Free-surface Gravity Waves Research Articles

Triad resonance of flexural gravity waves in the presence of shear current with constant vorticity

This study examines the formation of triads of flexural gravity wave in a homogeneous fluid within the context of blocking dynamics due to the presence of shear current. This study will enable us to understand the distribution of wave energy on an ice-covered sea surface. New classes of triads for flexural gravity waves are introduced depending on the direction of wave propagation with following and opposing currents. The study reveals that triad formation occurs due to the interaction of flexural gravity waves irrespective of the presence of compression and current, which has not been found in the case of free surface gravity waves. In addition, at most, three triads are formed in the case of flexural gravity waves in the presence of following and opposing currents prior to the threshold of blocking. In contrast, at least three triads are formed for any frequency within the primary and secondary blocking limits for certain values of compressive force and current speed. On the other hand, 11 triads are formed in the presence of uniform current speed as well as in the case of linear shear current with constant vorticity for a certain frequency within the blocking limit for higher values of compressive force and current speed.

Resonant triad interactions of gravity waves in cylindrical basins

We present the results of a theoretical investigation into the existence, evolution and excitation of resonant triads of nonlinear free-surface gravity waves confined to a cylinder of finite depth. It is well known that resonant triads are impossible for gravity waves in laterally unbounded domains; we demonstrate, however, that horizontal confinement of the fluid may induce resonant triads for particular fluid depths. For any three correlated wave modes arising in a cylinder of arbitrary cross-section, we prove necessary and sufficient conditions for the existence of a depth at which nonlinear resonance may arise, and show that the resultant critical depth is unique. We enumerate the low-frequency triads for circular cylinders, including a new class of resonances between standing and counter-propagating waves, and also briefly discuss annular and rectangular cylinders. Upon deriving the triad amplitude equations for a finite-depth cylinder of arbitrary cross-section, we deduce that the triad evolution is always periodic, and determine parameters controlling the efficiency of energy exchange. In order to excite a particular triad, we explore the influence of external forcing; in this case, the triad evolution may be periodic, quasi-periodic or chaotic. Finally, our results have potential implications on resonant water waves in man-made and natural basins, such as industrial-scale fluid tanks, harbours and bays.

Triad resonance of flexural gravity waves in a two-layer fluid within the framework of blocking dynamics

The present study deals with the formation of triads in flexural gravity waves in a two-layer density stratified fluid having a flexible plate-covered surface and an interface within the framework of blocking, which can be used as a mechanism for understanding the spectral distribution of wave energy. The physical model is considered in a two-dimensional Cartesian coordinate system, and the formations of triads are discussed geometrically and validated analytically. The study demonstrates the formation of at most eight triads of three different classes for flexural gravity waves before the threshold of blocking, whereas a maximum of six triads of three different classes have been reported in the case of free-surface gravity waves. However, at least twenty triads are formed for any frequency within the blocking limits for the compressive force lying within the threshold of blocking and buckling limit, irrespective of water depth. On the other hand, 24 triads are formed for a certain frequency in the left neighborhood of the primary blocking point, while thirty triads occur when the frequency is considered in the right neighborhood of the secondary blocking point for specific values of compressive force and density ratio in the case of deep water/short flexural gravity waves in both the layers. In addition, a new class of triad, which is rarely found in the case of surface gravity waves, occurs for a certain frequency within the blocking limits for a suitable choice of compressive force and density ratio in the case of waves in deep/intermediate upper layer depth.

On the sub-harmonic instabilities of three-dimensional interfacial gravity–capillary waves in infinite depths

In order to analyze the stability of gravity–capillary short-crested waves propagating at the interface of two fluids of infinite depths, a numerical procedure has been developed using a collocation method. The three-dimensional wave is generated by an oblique reflection of a uniform wave train arriving at a vertical wall at some angle of incidence θ measured from the perpendicular to the wall. Fixing the reflection at angle θ=45°, and wave steepness at h=0.25, we studied the influence of the density ratio μ and the inverse Bond number δ on the unstable area and the growth rate. It was observed that the unstable regions and the maximum growth rate decrease when μ and δ increase. Also, we found that the dominant instability belongs to the Class Ib as in the case of free-surface gravity waves in deep water.

Effect of incoming gravity waves on the energy extraction efficiency of flapping wing hydroelectric generators

The energy extraction performance of flapping wing hydroelectric generators is subject to the effects of the free surface and incoming gravity waves. The present work addresses this issue by conducting transient numerical simulations of a two-dimensional flapping wing system. The effects of the wave wing phase difference, encounter frequency, and wing submergence depth on the energy extraction performance are investigated. The results indicate that the phase difference is the primary factor affecting the energy extraction efficiency. The incoming gravity wave has the most positive effect on the energy extraction efficiency when it's in the range of 0°–90°. In addition, the energy extraction efficiency increases with the increase in the encounter frequency. Moreover, the periodicity is best when the oscillation frequency of the wing is equal to the encounter frequency. Finally, the influence of the submerged depth on the energy extraction efficiency is also discussed.

A new instability for Boussinesq-type equations

A wide class of problems for free-surface gravity waves fall into a weakly dispersive regime, in which wavelength is large compared to water depth, and wave phase speed differs by a small amount from the speed of shallow-water waves. The resulting problem is treated naturally using Taylor series expansions of dependent variables in the vertical coordinate, leading to a class of models that are collectively referred to here as Boussinesq-type models. Madsen & Fuhrman (J. Fluid Mech., vol. 889, 2020, A38) have recently shown that certain members of this broad class of models are subject to a high-wavenumber instability, which can grow rapidly when the elevation of the wave trough is sufficiently depressed below the mean water surface. This newly revealed instability may provide an explanation for the modelling community’s frequent observations of noisy behaviour in Boussinesq-type model calculations.

Economical Valuation of Wave Power Plant in the Baltic Sea Region at Pre-Flexibility Stage

Abstract The article deals with free surface gravity waves as one of the alternative energy sources and their use. The potential of the waves and their transformation devices are considered as the main energy use impact factors. The goal of the research is to perform a cost-benefit assessment. The following tasks have been set: to make an assessment of wave potential, to identify the conversion model and turbine, to determine economic criteria taking into account wave and price variability. The article also provides the description of the theoretical valuation of the costs of the mentioned turbine power plant.

A rapid numerical method for solving Serre–Green–Naghdi equations describing long free surface gravity waves

A new numerical method for solving the Serre–Green–Naghdi (SGN) equations describing dispersive waves on shallow water is proposed. From the mathematical point of view, the SGN equations are the Euler–Lagrange equations for a ‘master’ lagrangian submitted to a differential constraint which is the mass conservation law. One major numerical challenge in solving the SGN equations is the resolution of an elliptic problem at each time instant. This is the most time-consuming part of the numerical method. The idea is to replace the ‘master’ lagrangian by a one-parameter family of ‘augmented’ lagrangians, depending on a greater number of variables, for which the corresponding Euler–Lagrange equations are hyperbolic. In such an approach, the ‘master’ lagrangian is recovered by the augmented lagrangian in some limit (for example, when the corresponding parameter is large). The choice of such a family of augmented lagrangians is proposed and discussed. The corresponding hyperbolic system is numerically solved by a Godunov type method. Numerical solutions are compared with exact solutions to the SGN equations. It appears that the computational time in solving the hyperbolic system is much lower than in the case where the elliptic operator is inverted. The new method is applied, in particular, to the study of ‘Favre waves’ representing non-stationary undular bores produced after reflection of the fluid flow with a free surface at an immobile wall.

Large-scale numerical modeling of hydro-acoustic waves generated by tsunamigenic earthquakes

Abstract. Tsunamigenic fast movements of the seabed generate pressure waves in weakly compressible seawater, namely hydro-acoustic waves, which travel at the sound celerity in water (about 1500 m s−1). These waves travel much faster than the counterpart long free-surface gravity waves and contain significant information on the source. Measurement of hydro-acoustic waves can therefore anticipate the tsunami arrival and significantly improve the capability of tsunami early warning systems. In this paper a novel numerical model for reproduction of hydro-acoustic waves is applied to analyze the generation and propagation in real bathymetry of these pressure perturbations for two historical catastrophic earthquake scenarios in Mediterranean Sea. The model is based on the solution of a depth-integrated equation, and therefore results are computationally efficient in reconstructing the hydro-acoustic waves propagation scenarios.

A DEPTH-INTEGRATED EQUATION FOR LARGE SCALE MODELING OF TSUNAMI IN WEAKLY COMPRESSIBLE FLUID

Pressure waves generated by fast seabed movement in weakly compressible sea water, namely hydro-acoustic waves travel at the sound celerity in water (about 1500 m/s). These waves are precursors of the counterpart long free-surface gravity waves and contain significant information on the tsunamigenic source. Measurement of hydro-acoustic waves can therefore anticipate the tsunami arrival and significantly enhance the promptness and accuracy of tsunami early warning systems. In this paper derivation of a novel depth-integrated numerical model for reproduction of hydroacoustic waves is presented and the application of this computationally ecient model on two devastating historical tsunamis of Mediterranean Sea in real bathymetry analyzed to reveal the eect of variable bathymetry. On the basis of the model results, some hints for deep sea observatory are given.

Unsteady flow over a submerged source with low Froude number

In the low-Froude number limit, free-surface gravity waves caused by flow past a submerged obstacle have amplitude that is exponentially small. Consequently, these cannot be represented using an asymptotic series expansion. Previous studies have considered linearized steady flow past a submerged source in infinite-depth fluids, in which exponential asymptotics were used to determine the behaviour of downstream longitudinal and transverse free-surface gravity waves. Here, unsteady flow past a submerged source in an infinite-depth fluid is investigated, with the free surface taken to be initially waveless. The source is taken to be weak, and the flow is linearized about the undisturbed solution. Exponential asymptotics are applied to determine the wave behaviour on the free surface in terms of the two-dimensional plan-view, in order to show how the free surface waves evolve over time and eventually tend to the steady solution.

Extreme waves induced by strong depth transitions: Fully nonlinear results

Recent studies on free-surface gravity waves over uneven bathymetries have shown that “rogue” waves can be triggered by strong depth variations. This phenomenon is here studied by means of spectral simulations of the free-surface Euler equations. We focus on the case of a random, one-directional wave field with prescribed statistics propagating over a submerged step, and consider different depth variations, up to an almost deep-to-shallow transition (kpH ≈ 1.8 − 0.78, where kp is the characteristic wavenumber and H the water depth). Strongly non-Gaussian statistics are observed in a region localized around the depth transition, beyond which they settle rapidly on the steady statistical state of finite-depth random wave fields. Extreme fluctuations are enhanced by stronger depth variations. Freak-wave formation is interpreted as a general signature of out-of-equilibrium dynamics, associated with the spectral settling from the deep-water to the finite-depth equilibrium. We also document that during such a transition the wave spectrum shows remarkable similarity with Phillips ω−5-law for the strongest depth variations considered.

Steady gravity waves due to a submerged source

AbstractIn the low-Froude-number limit, free-surface gravity waves caused by flow past a submerged obstacle have amplitude that is exponentially small. Consequently, these cannot be represented using an asymptotic series expansion. Steady linearized flow past a submerged source is considered, and exponential asymptotic methods are applied to determine the behaviour of the free-surface gravity waves. The free surface is found to contain longitudinal and transverse waves that switch on rapidly across curves known as Stokes lines on the free surface. The longitudinal waves are present everywhere downstream of the singularity, while the transverse waves are restricted to two downstream wedges. As the depth of the source approaches the surface, the familiar Kelvin-wedge wave behaviour is recovered.

Dense gravity currents moving beneath progressive free-surface water waves

AbstractThe characteristics of dense gravity currents in coastal regions, where free-surface gravity waves are dominant, have yet to be studied in the laboratory. This paper provides a first insight into the dynamics of dense saline gravity currents moving beneath regular progressive free-surface water waves. The gravity currents were generated by releasing a finite volume of saline into a large wave tank with an established periodic wave field. After the initial collapse, the gravity currents propagated horizontally with two fronts, one propagating in the wave direction and the other against the wave direction. The fronts of the gravity currents oscillated with an amplitude and phase that correlated with the orbital velocities within a region close to the bed. To leading order, the overall length of the gravity current was found to be weakly affected by the wave action and the dynamics of the current could be approximated by simply considering the buoyancy of the released fluid. Other characteristics such as the position of the gravity current centre and the shape of the two leading profiles were found to be significantly affected by the wave action. The centre was displaced at constant speed dependent on the second-order wave-induced mean Lagrangian velocity. For long waves, the centre was advected downstream in the direction of wave propagation owing to the dominance of Stokes drift. For short waves, the gravity current centre moved upstream against the wave direction, as under these wave conditions Stokes drift is negligible at the bed. An asymmetry in the shape of the upstream and downstream current heads was observed, with the gravity current front moving against the waves being much thicker and the front steeper, similar to the case of a current moving in a stream.

A mild-slope model for membrane-coupled gravity waves

A depth averaged equation is derived via variational principle to study the effect of varying bottom on membrane-coupled gravity waves caused by a floating membrane with spatially varying material properties. Variation of gradients for bottom topography, membrane tension and membrane mass density is assumed to be small. The surface energy generated by the membrane deflection contributes to the total energy for membrane-coupled gravity waves. An alternative derivation is also provided by making use of integration by parts. The model equation for the uniformly tensional membrane with uniform mass density has been solved numerically, for the membrane-coupled gravity wave scattering in two dimensions, using mass conserving jump conditions applied at the locations of possible bottom slope discontinuities. This model also describes the reflection characteristics for capillary-gravity waves when the membrane mass density alone is neglected. Reflected amplitudes of membrane-coupled as well as capillary-gravity waves by different bottom topographies are shown graphically for the physical parameters of surface tension, membrane tension and uniform mass of membrane. Moreover, the phenomenon of Bragg resonance is discussed numerically for sinusoidal bottom variations. Further, the model has been utilized to study the effect of depth variations beneath a floating membrane of finite length on the free-surface gravity waves.

Control of Marangoni instability in a layer of variable-viscosity fluid

The Marangoni instability in a temperature-dependent-viscosity fluid layer with deformable free surface under the influence of a linear feedback control strategy is investigated. The exact analytical solution for the stationary modes is developed and the effects of the interaction between variable viscosity, controller gains, free surface deformation, gravity waves and heat transfer mechanism on the onset of Marangoni convection are determined. The destabilizing effect of exponential viscosity variation can be effectively suppressed through small controlled perturbations in the thermal boundary data.

Generation of unsteady waves by concentrated disturbances in an inviscid fluid with an inertial surface

The surface waves generated by unsteady concentrated disturbances in an initially quiescent fluid of infinite depth with an inertial surface are analytically investigated for two- and three-dimensional cases. The fluid is assumed to be inviscid, incompressible and homogenous. The inertial surface represents the effect of a thin uniform distribution of non-interacting floating matter. Four types of unsteady concentrated disturbances and two kinds of initial values are considered, namely an instantaneous/oscillating mass source immersed in the fluid, an instantaneous/oscillating impulse on the surface, an initial impulse on the surface of the fluid, and an initial displacement of the surface. The linearized initial-boundary-value problem is formulated within the framework of potential flow. The solutions in integral form for the surface elevation are obtained by means of a joint Laplace–Fourier transform. The asymptotic representations of the wave motion for large time with a fixed distance-to-time ratio are derived by using the method of stationary phase. The effect of the presence of an inertial surface on the wave motion is analyzed. It is found that the wavelengths of the transient dispersive waves increase while those of the steady-state progressive waves decrease. All the wave amplitudes decrease in comparison with those of conventional free-surface waves. The explicit expressions for the free-surface gravity waves can readily be recovered by the present results as the inertial surface disappears.

Generation of free-surface gravity waves by an unsteady Stokeslet

The interaction of unsteady Stokeslets with the free surface of an initially quiescent incompressible fluid of infinite depth is investigated analytically for two- and three-dimensional cases. The disturbed flows are generated by an unsteady singular force moving perpendicularly downwards away from the surface. The analysis is based on the assumption that the motion satisfies the linearized unsteady Navier–Stokes equations with linear kinematic and dynamic boundary conditions. Firstly, the asymptotic representation for the transient free-surface waves due to an instantaneous Stokeslet is derived for a large time with a fixed distance-to-time ratio. As is well known, the corresponding inviscid waves predicted by the potential theory do not decay to zero as the time goes to infinity. In the present study, the transient waves predicted by the viscous theory eventually vanish due to the presence of viscosity, which is consistent with reality from the physical point of view. Secondly, the asymptotic solutions are obtained for the unsteady free-surface waves due to a harmonically oscillating Stokeslet. It is found that the unsteady waves can be decomposed into steady-state and transient responses. The steady state can be attained as time approaches infinity. It is shown that the viscosity of the fluid plays an important role in the evolution of the singularity-induced waves.

Compact finite volume schemes on boundary-fitted grids

The paper focuses on the development of a framework for high-order compact finite volume discretization of the three-dimensional scalar advection–diffusion equation. In order to deal with irregular domains, a coordinate transformation is applied between a curvilinear, non-orthogonal grid in the physical space and the computational space. Advective fluxes are computed by the fifth-order upwind scheme introduced by Pirozzoli [S. Pirozzoli, Conservative hybrid compact-WENO schemes for shock–turbulence interaction, J. Comp. Phys. 178 (2002) 81] while the Coupled Derivative scheme [M.H. Kobayashi, On a class of Padé finite volume methods, J. Comp. Phys. 156 (1999) 137] is used for the discretization of the diffusive fluxes.Numerical tests include unsteady diffusion over a distorted grid, linear free-surface gravity waves in a irregular domain and the advection of a scalar field. The proposed methodology attains high-order formal accuracy and shows very favorable resolution characteristics for the simulation of problems with a wide range of length scales.

A Runge–Kutta discontinuous Galerkin method for linear free-surface gravity waves using high order velocity recovery

We present a higher order accurate discontinuous Galerkin finite element method for the simulation of linear free-surface gravity waves. The method uses the classical Runge–Kutta method for the time-discretization of the free-surface equations and the discontinuous Galerkin method for the space-discretization. In order to circumvent numerical instabilities arising from an asymmetric mesh a stabilization term is added to the free-surface equations. In combination with a higher order velocity recovery technique this stabilizes the numerical discretization with minimal effect on the accuracy of the wave computations. A stability analysis of the semi and fully-discrete scheme is presented, which suggests that for a suitable choice of the stabilization constant a relatively large time step can be chosen for accurate simulations over a long period of time. Numerical examples of a number of problems are also presented.