Multi-directional Irregular Waves Research Articles

Floating wind turbine response in uni- and multi-directional nonlinear waves by numerical and experimental investigations

Natural sea waves are typically multi-directional. However, the existing studies on the interaction of waves and structures mostly concentrate on uni-directional waves. In this study, using a higher-order boundary element method based on the nonlinear potential flow theory and the perturbation expansion technique, a numerical model is developed to investigate the hydrodynamic performance of a semi-submersible wind turbine foundation in uni- and multi-directional waves. Comprehensive validations with the wave-tank experiment are conducted. It is found that the significant platform response increases with the peak wave period in uni-directional irregular waves, while the high-frequency “energy” ratio changes little. The significant wave height hardly influences motion responses from either the time- or the frequency-domain perspective. In multi-directional irregular waves, the translational motions exhibit monotonicity with wave directionality. The energy concentration around the primary direction leads to a dominant wave-frequency motion and an increase in the high-frequency “energy” ratio. Although the individual modal motion responses are variable functions of wave nonlinearity, their averaged translational and rotational motions are nearly constant, indicating an energy transition or a trade-off relationship among the modal motions. In addition, unlike the uni-directional wave case, the low- and high-frequency “energy” ratios increase quadratically and decrease linearly with the significant wave height in multi-directional waves, respectively. All these findings demonstrate that wave directionality can change the wave–structure interaction properties and therefore needs to be adequately considered in engineering applications.

Analytical investigation on a wave energy converter-dual-arc breakwater integration system

A wave energy converter (WEC)-breakwater integration system consisting of a heaving WEC and a bottom-mounted dual-arc breakwater is proposed. The energy extraction performance and wave attenuation performance of the integration system were investigated using a theoretical model based on linear potential flow theory. The permeability of the dual-arc breakwater is considered in the theoretical model. The singularity at the edges of the perforated arc-shaped wall was solved using the multiterm Galerkin method. The unknowns in the expressions of velocity potential were solved using the eigenfunction expansion matching method. After successful validation, a comparative study between the models with dual-arc breakwater of different permeability was carried out first. Then, a parametric study on the configuration with a solid inner and a perforated outer arc-shaped wall is carried out. The performance of the proposed integration system is further investigated in multi-directional irregular waves, and it was found that the fluctuations of the average power capture width ratio are much lower compared with that under unidirectional regular wave conditions. The overall performance of the integration system is better with a larger directional spreading parameter.

Control co-design and uncertainty analysis of the LUPA’s PTO using WecOptTool

Control co-design has been shown to significantly improve the performance of wave energy converters (WEC). By considering the control and WEC design concurrently, the space searched by the optimization routine is greatly expanded which results in better performing devices. Recently, an open-source WEC co-design code, WecOptTool, was released to perform control co-design research and facilitate its adoption in the community. In this study, we use WecOptTool to perform control co-optimization and uncertainty analysis of the LUPA device. The Laboratory Upgrade Point Absorber (LUPA) is a new open-source laboratory-scale WEC that provides a platform for testing new concepts, innovating control schemes, and validating numerical models. The LUPA can be adjusted to different configurations, including changing the number of bodies, the degrees of freedom (DOF), the float and spar geometry, and the diameter of the drive sprocket pulley in the power take off (PTO) system, as well as providing different control algorithms and input waves. The drive sprocket diameter influences the torque vs speed of the generator, which allows for more flexibility in operating under different wave conditions or with different control schemes. In this study we optimize the drive sprocket diameter, while considering the optimal control algorithm for each potential design, to identify the optimal diameter for electric power production at the PacWave South WEC test site. This case study demonstrates several new capabilities of WecOptTool including a multi-body multi-DOF system and multi-directional irregular waves. The PTO dynamics are modeled using first principle methods for a parametrized model of the mechanical subcomponents in combination with generator model obtained using a power-invariant Park transform. The case-study will be made available to serve as a design tool along the LUPA hardware. Users can readily use this model to perform their own design optimization prior to testing with the physical LUPA device. Finally, we use the automatic differentiation capability of WecOptTool to perform a sensitivity and uncertainty analysis of the LUPA device.

Numerical simulation of a full spine of Edinburgh Duck modules in uni- and multi-directional irregular wave climates, with a view to design optimisation

Long, flexibly-jointed spines of Edinburgh Duck modules have the potential to enable the extraction of a large proportion of the wave energy from our seas and oceans. It is well-known that the ‘duck’ shape is able to efficiently absorb wave energy, and that jointed but controlled interconnections between ducks as part of a full spine can also benefit the performance. However, in order to progress further towards achieving optimal performance in real wave climates, a greater understanding of the significance of the spine configuration and scale, spine orientation, and directional, irregular wave conditions is required. By using an efficient hydrodynamic model of a ten-duck spine in conjunction with a constrained frequency-domain control strategy, this paper investigates the effects of the above factors on device performance (as a function of power extraction) in uni- and multi-directional versions of an irregular wave climate. A series of inferences are drawn from the simulations and discussed with regards to informing the direction of future duck spine designs.

Modulation effects of submarine internal wake waves on free surface divergence field

Disturbances of a submarine in a stratified flow stimulate an infinite mode of internal waves with multiple dispersion relationships. The surface convergence and divergence field are then modulated by such trailing waves, affecting the spatial distribution of microscale waves and leaving traces that are easily detected by Synthetic Aperture Radar. The present modified model combines the source-caused internal waves theory with the wind-waves spectrum model and coupling wave components of the full wave systems decomposed by the Fourier series method based on the second-order multi-directional irregular waves interactions theory. The effects of maximum Brunt–Vaisala frequency amplitude and depth, internal wave modes, submarine dive depth and velocity, wind speed, and wind direction are discussed in this study. It is shown that after the appearance of mode 0, the wake waves have an appreciable effect on the modulation of the free surface wind-wave field. However, this effect is not fixed to manifest as an enhancement of the free surface convergence or divergence field intensity. The free surface field has the maximum divergence intensity amplitude under the downwind sailing condition when the submarine is sailing at low speed. At high speed, the maximum divergence intensity amplitude is at an angle of 30° between the sailing direction and the wind direction, while the minimum divergence intensity is downwind.

Estimation of encountered wave elevation sequences based on response measurements in multi-directional seas

This paper presents a new approach for estimating encountered wave elevation sequences by use of measured ship responses, where wind-waves and swell may come from different directions, i.e. bi-directional waves. The main assumption of the approach, making use of Prolate Spheroidal Wave Functions (PSWF), is that the wave field is represented by multi-directional irregular waves. Thus, combining available measured responses, the phases and amplitudes of the multi-directional irregular waves are derived as the solution, by which the wave profiles can be estimated. Numerical investigations using artificially generated response measurements (sway, heave, pitch, vertical bending moment) of a bulk carrier in uni-directional and bi-directional long-crested as well as short-crested sea states are made. It is shown that the present approach can accurately estimate wave elevation sequences in such sea states.

Time-domain simulation of sum-frequency hydrodynamic performance of a TLP-type floating wind turbine in multidirectional seas

Previous studies (Ren et al., 2022) by the authors have confirmed that the second-order diffracted wave loads are very sensitive to the wave directional spreading of the so-called storm-swell mixed seas where the storm-driven local seas combined with long-period swells propagating from different angles. In this paper, particular attention is paid to estimate the sum-frequency hydrodynamic forces and motion responses of a TLP-type floating wind turbine (FWT) in such complex sea states. A time-domain potential flow model is developed for the second-order interaction between the multidirectional irregular waves and moored structures. The boundary value problems for the scattered potential are solved by a higher-order boundary element method (HOBEM). The fourth-order Adams–Bashforth–Moulton method is used to update the wave surface and the Newmark method combined with the Newton–Raphson iteration scheme is adopted to solve the body motion equations.For validation, a freely floating hemisphere in unidirectional bichromatic waves is investigated and good agreement is found by comparing with Kim and Yue's frequency-domain results. Numerical tests are conducted on the TLP-type FWT with different combinations of incident angles and spreading factors of the storm-swell mixed seas. The effects of the wave headings and directional spreading on the sum-frequency hydrodynamic response are discussed. It is noted that when the swells approach the storm seas from large relative angles, the sum-frequency vertical response of the FWT can be up to seven times greater than that in the collinear mixed seas. In addition, the present model can extend the rotor-floater-tether coupled dynamic simulator of FWT to more realistic seas such as storm-swell mixed seas.

Multi-directional irregular wave forces acting on a tandem pile group

In this study, the interactions between multi-directional irregular waves and a tandem pile group are experimentally investigated. A systematic statistical analysis of the force coefficients of the pile group and the wave force on a tandem pile group under multi-directional waves is first carried out. The results show that the wave force of the pile group is significantly affected by the wave directionality. Further, Reynolds-averaged Navier–Stokes equations are used to establish a 3D numerical wave tank by considering the interactions between the waves and the pile groups, and this is verified by using experimental data. The characteristics of the wave force on the first and the last pile in the tandem pile group were numerically examined. The flow field around the front first pile was affected by its rear pile such that the pressure at its rear increased. However, without the rear pile, the pressure difference between the upward and rearward zone of the last pile significantly increased compared with that of the first pile, and this caused the wave force of the last pile to be larger than that of the first pile.

Hydrodynamic interactions between waves and cylinder arrays of relative motions composed of truncated floating cylinders with five degrees of freedom

The hydrodynamic interactions between waves and a truncated floating circular cylinder array were examined in this work. Each cylinder of the array is allowed to oscillate with five degrees of freedom (DOFs), and there can be relative motions among different cylinders. The hydrodynamic responses of such arrays in the presence of ambient incident waves were studied using a semi-analytical technique, in which, the unknown coefficients of radiation–diffraction velocity potentials caused by array motions were expressed by the linear combination of each cylinder’s to-be-calculated oscillation amplitudes in each direction. Then, for each cylinder in the array, the amplitude of each degree of freedom was computed. Variations in the oscillation amplitudes of the cylinders in the array with wavenumbers for various incidence angles and column spacing, and amplitude distributions of free surface elevation for various wavenumbers were given. Additionally, extensions to the situation of multidirectional irregular waves were presented. Last but not least, we examined the effects of other DOFs (nonheave) on hydrodynamic responses and wave energy extraction performances (capture width w and interaction factor q) by comparing the results obtained by the 5-DOF and the 1-DOF (heave) models. In terms of response amplitude and wave energy extraction performance, the 5-DOF model’s results were significantly different from those of the 1-DOF model. Therefore, it is suggested to use the 5-DOF model to analyze such cylindrical arrays.

On the influence of multidirectional irregular waves on the PeWEC device

Wave energy is a promising renewable resource for its reliability and power density, and many technological milestones have been achieved. Significant efforts are made to design and optimize Wave Energy Converters (WECs); however, analyses are often limited to simplified conditions. Among such restrictive assumptions, waves are frequently described utilizing monodirectional spectra, thus leading to approximate evaluations, also in terms of absorbed power. In real sea conditions, the waves are multidirectional, and the analysis as a 2D superposition of multiple wave components should be investigated. In particular, linear waves can be analyzed as a sum of sine waves characterized by different amplitudes, frequencies, phases and directions. The case study device analyzed in this paper is PeWEC (Pendulum Wave Energy Converter), a rotating mass device that converts energy based on pitch motion, moored through a spread catenary mooring system. The sea states investigated are those of the island of Cyprus. The spectrum is defined as the combination between the JONSWAP frequency spectrum and the cos-2s directional spectrum. To compute the sea elevation components the Deterministic Amplitude Scheme (DAS) method is used. The forcing acting on the device, the mooring loads and the device motions are examined and compared to quantify the error produced by the monodirectional approximations. The time domain solver OrcaFlex is employed to investigate the interaction of the waves with the moored hull. Compared with the multidirectional analysis, the monodirectional approximation generates an overestimation of the pitch by 5% and of the surge by 3%, highlighting the importance of taking spreading into account if the device is directional.

Simulation of wave incident boundary for spatially inhomogeneous multidirectional irregular waves

Accurately simulating the incidence of the spatially inhomogeneous multidirectional irregular waves in numerical modeling of wave propagation on coastal areas is vitally important for engineering design. In this paper, the Inhomog-Bound-EEED, Inhomog-Bound-PTPD and Inhomog-Bound-DSFD approaches are established and their capabilities for simulating the inhomogeneous incident wave boundary are verified by the coupling of Boussinesq wave model with theoretical diffraction wave model and with SWAN wave model. The incident wave information of the Boussinesq wave model is simulated by the proposed methods based on the wave information provided by the wave diffraction or SWAN model. The simulated time series and significant wave heights by the Boussinesq wave model show good agreement with the results of the theoretical diffraction wave model, which verifies the validities of the proposed Inhomog-Bound-PTPD and Inhomog-Bound-DSFD approaches. The obvious consistency of the simulated results for the analyzed significant wave heights and directional distributions between the Boussinesq wave model and the theoretical diffraction wave model verifies the effectiveness of the Inhomog-Bound-DSFD method, and the method is further applied to the numerical coupling of the Boussinesq wave model and SWAN model. The proposed three methods have good applicability to simulate the spatially inhomogeneous multidirectional irregular wave incident boundary.

Mean and variance of the Eulerian and Lagrangian horizontal velocities induced by nonlinear multi-directional irregular water waves

In this paper, we study the mean and variance of the Eulerian and Lagrangian fluid velocities as a function of depth below the surface of directionally spread irregular wave fields given by JONSWAP spectra in deep water. We focus on the behaviour of these quantities in the bulk of the water, and using second-order potential flow theory we derive new simple asymptotic approximations for their decay in the limit of large depth below the surface. Specifically, we show that when the depth is greater than about 1.5 peak wavelengths, the variance of the Eulerian velocity decays in proportion to exp ⁡ ( − ( 135 4 ) 1 / 3 ( − k p z ) 2 / 3 ) , and the mean Lagrangian velocity decays in proportion to 1 ( − k p z ) 1 / 6 exp ⁡ ( − ( 135 4 ) 1 / 3 ( − k p z ) 2 / 3 ) . Here, k p is the peak wave number and z is the vertical coordinate measured positively upwards from the still water level. We test the accuracy of the second-order formulation against new fully nonlinear simulations of both short crested and long crested irregular wave fields and find a good match, even when the simulations are known to be affected substantially by third-order effects. To our knowledge, this marks the first fully nonlinear investigation of the Eulerian and Lagrangian velocities below the surface in irregular wave fields.

Experimental study of wave height, crest, and trough distributions of directional irregular waves on a slope

Presently, most studies of wave height distribution on slopes are based on normal incident unidirectional waves. In this study, the wave height, crest, and trough distributions of oblique incident unidirectional and multidirectional irregular waves on a 1:15 slope were experimentally investigated. The Composite Weibull Distribution fitted the experimental data well and could be applied to predict the wave height, crest, and trough distributions for unidirectional and multidirectional waves on the slope. The shape parameter with a constant value k2 = 3.6 in Composite Weibull Distribution proposed by Battjes and Groenendijk (2000) did not describe the actual variation of wave height distribution in the surf zone. The parameters' values in Composite Weibull Distribution were modified and the corresponding fitted formulas were proposed in this paper. The results showed that the fitting parameters' values of Composite Weibull Distribution between normal and oblique incident irregular waves, and between unidirectional and multidirectional irregular waves were slightly different. But the directional spreading had little effect on the fitting parameters’ values for narrow directional spreading waves. These findings for a 1:15 slope can be referenced as a supplement to the conclusions of Battjes and Groenendijk (2000).

Reduced order model for nonlinear multi-directional ocean wave propagation

In this paper, we introduce a reduced order model (ROM) for the propagation of nonlinear multi-directional ocean wave-fields. The ROM relies on Galerkin projection of Zakharov equations embedded in the high-order spectral (HOS) method, which describes the evolution of nonlinear waves. The dominant flow features of wave evolution are computed from proper orthogonal decomposition (POD) and these modes are used for the projection. The HOS scheme to compute the vertical velocity is treated in a novel way for an efficient implementation of POD-based ROM. We refer to this alternative formalism of HOS as HOS-simple. The final reduced order model (ROM) is derived from the Galerkin projection of HOS-simple. For the case of irregular waves, where the number of modes required are in the range of 200, the ROM has no significant advantage since both HOS and HOS-simple are much faster than real-time. The real advantage is demonstrated in multi-directional (or short-crested) irregular waves, where the ROM is the only model capable of achieving real-time computation, a major improvement to the standard HOS method. The potential use of the ROM in propagating short-crested waves from far-field to near-field for real-world applications involving wave probes in a wave tank/controlled environment as well as X-band radar in open ocean is also demonstrated.

On two approaches to the third-order solution of surface gravity waves

A third-order approximate solution for surface gravity waves in the finite water depth is studied in the context of potential flow theory. This solution corresponds to the steady-state part of a multidirectional irregular wave field and provides explicit expressions for the surface elevation, free-surface velocity potential, and velocity potential. The amplitude dispersion relation is also provided. Two approaches are used to derive the third-order analytical solution, resulting in two types of approximate solutions: the perturbation solution and the Hamiltonian solution. The perturbation solution is obtained by a classical perturbation technique in which the time variable is expanded in multiscale to eliminate secular terms. The Hamiltonian solution is derived from the canonical transformation in the Hamiltonian theory of water waves. By comparing the two types of solutions, it is found that they are completely equivalent for the first- and second-order solutions and the nonlinear dispersion, but for the third-order part only the sum–sum terms are the same. Due to the canonical transformation that could completely separate the dynamic and bound harmonics, the Hamiltonian solutions break through the difficulty that the perturbation theory breaks down due to singularities in the transfer functions when the quartet resonance criterion is satisfied. Furthermore, it is also found that some time-averaged quantities based on the Hamiltonian solution, such as mean potential energy and mean kinetic energy, are equal to those in the initial state in which the sea surface is assumed to be a Gaussian random process. This is because there are associated conserved quantities in the Hamiltonian form. All of these show that the Hamiltonian solution is more reasonable and accurate to describe the third-order steady-state wave field. Finally, based on the Hamiltonian solution, some statistics are given such as the volume flux, skewness, excess kurtosis, and non-uniqueness of the induced mean flow and mean surface.

On the statistical properties of inertia and drag forces in nonlinear multi-directional irregular water waves

Abstract

A generalized second-order 3D theory for coupling multidirectional wave propagation from a numerical model to a physical model. Part II: Experimental validation using an I-shaped segmented wavemaker

This paper provides the experimental validation of the generalized second-order three-dimensional (3D) coupling theory outlined by Yang et al. [Z. Yang, S. Liu, X. Ji, and H.B. Bingham, 2020. A generalized second-order 3D theory for coupling multidirectional nonlinear wave propagation from a numerical model to a physical model. Part I: Derivation, implementation and model verification, submitted for publication] using multidirectional nonlinear 3D irregular waves. Based on the second-order theory for two-dimensional (2D) irregular waves described by Yang et al. (2014a), this work provides a second-order dispersive correction for the physical wavemaker signal which improves the nonlinear 3D wave information transfer between the numerical and physical models compared to the first-order method of Zhang et al. (2007). The important coupling equation and its discretization schemes were presented in detail in Part I. For further validating the performance of the second-order 3D coupling model under complex waves and bathymetry conditions, in this Part II, a careful experimental validation has been carried out by establishing a flat coupling basin and a concave-convex coupling basin. In addition, a sequence of progressively more complex numerical target waves has been used and an I-shaped segmented piston-type wavemaker system has been considered for verification of the theory. The effectiveness, adaptability, and precision of the second-order 3D coupling model have been discussed in detail. The effects of the wave parameters and its nonlinearity on the coupling simulation have also been evaluated from the discrepancy error and a wave harmonic analysis. By defining a space synchronization error, the overall error of the spatial wave field has been analyzed. Coupling simulation experiments using oblique regular and irregular waves, unidirectional irregular waves, and multidirectional irregular waves show that the traditional first-order coupling model has limitations when coupling complex waves with strong nonlinearity. When controlling the waves using the second-order 3D coupling signal, the higher harmonics underlying the numerical waves are accurately captured and transferred into the physical model, and the unwanted second-order spurious free waves are substantially reduced. Results obtained from the experimental verification described in this paper demonstrate the strengths of the second-order 3D coupling theory, which are more evident with increased wave nonlinearity.

Interactions between multi-directional irregular waves and a pile group in a side-by-side arrangement: Probabilistic analysis

In a recent paper, Zhang et al. (2019) statistically analyzed the pile group effect of nine piles in a side-by-side arrangement exposed to multi-directional irregular waves. In this study, the corresponding data are further analyzed and their probabilistic distributions are examined. The Weibull distribution is used to fit the measured wave force, and the relationships between parameters A and B of the Weibull distribution are given, where it satisfactorily fitted the probability distribution of the measured wave force on the piles with the appropriate parameters. To explain the rules governing the probability distribution of the wave force, the relationships between this and the characteristics of force groupiness are studied. Variations in the parameters of the Weibull distribution and the average force coefficients are also examined using the proposed KCLD1/3 and relative spacing (S/D). The results show that the probability distribution of the irregular wave force on piles exposed to multi-directional irregular waves is strongly dependent on the wave directionality, KCLD1/3 numbers and relative spacing.

On the statistical properties of surface elevation, velocities and accelerations in multi-directional irregular water waves

Abstract

VERIFICATION OF MULTI DIRECTIONAL WAVE GENERATION BASED ON THREE-DIMENSIONAL NUMERICAL WAVE TANK

Although it is still tricky to stably solve multi-directional irregular waves using a three-dimensional numerical wave tank, several studies have been carried out in recent years with the development of computers (e.g., Wang et al., 2019). In order to calculate stable multidirectional irregular waves, it is necessary to devise the incident boundary conditions. In this study, the wave generation source model (Yamano et al., 2010), which can generate waves in the calculation domain, was applied to verify the stable multi-directional irregular wave generation based on 3D Navier-Stokes simulations. At first, it was verified whether unidirectional irregular waves could be generated or not. Next, multidirectional irregular waves were verified. The calculation time was also summarized.Recorded Presentation from the vICCE (YouTube Link):