Undisturbed Wave Research Articles

An experimental study of a quasi-impulsive backwards wave force associated with the secondary load cycle on a vertical cylinder

Steep wave breaking on a vertical cylinder (a typical foundation supporting offshore wind turbines) will induce slam loads. Many questions on the important violent wave loading and the associated secondary load cycle remain unanswered. We use laboratory experiments with unidirectional waves to investigate the fluid loading on vertical cylinders. We use a novel three-phase decomposition approach that allows us to separate different types of nonlinearity. Our findings reveal the existence of an additional quasi-impulsive loading component that is associated with the secondary load cycle and occurs in the backwards direction against that of the incoming waves. This quasi-impulsive force occurs at the end of the secondary load cycle and close to the passage of the downward zero-crossing point of the undisturbed wave. Wavelet analysis showed that the impulsive force exhibits superficially similar behaviour to a typical wave-slamming event but in the reverse direction. To monitor the scattered wave field and extract run-up on the cylinder, we installed a four-camera synchronised video system and found a strong temporal correlation between the arrival time of the Type-II scattered wave onto the cylinder and the occurrence of this quasi-impulsive force. The temporal characteristics of this quasi-impulsive force can be approximated by the Goda wave impact model, taking the collision of the Type-II scattered waves at the rear stagnation point as the impact source.

The effects of surfactants on plunging breakers

The effects of surfactants on a mechanically generated plunging breaker are studied experimentally in a laboratory wave tank. Waves are generated using a dispersively focused wave packet with a characteristic wavelength of $\lambda _0 = 1.18$ m. Experiments are performed with two sets of surfactant solutions. In the first set, increasing amounts of the soluble surfactant Triton X-100 are mixed into the tank water, while in the second set filtered tap water is left undisturbed in the tank for wait times ranging from 15 min to 21 h. Increasing Triton X-100 concentrations and longer wait times lead to surfactant-induced changes in the dynamic properties of the free surface in the tank. It is found that low surface concentrations of surfactants can dramatically change the wave breaking process by changing the shape of the jet and breaking up the entrained air cavity at the time of jet impact. Direct numerical simulations (DNS) of plunging breakers with constant surface tension are used to show that there is significant compression of the free surface near the plunging jet tip and dilatation elsewhere. To explore the effect of this compression/dilatation, the surface tension isotherm is measured in all experimental cases. The effects of surfactants on the plunging jet are shown to be primarily controlled by the surface tension gradient ( $\Delta \mathcal {E}$ ) while the ambient surface tension of the undisturbed wave tank ( $\sigma _0$ ) plays a secondary role.

Effect of currents on nonlinear waves in shallow water

Effect of various forms of currents on regular nonlinear waves in shallow water is investigated by use of a computational fluid dynamics approach. A range of wave conditions with different wave heights and wave periods are considered. Effect of three types of currents on these waves is investigated, namely (i) uniform current over the water depth, (ii) shear current from the seafloor to the still-water level, and (iii) a custom current profile that changes over the water depth. The current profiles are considered in both following and opposing directions of the incoming wave, forming in total 18 wave–current configurations. The Navier–Stokes equations for a laminar flow are solved computationally in two dimensions. A numerical wave–current maker is created to generate combined nonlinear waves and currents in shallow water. The effect of the currents on the change of the wave field, including quantitative change of the surface elevation, wave height, wavelength, horizontal particle velocity, and the velocity and pressure fields is presented and discussed. It is found that presence of the current can alter the wave field significantly, and the current profile and direction play a significant role in the change of the wave field. A following current in shallow water increases the peak of surface elevation, horizontal particle velocity and pressure, along with an increase in wavelength and wave height, while an opposing current reduces these. The change of wave height with current direction appears to be opposite to that observed in deep water in the literature. It is also concluded that a linear superposition of the undisturbed wave and current velocities can describe the horizontal particle velocity of the wave–current field for following currents (particularly under the wave trough) reasonably well, but larger differences are observed for opposing currents.

Identification and Investigation of Extreme Events Using an Arbitrary Lagrangian–Eulerian Approach With a Laplace Equation Solver and Coupling to a Navier–Stokes Solver

AbstractIncreased deployment of offshore wind turbines is seen as an important pathway to increase green renewable energy production. Improved and rapid identification of extreme events and evaluation of hydrodynamic loads due to such events is essential to reduce the cost of energy production. Numerical modeling to pre-screen sea states and to identify the crucial events to prioritize model tests will make a major contribution to reduce design times and costs for such structures. In this effort, a highly efficient and nonlinear numerical model based on the Laplace equations is used to generate undisturbed wave kinematics. Such a simulation is used to identify extreme wave events in a sea state realization, and further, the wave loading due to such events are evaluated using Morison formula. Events screened in this manner can then be transferred to a high-resolution model such as a Navier–Stokes equation-based solver to investigate the hydrodynamics in detail. The implementation and application of such an approach in the open-source hydrodynamic model REEF3D is presented in this work.

Required test durations for converged short-term wave and impact extreme value statistics–Part 2: Deck box dataset

In the assessment of wave-in-deck loads for new and existing maritime structures typically model tests are carried out. To determine the most critical conditions and measure sufficient impact loads, a range of sea states and various seeds (realisations) for each sea state are tested. Based on these measurements, probability distributions can be derived and design loads determined. In air gap model testing usually only few, if any, impact loads occur per 3-hour seed. This can make it challenging to derive reliable probability distributions of the measured loads, especially when only a few seeds are generated. In addition wave impact forces, such as greenwater loading, slamming, or air gap impacts are typically strongly non-linear, resulting in a large variability of the measured loads. This results in the following questions: How many impacts are needed to derive a reliable distribution? How is the repeatability of individual events affecting the overall distribution? To answer these questions wave-in-deck model tests were carried out in 100 x 3-hour realisations of a 10,000 year North Sea sea state. The resulting probability distributions of the undisturbed wave measurements as well as the measured wave-in-deck loads are presented in this paper with focus on deriving the number of seeds and exposure durations required for a reliable estimate of design loads.The presented study is Part 2 of a combined study on guidance for the convergence and variability of wave crests and impact loading extreme values. The data set of Part 1 ([1]) is based on greenwater loads on a sailing ferry and the data set of Part 2 on wave-in-deck loads on a stationary deck box.

Efficient Calculation of Hydrodynamic Loads on Offshore Wind Substructures Including Slamming Forces

Abstract Estimation of the hydrodynamic loads based on strip theory using the Morrison equation provides an inexpensive method for load estimation for the offshore industry. The advantage of this approach is that it requires only the undisturbed wave kinematics along with inertia and viscous force coefficients. Over the recent years, the development in numerical wave tank simulations makes it possible to simulate nonlinear 3-h sea states, with computational times in the order of real time. This presents the possibility to calculate loads using wave spectrum input in numerical simulations with reasonable computational time and effort. In the current paper, the open-source fully nonlinear potential flow model REEF3D::FNPF is employed for calculating the nonlinear wave kinematics. Here, the Laplace equation for the velocity potential is solved on a σ-coordinate mesh with the nonlinear free surface boundary conditions to close the system. A technique to calculate the total acceleration on the σ-coordinate grid is introduced which makes it possible to apply strip theory in a moving grid framework. With the combination of strip theory and 3-h wave simulations, a unique possibility to estimate the hydrodynamic loads in real time for all discrete positions in space within the domain of the numerical wave tank is presented in this paper. The numerical results for inline forces on an offshore wind mono-pile substructure are compared with measurements, and the new approach shows good agreement.

Estimation of Directional Wave Spectrum Using Measurement Array Pressure Data on Bottom-Mounted Offshore Structure in Incident and Diffracted Wave Field

Determination of the directional wave spectrum which offshore structures actually encounter is essential for multiple applications including wave-induced load and vibration evaluation, and hence becomes a fundamental task in ocean engineering. Due to the wave diffraction effect, wave field around an offshore structure is the mixture of incident wave components and diffracted wave components. Estimating directional wave spectrum in diffracted wave field significantly differs from the occasion in undisturbed waves since the amplitude and phase relationship between the incident and diffracted waves are coupled, and therefore making the conventional approach not applicable. In this study, the diffraction wave theory is introduced into the estimation of directional wave spectrum to consider the effect of diffracted waves using array pressure data from existing pressure gauges on structures. Considering the performance of the presented approach under scenarios with various gauge arrays, different directions, and spreading coefficients, multiple levels of background noise are evaluated and discussed, respectively. The presented approach is also deployed into an in-situ measurement application on a marine structure and compared with wave observation data to test its feasibility in engineering practice. In general, the presented approach can reasonably estimate the directional wave spectrum and show advantages over the conventional approach in which the diffraction effect is excluded.

Screening wave conditions for the occurrence of green water events on sailing ships

Design loads for extreme wave events on ships, such as slamming and green water, are hard to define. These events depend on details in the incoming waves, ship motions and structure layout, which requires high-fidelity tools such as CFD or experiments to obtain the correct loads. These tools (presently) do not have the capability to fully resolve the long-term statistics of rare events in all metocean conditions over the ship’s lifetime. The idea of ‘screening’ is to use lower-fidelity numerical methods to identify the occurrence of extreme load events based on an indicator. A good indicator has a strong correlation to the design load, but is easier to calculate. A high-fidelity tool can then be used to find the loads in these events. The low-fidelity statistics and the high-fidelity loads can be combined to define a design load and its probability. The present study compares different numerical screening indicators for green water loads on a containership against experiments. The quality of the identification of the critical events and the required computational time served as comparison metrics. This showed that screening both with potential flow tools and with coarse mesh CFD tools is feasible, provided the indicator, grid, time step and wave input settings are well chosen. The results from coarse mesh CFD are slightly better than from potential flow, but the computational costs are much higher. The results also show that the peaks and steepness of the relative wave elevation around the bow are suitable green water load indicators, as well as the undisturbed wave crests at the bow. Fine mesh CFD calculations were done for the identified events based on an example indicator, which resulted in a green water load distribution very close to that of the experiments. This study shows that screening could potentially reduce the required high-fidelity modelling time with up to ∼90% compared to common practice.

A Procedure to Calculate First-Order Wave-Structure Interaction Loads in Wave Farms and Other Multi-Body Structures Subjected to Inhomogeneous Waves

A typical assumption when performing analytical, numerical, and experimental studies in wave–structure interaction in multi-body problems such as for wave farms and very large floating structures is the homogeneity of the wave field. Important interactions between the floating elements are dependent on the direction, amplitude, and phase of the waves acting on each. Then, wave homogeneity is probably unrealistic in near-shore areas where these installations are to be deployed. In the present work, an existing interaction method, which allows the use of standard boundary element diffraction codes for solving the first order wave structure linear potential for each unique geometry in the problem, is shown to be able to account for inhomogeneous sea states across the domain of a multi-body problem requiring only minimal modification to its implementation. A procedure to use the method to include arbitrary incoming undisturbed wave conditions at each body is presented. A verification study was done by using an artificial numerical configuration to mimic an inhomogeneous wave field in a standard diffraction code, which was used as a reference. The results obtained using the interaction-method based procedure are shown to be in excellent agreement with the reference ones. Furthermore, an example of frequency inhomogeneity of the wave field in a wave farm is shown and the effects on the motion amplitudes and absorbed power are presented illustrating the applicability of the procedure.

Disturbing impact of multiple wind turbines on the indicated DVOR bearing

Abstract. Radio navigation aids, like the Doppler Very High Frequency Omnidirectional Radio Range (DVOR), provide their navigation service by specific radiation of electromagnetic waves depending on the direction in space and according to their specific antenna characteristics. Therefore, these navigation aids are reliant on undisturbed wave propagation in their operation range. Certainly, a propagation disturbance can be implicated by the presence of scattering objects like large buildings or wind turbines (WT) in the surrounding area of the DVOR, which potentially leads to deviations of the transmitted navigation content. In order to comply with the specified flight safety limits, especially respecting WTs, there is a necessity to predict the prospective bearing error due to installations not only of individual WTs but also of additional ones in a present wind farm or repowering projects. Accordingly, this paper is aimed at ascertaining the bearing deviations' dependency on the quantity of WTs in a realistic wind farm close to a DVOR, calculated in space areas of practical relevance.

Influence of Nonlinear Interaction on the Evolution of Waves in a Shallow Basin

The influence of nonlinear interaction of oppositely directed nonlinear waves in a shallow basin is studied theoretically and numerically within the nonlinear theory of shallow water. It is shown that this interaction leads to a change in the phase of propagation of the main wave, which is forced to propagate along the flow induced by the oncoming wave. The estimates of the undisturbed wave height at the time of interaction agree with the theoretical predictions. The phase shift during the interaction of undisturbed waves is sufficiently small, but becomes noticeable in the case of the propagation of breaking waves.

Peak forces on a point absorbing wave energy converter impacted by tsunami waves

Although a tsunami wave in deep sea can be simulated using linear shallow water theory, the wave dynamics of a tsunami running up a continental shelf is very complex, and different phenomena may occur, depending on the width and profile of the shelf, the topography of the coast, incident angle of the tsunami and other factors. How to simulate tsunami waves at an intermediate depth is studied in this paper by using three different simulation approaches for tsunamis, a soliton, a simulated high incident current and a dam-break approach. The surface wave profiles as well as the velocity- and pressure profiles for the undisturbed waves are compared. A regular Stokes 5th wave of the same amplitude is simulated for comparison. A wave energy converter model, previously validated with wave tank experiment, is then used to study the survivability of the Uppsala University wave energy device for the different waves. The force in the mooring line is studied together with the resulting force on a bottom mounted column, corresponding to the linear generator on the seabed.

Development of a Computational Fluid Dynamics Model to Simulate Three-Dimensional Gap Resonance Driven by Surface Waves

This paper expounds the process of successfully establishing a computational fluid dynamics (CFD) model to accurately reproduce experimental results of three-dimensional (3D) gap resonance between two fixed ship-shaped boxes. The ship-shaped boxes with round bilges were arranged in a side-by-side configuration to represent a floating liquefied natural gas offloading scenario and were subjected to NewWave-type transient wave groups. We employ the open-source CFD package openfoam to develop the numerical model. Three-dimensional gap resonance differs from its two-dimensional (2D) counterpart in allowing spatial structure along the gap and hence multiple modes can easily be excited in the gap by waves of moderate spectral bandwidth. In terms of numerical setup and computational cost, a 3D simulation is much more challenging than a 2D simulation and requires careful selection of relevant parameters. In this respect, the mesh topology and size, domain size and boundary conditions are systematically optimized. It is shown that to accurately reproduce the experimental results in this case, the cell size must be adequate to resolve both the undisturbed incident waves and near-wall boundary layer. By using a linear iterative method, the NewWave-type transient wave group used in the experiment is accurately recreated in the numerical wave tank (NWT). Numerical results including time series of gap responses, resonant amplitudes and frequencies, and mode shapes show excellent agreement with experimental data.

A note on the secondary load cycle for a monopile in irregular deep water waves

Laboratory experiments with a bottom hinged surface-piercing cylinder, exposed to irregular deep water waves, are used to investigate high-frequency forcing. The focus is on the secondary load cycle, a strongly nonlinear phenomenon regarding the wave load on a vertical cylinder, first identified by Grue et al. (1993 Preprint Series. Mechanics and Applied Mathematics, pp. 1–30. University of Oslo, available at http://urn.nb.no/URN:NBN:no-52740; 1994 Ninth International Workshop on Water Waves and Floating Bodies (ed. M. Ohkusu), pp. 77–81, available at http://iwwwfb.org). For a total of 2166 single wave events, the force above $3\unicode[STIX]{x1D714}$ (where $\unicode[STIX]{x1D714}$ is the governing wave frequency) is used to identify and split the strongly nonlinear forces into two peaks: a high-frequency peak closely correlated in time with the wave crest when the total load is positive and a high-frequency peak defining the secondary load cycle which occurs close in time to the wave zero downcrossing when the total load is negative. The two peaks are studied by regression analysis as a function of either the Keulegan–Carpenter number ($KC$) or the Froude number ($Fr$). Regarding the secondary load cycle, the best correlation is found with $Fr$. The speed of the travelling edge of the undisturbed wave approximates the fluid velocity. A threshold value separating between small and large forces is found for $KC\sim 4$–5, indicating effects of flow separation. Alternatively, the threshold occurs for $Fr\sim 0.3$–0.4, indicating local wave effects at the scale of the cylinder diameter. The findings suggest that both effects are present and important.

Fiber-Optic Current Sensor Tolerant to Imperfections of Polarization-Maintaining Fiber Connectors

We investigate the effect of polarization cross-coupling at polarization-maintaining (PM) fiber connectors on the accuracy of an interferometric fiber-optic current sensor. The sensor uses the Faraday effect in a fiber coil operated in reflection mode and an interrogator based on nonreciprocal phase modulation. PM connectors in the fiber link between the sensor's opto-electronic module and the fiber coil give rise to signal instability due to a limited and insufficiently stable polarization extinction ratio (typically <;25-30 dB). As a result the accuracy of the sensor can be well outside the allowed tolerances of applications in the electric power industry which often demands accuracy to within ±0.2%. We demonstrate that by means of a modified optical circuit the disturbing effects of polarization cross-coupling can be largely eliminated. The modified circuit introduces group delays for the cross-coupled light waves relative to the undisturbed waves much larger than the coherence length of the broadband light source. We theoretically and experimentally show that connector extinction ratios well below 20 dB are still uncritical. Furthermore, we verify the superiority of the modified circuit at changing connector temperature (and hence changing temperature-induced stress in the connector ferrules) and at repeated connector open-close operations.

A time domain boundary element method for wave added resistance of ships taking into account viscous effects

Linear boundary element methods generally provide accurately enough predictions of ship motions in small amplitude waves. However, when the wave-induced added resistance is investigated, nonlinear effects are to be accounted for. Higher order effects, such as second order wave-induced drift forces, depend largely on ships forward speed. This article presents a nonlinear time-domain Rankine source method to calculate the wave-induced added resistance of ships advancing at constant forward speed in regular head waves. The fully nonlinear steady flow was computed. Nonlinear Froude-Krylov and hydrostatic forces were obtained by integrating pressures over the instantaneous wetted surface (undisturbed incident wave). Radiation forces were computed in time domain using convolution integrals (Cummins approach); diffraction forces, were computed using the complex force amplitude resulting from the linear seakeeping problem of incident wave diffraction. Radiation and diffraction effects caused by the changing wetted surface were accounted for. An empirical approach to account for the viscous wave added resistance was introduced. Numerical results for a 14,000TEU container ship and a very large tanker were compared to results from a frequency domain method, CFD calculations, and experimental towing tank measurements.

Simulations in quantum tunneling

We study the timing effects of nonrelativistic wave packet tunneling through a barrier using a numerical simulation readily accessible to an undergraduate audience. We demonstrate that the peak of the transmitted packet can sometimes emerge from the barrier ahead of the peak of an undisturbed wave packet that does not encounter a barrier. Under the right circumstances, this effect can give the appearance that transmission through the barrier occurs at superluminal speeds. We demonstrate that this seemingly paradoxical effect is not all that puzzling. Rather, components from the front of the incoming wave packet are preferentially transmitted, forming a transmitted packet ahead of the average of the incoming wave packet but not ahead of the leading edge of that packet. Our studies also show how the timing depends on barrier height and width, consistent with expectations based on the different energy components of the wave packet.

Froude-Krylov forces from exact pressure integrations on adaptive panel meshes in a time domain partially nonlinear model for ship motions

A partially nonlinear time domain model for predicting ship motions is presented with Froude-Krylov forces being calculated on the exact instantaneous wet part of the floating body. An adaptive mesh technique following a quad-tree subdivision and cutting scheme is applied, in order to track the free-surface of the undisturbed wave, and its intersection with the hull, to generate instantaneous wet meshes. Analytical exact pressure integration expressions are evaluated on the, thus formed, wet panel mesh, regarding the undisturbed wave and hydrostatic pressures, which allow for considerably coarse meshes without any loss of accuracy. Diffraction and radiation forces are kept linear with the latter being introduced using linear unit impulse response functions. The body is impeded from drifting by artificial kinematic restrains in non-restoring modes.Roll decay motions of a military vessel's hull, and under head, beam and quartering waves are numerically predicted and compared with published experimental ones, for zero advance speed. The capabilities and limitations of the proposed method, taking into account the specific simulation setup applied, are addressed. An overall agreement with the experiments is witnessed, with the method showing a considerable improvement of numerical predictions of the ship motions in beam and quartering waves in comparison with those by a linear method.

On the relative intensity of Poisson’s spot

The Fresnel diffraction phenomenon referred to as Poisson’s spot or spot of Arago has, beside its historical significance, become relevant in a number of fields. Among them are for example fundamental tests of the super-position principle in the transition from quantum to classical physics and the search for extra-solar planets using star shades. Poisson’s spot refers to the positive on-axis wave interference in the shadow of any spherical or circular obstacle. While the spot’s intensity is equal to the undisturbed field in the plane wave picture, its intensity in general depends on a number of factors, namely the size and wavelength of the source, the size and surface corrugation of the diffraction obstacle, and the distances between source, obstacle and detector. The intensity can be calculated by solving the Fresnel–Kirchhoff diffraction integral numerically, which however tends to be computationally expensive. We have therefore devised an analytical model for the on-axis intensity of Poisson’s spot relative to the intensity of the undisturbed wave field and successfully validated it both using a simple light diffraction setup and numerical methods. The model will be useful for optimizing future Poisson-spot matter-wave diffraction experiments and determining under what experimental conditions the spot can be observed.

Prediction of non-linear ship responses in waves considering forward speed effects

This article describes the development of a non-linear time-domain boundary element method to determine non-linear ship responses (motions and loads) in waves. The general approach by Cummins was used to express the equation of motion in the time domain. Hydrodynamic forces were split into inertia, radiation, diffraction, Froude–Krylov and restoring components. Radiation forces were determined in time domain by convolution of the impulse response of the ship with the motion velocity. The Froude–Krylov and restoring forces were computed over the instantaneous wetted surface, taking into account ship motions, the undisturbed wave and stationary wave system. Diffraction forces were computed from the complex force amplitude resulting from the linear problem of the incident wave diffraction. The impulse response of the ship can be determined using the linear hydrodynamic damping coefficients or added masses. In this work, these two approaches were compared. Ship motions calculated with the developed method were compared with model test and RANSE-based simulations as well as with linear frequency domain results for three different ship types at various forward speeds. It was shown that the convolution integral represents the radiation forces satisfactorily. The convolution integral based on damping coefficients showed distinct advantages at zero speed. With increasing forward speed, the added mass-based convolution integral leaded in some cases to better results.