### 1. Introduction

*H =*3

*D*(

*D*= Hull diameter). The drag and lift forces of various underwater vehicles have been investigated through experiments and numerical analysis, and the standard depth for the fully submerged condition of each underwater vehicle varies among the reference papers.

*H =L*/2 (

*L*= hull length).

*H =*3

*D*(

*D*= hull diameter).

*H =*5

*D*.

*F*) from 0.099 to 0.349. The results showed that the residual resistance of the standard shape exceeded that of the tango shape at from 0.19 to 0.3.

_{n}*F*decreased, and that the effect of free surface is negligible when

_{n}*H*is greater than or equal to 3

*D*. Furthermore, they confirmed that if the Reynolds number increases at a low submergence depth, then the effect of underwater vehicle motion on the free surface becomes more evident.

*F*. However, we conducted this study based on our assumption that the depth can be expressed by a function of the

_{n}*F*if the underwater vehicles exhibit similar shapes. Many types of underwater vehicles exist, including underwear vehicles and submarines. However, for the underwater vehicle used in this study, we assumed that the characteristics of the vessel shape will not vary significantly.

_{n}*F*for an underwater vehicle with

_{n}*L*/

*D =*5.8 based on resistance test results obtained via model tests and CFD simulation. Experiments were conducted in a towing tank at Pusan National University (PNU). By comparing experimental fluid dynamics (EFD) and CFD analysis results, we demonstrate the reliability of the CFD analysis and propose a standard depth, at which the effect of free surface is negligible, for each

*F*by comparing the CFD analysis results with the no-free-surface condition and the resistance results at various depths with a free surface condition.

_{n}### 2. Resistance Tests

### 2.2 Conditions of Model Tests

*L*/

*D =*5.8.

*D*) and 1,250 mm (3.8

*D*). The total resistance was measured at intervals of 1 kn (0.514 m/s) at the vehicle speed range of 2 to 10 kn (1.03–5.14 m/s). Fig. 2 shows the towing carrier at PNU and the experimental setup used for the model tests. Sandpaper was attached as a turbulence stimulator at a length between perpendiculars (LPP) position of 0.05 on the model vehicle’s bow. The towing carrier and model vehicle were connected using two cylinders (diameter of cylinder = 5 cm), where a transparent polyvinyl chloride plate was attached to prevent disturbance from the free surface due to the connecting section of the load cell and the model vehicle.

### 2.3 Model Test Results

*D*) in the first experiment and at a depth of 1,250 mm (3.8

*D*) in the second experiment. The results are shown in Tables 2‒3 (

*R*is the Reynolds number,

_{n}*C*the residual resistance coefficient, and

_{R}*R*the total resistance of the model vehicle).

_{TM}*D*) based on a reference, which was unbounded by the effect of free surface.

*D*) indicate that as the speed increased, the residual resistance coefficient increased less as compared with the residual resistance coefficient at a depth of 850 mm (2.6

*D*). Fig. 3 shows a graphical comparison of the residual resistance coefficient values between the two depths.

### 3. Numerical Analysis and Correlation Derivation

### 3.1 Numerical Analysis Method

*ρ*is the density of the fluid,

*t*the time,

*u*the flow rate,

_{i}*p*the pressure,

*μ*the fluid viscosity coefficient,

*g*the gravitational acceleration,

_{i}*F*the body force per unit volume.

_{i}*k*−

*∊*model is the most typically used model in engineering. In this study, we used the realizable

*k*−

*∊*turbulence model, which showed improved performance for the boundary layer separation flow caused by the adverse pressure gradient. In the free surface analysis, we used the volume of fluid (VOF) method. The VOF method is a method that monitors the position of the free surface, which is the boundary between two fluids, based on the volume ratio of the two fluids, which have different densities in the grid (Jagadeesh and Murali, 2010).

*y*

^{+}

*=*50. Meanwhile, because the effect of shear force was significant around the underwater vehicle, six layers were established in the prism layer, and the wall function was applied (Byeon et al., 2018).

### 3.2 Numerical Analysis Results

#### 3.2.1 Comparison of resistance in no-free-surface condition

*D*depth and the CFD analysis results without the free surface condition. Table 4 and Fig. 5 show the differences in resistance coefficient and total resistance at vehicle speeds of 3, 6, 8, and 10 kn (1.54, 3.09, 4.12, and 5.14 m/s, respectively). Based on the result where the difference in total resistance at a low speed of 3 kn (1.54 m/s) was −0.25%, we conducted a CFD analysis under the same conditions for different vehicle speeds. The results show that as the speed increased, the differences in resistance coefficient and total resistance increased. The experimental results based on a low speed of 3 kn (1.54 m/s) were consistent with the CFD results for the no-free-surface condition because almost no wave-making resistance caused by free surface was present. Furthermore, the effect of wave-making resistance became dominant when approaching a high speed.

#### 3.2.2 Comparison of resistance in free surface condition

*V*= 5.14 m/s,

_{S}*V*= 2.71 m/s). A comparative analysis was performed for the EFD and CFD results under the following four depth conditions:

_{M}*H*= 3

*D*and

*H*= 5

*D*, which are suggested in a previous study;

*H*= 3.8

*D*, which is a depth at which the model test was conducted at PNU; and without any free surface condition (non-F.S.E.) (F.S.E.= free surface effect). Table 5 shows the results.

*V*= 5.14 m/s,

_{S}*V*= 2.71 m/s), which suggests that the mode test was not conducted at a sufficient depth.

_{M}#### 3.2.3 Case studies of submergence depth at different vehicle speeds

*D*at

*V*= 3 kn (1.54 m/s), 2.4

_{S}*D*at

*V*= 6 kn (3.09 m/s), 4.2

_{S}*D*at

*V*= 8 kn (4.12 m/s), and 6.0

_{S}*D*at

*V*= 10 kn (5.14 m/s). The graph in Fig. 6 shows the difference in the resistance coefficient between the EFD and CFD results based on the . Furthermore, Fig. 7 shows the waveform of the free surface in the z-direction for

_{S}*V*= 10 kn (5.14 m/s) and

_{S}*H*= 6.0 (the z-position of the free surface in the CFD simulation is 2.158 m.)

####
3.2.4 Correlation between *F*_{n} and appropriate submergence depth

_{n}

*F*and

_{n}*H*/

*D*, respectively, and their correlation is shown graphically in Fig. 8. The x-axis of the graph in Fig. 8 represents

*F*, and the y-axis represents the ratio of the submergence depth to the hull diameter (

_{n}*H*/

*D*). The results for the appropriate submergence depths at four

*F*of the underwater vehicle are shown in a trend line. Furthermore, a comparison between the current results and those obtained based on

_{n}*F*= 0.4 and

_{n}*H*= 5.0

*D*by Moonesun et al. (2013) suggest that the experiments were conducted at a deeper level than the required depth determined in this study. However, a deeper level may be required when

*F*is large (i.e., when the underwater vehicle exhibits a high vehicle speed or a small

_{n}*L*).

### 4. Conclusion

*L*/

*D*was 5.8, and the experiment was conducted twice at depths of 850 mm (2.6

*D*) and 1,250 mm (3.8

*D*), intervals of 1 kn (0.514 m/s), and a vehicle speed range of 2 to 10 kn (1.03–5.14 m/s). In terms of the residual resistance coefficient, the results obtained at a depth of 850 mm (2.6

*D*) were higher than those obtained at a depth of 1,250 mm (3.8

*D*) as the speed increased. Hence, we inferred that the wave-making resistance might have been caused by the free surface effect arising from the difference in depth. We conducted a CFD resistance analysis to verify the difference in the free space effect between the depth conditions. First, we compared the total resistance obtained via CFD simulation without a free surface and that obtained experimentally. The result showed that the difference in the total resistance was −0.25%, which is insignificant, at a low speed of 3 kn (1.54 m/s); however, the difference increased gradually to 13.75% at a high speed of 10 kn (5.14 m/s), and the difference in the residual resistance coefficient increased as well. In our opinion, this error occurred because the standard depth, at which the free surface imposed no effect, was not satisfied in the high-speed domain in the experiments; therefore, we examined the difference in the total resistance based on the submergence depth at specified speeds. We investigated four depths: depths of 3

*D*and 5

*D*, as suggested in a reference paper; a depth of 3.8, which was applied for an experiment performed in the PNU towing tank; and the no-free-surface condition. The CFD analysis results obtained based on these depths were compared with the EFD results. The results showed that as the depth from the free surface increased, the total resistance from the CFD results decreased and, consequently, the error from the model test result increased. Therefore, based on the two sets of results above, we concluded that the values measured in the model test were high owing to the effect of the free surface because the model test was not performed at a sufficient depth, and that the effect was more prominent in the high-speed domain.

*D*at

*V*= 3 kn (1.54 m/s), 2.4

_{S}*D*at

*V*= 6 kn (3.09 m/s), 4.2

_{S}*D*at

*V*= 8 kn (4.12 m/s), and 6.0

_{S}*D*at

*V*= 10 kn (5.14 m/s). Furthermore, the vehicle speed and submergence depth were non-dimensionalized into

_{S}*F*and

_{n}*H*/

*D*, respectively, and the correlation was formulated as

*y =*24

*x*

^{2}− 10

*x*+ 3.