2. Solving procedure

The *waveFoam* solver starts with the preprocessor, which is used to set up wave properties and computational meshes. The meshes of the NWT are generated by using the built-in tool *blockMesh* and *snappyHexMesh*. The N–S equations are discretized into a set of algebraic equations by integrating the boundary conditions over the whole solution domain and time domain. The physical parameters of the whole domain like the fluid pressure *p* and the fluid velocity **U**, etc., are calculated and updated at each timestep by calling solver *waveFoam*.

**Figure 2.** The boundary name of the NWT.

#### **3. Comparison Against Published Data**

A 3D numerical wave tank (NWT) is established with the above numerical methods of OpenFOAM and waves2foam. To validate the 3D NWT model, we compare the numerical results of wave interaction with a fixed and vertically suspended cylinder payload with the published data [16].

#### *3.1. Numerical Wave Tank*

A 3D numerical wave tank (NWT) is established, as shown in Figure 3. Its geometry has the outer dimensions 15 *m* × 4 *m* × 1.2 *m* with the water depth *h* = 0.505 *m* and the relaxation zone of 1.5 *L*, where *L* is the wavelength. A cylinder whose radius *a* = 0.125 *m* is stationary and vertical is suspended in the tank, leaving a 1 *mm* gap beneath to the bed of the tank. The length of the cylinder is 1 *m*. The cylinder is located at 7.5 *m* from the paddles in the center of the tank. A wave gauge WG2 is placed 2 *mm* in front of the upstream stagnation point of the cylinder to monitor the wave field around the cylinder, and a wave gauge WG1 is placed 0.77 *m* from the inlet to monitor the wave elevation.

**Figure 3.** Layout of the numerical wave tank.

Two regular wave cases [16] R1 and R2 are reproduced with our 3D NWT. The wave parameters are shown in Table 1, where *h* is the water depth, *k* is the wavenumber, *A* is the wave amplitude, and *T* is the wave period.

**Table 1.** Parameters of regular wave for validation.


*h* is the water depth, *k* is the wavenumber, *A* is the wave amplitude, and *T* is the wave period.

The mesh resolution in the computational domain affects the numerical solution. The built-in mesh generator *blockMesh* is used to generate meshes of hexahedral cells, then *snappyHexMesh* in OpenFOAM is used to generate the cylinder. The mesh consists of multilevel grids, as shown in Figure 4. In the areas around the payload, the grid cells have a resolution of Δ*x* in the horizontal direction and Δ*z* in the vertical direction, which are measured by the cells per wavelength and wave height.

Three different time steps are used here for the convergence examination. As shown in Figure 5, three cases are set to a fixed time-step, and the results are convergent. For each time-step, one, two, and three inner iterations (*nOuterCorrectors* in OpenFOAM) are used for convergence examination, the result is the same as Figure 5. For each inner iteration, the PIMPLE algorithm is called three times (*nCorrectors* in OpenFOAM). For the remaining cases in the paper, one *nOuterCorrectors* and three *nCorrectors* are used, the simulation time is 18 s, and the fixed time step is set to 0.005 s, the courant numbers during the simulation are all less than 0.1.

The time history of horizontal force Fx on the cylinder payload with three different mesh schemes for the regular waves are shown in Figure 6. From this grid convergence examination, it can be seen that the results of Mesh 2 and Mesh 3 are convergent and Mesh 2 uses much less time; thus, the intermediate Mesh 2 is selected in this paper. For the Mesh 2 scheme, multilevel grids are used just as Figure 4 shows: in total, 470 cells in the x-direction, 125 cells in the y-direction, and 100 cells in the z-direction. The mesh around the inlet, outlet, and object is dense, and the rest transitions smoothly. The mesh around the cylinder and free surface is uniform: 110 cells per wavelength and 110 cells in total are set in the *x*-direction, 30 cells per wave height, and 60 cells in total are set in the z-direction and 60 cells in total in 1 m are set in the y-direction.

The simulations are run on purchased Dell T7920 workstation with Intel Xeon (R)E5 2699v4 CPU, 128GB RAM, and 44 cores. The comparison of the computation cost, the total cell numbers, the number of cores, and the simulated time under three different mesh schemes are illustrated in Table 2.


**Table 2.** Mesh parameters and computation cost.

*L* is the wavelength. *H* is the wave height.

**Figure 4.** Mesh around the cylinder in the 3D NWT.

**Figure 5.** Time history of the surface elevation at WG1 for the wave R1 with three different time-steps.

**Figure 6.** Time history of horizontal force Fx on the cylinder payload with three different mesh schemes for regular wave R1.

#### *3.2. Comparison with the Published Data*

Before the simulation, the surface elevation at WG1 for the wave R1 is compared with theory results, the result is as Figure 7 shows, the surface elevation agrees well with the theory. The free surface elevation and horizontal force are compared with published data. The time histories of the free surface elevation at WG2, and the corresponding amplitude spectra obtained by applying the FFT algorithm to the time histories are shown in Figure 8. The surface elevation is normalized by the wave amplitude *A*, and the time is normalized by the wave period *t*. The time series of the horizontal force on the cylinder and the corresponding amplitude spectra are presented in Figure 9. The force is normalized by 0.5ρ*gAS*, where ρ is density of the water, and *S* is the cross-sectional area of the payload in the water perpendicular to the wave propagation direction. It can be seen that the results obtained with our NWT model match with the published data [16]. It is validated that our present 3D NWT numerical model can be used to calculate the wave load exerted on the payload with a reasonable degree of accuracy.

**Figure 7.** Surface elevation at WG1 for the wave R1 compared with theory result.

**Figure 8.** Time series of free surface elevation and amplitude spectra at WG2 for regular wave R1 and R2. (**a**) Results of R1; (**b**) results of R2.

**Figure 9.** Time series of horizontal force on the cylinder and amplitude spectra for regular wave R1 and R2. (**a**) Results of R1; (**b**) results of R2.

#### **4. Numerical Results**

The 3D NWT established in Section 3 is applied in a series of simulation experiments in this section. Our study focuses on the influence of different postures of the payloads on wave forces and moments exerted on the payloads while suspending in the sea. We assume that the payloads are fixed and stationary while suspended in the sea without considering their translational and rotational motions caused by wave forces and rotational moments. A cylinder payload and a cuboid payload with different postures that are fixed and suspended in the regular wave R1 are simulated respectively. *Fx*, *Fy*, *Fz* three forces along the axes *ob***x***w*, *ob***y***w*, *ob***z***w*, and *Mx*, *My*, *Mz*, three rotational moments about the axes are computed for two payloads. The influence of the postures of the payloads on wave forces and rotational moments exerted on the payloads are analyzed.

#### *4.1. Case 1: A Cylinder Payload Fixed Suspending in the 3D NWT*

The same cylinder payload in Section 3 is used here. In this subsection, in addition to the vertical suspension, several postures of the cylinder in the NWT are considered. The posture of the cylinder payload in the 3D NWT is shown in Figure 10. The wave condition is the same as the regular wave R1.

(**a**)

(**b**)

**Figure 10.** Two different descriptions of frames and the postures of the cylinder payload in the 3D NWT. (**a**) The overall description of two frames and a cylinder payload in the 3D NWT; (**b**) description of Euler angles in the top, side, and front view.

To validate the 3D NWT, the horizontal wave force *Fx* is compared with results computed by Morison's equation [27]

$$F\_x = \rho \mathbf{C}\_m V \dot{u} + \frac{1}{2} \rho \mathbf{C}\_d \mathbf{S} u |u| \tag{9}$$

where *Cm* is the added mass coefficient (*Cm* = 1.15), *Cd* is the drag coefficient (*Cd* = 1), *V* is the volume of the payload in the water, *S* is the cross-sectional area of the payload in the water perpendicular to the wave propagation direction, and . *u* is the horizontal acceleration.

As shown in Figure 11, the normalized first-harmonic forces and moments are obtained by applying the FFT algorithm to the time histories. The first-harmonic forces are normalized by 0.5ρ*gAS* and the first-harmonic rotational moments are normalized by ρ*gdAS* where *d* is the draft of the cylinder. In the latter study, the same normalization method is used for the forces and moments.

**Figure 11.** Normalized forces and moments on the cylinder payload versus a single posture angle. (**a**) the pitch angle θ; (**b**) the roll angle φ.

1. Pitch angle θ = (0◦, 15◦, 30◦, 45◦, 60◦, 75◦, 90◦), roll angle φ = 0◦.

In Figure 11a, it can be seen that the horizontal force *Fx* obtained with the 3D NWT matches with that computed by Morison's equation. The horizontal force *Fx* decreases with the pitch angle and *Fz* increases with the pitch angle. This could be explained by the decrease of the projection area of the cylinder on the surface *yoz* and increase of the projection area of the cylinder on the surface *xoy*.

It is obvious that *Fy*, the lateral force, and *My*, *Mz* the rotational moments about x-axis and z-axis, are much less than the others, and can be neglected regardless of pitch angle θ. The numerical results match the physical phenomena and can be explained easily with the force analysis. In the case of ψ = 0◦, φ = 0◦ the rotational moment *My* exerted on the cylinder about y-axis depends on the horizontal force *Fx* and the vertical force *Fz*. *My* increases with the increase of θ from 0◦ to 60◦, then it decreases with the increase of θ from 60◦ to 90◦. The maximum moment with θ = 60◦ is 10 times larger than that of vertical suspension.

2. Roll angle φ = (0◦, 15◦, 30◦, 45◦, 60◦, 75◦, 90◦), pitch angle θ = 0◦.

In Figure 11b, it can be seen that the horizontal force *Fx* obtained from the 3D NWT matches with that computed with Morison's equation only for a limited range near to φ = 0◦. The reason is that the two coefficients *Cm*, *Cd* change with the increase of the roll angle φ. The values of *Cm*, *Cd* at φ = 0◦ no longer work with the increase of φ.

It can be seen that the lateral force *Fy*, the rotational moments *My*, *Mz*, about the y-axis and z-axis are not zero but small values. Both the horizontal force *Fx* and the vertical force *Fz* increase with the roll angle φ, but *Fx* decreases when the angle is 75◦. The projection area of the cylinder on the surface *xoy* increases with the increase of θ from 0◦ to 90◦, and the vertical force exerted on the cylinder also increases. The increase is quicker with the roll angel φ from 60◦ to 90◦. The rotational moment *Mx* increases with the increase of φ from 0◦ to 60◦, and then decreases with the increase of φ from 60◦ to 90◦.

From the above simulations of the two cases, we can see that the changes of force could be explained by the change of the corresponding projection area. Additionally, the moment around a certain axis changes drastically with the change of angle. For example, when the cylinder rotates around the y-axis (pitch angle), the max value of *My* is 10 times larger than the initial value. For the vertical cylinder payload, there is no angle where all the forces and moments are minimal, but the initial posture could be an optimal selection.

#### *4.2. Case 2: A Cuboid Payload Fixed Suspending in the 3D NWT*

A cuboid payload is fixed and suspended in the 3D NWT as shown in Figure 1. The size of the cuboid is 1*m* × 0.5*m* × 0.5*m*, and the draft is *d* = 0.25*m*. The cuboid's posture in the 3D NWT is represented by the three Euler angles. A series of simulations are done with different postures of the suspending cuboid in the 3D NWT.

1. Yaw angle ψ = (0◦, 15◦, 30◦, 45◦, 60◦, 75◦, 90◦), pitch and roll angle θ = 0◦, φ = 0◦.

The normalized forces and moments on the cuboid payload versus its yaw angle ψ are shown in Figure 12a. With the normal incident regular waves, the horizontal force *Fx* decreases with the increase of ψ from 0◦ to 90◦. The projection area on the surface *yoz* is the biggest when ψ = 0◦. The projection area on the surface *yoz* decreases with the increase of ψ from 0◦ to 90◦. It can be seen only *Fx* changes drastically with the angle, which is not the same as the results of cylinder where the moment changes drastically.

2. Pitch angle θ = (0◦, 15◦, 30◦, 45◦, 60◦, 75◦, 90◦), yaw and roll angle ψ = 0◦, φ = 0◦.

The normalized forces and moments on the payload versus the pitch angle θ are shown in Figure 12b. It is obvious that both the lateral force *Fy* and the rotational moment *Mx* and *Mz* are near to zero no matter the pitch angle. The horizontal force *Fx*, the vertical force *Fz*.and the rotational moment *My* are all symmetrical around 45◦. This could be easily explained by the change in the projection area. When the pitch angle θ increases from 0◦ to 45◦, the horizontal force *Fx* decreases. Additionally, it increases when θ increases from 45◦ to 90◦. At θ = 0◦, 90◦, the regular waves which are normally incident to the cuboid's face with the largest surface area exert the maximum horizontal force on the cuboid. It can be seen that there is also not a drastically changed moment with the change of the angle.

3. Roll angle φ = (0◦, 15◦, 30◦, 45◦, 60◦, 75◦, 90◦), pitch, yaw angle θ = 0◦, ψ = 0◦.

The normalized forces and moments on the cuboid payload versus the roll angle φ are shown in Figure 12c. It can be seen that compared with *Mx*, the changes of other forces and moments are small, and *Mx* increases very quickly with the increase of φ from 0◦ to 30◦, and decreases with φ from 30◦ to 90◦. The phenomenon of a drastically changed moment is similar to the results of the cylinder.

**Figure 12.** Normalized forces and moments on the cuboid payload versus a single posture angle. (**a**) the yaw angle ψ; (**b**) the pitch angle θ; (**c**) the roll angle φ.

For the two cases of the cylinder, the moment changes drastically with the angle. For the three cases of the cuboid payload, when the cuboid payload rotates around the x-axis (roll angle), the change of force and moment is similar to the cylinder cases. The changes of force could be explained by the change of the corresponding projection area, and the moment around a certain axis changes drastically with the change of angle. However, results when rotating around the z-axis (yaw angle) and y-axis (pitch angle) show no drastically changed moment.

To show that the difference could be brought by the initial posture, we plot the normalized forces and moments on the cylinder payload versus its yaw angle ψ when the roll angle φ = 90◦, just as Figure 13 shows. It can be seen that the result is similar to Figure 12a,b. When the roll angle φ = 90◦, the cylinder is horizontally placed, its length side along the y-axis. When this happens, the phenomenon of a drastically changed moment disappears.

For further study, we also plot the result when the vertically placed cuboid payload rotates around the y-axis and z-axis. Just as Figure 14 shows, the result is similar to Figure 12c. When the cuboid is vertically placed and the roll angle φ = 90◦, its long side along the z-axis. However, in Figure 14b, there is an exception, the drastically changed moment is not around the z-axis but the y-axis, other results are all as expected.

All three above figures show that the drastically changed moment is brought about by the initial posture. The moment around a certain axis changes drastically with the change of angle when the payload is vertically placed (which means the long side of the payload is vertical to the water surface) such as in Figure 14, and this phenomenon could happen when the horizontally placed payload changes to the vertical posture, such as in Figure 12c.

**Figure 13.** Normalized forces and moments on the cylinder payload versus its yaw angle ψ when the roll angle φ = 90◦.

**Figure 14.** Normalized forces and moments on the cuboid payload versus a single posture angle when the roll angle φ = 90◦. (**a**) the pitch angle θ; (**b**) the yaw angle ψ.

4. Yaw ψ and roll φ concurrently change from 0◦ to 90◦, pitch θ = 0◦.

The force and moment exerted on the cuboid versus the yaw angle ψ and the roll angle φ is shown in Figures 15 and 16. For the horizontal force *Fx*, it decreases with the yaw angle ψ regardless of the roll angle φ. When the roll angle is 0◦, the cuboid is horizontally placed, with its long side vertical to the wave direction when the yaw angle is 0◦. When the roll angle increases from 0◦ to 90◦, the long side gradually changes to the vertically placed position; thus, the gradient along the yaw angle decreases with the increase of the roll angle. For the lateral force *Fy*, the result is symmetrical about yaw angle and roll angle.

The results of *Mx* could also be explained by the conclusion raised above. When the yaw angle is 0◦, the roll angle increases to 90◦, the cuboid changes from horizontally placed to be vertically placed, and the *Mx* changes drastically with the roll angle. When the yaw angle is 90◦, the cuboid payload could not change to be vertically placed with the change of roll angle, and the phenomenon of a drastically changed moment disappears. When the yaw angle change from 0◦ to 90◦, the phenomenon gradually disappears.

The result of *My* and *Mz* demonstrate the exceptional condition in Figure 14b. When the roll angle is 0◦, the cuboid is horizontally placed, and there is no drastically changed moment. When the roll angle is 90◦, the cuboid is vertically placed and the moment *My*, instead of *Mz*, drastically changes with the yaw angle. There is also a transition when the roll angle increases from 0◦ to 90◦. The amplitude of *Mz* is much less than the others, and can be neglected.

**Figure 15.** Normalized force on the cuboid payload versus yaw ψ and roll φ. (**a**) the horizontal force *Fx*; (**b**) the lateral force *Fy*; (**c**) vertical force *Fz*.

**Figure 16.** Normalized moment on the cuboid payload versus yaw ψ and roll φ. (**a**) the horizontal moment *Mx*; (**b**) the lateral moment *My*; (**c**) vertical moment *Mz*.

#### *4.3. Parameter Studies*

The parameter studies are done to analyze the effects of the cuboid's size and wave parameters on the forces and moments exerted on the cuboid payload. Here, in order to focus on the effects of cuboid's size and wave parameters, no posture angles are considered and the cuboid is horizontally placed.

#### 1. Cuboid's size effects on forces and moments

The normalized forces and moments on the cuboid payload with different drafts, lengths, and widths are shown in Figure 17a–c. The results show that the horizontal force *Fx* increases with the increase of the payload draft and length. The vertical force *Fz* decreases slowly with the increase of the draft and increases with the length and width. The rotational moment *My* increases slowly with the increase of the payload draft *d*. The change of other forces and moments can be neglected.

**Figure 17.** *Cont.*

**Figure 17.** Normalized forces and moments on the cuboid payload with different size parameters. (**a**) with different drafts; (**b**) with different lengths; (**c**) with different widths.

2. Wave's parameters effects on forces and moments

The normalized forces and moments on the cuboid payload with different drafts, lengths, and widths are shown in Figure 18a,b. It can be seen that both the horizontal force *Fx* and the vertical force *Fz* increase with the increase of the wave amplitude and wavelength. The change of other forces and moments could be neglected.

**Figure 18.** *Cont.*

**Figure 18.** Normalized forces and moments on the cuboid payload with different wave parameters. (**a**) different wave amplitudes; (**b**) different wavelengths.

From the above parameter simulations, we can see that the horizontal force *Fx* and the rotational moment *My* exerted on the cuboid payload increase with the increase of its draft *d*, and its length *l*. The vertical force *Fz* and the rotational moment *Mx* increase with its width *B*. The increase of the wave amplitude *A* and wave length *L* cause the increase of the horizontal force *Fx* and the vertical force *Fz*.

#### **5. Conclusions**

In order to investigate regular wave interaction with a fixed suspending payload with different postures, a three-dimension NWT based on OpenFOAM and waves2foam is established. Regular wave interaction with a vertically suspended cylinder is investigated. The free surface elevation, horizontal wave force, as well as the corresponding amplitude spectra obtained by the FFT algorithm, are compared with the theory result and the results reported in [16] for validation. Then, the representation of the payload's posture in the regular wave is given. The forces and moments exerted on a suspended cylinder and a suspended cuboid with different postures are investigated separately. Finally, parameter studies in the case of payload's size wave parameters are considered.

It can be concluded that the moment around a certain axis changes drastically with the change of the same angle when the payload is initially vertically placed (which means the long side of the payload is vertical to the water surface). For example, the moment around the y-axis could change drastically when rotating around the y-axis. This phenomenon could also happen when the horizontally placed payload (which means the long side parallel to the sea level) changes to the vertical posture. There is an exception: when rotated around the z-axis, the drastically changed moment is not around the z-axis but the y-axis. Therefore, for the rectangular shape payload, it is better to keep the payload horizontally placed to prevent the drastic change of the moment. Additionally, the projection area of the payload vertical to the direction of force affects the corresponding force. It is better to keep the short side vertical to the incident direction of the wave; thus, a minimal horizontal force can be obtained. Through the simulations, some certain posture of the payload with the minimum forces and moments can be reached. It can guide the design of control strategies for the safe operation of offshore cranes, such as keeping the payload to a certain posture that suffers minimal force and moment or changing the controller weight of some forces and moments under specific circumstances.

**Author Contributions:** Conceptualization, M.Y. and X.M.; data curation, M.Y.; formal analysis, M.Y. and X.M.; funding acquisition, X.M.; investigation, M.Y.; methodology, M.Y.; project administration, X.M. and Y.L.; resources, M.Y.; Software, M.Y., W.B. and Z.L.; supervision, X.M.; validation, M.Y.; Visualization, M.Y.; writing—original draft preparation, M.Y.; writing—review and editing, M.Y., X.M. and W.B. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Joint Fund of the National Nature Science Foundation of China and Shandong Province, grant number No. U1706228.

**Conflicts of Interest:** The authors declare no conflicts of interest.

## **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
