**1. Introduction**

The shallow-water effect refers to the situation that the hydrodynamic performance of a ship clearly changes when the water depth is less than a certain critical value. For a channel that produces the shallow water effect, we divide it into two categories: (1) one is the channel that only considers the limited water depth and its effect on the hydrodynamic performance of the ship, which is called the shallow water channel; and (2) the other is the channel in which both the water depth and the width are limited, causing an effect on the hydrodynamic performance of the ship, which is called the restricted channel. The hydrodynamic characteristics of ships in navigation will change and differ significantly when encountering restricted channels versus shallow water channels. Generally, the effects of shallow water can be summarized into the following three aspects: (1) the change of attitude manifested as the change of trim and sinkage; (2) the increase of ship resistance; and (3) worse maneuverability of the ship. For some high-speed vessels, the water depth needs to be rather deep to avoid the influence of shallow water, which cannot always be guaranteed. Especially in recent years, the shallow-water effect of vessels becomes more and more evident with the increase in tonnage of ships. The obvious sinkage of vessels in shallow water is caused by many bottom touching accidents, which makes it difficult for the safe navigation of vessels encountering shallow-water channels. In addition, the obvious increase of ship resistance in shallow water leads to a worse speed–power

characteristic, which directly affects the operational efficiency of the vessel. An accurate prediction of ship resistance in shallow or confined water is crucial. Model tests are the most common way to predict the resistance of a ship, and the resistance of the full-scale ship can be obtained utilizing extrapolation. Even though much practice has proved the reliability of the extrapolation approach, the Reynolds number similarity between the ship model and the full-scale ship cannot be achieved, which results in significant differences between the model-scale and full-scale ship flow. The ITTC-57 correlation line [1] used to build a relationship between the resistance of a scaled model and the full-scale ship may not be accurate in shallow water. Zeng et al. [2] mentioned in their paper "This is probably due to the backflow and/or a different wetted surface". Raven [3] also suggested considering the scale effect in the extrapolation. The rapid development of computational performance and numerical methods promotes the development of computational fluid dynamics (CFD) [4], and the Unsteady Reynolds Average Navier-Stokes equation (URANS) CFD solver is becoming another practical tool used for predicting the hydrodynamic characteristics of ships. In addition to saving time and money, another advantage of using a CFD solver is that it is easier to obtain the local flow characteristics. URANS simulations were conducted for a KCS [5]; the effective power and the increase of resistance in a series of designed head waves are predicted. In addition, the effect of speed loss on the reduction of effective power is explained. Yang et al. [6] presented a study on the air cavity under a stepped planing hull based on the finite volume method (FVM), and a mesh convergence study was conducted to ensure the accuracy of the simulation. Cucinotta et al. [7,8] analyzed the performance of a multi stepped air cavity planing hull using URANS CFD code. Duy et al. [9] investigated the stern flow field for several transom configurations of a KCS using a viscous CFD solver. Jachowski et al. [10] predicted the squat of a KCS scaled model in shallow water using the CFD method; the results show quite good agreement with the empirical method of Hooft [11]. Numerical prediction of the resistance of a barge ship with different calculation velocities at different water depth-to-ship draft ratios (T/H) was conducted [12]. It can be seen in the study that the increase of resistance becomes increasingly obvious with the decrease of water depth and the increase of velocity, which means that the increase of resistance is related to both velocity and water depth. JI et al. [13] conducted a 3D numerical simulation to research the relationship of the sediment movement induced by the compounding effects of ship-generated waves, water flow due to ship propellers, and the influence of ship and channel characteristics. Linde et al. [14] proposed a 3D hydrodynamic numerical model to predict ship resistance and sinkage of an inland ship in restricted waterways; the results showed that the ship resistance is more sensitive to water depth than channel width. Du et al. [15] studied inland vessels in the fully-confined waterway, and the characteristics of resistance and waves were analyzed. CFD simulations of the pure sway tests in a shallow water towing tank were conducted for the DTC container ship model using URANS solver [16]. From the study, the ability of URANS CFD solver to simulate the pure sway tests in a shallow water towing tank was proved by comparing with the test data. Researchers also conducted the maneuvering tests of a scaled ship model with different water depths and speeds [17], the results show that the shallow water effect has an adverse effect on ship maneuverability, which is manifested by the increase of turning diameter and the decrease of course stability. Simulations of straightforward, turning and zig-zag motions for a cargo ship were carried out [18]; as the depth–draft ratio decreases, the ship's resistance increases and the maneuverability becomes worse.

Furthermore, some researchers proposed methods for resistance correction in different water depths. The earliest method that can be found for correcting shallow water resistance was proposed by Schlichting [19]. A further method of Lackenby [20] was proposed by the reanalysis of Schlichting's method, which was recommended by the International Towing Tank Conference in 2014 [21]. Methods proposed at that time was based on less experimental data due to limited resources. In recent years, with the development of numerical methods and experimental technology, researchers derived some new methods based on numerical calculations and experimental data. Jiang introduced a mean effective speed, which can be calculated by the mean sinkage, ship speed, and water depth [22]. In his study, Jiang found that the model resistance is almost a unit function of the effective speed and independent

of the water depth. Raven [23] proposed a new method to correct the resistance in shallow water after theoretical analysis and numerical calculations, which was recommended by the ITTC in 2017 [24].

For the great importance of estimating ship resistance in different water depths, it is significant to predict the hydrodynamic characteristics of a ship in different water depths. The existing relevant literature mainly focuses on the very shallow water; most of their water depths/draft ratios are less than 2. However, the limited water depths of a larger value play a very important role in reality. This paper studied the influence of water depth on ship resistance; several large limited water depths were chosen to conduct the towed resistance simulations for a Kriso container ship (KCS). Resistance and attitude in different water depths for the scaled model and the full-scale KCS were calculated. Before performing the calculation at different water depths, analysis of numerical uncertainties was carried out. For model scale, the sensitivity of grid spacing from the bottom of the ship to the tank bottom was also studied. Numerical tanks with large width were established to ignore the influence of limited width. The chosen water depths were slightly larger to match the constraints of Raven's method. The in-house CFD solver HUST-Ship was employed to carry out the calculations. The simulation results were compared with the predicted results of Raven's model.
