Pile group is a commonly used structure in coastal and ocean engineering. The wave action on pile group structures has always been the focus of scholars' research. Because of the vortex shedding around the piles, small scale piles are different from large scale piles. Except inline force, transverse force of a small scale piles cannot be ignored. In order to explore the interaction between different piles, experimental investigations of the interaction of irregular waves with small scale, vertical bottom-mounted pile group which has 9 piles in side by side arrangement have been carried out. Considering the comprehensive influence of the relative pile diameter and KC1/3 number, a new parameter KCLD1/3 is proposed. The influence of relative spacing on the wave force of the pile group is analyzed. The change of pile group coefficient, inline force and resultant force with KCLD1/3 parameter and relative spacing are discussed.
Pile group structures are widely used in the area of coastal and offshore engineering such as crossing bridge and offshore wind turbine platform. However, there are many uncertain issues in the wave force of such piles. Accurate analysis of wave force is essential for designing pile group-supported marine structures. When the distance between the piles in the pile group structures is small, wave force on a single slender pile is significantly affected by the neighboring piles. The formula which is based on the concept of Morison et al. (1950) for calculating the wave force of a single isolated pile is not applicable.
So, a lot of laboratory tests had been conducted to study the interference effects of neighboring piles under the action of irregular waves. Chakrabarti (1981, 1982) measured inline forces on instrumented sections of the piles, the inertia and drag coefficients (Cmand Cd) are determined based on experimental data by applying for instance the least square fit. These coefficients are shown as functions of the KC number which is suggested by Keulegan (1958). The total forces on the piles were computed from the mean curves of the inertia and drag coefficients. The correlation between the maximum calculated forces and the corresponding measured maximum forces is good. However, any relationship with the Reynolds number could not be established primarily because of the small range of Reynolds number covered by the test. Sundar et al. (1998) found that the variations of Cd and Cm with KC for inclined cylinders are significantly wide. Boccotti et al. (2012, 2013) revealed that the inertia and drag coefficients are given as a function of KC number and Reynolds number Re for KC in (0, 20) and Re in (2*104, 2*105). Calculation of wave force of pile group by the Morison equation depends on inertia coefficient, Cm, and drag coefficient, Cd. However, the inertia and drag coefficients are not easy to be determined.
Wave transformation of multidirectional waves were measured in a three-dimensional laboratory model of a coral reef with a steep slope and a horizontal reef flat. The experimental results showed that when the waves broke on the reef flat, the relative breaking wave height (H/d)b was influenced by the deep-water wave steepness and the relative water depth db/Lo, thus the criterion for the breaking waves on the reef flat was proposed. The nonlinear wave parameters and representative wave heights were analyzed and it was confirmed that the wave transformation on this terrain was influenced by the incident wave directions and directional spreading parameters slightly.
Coral reefs are present in many tropical and subtropical regions, they are usually described as a steep slope and a reef flat with shallow water. As wave propagates from deep water to shallow region, the wave profile becomes steep, and the wave breaks when the wave height reaches certain limit. It is essential to study the wave propagation characteristics on reef terrain for the evaluation of coastal processes and the design of coastal structures on the reef flats.
Wave breaking is one of the most important phenomena in wave propagation, most of the previous studies of wave breaking were conducted on the common coastal slopes. Munk (1949) described the relationship between the breaking wave height and deep-water wave steepness based on an evaluation of the breaking wave energy using the theoretical solitary wave equation. According to Airy wave theory, Komar and Gaughan (1972) modified the factor in Munk’s formula using the available laboratory and field data. Goda (1970) proposed the formula about the relationship between the breaking wave height with the relative water depth and the gradient of slope. And according to the empirical formula of Goda (1970), some researchers proposed the modified breaking criteria by analyzing amount of experimental and field data (Kamphuis, 1991; Rattanapitikon and Shibayama, 2007; Seyama and Kimura, 1988). However, the criteria of wave breaking on coral reef terrain are rare.
The numerical wave flume using first order wavemaker theory has been well established and widely used for a long time. But the existing numerical models based on the first order wave-maker theory will lose accuracy as the nonlinear effects enhance. Because of the different propagation velocities of the spurious harmonic waves and the primary waves, the simulated waves with the first order wave-maker theory have an unstable wave profile. In this paper, a numerical wave flume with piston-type wave maker is established. The comparison of the surface elevation using first order and second order wave-maker theory proves that second order wave-maker theory can make stable wave profile in both temporally and spatially. Harmonic analysis is applied to prove the superiority of second order wave-maker theory.
Piston-type wave-maker has been widely used to generate waves in laboratory flume or basin. Havelock (1929), Svendsen (1985) and Dean & Dalrymple (1991) had well established first order wave-maker theory. Flick & Guza (1980) and Ursell et al. (1960) had verified the first order wave-maker theory by experiments in the laoratory (see also Galvin, 1964; Keating & Webber, 1977). Small-amplitude assumption is the basic assumption of first order wave-maker theory. The small-amplitude waves will decompose into a primary wave and spurious superharmonic wave, which will affect the stability of the wave profile (see Gōda and Kikuya, 1964; Multer and Galvin, 1967; Iwagaki and Sakai, 1970), when the motion of the wave-maker is sinusoidal. In early 1847, Stokes found the superharmonic wave by regular wave in terms of a perturbation series using the wave steepness as the small ordering parameter. But the problem of generated nonlinear wave was gave a solution by Fontanet (1961). He found the spurious superharmonic wave by piston-type wave-maker with sinusoidal motion in Lagrangian coordinates and suggested that it can be restrained using wave paddle control signal with an addition component.
Further, Moubayed & Williams (1994) extended second order wave-maker theory from the regular wave to the bichromatic wave. For the irregular waves, second order wave-maker theory has both sum and difference frequencies in the interaction terms. Longuet-Higgins & Stewart (1962, 1964) deduced the subharmonics generated by wave components interaction under the narrow band assumption. Flick & Guza (1980) pointed out that spurious long wave will be generated by a first order bichromatic control signal. Barthel et al. (1983) used the second order difference frequencies wave paddle control signal to restrain spurious long wave and expended the theory to the flap-type wave-maker. Schaffer (1996) derived second order wave-maker theory including sum frequencies and difference frequencies components without the narrow band assumption. The theory was applied to the piston-type and flap-type wave-makers and was verified by experiments. Schaffer & Steenberg (2003) extended the second order wave-maker theory to multidirectional waves.
In this paper, a numerical wave tank is developed based on High Order Spectral (HOS) method, considering the wave-maker boundary. Comparison of the numerical results with the experimental data shows that the model can simulate the wave generation and propagation, even for the freak wave. Using this model, the 2D irregular wave trains with single peak spectrum and the bimodal spectrum are simulated for a long time. Freak waves are observed in the numerically simulated random wave trains. As a result of the existence of high frequency energy, the probability of the freak wave generation in the bimodal spectrum is greater than that of the single peak spectrum, which provides a basis for further research of freak waves.
The freak wave (also called extreme wave, rouge wave) is defined as the wave whose height is more than twice the significant wave height, which represents the extreme oceanic conditions. It has larger wave height and stronger nonlinear, and can cause serious damage to the oceanic structures and ships. In recent years, as frequent human activities moving towards the deep ocean, the harsh marine environment becomes a threat to oceanic engineering. The accurate prediction of the occurrence of freak waves, which are at the tail of the probability curve, is essential for the design of offshore structures. It is also helpful to deeply understand the mechanism of freak wave generation.
In the past years, many researchers paid more attentions to the generation of freak wave and its influencing parameters. Some investigations showed that the large wave heights do not obey the Rayleigh distribution. Haring et al. (1976) and Forristall (1978) found that the large wave heights observed in storms are less than those predicted by the Rayleigh distribution. While, measured data in the field of Dankert (2003) showed the Rayleigh distribution underestimated the occurrence of freak wave. Some later research found that the occurrence of freak wave was affected by many parameters, such as wave steepness, the bandwidth of frequency spectrum, wave directional distribution, and so on. Janssen (2003) studied the freak wave using the Zakharov equation. The results showed that freak waves are well described by the so-called Benjamin-Feir Index (BFI), which is the ratio of wave steepness and the width of the spectrum. The relationship between BFI and large waves was experimentally verified by Onorato et al. (2005) who found experimental evidence that the tail of the probability density function for wave height is strongly dependent on the BFI. For a small BFI, the probability distributions are consistent with the Rayleigh distribution, but for a large BFI, Rayleigh distribution clearly underestimates the probability of large events. Further, Waseda et al. (2009), Latheef and Swan (2013) and Li et al. (2015) investigated the influence of wave direction distribution on the freak wave occurrence. Their results showed that the probability of freak wave occurrence decreased with increasing directional spreading.
Real sea waves are multi-directional, and it's quite different with unidirectional waves. In present, based on the linear theory of wave interaction with an array of circular bottom-mounted vertical cylinders, systematic calculations are made to investigate the effects of the wave directionality on wave loads in real conditions. The time series of multi-directional wave loads can be simulated. The effect of wave directionality on the normal and transverse wave force on an array of tandem cylinders is investigated. It was found that the wave directionality has a significant influence on the transverse force. The biggest transverse force is found to occur on the rear cylinder rather than the front one. This is quite different from the results in unidirectional waves and should be paid much more attention in the design of offshore structures.
Circular cylinder are frequently used as the foundation of offshore structures, for example, include bridges, wind turbine foundations, offshore platforms and floating airports. In the designing of ocean engineering, wave loading is an important factor. For wave interaction with a large-scale cylinder (D/λ>0.15, where D is the diameter of a cylinder and λ is the wave length), wave diffraction is important and should be considered. A superposition eigenfunction expansion method was used by MacCamy and Fuchs (1954) to obtain a linear solution, based on the assumption that the incident wave has a small steepness. For the case of waves acting upon an array of cylinders, the effect of a given cylinder on the incident wave will produce a scattered wave which will in turn be scattered by adjacent cylinders. Thus the computation of the velocity potential must account for the diffraction of the incident wave field by each body and the multiple scattering from other bodies. An exact solution for the diffraction of linear water waves by arrays of bottom-mounted, vertical circular cylinders was first given by Spring and Monkmeyer (1974) using a direct matrix method. It represented an extension of the single cylinder case presented by MacCamy and Fuchs (1954). An accurate algebraic method was developed by Kagemoto and Yue (1986) to calculate the hydrodynamic properties of a system of multiple three-dimensional bodies in water waves. Subsequently, a simplified expression for the velocity potential in the vicinity of a particular cylinder was developed by Linton and Evans (1990) who led to simple formulae for the first-order and mean second-order wave forces on multiple cylinders as well as the free surface profile.
A series of experiments with normal and oblique wave are carried out in a laboratory wave basin to observe the wave breaking on lens topography, which has a focusing effect on the wave. The formulas of wave breaking on slope topography are first summarized in this paper. To investigate the difference between wave breaking parameters of these two topographies, the formulas are categorized into four types and compared with experimental data on lens topography. Two different water levels were adopted in this experiment to distinguish the positions of wave breaking. For both water levels, the experimental results indicate that the breaking wave height on lens topography is larger than the estimations by formulas on slope topography. But the wave incident angel α has little influence on breaking wave parameters.
Wave breaking is an important phenomenon in coastal engineering. Many breaking criteria have been proposed to calculate the wave parameters at the breaking point. McCowan (1894) proposed the wave limiting height based on the assumption that a solitary wave breaks when the crest angle approaches a limiting value or the velocity at crest surpasses the celerity of the wave, as following:
H b/db = 0.78 (1)
in which Hb is the wave height at the breaking point, db is the water depth. The limiting steepness of deepwater waves was found by Michell (1893):
H b/Lb = 0.142 (2)
in which Lb is the breaking wave length. Miche (1944) proposed following formula:
H b/Lb = 0.142tanh 2πdb*3)
which can be seen as the extension of Michell’s formula. Based on wave energy conservation, Munk (1949) calculated the limiting wave height ratio:
A wave-basin experiment was conducted considering broad directional spreading in different water depth. Longtime surface elevation data were recorded. Freak waves generated by different mechanisms were observed and the effects of the directionality spreading on freak wave occurrence probability were investigat ed. The results showed that both modulation instability and linear directional focusing can generated the freak wave in multi-directional wave train. The directionality spreading had significant effects on freak wave occurrence probability. In relatively deep water, the waves with a narrow directional spreading range (less than 11) and larger BFI values were unstable. The freak wave occurrence probability increased owing to the modulation instability. While, in shallow water depths or with wide directional spreading, freak waves occurrence probability decreased. This indicates that freak waves occur more easily in deep water when the wave directional spreading is narrower.
Based on the multiple-cylinder diff raction solution, systematic calculations are made about multidirectional random wave loads on cylinders. The time series of multidirectional wave loads can be calculated. The effect of wave directionality on the wave runup and wave loading on large-scale cy linders is investigated. For multidirectional random waves, as the wave directional spreading parameter s becomes small, the wave runup on most of points around cylinders would be larger. This means that the wave runup on multidirectional wave condition is larger than it for unidirectional wave. The effect of wave directionality on the transverse force is more obvious than on the normal force. The transverse force on the front cylinder is not always larger than that on the back cylinder, especially, for smaller s . As a consequence of the results, the engineering project design based on unidirectional waves cannot always give a conservative estimate.
A second-order coupling model is extended to consider 3D nonlinear wave propagation in a complete coupling numerical and physical wave basin. The first-and second-order solutions for the wave paddle position are obtained by combining the second order coupling theory with the nonlinear shallow water wave generation theory, in which the wave direction, nonuniformity and nonlinearity along with the numericalphysical-boundary can be completely included. Several comparisons between numerical and physical waves have been conducted. Comparisons of the proposed second order coupling model are also made to the existing first order coupling model. In all experiments, the results support the proposed model, which also show that it is capable of simultaneously processing large-scale numerical wave propagation with physical model including local complex wave phenomena.
A wave basin experiment is carried out to investigate the interaction between multi-directional irregular waves with a large vertical bottom-mounted cylinder. In the experiment, the ratios of the significant wave height and water depth (H1/3/d) are 0.08, 0.12 and 0.16, the relative size between cylinder and wavelength (ka) varies from 0.44 to 1.26. In this paper, the wave run-up on the cylinder is presented. The experimental results show that the directional spreading parameter has significant effects on the wave run-up on cylinder. The maximum value of the wave run-up increases with the directional spreading concentration parameter increasing. The minimum value of the wave run-up also depends on the directional spreading parameter. For smaller directional spreading parameter, it is at 180° or 135°. At different position of the cylinder, the effect of the wave directionality on the wave run-up is different. The results could provide an important basis for the design of appropriate engineering and be regard as a reference for numerical verification.