Vertical Wave Forces Research Articles

Experimental study of wave diffraction loads on a vertical circular cylinder with heave plates at deep and shallow drafts

This study focuses on wave diffraction loads influenced by heave plates at deep and shallow submerged depths. An extensive experimental campaign is conducted on a fixed vertical and free-surface piercing circular cylinder with five different circular plates, including a perforated plate. The horizontal and vertical wave forces are studied in monochromatic and bichromatic waves. The results show that the heave plate significantly influences the forces varying in time and representative harmonics, including sum- and difference-frequency components. The study found that wave steepness and submerged depth of the plate contribute to the nonlinearity of the loads and that the porous plate reduces the loads compared to the solid plate. Corresponding results from a boundary element method (BEM) solver, HydroStar, provide a reasonable prediction of the harmonics. However, the first harmonic of the vertical force is underpredicted at low frequencies, specifically where the BEM results indicate the cancellation of the forces. Dedicated computational fluid dynamics (CFD) simulations are carried out to investigate the discrepancy, and it is observed that flow separation occurs around the edge of the plate. A simplified approach based on Morison’s equation is employed to model the flow separation effects using a quadratic drag force calculated with a constant drag coefficient. The approach shows that viscous drag is the dominant contributor to vertical wave forces on the heave plate in the low-frequency regime.

Numerical Study of Hydrodynamic Characteristics of a Three-Dimensional Oscillating Water Column Wave-Power Device

The impact of wave-induced forces on the integrity of stationary oscillating water column (OWC) devices is essential for ensuring their structural safety. In our study, we built a three-dimensional numerical model of an OWC device using the computational fluid dynamics (CFDs) software OpenFOAM-v1912. Subsequently, the hydrodynamic performance of the numerical model is comprehensively validated. Finally, the hydrodynamic performance data are analyzed in detail to obtain meaningful conclusions. Results indicate that the horizontal wave force applied to the OWC device is approximately 6.6 to 7.9 times greater than the vertical wave force, whereas the lateral wave force is relatively small. Both the horizontal and vertical wave forces decrease as the relative water depth increases under a constant wave period and height. In addition, the highest dynamic water pressure is observed at the interface between the water surface and device, both within and outside the front wall of the gas chamber. The dynamic water pressure at different locations on the front chamber increases and subsequently decreases as the wave frequency increases.

Flexural-gravity wave forces acting on a submerged spherical object over a flexible sea bed

ABSTRACT A study has been conducted to analyse the impact of an ice sheet on the flexural-gravity wave force acting on a neutrally buoyant, spherical object immersed in a sea in accordance with the linear wave hypothesis. The bottom of the sea is presumed to be protected by a surface composed of some sort of flexibility, and the top is enclosed by an ice sheet. The solution to a diffraction problem is established using the multipole method. The estimated vertical and horizontal flexural-gravity wave forces are visualized in relation to different immersion depths of the spherical body, depths of the fluid and various flexural rigidities of the ice plate and the flexible seabed. The approach adopted here is likely to be of immense significance in the construction of various spherical gadgets like a seabed pressure detector device for tsunami recognition or communication in crisis for a submarine in earthquake zones.

Analysis of Wave-Induced Forces on a Floating Rectangular Box with Analytical and Numerical Approaches

A three-dimensional mathematical hydrodynamic model associated with surface wave radiation by a floating rectangular box-type structure due to heave, sway, and roll motions in finite water depth is investigated based on small amplitude water wave theory and linear structural response. The analytical expressions for the radiation potentials, wave forces, and hydrodynamic coefficients are presented based on matched eigenfunction expansion method (MEFEM). The correctness of the analytical results of wave forces is compared with the construction of a numerical model using the open-source boundary element method code NEMOH. In addition, the present result is compared with the existing published experimental results available in the literature. The effects of the different design parameters on the floating box-type rectangular structure are studied by analyzing the vertical wave force, horizontal wave force, torque, added mass, and damping coefficients due to the heave, sway, and roll motions, and the comparison analysis between the forces is also analyzed in detail. Further, the effect of reflection and transmission coefficients by varying the structural width and drafts are analyzed.

Dynamic Behavior Assessment of OC4 Semi-submersible FOWT Platform Through Morison Equation

This paper proposes an effective inertia coefficient (EIC) in the Morison equation for better wave-force calculations. The OC4 semi-submersible floating offshore wind turbine (FOWT) platform was considered to test the feasibility. Large diffraction at large Keulegan– Carpenter (KC) numbers and the interaction between columns can result in errors in estimating the wave force using the Morison equation with a theoretical inertia coefficient, which can be corrected by the EIC as a function of the wave period and direction. The horizontal and vertical wave forces were calculated using the Morison equation and potential theory at each column, wave period, and wave direction. The EICs of each column were then obtained, resulting in a minimal difference between the Morison inertia force and the wave excitation force by the potential theory. The EICs, wave forces, phase angles, and dynamic motions were compared to confirm the feasibility of an EIC concept under regular and random waves.

Gravity wave interaction with a composite pile-rock breakwater

Abstract Surface gravity wave interaction with a novel composite pile-rock breakwater having a stack of porous plates fixed on its top is investigated in the present study. A novel numerical code based on dual-boundary-element-method is developed to understand the wave scattering and force coefficients within framework of linearized potential flow theory. Out of the four different proposed configurations (pile-rock alone, vertical, horizontal, and H-shaped porous plate assembly with pile-rock), it is found that a novel H-shaped porous plates with submerged pile-rock are very effective in attenuating the wave energy. The parametric study for the H-shaped configuration with several key aspects like porosity of the permeable plates, submergence depth of the horizontal plate, pile-rock relative height and width of the pile-rock barriers are investigated. Increasing relative rock barrier width from 0.25-0.75 offers only a marginal reduction in wave transmission but increases the vertical wave force on the H-plate barrier almost twice. By changing relative submergence of the horizontal porous plate from, it is possible to reduce wave transmission by about 10% but at the expense of increasing vertical wave force almost 50%-75%. Increasing the pile-rock height helps to reduce the wave transmission but significantly increases horizontal wave force and moment on perforated H-shaped barrier. The results of the parametric study can be used for optimizing the dimensions of pile-rock cum porous plate wave barrier for a wide range of field conditions.

Determination of Formulae for the Hydrodynamic Performance of a Fixed Box-Type Free Surface Breakwater in the Intermediate Water

A two-dimensional viscous numerical wave tank coded mass source function in a computational fluid dynamics (CFD) software Flow-3D 11.2 is built and validated. The effect of the core influencing factors (draft, breakwater width, wave period, and wave height) on the hydrodynamic performance of a fixed box-type free surface breakwater (abbreviated to F-BW in the following texts) are highlighted in the intermediate waters. The results show that four influence factors, except wave period, impede wave transmission; the draft and breakwater width boost wave reflection, and the wave period and wave height are opposite; the draft impedes wave energy dissipation, and the wave height is opposite; the draft and wave height boost the horizontal extreme wave force; four influence factors, except the draft, boost the vertical extreme wave force. Finally, new formulas are provided to determine the transmission, reflection, and dissipation coefficients and extreme wave forces of the F-BW by applying multiple linear regression. The new formulas are verified by comparing with existing literature observation datasets. The results show that it is in good agreement with previous datasets.

On the hydrodynamic behaviour of second-order harmonic induced sloshing-mode resonance in moonpool

Fluid resonance in moonpool formed by two-identical rectangular hulls under the regular-wave action is investigated by employing the Computational Fluid Dynamics (CFD) package OpenFOAM®. In this study, the second-order harmonic induced sloshing-mode resonance occurs simultaneously with the first-order induced piston-mode resonance, generating a significant peak value on the free surface amplitudes in moonpool at the corresponding frequency. One period of piston-mode oscillation and two periods of sloshing-mode oscillation can be identified during one incident wave period. It is confirmed that the piston- and sloshing-mode resonances above are generated by the wave frequency and double frequency of incident waves, respectively. The non-monotonic variations of normalized resonant amplitudes in moonpool with incident wave amplitudes can be observed, which is the combined influence of free surface nonlinearity and energy dissipation on wave responses in moonpool. Finally, the second-order harmonic induced sloshing-mode resonance can also affect horizontal wave forces; while it has insignificant effect on vertical wave forces.

Surface gravity wave interaction with a floating dock in the presence of a submerged composite wavy porous plate

A numerical model based on the multi-domain boundary element method is adopted to study surface gravity wave interaction with a finite floating dock in the presence of a submerged composite wavy porous plate in the two-dimensional Cartesian coordinate system. Physical parameters of interest such as reflection coefficient, transmission coefficient, dissipation coefficient, horizontal and vertical wave force coefficients of the wavy plate, vertical force and moment coefficients about the centre of the dock are calculated. The influence of structural parameters such as the number of ripple wavelengths, relative ripple amplitude, structural porosity and relative submergence depth of wavy plate, relative space between the plate and dock are computed and studied. An insight into the comparison of various types of plates such as flat plate, wavy plate and composite wavy plate is also presented. The study reveals that the mitigation of wave force on the dock is significant due to a wavy porous plate. Besides, the study exhibits that placing the wavy porous plate at a finite distance from the floating dock has a significant role in dissipating incident wave energy compared to that placing the submerged plate below the dock. Bragg reflection occurs due to the presence of the submerged wavy plate, which diminishes with an increase in structural porosity and depth of submergence of the wavy plate.

Estimating hurricane-induced vertical surge and wave loads on elevated coastal buildings based on CFD simulations and ensemble learning

This study focuses on developing Machine Learning models based on CFD simulations to estimate hurricane induced vertical wave and surge forces on elevated coastal buildings caused by regular waves. In particular, this study aims at addressing the existing gap in designing elevated coastal buildings, which stems from lack of a generalizable equation for estimating vertical hurricane wave and surge forces for a variety of building geometries and wave conditions. To this end, based on a validated Computational Fluid Dynamics (CFD) model, six key variables that affect the wave forces on buildings are considered, which include building dimensions (length and width of rectangular plan buildings) and wave characteristics. In addition, the effect of building length in relation to wave length on the resulting vertical forces is explicitly considered, while the effect of building width is considered implicitly. Subsequently, a Latin Hypercube Sampling (LHS) strategy is implemented for generating 256 random samples, each representing a unique building geometry with a rectangular plan and wave condition scenario. A pipeline is then developed for generating CFD models corresponding to each sample, which are then deployed on a high-performance computing (HPC) cluster, creating a dataset of vertical surge and wave forces. Based on the key design variables, an ensemble of Machine Learning (ML) models is developed to predict vertical wave forces. For the purpose of design and risk assessment, different force levels may be of interest. Therefore, different models were trained to predict multiple levels of forces on elevated buildings, including maximum, average, 25th, 75th, and 99th percentiles of peak forces. The developed model for each of these force levels is then investigated to identify the main building geometry and wave conditions features that affect the resulting forces, which can inform the decisions of the designers in the early-stage phases. For further assistance in the early-stage design, based on a subset of identified important features, a simple expression for each force level is provided to narrow down the design space to top performing early design candidates, while the main estimators can be used for the final design of elevated buildings. Results show that based on CFD simulations, highly accurate and generalizable predictive models with error rates of about 4% on the test set can be developed.

A study on the interaction between circular or elliptical cross-section submerged floating tunnels and elevation internal solitary waves

Recently, submerged floating tunnels (SFTs) have been widely studied around the world, as it is viewed as one of the alternatives to sea bridges and underwater tunnels for wide and deep sea-crossings. Being submerged in the water, SFTs face the potential safety risk from the internal waves, which contain huge amounts of energy and span across a substantial portion of the water column. To better understand the nonlinear interaction between elevation internal solitary waves (ISWs) and SFTs with different cross-sections in stratified fluids, a numerical model based on the Navier-Stokes equations and density transport equation has been developed. The accuracy of this numerical model is then confirmed through comparisons with both theoretical and experimental results. By applying this model, a comparative study focusing on the three wave forces acting on fixed circular and elliptical SFTs, including horizontal wave force, vertical wave force variation and moment, has been conducted with the consideration of different submergences, incident wave amplitudes and density ratios. Subsequently, a typical tether-constrained SFT system under the wave forces of elevation ISW is studied. The results show motion responses of sway, heave and roll are relatively small compared to the size for both circular and elliptical SFTs, which implies that the elevation ISW-SFT interaction can be effectively limited by the mooring lines. Nevertheless, the motion responses under mooring line constraint are close to the allowable deflection limit for floating bridges. Therefore, the potential for fatigue failure should still be carefully considered, particularly when the SFT encounters internal soliton trains.

Fast Prediction of Solitary Wave Forces on Box-Girder Bridges Using Artificial Neural Networks

The extreme shallow-water waves during a tropical cyclone are often simplified to solitary waves. Considering the lack of simulation tools to effectively and efficiently forecast wave forces on coastal box-girder bridges during tropical cyclones, this study investigates the impacts of solitary waves on box girders and accordingly develops a fast prediction model for solitary wave forces. Computational fluid dynamics (CFD) simulations are used to simulate the hydrodynamic forces on the bridge deck. A total of 368 cases are calculated for the parametric study by varying the submergence coefficients (Cs), relative wave heights (H/h) and deck aspect ratios (W/h). With the CFD simulation results as the training datasets, an artificial neural network (ANN) is trained utilizing the back-propagation algorithm. The maximum wave forces first increase and then decrease with the Cs, while they monotonically increase with H/h. For relatively large H/h and small Cs values, the relationship between the maximum wave forces and W/h presents strong nonlinearities. The observed correlation coefficients between the ANN predictions and the CFD results for the vertical and horizontal wave forces are 98.6% and 98.1%, respectively. The trained ANN-based model shows good prediction accuracy and could be used as an efficient model for the tropical cyclone risk analysis of coastal bridges.

Numerical simulation and fragility analysis of coastal bridges with tension-compression bearings under extreme waves

The bearing connection between the bridge girder and substructure plays a crucial role in resisting wave forces. Previous works pointed out that the box-girder bridge with laminated rubber bearings is susceptible to unseating, sliding, and overturning under extreme waves. This study develops a tension-compression (TC) bearing system to enhance the structural resistance against wave impact. A three-dimensional (3D) finite element (FE) model considering the nonlinear rubber material is established for investigating the dynamic behavior of coastal box-girder bridges, where the wave forces are imported from the computational fluid dynamics (CFD) simulations. By tracking the time histories of the reaction forces and efficient stress, four damage status levels are identified to assess the vulnerability of coastal bridges. To improve calculation efficiency, the Kriging-based surrogate model is adopted for the fragility analysis of coastal box-girder bridges under extreme waves. The results indicate that the lower plate of the bearing on the seaward side (R1) is more prone to yield under the uplifting wave forces, while the upper plate of the bearing on the landward side (R2) is more sensitive to the horizontal forces. The vertical wave force is the key influential factor of bridge damage under extreme waves. In addition, the structural capacity of the bridge with the proposed bearing system is enhanced compared with the one using the traditional laminated rubber bearing.

Influences of the pile-restrained floating breakwater on extreme wave forces of coastal bridge with box-girder superstructure under the action of two-dimensional focused waves

In the complex marine environment, coastal bridges are vulnerable to damage under extreme wave action, and researchers have focused on taking effective measures to reduce structural damage and extreme wave forces in recent years. In this study, a wave-structure coupling method was utilized to numerically investigate the focused wave forces on the two-dimensional coastal bridge with a box-girder superstructure under the influence of the pile-restrained floating breakwater. After a series of validations in the focused wave generation and wave-structure interactions, the effectiveness of floating breakwaters in protecting the box-girder superstructure from extreme wave damage and the influence of control parameters of floating breakwaters on extreme wave forces were conducted. The numerical results demonstrate that the focused wave forces on the box-girder superstructure are effectively reduced after the installation of the pile-restrained floating breakwater. Compared with the floating breakwater with linear stiffness, the maximum horizontal and vertical wave forces are reduced more significantly by the floating breakwater with the displacement constraint. According to the wave parameters of the coastal bridge site, the appropriate size of the pile-restrained floating breakwater and the distance between the breakwater and the superstructure are suggested to improve the reduction effect of focused wave forces on the box-girder superstructure.

Analytical study on an oscillating buoy integrated with a perforated structure with a quadratic pressure drop condition: A case study on a WEC-perforated breakwater integration system

A general analytical method for solving the linear wave interactions with the integration systems comprising an oscillating buoy and perforated structure is presented, with a quadratic pressure drop condition on the perforated surface. Using the method of variable separation and eigenfunction expansion, the motion response of the buoy and total velocity potential can be solved using a two-loop iterative method. The method is validated against the published results. The integration system with a heaving WEC arranged inside a perforated breakwater was investigated particularly. Under the quadratic pressure drop condition, the symmetry of the hydrodynamic coefficients and Haskind relation, are found no longer satisfied, and the dissipation through the perforated wall also contribute to the damping coefficient. Effects of the damping of the power take-off system, front wall porosity, wave amplitude, and position of the WEC are investigated in particular. It was found that the horizontal and vertical wave forces on the WEC increase significantly as wave amplitude increases; however, the hydrodynamic efficiency decreases. When the WEC is near the solid rear wall, the horizontal wave forces on the WEC and solid rear wall increase significantly at the frequencies of piston mode gap resonance and heave motion resonance.

Experimental and theoretical study of internal solitary wave loads on a submerged slender body

Based on the extended Korteweg de Vries (eKdV) theory and Morison's equation, a theoretical model of internal solitary wave (ISW) loads on a submerged slender body is investigated, and experimental measurements are further carried out in a large stratified fluid flume. The results show that the theoretical results are in good agreement with the experimental results. The horizontal force of the ISW on the slender body is consists of the horizontal dynamic pressure force and the friction force, in which the horizontal dynamic pressure force plays a dominant role. The vertical force includes the vertical wave force and the reduced gravity during the interaction between a slender body and an ISW. The magnitude of the force increases with increasing ISW amplitude, while its direction remains unchanged, and the force is closely related to the submerged depth of the slender body in the density pycnocline, which affects both the magnitude and the direction of the force.

Hydrodynamic analyses of multiple porous plates attached to the front of a vertical composite breakwater

Based on the dual boundary element method, hydrodynamic performance analyses including the wave transmission, reflection, dissipation, forces, and moments were conducted on a vertical composite breakwater comprising a submerged rubble-mound breakwater with a vertical wall affixed to its top with horizontal porous plates. These analyses were performed over a wide range of relative water depths, horizontal plate configurations, horizontal plate quantities, widths, porosities, and plate depths of immersion and inclinations. The effects of altering the immersion depth of the vertical barrier and the submerged rubble-mound breakwater height were also investigated. Adding more horizontal plates to the vertical wall was found to help reduce wave transmission, energy dissipation, horizontal wave force, and wave-induced moments; however, this also increased the vertical wave force. Further, increasing the horizontal plate's relative width from 0.25 to 1.0 did not reduce the wave transmission significantly; this also reduced the horizontal wave force considerably, albeit at the expense of increasing the vertical wave force. Based on the results, a medium porosity of approximately 20% is recommended for the horizontal plates to ensure the optimal overall performance. Notably, the results of this study are expected to be useful for the optimal designs of this type of wave barrier.

Numerical Investigation into the Effects of a Viscous Fluid Seabed on Wave Scattering with a Fixed Rectangular Obstacle

We study numerically the effects of a viscous fluid seabed on wave scattering with a solid obstacle of rectangular shape fixed at the free surface, on the seafloor, or internally within the water layer. The computational model is based on OpenFOAM and it is validated using existing analytical solutions for waves encountering an obstacle on a solid bed and available experimental data for waves propagating over a muddy seabed with no obstacles. With the consideration of a solid obstacle on a viscous fluid bottom, we examine the corresponding transformations of incident, reflected, and transmitted wave components. The velocity field near the obstacle and the wave forces exerted on the obstacle are also analyzed. Our simulations show that all wave components experience significant amplitude attenuation caused by the viscous fluid bed. For both surface and bottom obstacles, the presence of an obstacle enhances the damping of reflected waves. When an internally submerged obstacle is considered, transmitted waves are the most affected due to a prominent vortex generated in the lee of the obstacle. Patterns of the velocity field in the vicinity of the obstacle are shown to be controlled mainly by the obstacle with some modulations in magnitude and wavelength contributed by the viscous fluid bed. In view of the vertical wave force on the obstacle surface, both a phase shift and decrease in magnitude are observed. These findings enhance our understanding of the underlying physical processes in the wave–obstacle–mud problems. More studies are still needed in order to provide the necessary technical tools for the engineering design of coastal structures in muddy marine environments.

Numerical simulations of a twin-square marine structure with different submerged depths and gap ratios in waves

The wave-structure interactions for a twin-square structure with different submerged depths and gap ratios are investigated using a 2D numerical tank based on the Unsteady Reynolds-averaged Navier-Stokes equations with the k−ω SST turbulence model. The volume of fluid method is employed to capture the characteristics of the free surface. The wave past a horizontal circular structure is simulated to validate the present numerical model by comparing the numerical results with the experimental data measured by Dixon et al. (1979). Then, the hydrodynamic characteristics of a twin-square structure in waves are investigated using the validated numerical model, including wave forces, free surface elevations and vortex structures. The influences of the twin-square structure’ submerged depth and gap ratio on its hydrodynamic characteristics are studied. The numerical results show that both submerged depth and gap ratio have significant influences on the wave forces acting on and the flow field around the structure. The structure is easily subjected to the wave impact loads when it is located above the still water level. When the gap ratio is less than 0.6, the vertical wave forces on the structure decrease with the increase of the gap ratio. The influence of the gap ratio is considered to be negligible when the gap ratio is larger than 0.6.

Solitary Wave Interaction with a Floating Pontoon Based on Boussinesq Model and CFD-Based Simulations

A mathematical model of solitary wave interaction with a pontoon-type rigid floating structure over a flat bottom is formulated based on Boussinesq-type equations under weakly nonlinear dispersive waves. Based on the higher-order Boussinesq equations, the solitary wave equation is derived, and a semi-analytical solution is obtained using the perturbation technique. On the other hand, brief descriptions of the application of wave2Foam and OceanWave3D on the aforementioned problem are presented. The analytical solitary wave profiles in the outer region are compared with Computational Fluid Dynamics (CFD) and OceanWave 3D model simulations in different cases. The comparison shows a good level of agreement between analytical, wave2Foam, and OceanWave3D. In addition, based on the wave2Foam and coupled OceanWave3D model, the horizontal, vertical wave forces, and the pressure distributions around the pontoon are analysed. Further, the effect of the Ursell number, pontoon length, and water depth on the solitary wave profiles are analysed based on the analytical solution. The paper validates each of the three models and performs intercomparison among them to assess their fidelity and computational burden.