### 1. Introduction

*x*) between the two thrusters or the operating direction (

*ϕ*) of the azimuth thruster located at the front and proposed a simple estimation formula (Fig. 1). Nienhuis (1992) studied how the wake characteristics of an azimuth thruster located at the bottom of the hull change depending on the side curvature when the wake flows toward the side of the vessel. Cozijn et al. (2010) conducted a study similar to Nienhuis (1992) using the Particle image velocimetry test equipment and precisely measured and analyzed the wake. Since the 2000s, studies using numerical analysis methods have been conducted. Song et al. (2013) simulated the thruster-hull interference effect and the thruster-thruster interference effect using a numerical analysis method with the WTIV as a target vessel. They conducted a quantitative study to compare the thrust loss with the model test results. Ottens et al. (2011) numerically predicted the thrust loss due to the thruster-hull interference effect for semi-submergible. In addition, the duct or propeller axis of the azimuth thruster is rotated (tilt) downward by about 5°–7° to minimize the thrust loss due to the thruster-hull mutual interference effect and Palm et al. (2010) studied the resulting changes in thrust performance and mutual interference effect. Dang and Laheij (2004) reported that although there are numerous variables, such as vessel type, hull shape, stern appendage, and special thruster shape, the loss of thrust due to the thruster-hull interference effect can comprise up to approximately 40% of the thrust produced by an azimuth thruster in a single state. In addition, the actual DP is usually conducted under the bollard condition, in which only the thruster operates while offshore facilities or special vessels are stopped. Although there are points to note in implementing this condition in numerical analysis, research on the imposition of boundary conditions is lacking (Song et al., 2022).

### 2. Numerical Analysis Method

### 2.1 Definition of Numerical Analysis Method

### 2.2 Definition of Target Vessels

*B/T*) because WTIV has a relatively small draft compared to the width, resulting in a

*B/T*value of 11.8. By contrast, FPSO has a relatively small value of 4.84, meaning that the draft is larger than WTIV. Fig. 2 shows target vessels, respectively.

### 2.3 Performance Comparison of the Azimuth Thruster in a Single State

*K*) of the azimuth thruster is the sum of the thrust produced by the propeller (

_{TT}*K*), the thrust produced by the duct (

_{TP}*K*), and the resistance (

_{TD}*R*) of the remaining components (Housing, Leg, and Support). Numerical analysis was performed to construct a grid system so that the total thrust value calculated from the results of the single state of the azimuth thruster showed a <3% difference compared to the measured value of the model test. This analysis was performed using the direct rotation method (sliding mesh) that directly rotates the propeller, and Y

_{1}

^{+}, the dimensionless grid size, was set to 50 near the wall to use the wall function. The total grid was approximately 1.5 million; other numerical analysis conditions are listed in Table 1. To perform the single state analysis, the computational domain was defined as 7D × 4D × 4D based on the propeller diameter, and the velocity inlet condition, pressure outlet condition, and symmetry boundary condition were imposed, as shown in Fig. 3 (Song et al., 2013). In addition, the advance ratio (

*J*) considered was from 0.1 to 0.5. Fig. 4 shows the model azimuth thruster installed on the towing carriage, the shape defined for numerical analysis, and the grid system around the azimuth thruster. The results of a single test and numerical analysis are shown in Fig. 5 and Table 4 (Song et al., 2013).

*J*= 0, the thrust coefficient under the bollard condition can be estimated and used as a reference thrust value when predicting the thrust reduction due to the actual thruster-hull interference effect.

### 2.4 Analysis Condition Definition

*L*) of the target vessel and had sizes of 3.0

*L*, 3.0

*L*, and 1.2

*L*in the X, Y, and Z directions, respectively (Song et al., 2013; Song et al., 2022). In this study, the free water surface was not considered, and the area below the waterline of the vessel was defined as the analysis area. Symmetric boundary conditions were also applied to the top and bottom surfaces of the computational area.

#### 2.4.1 Target vessel 1: WTIV

#### 2.4.2 Target vessel 2: FPSO

### 2.5 Definition of Boundary Condition

*J*) was zero, such as the DP operating situation, and the vessel was stationary. As mentioned by Funeno (2009), numerical analysis under the bollard condition is unstable with poor convergence, so an artificially small advance ratio must be defined for numerical stability. The value was defined as 0.05 in this study. With the symmetry boundary condition defined on the top and bottom of the computational domain, each of the four sides shown in Fig. 9 was intended to be imposed in the following two forms, as shown in Fig. 6. The first boundary condition imposition method always set the velocity-inlet condition on the bow-direction surface and the pressure-outlet condition on the stern-direction surface regardless of the direction of the azimuth thruster attached to the hull. A symmetry boundary condition was imposed on the remaining two surfaces, called Fixed BC. The second boundary condition imposition method was to impose inflow and outflow conditions to the side in the same operating direction as the azimuth thruster attached to the hull. This was called Directional BC. Table 7 lists the boundary conditions by selecting some of the azimuths mentioned in Tables 5 and 6 for one of the target vessels. In particular, when the inflow conditions were given for each azimuth angle, the velocity in the X direction and Y direction was decomposed into components and defined as much as the azimuth angle of the azimuth thruster.

### 3. Numerical Analysis Result

### 3.1 Target Vessel 1: WTIV

*Fx_Hull*) and the force in the Y direction (

*Fy_Hull*) measured across the entire hull, including the thruster, when the model azimuth thrusters attached to the port side operates in each azimuth direction under different boundary conditions, and the resulting force (

*F_Total*) is shown according to the operating direction of the propeller. The resulting force (

*F_Total*) is defined using Eq. (1).

*Fx_Hull*and

*Fy_Hull*among the numerical analysis results calculated from the model test and imposed boundary conditions. Table 9 presents the degree of conformity between the

*F_Total*values and the model test. The numerical analysis results generally showed a similar trend to the test results. As shown in Fig. 10(c), for the force (

*F_Total*), the thrust loss occurs due to the thruster-hull interference effect between approximately 75° and 240°. Comparing the differences according to the method of imposing boundary conditions, the resulting force (

*F_Total*) calculated from Fixed BC showed a result closer to the model test result than the result calculated from Directional BC.

### 3.2 Target Vessel 2: FPSO

*Fx_Hull*) and the force in the Y direction (

*Fy_Hull*) measured across the entire hull, including the thruster, when the model azimuth thrusters located independently at the port and center operates in each azimuth direction under different boundary conditions and the resulting force (

*F_Total*) is shown according to the operating direction of the propeller. Table 11 lists the specific values. Numerical analysis showed that the force in the X direction (

*Fx_Hull*) was similar overall, even though there was a difference in the attachment position of the azimuth thrusters or the method of imposing boundary conditions. On the other hand, for the force in the Y direction (

*Fy_Hull*), there was no significant difference according to the attachment position of the azimuth thrusters under the same boundary conditions. On the other hand, even when the azimuth thrusters were installed in the same position, the results showed a large difference according to the imposed boundary conditions method. In particular, in a situation where the Directional_BC condition was imposed, the magnitude of the force in the Y direction (

*Fy_Hull*) applied to the entire target vessel, including the hull and thruster, was reduced compared to the result of the Fixed_BC condition, and was approximately twice as large under an azimuth angle of 75°. Thus, the distribution of the resulting force (

*F_Total*) of the entire target vessel according to the azimuth also showed a difference following the boundary condition imposition method. For the Fixed_BC condition, a constant resulting force was predicted regardless of the attachment position of the azimuth thruster. In contrast, the overall resulting force (

*F_Total*) decreased as the azimuth increased for the Directional_BC condition.

### 4. Discussion

*C*,

_{X}*C*, and the resulting force (

_{Y}*C*) for the target vessels, WTIV and FPSO, by classifying the current loads according to each azimuth in the X and Y directions. At this stage, the applied flow rate was the flow rate when the advance ratio was defined as 0.05, and the own current load of the vessel was calculated without the azimuth thruster attached. Fig. 14 shows the total dimensionless loads calculated for each azimuth based on the value of the resultant force (

_{Total}*C*) of the current load when the azimuth is 0° in each target vessel are shown together. As shown in Tables 13 and 14 and Fig. 14, the degree of increase in load varied greatly depending on the vessel type. In the case of WTIV, based on the resulting force (

_{Total}*C*), when the azimuth was 0°, it increased to approximately four times when the azimuth was 90°. On the other hand, in the case of the FPSO, based on the resulting force (

_{Total}*C*) when the azimuth was 0°, the load value was up to approximately 20 times greater when the azimuth was 75°. Moreover, the size of the load value itself according to the direction of the current also showed a large difference depending on the type of vessel. It was shown that FPSO is approximately 10 times larger than WTIV. These results showed that even if the azimuth thruster attached to the target vessel constantly calculates the thrust during numerical analysis. The total thrust (

_{Total}*F_Total*) effective for the vessel is greatly reduced as the current load of the vessel becomes excessively large. In particular, the magnitude of the current load may vary greatly according to the vessel characteristics, e.g., the vessel type, hull shape, appendage, and draft. Fig. 15 shows the distribution of the dimensionless pressure coefficient of the hull calculated under the condition of the current at the 45° azimuth from the port side of each target vessel. The distribution form has different characteristics depending on the target vessel. Therefore, when performing numerical analysis to predict thrust loss due to thruster-hull mutual interference, Fixed_BC can be imposed on the side of the computational area rather than Directional_BC, which unnecessarily causes a large current load of the target vessel to be induced and unrealistically large thrust loss to be estimated. When Fixed_BC is imposed, a corresponding flow rate with a small advance ratio artificially defined for numerical stability was additionally imposed, and the effect of the current load can be minimized because this flow rate was very low. As shown in Table 10, when the target vessel is WTIV, the loss of thrust estimated through numerical analysis matches well with the model test results. Even when the target vessel is FPSO, the thrust loss according to the azimuth was estimated to be within 10% regardless of the attachment position of the azimuth thruster, as shown in Table 12. Moreover, when Fixed_BC was imposed as a boundary condition, FPSO, one of the target vessels, was predicted to have relatively less thrust loss than the other target vessel, WTIV. The causes are as follows. The azimuth thruster was installed at the lower part of the head box protruding downward from the hull, so the distance from the hull was relatively far. Second, the duct of the azimuth thruster applied in this study rotated downward (tilt), which can reduce the Coanda effect. Third, the optimal arrangement of azimuth thruster that minimizes interference between the wake direction of the azimuth thruster and the hull, the selection of an azimuth thruster with appropriate capacity, and the advancement of DP control algorithms are fundamentally needed to prevent excessive thrust loss due to thruster-hull mutual interference in offshore facilities or special vessels.

### 5. Conclusion

When estimating the thrust loss due to thruster-hull interference by the numerical analysis, it is practical to impose inflow conditions and pressure outflow conditions in the computational domains in the bow and stern directions, such as the boundary condition at 0° azimuth. On the other hand, if the velocity inflow condition and the pressure outflow condition are imposed according to the direction of the operating direction of the azimuth thruster, an unintended current load of the hull may occur, which results in excessive thrust loss may be excessively predicted.

Even if the aforementioned boundary conditions were imposed during the numerical analysis, the thrust loss due to the thruster-hull interference effect can show a difference depending on the vessel type, hull shape, various appendages, and arrangement of the azimuth thruster. In the case of the WTIV, a thrust loss of approximately 30% was expected. On the other hand, in the case of the FPSO, a thrust loss of less than 10% was expected at the azimuth angle of the azimuth thruster considered, and the differences in the azimuth thruster attachment position (port or center of the vessel) were up to approximately 5%.