Linear Potential Theory Research Articles

Modelling of a Three-Body Hinge-Barge Wave Energy Device Using System Identification Techniques

In order to increase the prevalence of wave energy converters (WECs), they must provide energy at competitive prices, especially when compared with other renewable energy sources. Thus, it is imperative to develop control system technologies that are able to maximize energy extraction from waves, such that the delivered energy cost is reduced. An important part of a model-based controller is the model that it uses. System identification techniques (SITs) provide methodologies to get accurate dynamic models from input-output data. However, even though these techniques are well developed in other application areas, they are seldom used in the context of WECs. This paper proposes several strategies based on SIT to get a linear time-invariant model for a three-body hinge-barge wave energy device using experimental data. The main advantage of the model obtained with this methodology, against other methods such as linear potential theory, is that this model remains valid even for relatively large waves and WEC displacements. Other advantages of this model are its simplicity and the low computational resources that it needs. Numerical simulations are carried out to show the validation of the obtained model against recorded experimental data.

Comparison of wave–structure interaction dynamics of a submerged cylindrical point absorber with three degrees of freedom using potential flow and computational fluid dynamics models

In this paper, we compare the heave, surge, and pitch dynamics of a submerged cylindrical point absorber, simulated using potential flow and fully resolved computational fluid dynamics (CFD) models. The potential flow model is based on the time-domain Cummins equation, whereas the CFD model uses the fictitious domain Brinkman penalization technique. The submerged cylinder is tethered to the seabed using a power take-off (PTO) unit, which restrains the heave, surge, and pitch motions of the converter and absorbs energy from all three modes. It is demonstrated that the potential theory overpredicts the amplitudes of heave and surge motions, whereas it results in an insignificant pitch for a fully submerged axisymmetric converter. It also underestimates the slow drift of the buoy, which the CFD model is able to capture reliably. Furthermore, we use fully resolved CFD simulations to study the performance of a three degrees of freedom cylindrical buoy under varying PTO coefficients, mass density of the buoy, and incoming wave heights. It is demonstrated that the PTO coefficients predicted by the linear potential theory are sub-optimal for waves of moderate and high steepness. The wave absorption efficiency improves significantly when a value higher than the predicted value of the PTO damping is selected. Simulations with different mass densities of the buoy show that converters with low mass densities have an increased tension in their PTO and mooring lines. Moreover, the mass density also influences the range of resonance periods of the device. Finally, simulations with different wave heights show that at higher heights, the wave absorption efficiency of the converter decreases and a large portion of available wave power remains unabsorbed.

Analysis of free surface oscillations of a droplet due to ultrasonic wave impingement

An analytical approach based on the linear potential theory is employed to enlighten the fundamental physics of atomization of droplets with an impinging sound wave, with a particular application in surface acoustic wave (SAW) atomization. When a plane sound wave, originated from the gas or the liquid side (resembling SAW), impinges on a liquid droplet, capillary waves are generated. It is shown that, for both cases, spatial phase-locking between the sound spherical modes and the free surface oscillations occurs. Hence, capillary waves will have the same spatial modes of the sound wave. The frequency spectrum analysis shows that the phase-locking causes two types of waves: the natural capillary waves with a wide range of frequencies, two to five orders of magnitude smaller than the impinging sound wave, and the forced wave, with a frequency equal to that of the sound wave. Since the instability of these surface waves leads to separation of droplets from the surface and the size of these droplets is correlated with the wavelength of the surface waves, this well explains the previous observations that droplets with a wide range of sizes are generated in the SAW experiments. Finally, a correlation is also proposed for predicting the atomized droplet size, which gives the size order for the generated droplets in SAW with good accuracy. The correlation could also suggest the possible size for remote atomization of the droplets by sound wave propagated in gas.

Influence of structural parameters on pitching stability of a vertical shiplift

To analyze the pitching stability of the fully balanced hoist vertical shiplift, this paper develops a fluid velocity potential and dynamical model of the vertical shiplift based on an improved energetic exponent method, to study the influence of the primary structural parameters on the pitching stability of an actual shiplift installed in GuoPitan shiplift (GuiZhou province, China). With the linearized potential theory and the undetermined coefficient method, the numerical analytical expression of the velocity potential is obtained by discussing the pitching motion of ship chamber. Applying the Lagrange equation of the second kind to establish a dynamical model of the shiplift having multiple degrees of freedom, the energetic calculated model for ship chamber in respect to the pitching motion is proposed. To accurately analyze the variation of the pitching stability, this paper develops the slope of energetic varied curve, based on the orthogonal polynomial approximation and the subsection averaging method. Also, the influence of the parameters is studied, including the distance of lifting point, the synchronous synchronous shaft stiffness, and the hoist radius, with the optimized design of these three structural parameters: a = 40.0 m, C = 2.0 × 109 N/m, and R = 1.8 m. The changing range of the absolute value of energetic exponent for the distance of lifting points is the largest of the three structural parameters (5.00–12.80), indicating that the distance of lifting points has the most remarkable influence on the pitching stability. Moreover, by the Runge–Kutta algorithm, the response of the angle displacement is analyzed before and after optimization, and then the reliability of results is demonstrated. The four angle displacements at the peak values close to 300 s decrease by 11.11%, 13.33%, 15.91%, and 10.86% respectively after optimization. It reveals that the pitching stability of shiplift is enhanced evidently after optimization.

Theoretical Analysis of a Vertical Cylindrical Floater in Front of an Orthogonal Breakwater

This study investigates the effect of an orthogonal-shaped reflecting breakwater on the hydrodynamic characteristics of a vertical cylindrical body. The reflecting walls are placed behind the body, which can be conceived as a floater for wave energy absorption. Linear potential theory is assumed, and the associated diffraction and motion radiation problems are solved in the frequency domain. Axisymmetric eigenfunction expansions of the velocity potential are introduced into properly defined ring-shaped fluid regions surrounding the floater. The hydrodynamic interaction phenomena between the body and the adjacent breakwaters are exactly taken into account by using the method of images. Results are presented and discussed concerning the exciting wave forces on the floater and its hydrodynamic coefficients, concluding that the hydrodynamics of a vertical cylindrical body in front of an orthogonally shaped breakwater differ from those in unbounded waters.

An open-source library for hydrodynamic simulation of marine structures

The work focuses on an open and collaborative approach for hydrodynamic simulations of multibody operations. It builds on Vessel.js, an existing web-based ship design library, by modeling the interaction between entities and creating multibody models able to output different responses. To develop the cases here studied, the simulations are decomposed into single elements to understand their behavior separately before making them interact with other elements to create a multibody simulation. In the process, different hydrodynamic models are used to analyze the bodies according to the requirements of the simulations and the needed level of complexity. The simulations are coded in JavaScript and visualized in a web environment, with the option of using external hydrodynamic analyzes, which in this work were exemplified using a commercial software that adopts the linear potential wave theory. The paper concludes with a discussion about future applications of methods and simulations.

Performance estimation of resonance-enhanced dual-buoy wave energy converter using coupled time-domain simulation

This paper presents the modeling methodology and performance evaluation of the resonance-enhanced dual-buoy WEC (Wave Energy Converter) by HEM (hydrodynamic & electro-magnetic) fully-coupled-dynamics time-domain-simulation program. The numerical results are systematically compared with the authors’ 1/6-scale experiment. With a direct-drive linear generator, the WEC consists of dual floating cylinders and a moon-pool between the cylinders, which can utilize three resonance phenomena from moon-pool dynamics as well as heave motions of inner and outer buoys. The contact and friction between the two buoys observed in the experiment are also properly modeled in the time-domain simulation by the Coulomb-friction model. Moon-pool resonance peaks significantly exaggerated in linear potential theory are empirically adjusted through comparisons with measured values. A systematic comparative study between the simulations and experiments with and without PTO (power-take-off) is conducted, and the relative heave displacements/velocities and power outputs are well matched. Then, parametric studies are carried out with the simulation program to determine optimum generator parameters. The performance with various wave conditions is also assessed.

Effects of ice-floe on surface wave interaction with an irregular flexible seabed

The hydro-elastic response of incident wave with a horizontal floating ice-floe on a sea is studied by using linear potential theory. The lower side of the sea is bounded by an adaptable layer with small base distortion. Here, the ice-floe and adaptable bottom surface are modeled as lean uniform elastic plates. In such circumstances, there are two types of time-harmonic proliferating waves that exist in the fluid region. The waves with smaller wavenumber propagate beneath the ice-floe (ice-floe wave mode) and the other propagates along the adaptable sea-bed (adaptable base wave mode). Utilizing perturbation analysis, the initial problem is transferred to a first-order problem. The reduced problem is resolved by using the Fourier transform approach. The first-order approximations of the reflection and transmission coefficients are acquired for both the wave modes. For a particular example of a sinusoidal patch with two distinct wavenumbers is taken up to validate the results. The key benefit of the recent study is that the estimations of reflection and transmission coefficients in the case of both the wave modes are shown to establish the energy relation almost successfully.

A non-overlapping Schwarz domain decomposition method with high-order finite elements for flow acoustics

A non-overlapping domain decomposition method is proposed to solve large-scale finite element models for the propagation of sound with a background mean flow. An additive Schwarz algorithm is used to split the computational domain into a collection of sub-domains, and an iterative solution procedure is formulated in terms of unknowns defined on the interfaces between sub-domains. This approach allows to solve large-scale problems in parallel with only a fraction of the memory requirements compared to the standard approach which is to use a direct solver for the complete problem. While domain decomposition techniques have been used extensively for Helmholtz problems, this is the first application to aero-acoustics. The optimized Schwarz formulation is extended to the linearized potential theory for sound waves propagating in a potential base flow. A high-order finite element method is used to solve the governing equations in each sub-domain, and well-designed interface conditions based on local approximations of the Dirichlet-to-Neumann map are used to accelerate the convergence of the iterative procedure. The method is assessed on an academic test case and its benefit demonstrated on a realistic turbofan engine intake configuration.

Analytical and experimental studies on water wave interaction with a submerged perforated quarter-circular caisson breakwater

Water wave interaction with a submerged perforated quarter-circular caisson breakwater is investigated in this study. Based on linear potential theory, analytical solutions of the problem are developed for normally and obliquely incident waves, respectively. In the solving procedure, the entire fluid domain is divided into an external region and two quarter-circular inner regions. The series solutions of the velocity potential in the external region are given through multipole expansions, and those in the quarter-circular regions are developed by the separation of variables. A quadratic pressure drop condition is imposed on the perforated quarter-circular arc to model the wave energy dissipation. The expansion coefficients in velocity potentials are determined by matching the conditions on the boundaries between adjacent regions. The calculation methods of the reflection, transmission and energy loss coefficients as well as the horizontal and vertical wave forces on the breakwater are given. Laboratory experiments are carried out to measure the reflection and transmission coefficients of the breakwater. The convergence of the analytical solutions is good, and the calculation results with at least three-figure effective accuracy are obtained. The analytical results agree reasonably with the experimental data. The hydrodynamic quantities of the breakwater are carefully examined by presenting calculation examples. The analysis results can provide valuable references for the application of perforated quarter-circular caisson breakwaters.

Oblique wave scattering by porous breakwaters/seawalls: Novel analytical solutions based on contour integral without finding complex roots

This article develops novel analytical solutions to oblique wave scattering by porous rubble-mound structures based on linear potential theory. The novelty of the solutions is that the usual complex root finding algorithm is avoided by applying a contour integral technique. In the solving procedure, the whole fluid domain is divided into multiple regions, and the series solution of the velocity potential in each region is obtained using the separation of variables. A linear equation system for the expansion coefficients is achieved by matching the interface conditions between adjacent regions. The matrix elements, which initially include the complex roots of complex dispersion relations, are evaluated by applying the contour integral technique. The evaluation process is free of complex roots, and thus the difficulties arising in solving the complex dispersion relations in traditional solutions are avoided. Case studies show that the series solution converge rapidly, and the calculation results of the new solutions agree well with those of previous analytical solutions with complex roots and multi-domain boundary element method (BEM) solutions. Besides providing a simple and elegant approach for analyzing the hydrodynamic performance of porous structures, the new solutions can be extended to water wave problems involving other dissipative coastal structures.

Analytical study of exciting forces acting on a rigid sphere in a fluid with flexible base surface

AbstractA three‐dimensional problem of wave diffraction by a neutrally buoyant immersed sphere in a fluid having a flexible bottom surface is studied within the framework of linear potential theory. The flexible base surface of the fluid is modelled as a narrow flexible sheet and it follows on the Euler‐Bernoulli bar condition. The impact of the free‐surface tension is ignored. In such circumstances, the time‐harmonic proliferating waves with two distinct wavenumbers propagate in the fluid for any specific frequency. The proliferating waves having higher wavenumber spread along the flexible bottom surface, say, flexural mode and the proliferating waves having smaller wavenumber spread along the free‐surface of the fluid, say, free‐surface mode. By applying the multipole technique, we acquire the analytical solution for the issue of wave diffraction by a rigid sphere immersed in a fluid having flexible bottom surface. This technique reduces the boundary value problem to the solution of an infinite number of algebraic linear non‐homogeneous equations which can easily resolve numerically by any conventional technique. The exciting forces acting on the submerged sphere along the horizontal and vertical directions in the fluid are estimated in the case of both the flexural and free‐surface wave modes. These forces are plotted graphically for several immersion depths of the rigid sphere and the rigidity factors of the flexible base surface of the fluid.

Nonlinear vibrations of fluid-conveying FG cylindrical shells with piezoelectric actuator layer and subjected to external and piezoelectric parametric excitations

Nonlinear vibrations of a fluid-conveying functionally graded (FG) cylindrical shell with piezoelectric actuator layer and subjected to external excitation and piezoelectric parametric excitation are analyzed theoretically. Considering the electric-thermo-fluid-structure interaction effect, a nonlinear dynamic model of fluid-conveying FG cylindrical shell with piezoelectric actuator layer is developed based on Hamilton’s principle and von-Karman geometrical nonlinearity. The inviscid, incompressible, isentropic and irrotational fluid is coupled into governing equations using the linearized potential theory. The nonlinear coupled differential governing equations of system are obtained by using Galerkin’s method with two modes. The developed coupled model is validated by comparing with the prior data and good agreements are observed. Multiple scales method is used to obtain nonlinear dynamic behaviors of the coupled system in case of 1:2 internal resonance, primary parametric resonance and 1/2 subharmonic resonance. The effects of three coupling cases between two modes on the dynamic behaviors of system are explored. Considering strongly coupled effect between two modes, the effect of external excitation, damping coefficient, piezoelectric harmonic voltage, fluid flow velocity, temperature and volume fraction exponent of FG cylindrical shell on the frequency-response curves are investigated. The influence of detuning parameters on force-amplitude response and voltage-amplitude response of system are also discussed.

An Analytical Approach for the Two-Dimensional Plunging Breaking Wave Impact on a Vertical Wall with Air Entrapment

This study investigates an idealized formulation of the two-dimensional impact of a breaking wave on a vertical impermeable wall. An overturning-like wave is assumed, which is close to the concept of a plunging breaker. It is assumed that during the collision an air pocket is entrapped between the wave and the wall. The air pocket width is assumed to be negligible and the compression effects are omitted. The problem is considered in the two-dimensional space (2D) using linear potential theory along with the small-time approximation. We use a perturbation method to cope with the linearized free-surface kinematic and dynamic boundary conditions. We impose the complete mixed boundary value problem (bvp) and we solve for the leading order of the velocity potential. The problem derived involves dual trigonometrical series and is treated analytically. The main assumption made is that, within the air pocket, the pressure is zero. Results are presented for the velocity potential on the wall, the velocity, and the free-surface elevation.

Wave Power Absorption by Arrays of Wave Energy Converters in Front of a Vertical Breakwater: A Theoretical Study

The present paper deals with the theoretical evaluation of the efficiency of an array of cylindrical Wave Energy Converters (WECs) having a vertical symmetry axis and placed in front of a reflecting vertical breakwater. Linear potential theory is assumed, and the associated diffraction and motion radiation problems are solved in the frequency domain. Axisymmetric eigenfunction expansions of the velocity potential are introduced into properly defined ring-shaped fluid regions surrounding each body of the array. The potential solutions are matched at the boundaries of adjacent fluid regions by enforcing continuity of the hydrodynamic pressures and redial velocities. A theoretical model for the evaluation of the WECs’ performance is developed. The model properly accounts for the effect of the breakwater on each body’s hydrodynamic characteristics and the coupling between the bodies’ motions and the power take-off mechanism. Numerical results are presented and discussed in terms of the expected power absorption. The results show how the efficiency of the array is affected by (a) the distance between the devices and the wall, (b) the shape of the WEC array configuration, as well as (c) the angle of the incoming incident wave.

Nonlinear waves in a floating thin elastic plate, predicted by a coupled SPH and FEM simulation and by an analytical solution

This paper highlights linear and nonlinear waves under the influence of surficial structural stiffness and inertia. The main focus is on the waves in a thin elastic plate floating on the water surface. A numerical simulation method, combining smoothed-particle hydrodynamics and the finite element method, is developed to predict the waves propagating in the thin elastic plate. For validation, the simulation results are compared to those obtained through a numerical method, based on the linear potential theory for linear regular wave cases. When the incident wave amplitude was increased, the waves' nonlinearity was observed in the simulation results. The nonlinear waves' characteristics are elucidated through a mathematical solution derived in a similar manner to Stokes wave theory. A significant difference is confirmed through comparisons with the conventional, nonlinear, free-surface wave. The positive peak was higher than the negative peak due to nonlinearity in one frequency range, while it was the opposite in the other frequency range. The nonlinear wave's amplitude increased divergently at the frequency between the two frequency ranges. Overall, it is shown that the mathematical solution effectively explains the characteristics of the nonlinear waves in the elastic plate observed in the numerical simulations.

Numerical investigation of the effects of compressibility on the flutter of a three-dimensional, cantilevered plate in an inviscid, subsonic, open flow

In this work we carry out a numerical study of the influence of the upstream Mach number on the flutter of a three-dimensional, cantilevered, flexible plate subject to a subsonic, inviscid, open flow. We have assumed a linear elastic model for the plate and that the fluid flow is governed by the linearized potential theory. The fluid equations are solved by means of a doublet point method to obtain the generalized aerodynamic forces as a function of the plate displacements. Then, these generalized forces are coupled to the equation of motion of the plate, which is discretized with a spectral Galerkin method, and an eigenvalue analysis is performed to find the flutter point. The obtained results are in good agreement with those of related theoretical and experimental studies found in the literature.

Iterative multipole solution for wave interaction with submerged partially perforated semi-circular breakwater

This study develops an iterative multipole solution for water wave interaction with a submerged partially perforated semi-circular breakwater based on the linear potential theory. Multipole expansion method and separation variable method are used for getting the velocity potential expressions of the flow fields, and the expansion coefficients in velocity potentials are determined by iteratively solving the non-linear pressure loss boundary condition on the partially perforated wall. The calculation results of the present multipole solution are in good agreement with the numerical results of multi-domain BEM (boundary element method) solution. The effects of the radius, porosity and perforated position on the reflection coefficient, transmission coefficient and wave force of breakwater are thoroughly discussed. Some meaningful suggestions are presented for engineering design.

Optimal performance studies of a horizontal eccentric cylinder-type WEC in western sea of Jeju island with linear inviscid and nonlinear viscous numerical model

This study aims to propose an optimal horizontal eccentric cylinder-type wave energy converter (WEC) prototype suitable for western sea of Jeju island by using linear inviscid and nonlinear viscous numerical model, appropriately. Parametric studies with respect to draft and position of center of rotation of horizontal eccentric cylinder-type WEC were carried out through a frequency domain analysis based on the linear potential theory to propose an optimal design of WEC. Dynamic behaviors of the optimized WEC in 1/20 scale induced by regular waves with different wave heights was not only experimentally investigated but also compared with a numerical result with OpenFOAM, one of computational fluid dynamic models. A response amplitude operator (RAO) induced by a higher wave height is smaller than RAO induced by a lower wave height. A non-linear dynamic behavior is investigated from spectrum analysis. A free decay test with different initial pitch excursions shows that nonlinear effects due to viscosity. The extracted power of the optimized WEC in full scale can be estimated from the simulation of full scale model induced by the wave climate in western sea of Jeju island.

Mooring Drag Effects in Interaction Problems of Waves and Moored Underwater Floating Structures

In contrast to either considering structures with full degrees of freedom but with wave force on mooring lines neglected or with wave scattering and radiation neglected, in this paper, a new analytic solution is presented for wave interaction with moored structures of full degrees of freedom and with wave forces acting on mooring lines considered. The linear potential wave theory is applied to solve the wave problem. The wave fields are expressed as superposition of scattering and radiation waves. Wave forces acting on the mooring lines are calculated using the Morison equation with relative motions. A coupling formulation among water waves, underwater floating structure, and mooring lines are presented. The principle of energy conservation, as well as numerical results, are used to verify the present solution. With complete considerations of interactions among waves and moored structures, the characteristics of motions of the structure, the wave fields, and the wave forces acting on the mooring lines are investigated.