In this study, an experiment was performed using a towing tank that can directly tow a submerged body in a single fluid environment. The vertical displacement of the generated free surface waves was measured, and their propagation characteristics were analyzed according to the submerged depth of the submerged body and the towing velocity. Fig. 2 shows the cylindrical submerged body model connected to the rail support on the side wall of the two-dimensional towing tank with a wire. In the towing-tank experiment (Kim et al., 2022a), the two-dimensional submerged-body towing tank of Inha University was used. The specifications of the cylinder and towing tank are presented in Table 1. It was possible to clearly observe the displacement and propagation of the free surface because the walls of the tank are made of a transparent acrylic material, its frames are made of aluminum, and its bottom is made of polyvinyl chloride. The cylindrical submerged body was divided into upper and lower hemispheres to allow a wire to pass through, and the cylinder was formed by fastening four 3-mm-diameter bolts. The width of the submerged body was set equal to that of the two-dimensional towing tank to minimize the interference of the tank walls with the formation and propagation of free surface waves. Fig. 3 shows an overview of the towing-tank experiment. For the submerged-body towing system, the submerged body was connected to the wire installed underwater and was directly towed using the wire drum connected to the servomotor. The system did not directly interfere with the free surface, because it towed the submerged body with the wire. The system made it possible to observe the exact free surface displacement caused by the submerged-body motion, ensuring the reliability of the experimental results. Additionally, the submerged body could be accelerated from the stationary state and towed at a constant velocity.

The submerged-body towing system was used to tow the submerged body at a constant velocity. Fig. 4 shows the wire drum connected to the alternating-current (AC) servomotor installed in the towing tank. It towed the submerged body at a precise velocity along the rail installed on the tank wall by circulating the wire shown in Fig. 3. The wire connected to the submerged body passed through the centerline of the body. The tension of the wire acted on the center of gravity of the submerged body, causing translational motion. For the towing system, the towing velocity and towing distance were adjusted through the Programmable Multi Axis Controller (PMAC) software.

The free surface waves generated by the submerged body were accurately measured using a servo-type wave gauge (PCA-WH60S). This wave gauge can measure the vertical displacement of a free surface every 0.01 s (100 Hz) with a precision of up to 0.056 mm using water as a conductor. Proper tension must be maintained in the underwater wire for stable towing of the submerged body. In the experiment, the overall tension of the towing system was controlled by adjusting the pulley connected to the wire in the vertical direction.

To validate the experimental results, a numerical analysis was performed using a two-dimensional numerical towing tank modeled according to the experimental towing tank. Fig. 5 shows a schematic of the numerical towing tank, whose specifications are those presented in Table 1. However, the numerical towing tank did not need the underwater rail or wire, because it could move the submerged body precisely at the fixed velocity every hour. While the cylindrical submerged body was accelerated from the stationary state and subjected to the constant-velocity translational motion in the same manner as the experimental conditions, the displacement and propagation pattern of the free surface were calculated in the time domain. The two-dimensional numerical towing tank was detailed in previous works (Seong et al., 2022; Kim et al., 2022b). It was briefly described in this paper because the accuracy and reliability of the solution were verified.

In the experiment of this study, the vertical displacement of free surface waves was analyzed, and its characteristics were identified according to the velocity and submerged depth of the submerged body. Table 2 presents the various submerged depth, velocity, and Froude number conditions examined. The waves generated in front of the submerged body by its motion were measured to identify their propagation characteristics, and the maximum wave height that occurred behind the body was measured and analyzed. Fig. 7 presents the vertical displacement of the free surface according to the body velocity in time series for submerged-body depths of d = 1D and d = 1.75D. As shown, the displacement increased as the body velocity increased, and the maximum wave height occurred behind the body.

The number of free surface waves that propagated to the front of the submerged body increased as the body approached the measuring point (the position of the wave gauge) at a low velocity. However, when the submerged-body depth was large (d = 1.75D), the maximum wave height and the number of free surface waves that propagated to the front of the body decreased. In addition, when the submerged-body depth was small (d = 1.0D), the maximum free surface displacement occurred after the submerged body passed the wave gauge owing to the occurrence of the maximum wave depression with the rapid free surface increase, and the shape of the wave was mostly lost behind the body. In contrast, when the submerged-body depth was large, the overall free surface displacement decreased, as only the maximum wave depression occurred, without a rapid increase in surface displacement. This appears to be because the jet-like flow increased behind the submerged body, which led to an increase in the displacement of the free surface as the body velocity increased (increase in Froude number) or the submerged depth decreased at a constant velocity, as reported by Shin and Cho (2021). Therefore, it can be said that the jet flow, which depends on the submerged depth and velocity of the submerged body, affects the maximum wave height.

The cause of the loss of free surface waves behind the submerged body can be inferred from the work of Hyun and Shin (2000). It is judged that as the submerged-body depth decreases, the displacement of the free surface rapidly decreases owing to wave breaking that occurs behind the body. Conversely, if the submerged-body depth increases (d = 1.75D), the degree of wave breaking that occurs behind the body is weak owing to the sufficient distance between the body and free surface, and there is room for an increase in the displacement of the free surface.

Fig. 8 shows the results of measuring and comparing the wave periods observed in front of the submerged body in order. Here, Nth Wave = 1 represents the first period measured by the wave gauge as the wave generated at the time of the acceleration of the submerged body propagated forward. As indicated by Fig. 7, the number of waves measured in front of the submerged body increased as the submerged-body depth decreased. Therefore, the number of waves measured was small when the submerged depth was large (d = 1.75D). Long-period waves were generated at the beginning when the submerged body accelerated, and then the period of the waves decreased slightly. As the submerged body approached the measuring point (the position of the wave gauge), the measured wave period decreased, but the rate of reduction gradually decreased and tended to converge to a certain value. This is because various waves were generated at different velocities until the submerged body reached a certain velocity after acceleration, and fast long-period waves among them reached the measuring point first, followed by short-period waves. The value to which the wave period converged was similar to that given by a dispersion relation (Eq. 8) that corresponds to deep water conditions. In the comparison graph, the part where the period decreases and increases suddenly represents the wave period corresponding to the maximum wave height caused by the submerged-body motion. The wave period was longer when the submerged-body depth was larger; i.e., long-period waves were generated at the maximum surface displacement that occurred behind the submerged body. In addition, the period measured in front of the submerged body was generally longer when the submerged-body depth was larger.

Here, T represents the wave period, w represents the wave frequency, and k represents the wavenumber.

Fig. 9 shows the maximum wave height according to the submerged depth and velocity of the submerged body. As in the previous analysis, the maximum wave height tended to increase as the body velocity increased. At 0.3 m/s, which was the lowest body velocity, there was no significant difference in wave height with respect to the submerged depth. However, as the velocity increased, the wave height rapidly increased at d = 1.0D, which was the smallest submerged depth, and then decreased at a maximum velocity of 0.7 m/s. This indicates that the wave height increases with the body velocity when the submerged depth is deep enough, but the maximum wave height decreases when the velocity increases further and wave breaking occurs behind the body. If the submerged-body depth is sufficiently large, the wave-height increase is small when the velocity is low; however, the wave height is likely to increase if the velocity increases. Under the experimental conditions of this study, even if the body velocity is higher than 0.7 m/s at a submerged depth of d = 1.75D, the maximum wave height will increase further because the effect of wave breaking that occurs behind the cylinder is expected to be insignificant. This phenomenon can be more accurately understood by comparing the wave steepness at the maximum wave height, as shown in Fig. 10. Wave steepness is expressed by an equation in which the wave height is divided by the wavelength ( H2Δx,Δx=UΔt). Theoretically, when the wave steepness is close to 0.1, wave breaking occurs. When the body velocity was 0.7 m/s, the wave steepness was close to 0.1 at submerged depths of d = 1.0D and d = 1.5D. Under these conditions, the magnitude of the wave height no longer increases, because wave breaking occurs in free surface waves.

The results of the submerged-body towing experiment performed in the two-dimensional towing tank were compared with the calculation results obtained using the numerical towing tank. The benefit of the numerical technique is that the entire experimental tank can be modeled to perform calculations under the same conditions as the experiment by controlling the body velocity and reflected wave. Fig. 11 presents a comparison of the free surface displacement generated when the submerged body moved at the highest velocity (U = 0.7 m/s). As shown, the numerical analysis results generally agreed with the experimental results for the maximum wave depression of the free surface caused by the submerged-body motion. However, there was a significant difference in the free surface displacement generated behind the cylinder. This is because wave breaking occurred behind the body, and the shape of the wave was lost, as the wave steepness of the maximum wave height corresponded to the wave-breaking condition, as mentioned in the analysis of the experimental results. In contrast, in the potential flow-based numerical analysis, the waves generated by the submerged body-motion were maintained without breaking, because fluid viscosity and wave breaking were not considered. The maximum wave depression that occurs immediately after the passing of the submerged body and before the occurrence of wave breaking is the free surface displacement generated by the fluid pressure change caused by the cylinder motion. For the potential flow-based calculation, the fluid pressure change can be implemented, but it is difficult to implement the complex physical phenomena caused by fluid viscosity and wave breaking behind the submerged body. In addition, the free surface displacement that occurred in front of the submerged body was smaller than that in the experiment. This appeared to be because linear free surface boundary conditions were applied in the potential flow-based calculation; thus, the nonlinear geometry of the free surface was not reflected.

Fig. 12 presents a comparison of the magnitudes of the maximum wave depression of the free surface with respect to the velocity of the submerged body. The magnitude of the maximum wave depression corresponds to the length of the arrow in Fig. 11. The results of the experiment and numerical analysis were generally similar, and the magnitude of the maximum wave depression increased as the body velocity increased. This indicates that the formation of the maximum wave depression is affected more significantly by the kinematic factors of the submerged body and the resulting fluid pressure change than by the fluid viscosity. Therefore, the maximum wave height was accurately predicted in the potential flow-based numerical analysis, although the maximum wave height measured in the experiment slightly exceeded the numerical analysis result. This appears to be because some of the effects of fluid viscosity that could not be considered in the potential flow-based calculations were included, and the nonlinear waveforms could not be properly implemented by applying linear free surface boundary conditions.

Fig. 12 presents a comparison of the measured and calculated periods of the free surface waves that propagated in front of the submerged body due to the body motion. As shown, the experimental and numerical analysis results had similar tendencies at all the body velocities. As the submerged body began translational motion and accelerated, waves with the longest period were generated and propagated first, followed by waves with gradually decreasing periods. The period of the waves generated by the submerged body at a constant velocity ultimately converged to the value given by Eq. (9), and it was confirmed that the convergence of the measurement and calculation results improved as the body velocity increased. Under all the velocity conditions, long-period waves were generated when the maximum wave depression occurred. Additionally, the rate of the wave-period increase decreased as the body velocity increased. This indicated that less long-period waves were generated as the body velocity increased, because the body quickly exited the acceleration region and entered the constant-velocity region owing to the submerged-body acceleration time being fixed at 0.22 s. Therefore, the agreement between the numerical analysis results and experimental results improved. However, the wave-period increase point differed between the experimental and numerical results, as shown in Fig. 13(a), because the numerical results and the measured data were different (Fig. 7(a)).