Theoretical approximation of focusing-wave induced load upon a large-scale vertical cylinder

Until now, most researches into the rogue-wave-structure interaction have relied on experimental measurement and numerical simulation. Owing to the complexity of the physical mechanism of rogue waves, theoretical study on the wave-structure issue still makes little progress. In this paper, the rogue wave flow around a vertical cylinder is analytically studied within the scope of the potential theory. The rogue wave is modeled by the Gauss envelope, which is one particular case of the well-known focusing theory. The formulae of the wave-induced horizontal force and bending moment are proposed. For the convenience of engineering application, the derived formulae are simplified appropriately, and verified against numerical results. In addition, the influence of wave parameters, such as the energy focusing degree and the wave focusing position, is thoroughly investigated.


Introduction
Rogue waves are usually defined as waves whose height exceeds the significant wave height in two times (Sand et al., 1990;Kjeldsen, 2005). Rogue waves occur under various conditions, and live only for a short time, making them difficult to be predicted. By now, existing real-sea records about rogue waves still have not been enough, and the physical mechanisms of rogue waves still need to be improved. Although being a small probability event, encountering a rogue wave can be disastrous for ships and ocean structures. In January 1995, a 26m-high rogue wave, namely the famous "New Year Wave", hit into the "Draupner" jacket platform in the North Sea, and caused severe destruction (Haver and Andersen, 2000). Therefore, it is of great importance to study the nature of rogue waves and their interaction with marine structures.
The physical models of rogue waves can be classified into two categories, i.e., linear models and nonlinear models. Kharif and Pelinovsky (2003) gave a comprehensive summery of the existing physical mechanisms of rogue waves. Even at present time, linear models are still widely adopted. In laboratories and numerical wave tanks, the linear superposition of wave trains is probably still the only way to generate a rogue wave with randomness features, like the "New Year Wave" (Zhao et al., 2010;Fochesato et al., 2007;Sun et al., 2009;Hu et al., 2014a). Linear models are also utilized for the study of rogue-wave-structure interactions (e.g., Clauss et al., 2003Clauss et al., , 2004Soares et al., 2006). Recently, more researchers have turned to nonlinear models such as the modulational instability model, the nonlinear Peregrine breather model and its higher-order solutions, for instance, Chabchoub et al. (2011) and Hu et al. (2015a). Apart from their formation models, some researchers also study the influence of rogue waves on marine structures, which includes the entire floating or submerging bodies as well as the local structures such as pillars or plates, for instance, Sundar et al. (1999), Sparboom et al. (2001), Clauss et al. (2003Clauss et al. ( , 2004 and Soares et al. (2006). As has been mentioned, in these sorts of studies the target rogue waves are usually generated using linear models, despite nonlinear models have been tried recently (Hu et al., 2015b). Theoretical study of these problems can be seldom found, due to the fact that rogue waves are complicated in nature.
In this paper, we approximate the issue of the interaction between rogue wave and structures in an analytical way. To reduce the difficulty of solution, a single vertical cylinder is adopted, since cylinders are the fundamental components which can be frequently found in marine and offshore structures. Furthermore, we utilize the linear theory to model rogue wave. Although omitting nonlinear effects will induce some errors, the main features of rogue waves, such as large-height, short-living and sudden occurrence, can still be reflected by linear models, that is why they are still widely used in studying rogue-wave-structure interaction. Besides, as the first-order solution of nonlinear equations, the linear solution is still valuable and gives the fundamental approximation of the complex rogue-wave-structure problem.

Rogue wave solution based on the Gauss envelope
The Gauss-envelope-based model can explicitly give the expression of the surface elevation as well as the velocity potential in relatively simple forms. It can explain most features of rogue waves. As a result, the Gauss-envelope-based rogue wave has been thoroughly studied (Kharif and Pelinovsky, 2003;Hu et al., 2014b). Hu et al. (2014b) discussed the evolution of Gauss-envelope-based rogue waves in detail from the point of view of the inversion problem of initial disturbances. Here, we list the corresponding governing equations, and give the expression of the surface elevation and velocity potential without going into the detail of derivation.
The governing equations are written as: and denote the velocity potential and free surface elevation of the incident rogue wave, respectively. h is the water depth. g the is gravitational acceleration. z and x are the horizontal and vertical coordinate, respectively. t represents the time. In Eq. (1e), the right side of the equation represents the Gauss envelope, where x 0 is the focusing position of the rogue wave, k 0 is the dominant wave number of the carrier waves, and σ describes the degree of en-ergy focusing, i.e., a large σ indicates that the energy spectrum band is narrower. A 0 is the envelope amplitude and Re[ ] stands for the real part.
The solution of Eq. (1) is as follows: (2) where satisfies the dispersion relationship , and i is the imaginary unit.

Rogue-wave-induced flow around a vertical cylinder
3.1 Governing equations A vertical cylinder is placed into the flow field, with its central axis coinciding with the z-axis. R is the radius of the cylinder. The incident rogue wave is travelling along the direction of from to . Here, the diameter of the cylinder is supposed to be large enough such that the potential theory applies. Therefore, the governing equations of the above problem, within a cylindrical coordinate system, is formulated as: (4) can be expressed as the summation of the incidentwave potential and the diffraction potential . For a linear system, the entire system is thereby split into two sub-systems. One subsystem is the governing equation of the incident wave, which has already been listed in Eq.
(1). The other is the diffraction potential one, and is written as: In Eq. (5), a radiation condition is used to make the solution unique and physically meaningful.

Solution
The solution of incident potential is given by Eq.
(3). Before processing the diffraction issue, Eq. (3) is transformed into an integral form as follows: For this kind of problem, a widely-used identical equation is (Mei, 1989) where J n ( ) is the first-type Bessel function. With Eq. (7), the integral form of the incident potential is finally written as: Φ D Using Eq. (8) rather than Eq.
(3) will significantly simplified the deduction of . The right side of Eq. (8) can be understood as the superposition of a series of linear regular waves. Following this principle, the diffraction potential can also be formulated as the superposition of components which correspond to the regular wave components on the right side of Eq. (8).
For a linear regular wave, the diffraction potential induced by an upright cylinder is α n where H n ( ) is the Hankel function of the first type, and is the unknown to be determined. Since one can find Eq. (9) in almost any course book on wave loads, we do not give the detailed derivation of Eq. (9). As mentioned, the entire diffraction potential is the superposition of elemental components, which is written as: One can easily verify that Eq. (10) satisfies all the subequations of Eq. (5), except for Eq. (5e). By substituting Eq. (8) and Eq. (10) into Eq. (5e), the unknown is thereby obtained, and the diffraction potential is Finally, the whole potential is written as: 3.3 Rogue-wave-induced load With Eq. (12), the pressure and surface elevation can be determined by using the formulae as: ; To compute the pressure upon the cylinder surface, the following equation (Abramowitz and Stegun, 1964) would be useful.
With Eq. (15), the local pressure upon the cylinder surface is finally written as: The wave force upon a cylinder segment with unit height is thus written as: Although Eq. (17) shows a much simpler form compared with Eq. (16), it is still difficult to be applied to realistic engineering problems, since it is an integral formula and the function is hard to be computed if no special math tool is adopted. To further simplify Eq. (17), the identical equation , and the asymp- ) ω totic expansion of the Hankel function and are employed (This is rational with the fact that the diameter of the cylinder or kR is large enough). In addition, considering the energy focusing feature of rogue waves, we expand into the Taylor series with respect to k, in order to make the equation integrable. As a result, the simplified wave force is written as: where and .
By integrating Eq. (18) over the entire cylinder along z-direction, the global horizontal force and bending moment are formulated as: 4 Numerical validation σ σ The water depth is taken as 3 m, and the cylinder radius is 3 m. As it can be seen from Eq. (2) or Eq. (3), the characteristics of the Gauss envelope are controlled by four parameters: the wave number k 0 of the carrier, amplitude A 0 , energy focusing degree and focusing location x 0 . Since A 0 influences the rogue wave or the wave load in a linear manner, in this section, A 0 is 0.03 m, and the formula is validated with various k 0 , x 0 and . The results predicted by the theoretical formulae are compared with ones obtained by WADAM, a widely-used frequency-domain solver.
Two grid sets are adopted: 36×18 (coarse) and 180×60 (dense), along the circumferential and vertical direction, respectively. The water density is 1025 kg/m 3 . The Gauss-envelope rogue wave, lying within a frequency range of 0.2-10 rad/s, is divided into 50 unit waves with various amplitudes (this can be accomplished by discretizing Eq. (6)). To ensure the selected parameters are reasonable, Fig. 1 combines 50 unit waves, and compares the resulted surface elevation with that predicted by Eq. (2), which has already been validated by Mei (1989) and Hu et al. (2014b).
From Fig. 1, it is found that the superposition method is in a good agreement with the theoretical formula, when -20 s<t<20 s. This indicates that the frequency range and interval amount are valid. Fig. 2 displays the RAOs obtained by WADAM and by the theoretical formulas derived from the potential theory (Li, 1994). Fig. 2 proves that the parameters used in the numerical simulation are appropriate, and that a coarse grid set (36×18) is enough for this problem.
Figs. 3 and 4 show the time histories of the horizontal wave force and bending moment, respectively, where 'numerical' represents results obtained by WADAM and 'the-oretical ' by Eq. (19) or Eq. (20). From Figs. 3 and 4, it is observed that the theoretical results agree well with the numerical ones, which verifies the formulae proposed in this paper. It should be mentioned that in subplots (a)-(c) the peak values of wave load happen before t=0. This event is mainly caused by the phase difference between the incident

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Discussions
σ From Figs. 3 and 4, it can be observed that parameters , k 0 and x 0 affect the value and the appearing moment ofx the maximum load peak, as well as the contour of the envelope. For the purpose of systematically analyzing this effect, Eqs. (19) and (20) are normalized. By introducing non-dimensional variables , , , and , the normalized (or non-dimensional) wave force and wave moment are written as: where the non-dimensional coefficient and . It is worth noting that of Eq. (26) and of Eq. (27) tend to zero under the shallow water condition, which makes the normalization meaningless. One way to avoid this is to utilize and instead of and in shallow water. At deep water depth, the normalization does not need be changed, i.e., and . One can also find from Eqs. (26) and (27) that and have the same features at certain water depth, and therefore only is dis-cussed in the following sections.
5.1 Further discussion on the expressions of wave force and wave moment Before analyzing the influence of various parameters, Eq. (26) and Eq. (27) are simplified and formulated as: where As it can be seen from Eqs. (28) and (29), the expressions of the wave force and wave moment are far more complex than a sine or cosine function. Parameter and decide the amplitudes of the "carrier-wave" component of the wave force and wave moment. and decline with the growth of time, at a decreasing rate of . One interesting thing is that the focusing location does not affect carrier amplitudes and , by contrast its influence can be reflected by , which describes the evolution of the wave envelope during the wave-cylinder interaction. Usually, the focusing waves propagate in both the time and spatial domain, yet since the cylinder is located at , only the time evolution of the envelope matters. Generally, the wave force and moment are big when or . However, the wave force and moment technically do not reach their maximum values at , because also affects other parameters such as . reflects the frequency shifts due to the focusing and de-focusing effects of the rogue wave. When (or when the wave force and moment almost reach , indicating the angular frequency tends to . By contrast, when is away from 0, the frequency growth (noticing and thus ) cannot be neglected. The cylinder-induced phase deviation is reflected by the term , which is determined by . reflects the time-varying phase shifts induced by the propagation of the focusing wave. is not affected by the cylinder size, and tends to when .

Influence of water depth
Usually, the value of the maximum wave load is of most concern in engineering problems. From Eqs. (26) and (27), one can find that the water depth affects the terms , , and . The former two terms affect the results in a liner manner, yet the influence of the latter two terms is complicated. In deep water , and ; in shallow water, by changing the strategy of the normalization the two terms are also equivalent to 1. At shallow water depth , , and, x 0 +t and in deep water, , and . One important feature of shallow water is that the maximum wave force or wave moment is not affected by the focusing position , since and in shallow water. However, changes the occurrence moment of the maximum peak, as must be changed such that equals the value when the maximum peak is reached.
σσ Fig. 5 compares non-dimensional wave force obtained in deep and shallow water, changing with the variation of the focusing parameter . One can observe that wave force in deep water is slightly larger than that in shallow water. However, with the growth of the focusing degree , the in-R fluence of the water depth becomes weak. Fig. 6 compares non-dimensional wave force obtained in deep and shallow water, changing with the variation of the dimensionless radius . It is found that the influence of the water depth becomes strong with the growth of the cylinder radius.

R
Since the influence of the water depth has been investigated in Section 5.2, we study the influence of various parameters only in deep water in this section. Fig. 7 displays the dimensionless wave force changing with various . In Fig.  7, the wave force increases with the growth of , yet the increasing rate becomes small when is large. x 0 x 0 = −5x 0 Fig. 8 shows the dimensionless wave force changing with various . It is found that the maximum force peak appears around , and becomes small when is away from -5. This reveals that the wave force can be significant if the focusing location is near the frontal edge (i.e., the frontier facing the propagation direction of the rogue wave) of the cylinder, otherwise the wave force will be reduced. By comparing sub-plots from (a) to (d), it can be found thatσ σ the influence of the focusing location is strong for small , yet becomes weak with the growth of . σ Fig. 9 illustrates the dimensionless wave force changing with various . In Fig. 9, strong focusing parameter leads to greater wave force, which means the wave force increases with the growth of the energy focusing degree. However, the increasing rate becomes small when the focusing degree is already strong.

Conclusions
In this paper, the rogue-wave flow around a vertical cylinder is studied, within the scope of the potential theory. The target rogue wave is modeled by the Gauss envelope, and the rogue-wave-induced wave load, including the horizontal force and bending moment, is given analytically. The proposed formulae are verified by making comparison against the numerical results obtained with the commercial solver WADAM. In addition, the influence of wave parameters, such as the energy focusing degree and wave focusing position, is thoroughly investigated. During the study, some meaningful conclusions are drawn as follows.
(1) The proposed formulae predict the results which agree well with the numerical simulation of WADAM, and this verifies the validity of the theoretical formulae.
(2) The focusing location of the rogue wave affects the magnitude of wave load. This influence becomes remark-able under deep water condition, whereas negligible at shallow water depth. Usually, the wave load reaches its maximum value when the focusing position of rogue wave approaches the wave-meeting edge of the cylinder.