**1. Introduction**

With ever-increasing marine exploration and subsea resource exploitation, offshore cranes which are mounted on vessels and carry out lifting/lowering have been widely used in marine operations. While working on the sea, offshore cranes suffer from persistent disturbances induced by ocean waves. During lifting or lowering, the payloads may be subject to large hydrodynamic forces, which could cause payload damages or cable breaks. This would further cause accidents and impair the safety of life and property [1].

In order to lift/lower payloads on the sea safely and efficiently, the capability to estimate the hydrodynamic loads on payloads is of vital importance. The hydrodynamic loads on stationary structures in waves have been studied for the safe and cost-effective design of coastal and offshore structures in the past decades. Compared to physical experiments, which need to establish scaled models, numerical modeling is more practical. The numerical models based on potential flow theory and Navier-Stokes (N-S) equations are two main categories for the simulation of wave-structure interactions.

The potential flow model is applied for wave interaction with large structures where viscous and turbulence effects can be ignored, such as the second-order potential flow theory model [2,3] and the fully nonlinear potential flow theory model [4]. With the assumption that the flow is inviscid and flow irrotational, it is challenging for the potential flow theory to capture the nonlinear free surface correctly when wave breaking occurs. Computational Fluid Dynamics (CFD) based on Navier–Stokes (N–S) equations is used for highly nonlinear wave–structure interactions in the case of breaking wave impacts and evolution of vortices. Various methods or models have been considered for wave–structure interaction, such as the Institute of Environmental Hydraulics of Cantabria Field Operation and Manipulation (IHFOAM) model, which solves Volume-Averaged Reynolds-Averaged Navier–Stokes equations (VARANS) [5,6], the multiple-layer σ-coordinate model [7], the Immersed Boundary Method [8], the Smooth Particle Hydrodynamics method [9,10], and the Constrained Interpolation Profile method [11].

OpenFOAM, a free open-source C++ toolbox for the development of customized numerical solver (such as the naoe-FOAM-SJTU solver [12]) based on CFD, has been applied in coastal and offshore engineering recently. Regular wave interaction with two tandem cylinders is studied with OpenFOAM [13], and an improved model named IHFOAM is used to study wave interaction with porous coastal structures [14,15]. The performance of OpenFOAM for nonlinear wave interactions with offshore structures is assessed, with up to eighth order harmonics correctly modeled [16].

In addition to the normal incident wave interaction with structures, many researchers have also investigated the interaction of oblique waves with stationary structures, such as perforated caissons [17], bridge decks [18], and various other structures [19–24]. The stationary nature of the structure makes it hard to rotate around different axes, the above oblique papers only focus on the situation of one single posture angle. Compared with stationary structures, the payloads can move with much more freedom while lifting or lowering payloads on the sea. Here, we want to reach a general conclusion when considering different posture angles, and to the authors' knowledge, there has been no previous research about the general postures' study of the payload.

Importantly, the posture of the payload has an impact on the force and moment exerted by the wave; additionally, the force and moment can also change the posture. This paper focuses on studying the influence of different postures of the payloads on wave forces and moments exerted on the payloads; thus, we assume that the payload is fixed without linear motion and rotation. A cylinder payload and a cuboid payload, both fixed and suspended with different postures in regular waves, are investigated, respectively. By carrying out a series of simulations, the influence of the payloads' posture angles relative to the regular waves on the hydrodynamic forces and moments exerted on the payloads are analyzed. It can be concluded from the results that the rotating rectangular payload (cuboid or cylinder) suffers a drastically changed moment when it is initially vertically placed, and the direction of the moment is the same as axis' rotation except for one situation. The projection area of the payload vertical to the force affects the corresponding force. The analysis could provide help for developing control strategies for offshore cranes, such as choosing the appropriate payload posture during water entry, and then using a controller to keep the payload on a certain posture that suffers minimal forces or moments during water entry.

### **2. Numerical Methods**
