are suitable for general transient wave loading and for the shear modulus of the soil skeleton being of the same order of magnitude as the effective bulk modulus of elasticity of the pore fluid. The present theory is first validated by checking the solutions with the experimental data for the pore pressure induced by periodic-wave loading. The present analytical solutions are then used to investigate the seabed responses under transient waves, including linear periodic wave, a solitary wave and a bore. The effects of the wave-induced effective stresses on the bed failure potential are further analysed. The results show that the shear failure potential and its duration are highly dependent on the soil properties, such as saturation degree, shear modulus and permeability. Sensitivity analyses are presented.

The depth-integrated wave–current models developed by Yang & Liu (J. Fluid Mech., vol. 883, 2020, A4) are extended to investigate currents with an arbitrary vertical profile in the water column. In the present models, horizontal velocities are decomposed into two components. The first part deduces the prescribed current velocity when waves are absent. The second part is approximated in a polynomial form. The resulting depth-integrated wave–current models are obtained by applying the weighted residual method. In the absence of currents, the present models are identical to those in Yang & Liu (J. Fluid Mech., vol. 883, 2020, A4) and are validated with several three-dimensional (3D) benchmark laboratory experiments. A theoretical analysis is conducted to study the frequency dispersion relation of linear waves on currents with an exponential vertical profile and the results are compared with numerical solutions of the Rayleigh equation. Using the new models, validations and investigations are then conducted for periodic waves and solitary waves on currents with an arbitrary profile in one-dimensional horizontal (1DH) space. Furthermore, the new models are applied to wave–current interactions in two-dimensional horizontal (2DH) space. Two scenarios are considered: (1) wave propagation over a vortex-ring-like current and (2) obliquely incident wave propagation over a 3D sheared current on a varying bathymetry. The vertical and horizontal shear of the current have significant effects on modifying various wave properties, which are well captured by the present models. However, the time-averaged velocity under wave–current interaction shows small differences with the prescribed current velocity, except in the region between the wave trough and crest.

This study investigates water waves generated by a bottom obstacle translating at a subcritical speed in constant water depth, using a combination of analytical and numerical approaches. The newly derived analytical solutions reveal two types of waves – the transient free waves that propagate radially outwards, and the trapped wave that stays on top of the translating bottom obstacle. Closed-form asymptotic solutions for both the free surface and the flow velocities are derived in the far field, and near the leading wave or in the shallow water limit. The far-field leading waves are mathematically shown to be insensitive to the exact shape of the obstacle. Numerical long-wave models are employed to examine effects unaccountable by the linear analytical solutions. Nonlinear effects are found to cause only small deviations from the linear solutions. The effects of the obstacle's acceleration and deceleration are also examined numerically. Overall, the idealised linear analytical solutions predict well the characteristics of water waves generated by a bottom obstacle, and therefore can serve as the cornerstone of a theory-based model for quickly predicting the tsunamis generated by a submarine landslide.

Non-decaying bores are generated in a laboratory using a dam-break system with different reservoir length and depth ratios at the dam-break gate. The experimental data show the dependency of inundation depth, run-up height and flood duration on the reservoir length and the bore strength at the beach toe. Employing the method of characteristics, the relationship between the reservoir length and the bore characteristics at beach toe is obtained. Numerical simulations are carried out for a series of dam-break generated bores, extending the range of physical parameters used in the laboratory experiments. The accuracy of the numerical results is confirmed by the experimental data. Predictive formulae are then obtained for the inundation depth, run-up height and flood duration in terms of the bore characteristics at beach toe and beach slope. Finally, the minimum bore lengths at beach toe necessary to produce the maximum inundation depths and the flooding plateau are identified in the parameter space. These relations can be employed to design dam-break experiments for generating bores with a target length or duration.

Direct numerical simulations are conducted to study the receptivity and transition mechanisms in a solitary wave boundary layer developing over randomly organized wave-like bottom topography. The boundary layer flow shows a selective response to broadband perturbations from the bottom, and develops streamwise-elongated streaks. When the streaks reach high amplitudes, they indirectly amplify streamwise-elongated vortices through modulating small-scale fluctuations and pressure fields. These stronger vortices in turn stir the boundary layer more effectively and further intensify streaks via the lift-up mechanism. This nonlinear feedback loop increases the sensitivity of the boundary layer to the roughness level and yields dramatic variations among cases sharing the same Reynolds number with differing roughness height. Three different local breakdown scenarios are observed depending on the amplitude of the streaks: (i) two-dimensional wave instabilities in the regions with weak streaks; (ii) inner shear-layer instabilities in the regions with moderate-amplitude streaks; and (iii) rapidly growing outer shear-layer instabilities in the regions with highly elevated high-amplitude streaks. Inner instabilities have the slowest growth rate among all transition paths, which confirms the previous predictions on the stabilising role of moderate-amplitude streaks (Önder & Liu, J. Fluid. Mech., vol. 896, 2020, A20).

The stability and transition in the bottom boundary layer under a solitary wave are analysed in the presence of finite-amplitude disturbances. First, the receptivity of the boundary layer is investigated using a linear input-output analysis, in which the environment noise is modelled as distributed body forces. The most ‘dangerous’ perturbations in a time frame until flow reversal are found to be arranged as counter-rotating streamwise-constant vortices. One of these vortex configurations is then selected and deployed to nonlinear equations, and streaks of various amplitudes are generated via the lift-up mechanism. By means of secondary stability analysis and direct numerical simulations, the dual role of streaks in the boundary-layer transition is shown. When the amplitude of streaks remains moderate, these elongated features remain stable until the adverse-pressure-gradient stage and have a dampening effect on the instabilities developing thereafter. In contrast, when the low-speed streaks reach high amplitudes exceeding 15 % of the free stream velocity at the respective phase, they become highly unstable to secondary sinuous modes in the outer shear layers. Consequently, a subcritical transition to turbulence, i.e. bypass transition, can be initiated already in the favourable-pressure-gradient region ahead of the wave crest.

The study of sediment transport requires in-depth investigation of the complex effects of sediment particles in fluid turbulence. In this paper we focus on intense sediment transport flows. None of the existing two-phase models in the literature properly replicates the liquid and solid stresses in the near bed region of high concentration of sediment. The reason for this shortcoming is that the physical processes occurring at the length scale of the particle collisions are different from those occurring at larger length scales and therefore, they must be modelled independently. We present here a two-level theoretical derivation of two-phase, Favre averaged Navier–Stokes equations (FANS). This approach treats two levels of energy fluctuations independently, those associated with a granular spatial scale (granular temperature and small-scale fluid turbulence) and those associated with the ensemble average (turbulent kinetic energy for the two phases). Although similar attempts have been made by other researchers, the two level approach ensures that the two relevant length scales are included independently in a more consistent manner. The model is endowed with a semi-empirical formulation for the granular scale fluid turbulence, which is important even in the dense collisional shear layer, as has been recently recognized. As a result of the large and small scale modelling of the liquid and solid fluctuations, predictions are promising to be reliable in a wide range of flow conditions, from collisional to turbulent suspensions. This model has been validated for steady state flows with intense, collisional or mixed collisional–turbulent sediment transport, using various sources of detailed experimental data. It compares well with the experimental results in the whole experimental range of Shields parameters, better than previous models, although at the cost of increased complexity in the equations. Further experiments on turbulent suspensions would be necessary to definitely assess the model capabilities.

This paper presents a suite of analytical solutions, for both the free-surface elevation and the flow velocity, for landslide-generated water waves. The one-dimensional (horizontal, 1DH) analytical solutions for water waves generated by a solid landslide moving at a constant speed in constant water depth were obtained for the linear and weakly dispersive wave model as well as the linear and fully dispersive wave model. The area enclosed by the landslide was shown to have stronger lasting effects on the generated water waves than the exact landslide shape. In addition, the resonance solution based on the fully dispersive wave model was examined, and the growth rate was derived. For the 1DH linear shallow water equations (LSWEs) on a constant slope, a closed-form analytical solution, which could serve as a useful benchmark for numerical models, was found for a special landslide forcing function. For the two-dimensional (horizontal, 2DH) LSWEs on a plane beach, we rederived the solutions using the quiescent water initial conditions. The difference between the initial conditions used in the new solutions and those used in previous studies was found to have a permanent effect on the generated waves. We further noted that convergence rate of the 2DH LSWE analytical solutions varies greatly, and advised that case-by-case convergence tests be conducted whenever the modal analytical solutions are numerically evaluated using a finite number of modes.

The swash of solitary waves on a plane beach is studied using large-scale experiments. Ten wave cases are examined which range from non-breaking waves to plunging breakers. The focus of this study is on the influence of breaker type on flow evolution, spatiotemporal variations of bed shear stresses and run-up. Measurements are made of the local water depths, flow velocities and bed shear stresses (using a shear plate sensor) at various locations in the swash zone. The bed shear stress is significant near the tip of the swash during uprush and in the shallow flow during the later stages of downrush. In between, the flow evolution is dominated by gravity and follows an explicit solution to the nonlinear shallow water equations, i.e. the flow due to a dam break on a slope. The controlling scale of the flow evolution is the initial velocity of the shoreline immediately following waveform collapse, which can be predicted by measurements of wave height prior to breaking, but also shows an additional dependence on breaker type. The maximum onshore-directed bed shear stress increases significantly onshore of the stillwater shoreline for non-breaking waves and onshore of the waveform collapse point for breaking waves. A new normalization for the bed shear stress which uses the initial shoreline velocity is presented. Under this normalization, the variation of the maximum magnitudes of the bed shear stress with distance along the beach, which is normalized using the run-up, follows the same trend for different breaker types. For the uprush, the maximum dimensionless bed shear stress is approximately 0.01, whereas for the downrush, it is approximately 0.002.

A cross-sectionally averaged one-dimensional long-wave model is developed. Three-dimensional equations of motion for inviscid and incompressible fluid are first integrated over a channel cross-section. To express the resulting one-dimensional equations in terms of the cross-sectional-averaged longitudinal velocity and spanwise-averaged free-surface elevation, the characteristic depth and width of the channel cross-section are assumed to be smaller than the typical wavelength, resulting in Boussinesq-type equations. Viscous effects are also considered. The new model is, therefore, adequate for describing weakly nonlinear and weakly dispersive wave propagation along a non-uniform channel with arbitrary cross-section. More specifically, the new model has the following new properties: (i) the arbitrary channel cross-section can be asymmetric with respect to the direction of wave propagation, (ii) the channel cross-section can change appreciably within a wavelength, (iii) the effects of viscosity inside the bottom boundary layer can be considered, and (iv) the three-dimensional flow features can be recovered from the perturbation solutions. Analytical and numerical examples for uniform channels, channels where the cross-sectional geometry changes slowly and channels where the depth and width variation is appreciable within the wavelength scale are discussed to illustrate the validity and capability of the present model. With the consideration of viscous boundary layer effects, the present theory agrees reasonably well with experimental results presented by Chang et al. (J. Fluid Mech., vol. 95, 1979, pp. 401–414) for converging/diverging channels and those of Liu et al. (Coast. Engng, vol. 53, 2006, pp. 181–190) for a uniform channel with a sloping beach. The numerical results for a solitary wave propagating in a channel where the width variation is appreciable within a wavelength are discussed.

We address two shortcomings in the article by Liu, Park & Cowen (J. Fluid Mech., vol. 574, 2007, pp. 449–463), which gave a theoretical and experimental treatise of the bottom boundary-layer under a solitary wave.

In this paper we present a set of wave flume experiments for a solitary wave reflecting off a vertical wall. A particle tracking velocimetry (PTV) technique is used to measure free-surface velocity and the velocity field in the vicinity of the moving contact line. We observe that the free surface undergoes the so-called rolling motion as the contact line moves up and down the vertical wall, and fluid particles on the free surface almost always flow toward the wall except at the end of the reflection process. As the contact line descends along the wall, wall boundary layer flows move in a downward direction and therefore the boundary layer acts like a conduit through which the surface-rolling-induced flow escapes from the meniscus. However, during the last phase of the reflection process flow reversal occurs inside the wall boundary layer. An approximate analytical solution is developed to explain the flow reversal feature. Very good agreement between the approximate theory and measured data is obtained. Because of the flow reversal, boundary layer flows collide with the surface-rolling-induced flows. The collision gives rise to a jet ejecting from the meniscus into the water body, which later evolves into a small eddy. It is noticed that the fluid particles in different regions such as the free stream, the free-surface boundary layer and the wall boundary layer, can be transported to other regions by passing through the meniscus.

In this paper, a multi-layer model is developed for the purpose of studying nonlinear internal wave propagation in shallow water. The methodology employed in constructing the multi-layer model is similar to that used in deriving Boussinesq-type equations for surface gravity waves. It can also be viewed as an extension of the two-layer model developed by Choi & Camassa. The multi-layer model approximates the continuous density stratification by an -layer fluid system in which a constant density is assumed in each layer. This allows the model to investigate higher-mode internal waves. Furthermore, the model is capable of simulating large-amplitude internal waves up to the breaking point. However, the model is limited by the assumption that the total water depth is shallow in comparison with the wavelength of interest. Furthermore, the vertical vorticity must vanish, while the horizontal vorticity components are weak. Numerical examples for strongly nonlinear waves are compared with laboratory data and other numerical studies in a two-layer fluid system. Good agreement is observed. The generation and propagation of mode-1 and mode-2 internal waves and their interactions with bottom topography are also investigated.

We study the effects of emergent coastal forests on the propagation of long surface waves of small amplitude. The forest is idealized by an array of vertical cylinders. Simple models are employed to represent bed friction and to simulate turbulence generated by flow through the tree trunks. A multi-scale (homogenization) analysis similar to that for seepage flows is carried out to deduce the effective equations on the macro-scale. The effective coefficients are calculated by numerically solving the micro-scale problem in a unit cell surrounding one or several cylinders. Analytical and numerical solutions for wave attenuation on the macro-scale for different bathymetries and coastal forest configurations are presented. For a transient incident wave, analytical results are discussed for the damping of a leading tsunami. For comparison series of laboratory data for periodic and transient incident waves are also presented. Good agreement is found even though some of the measured waves are short or nonlinear.

Theoretical formulation describing the wave generation by a slender solid landslide is presented. The slide motion is coupled with the surface wave dynamics and both the fluid viscous (lubricative) resistance and the solid friction are considered.

Numerical results are obtained for a cosine function shape slide. Numerical results show clearly the importance of the dynamic nonlinear coupling. The effects of lubricative resistance, solid friction, initial landslide speed and the slide height are discussed.

Oscillatory pipe flows of aqueous Carbopol solutions are investigated both experimentally and analytically. Using the PIV technique, the velocity profiles are measured and compared with the numerical solutions based on an elasto-viscoplastic rheological model, in which an elastic spring is serially connected to a regularized Bingham viscoplastic model. The rheological parameters, such as shear modulus of elasticity, yield stress and viscosity, are estimated from steady-shear measurements. Good agreement between the experiments and the model results is observed. It is apparent that the elasticity plays an important role in the unsteady flows of the soft yield-stress fluid studied herein.