**1. Introduction**

The majority of existing ship seakeeping investigations are concerned with ship motion behavior in uni-directional or long-crested waves [1]. As a result, a lot of effort, which includes potential flow, computational fluid dynamics (CFD) and model test methods, is being dedicated to predicting ship motions in 2D regular or even irregular waves in the past few decades [2]. However, realistic sea states are multi-directional or short-crested, with wave components propagating in different directions. Therefore, an in-depth understanding of the multi-directional wave interactions with ship is essential in accurate prediction of ship hydrodynamics in realistic sea wave conditions.

In some places, e.g., the popular tourist destination of the Isle of Rhe, cross wave can be often observed due to the complex and extreme patterns of weather (see Figure 1a). The cross wave can be regarded as the superposition of wind or swell waves coming from two orthogonal directions. An example of the cross wave phenomenon in the ocean is shown in Figure 1. Until now, the problem for the interaction of a ship with a cross wave is not clear and has not been investigated, although people realize that the cross wave may pose a great threat to the safety of the passing ship. Therefore, this paper aims at investigating ship seakeeping behavior in wind-driven bi-directional cross sea waves.

**Figure 1.** Phenomenon of cross sea wave: (**a**) swell wave; (**b**) wind wave.

Some researchers have investigated the interaction of multi-directional wave with fixing offshore structures [3–5]. However, studies on the seakeeping behavior of free running ships or other floating structures in multi-directional wave are quite limited. Renaud et al. [6] investigated the effect of wave directionality on second-order slow-drift loads and motion response of a Liquefied Natural Gas (LNG) carrier in regular cross waves by a potential flow code and tank model test. Chen et al. [7] numerically investigated ship motions in long-crested and short-crested irregular waves by 3D time domain potential flow theory. Jiao et al. [8] comparatively studied the wave-induced ship motion and load responses in long- and short-crested irregular waves by theoretical and experimental methods. To summarize, the existing investigations on the hydrodynamic behavior of ships in multi-directional waves are mainly conducted by potential flow theory or model experiments, while related investigation by CFD has not been found.

In recent years, CFD, which captures most complexities of the fluid physics with few assumptions, has been widely used as a potential tool in ship design and evaluation [9–11]. Although tremendous advances have been made in the CFD simulations of ship motion responses in uni-directional wave [12–14], CFD investigations on ship motions induced by multi-directional waves are rarely seen. Recently, fundamental work of multi-directional wave simulation by CFD tool has been conducted by some researchers. For example, Wang et al. [15] simulated 3D directional irregular wave by open-source CFD model REEF3D. Cao and Wan [16] developed a multi-directional nonlinear numerical wave tank by Naoe-FOAM-SJTU solver. Wang et al. [17] calculated the wave forces on a large cylinder in multi-directional irregular waves using a CFD tool. These works focus on the simulation of multi-directional wave and also provide some foundations for the research of ship seakeeping behavior in multi-directional waves using CFD.

This paper focuses on investigating ship seakeeping behavior in bi-directional cross progressive waves by the CFD method, which will also provide some insights into ship motion behavior in multi-directional waves. For this purpose, a cross progressive wave is simulated in a rectangular numerical wave tank, and corresponding ship hydrodynamic behavior are estimated by solving Unsteady Reynolds Averaged Navier–Stokes (URANS) equations. Ship motion response and slamming pressure behavior under different cross wave states are analyzed. This study also provides some useful guidance for the safe operation of a ship when sailing in bi-directional cross seas.

The structure of the rest of this paper is arranged as follows: The description of the adopted S175 containership model, CFD numerical scheme, and calculation conditions are reported in Section 2. In Section 3, the characteristics of a cross wave are analyzed and also demonstrated by CFD simulation results. The seakeeping behavior of a ship in different cross wave conditions are systematically analyzed and discussed in Section 4. The slamming behavior of a ship in typical cross wave conditions is analyzed in Section 5. Finally, the main conclusions and future perspectives are reported in Section 6.
