Free Surface Waves Research Articles

Passive Suppression of Resonance Vibrations of a Plate and Parallel Plates Assembly, Interacting with a Fluid

We present the results of a series of experiments, which study the effect of the fluid layer height, viscosity and free surface waves on the vibration damping of a thin rectangular plates interacting with a fluid. The suppression technique is based on the shunt circuit connected to the piezoelectric elements. The possibility of using the previously proposed mathematical formulation and approach for calculating the optimal values of the parameters of a RL-circuit in systems with hydrodynamic dissipation is estimated. The experimental results presented confirm its reliability. It has been demonstrated that the efficiency of vibration damping of a single plate structure changes insignificantly with increasing height of the fluid layer except for some interval. At a small amount of the liquid, the waves on its free surface disturb the piezoelectric element reducing its efficiency. Using as an example an assembly of two parallel plates damping of “in-phase” and “out-of-phase” modes has been shown using one location of piezoelement.

Theoretical analysis of nonlinear fluid–structure interaction between large-scale polymer offshore floating photovoltaics and waves

The present research analyzes the nonlinear fluid–structure interaction (FSI) of free surface waves with large-scale polymer offshore floating photovoltaics (LPOFPV). The floating structure is modeled as a nonlinear Euler Bernoulli–von Kármán (EBVK) beam coupling with water beneath. The EBVK theory takes the in-plane force into consideration to account for the moderately large deflection slopes in LPOFPV. A multi-time-scale perturbation method leads to hierarchic partial differential equations by introducing the wave steepness squared as the perturbation. The analytical solution of the proposed nonlinear FSI model is obtained up to the second order. Pontoon structures and LPOFPV are studied and compared. The asymptotic solution provides the expressions of the propagating wave through the coupled system and its frequency–amplitude dispersion relation in a closed-form. A property of the solution is that the progressive plane wave through the coupled system remains linear for small dimensionless amplitudes, and features a second order correction for moderately large dimensionless amplitudes. Furthermore, it is also theoretically proven that no resonance occurs in the considered infinite problem. The proposed approach can be extended to the nonlinear coupling between a EBVK beam and Stokes waves.

Boussinesq modelling of near-trapping in a four-cylinder array

In this paper, complex wave interactions with a four-cylinder array is considered, which has clear relevance to supporting columns of certain nearshore/offshore structures and supports so-called near-trapping phenomenon when subjected to water waves of appropriate period and direction. High free surface elevations and wave loading are expected and thus endanger structure safety. A Boussinesq model is established based on the cut-cell technique for dealing with complex geometries with curved surfaces. The model is validated against the experimental measurements for capturing the near-trapping by a four-cylinder structure in a square arrangement, and is extended to investigate the near-trapping for various scenarios. It is found that the wave periods at which the near-trapping occurs increase with the center-to-center spacings between two adjacent cylinders. Moreover, the re-reflection modes that the waves inside the cylinder array are reflected back and forth among neighboring cylinder pairs can be clearly seen from the flow field streamlines. The energy is trapped inside the array and decays slowly due to wave radiation. This trapped wave would eventually meet with the subsequent incident wave crest, leading to high free surface elevations at near-trapping periods. In view of this, two scenarios are designed to avoid/minimize the near-trapping by breaking the geometry symmetry, hence, the re-reflections modes.

On Stokes wave solutions

Using a perturbation method with the aid of MATLAB®Symbolic Toolbox, a new set of Stokes wave solutions, up to the fifth order, are derived. The new solutions are expressed in terms of free surface profile, velocity potential and wave celerity (or frequency dispersion relation). The solutions in the deep water limit are also deduced and discussed. The new solutions are compared with the existing solutions up to the fifth order. Differences appear among the existing and the present solutions, starting at the third order. The causes for the differences are identified. The characteristic differences were highlighted in the figures for high order solutions of Stokes waves. Numerical examples are provided to quantify the differences between the new fifth-order solutions and Fenton’s, and Skjelbreia and Hendrickson’s solutions.

Multiple-scales analysis of wave evolution in the presence of rigid vegetation

The study of free-surface flows over vegetative structures presents a challenging setting for theoretical, computational and experimental analysis. In this work, we develop a multiple-scales asymptotic framework for the evolution of free-surface waves over rigid vegetation and a slowly varying substrate. The analysis quantifies the balance between the competing effects of vegetation and shoaling, and provides a prediction of the amplitude as the wave approaches a coastline. Our analysis unifies and extends existing theories that study these effects individually. The asymptotic predictions are shown to provide good agreement with full numerical simulations (varying depth) and published experimental results (constant depth).

Numerical investigation of local scour around a vibration pipeline under free surface wave condition

A two-dimensional finite element numerical model was developed to predict local scour around a vibration submarine pipeline under free surface wave condition. The flow governing equations are the two-dimensional Navier-Stokes equations which were closed by using the Shear-Stress Transport (SST) k-ω turbulence model. Both suspended load and bed load sediment transportations are considered in the scour model. The Arbitrary Lagrangian-Eulerian (ALE) method was used to track the moving boundaries induced by the vibration pipeline, the free wave surface, and the evolution of the seabed due to local scour. Satisfactory results were obtained after validations carried out. The numerical model is then used to investigate local scour around a vibration pipeline under free surface wave conditions. Present numerical results show that vibration makes the scour depth below the pipeline deeper, scour scope around the pipeline wider.

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.

Interference effect on tsunami generation by segmented seafloor deformations

Most tsunamis are triggered by seismic events, involving an underwater earthquake with a sharp vertical displacement of the seafloor. Generally, seabed deformations are not unipolar, but instead bipolar or even more complex. To model tsunamis generated by segmented seafloor deformations, an analytical model and a numerical model including source kinematics are presented. Specifically, a parametric analysis, varying the number of segments and the relative sizes of each segment, is performed to investigate their influence on the wave generation process. Destructive interference is observed between the free surface waves generated by adjacent segments moving in opposite directions. The relatively large horizontal velocities compared to vertical velocities within the rotating flow generated in the case with two opposing segments correspond to relatively low wave amplitudes and the shift in the location of the largest wave. When the net uplift or subsidence volume is smaller, the associated destructive interference is more pronounced. In addition, a larger number of opposing segments produce a more complex velocity field within the tsunamigenic region, causing more destructive interference on the free surface. The role of destructive interference induced by segmented seabed deformations should be considered in assessing tsunami hazards caused by complex fault ruptures.

Weakly nonlinear waves in stratified shear flows

<p style='text-indent:20px;'>We develop a Korteweg–De Vries (KdV) theory for weakly nonlinear waves in discontinuously stratified two-layer fluids with a generally prescribed rotational steady current. With the help of a classical asymptotic power series approach, these models are directly derived from the divergence-free incompressible Euler equations for unidirectional free surface and internal waves over a flat bed. Moreover, we derive a Burns condition for the determination of wave propagation speeds. Several examples of currents are given; explicit calculations of the corresponding propagation speeds and KdV coefficients are provided as well.</p>

Research into changes in ship resistance in the shallow water by numerical method

Vietnam has a long coastline and dense network of rivers and canal systems with various and complicated water depths of a waterway. According to data from the Department of Inland Waterways in 2019, Vietnam's waterways transport industry reached 250 million tons/year. In which coastal transport accounted for over 60 million tons/year, promoting the shipping industry and economic integration between regions, making the most of Vietnam's natural coastal conditions. In most general cases, the vessels working in the different water depths of the operational area will significantly affect the ship and wave field resistance. For the transport vessels with low speed, the research of the effect of resistance and wave field has an essential role in ensuring the primary maritime aim of energy saving and energy efficiency. Nowadays, computation fluid dynamics is a practical approach for predicting the ship's hydrodynamic features. However, many computation fluid dynamics types of research have investigated the ship resistance in deep water, which can not consider the effect of changing ship resistance with different water depths. In shallow water, the increasing pressure gradient field leads to an increase in the overall resistance. Therefore, in this study, the volume of fluid (VOF) multiphase model can show the free surface effect and wave elevation in shallow waters. The results clearly showed the success of the computational modeling of multiphase flows around the vessel (air and water). The wavefield in the computation fluid dynamics results is also suitable for related published research experiments. Besides, the resistance and wave elevation around the hull is significantly changing at different depths. In the preliminary design stage, it can apply this work to propose a suitable hull form of transport vessels for Vietnam's waterways.

Energy-harvesting characteristics of flapping wings with the free-surface effect

Flapping-wing devices working in an energy-harvesting mode have the advantage of environmental adaptation. To analyze the energy-extraction characteristics of a flapping wing with the free-surface effect, transient-numerical studies were carried out based on the homogeneous two-phase volume-of-flow model, the shear stress transport k–ω turbulence model, and dynamic-grid technology. These studies took into consideration the influences of the dimensionless submergence depth Sd along with the Froude number Fr. The following results were obtained. (1) In the subcritical condition of Fr < 1, there was a critical hydrofoil submergence depth. When it was greater than this critical value, the existence of the free surface was able to promote the energy-extraction efficiency. On the contrary, the closer the flapping wing was to the free surface, the lower its energy-harvesting efficiency was. (2) When the hydrofoil submergence depth was small, the energy-harvesting efficiency first increased and then decreased with the increase in Fr. Furthermore, the smaller Sd was, the faster the energy-extraction efficiency of the flapping wing decreased. While Sd was large, for example Sd > 9, the energy-extraction efficiency first increased and then gradually approached the unbounded-flow condition as Fr increased, but it was always lower than the unbounded-flow case. (3) Compared with the case of unbounded flow, the existence of the free surface affected the motion of the leading-edge vortex, thereby changing the magnitude and direction of the lift force and pitch moment. The relative position of the free-surface wave crest to the wing also affected the pressure distribution around the flapping-wing surface, which in turn affected the energy-harvesting properties. Additionally, Fr affected the formation and shedding of the vortex around the flapping wing, and the movement synchronization between the leading-edge vortex and the flapping wing was extremely important to the energy-harvesting performance.

Nonlinear liquid sloshing in an upright circular container: Modal responses and higher-order harmonics

Nonlinear sloshing in an upright circular container near the lowest natural frequency is analyzed by using a fully nonlinear overset-mesh-based harmonic polynomial cell method, two weakly nonlinear Narimanov–Moiseev-type multimodal models and a linear multimodal method. Modal responses are extracted from the fully nonlinear results based on a simple but accurate least-square procedure using the time series of free-surface wave elevations, which provides new ways to delve into the underlying modal responses and energy transfer between modes, as well as to verify the validity of ordering assumptions in the weakly nonlinear models. Wavelet analyses are also performed for the wave elevations and generalized coordinates of the modes to better understand the time-frequency information of the higher harmonics of the sloshing responses and energy transfer in a nonlinear process. Planar harmonic sloshing state, swirling harmonic sloshing state, and periodically modulated sloshing state are analyzed. It is found that the energy is more dispersed among different modes in the periodically modulated sloshing state, which means higher natural modes are consequential. In general, energies are found to transfer from lower to higher natural modes and between symmetric and antisymmetric natural modes. The results also show that the O(ε1/3) and O(ε2/3) responses are dominated by only first and second harmonics, respectively, while the O(ε) response contains non-negligible first and third harmonic contribution. At last, the influence of initial disturbance is examined, demonstrating that different initial disturbances may lead to the different rotation direction of the swirling waves and the sloshing-wave responses in the transient stage, while the main characteristics of the sloshing waves are robust and independent of initial conditions.

CFD analysis of added mass, damping and induced flow of isolated and cylinder-mounted heave plates at various submergence depths using an overset mesh method

Fluid–structure interaction processes of heave damping plates in forced vertical motion are investigated using a CFD numerical model. First, a single isolated disk is considered with two submergence depths, including a case where the disk is very close to the mean water level with d/D=1/12 (d is the submergence depth and D the diameter of the disk). Then, the case of a disk attached to a vertical cylinder is analyzed, corresponding to one leg of a semi-submersible floater. The open source software OpenFOAM® is used to model the flow around the disk and to extract the relevant hydrodynamic coefficients of the structures. The dynamics of the structure and induced flow and free surface waves are tackled with the overset mesh tool implemented in OpenFOAM®. Considering the rotational symmetry of the problem, an axisymmetry model is used with wedge symmetry boundary conditions. The results are compared with experimental measurements and linearized potential flow modeling approaches with empirical correction. The CFD results of the models of both structures predict well the experimental measurements, including large oscillation periods and various amplitudes of heaving motion, where the potential model results are deteriorated.

The Close Relationship between Internal Wave and Ocean Free Surface Wave

A numerical model was used to simulate the propagation of internal waves (IW) along the surface layer. The results show that strong water exchange during IW propagation results in strong free surface flow and produces small but distinct free surface waves. We found a close relationship between the internal and ocean surface waves. Our intuitive reaction is that by training the relationship between the water surface wave height and the internal wave waveform, the internal wave waveform can be reversed from the water surface wave height value. This paper intends to validate our intuition. The artificial neural network (ANN) method was used to train the Fluent simulated results, and then the trained ANN model was used to predict the inner waves below by the free surface wave signal. In addition, two linear internal wave equations (I and II) were derived, one based on the Archimedes principle and the other based on the long wave and Boussinesq approximation. The prediction by equation (II) was superior to the prediction of equation (I), which is independent of depth. The predicted IW of the proposed ANN method was in good agreement with the simulated results, and the predicted quality was much better than the two linear wave formulas. The proposed simple method can help researchers infer the magnitude of IW from the free surface wave signal. In the future, the spatial distribution of IW below the sea surface might be obtained by the proposed method without costly field investigation.

Delayed detached eddy simulation method for breaking bow waves of a surface combatant model with different trim angle

Sailing attitudes vary with different loading conditions for ships sailing in open sea, and the trim angle, either by bow or by stern, exerts considerable impact on the flow field around the ship. Currently, attention has primarily been focused on hull resistance and wave patterns while few studies have touched upon breaking bow wave under different trim conditions. This study aims to analyze the breaking bow wave of a surface combatant model with a length of 5.72 m DTMB (name of a surface combatant model) under different sailing attitudes at Fr = 0.35. The Delayed Detached Eddy Simulation (DDES) approach was adopted to study the breaking bow wave features such as plunging jet and air entrainment. The interface-compression algebraic Volume Of Fluid (VOF) method in OpenFOAM was used to capture the free surface. The Computational Fluid Dynamics (CFD) approach was firstly validated by comparing hull resistance and wave contour with the available experimental data. The study then delves into the influence of trim angles on breaking bow waves through simulations of the different conditions, i.e., 1 deg trim by bow, test condition and 1 deg trim by stern. The wave contour, breaking wave profile, vorticity field and the wake field at several transverse sections are then compared in three trim conditions, followed by an analysis of the initial evolution of bow wave breaking based on the turbulent kinetic energy and the vorticity field at four cross sections. The results suggest that trim by bow makes free surface sharper and wave amplitude larger in the breaking bow wave region. The reason is that trim by bow enlarges the attack angle of the bow, thus energizing the bow wave and generating a more violent free surface of bow wave breaking.

Wave Elevation for an Array of Columns Under Wave–Current Interactions

Abstract A wave and current diffraction model is developed based on the potential flow theory and a high-order boundary element method with the successful treatment of singular and nearly singular integrals. The wave–current diffraction from four mounted cylindrical columns are computed, and the free surface wave elevations among the columns are investigated. The influences of the current speed, wave direction, and column spacing on the wave elevation are examined. Ultimately, the presence of a current has a significant influence on the magnitude, spatial location, and occurrence frequency of the maximum wave elevation.

Combination of the finite element method and particle-based methods for predicting the failure of reinforced concrete structures under extreme water forces

We present a combination of the Finite Element Method (FEM), the Particle Finite Element Method (PFEM), and the Discrete Element Method (DEM) for modeling and analyzing the failure of reinforced concrete structures under impulsive wave forces originating from free-surface flows in critical water hazards. The free-surface water flow is modeled with the PFEM, while the structural behavior and the fractures induced by the water forces in the structure are modeled with a coupled FEM–DEM technique. The concrete material behavior is simulated with a standard isotropic damage model. The reinforcing bars are modeled by a rule of mixtures procedure, for simplicity. The possibilities of the new integrated PDFEM approach for predicting the evolution of free-surface tsunami-type waves and their devastating effect on constructions are validated with experiments on the failure of reinforced concrete plates under large impacting waves, performed in a laboratory facility in Japan.

Optimizing wave generation and wave damping in 3D-flow simulations with implicit relaxation zones

In finite-volume-based flow simulations with free-surface waves, wave reflections at the domain boundaries can cause substantial errors in the results. ‘Implicit relaxation zones’ can be used to minimize these reflections, but only if the relaxation zone’s case-dependent parameters are optimized. This work proposes an analytical approach for optimizing these parameters. The analytical predictions are compared against results from 2D-flow simulations for different water depths, wave steepnesses, flow solvers, and relaxation functions, and against results from 3D-flow simulations with strongly wave-reflecting bodies subjected to nonlinear free-surface waves. The present results demonstrate that the proposed approach satisfactorily predicts both the optimum parameter settings and the upper-limit for the corresponding reflection coefficients CR. When optimized as proposed, simulation results for CR were mostly below or equal to the analytical predictions, but never more than 2.2% larger. Therefore, the proposed approach can be recommended for engineering practice. Furthermore, it is shown that implicit relaxation zones can be considered as a special-case of ‘forcing zones’, a family of approaches which includes absorbing layers, damping zones and sponge layers amongst others. The commonalities and differences between these approaches are discussed, including to which extent the present findings are applicable to these other approaches and vice versa.

Geometry Optimization of Heaving Axisymmetric Point Absorbers under Parametrical Constraints in Irregular Waves

The objective of this work is to identify the maximum absorbed power and optimal buoy geometry of a heaving axisymmetric point absorber for a given cost in different sea states. The cost of the wave energy converter is estimated as proportional to the displaced volume of the buoy, and the buoy geometry is described by the radius-to-draft ratio. A conservative wave-height-dependent motion constraint is introduced to prevent the buoy from jumping out of the free surface of waves. The constrained optimization problem is solved by a two-nested-loops method, within which a core fundamental optimization process employs the MATLAB function fmincon. Results show that the pretension of the mooring system should be as low as possible. Except for very small energy periods, the stiffness of both the power take-off and mooring system should also be as low as possible. A buoy with a small radius-to-draft ratio can absorb more power, but at the price of working in more energetic seas and oscillating at larger amplitudes. In addition, the method to choose the optimal buoy geometry at different sea states is provided.

Heat transfer in the seabed boundary layer

We present a theoretical model of the temperature distribution in the boundary layer region close to the seabed. Using a perturbation expansion, multiple scales and similarity variables, we show how free-surface waves enhance heat transfer between seawater and a seabed with a solid, horizontal, smooth surface. Maximum heat exchange occurs at a fixed frequency depending on ocean depth, and does not increase monotonically with the length and phase speed of propagating free-surface waves. Close agreement is found between predictions by the analytical model and a finite-difference scheme. It is found that free-surface waves can substantially affect the spatial evolution of temperature in the seabed boundary layer. This suggests a need to extend existing models that neglect the effects of a wave field, especially in view of practical applications in engineering and oceanography.