Calculating the Wave Force on Partially Immersed Large-Scale Horizontal Cylinders

Large-scale interceptors constitute the main structure of offshore self-driven floating marine litter collection devices, and the structural stability of such interceptors under the action of waves directly influences the overall safety of the device. When the ratio of the diameter of a horizontal cylinder in such interceptors to the incident wavelength is larger than 0.25, the wave force can be calculated by using the diffraction theory, by considering the problem as that of the interaction between the waves and a partially immersed large-scale horizontal cylinder. In this study, an analytical approach to calculate the wave force on a partially immersed large-scale horizontal cylinder was formulated by using the stepwise approximation method. Physical model tests were conducted to investigate the effects of different factors (wave height, period, and immersion depth) on the wave force on a large-scale horizontal cylinder under conditions involving short-period waves. The results show that both horizontal and vertical wave forces on the cylinder increase as the wave height (immersion depth) increases in most cases. The vertical wave force decreases with the decrease of the period. For the horizontal wave force, it increases with the decrease of the period when the wavelength is larger than the diameter of the cylinder and decreases with the decrease of the period when the wavelength is smaller than the diameter of the cylinder.


Introduction
With the development of the marine industry and the increase in marine activities, the accumulation of floating marine litter is increasingly affecting the coastal marine environment. Such litter is mainly distributed in offshore areas such as tourist and recreation areas, agricultural and fishery areas, ports and shipping areas, and seaside resorts, and they considerably affect the coastal landscapes, ship navigation, marine life, and human health (Jeftic et al., 2009;Rech et al., 2014). In recent years, the clean-up of floating marine litter has become a critical objective in the field of international marine environmental protection (Ryan and Moloney, 1993;Law et al., 2010). The development of self-driven equipment to collect coastal floating litter has facilitated the management of such litter, overcome the limitations (low efficiency, time-and labour-intensive) of manual clean-up, and provided an efficient and automated cleaning mode. In particular, the self-driven equipment for offshore floating litter collection consists of a collector and an interceptor, generally arranged along the bay in a "duck-billed" shape. Under the action of waves and tidal forces, the floating mar-ine litter is aggregated by the interceptor and transferred to the collector to enable all-weather, energy-free, and automatic collection (Fig. 1). In a coastal region with poor shielding conditions and high wind waves, a large-scale horizontal cylinder structure should be used as the interceptor. In general, the cross-sectional diameter of a large-scale interceptor is set between 1 m and 3 m; such an interceptor is suitable for application in open sea areas with large wind waves and can also realise wave dissipation. The interceptor is the main structure in floating marine litter collection equipment. In particular, the structural stability of the interceptor under wave action directly determines the overall safety of the equipment. Therefore, to optimise the structural design of such equipment, this study focuses on the wave force on the horizontal cylindrical structure of a large-scale interceptor.
The cross-sectional diameter of cylinders in large-scale interceptors ranges from 1 m to 3 m. Therefore, under shortperiod conditions, D/L>0.25, the problem can be considered as the interaction between waves and partially immersed large-scale horizontal cylinders. If the viscosity and compressibility of water are ignored, the interaction between a wave and a large-scale cylinder (D/L>0.25) can be examined in a simplified manner by solving the Laplace equation satisfying certain boundary conditions and initial conditions. In 1954, MacCamy andFuchs (1954) used the eigenfunction expansion method to obtain an analytic solution to calculate the wave load on vertical cylinders, and this approach has been widely used in engineering practice. In addition, the calculation of the wave load on large-scale horizontal cylinders has been extensively investigated (Dean, 1959;Levine, 1965;Mandel and Goswami, 1985;Linton, and McIver, 1995;Porter and Evans, 2009;Loh et al., 2018). However, until now, directly determining the wave force on horizontal cylinders (especially partially immersed ones) has remained challenging.
Using a multipole expansion method, Ursell (1949) pioneered a theoretical analysis method to solve the heave problem of a horizontal cylinder. The so-called multipole expansion method can be used to express the velocity potential in terms of the sum of an infinite series of singularities of the Laplace equation for the inside of the object, to ensure that the equation can satisfy the free water surface condition, submarine condition, and radiation condition at infinity. These singularities, i.e., the so-called "multipoles", involve two parts: the no-wave potential that dissipates to zero in the far field away from the object, i.e., the non-propagating mode, and the oscillation potential that propagates towards the far field along the radial direction of the object, i.e., the propagating mode. The method for treating a fully or partially immersed horizontal cylinder depends on the location of the singularities.
Specifically, if the singularities lie below the free water surface, the equation involves complete no-wave terms and oscillation terms. Fu et al. (2012) used the multipole expansion method to perform a theoretical analysis of the wave forces on horizontal cylinders at a finite water depth. The authors decomposed the velocity potential into symmetric and antisymmetric potentials and provided multipole linear combination expressions for the velocity potential. The undetermined coefficients in the expression of the velocity potential were determined using the boundary conditions of the object surface and the orthogonality of trigonometric functions. Finally, an analytic expression for the wave force on a submerged horizontal cylinder was derived. In addition, the researchers compared the theoretical results with the experimental data and observed a high degree of goodness of fit. To address the scattering and radiation problems of the interaction between linear oblique waves and completely submerged long horizontal cylinders at a finite water depth, Shen et al. (2007) used the multipole expansion method and analyzed the effects of the number, arrangement, and spacing of cylinders on the wave force, hydrodynamic coefficient, and transmission and reflection coefficients.
When the singularities are located on the free water surface, certain non-wave terms become zero, and only certain even-and odd-numbered terms remain in the non-wave potential. Furthermore, the oscillation terms are different from those in the case of submerged singularities. A source point located on the free water surface can be used to solve the floating problem. Therefore, the multipole expansion method can be used to determine the wave force on a partially immersed horizontal cylinder. Considering this aspect, Dean (1959) provided an analytical solution for the interaction between waves and a fixed partially immersed cylinder at an infinite water depth. Mandel and Goswami (1985) provided an asymptotic analytical solution for the interaction between waves and a fixed partially immersed cylinder at a finite water depth and noted that the water depth considerably influences the reflection and transmission coefficients.
The horizontal cylinder in the large-scale interceptor considered in this work is only partially immersed in water, and the multipole expansion method cannot be applied to calculate the wave force on such structures. At present, a feasible approach to obtain a solution is to match the numerical calculations and analytical methods. In this approach, the boundary element method based on Green's functions is adopted for the inner domain, analytic expressions are used for the outer domain, and the two results are matched to jointly solve the hydrodynamic problem of an arbitrarily shaped object (Taylor and Hu, 1991;Nossen et al., 1991;Liu et al., 2016;Dişibüyük et al., 2017;Das and Sahu, 2019). However, this approach involves a large computational burden, which is not desirable for solutions based on an analytical theory. Therefore, the calculation of the wave force on large-scale partially immersed horizontal cylinders requires further systematic investigation. This study focuses on the wave force on large-scale interceptors under short-period conditions considering different factors (period, wave height, and immersion depth). An analytical approach to calculate the wave force on a large-scale partially immersed horizontal cylinder (D/L>0.25) is developed using the stepwise approximation method, which expands the existing scope of research pertaining to the interaction between waves and a partially immersed structure and provides theoretical guidance regarding the quantitative determination of the distribution characteristics of the wave force around a horizontal cylinder.

Analytical theory to calculate the wave force on a large-scale horizontal cylinder
Assuming that a fluid is in viscid and incompressible, the interaction between waves and large-scale cylinders can be examined by solving the Laplace equation that satisfies certain boundary conditions and initial conditions. As shown in Fig. 2, the centre of the cylinder cross-section is considered the origin of the coordinate system, the x and z axes respectively represent the wave propagation and vertical directions, h is the water depth, a is the cylinder radius, and d is the immersion depth.
To facilitate the solution, the entire region is divided into three subregions: , and the corresponding velocity potentials are , , and .
In Region I = (−∞, −x 0 ), the fluid motion can be described by the velocity potential ω ϕ 1 where t is time, is the angular frequency, and (x, z) is a complex spatial velocity potential independent of time which satisfies the Laplace equation can be expressed as the superposition of the incident wave ϕ I (x, z) and the radiation wave (x, z,), that is The incident wave with amplitude A propagating along x direction can be written as: where , k is the wave number, which satisfies the dispersion relation = gk tanh(kh), g is the gravity acceleration.
The radiation potential can be written as follows: With the method of separation of variables, the solution for Laplace's equation (5) can be expressed as: where k n , the eigenvalues of the n-th wave modes in Region I, can be given by Therefore, the velocity potential of Region I = (−∞, −x 0 ) can be expressed as: The coefficient A m needs to be determined considering a specific problem.
where r is in the radial direction. The stepwise approximation method is used to solve the problem in this region. As shown in Fig. 3, a series of steps are used to fit the immersed portion of the large-scale horizontal cylinder. The immersion depth is divided into n equal parts, and, correspondingly, Region II can be divided into 2n−1 subregions: II 1 , II 2 ,..., II i ,..., II 2n−1 . Because of the symmetry of the cylinder, the immersion depth and width of the i-th step II i are identical to those of the (2n−i)-th step II 2n−i , and the ordinate is where i = 1, 2, ... , n, and the abscissa of the two endpoints are and The velocity potential of each region satisfies and where i = 1, 2,…, n, and j = 1, 2,…, n−1.
Similarly, the method of separation of variables is used to obtain the eigenfunction expansion formula of the velocity potential in each subregion, as follows: The velocity potential (x, z, t) =Re[ϕ 3 (x, z) e −iωt ] for Region III = (x 0 , ∞) satisfies and the radiation condition at infinity is as follows: ∂ϕ The velocity potential of Region III = (x 0 , ∞) can be expressed as: The entire physical region is divided into three large regions, namely, the constant depth regions of the open sea, I: x ≤−x 0 and III: x ≥ x 0 , as well as the region in which the cylinder is located, II: −x 0 ≤ x<x 0 , which is further divided into 2n−1 subregions in accordance with the stepwise approximation method. According to the fluid continuity conditions, the water surface elevations and flow rates at the junction of adjacent regions are equal, i.e., the following conditions are satisfied at x =−x 0 , −x i , …, x 2n−j and x 0 , respectively: Once the coefficients to be determined are obtained from the matching conditions, the velocity potential function of the entire region can be determined. In this work, the infinite summation stops when the error is smaller than 10 −6 , which can be satisfied by setting m = 30.
The wave force on an object can be determined by integrating the wave pressure on the object surface over the object area: where n is the unit normal vector of the object surface. By substituting the velocity potential into the linearized Bernoulli equation, the components of the wave force can be obtained, as follows:

Test equipment and setup
The tests were conducted in the wave tank in the Harbour and Navigation Laboratory of Xiamen University of Technology. The water tank, with dimensions of 35 m×0.7 m×0.8 m (length×width×height) and an effective water depth of 0.6 m, was placed on a level ground. A piston wave maker was employed in the wave tank, which could produce waves with a maximum height of 0.2 m. A wave dissipater was installed on the other end of the water tank. The test model was placed in the water tank, approximately 13 m from the wave maker, as shown in Fig. 4.   Fig. 4. Layout for the physical model test.
In the wave force test, the force on the model was measured using L3 three-component force sensors which have a vertical and horizontal force measurement range of 60 kg and 20 kg, respectively, with an error smaller than 0.5% of the corresponding range. Before the measurement, all the sensors were calibrated to have a linear confidence level of more than 0.999. The total force sensor was attached to the model to form one unit and fixed by a custom rigid cross rod. The vertical rod could slide and was marked with a scale to ensure that the immersion depth of the model could be arbitrarily adjusted, and the bottom of the rod was a rigid fixture, which could firmly hold the model, as shown in Fig. 5. The test was conducted at an acquisition frequency of 0.01 Hz for an acquisition duration of 1 min. Each test was repeated three times, and the results were averaged. Based on a comprehensive consideration of the model preparation conditions and wave tank test conditions, the scale ratio of the model was set as = 1:10 in the test. According to the specifications of the JTJ/T 234-2001 Wave Model Test Regulation, the time scale of the test was set as λ t = .
The dimensions of the model are listed in Table 1. The wave conditions of the model are presented in Table 2. In the test, the water depth was fixed, and the other parameters were modified accordingly.   Fig. 6 shows that due to more part of columns above the static water surface, the horizontal wave force in the positive direction on the large-scale cylinder is larger than that in the negative direction. Fig. 8 gives the trend of dimensionless maximum wave force with wave height, with an increase in the wave height, the horizontal wave force increases sharply, while that the vertical wave forces first increase and then decrease. This phenomenon occurs because as the wave height increases, the reflection decreases, and the wave transmission increases the difference between the front and rear water surface elevations.
Considering that the frequency characteristics of horizontal force under different wave heights are obviously different, the frequency spectra are given in Fig. 9. It can be seen from the spectrum that the increase of wave height produces high frequency value. In addition, with the increase of wave height, the nonlinearity of horizontal wave force exerted on the cylinder increases, and new frequency components are produced. As shown in Fig. 9, the frequency first exhibits a secondary peak and later exhibits a larger peak at wave heights of 0.09 m and 0.12 m.

Effect of immersion depth on wave force
Figs. 10 and 11 show the evolution of the wave force on the large horizontal cylinder with time under different immersion depths. The parameters were set as follows: period T = 0.9 s, water depth h = 0.5 m, wave height H = 0.06 m, horizontal cylinder cross-section diameter D = 0.3 m and   To more deeply understand the influence of period, Fig. 15 shows the variation trend of the maximum value of the wave force with period (or wavelength). The vertical wave force decreases with the decrease of the period. For the horizontal wave force, when the wavelength is larger than the diameter of the cylinder, it increases with the decrease of period; otherwise, it is the opposite.

Conclusions
When the ratio of the diameter of a horizontal cylinder in a large-scale interceptor to the incident wavelength, D/L, is larger than 0.25, the wave force can be calculated by using the diffraction theory. Based on the stepwise approximation method, this paper proposes an analytical approach to calculate the wave force on a partially immersed large-scale horizontal cylinder. Physical model tests were performed to investigate the effect of different factors (wave height, period and immersing depth) on the variation pattern of the wave force on a large-scale horizontal cylinder under shortperiod conditions (D/L>0.25). The proposed theory expands the existing scope of theoretical research pertaining to the interaction between waves and partially immersed structures and provides a reference to predict the wave forces on  large-scale horizontal cylinders in practical engineering projects. The following conclusions can be drawn from this study.
(1) For a large-scale horizontal cylinder, the maximum horizontal wave force in the positive direction is larger than that in the negative direction. With an increase in the wave height, the horizontal wave force increases sharply, while the vertical wave forces first increase and then decrease.
(2) Both the horizontal and vertical wave forces on a cylinder increase as the wave height (immersion depth) increases in most cases.
(3) The vertical wave force decreases with the decrease of the period. For the horizontal wave force, when the wavelength is larger than the diameter of the cylinder, it increases with the decrease of the period; otherwise, it is the opposite.  LIU Bi-jin, FU Dan-juan China Ocean Eng., 2021, Vol. 35, No. 2, P. 291-300 299