An Experimental Study of Wave Impact Loads on an FPSO Bow in 2D Wave-Tank

2.1 Model and 2D Wave Tank

Due to spatial constraints in a two-dimensional wave tank, it is difficult to conduct model tests on the entire bow of an FPSO. In order to put the entire bow into the two-dimensional tank, the model must be small, but the wave impact load is small in a small model, making it difficult to measure. In addition, the size of the area to be measured is the same in the model scale, but the curvature of the area to be measured increases as the model becomes smaller. For this reason, a model was produced for a part of the bow of the FPSO. The scale of the model was set to 1/120. The width of the partial model was set to 0.2 m, considering the width (0.6 m) of the 2D wave tank. Fig. 2 presents the front view and side view of the model as well as the internal structure to install sensors. The bow section where the sensors are installed is positioned at an angle of 60 degrees to the water surface.

Fig. 3 shows a schematic diagram of the 2D wave tank. The tank had a length, width, and height of 29.0 m, 0.6 m, and 1.0 m, respectively. The tank was filled with water up to a height of 0.5 m, and it is equipped with a piston-type wave maker and a slope beach type wave absorber. The model was installed 17.8 m from the wave maker.

2.2 Measurement System

The speed and slope of the waves and wave impact loads were physical quantities measured in the model test. Two methods were used to measure the speed and slope of the waves. The first method involves arranging wave probes in a line to measure wave height, from which wave speed and slope of the waves are determined. The second method uses a grid attached to one side of the two-dimensional wave tank, capturing images with a high-speed camera to measure wave speed and slope of the waves. The waves measured in this study are those where breaking waves occur. When breaking waves happen, a significant amount of air bubbles forms on the wave surface, making it difficult to measure wave height. According to Nam et al. (2022), capacity-type wave probes measure the wave elevation that includes air bubbles, while resistance-type wave probes measure the wave elevation that does not include air bubbles. It is expected that the part including air bubbles has a minimal effect on wave impact loads. Therefore, this study used a resistance-type wave probe that does not account for the part including air bubbles.

Seven wave probes were placed to measure the slope of the waves, as shown in Fig. 4. Fig. 4(a) shows the arrangement of the wave probes and the conceptual diagram of the grid attached to the wall of the 2D wave tank, while Fig. 4(b) presents a snapshot of the installed wave probes. Reducing the gap between the wave probes is better for measuring the wave slope, but significant noise occurred when the gap was reduced due to the interference between the wave probes. Wave probes with different frequencies (4.7 and 4.8 kHz) were alternately arranged to reduce the noise of the wave probes. The frequencies of the wave probes were not the frequencies measured by DAQ but those used for data measurement inside the wave probes. The finally determined gap between the wave probes was 50 mm based on the model scale and 6 m based on the real ship. The width of the grid cells attached to the wall of the 2D wave tank walls was 20 mm.

Before the formal model test, the measured values of the wave probes and the photographs taken by the high-speed camera were compared. The left-hand side of Fig.5 shows the time series of the measured values, while the right-hand side shows a photograph of the moment when the largest value was measured in the W4 wave probe. For a visual comparison, the measured values and the y-axis of the photograph are expressed in the same size, and Z = 0 m and Z = 0.2 m were marked with red dotted lines. The sync signal shown in the left-hand image of Fig. 5 was used to synchronize the measured values of the wave probes and the high-speed camera. Although it is difficult to estimate the accurate values from the photograph, the values measured by the wave probes were similar to those captured by the high-speed camera. The distortion caused by the view angle of the camera lens can be identified in the additionally taken photographs. Points A and B in the right-hand image of Fig. 5 have the same vertical height. The vertical height of A appears to be higher than that of B because A is located closer to the camera, owing to the view angle of the lens. As the grid on the wall of the wave tank is attached to the side where B is located, it is necessary to observe the water surface close to the side where the grid is attached when estimating the wave elevation from the photograph taken. The position of the water surface can be estimated relatively easily from the photograph because the wave measured in Fig. 5 did not involve wave breaking. When wave breaking occurs, however, many bubbles form on the surface, making it difficult to estimate the position of the water surface from the photograph. On the other hand, the wave elevation measurements using wave probes are free from this problem.

A strain gauge type KISTLER 4567A sensor was used to measure wave impact loads. This study referred to Kim et al. (2014) for the sensor selection, and the pressure value was obtained by dividing the measured force by the measurement area.

The Hammering test was conducted to identify the natural frequency of the measurement system and model. The test was conducted under the condition of installing the model in the 2D wave tank (wet condition). An impact was applied to the model and sensor using a rubber hammer. Fig. 6 shows the force measured when an impact was applied to the sensor and the results of the Fast Fourier transform. The measured values show the general characteristics of the impact load; the natural frequency of the model was 700 Hz.

2.3 Frequency-Focused Waves

To understand the effect of wave slope change and resulting breaking waves on wave impact loads, three types of frequency-concentrated waves (steep, spilling, and plunging) were generated. A steep wave is a wave that does not cause wave breaking, and a spilling wave is a wave that causes an increase in wave slope and weak wave breaking at the wave crest. A plunging wave is a wave that causes an increase in wave slope, resulting in curling of the wave crest and severe wave breaking. The three types of frequency-focused waves were generated by referring to Ha et al. (2020). The method of generating frequency-focused waves and the characteristics of each wave were well described by Ha et al. (2020). Fig. 7 shows the displacement of the wave maker to generate the three types of frequency-focused waves.

3.1 Wave Elevation Measurement Results

The waves were measured in the absence of the model to measure the characteristics of the waves before measuring wave impact loads. The data related to wave elevation measurements were summarized on the model scale for comparison with the photographs taken by the high-speed camera. Fig. 8 shows the steep wave measured at the W4 wave probe. The highest wave elevation was measured between 41 and 41.5 seconds. Five repeated tests were conducted to examine the repeatability of the wave; excellent repeatability was obtained. Because the actual area of interest was 41 to 41.5 seconds when the highest wave elevation occurred, this part was magnified in Fig. 9. Fig. 9 shows the measurement results of the three waves. The time series of the measured wave elevations can be seen, and they are not the shape of the wave elevations. The slope of the measured values was the speed in the vertical direction (dζ/dt) because the time series of the wave elevations were measured at the W4 wave probe. To confirm the shape of the waves, multiple wave probes are necessary. Examining the characteristics of each wave in the time series reveals that before the occurrence of large wave heights, the measured values drop below the still water level in the order of steep, spilling, and plunging waves. Additionally, it can be observed that the slope of the time series, or vertical velocity, is asymmetrical based on the wave crest.

Fig. 10 shows the time series of wave heights measured from seven wave probes. Using seven wave height data measured at the same time, the wave shape can be obtained as shown in the left figure of Fig. 11. Before analyzing the measured wave height, the measured wave height and the photograph taken by a high-speed camera were compared. In Fig. 5, since the time series of the measured values and the photograph were compared, only the highest value of the measured value and the wave height estimated from the photograph could be compared. However, Fig. 11 shows the wave shape measured from seven wave probes, so it can be directly compared with the wave shape captured in the photograph. For a visual comparison, the measured values and the y-axis of the photographs are expressed in the same size. Z = 0 m and Z = 0.2 m are marked with red dotted lines. In addition, numbers were marked on the grid to compare the shapes of the measured waves with those captured in the photographs. When the measurements of the steep wave were examined, the shape of the wave captured in the photograph was in good agreement with the shape of the wave measured by the wave probes. The shape of the measured wave was almost symmetrical in the horizontal direction with respect to the highest wave elevation, and the wave slope was similar. The wave slope can be obtained directly from the shape of the measured wave. The spilling wave contains many air bubbles because of the occurrence of wave breaking. The shape of the measured wave is similar to the shape of the wave captured in the photograph, but it is difficult to define the boundary of the wave in the part where wave breaking occurred. Judging from the values measured by the wave probes, the shape was asymmetrical with respect to W4, at which the highest wave elevation was measured. For the plunging wave, wave breaking also occurred, as with the spilling wave, and the significant asymmetry of the wave shape was measured. The plunging wave exhibited the highest wave slope, followed by the spilling and steep waves.

The speed of the wave in the horizontal direction can be measured using three methods. The first method is to obtain the horizontal speed of the wave through the images captured using a high-speed camera. This method was the most intuitive, but the subjective judgment of the interpreter might have been involved. This is because the position of the wave crest must be determined through the eyes for this method, while the two methods presented later use the data measured by wave probes. The horizontal speed was obtained using the time required to pass through two grid cells attached to the wall of the 2D wave tank as follows.

In the above equation, the numerator is the width of two grid points, while the denominator represents the time points at which the wave crest starts (T_S) and finishes (T_E) passing through the two grid points.

Table 1 lists the horizontal speed of the wave. To account for subjective judgment, an additional error of one frame (1/1000 s) was considered, which corresponds to an error of about 5% in speed. The spilling wave exhibited the highest horizontal speed, while the steep wave showed the lowest.

The second method involves obtaining the horizontal speed of the wave from the time of the wave crest measured at W1 and W7, using the time series of wave heights shown in Fig. 10. This can be expressed mathematically as follows:

where P_W1 and P_W7 are the positions of the W1 and W7 wave probes, respectively, and T_W1 and T_W7 are the times at which W1 and W7 are passed, respectively

This method has the advantage of requiring only two wave probes; however, it has the limitation of providing an average speed of the wave passing between the two probes rather than a localized wave speed. The horizontal speed of the waves obtained using the second method are summarized in Table 2. No significant differences were observed among the speeds of the three waves. The spilling wave had the highest horizontal speed, and the steep wave had the lowest speed. The speed of the steep wave was approximately 10% lower than that of the spilling wave.

The third method is to obtain the horizontal speed of the wave from the slope and the vertical speed of the wave as follows.

The vertical speed (dζ/dt) can be obtained by differentiating the time series measured by the wave probes. In addition, the slope of the wave (dζ/dx) can be obtained from the wave shape. Table 3 lists the slopes, vertical speeds, and horizontal speeds of the waves obtained from the wave probes. The plunging wave exhibited the highest wave slope, followed by the spilling and steep waves. The vertical speed also increases in the order of steep, spilling, and plunging waves, just like the wave slope. However, the horizontal speed obtained from the wave slope and vertical speed showed the smallest value in plunging waves and then increased in the order of steep, spilling waves.

The horizontal speeds obtained using the three methods were similar values, although with some degree of variation. The characteristics of each method are as follows. The first method requires photographing the wave cross-section under special conditions, such as a two- dimensional tank, to obtain the horizontal speed. In addition, there is a disadvantage in that a person must directly determine the position to obtain the horizontal speed. The horizontal speed obtained through the second method is the average speed of the wave crest passing between the W1 and W7 wave probes. The method requires only two wave probes, but only the average speed can be obtained. The third method can obtain the slope and horizontal speed of the wave, but several wave probes must be installed.

3.2 Wave Impact Loads

The comparison of wave impact loads was conducted sequentially for the three types of waves (steep, spilling, plunging). First, the images of the moment when wave impact loads occurred were examined under the conditions of each wave. The time series of the wave impact loads were then examined, and the magnitude and spatial distribution of the wave impact loads obtained through repeated tests were analyzed. Finally, the wave impact loads estimated from the speeds of the waves were compared with those measured in the experiment in Table 4. Regarding the values used for the wave impact loads in Table 4, the slope and speed of the waves obtained through the third method were used.

Many studies have been conducted on the wave impact loads of wedges or cylinders, and the impact pressure and speed can be expressed as follows:

where, ρ: 1000 kg/m³ where C_p is the pressure coefficient. Several studies have been conducted to estimate the pressure coefficient. In this study, the most basic π/tanθ presented by Von Karman (1929) was applied. The wave impact loads estimated using the above equation and the slope and speed of the waves measured earlier were compared with those measured in the model test. The wave impact loads were analyzed on the actual ship scale for the time and pressure.

Fig. 12 shows photographs of the moment when the wave impact loads of the steep wave occurred. The moment when the wave impact loads occurred was set to T₀, and the photographs of T₀−0.5 s, T₀+0.5 s, and T₀+1.0 s are shown. Fig. 13 shows the time series of the measured wave impact loads. In the time series, the X-axis is the time of the actual ship scale. The time of 0 and the T₀ in Fig. 12 were synchronized. In Fig. 13, the Y-axis represents the wave impact pressure on the actual ship scale. Fig.14 shows the average values of five repeated tests conducted under steep wave conditions. The spatial distribution of the wave impact loads was examined by expressing the magnitudes of the pressures at the same positions as the measured positions using the color and size of the circle. The X and Y axes represent the positions of the measurement points, which were non-dimensionalized with the draft.

Fig. 13 shows the time series of the wave impact loads under the steep wave condition. The loads measured by the sensors installed in the center show that wave runup at the bow increased the pressure at P#4 in the lower part, and the pressure increased in the order of P#3 and P#2. The measured pressure is not an impact load but the hydrostatic pressure caused by wave runup. No significant wave impact load also occurred in Fig. 14, which shows the average of five repeated tests. An impact load of approximately 760 kPa is expected to occur if the speed of the wave measured from the steep wave and the angle of the FPSO bow is applied to Eq. (4). In the measured values, however, only the hydrostatic pressure was measured. An examination of Fig. 12 showed that wave runup occurred at the bow before the wave crest touched the FPSO bow. It is inferred that the wave runup acts as a cushion, preventing the occurrence of wave impact loads and only resulting in increased hydrostatic pressure. Wave runup occurs under conditions like those of steep waves, where no breaking takes place, allowing the wave to ascend along the hull surface with minimal energy loss and a smooth free surface, without breaking (Hu et al., 2017).

Figs. 15 to 17 summarize the results of the spilling wave. The time series in Fig. 16 indicates that the highest pressure was recorded at P#2, while significant wave impact loads were observed at P#9, P#3, and P#6. Photographs of the moment when large wave impact loads occurred showed that wave runup barely occurred on the surface of the FPSO bow, unlike the steep wave, and that many air bubbles were generated by the occurrence of wave breaking. This tendency occurred in all of the five repeated tests. Fig. 17 shows the average values of the wave impact loads obtained from five repeated tests. The largest value (1,000 kPa) occurred at P#2. The wave impact load estimated using the speed and angle measured from the spilling wave was approximately 1,120 kPa, which was similar to the value obtained from the model test.

Figs. 18 to 20 summarize the results of the plunging wave. The time series in Fig. 19 showed that large pressures were measured at P#2, P#3, P#6, and P#9. The plunging wave caused large wave impact loads in many sensors compared to the spilling wave. The photographs of the moment when large wave impact loads occurred show that wave runup barely occurred on the surface of the FPSO bow as with the spilling wave. Moreover, many air bubbles were generated by the occurrence of wave breaking. This tendency occurred in all of the five repeated tests. Among the average wave impact loads obtained from the five repeated tests, the highest value was 1,030 kPa at P#3, and the second highest was 977 kPa at P#2. When the spatial distribution of the wave impact loads in Fig. 20 was examined, large wave impact loads occurred in a large area, as with the results in the time series of Fig. 19. The wave impact load estimated using the speed and angle measured from the plunging wave was approximately 1,290 kPa. This load was similar to the value obtained from the model test, even though it was slightly higher.

From the measurement of wave impact loads for the three types of frequency-focused waves, the following characteristics can be identified. First, wave runup should not occur for wave impact loads. If wave runup occurs before the wave crest hits the bow flare, it acts as a cushion, resulting in the measurement of only the hydrostatic pressure caused by the runup. Further research is required on conditions that cause wave runup. Second, both the horizontal speed and the slope of the wave are important factors for wave impact loads. The results of the third method (Method III) for estimating the speed and slope of the waves showed that the horizontal speed of the spilling wave was higher than that of the plunging wave. On the other hand, the results of the measured wave impact loads showed that the wave impact load of the plunging wave was larger than the spilling wave. Moreover, it occurred over a wider area. This is the effect of the pressure coefficient C_p depending on relative angle. These results are the same as the parameters for predicting the wave impact loads presented by Stansberg (2008). The use of conditions that do not cause wave runup and the horizontal speed and slope of the waves make it possible to predict the wave impact loads more accurately.