# Interaction of a Solitary Wave with Vertical Fully/Partially Submerged Circular Cylinders with/without a Hollow Zone

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Flow-Field Equations

^{*}, the undisturbed water depth, the velocity scale, $\sqrt{g{H}^{*}}$ (here $g$ is a constant due to gravity), and $\sqrt{{H}^{*}/g}$ as the time scale. In this model, the right-hand referencing frame has its x-axis pointing in the positive (right) direction, the y-axis is expanding laterally, and the z-axis is pointing up; the coordinate origin is located at the water level of the undisturbed fluid region. The non-dimensionalized initial-boundary-value problem can be formulated as the governing equation, initial condition, and associated boundary conditions. These equations can be referenced in many books on classical wave mechanics, such as Stokers [62]. These equations are listed as follows:

_{0}= the incident wave height, and X

_{0}= the wave’s starting position. Wu [65] derived the relation between the potential function and its average value in the flow region as:

## 3. Validations

_{0}= 0.4, and a cylindrical diameter D = 1.63. As the cylinder is bottom-mounted, H = H

_{s}= 1.0. This case is typically adopted to test a 3D nonlinear water wave model. Mo [37] examined this case both experimentally and numerically, and provided abundant data for a solitary wave encountering a single cylinder or multiple piles. First, to verify our model, it is compared with that used by Mo. The calculation range is (x

_{max}, x

_{min}) = (−30, 30) and (y

_{max}, y

_{min}) = (−30, 30). This study used six wave-gauge positions (see Figure 2) around the cylinder (as shown in Figure 3). In Figure 3, ${t}^{\prime}$ is the time shift with respect to the peak time at Gauge 4. Although our model does not consider fluid viscosity, generally, it can efficiently identify the consistent trends at all six gauge positions.

^{*}= 7.62 cm, and according to its dimensionless channel experimental conditions, its length was (−40, 40), width was (−2.5, 2.5), wave gauge positions G1 and G2 were −1.411 and 1.411, respectively, and the cylinder diameter D was 1.5. The wave gauge measurements reported by the authors are for the case of H

_{s}= 0.5. Compared with Chen and Wang’s measured results shown in Figure 4, the results of the numerical simulation obtained in this study, which are consistent with their experimental results, are shown in Figure 4a,b with A

_{0}= 0.19 and Figure 4c,d with A

_{0}= 0.31. Because their channel width was W = 5 (only five times the still water depth), the cylinder with D = 1.5 was very close to the side wall of the channel. Figure 5 compares the difference in results obtained for W = 5 and 60, and shows that the wide channel W = 60 causes the scattering dispersive waves around the cylinder to be relatively small and weak.

## 4. Results and Discussion

#### 4.1. Solitary Wave Hitting a Circular Cylinder with Different Drafts and Sizes

_{0}= 0.3 passes through a cylinder of D = 4, for which different immersion depths (H

_{s}) are simulated. Figure 6 shows a plot of the wave motions for a circular cylinder that is semi-immersed at H

_{s}= 0.5, with the left side of each figure showing the top view to enable observation of the evolutions of the crest lines, and the right side showing a 3D perspective. For example, in Figure 6 at t = 0, the distance of a solitary wave from the center of the cylinder 15 times the water depth. In this figure, it can be seen that when t = 3, the solitary wave approaches the cylinder but is not significantly affected by it. When t = 12, the wave touches the cylinder’s surface, and runs up the front of the cylinder. Then, most of the wave passes through the cylinder to form a transmitted wave. The wave is also scattered by the cylinder, and the water elevation initially runs up the front of the cylinder and then drops and forms a system of diffraction waves around it. At t = 18, the wave passed through the cylinder, and the crest lines behind the cylinder are just slightly lower than those on both sides, such that the cylinder does not seriously eradicate the crest lines. After the wave has completely passed through the cylinder (such as at t = 30), the shape of the crest line that was damaged by the cylindrical structure returns to that of the original incident wave and continue to propagate forward. That is, the crest lines gradually become straight again. Eventually, a system of cylindrical diffraction waves radiates around the cylinder.

_{s}. Figure 7a shows the water level histories of the front cylinder at point A (−2, 0), and Figure 7b shows those of point B (2, 0) behind the cylinder. This figure shows the differences among H

_{s}= 1.0 (complete immersion), 0.7, and 0.5. At an incident wave height of 0.3, although the H

_{s}value changes, there is no obvious difference in the overall phenomenon. The maximum water level at point A can increase to about 0.5, which is about 1.7 times the height of the original wave. Then, the water level at point A drops below the still water level and gradually returns to the still water level. The evolution at point B gradually increases and then decreases, but all the recorded water levels at point B during this interaction are higher than the still water level.

_{s}for a deep immersion reveals that it produces a large runup and depression at point A and a large increase in the runup at point B. When H

_{s}= 1.0, the maximum runup elevation of point B nearly reaches the height of the incident wave. However, if H

_{s}is smaller, the runup of point B is slightly reduced, and the water level starts to rise earlier. This means that a larger gap (shallower H

_{s}) allows more wave-induced flow to pass under the cylinder to form earlier wave surges at the rear of the cylinder. As such, the water level rise occurs earlier, but the volume of water squeezing and rising from the lateral sides of the cylinder after diffraction is weak, so the maximum runup at point B is low.

_{s}= 1.0, 0.7, and 0.5. In this figure, the free-surface profiles at the symmetry plane (y = 0) are plotted at various moments. When t = 9, it can be seen that the wave touches the cylinder, so there is a rising runup at the front of the cylinder. Prior to this time, there is no obvious difference among the three immersions (H

_{s}= 1.0, 0.7, and 0.5). However, when t = 12, the wave rises higher. For example, in the case of H

_{s}= 0.5, there is a gap between the bottom of the cylinder and the seabed, so the current caused by the wave will pass through the gap. Therefore, the larger the gap (i.e., shallower H

_{s}), the stronger the transmitted wave that initially emerges at the rear of the cylinder. However, the rear water level changes over time. By t = 15, a wave reflection and transmission mechanism appears. However, when t = 18 and H

_{s}= 1.0, complete diffraction occurs (no current moves through the gap), so more diffraction waves will accumulate from the surrounding lateral areas of the cylinder, which will result in greater surge behind the cylinder, such that the transmitted wave behind the cylinder will become large. That is, at this time, the large gap creates a weaker runup at the rear of the cylinder. At t = 30, we can see that the wave bypasses the cylinder and is destroyed but still appears as a solitary wave, even though the wave height is slightly lower than the incident wave height (as denoted by the horizontal dashed line in Figure 8f, which represents the original wave height). A detailed comparison of the cases of H

_{s}= 0.5 and H

_{s}= 1.0 reveals that the transmission wave height at H

_{s}= 1.0 is slightly high and is accompanied by some trailing waves, but the reflected waves produce more trailing waves for the case of H

_{s}= 0.5. Generally, there is no significant difference between them.

_{s}, the larger are both the front and rear runups. This echoes the results shown in Figure 7.

#### 4.2. Solitary Wave Hits a Circular Cylinder with a Hollow Zone

_{0}), other possible influencing parameters are the still water depth H (H = 1 after normalization), the cylindrical immersion depth H

_{s}, and the outer diameter r

_{1}and inner diameter r

_{2}of the concentric circular cylinder (cylinder thickness dr = r

_{1}− r

_{2}).

_{0}= 0.3 as an example, mainly to analyze the influence of r

_{1}, dr, H

_{s}, and W on wave oscillation in the hollow circular cylinder. Figure 11 shows a hollow cylinder with r

_{1}= 2, r

_{2}= 1 (dr = 1), immersed at H

_{s}= 0.5, with a channel width W = 10 to determine the characteristics of the change in the surrounding water level when a solitary wave passes through this cylinder. Figure 11a shows plots of the time history of the water level at three positions, a, b, and c, in the transverse direction. It is apparent that the wave runup and rundown at points a and c are similar to those obtained in the previous water-level analysis of a wave passing through a single solid cylinder at its front and rear positions, whereas Point b shows a change in the water-level oscillation in the hollow area, with its oscillation amplitude obviously larger than those of Points a and c. Figure 11b shows the water-level changes at Points e and f in the longitudinal direction. This figure shows that Point f in the hollow area has a significant oscillation amplitude similar to that of Point b, which indicates that the hollow area causes waves to generate an oscillation effect in the hollow column area.

_{2}), Figure 12 shows the case of r

_{1}= 3, H

_{s}= 0.5, and W = 20. That is, the outer diameter is held constant while the thickness (or inner diameter) is varied to observe how the water oscillates in the hollow column. A comparison of Figure 12a,b reveals that in Figure 12a, when r

_{2}= 1, the difference in the water levels at points a, b, and c of the hollow area is small. However, in Figure 12b, when r

_{2}= 2 (which is larger than that in Figure 12a), the water levels of the hollow area are significantly different at points a’, b’, and c’. This result indicates that when r

_{2}= 1 (Figure 12a), the water level of the hollow area oscillates more uniformly. Thus, if the hollow area is small (not greater than the still water depth), the hollow water column will fluctuate more uniformly. This phenomenon can be conceptualized and anticipated and can also be observed in the 2D water-level color contour map in Figure 13. Figure 13a–g and Figure 13a’–g’ correspond to Figure 12a,b, respectively. The planar view shows the overall changes in the reflection, transmission, and diffraction of the wave as it encounters the hollow cylinder. This result is indicated by the color change in the hollow zone. The hollow area in Figure 13a–g is small (r

_{2}= 1) and always shows a single color in subsequent figures, which indicates that its water level is uniform in the hollow zone. In contrast, the color of the hollow area in Figure 13a’–g’ is not uniform.

#### 4.2.1. Fixed Thickness, with Changes in the Outer and Inner Diameters

_{1}= 2, 3, and 4. Figure 8a–c show the effect of different H

_{s}values on the rise and fall of the water column. These figures reveal that when dr is fixed and the outer diameter becomes larger, there will be a larger hollow area, which is not conducive to the formation of a uniform water level over the entire hollow area. A smaller r

_{1}value results in a better oscillation effect, and the larger is H

_{s}(e.g., H

_{s}= 0.7), the more significant the oscillation, i.e., a water column with a smaller r

_{1}and a larger H

_{s}has a better oscillation effect.

#### 4.2.2. Fixed Inner Diameter, with Changes in the Outer Diameter and Thickness

_{2}but changing the thickness. Similar to the analysis shown in Figure 14, Figure 15 shows the water levels for a fixed inner diameter r

_{2}= 1 and channel width W = 15. The inner diameter of the hollow area is fixed so the influence of the outer diameter r

_{1}can be observed. Figure 15a shows that when the cylinder is not deeply immersed (e.g., H

_{s}= 0.3 in Figure 15a), a larger outer diameter helps to drive the amplitude to generate more regular motions. In Figure 15a, for the case of H

_{s}= 0.3, the larger is r

_{1}, the larger the amplitude (except for the main wave). If the immersion depth is deep, this phenomenon does not occur, but a smaller outer diameter brings about larger amplitudes. That is, if the immersion depth is not deep, the outer diameter can greatly improve the amplitude height and regularity, but a deeper immersion depth and smaller outer diameter can generate larger amplitudes.

#### 4.2.3. Fixed Outer Diameter, with Changes in the Inner Diameter and Thickness

_{1}= 4 and channel width W = 20, here, the effect of changing the thickness is analyzed. Figure 16 shows the case of a fixed outer diameter, which reveals that the greater the thickness (that is, the smaller the hollow area), the greater the oscillation, and if H

_{s}is larger (e.g., Figure 16c), this phenomenon will be more obvious. However, the greater the thickness of the structure, the greater the manufacturing cost, which is not practical. Therefore, it is also recommended that the outer diameter not be too large.

#### 4.3. Influence of W on the Solitary Wave Hitting a Circular Cylinder with a Hollow Zone

_{1}= 2 and r

_{2}= 1 (dr = 1) are fixed, the influence of W can be analyzed. Figure 17 shows that when H

_{s}= 0.3 or 0.5, there is no obvious influence by W, but when H

_{s}= 0.7, W = 15 produces larger oscillations than either W = 10 or 20. This result is somewhat confusing. To understand why W = 15 produces larger amplitudes, other W values were analyzed (W = 10–20) for the case of H

_{s}= 0.7. Figure 18 shows that there are larger amplitudes when W = 16. Under the conditions of a fixed hollow cylinder size and immersion depth, there is an optimal W value that generates the maximum amplitude effect.

## 5. Conclusions

_{0}= 0.3 was used as the typical incident condition, and the model was first compared with the experimental data obtained by other researchers for verification, and was then applied to explore the wave–cylinder interactions. The conclusions are summarized below.

_{0}= 0.3 passing through a cylinder of D = 4, the peak of the wave passing through the cylinder was determined to be slightly lower than the peaks on both sides that are undisturbed by the cylinder. When the wave has completely passed the cylinder, the waveform destroyed by the cylinder can nearly be restored. The observed runup elevations of the cylinder reveal that the maximum water level in front of the cylinder is about 0.5, which increases to approximately 1.7 times the original wave height (0.3) and then drops and gradually returns to the still water level. The runup evolutions behind the cylinder gradually increase and then decrease, but the recorded water elevations at the front and rear points remained higher than the still water level. This phenomenon changes when there is a gap between the bottom of the cylinder and the seabed. We analyzed the effect of the size of the gap between the bottom of a single cylinder and the seabed on the runup and rundown evolutions. Overall, the main effect of the immersion depth H

_{s}is that when the H

_{s}value is deep, greater runup and rundown occur in front of the cylinder, and the maximum water elevation behind the cylinder is larger. However, the water level behind the cylinder is always higher than the still water level. Regarding the effect of cylinder size D, the simulation results indicate that the larger the D value, the larger the front runup and the smaller the rear runup.

_{0}, r

_{1}, r

_{2}(dr), H

_{s}, and W, for which A

_{0}= 0.3; r

_{1}= ~1–4; r

_{2}= ~1–3; H

_{s}= 0.3, 0.5, and 0.7; and W = 10, 15, and 20 were considered to analyze the oscillation in the hollow area under these conditions. The analysis showed that if r

_{2}is small (e.g., r

_{2}=1), the hollow area obtains more uniform water level oscillation in the water column; the smaller the outer diameter (r

_{1}), the better the oscillation effect; and the deeper the immersion depth (i.e., larger H

_{s}), the more significant is this phenomenon. For long waves, when H

_{s}is large and r

_{1}and r

_{2}are small (r

_{1}> r

_{2}), large wave fluctuations can be produced in the hollow area. For the conditions considered in this study, when r

_{1}= 2, r

_{2}= 1, and W = 16, and the immersion depth is deeper, more uniform, and larger amplitude oscillation waves were observed in the hollow area. Therefore, the inner diameter should not exceed the depth of the water, and the outer diameter should not be too large (if the thickness can withstand the external forces, a less thick wall should be used). However, the effect of the W value was found to become more obvious when only H

_{s}is larger. All results show that the larger is H

_{s}, the larger the oscillating wave generated in the hollow area.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Grid Generations

_{KC}is the vertical position of the bottom of the cylinder (see Figure 1), and $z(i,j,KM)$ is equal to free-surface elevation, that is ${\zeta}_{i,j}$.

#### Appendix A.2. Equation Transformation

^{ij}(i, j = 1, 2, 3) and f

^{i}(i = 1, 2, 3) are grid geometric coefficients. Since x and y are uniform grids, and the gridlines in the z-direction are vertical and straight, the grid geometric coefficients in the formula can be simplified as:

#### Appendix A.3. Numerical Discretization

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**Figure 1.**Schematic diagram of a wave passing through a vertical circular cylinder. (

**a**) side view and (

**b**) top view.

**Figure 2.**Plane view of wave-gauge locations for the experiment by Mo et al. (2010) in a dimensionless scale.

**Figure 4.**Comparisons of wave elevations at two gauges for a solitary wave passing through a non-touching seabed cylinder (H

_{s}= 0.5).

**Figure 5.**Comparisons of wave elevations at two gauges for a solitary wave passing through a non-touching seabed cylinder (H

_{s}= 0.5). The results of W = 5 and 60 calculated and compared with the experimental condition with W = 5.

**Figure 6.**Wave patterns of a solitary wave passing through a partially immersed cylinder for A

_{0}= 0.3, D = 4, and H

_{s}= 0.5. The left is a plane view, and the right is a 3D perspective.

**Figure 7.**Front and rear wave elevations of a solitary wave passing through a vertically immersed circular cylinder in water with different H

_{s}for A

_{0}= 0.3 and D = 4.

**Figure 8.**The solitary wave A

_{0}= 0.3 passes through an upright circular cylinder with D = 4 for H

_{s}= 1.0, 0.7, and 0.5, comparing the wave profiles on y = 0.

**Figure 9.**Front and rear wave elevations of a solitary wave passing through a vertically immersed circular cylinder in water for A

_{0}= 0.3; H

_{s}= 0.5 with different D and H

_{s}.

**Figure 10.**Schematic diagram of waves passing through a hollow cylinder: (

**a**) side view on y = 0 and (

**b**) top view.

**Figure 11.**A solitary wave of height A

_{0}= 0.3 passes through a hollow circular cylinder generating the time histories of the water level at surrounding points.

**Figure 12.**A solitary wave of height A

_{0}= 0.3 passes through the hollow circular cylinder recording wave elevations at points in the hollow area. A comparison of the wave elevations in the hollow area of the hollow cylinder with different inner diameters.

**Figure 13.**The wave-elevation contours at various times for a solitary wave A

_{0}= 0.3 propagating in the channel width W = 20 through the hollow cylinder with immersion depth H

_{s}= 0.5. The cylinder’s conditions are (

**a**–

**g**): r

_{1}= 3, r

_{2}= 1 and (

**a**’–

**g**’): r

_{1}= 3, r

_{2}= 2.

**Figure 14.**Influence of outer diameter (fixed wall thickness) is considered: A solitary wave of height A

_{0}= 0.3 passing through a hollow cylinder has the same cylindrical plate thickness (dr = 1) but different outer diameters (r

_{1}). The water level of the hollow center point varies with time. The immersion depths (H

_{s}) are not equal: (

**a**) H

_{s}= 0.3, (

**b**) H

_{s}= 0.5, and (

**c**) H

_{s}= 0.7.

**Figure 15.**The influence of outer diameter r

_{1}(fixed r

_{2}) is considered: A solitary wave height A

_{0}= 0.3 passes through a hollow cylinder with the same inner diameter (r

_{2}= 1) but different outer diameters (r

_{1}) and the water level at the central point of the hollow is recorded over time. The immersion depths (H

_{s}) are not equal: (

**a**) H

_{s}= 0.3, (

**b**) H

_{s}= 0.5, and (

**c**) H

_{s}= 0.7.

**Figure 16.**The influence of cylinder thickness (dr) is considered: A solitary wave of height A

_{0}= 0.3 passes through the hollow cylinder with the same outer diameter (r

_{1}= 4) but different cylinder plate thicknesses (dr). The water levels at the hollow center point are recorded over time. The immersion depths (H

_{s}) are not equal: (

**a**) H

_{s}= 0.3, (

**b**) H

_{s}= 0.5, and (

**c**) H

_{s}= 0.7.

**Figure 17.**The influence of channel width W is considered: A solitary wave of height A

_{0}= 0.3 passes through a hollow cylinder with the same outer diameter (r

_{1}= 2) and inner diameter (r

_{1}= 1) but different channel widths (W). Observe the wave elevations at the center point varying with time. The immersion depths (H

_{s}) are not equal: (

**a**) H

_{s}= 0.3, (

**b**) H

_{s}= 0.5, and (

**c**) H

_{s}= 0.7.

**Figure 18.**Comparisons of time histories of wave elevations at the central point for a solitary wave with A

_{0}= 0.3 passing through a hollow cylinder with the same outer diameter (r

_{1}= 2), inner diameter (r

_{1}= 1), and immersion depth H

_{s}= 0.7 but different channel widths (W).

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## Share and Cite

**MDPI and ACS Style**

Chang, C.-H. Interaction of a Solitary Wave with Vertical Fully/Partially Submerged Circular Cylinders with/without a Hollow Zone. *J. Mar. Sci. Eng.* **2020**, *8*, 1022.
https://doi.org/10.3390/jmse8121022

**AMA Style**

Chang C-H. Interaction of a Solitary Wave with Vertical Fully/Partially Submerged Circular Cylinders with/without a Hollow Zone. *Journal of Marine Science and Engineering*. 2020; 8(12):1022.
https://doi.org/10.3390/jmse8121022

**Chicago/Turabian Style**

Chang, Chih-Hua. 2020. "Interaction of a Solitary Wave with Vertical Fully/Partially Submerged Circular Cylinders with/without a Hollow Zone" *Journal of Marine Science and Engineering* 8, no. 12: 1022.
https://doi.org/10.3390/jmse8121022