Direct time domain analysis of floating structures with linear and nonlinear mooring stiffness in a 3D numerical wave tank

In this paper, motion response of a moored floating structure interacting with a large amplitude and steep incident wave field is studied using a coupled time domain solution scheme. Solution of the hydrodynamic boundary value problem is achieved using a three-dimensional numerical wave tank (3D NWT) approach based upon a form of Mixed-Eulerian–Lagrangian (MEL) scheme. In the developed method, nonlinearity arising due to incident wave as well as nonlinear hydrostatics is completely captured while the hydrodynamic interactions of radiation and diffraction are determined at every time step based on certain simplifying approximations. Mooring lines are modelled as linear as well as nonlinear springs. The horizontal tension for each individual mooring line is obtained from the nonlinear load-excursion plot of the lines computed using catenary theory, from which the linear and nonlinear line stiffness are determined. Motions of three realistic floating structures with different mooring systems are analyzed considering various combinations of linear and approximate nonlinear hydrodynamic load computations and linear/nonlinear mooring line stiffness. Results are discussed to bring out the influence and need for consideration of nonlinearities in the hydrodynamics and hydrostatics as well as the nonlinear modelling of the line stiffness.     

强非线性和色散性Boussinesq方程数值模型检验

采用同位网格有限差分法,建立了强非线性和色散性Boussinesq方程数值计算模型。以稳恒波Fourier近似解给定入射波边界条件,对均匀水深深水和浅水域不同非线性的行进波、缓坡地形上深水至浅水域的浅水变形波、以及缓坡和陡坡地形上的波浪水槽实验进行了数值计算,并将计算结果与解析解、解析数值解以及实验值进行了较为详细的比较,从而检验了模型的色散性、非线性以及不同底坡下非线性波的浅水变形性能。     

A comparative linear and nonlinear ship motion study using 3-D time domain methods

This paper reports on 3D time domain seakeeping computations at different level of modeling of nonlinear effects. Four successively improved levels of computations are considered: (i) a linear computation, (ii) Froude-Krylov nonlinear computation, (iii) body nonlinear computation where the perturbation potential is computed based on the instantaneous hull under the mean free-surface, and finally (iv) the body-exact nonlinear computation, where the perturbation potential is determined based on the wetted hull under the incident wave profile after suitable mapping of the hull into a computational domain. Computations are carried out for a Wigley hull (having less ‘geometric’ nonlinearity due to vertical sides), and a S175 hull at different forward speeds. The results are obtained for both regular and irregular waves.     

Time-domain wave diffraction of two-dimensional single and twin hulls

The nonlinear diffraction of 2D single and twin hulls are studied by employing a mixed Eulerian–Lagrangian model based on a higher-order cubic-spline boundary element solver. Two types of simulations are considered. In the first, waves are generated by a piston-type wave-maker in a rectangular tank and in the second case a nonlinear incident wave is assumed to exist in the tank in which the body is introduced. For the application of this model, the full nonlinear diffraction problem is recast in terms of a perturbation wave-field. Computations are performed for rectangular and triangular hull geometries. Computed results show significant nonlinearities, particularly in the heave force. The twin hull results show the influence of wave interference on the diffraction forces. This interference influences the surge force considerably, but heave force is less affected.     

Nonlinear effects of the distribution of amplitudes of sea waves

The effect of nonlinearities, such as wave-breaking and vertical asymmetry associated with sea waves, on the distribution of wave amplitudes is explored. Semiclosed theoretical expressions are derived to describe the distributions of breaking-limited crest and trough amplitudes for Stokes-type nonlinear sea waves. These are compared with the conventional Rayleigh distribution appropriate to linear wave amplitudes. The construction of nonlinear wave envelopes with the fast Fourier transform technique is described. The technique can be utilized to enlarge the data base in empirical analyses of field records which typically contain limited information on amplitude characteristics. The theoretical distributions and the proposed data enlargement technique are demonstrated with the analysis of a nonlinear wave record.     

Statistics of nonlinear wave crests and groups

Groups of large nonlinear waves with sharper higher crests can pose hazards to ships, induce harbor resonance and cause wave-overtopping of fixed and floating structures. Past interest in wave groups has mostly been focused on the statistics and modeling of linear wave groups. Studies on nonlinear wave groups are surprisingly few, and address deep water waves only. Here, statistics of nonlinear wave crests and wave-crest groups in deep and transitional water depths are considered, using an appropriate second-order representation for crest heights and the continuous wave-envelope approach. In particular, theoretical expressions describing the statistics of nonlinear wave crests and their groups are posed in the form of a simple second-order transformation of well-known results on linear waves. Predictions from the transformation so posed compare well with nonlinear wave data gathered in the North Sea, and demonstrate that nonlinearities do affect the statistics of large wave crests and their groups significantly.     

Comments on `Theory and application of calibration techniques foran NDBC directional wave measurements buoy' by K.E. Steele, et al.:nonlinear effects

For original paper see ibid., vol. OE-10, no.4, p.382-96 (1985). The authors of the above mentioned paper present an extensive set of linear calibration techniques that are applied to National Data Buoy Center wave-buoy sensor spectral output before calculating and disseminating directional wave spectra. The commentators identify and estimate the nonlinear effects that produce biases still present in the output, due both to wave nonlinearities themselves and to constraints on the buoy and mooring system to the driving forces. Simple models show that these nonlinearities can produce spectral energy biases of 5-15% at and above the spectral peak frequency, and even greater errors below it. NDBC presently records wave data from vertically stabilized and fixed accelerometers and slope sensors. Calculations show that these sensors all incur bias due to wave nonlinearities: this is greater for vertically stabilized accelerometers and least for slope sensors. Effects of the resulting inconsistencies between the different sensors are most pronounced below the spectral peak where the nonlinear terms dominate; these effects are illustrated with measured data     

Freely floating-body simulation by a 2D fully nonlinear numerical wave tank

Nonlinear interactions between large waves and freely floating bodies are investigated by a 2D fully nonlinear numerical wave tank (NWT). The fully nonlinear 2D NWT is developed based on the potential theory, MEL/material-node time-marching approach, and boundary element method (BEM). A robust and stable 4th-order Runge–Kutta fully updated time-integration scheme is used with regriding (every time step) and smoothing (every five steps). A special φ_n-η type numerical beach on the free surface is developed to minimize wave reflection from end-wall and wave maker. The acceleration-potential formulation and direct mode-decomposition method are used for calculating the time derivative of velocity potential. The indirect mode-decomposition method is also independently developed for cross-checking. The present fully nonlinear simulations for a 2D freely floating barge are compared with the corresponding linear results, Nojiri and Murayama’s (Trans. West-Jpn. Soc. Nav. Archit. 51 (1975)) experimental results, and Tanizawa and Minami’s (Abstract for the 6th Symposium on Nonlinear and Free-surface Flow, 1998) fully nonlinear simulation results. It is shown that the fully nonlinear results converge to the corresponding linear results as incident wave heights decrease. A noticeable discrepancy between linear and fully nonlinear simulations is observed near the resonance area, where the second and third harmonic sway forces are even bigger than the first harmonic component causing highly nonlinear features in sway time series. The surprisingly large second harmonic heave forces in short waves are also successfully reproduced. The fully updated time-marching scheme is found to be much more robust than the frozen-coefficient method in fully nonlinear simulations with floating bodies. To compare the role of free-surface and body-surface nonlinearities, the body-nonlinear-only case with linearized free-surface condition was separately developed and simulated.     

An Oscillating Wave Energy Converter with Nonlinear Snap-Through Power-Take-Off Systems in Regular Waves

Floating oscillating bodies constitute a large class of wave energy converters, especially for offshore deployment. Usually the Power-Take-Off (PTO) system is a directly linear electric generator or a hydraulic motor that drives an electric generator. The PTO system is simplified as a linear spring and a linear damper. However the conversion is less powerful with wave periods off resonance. Thus, a nonlinear snap-through mechanism with two symmetrically oblique springs and a linear damper is applied in the PTO system. The nonlinear snap-through mechanism is characteristics of negative stiffness and double-well potential. An important nonlinear parameter is defined as the ratio of half of the horizontal distance between the two springs to the original length of both springs. Time domain method is applied to the dynamics of wave energy converter in regular waves. And the state space model is used to replace the convolution terms in the time domain equation. The results show that the energy harvested by the nonlinear PTO system is larger than that by linear system for low frequency input. While the power captured by nonlinear converters is slightly smaller than that by linear converters for high frequency input. The wave amplitude, damping coefficient of PTO systems and the nonlinear parameter affect power capture performance of nonlinear converters. The oscillation of nonlinear wave energy converters may be local or periodically inter well for certain values of the incident wave frequency and the nonlinear parameter , which is different from linear converters characteristics of sinusoidal response in regular waves.     

Nonlinear features in wave-resolving microwave radar observationsof ocean waves

Transformation of sea-surface Doppler microwave backscatter observations from the space-time domain to the wavenumber-frequency domain separates linear wave energy from nonlinear effects. Here observations and modeling are used to investigate the sources of these nonlinearities. Wave breaking and electromagnetic shadowing are examined with emphasis on their relative effects both inside and outside the region of the wavenumber-frequency spectrum associated with the linear dispersion equation. Shadowing significantly reduces the variance levels within the linear spectral region. In addition, shadowing is less directly related to changes in variance outside this region, i.e., that region associated with nonlinearity in the wave field. Wave breaking has less of an effect on the variance within the linear region than shadowing. However, the modeled wave breaking does have a greater tendency to increase variance levels at frequencies less than that of the linear wave field, for any given wavenumber. Aliasing and emphasis of crest backscatter are also explored to explain features seen in some wavenumber-frequency intensity images. Two-dimensional data allow the linear wave spectrum to be separated from nonlinear effects. This results in improved wave height spectrum estimation     

Time domain three-dimensional fully nonlinear computations of steady body–wave interaction problem

Three-dimensional fully nonlinear waves generated by moving disturbances with steady forward speed without motions are solved using a mixed Eulerian–Lagrangian method in terms of an indirect boundary integral method and a Runge–Kutta time marching approach which integrates the fully nonlinear free surface boundary conditions with respect to time.A moving computational window is used in the computations by truncating the fluid domain (the free surface) into a computational domain. The computational window maintains the computational domain and tracks the free surface profile by a node-shifting scheme applied within it. An implicit implement of far field condition is enforced automatically at the truncation boundary of the computational window.Numerical computations are applied to free surface waves generated by Wigley and Series 60 hulls for the steady problem. The present numerical results are presented and compared with existing linear theory, experimental measurements, and other numerical nonlinear computations. The comparisons show satisfactory agreements for these hydrodynamic problems.     

Nonlinear wave effect on the slow drift motion of a floating body

The slow drift motion of a floating body in a two-dimensional wave field has been investigated using a time-domain, fully nonlinear numerical model with non-reflective open boundaries. Preliminary computations were conducted for incident bichromatic waves, in which wave theories with different orders were applied in generating the waves required. The results show that the use of low-order theories generates undesirable free waves, and that fourth-order terms contribute markedly to low-frequency input. The motion of a rectangular floating body in response to nonlinear bichromatic waves was computed. The numerical results for small-amplitude incident waves agree reasonably well with the second-order approximation for both the steady and difference-frequency (Δσ) components in the body's motion. For relatively large waves, however, the 2Δσ component becomes predominant compared with the Δσ component. The motion of the body in irregular waves with different wave parameters has also been presented in order to discuss the validity range of a second-order approximation.     

Simulating nonlinear waves on the free surface in surf zones with two-dimensional sloping beach

Unsteady nonlinear wave motions on the free surface in shallow water and over slopes of various geometries are numerically simulated using a finite difference method in rectangular grid system. Two-dimensional Navier–Stokes equations and the continuity equation are used for the computations. Irregular leg lengths and stars are employed near the boundaries of body and free surface to satisfy the boundary conditions. Also, the free surface which consists of markers or segments is determined every time step with the satisfaction of kinematic and dynamic free surface conditions. Moreover, marker-density method is also adopted to allow plunging jets impinging on the free surface. Either linear or Stokes wave theory is employed for the generation of waves on the inflow boundary. For the simulation of wave breaking phenomena, the computations are carried out with various wave periods and sea bottom slopes in surf zone. The results are compared with other existing computational and experimental results. Agreement between the experimental data and the computation results is good.     

Wave-height distributions and nonlinear effects

Theoretical distributions proposed for describing the crest-to-trough heights of linear waves are reviewed briefly. To explore the effects of nonlinearities, these are generalized to second-order waves, utilizing quasi-deterministic results on the expected shape of large waves. The efficacy of Gram–Charlier models in describing the effects of third-order nonlinearities on the distributions of wave heights, crests and troughs are examined in detail. All models and a fifth-order Stokes–Rayleigh type model recently proposed are compared with linear and nonlinear waves simulated from the JONSWAP spectrum representative of long-crested extreme seas, and also with oceanic data gathered in the North Sea. Uncertainties arising from the variability of probability estimates derived from sample populations of limited size are considered. Ultimately, the comparisons show that nonlinearities do not have any discernable effect on the crest-to-trough heights of oceanic waves. Most of the linear models considered yield similar and reasonable predictions of the observed data trends. Gram–Charlier type distributions seem neither effective nor particularly useful in describing the statistics of large wave heights or crests under oceanic conditions. However, they do surprisingly well in predicting unusually large wave heights and crests observed in some 2D wave-flume experiments and 3D numerical simulations of long-crested narrow-band random waves.     

Long waves dissipation and harmonic generation by coastal vegetation

In this paper, we study the harmonic generation and energy dissipation as water waves propagating through coastal vegetation. Applying the homogenization theory, linear wave models have been developed for a heterogeneous coastal forest in previous works (e.g. [17], [10], [11]). In this study, the weakly nonlinear effects are investigated. The coastal forest is modeled by an array of rigid and vertically surface-piercing cylinders. Assuming monochromatic waves with weak nonlinearity incident upon the forest, higher harmonic waves are expected to be generated and radiated into open water. Using the multi-scale perturbation theory, micro-scale flows in the vicinity of cylinders and macro-scale wave dynamics are separated. Expressing the unknown variables (e.g. velocity, free surface elevation) as a superposition of different harmonic components, the governing equations for each mode are derived while different harmonics are interacting with each other because of nonlinearity in the cell problem. Different from the linear models, the leading-order cell problem for micro-scale flow motion, driven by the macro-scale pressure gradient, is now a nonlinear boundary-value-problem, while the wavelength-scale problem for wave dynamics remains linear. A modified pressure correction method is employed to solve the nonlinear cell problem. An iterative scheme is introduced to connect the micro-scale and macro-scale problems. To demonstrate the theoretical results, we consider incident waves scattered by a homogeneous forest belt in a constant shallow depth. Higher harmonic waves are generated within the cylinder array and radiated out to the open water region. The comparisons of numerical results obtained by linear and nonlinear models are presented and the behavior of different harmonic components is discussed. The effects of different physical parameters on wave solutions are discussed as well.     

An experimental and numerical study of the freak wave speed

The propagation speed is one of the most important characteristics for describing freak waves. The research of freak wave speed is not only helpful for understanding the generation mechanism and evolution process of freak waves, but also applicable to the prediction. A stable and accurate method is proposed for the calculation of the freak wave speed, in which physical model tests are carried out to measure the motion of the largest wave crest along the wave tank. The linear regression relationship between the spatial position of the largest wave crest and instantaneous moment is established to calculate the speed of totally 248 cases of experimental freak waves and 312 supplementary cases of numerical freak waves. Based on the calculate results, a semitheoretical and semiempirical formula is proposed by using a regression analysis method to predict the speed of the freak wave, and the nonlinear characteristic of the freak wave speed is also investigated.     

考虑多种非线性因素的张力腿平台动力响应

建立了一种考虑多种非线性因素的张力腿平台(TLP)分析模型,其中包括六自由度有限位移,各自由度之间的耦合,瞬时湿表面,瞬时位置,自由表面效应及粘性力等因素引起的非线性。推导出TLP六自由度非线性运动方程。对某典型张力腿平台ISSC TLP进行了时域上的数值计算,求得该平台在规则波作用下的六自由度运动响应。用退化到线性范围的解与已有解进行了对比,吻合良好。数值结果表明,综合考虑非线性因素后响应明显改变,建议在设计TLP平台的时候,考虑非线性因素的影响。     

Experimental Study of Nonlinear Behaviors of A Free-Floating Body in Waves

Nonlinear behaviors of a free-floating body in waves were experimentally investigated in the present study. The experiments were carried out for 6 different wave heights and 6 different wave periods to cover a relatively wide range of wave nonlinearities. A charge-coupled device (CCD) camera was used to capture the real-time motion of the floating body. The measurement data show that the sway, heave and roll motions of the floating body are all harmonic oscillations while the equilibrium position of the sway motion drifts in the wave direction. The drift speed is proportional to wave steepness when the size of the floating body is comparable to the wavelength, while it is proportional to the square of wave steepness when the floating body is relatively small. In addition, the drift motion leads to a slightly longer oscillation period of the floating body than the wave period of nonlinear wave and the discrepancy increases with the increment of wave steepness.     

Experimental study of nonlinear behaviors of a free-floating body in waves

Nonlinear behaviors of a free-floating body in waves were experimentally investigated in the present study. The experiments were carried out for 6 different wave heights and 6 different wave periods to cover a relatively wide range of wave nonlinearities. A charge-coupled device (CCD) camera was used to capture the real-time motion of the floating body. The measurement data show that the sway, heave and roll motions of the floating body are all harmonic oscillations while the equilibrium position of the sway motion drifts in the wave direction. The drift speed is proportional to wave steepness when the size of the floating body is comparable to the wavelength, while it is proportional to the square of wave steepness when the floating body is relatively small. In addition, the drift motion leads to a slightly longer oscillation period of the floating body than the wave period of nonlinear wave and the discrepancy increases with the increment of wave steepness.