Hydrodynamic Analysis of Two Coaxial Moonpool Floaters Using Theoretical Methodologies

Abstract

The present contribution aims at presenting a semi-analytical solution methodology of the linear hydrodynamic diffraction and radiation problems induced by two coaxial moonpool-type floaters subjected to incident waves. The flow field around the floaters is subdivided into ring-shaped fluid regions, in each of which axisymmetric eigenfunction-expansions of the velocity potential are made. The applied theoretical formulation is complemented by numerical-panel methodologies, using pulsating singularities distributed over the floaters’ wetted surface. Extensive numerical results in terms of exciting forces and hydrodynamic coefficients are given by applying frequency-domain techniques. The effect of the water trapped between the solids’ vertical walls on the floaters’ hydrodynamics is investigated and discussed. The presented analysis led to some remarkable trapping phenomena in the enclosed water areas, broadening the resonance-frequency bandwidth of the floaters when compared to a single moonpool body.

1. Introduction

A moonpool is feature of marine drilling platforms, drillships, diving vessels, etc., which consists of an opening from deck to keel, used for lowering tools and instruments into the sea. Moonpools with an enclosed air chamber are a class of wave energy converters (WECs) called oscillating-water-column (OWC) devices. As the incident waves cause pressure fluctuations in the confined air chamber, the rotation of an air turbine at the top of the oscillating chamber will produce power [1,2,3,4]. Moonpools open to the atmosphere above the internal free-surface are also applied as WEC devices, since wave resonances occur at discrete resonant-sloshing frequencies [5,6,7,8].

Accurate modeling of moonpool floaters requires that both the wave-body interactions and the oscillations of the internal water column are modeled in a coupled hydrodynamic formulation. To date, much work has been devoted to moonpool floaters. Garrett [9] examined for the first time a thin-walled bottomless, partially immersed cylindrical hull, interacting with gravity waves. The diffracted waves around and inside the body were expanded in Bessel functions, and the corresponding diffraction potentials were determined using a least-squares minimization method. In [10] the wave loads, and wave motions of a hollow cylindrical-shell structure of infinite thickness was examined theoretically and experimentally, whereas in [11,12] the wave loads and the hydrodynamic coefficients of a floating, bottomless cylindrical-body with finite wall-thickness were presented. Herein, for the solution of the diffraction and the radiation problem, the flow field around the body was subdivided into ring-shaped fluid regions, in each of which an axisymmetric eigenfunction-expansion for the velocity potential was made. In [13] a general formulation of the hydromechanical problem for an open-bottom floating structure, with an air-chamber above the inner free-surface which was equipped with a turbine generator, was presented. The formulation used the idealization of the flow around and inside the structure by means of macroelements, which were of rectangular shape for cross-sections, and co-axial rings for vertical bodies of revolution. A square moonpool was examined in [14], evaluating the free-surface response inside the body. The work was based on some simplifying assumptions, i.e., the floater was to be motionless, the water depth and the length of the floater were to be infinite. The latter study was extended in [15] to finite water-depths and the body’s limited horizontal dimensions. In [16], the motions and the time-dependent part of the second-order wave forces (i.e., drift forces) on a floating cylindrical OWC device, using the Pinkster and van Oortmerssen method [17] was presented, whereas in [18] the latter method was applied to a floating axisymmetric OWC with an arbitrary geometry. Additional studies on moonpool floaters dealt with the diffraction and radiation problems of a cylindrical moonpool body partially opened at the bottom [19], whereas in [20] a cylindrical moonpool with a restricted entrance was examined, concluding that the resonant period of water motion in the moonpool was affected by the size of the moonpool entrance. Moreover, in [21,22] a moored cylindrical OWC-device was numerically and experimentally investigated. The inner air-pressure, the power capture efficiency, and the translational and rotational motions of the device, were compared for various orifices’ diameters representing different damping values of the device’s power take-off mechanism. In addition, the natural frequencies of a moonpool floater were studied in [23,24], whereas in [25,26] this theory was applied to OWCs. Recently, a semi-analytical model was proposed in [27], for the solution of the diffraction problem from a truncated cylinder with a moonpool of an arbitrary cross-section. The model was adopted to examine an elliptic platform with different eccentricities and a square platform with different outer-hull widths. In addition, in [28] the limitations of the boundary-element-method solvers considering a moonpool-type floater were exploited, assuming as non-negligible the viscous effects near the sharp edges of the body.

All these aforementioned research works look at a hollow floating body that includes a moonpool. Nevertheless, moonpools with an annulus fluid area formed between an exterior, partially immersed, toroidal body and a coaxial, interior truncated-cylinder are also reported in the literature. Specifically, in [29] a semi-analytical method was developed for the solution of the linearized hydrodynamic-radiation problem of two independently heaving concentric, surface-piercing, truncated circular cylinders. It was found that as the external cylinder’s draught increases, the pumping-resonance locations, occurring in the annular fluid region formed between the internal and the external bodies, are shifted to lower wave frequencies. The latter study was extended in [30] to consider both the exciting forces on a coaxial moonpool floater and the hydrodynamic coefficients in surge and pitch motions. Furthermore, in [31] an external cylindrical body, supplemented by an internal body configured as a compound vertical cylinder, was experimentally and theoretically investigated. Mavrakos and Chatjigeorgiou [32,33] presented a solution for the second-order diffraction problem for a piston-like arrangement that consisted of two cylindrical structures with an annular moonpool between them, whereas in [34] the hydrodynamic performance of a two-concentric-cylindrical-body WEC was investigated. In the latter study, the power take-off (PTO) efficiency was enhanced by the optimization of the PTO damping and the intentional mismatching of the heave natural-frequencies of the two buoys. Recently, coaxial moonpools were proposed to operate as OWC devices in the open sea or nearshore, combined with a wind turbine, which is supported on the devices’ interior coaxial body, and as part of a jacket platform [35,36,37,38,39].

Another class of moonpool floater which is reported in the literature is comprised of two coaxial, free-surface-piercing, toroidal cylinders with vertical symmetry axes. This structure has been investigated in conjunction with the trapped-wave modes that can be performed. In particular, in [40], the wave trapping by two concentric surface-piercing circular cylindrical-shells of zero thickness was investigated. The latter study was extended by [41] to the case of vertically axisymmetric structures with finite volumes and two coaxial interior free-surfaces. Additionally, in [42] a semi-analytical formulation was presented for the solution of the diffraction problem of a moonpool floater with two interior free-surfaces. The formulation was validated against computational fluid dynamics (CFD) results. From the study, it was concluded that the two methodologies attain accurate results. However, in the vicinity of inner fluid-pumping resonances, the semi-analytical method overpredicts the piston-mode amplitude.

In the present work, an outline of a semi-analytical method is given which is suitable for solving the linearized diffraction and radiation problems around two independently moving coaxial, free-surface-piercing, toroidal cylinders with vertical-symmetry axes. The results from the theoretical formulation are compared with numerical simulations for different types of floater configurations. The present work extends the work presented in [42], to evaluate the exciting forces on the coaxial bodies and their hydrodynamic coefficients in surge and pitch motions. The method forms a useful tool towards estimating the resonance locations and assessing the effect of the bodies’ geometry on their hydrodynamic characteristics, originating in the need to increase the power efficiency of a WEC by the relative motion of the two oscillators.

The work is organized as follows: Section 2 formulates the solution of the corresponding diffraction and radiation problems, while in Section 3 the outcomes of the two applied formulations are compared and discussed. Finally, Section 4 is dedicated to the conclusions of the work.

2. Semi-Analytical Method

2.1. Velocity-Potential Representation

Two coaxial, free-surface-piercing toroidal cylinders, with vertical-symmetry axes are considered, placed in a water region of constant water depth, $d$ (see Figure 1). The inner toroidal body has a radius of $a_2$, whereas the radius of the internal water surface is$a_1$. In addition, the distance of the body’s bottom from the seabed is denoted by $h_1$. Similarly, the radius and the distance of the outer toroidal-body bottom from the seabed are $a_4$ and $h_2$, respectively. In addition, the radius of the formed annulus water-area between the two solids is denoted by $a_3$. The moonpool is subjected to the action of monochromatic incident-waves of frequency $\omega$ and linear amplitude $A$. A cylindrical coordinate system is introduced, located on the seabed, pointing upwards and coinciding with the floater’s vertical axis. Viscous effects are neglected, and it is assumed that the fluid is incompressible. Additionally, the motions of the moonpool and the fluid are assumed to be small, so that linear diffraction and radiation theory can be considered.

In the solution to the diffraction and radiation problems, the method of matched axisymmetric-eigenfunction-expansions is applied. According to the method, the flow around the moonpool is subdivided into coaxial ring-shaped fluid regions, denoted by $I,II,III,IV,V$ (see Figure 1), where appropriate series representations of the fluid’s velocity potential can be established. Using the Galerkin method, the potential solutions are matched by the requirement of continuity of the hydrodynamic pressure and the radial velocity at the vertical boundaries of adjacent fluid regions, and by satisfying proper kinematic conditions on the bodies’ wetted surfaces.

In the considered moonpool configuration, each toroidal floater $p, p=1, 2$ performs a two-degrees-of-freedom motion under the action of regular waves, i.e., one translation motion (surge ${\xi }_1^p$) and one rotation motion (pitch ${\xi }_5^p$). Hence, the velocity potential can be expressed as: $\Phi \left(r,\theta ,z;t\right)=Re\left[\phi \left(r,\theta ,z\right)e^{-i\omega t}\right]$:

$$\phi \left(r,\theta ,z\right)={\phi }_D\left(r,\theta ,z\right)+{\displaystyle \sum }_{p=1,2}{\displaystyle \sum }_{j=1,5}{\dot{\xi }}_{j0}^p{\phi }_j^p\left(r,\theta ,z\right)$$

(1)

In Equation (1) ${\phi }_D$ is the diffraction potential, and ${\phi }_j^p \left(j=1, 5;p=1, 2\right)$ is the radiation potential resulting from the forced oscillation of the $p$ cylinder in the $j$th mode of motion, with unit velocity amplitude ${\dot{\xi }}_{j0}^p$. Here, ${\phi }_D={\phi }_0+{\phi }_7$, where ${\phi }_0$ is the velocity potential of the undisturbed incident-harmonic wave, and ${\phi }_7$ is the scattered potential for the cylinders restrained in waves.

$${\phi }_0\left(r,\theta ,z\right)=-i\omega A\frac{Z_0\left(z\right)}{Z_0^{\prime }\left(d\right)}{\displaystyle \sum }_{m=0}^{\infty }{\epsilon }_m{i}^m{J}_m\left(kr\right)\cos \left(m\theta \right)$$

(2)

In Equation (2), $A$ denotes the wave amplitude, ${\epsilon }_m$ is the Neumann’s symbol (i.e., ${\epsilon }_0=1; {\epsilon }_m=2$ for $m>0$), $J_m$ is the $m$th order Bessel function of the first kind,$Z_0$ is equal to

$$Z_0\left(z\right)=N_0^{-1/2}\cosh \left(kz\right)={\left[\frac{1}{2}\left[1+\frac{\sin \mathrm{h}\left(2kd\right)}{2kd}\right]\right]}^{-\frac{1}{2}}\cosh \left(kz\right)$$

(3)

and $Z_0^{\prime }\left(d\right)$ denotes $Z_0$ derivative at $z=d$. The wave number, $k$, is related to the wave frequency, $\omega$, by the dispersion equation ${\omega }^2=kg\tan \mathrm{h}\left(kd\right).$

Based on Equation (2), the diffraction potential at each fluid region $\ell =I,II,III,IV,V$ can be written as

$${\phi }_D\left(r,\theta ,z\right)=-i\omega A{\displaystyle \sum }_{m=0}^{\infty }{\epsilon }_m{i}^m{\Psi }_{Dm}^{\ell }\left(r,z\right)\cos \left(m\theta \right)$$

(4)

whereas the radiation velocity potential ${\phi }_j^p$ at each fluid domain, $\ell$, can be expressed as

$${\phi }_j^p\left(r,\theta ,z\right)={\displaystyle \sum }_{m=0}^{\infty }{\Psi }_{jm}^{\ell ,p}\left(r,z\right)\cos \left(m\theta \right), j=1, 5$$

(5)

In the unknown functions ${\Psi }_{jm}^{\ell ,p}$ (see Equations (4) and (5)) the first subscript $j=D,1,5$ denotes the representative boundary-value problem, while the second one denotes the $m$ values that should be taken into consideration.

It should be mentioned that the fluid flow, due to the forced oscillation of the cylinders in still water, is symmetric about the $\theta =0$-plane and antisymmetric about the $\theta =\frac{\pi }{2}$ -plane for surge and pitch. Hence, Equation (5) can be rewritten as

$${\phi }_j^p\left(r,\theta ,z\right)={\Psi }_{j1}^{\ell ,p}\left(r,z\right)\cos \left(\theta \right), j=1, 5$$

(6)

Furthermore, the potentials ${\phi }_D, {\phi }_j^p, j=1, 5$ have to satisfy the Laplace equation in the entire fluid domain, the mean water-surface condition, $z=d$, and the boundary condition of the seabed, $z=0$ [42]. Furthermore, ${\phi }_D, {\phi }_j^p$ have to fulfill the following kinematic conditions on the mean bodies’ wetted-surfaces:

$$\frac{\partial {\phi }_D}{\partial n^q}=0$$

(7)

$$\frac{\partial {\phi }_j^p}{\partial n^q}={\delta }_{pq}{n}_j^p$$

(8)

In Equations (7) and (8), $\frac{\partial ()}{\partial n^q}$ stands for the derivative in the direction of the outward unit-normal-vector, $n^q$, to the mean wetted-surface of the $q$ toroidal cylinder, $n_j^p$ are its generalized components, defined as:($n_1^p,n_2^p,n_3^p$)$=n^p$ and ($n_4^p,n_5^p,n_6^p$)$=r^p\times n^p$ where $r^p$ is the position vector with respect to the origin of the coordinate system, and ${\delta }_{pq}$ is the Kronecker delta, ${\delta }_{pp}=1, p=q$ and ${\delta }_{pq}=0, p\ne q$ [30].

In addition, a radiation condition should be applied, stating that propagating disturbances must be outgoing.

In order to evaluate the unknown functions, ${\Psi }_{jm}^{\ell ,p}$, the method of separation of variables for the Laplace differential equation is applied. Hence, appropriate series representations of the functions${\Psi }_{jm}^{\ell ,p}, j=D,1, 5;p=1, 2; \ell =I, II, III, IV, V$ in each fluid domain around the coaxial cylinders can be established. The latter are presented in Appendix A. These expressions satisfy the kinematic boundary conditions at the horizontal walls of the cylinders, the radiation condition at infinity, and the linearized condition on the free-surface and the seabed. In continuation, applying the Galerkin method, the five potential solutions, ${\Psi }_{j1}^{\ell ,p},$ from each representative boundary value problem $j=D,1, 5$ are matched by the continuity requirements of both the velocity potential and its derivatives, $\frac{\partial ()}{\partial r}$, at the common boundaries of neighboring fluid regions, as well as by fulfilling the kinematic conditions at the vertical walls of the cylinders. The reader is referred to [11,12,29,42] for a detailed presentation of the method.

2.2. Hydrodynamic Loads

From the applied semi-analytical methodology, the diffraction and radiation velocity-potentials can be calculated. Hence, the exciting forces, $F_k, k=1, 3$ and moments, $F_5,$ acting on the moonpool floater are evaluated by the integration of the hydrodynamic pressure, $\mathrm{p}$, over the mean wetted-surface, $S_0^p$, of the $p$ body:

$$\begin{gathered} F_k\left(t\right)=f_k{e}^{-i\omega t}=-\underset{S_0^p}{{\displaystyle \iint }}\mathrm{p}{n}_{k}dS=-i\omega \rho e^{-i\omega t}\underset{S_0^p}{{\displaystyle \iint }}{\phi }_D{n}_{k}dS= \\ -{\omega }^2\rho A{e}^{-i\omega t}{\displaystyle \sum }_{m=0}^{\infty }{\epsilon }_m{i}^m\underset{S_0^p}{{\displaystyle \iint }}{\Psi }_{Dm}^{\ell }\left(r,z\right)\cos \left(m\theta \right)n_{k}dS, \mathrm{for} k=1, 3 \end{gathered}$$

(9)

$$F_5\left(t\right)=M_1\left(t\right)+M_3\left(t\right)$$

(10)

$$\begin{gathered} M_k\left(t\right)=f_k{e}^{-i\omega t}=-\underset{S_0^p}{{\displaystyle \iint }}\mathrm{p}(r\times n_k)dS=-i\omega \rho e^{-i\omega t}\underset{S_0^p}{{\displaystyle \iint }}{\phi }_D(r\times n_k)dS= \\ -{\omega }^2\rho A{e}^{-i\omega t}{\displaystyle \sum }_{m=0}^{\infty }{\epsilon }_m{i}^m\underset{S_0^p}{{\displaystyle \iint }}{\Psi }_{Dm}^{\ell }\left(r,z\right)\cos \left(m\theta \right)(r\times n_k)dS, \mathrm{for} k=2 \end{gathered}$$

(11)

In Equations (9) and (11), $n_k$ are the generalized normal vector components and $r$ the position vector of a point on the wetted surface with respect to the reference co-ordinate system of the cylinder. Analytical representations of the exciting forces and moments on the moonpool are presented in [42].

Similarly, the hydrodynamic reaction forces on the $p$ toroidal cylinder, $p=1, 2,$ in the $i$th, $i=1, 5$, direction, due to the forced oscillation of the $s$ cylinder, $s=1, 2,$ in the $j$th $j=1, 5$ direction are written as

$$F_{i,j}^{p,s}=f_{i,j}^{p,s}{e}^{-i\omega t}=-\underset{S_0^p}{{\displaystyle \iint }}{p}_j^s{n}_i^{p}dS=-\rho {\omega }^2{e}^{-i\omega t}\underset{S_0^p}{{\displaystyle \iint }}{\Psi }_{j1}^{\ell ,p}\left(r,z\right)n_i^{p}dS$$

(12)

Here, $n_i^p$ denotes the generalized normal-vector components defined in Equation (8).

The hydrodynamic reaction forces can be rewritten as a function of the hydrodynamic mass and damping coefficients of the $p$ floater in the $i$th direction, due to the forced motion of the $s$ floater in the $j$th direction, denoted as ${\alpha }_{i,j}^{p,s}, b_{i,j}^{p,s},$ respectively. Hence, it holds that

$$f_{i,j}^{p,s}={\omega }^2{\alpha }_{i,j}^{p,s}+i\omega b_{i,j}^{p,s}$$

(13)

From the applied semi-analytical methodology, the following relations for the non-dimensional surge hydrodynamic-coefficients are derived:

$$\begin{gathered} \frac{{\alpha }_{1,1}^{1,s}}{\rho a_2^3}+i\frac{b_{1,1}^{1,s}}{\rho \omega a_2^3}=-\pi \frac{d}{a_2}\left\{\frac{N_0^{-1/2}}{k{a}_2}\left(F_{1,0}^{*III,s}-\frac{a_1}{a_2}{F}_{1,0}^{V,s}\right)\left(\sinh \left(kd\right)-\sinh \left(k{h}_1\right)\right) \\ +{\displaystyle \sum }_{i=1}^I\frac{N_i^{-1/2}}{{\mathrm{a}}_i{a}_2}\left(F_{1,i}^{*III,s}-\frac{a_1}{a_2}{F}_{1,i}^{V,s}\right)\left(\sin \left({\mathrm{a}}_{i}d\right)-\sin \left({\mathrm{a}}_i{h}_1\right)\right)\right\} \end{gathered}$$

(14)

$$\begin{gathered} \frac{{\alpha }_{1,1}^{2,s}}{\rho a_4^3}+i\frac{b_{1,1}^{2,s}}{\rho \omega a_4^3}=-\pi \frac{d}{a_4}\left\{\frac{N_0^{-1/2}}{k{a}_4}\left(F_{1,0}^{I,s}-\frac{a_3}{a_4}{F}_{1,0}^{III,s}\right)\left(\sinh \left(kd\right)-\sinh \left(k{h}_2\right)\right) \\ +{\displaystyle \sum }_{l=1}^L\frac{N_l^{-1/2}}{{\mathrm{a}}_l{a}_4}\left(F_{1,l}^{I,s}-\frac{a_3}{a_4}{F}_{1,l}^{III,s}\right)\left(\sin \left({\mathrm{a}}_{l}d\right)-\sin \left({\mathrm{a}}_l{h}_2\right)\right)\right\} \end{gathered}$$

(15)

In Equations (15) and (16), ${\mathrm{a}}_l$ are the positive real roots of the transcendental equation (see Appendix A, Equation (A4)). In addition, the terms $N_0^{-1/2},N_l^{-1/2}$ are presented in Equation (3) and Appendix A (see Equation (A3)).

Regarding the hydrodynamic coefficients in pitch, these can be written as:

$$\frac{{\alpha }_{5,5}^{1,s}}{\rho a_2^5}+i\frac{b_{5,5}^{1,s}}{\rho \omega a_2^5}=-\pi \frac{d^2}{a_2^2}\left(M_h^1+M_v^1\right)$$

(16)

$$\frac{{\alpha }_{5,5}^{2,s}}{\rho a_4^5}+i\frac{b_{5,5}^{2,s}}{\rho \omega a_4^5}=-\pi \frac{d^2}{a_4^2}\left(M_h^2+M_v^2\right)$$

(17)

where

$$\begin{gathered} M_h^1=\frac{N_0^{-\frac{1}{2}}}{{\left(k{a}_2\right)}^2}\left(F_{5,0}^{*III,s}-\frac{a_1}{a_2}{F}_{5,0}^{V,s}\right)\left(k\left(d-e\right)\sinh \left(kd\right)-k\left(h_1-e\right)\sinh \left(k{h}_1\right) \\ -\cosh \left(kd\right)+\cosh \left(k{h}_1\right)\right) \\ +{\displaystyle \sum }_{i=1}^I\frac{N_i^{-1/2}}{{\left({\mathrm{a}}_i{a}_2\right)}^2}\left(F_{5,i}^{*III,s}-\frac{a_1}{a_2}{F}_{5,i}^{V,s}\right)\left({\mathrm{a}}_i\left(d-e\right)\sin \left({\mathrm{a}}_{i}d\right)-{\mathrm{a}}_i\left(h_1-e\right)\sin \left({\mathrm{a}}_i{h}_1\right) \\ -\cos \left({\mathrm{a}}_{i}d\right)+\cos \left({\mathrm{a}}_i{h}_1\right)\right) \end{gathered}$$

(18)

$$\begin{gathered} M_v^1=g_5^{1,s}+\frac{1}{4}{F}_{5,0}^{IV,s}\left(1-{\left(\frac{a_1}{a_2}\right)}^2\right)+\frac{1}{4}{F}_{5,0}^{*IV,s}\left(\frac{a_1}{a_2}\right)\left(1-{\left(\frac{a_1}{a_2}\right)}^2\right) \\ +2{\displaystyle \sum }_{n=1}^N{\left(-1\right)}^n{\left(\frac{h_1}{n\pi a_2}\right)}^2\left(F_{5,n}^{IV,s}\left({\Lambda }_{5n}^{IV}-\frac{a_1}{a_2}{D}_{5n}^{IV}-1\right) \\ +F_{5,n}^{*IV,s}\left({\Lambda }_{5n}^{*IV}-\frac{a_1}{a_2}{D}_{5n}^{*IV}+\frac{a_1}{a_2}\right)\right) \end{gathered}$$

(19)

$$\begin{gathered} M_h^2=\frac{N_0^{-1/2}}{{\left(k{a}_4\right)}^2}\left(F_{5,0}^{I,s}-\frac{a_3}{a_4}{F}_{5,0}^{III,s}\right)\left(k\left(d-e\right)\sinh \left(kd\right)-k\left(h_2-e\right)\sinh \left(k{h}_2\right) \\ -\cosh \left(kd\right)+\cosh \left(k{h}_2\right)\right) \\ +{\displaystyle \sum }_{l=1}^L\frac{N_l^{-1/2}}{{\left({\mathrm{a}}_l{a}_4\right)}^2}\left(F_{5,l}^{I,s}-\frac{a_3}{a_4}{F}_{5,l}^{III,s}\right)\left({\mathrm{a}}_l\left(d-e\right)\sin \left({\mathrm{a}}_{l}d\right) \\ -{\mathrm{a}}_l\left(h_2-e\right)\sin \left({\mathrm{a}}_l{h}_2\right)-\cos \left({\mathrm{a}}_{l}d\right)+\cos \left({\mathrm{a}}_l{h}_2\right)\right) \end{gathered}$$

(20)

$$\begin{gathered} M_v^2=g_5^{2,s}+\frac{1}{4}{F}_{5,0}^{II,s}\left(1-{\left(\frac{a_3}{a_4}\right)}^2\right)+\frac{1}{4}{F}_{5,0}^{*II,s}\left(\frac{a_3}{a_4}\right)\left(1-{\left(\frac{a_3}{a_4}\right)}^2\right) \\ +2{\displaystyle \sum }_{n=1}^N{\left(-1\right)}^n{\left(\frac{h_2}{n\pi a_4}\right)}^2\left(F_{5,n}^{II,s}\left({\Lambda }_{5n}^{II}-\frac{a_3}{a_4}{D}_{5n}^{II}-1\right) \\ +F_{5,n}^{*II,s}\left({\Lambda }_{5n}^{*II}-\frac{a_3}{a_4}{D}_{5n}^{*II}+\frac{a_3}{a_4}\right)\right) \end{gathered}$$

(21)

Here, the functions $g_5^{1,s},g_5^{2,s}$ are defined as:

$$g_5^{1,s}=\left\{\begin{array}{cc}\frac{h_1\left(a_2^4-a_1^4\right)}{8a_2^3{d}^2}-\frac{a_2^6-a_1^6}{48h_1{d}^2{a}_2^3}, \hfill & \mathrm{for} s=1\hfill \\ 0,\hfill & \mathrm{for} s=2\hfill \end{array}\right\}$$

(22)

$$g_5^{2,s}=\left\{\begin{array}{cc}0,\hfill & \mathrm{for} s=1\hfill \\ \frac{h_2\left(a_4^4-a_3^4\right)}{8a_4^3{d}^2}-\frac{a_4^6-a_3^6}{48h_2{d}^2{a}_4^3},\hfill & \mathrm{for} s=2\hfill \end{array}\right\}$$

(23)

The terms ${\Lambda }_{5n}^{II},{\Lambda }_{5n}^{*II},{\Lambda }_{5n}^{IV},{\Lambda }_{5n}^{*IV}$ and $D_{5n}^{II},D_{5n}^{*II},D_{5n}^{IV},D_{5n}^{*IV}$ are presented in Appendix B.

3. Numerical Results

The present section is dedicated to the presentation, validation, and discussion of the theoretical formulation outcomes. The computer software DIFFRAC-R [43] was extended [44] to evaluate the exciting forces and the hydrodynamic coefficients in surge and pitch for two independently moving, coaxial moonpool-floaters. The most time-consuming part of the theoretical methodology is the calculation of the Fourier coefficients in each fluid domain, since it affects the accuracy of the solution. For the following calculations, 80 terms have been retained for the series expansions of the velocity potential in the fluid domains $I,III$ and $V$, whereas 150 terms are used for the velocity representations in the $II$ and $IV$ fluid domains. It should be mentioned that within these sets the CPU time for each wave-frequency is less than a second.

The dimensions of the two coaxial floaters have been varied in order to gain a closer insight into the hydrodynamic behavior of this type of floater. Initially, we consider the following geometric characteristics, $d/a_4=14/3,$ $a_3/a_4=4/5$, $a_2/a_4=3/5$, $a_1/a_4=2/5$, and $h_1/a_4=4.3, h_2/a_4=4.16$ (see Figure 1). The results from the theoretical formulations are verified against the numerical outcomes from the numerical panel software HAQi [45], which uses the sink-sources technique. According to this technique, the velocity potential at each point of the wave fields is obtained as a superposition of the potentials, due to pulsating singularities distributed over the floaters’ wetted surfaces [46]. In the latter software, a total of 1560 elements have been applied for the discretization of the floaters’ wetted surface. Furthermore, apart from the aforementioned configuration of two coaxial moonpools, an additional configuration is examined, which consists of a moonpool floater and a solid, coaxial, surface-piercing truncated circular-cylinder, i.e., $a_1=0$ (Figure 1). This is selected in order to examine the effect of the number of oscillating water column chambers (i.e., one or two) on the floater’s hydrodynamics.

Subsequently, the results from the semi-analytical solution of the radiation problem are presented and compared against the numerical ones. It should be noted that the accuracy of the present methodology for the diffraction problem has been verified in [42]. In Figure 2, the hydrodynamic parameters in surge of the two coaxial moonpool floaters are presented. Specifically, the non-dimensional terms of $A_{1,1}^{1,1}=\frac{{\alpha }_{1,1}^{1,1}}{{\rm N}},A_{1,1}^{1,2}=\frac{a_{1,1}^{1,2}}{N}, A_{1,1}^{2,2}=\frac{a_{1,1}^{2,2}}{N}$ and ${{\rm B}}_{1,1}^{1,1}=\frac{b_{1,1}^{1,1}}{N_2},{{\rm B}}_{1,1}^{1,2}=\frac{b_{1,1}^{1,2}}{N_2}, {{\rm B}}_{1,1}^{2,2}=\frac{b_{1,1}^{2,2}}{N_2}$ are depicted. Here, the factors $N=\rho {\alpha }_4^3$ and $N_2=\omega \rho {\alpha }_4^3$, whereas $\rho$ is the fluid density. Both hydrodynamic mass and damping coefficients exhibit a peculiar behavior at $\omega ~1.15$ rad/s, which corresponds to the seiche-resonance mode occurring in the annular region between the outer and the inner torus. Similar behavior was also reported by [42], who examined the exciting forces on coaxial floating toroidal-bodies. Furthermore, it should be noted that negative values of the added mass coefficients $A_{1,1}^{1,1}$ and $A_{1,1}^{2,2}$ are obtained in the vicinity of the resonant frequency. A similar phenomenon was also observed by Ogilvie [47], concerning a submerged cylindrical body. On the contrary, however, the corresponding damping coefficients $B_{1,1}^{1,1}$ and $B_{1,1}^{2,2}$ attain positive values. It is also noted that a secondary resonance appears at $\omega ~1.75$ rad/s. This phenomenon is more pronounced in $A_{1,1}^{1,1},B_{1,1}^{1,1}$, although it is also evident in $A_{1,1}^{1,2},B_{1,1}^{1,2}$. On the contrary, however, it is absent in $A_{1,1}^{2,2}$ and $B_{1,1}^{2,2}$. In the latter wave-frequency, resonance-pitch oscillations of the interior water basin inside the inner torus body occurred. Regarding the comparison of the theoretical and numerical method, it can be concluded that both methods attain similar results, and describe accurately the hydrodynamics of the two coaxial torus bodies. However, small discrepancies do occur, especially in the neighborhood of resonance locations.

Figure 3 illustrates the variation of the hydrodynamic added-mass and radiation-damping coefficients in pitch, i.e., $A_{5,5}^{1,1}=\frac{{\alpha }_{5,5}^{1,1}}{{{\rm N}}_3},A_{5,5}^{1,2}=\frac{a_{5,5}^{1,2}}{{{\rm N}}_3}, A_{5,5}^{2,2}=\frac{a_{5,5}^{2,2}}{{{\rm N}}_3}$ and ${{\rm B}}_{5,5}^{1,1}=\frac{b_{5,5}^{1,1}}{N_4},{{\rm B}}_{5,5}^{1,2}=\frac{b_{5,5}^{1,2}}{N_4}, {{\rm B}}_{5,5}^{2,2}=\frac{b_{5,5}^{2,2}}{N_4}$, where $N_3=\rho {\alpha }_4^5$ and $N_4=\omega \rho {\alpha }_4^5$. It can be seen that the pitch hydrodynamic-coefficients attain, in general, a similar variation pattern as the added-mass and damping coefficients in surge (see Figure 2). Specifically, concerning the resonant pitch oscillations at the vicinity of $\omega ~1.15 \mathrm{and} \omega ~1.75$ rad/s, these are also notable in the presented pitch hydrodynamic-components. In addition, the excellent correlation between the theoretical and the numerical results should be noted.

In continuation, the hydrodynamic coefficients for the second configuration (i.e., moonpool floater and coaxial truncated circular-cylinder) are presented. The added-mass and damping components are normalized by the factors $N=\rho {\alpha }_3^3$, and $N_2=\omega \rho {\alpha }_3^3$ for the surge coefficients, and for the pitch coefficients by the factors $N_3=\rho {\alpha }_3^5$ and $N_4=\omega \rho {\alpha }_3^5$. Figure 4 shows the surge hydrodynamic-coefficients, i.e., $A_{1,1}^{1,1}=\frac{{\alpha }_{1,1}^{1,1}}{{\rm N}},A_{1,1}^{1,2}=\frac{a_{1,1}^{1,2}}{N}, A_{1,1}^{2,2}=\frac{a_{1,1}^{2,2}}{N}$ and ${{\rm B}}_{1,1}^{1,1}=\frac{b_{1,1}^{1,1}}{N_2},{{\rm B}}_{1,1}^{1,2}=\frac{b_{1,1}^{1,2}}{N_2}, {{\rm B}}_{1,1}^{2,2}=\frac{b_{1,1}^{2,2}}{N_2}$. It can be seen that in the proximity of $\omega ~1.15$rad/s, the hydrodynamic components also exhibit in this case a peculiar behavior. The latter is due to the anti-symmetric resonant motion (seiche mode) of the fluid trapped in the annular fluid area between the interior cylinder and the exterior toroidal body. Comparing the two examined configurations, it can be stated that the surge hydrodynamic-components from the wave radiation exerted on both configurations are quite similar. A major difference is the absence of the secondary resonance in the second configuration, due to the absence of fluid inside the inner body. As far as the comparison between the theoretical and numerical methodologies is concerned, it can be seen that both methods predict accurately the surge hydrodynamics of the second configuration. Some discrepancies which are attained near resonant-wave frequencies can be considered negligible.

Figure 5 shows the non-dimensional hydrodynamic mass and damping coefficients in pitch for the second configuration, plotted against the wave-frequency, ω, i.e., $A_{5,5}^{1,1}=\frac{{\alpha }_{5,5}^{1,1}}{{{\rm N}}_3},A_{5,5}^{1,2}=\frac{a_{5,5}^{1,2}}{{{\rm N}}_3}, A_{5,5}^{2,2}=\frac{a_{5,5}^{2,2}}{{{\rm N}}_3}$ and ${{\rm B}}_{5,5}^{1,1}=\frac{b_{5,5}^{1,1}}{N_4},{{\rm B}}_{5,5}^{1,2}=\frac{b_{5,5}^{1,2}}{N_4}, {{\rm B}}_{5,5}^{2,2}=\frac{b_{5,5}^{2,2}}{N_4}$. A similar variation pattern with the surge hydrodynamic-components is depicted. Furthermore, the resonance-pitch oscillation (an anti-symmetric mode of motion) occurred in the vicinity of $\omega ~1.15$rad/s, as in the case of the surge coefficients (see Figure 4). From the comparison of the theoretical and numerical methodologies, it can be concluded that panel predictions follow the theoretical method’s trends, with minor discrepancies. However, it is evident that there are some variations between the results of the two methods in $A_{5,5}^{1,1}$ which can be justified by the discretization of the panels of the inner body.

Next, three different draughts for the outer and inner moonpool floaters are investigated. Specifically, firstly it is assumed that $d/a_4=14/3,$ $a_3/a_4=4/5$, $a_2/a_4=3/5$, $a_1/a_4=2/5$, and $h_1/a_4=4.3, h_2/a_4=3.3, 3.73, 4.16$. Figure 6 shows the non-dimensional exciting-wave forces and moments on the outer and inner moonpool floaters for various examined $h_2$values. The results are normalized by the terms $\rho g{\alpha }_4^{2}A$ and $\rho g{\alpha }_4^{3}A$, for the forces and moments, respectively. Regarding the horizontal-exciting forces (see Figure 6a), it can be seen that $f_1$ appears to exhibit a peculiar behavior at $\omega ~1.15$ rad/s and $\omega ~1.75$ rad/s. This is due to the resonance-pitch oscillations of the interior basin of each floater. It should be further noted that this peculiar behavior seems to be affected by the outer-floater draught, i.e., as $h_2/a_4$ decreases, the resonance frequency is transferred at lower values of ω. In addition, it can be observed that as the ratio $h_2/a_4$ decreases (i.e., the draught of the outer-floater increases), the horizontal-exciting forces on the outer body increase. The opposite holds for the inner floater. With regard to the vertical forces (see Figure 6b) on the two floaters, peaks appear at the wave frequencies where pumping resonances of the fluid motion in the interior water area occur. Furthermore, as the ratio $h_2/a_4$ decreases, these resonances are shifted at lower values of ω, regardless of the examined floater. As far as the horizontal-exciting moments are concerned (see Figure 6c) a similar variation pattern to $f_1$is attained, concerning the resonance locations. In addition, it can be seen that the values of $f_5$ on the inner and outer floater vary inversely as the ratio $h_2/a_4$ decreases.

As far as the non-dimensional hydrodynamic mass and damping coefficients of the two floaters are concerned, these are depicted in Figure 7. Here, the values of $A_{1,1}^{1,1}=\frac{{\alpha }_{1,1}^{1,1}}{{\rm N}}, A_{1,1}^{2,2}=\frac{a_{1,1}^{2,2}}{N},$ ${{\rm B}}_{1,1}^{1,1}=\frac{b_{1,1}^{1,1}}{N_2}, {{\rm B}}_{1,1}^{2,2}=\frac{b_{1,1}^{2,2}}{N_2}$ and $A_{5,5}^{1,1}=\frac{{\alpha }_{5,5}^{1,1}}{{{\rm N}}_3}, A_{5,5}^{2,2}=\frac{a_{5,5}^{2,2}}{{{\rm N}}_3}, {{\rm B}}_{5,5}^{1,1}=\frac{b_{5,5}^{1,1}}{N_4}, {{\rm B}}_{5,5}^{2,2}=\frac{b_{5,5}^{2,2}}{N_4}$ are presented indicatively. From the depicted results, it can be observed that the draught of the outer floater affects the surge-hydrodynamic characteristics of both bodies (see Figure 7a–d). Specifically, the seiche-resonance mode occurring in the annular region between the outer and the inner torus, in the vicinity of $\omega =1.15$ for $h_2/a_4=4.16$, is reallocated at lower values of $\omega$ as $h_2/a_4$decreases. Nevertheless, small discrepancies are notable between the $h_2/a_4=3.73$ and $h_2/a_4=3.3$ cases (i.e., resonance frequencies $~$1.13 rad/s and $~$1.12 rad/s, respectively). This holds true for the hydrodynamic mass $A_{1,1}^{1,1}$ and $A_{1,1}^{2,2}$, as well as for the hydrodynamic-damping coefficients ${{\rm B}}_{1,1}^{1,1},$ ${{\rm B}}_{1,1}^{2,2}$. Furthermore, a similar variation pattern is also attained for the depicted secondary resonances in $A_{1,1}^{1,1},$ ${{\rm B}}_{1,1}^{1,1}.$ The latter occur due to the resonance-pitch oscillations of the interior water basin inside the inner torus body (see discussion of Figure 2). Specifically, the peak in the neighborhood of $\omega =1.75$ rad/s, where $h_2/a_4=4.16$,is minorly shifted at $\omega =~1.73$ rad/s, $~$1.72 rad/s for $h_2/a_4=3.73, 3.3$respectively. With regard to the pitch-hydrodynamic characteristics of both bodies, it can be seen (Figure 7e–h) that the variation of $A_{5,5}^{1,1}, A_{5,5}^{2,2},{{\rm B}}_{5,5}^{1,1},{{\rm B}}_{5,5}^{2,2}$ follows a similar behavior to the added-mass and damping coefficients in surge. Specifically, concerning the resonant-pitch oscillations at the vicinity of $\omega =1.15; 1.75$ rad/s, these are reallocated to lower values of $\omega$ as$h_2/a_4$ decreases. In addition, it can be seen that the hydrodynamic characteristics in surge and pitch of the outer floater increase by analogy, with its draught.

Thereinafter, the draught of the outer floater is assumed as $h_2/a_4=4.16,$ and three different draughts of the inner moonpool-floater are investigated, i.e.,$h_1/a_4=3.43, 3.86, 4.3$. All other floaters’ dimensions are kept constant. In Figure 8, the non-dimensional exciting forces and moments on the outer and inner moonpool floaters for various examined $h_1$values are presented. It can be seen that the draught of the inner floater seems to affect the surge-exciting forces on both floaters (see Figure 8a). In particular, $f_1$ on the outer floater increases by analogy with the ratio $h_1/a_4$.In addition, as $h_1/a_4$ decreases, the resonance frequency is transposed to lower values of ω. On the contrary, however, $f_1$ on the inner floater increases as the ratio $h_1/a_4$decreases. Additionally, the annular water area pitch-resonance is transposed at lower wave-frequencies as $h_1/a_4$decreases (i.e., from ω~1.18 rad/s for $h_1/a_4=4.3$ to ω~1.11 rad/s and ~1.10 rad/s for $h_1/a_4=3.86, 3.43,$ respectively). A similar trend seems to hold for $f_3$ (see Figure 8b), where the wave frequencies in which the pumping resonance of the fluid occurs are shifted at lower values for the inner floater, whereas these ω values remain unaffected by the $h_1/a_4$ratio for the outer floater. Regarding the pitch-exciting moments on the two bodies, $f_5$ on the inner body is significantly enhanced as its draught increases, while the corresponding counterpart on the outer body seems to be less affected by the$h_1/a_4.$

Figure 9 depicts the $A_{1,1}^{1,1}=\frac{{\alpha }_{1,1}^{1,1}}{{\rm N}}, A_{1,1}^{2,2}=\frac{a_{1,1}^{2,2}}{N},$ ${{\rm B}}_{1,1}^{1,1}=\frac{b_{1,1}^{1,1}}{N_2},{{\rm B}}_{1,1}^{2,2}=\frac{b_{1,1}^{2,2}}{N_2}$ and $A_{5,5}^{1,1}=\frac{{\alpha }_{5,5}^{1,1}}{{{\rm N}}_3},A_{5,5}^{2,2}=\frac{a_{5,5}^{2,2}}{{{\rm N}}_3}, {{\rm B}}_{5,5}^{1,1}=\frac{b_{5,5}^{1,1}}{N_4}, {{\rm B}}_{5,5}^{2,2}=\frac{b_{5,5}^{2,2}}{N_4}$ for different values of $h_1$. It can be seen that both surge- and pitch-hydrodynamic added-mass and damping coefficients of the inner floater increase as its draught increases. In addition, both resonances (primary and secondary) are shifted at lower values of wave frequencies as $h_1/a_4$ decreases (i.e., from ~1.16 rad/s for $h_1/a_4=4.3$ to ~1.1 rad/s and ~1 rad/s for $h_1/a_4=3.86, 3.43$, respectively, and from ~1.74 rad/s for $h_1/a_4=4.3$ to ~1.73 rad/s and ~1.72 rad/s for $h_1/a_4=3.86, 3.43$, respectively). Regarding the corresponding hydrodynamic characteristics of the outer floater, these behave analogously to the ratio $h_1/a_4$ (as $h_1/a_4$ decreases, the resonance frequency, ω, also decreases). The absence of secondary resonance from the hydrodynamic characteristics of the outer floater should also be noted.

Following this, three different radii of the outer oscillating-water-column are examined. In particular, it holds that $d/a_4=14/3$, $a_2/a_4=3/5$, $a_1/a_4=2/5$, and $h_1/a_4=4.3, h_2/a_4=4.16$, whereas $a_3/a_4=0.7, 0.8, 0.9.$ Figure 10 and Figure 11 show the exciting forces/moments and the aforementioned hydrodynamic coefficients of the two coaxial moonpool floaters, respectively, for several $a_3$ values. Clearly, as the radius of the fluid area increases, the horizontal-exciting forces and moments on the outer body are reduced. However, the opposite trend seems to hold for body 1. In addition, it should be noted that $f_3$ attains a secondary resonance for $a_3/a_4=0.7$ (see Figure 10b). This occurs at ω~1.15 rad/s, and it is valid for both bodies. The latter peculiar behavior can be attributed to the coupling effects between the two fluid surfaces, which are amplified as the radial extent of the annulus fluid area, enclosed by the external and internal torus, decreases. As far as the presented hydrodynamic coefficients are concerned, it can be seen that the radius of the annular fluid surface seems to have a minor effect on the pitch secondary-mode of the fluid motion inside body 1. On the contrary, the resonance frequency of the fundamental pitch fluid-motion inside both bodies is transferred at lower values of ω as the ratio $a_3/a_4$ increases.

4. Conclusions

The present study deals with a semi-analytical model for the investigation of the hydrodynamics of two coaxial moonpool floaters moving independently. In the framework of linear wave theory, a 3D theoretical solution based on eigenfunctions expansion is applied for the evaluation of the velocity potential of the flow field around and inside each moonpool. Several geometric characteristics are examined in order to estimate the resonance locations and determine the effect of the bodies’ geometry on their hydrodynamic characteristics. The main conclusions drawn from the theoretical predictions are:

• The presence of the water surface inside body 1 (the inner moonpool) attains a secondary resonance, due to the resonance-pitch oscillations of the interior water basin inside the inner torus body. This behavior is notable in both the exciting forces and hydrodynamic parameters of body 1. On the contrary, however, this is absent in the corresponding values of body 2.
• The two fundamental modes of the fluid motions, namely heave (pumping) and pitch (seiche), in the annulus fluid area are depicted in both bodies 1 and 2.
• The resonance frequencies of both moonpools are strongly affected by the examined draughts of the outer and inner solids. Specifically, by increasing the draught, a transition of the resonance location to lower values of ω is obtained.
• The resonance frequencies of both bodies are also affected by the radius of the annulus fluid area, since these are shifted to lower values of ω as the radius increases.
• The vertical-exciting force on each moonpool attains a secondary resonance in the case of a reduced radial-extent of the annulus fluid area. This can be attributed to coupling effects between the two fluid surfaces, which are amplified as the radial distance of the fluid area decreases. Nevertheless, this behavior should be further examined in the future, with the evaluation of the heave hydrodynamic-coefficients in two independently moving coaxial moonpools.
• The accuracy of the presented semi-analytical method has been validated against the numerical results for both examined configurations (i.e., two coaxial moonpools and a moonpool and a coaxial truncated-cylinder). It can be concluded that the comparisons between the two methods are excellent, and hence the presented theoretical model predicts accurate results.

The present research will be continued further by solving the heave-radiation problem of independently moving moonpools, as well as by comparing the present results with computation-fluid-dynamics simulations, in order to investigate the effect of viscosity on the piston-mode amplitude in the vicinity of resonances, compared to their counterparts derived from the linear-potential methodology. Furthermore, the large amplifications of the fluid’s motion in the vicinity of resonance locations introduce uncertainties with respect to the validity of the applied first-order formulation. Hence, the relative significance of the contribution of second-order effects to the wave loading on the considered moonpool floaters should be further investigated.