1 Introduction

There are many circumstances in which it is important to quantify the hydrodynamic in- teractions,occurring as a result of wave motion,among the members of an array of flexible cylindrical structures. One example of such offshore structures is the tension leg platform, which is usually supported by concrete piles as the foundation or the tension legs used as the anchoring device. Another example is the wave energy devices whose oscillations in waves can be used to extract the wave energy for electricity generation or desalination.

MacCamy and Fuchs ^[1] developed an analytical solution for the diffraction of incident regular waves by a single cylinder in water of finite water depth. Linton and Evans ^[2] provided a major simplification to the solution for the case of a general array which allowed the evaluation of near- field quantities such as loads and runup on the cylinders in a much more straightforward manner. Walker and Taylor ^[3] used the method for applications to linear arrays with particular emphasis on force and free surface magnification effects. Williams and Li ^[4] developed the method for porous side wall cylinder problems and found that the porosity of the cylinders might have a significant influence on the diffracted wave field and hydrodynamic loads experienced by the structures. Ning et al. ^[5] and Teng and Ning ^[6] solved the wave radiation problem of a cylinder in front of a vertical wall based on the image principle. Huang and Zhan ^[7] discussed the diffraction of nonlinear water waves by porous structures.

The dynamic behaviors of ocean structures in waves are widely studied to ensure their safety and economy. Williamson ^[8] predicted resonant in-line vibrations of a cylinder by deducing an equation of motion involving the Morison equation for flow-induced in-line forces. Lipsett and Willisamson ^[9] studied the two-dimensional response of a flexibly mounted rigid cylinder in an oscillatory flow. Li et al. ^{[10, 11]} carried out both theoretical analyses and experimental measurements. The theoretical predictions were based on the Morison equation which was solved by the incremental harmonic balance method.

Studies on wave incidents upon an array of cylinders,with relative motion occurring among the members of the array,are scarce. Teng et al. ^[12] used the eigenfunction expansion method and the image principle to solve the hydrodynamic problem of a cylinder in front of a vertical wall. The radiation from the cylinder resulting from surge,sway,roll,or pitch was analyzed. Zeng and Liang ^[13] discussed the diffraction-radiation problem of arrays of bottom-seated ver- tical circular cylinders with relative motion. The calculations indicated that the influences of the relative movements among the cylinders on the responses were obvious in some range of wave numbers.

The prediction of the response of a flexibly mounted circular cylinder array to wave forces is a difficult problem due to the complexity of the fluid-structure interaction mechanism. The aim of this study is to assess the accuracy of the proposed numerical approach to predict the responses of three elastically mounted cylinders under regular waves. The interaction problem is separated into two parts,i.e.,hydrodynamic and structural oscillation problems. The problems of wave diffraction and radiation caused by the oscillation of the cylinder are solved by the eigenfunction expansion method. The unknown oscillations are determined by the motion equations of the cylinders. Experiments are designed and performed in a wave flume,and a comparison of the calculated results with experimental values is made to verify the numerical approach.

2 Formulation of interaction problem

The interaction of linear waves with three bottom elastically mounted vertical cylinders is investigated in this paper. All circular cylinders with the radius a are situated in water with the uniform depth H. The global Cartesian coordinate system (x,y,z) is defined with the origin located on the still water level and the z-axis directed vertically upwards. The center of each cylinder,at (X _j ,y_j ) (j = 1,2,and 3),is taken as the origin of a local polar coordinate system (r_j ,θ_j ),where θ_j is measured counterclockwise from the positive x-axis. The center of the kth cylinder has polar coordinates (R jk ,α jk ) (j,k = 1,2,and 3) relative to the jth cylinder. The geometry of this problem is shown in Fig. 1. The movements of the cylinder array are restricted along the x-direction.

As usual,it is assumed that the fluid is inviscid and incompressible,its motion is irrotation, and the amplitude of the fluid oscillation is small. The cylinders are assumed to have motion with one degree of freedom in the presence of an incident wave in the x-direction with the angular frequency ω. We can introduce a velocity potential Φ(r,θ,z) as follows:

The boundary value problem to be solved here is The free surface boundary condition is The sea bed boundary condition is The cylinder surface boundary condition is where X _j is the oscillation of the jth cylinder in the x-direction which will be decided later by solving the motion equations.

Because of the linearity of the motion,the complex velocity potential φ(r,θ,z) can be de- composed into two components,i.e.,the velocity potential due to the diffraction of an incident wave acting on a fixed vertical cylinder array φ D and the velocity potential of the wave radiation due to the movement of the elastically mounted vertical cylinder array φ_R . That is to say,φ can be written as

2.1 Diffraction problem

In this case,the boundary value problem is

The radiation condition is The frequency ω and the parameter κ0are related by the following dispersion relation: For the diffraction problem,the cylinders are considered to be at rest. The diffraction potential φ_D can be written as where φ_s and φ_I denote the scattering potential and the incident wave potential,respectively.

The incident plane wave potential can be expressed in the jth local polar coordinate system by

where I_j is a phase factor associated with the jth cylinder expressed by

where H₀ represents the wave height of the incident wave.

Equation (14) in turn can be rewritten as

in which j_n denotes the Bessel function of the first kind of order n.

Following Linton and Evans ^[2] ,the general form of the scattered wave emanating from the jth cylinder can be written as

For some set of complex numbers A^j _n in Eq.(16),

where H_n is the Hankel function of the first kind of order n. The introduction of the factor Z^j_n simplifies the results which will eventually be obtained. The total potential can therefore be written as

To account for the interactions among the cylinders,it is necessary to evaluate the scattering potential φ^λ_s in terms of the representation of the incident wave potential φ^λ_I at the jth (j = 1,2,3,_j 6= λ) cylinder. This can be accomplished by using Graf’s addition theorem for the Bessel functions to give

For_j = 1,2,3,_j 6= λ,Eq.(18) is valid for r_j ＜ R_λj ,which is true on the boundary of the jth cylinder for all λ. The diffraction potential thus can be written as

which is valid if r_j ＜ R_λj for all λ,i.e.,this expansion is valid only near the jth cylinder. Then, we apply the boundary conditions as follows:

Some algebra leads to the following infinite system of equations:

By replacing m by −m in the final term in Eq.(19) and substituting Eq.(21) into it,the most general local solution near the jth cylinder can be expressed as

In order to evaluate the constants A^j_n ,Eq.(21) is truncated to a 3(2M + 1) system of equations with 3(2M + 1) unknowns as follows:

where k = 1,2,3,and −M ＜ m ＜ M.

An expression for the force on the jth cylinder is obtained by integrating the pressure over the surface of the cylinder. The horizontal force in the x-direction is

By linearizing Bernoulli’s equation,the following expression for pressure is obtained:

which leads to

Substituting Eq.(22) to Eq.(26) give

2.2 Radiation problem

In this case,the boundary value problem is

The radiation condition is

The radiation analysis has been undertaken by Teng et al. ^[12] . For the radiation problem, there is now no incident wave. The oscillations of the cylinders change the boundary conditions on the cylinder surface.

A well-known eigenfunction expansion in the local coordinate system for such a radiating wave from the jth cylinder is

where_j = 1,2,3,B^j_ml (m,l = 0,1,··· ,∞) are the unknown complex coefficients to be deter- mined,K_m denotes the modified Hankel function of the second kind with order m,and the vertical eigenfunction Z_l(z) is defined as

The eigenvalues κ_l are the real roots of the following dispersion equations:

In order to express the radiation potential in the local coordin

in which r_k ≤ R_jk ,I_n is the nth-order modified Bessel function of the first kind,R_jk is the distance between the centers of the kth and jth cylinders,α_jk is the angle defined positive anticlockwise from the positive x-axis to the line joining the centers of the jth and kth cylinders.

Therefore

where_j = 1,2,3. The total radiation potential can be expressed as a summation of waves from each cylinder as follows:

By replacing_n by −n,we obtain

Applying Eq.(29),we have Utilizing the orthogonality properties of eigenfunctions Z_l(κlz),cos(nθ_k),and sin(nθ_k) leads to the following relationships between the coefficients B^k_ml : where

and in which

In order to evaluate the coefficients B^k _nl ,the equation is truncated to a (2M + 1)3(L + 1) system of equations with (2M + 1)3(L + 1) unknowns,where k = 1,2,3,L = maxl,and M = maxm.

This system may be solved by standard matrix techniques. In this manner,the velocity potentials may be determined as functions of the oscillations of the cylinders,X _j (j = 1,2,3), which will be determined by the motion equations of the cylinders later.

With Eqs.(42) and (43),the formula for the velocity potential near the jth cylinder can be reduced to

where the following Wornskian relations for Bessel functions are used: The radiation forces on the jth cylinder can be computed from the radiation potentials as follows: where X _j (j = 1,2,3) is the oscillation in the x-direction of the jth cylinder.

Let

Then,we can obtain the radiation force F^R_kx due to the unit oscillation of the jth cylinder. The added mass and damping of the radiation problem,µ jk and λ_jk ,are given,respectively,by Then,the radiation force can be written as

2.3 Wave elevation

An expression for wave elevation is derived here. Considering Bernoulli’s equation,the dynamic boundary condition for linearized pressure can be written as

on z = 0. Rearranging η and substituting Eqs.(44) and (22) for the radiation and diffraction potentials,we can obtain the free surface elevation.

3 Solution of wave structure interaction problem

The equations used to describe the x-direction response of the oscillating vertical circular cylinders in waves can be written as follows:

where_j = 1,2,3,m_j is the mass of the cylinder in air,c_j is the structural damping of the cylinder in air,k_j is the effective spring stiffness,and X _j is the cylinder displacement. The over dot in Eq.(52) represents differentiation with respect to t.

The horizontal x-direction displacement response of the cylinders is harmonic,i.e.,

where_j = 1,2,3.

Substituting Eq.(53) into Eq.(50),we have

where_j = 1,2,3.

Substituting Eqs.(53) and (54) into Eq.(52),we obtain

where_j = 1,2,3. It has been mentioned in an earlier section that the hydrodynamic parameters, µ_kj and λ_kj ,are determined by the radiation problem. Using Eq.(55),the oscillations of the cylinders,X _j (j = 1,2,3),can be determined by standard matrix techniques.

The radiation velocity potential can then be determined by Eq.(44). Therefore,the total velocity potential can be determined by Eq.(6).

4 Laboratory experiments

To investigate the responses of the elastically mounted vertical cylinder array in waves,laboratory experiments are conducted in a wave flume with a piston type wave generator installed at one end and an absorber system installed at the other end to minimize the wave reflections.

4.1 Experimental set-up

The physical dimensions of the wave flume are 25 meters long,1.5 meters wide,and 1.5 meters deep. The experimental set-up is shown schematically in Fig. 2. Capacitance wave gages are used to record the free surface and oscillation around the vertical cylinders. The principle of the wave gauges is that the capacitance changes with the wetted length of the measuring wire during the passage of wave trains.

4.2 Measurement of parameters c and k

The three cylinders used in the study of the in-line response are made of a plastic tube with an outside diameter of 215mm and a length of 1000mm (see Fig. 3). It is positioned vertically with its bottom at a distance of 600mm below the water surface and mounted on two steel plates having a cross-section of 30mm×2mm. The movements of the cylinder array are restricted along the x-direction. An accelerometer is placed on the top of each cylinder. The type of accelerometers installed on the top of the elastically mounted cylinders is Kyowa ASQ- 1BL. It is a closed-loop system with high accuracy,stability,and reliability. The responding signal conditioner is Kyowa VAQ-500A.

The structural damping c and spring stiffness k are first determined by allowing each cylinder to oscillate freely in air. The required parameters can be evaluated as follows:

where x(t) is the value in the curve at time t.

4.3 Responses of elastically mounted array of cylinders in regular waves

The experiments are performed at a water depth of 0.6m,an incident wave height of 10mm, and wave periods ranging from 0.5s to 2.0s. The sampling frequency is 100Hz. The response amplitudes of the cylinders in regular waves are measured,and the zero-upcrossing technique is used to analyze the experimental results.

A picture of an arrangement of cylinders is shown in Fig. 5,where 1,2,and 3 means the different cylinders. As shown in Fig. 5,three cases are studied here.

5 Numerical results and discussion 5.1 Wave elevation around cylinders

After the diffraction force and radiation coefficients are determined,the motion equations can be used to obtain the oscillations of each cylinder. The free surface profiles can then be plotted. A computer program has been written to implement the above analysis. The typical diffraction and radiation wave profiles for Case (c) are shown in Fig. 6.

5.2 Validation of experimental results

Both the experimental and numerical results for the amplitude of the response of the cylinder array at the excitation frequency in regular waves at a wave height of 0.01m are presented in Figs.7-9. Each cylinder primarily oscillates at the same frequency as the incident waves do. It is observed that the computed values agree well with the experimental results.

It can be seen in Figs.7-9 that when the incident wave frequency approaches the natural frequency of the cylinder array,the flow field is strongly influenced. At 1.39Hz,the maximum amplitude of the response of the first cylinder is 3 times larger than that of either the second or third cylinder. The response curves differ significantly from that of a single cylinder. If a single cylinder has only one peak at its natural frequency,it is not the case for the three- cylinder array. This is because the interactions between the cylinders modify the diffraction and radiation forces. It is also found that the theoretical frequency response curves are overestimated in the high frequency region due to the use of an inviscid model in which energy dissipation has not been taken into account.

Fig. 7 and Fig. 8 show how the response curves evolve with a decrease in the distance between the second and third cylinders. It can be found that as the distance between the two cylinders decreases,keeping the same distance between the first cylinder and any other cylinder,the response curve of the first cylinder increases at the low frequency zone. For the second and third cylinders,the response peak values decrease slightly in the high frequency zone. The reason is that the smaller the distance between the second and third cylinders is,the larger the sheltering effects of the first cylinder are.

As we can see from Fig. 8 and Fig. 9,keeping the same distance between the second and third cylinders,the main peak values decrease as the distance between the first cylinder and any other cylinder increases.

It may be noted from Fig. 7,Fig. 8,Fig. 9 that the numerical simulation results are slightly shifted to the low frequency zone compared with the experimental results. This can be explained by the fact that the masses of the oscillating cylinders used in the numerical model are a little larger than the actual values because some bottom parts of the experimental cylinders are fixed to the bottom of the wave flume.

6 Conclusions

This paper has examined the interaction of regular waves with an elastically mounted three- cylinder array. A semi-analytic method based on eigenfunciton expansion is proposed to predict the hydrodynamic forces on the three vertical cylinders of the array. The responses of the cylin- ders induced by wave excitation are determined by the equations of motion coupled with the solutions of the wave radiation and diffraction problems. Experiments for three different ar- rangements of cylinders are then designed and performed in a wave flume to determine the accuracy of the proposed prediction method for regular waves. The results show that the cal- culated results agree with the experimental values. It is found that that the response curves differ significantly from those of a single cylinder because the interactions between the cylin- ders modify the diffraction and radiation forces. The layout of the cylinder array results in a significant reduction in the amplitude of responses of the first and sheltered cylinders. It is also found that the theoretical frequency response curves are overestimated in the high frequency region.