Large Scale Experimental Study on Waves and Submerged Horizontal Cylinders

Abstract

A large-scale physical model experiment was conducted to study the interaction between regular waves and submerged horizontal cylinders, breaking through the bottleneck of a low Reynolds number in a traditional small water tank. The surface pressure distribution and overall force of submerged horizontal cylinders at different submerged depths in waves with different heights and periods under a high Reynolds number were analyzed, and the variation in first- and second-order wave forces with wave height periods was analyzed. Subsequently, the influence of the cylinder on the wave field was analyzed, and the reflection coefficient was obtained through the three-point separation method. According to the analysis of the experimental results, the horizontal and vertical forces acting on the submerged cylinder per linear meter are similar, with the horizontal force slightly greater than the vertical force; at the same time, the submerged cylinder has a certain reflection effect on long-period waves. The summary, analysis, and display of relevant experimental results provide validation support for the current high-precision mathematical model as much as possible.

1. Introduction

Horizontal cylinders are widely used in ocean engineering, and waves are one of the main loads on ocean engineering structures. Under long-term wave action, marine structures are prone to fatigue failure and other problems. Therefore, research on the effect of wave forces on horizontal cylinders has important practical significance and has always been a hot research topic for scholars.

The theoretical research on horizontal cylindrical wave forces is mainly based on potential flow theory. Dean [1] proposed an analytical solution for linearly solving regular waves and submerged horizontal cylinders and found that the reflection coefficient is 0 in deep water. Ursell [2] subsequently confirmed this conclusion using the multipole expansion method. Ogilvie et al. [3] applied Ursell’s method [2] to further calculate the second-order wave force on a submerged cylinder and found that in deep water, the first-order horizontal wave force is equal to the first-order vertical wave force, and the second-order average drift force is zero. However, theoretical analysis usually simplifies the structural form and adopts the assumption of waves with small amplitude, which has certain limitations for strong nonlinear large amplitude sea conditions.

In recent years, with the development of numerical simulation, nonlinear potential flow wave tanks have been used to study the effect of high-order wave forces on submerged cylinders [4,5,6]. Jin et al. [7] compared and analyzed the hydrodynamic characteristics of submerged structures with different shapes using high-order boundary element methods. Yan et al. [8] established a completely nonlinear wave tank and studied the interaction between strongly nonlinear waves and submerged cylinders. They analyzed the changes in high-order free harmonics and locked-in waves behind the cylinder with factors such as incident wave nonlinearity, cylinder size, and submerged water depth. Although numerical calculations based on potential flow theory have been widely applied, they may overestimate the first harmonic wave force under the action of large amplitude waves due to the neglect of fluid viscosity [9]. Numerical calculations based on viscous flow theory can to some extent more realistically reproduce the physical process of the interaction between a horizontal cylinder and waves. The numerical methods used include the time-varying Reynolds equation [10], the Reynolds average equation [11,12,13,14], and the smooth particle method [15], which promote the understanding of the development process of vortices around the cylinder and the related energy dissipation mechanism, however, the simulation results of viscous flow are extremely sensitive to model settings (such as grid quality and turbulence model), and physical model measurement results need to be used in advance for calibration and verification. Gao et al. [16,17,18] applied the Computational Fluid Dynamic (CFD) method to simulated the wave–structure interaction to investigated the subtle changes in the flow field near the structure. Although some model experiments have been conducted, they were mainly based on traditional small wave tanks [14,19,20,21,22,23,24]. Due to experimental conditions, the model scale and the Reynolds number are both small. The accuracy of the simulated high Reynolds number operating conditions using the model validated by small Reynolds number experiments still needs further verification.

In the design of ocean engineering, the Morrison equation is used to calculate the wave force on the cylinder, and it is necessary to determine the drag force coefficient Cd and inertia force coefficient Cm in the equation. Li [25] obtained relevant results for Cd, Cm, and KC numbers under various environmental conditions based on experiments. Chen and Zhu [26] fitted experimental and empirical data to the variation of the inertial force coefficient and resistance coefficient of a cylinder with a Reynolds number and Churchill number, and obtained the relevant coefficients. Hu et al. [27] used an experimental method of forced oscillation of a horizontal cylinder in a towing tank to study the hydrodynamic characteristics of a horizontal cylinder with different immersion depths under flow, wave, and wave current coupling. They obtained the relationships between the drag force coefficient and the added mass coefficient with a Reynolds number, KC number, and Vr at different immersion depths. It should be emphasized that these experiments were still carried out in traditional small wave tanks, and currently there are relatively few large-scale wave tank experiments in the corresponding research.

For this purpose, this paper conducted a series of large-scale physical model tests on waves and submerged cylinders. Since the cylinder itself has a larger dimension, and there will be obvious diffraction phenomena after waves pass through, pressure sensors are placed on the surface of the cylinder to measure the dynamic force on the cylinder, and wave probes are arranged in front of the cylinder to measure the reflection of waves passing through the cylinder. The analysis of the pressure distribution, wave force on the submerged cylinder, as well as the reflection of waves by the large cylinder provided support for the verification of high-precision mathematical models and the calculation of horizontal cylinder wave forces at present.

2. Physical Model Tests

The experiment was conducted in a large-scale wave tank at the Tianjin Research Institute of Water Transport Engineering, Ministry of Transport (TIWTE). The large wave tank is 456 m long, 5 m wide, and 8 m high. The model is located about 230 m away from the wave maker. To display the meanings of parameters in this paper more clearly,

Table 1. The introduction of different parameters.

Parameter	Meaning	Parameter	Meaning
D	Diameter of submerged cylinder	KC	Keulegan–Carpenter number
ds	Submerged depth	L	Wave length
d	Water depth	ηr	Reflected wave elevation
WP	Wave probe	CR	Reflection coefficient
H	Wave height	CM	Inertia force coefficient
T	Wave period	F(1)/F(2)	First-/Second-order wave force
k	Wave number	ρ	Water density
Re	Reynolds number	g	Gravity acceleration

introduces the meanings of the relevant symbols. The specific layout is shown in Figure 1, and the photo of the model after installation is shown in Figure 2. The diameter of the submerged cylinder is 1.0 m, considering two submerged depths. Three wave probes are arranged in front of the submerged cylinder to measure the reflection coefficient of the waves caused by the cylinder. When waves acted on submerged cylinders, the contribution of viscous shear force to the total wave force was very small [9], therefore, 16 pressure sensors were arranged at equal angles along the circumference of the cylinder surface to measure the pressure time history curve of each point, and then multiplied by their respective force area and azimuth to obtain the horizontal and vertical forces of the cylinder. The distribution of pressure sensors is displayed in Figure 3, and the expressions of the wave force on the cylinder are shown below.

$$F_x={\displaystyle \sum }_{i=1}^{16}{p}_i{s}_i\cos {\theta }_i$$

(1)

$$F_y={\displaystyle \sum }_{i=1}^{16}{p}_i{s}_i\sin {\theta }_i$$

(2)

The experimental groups are shown in

Table 2. Model test table.

No.	d/m	H/m	T/s	kA	H/gT2	d/gT2	Re (ds/d = 0.25)	Re (ds/d = 0.50)	KC (ds/d = 0.25)	KC (ds/d = 0.50)
1	4.2	0.123	2.25	0.0489	2.47 × 10−3	8.43 × 10−2	1.11 × 105	5.07 × 104	0.169	0.075
2	4.2	0.236	2.25	0.0938	4.74 × 10−3	8.43 × 10−2	2.13 × 105	9.20 × 104	0.323	0.144
3	4.2	0.301	2.25	0.1196	6.04 × 10−3	8.43 × 10−2	2.72 × 105	1.21 × 105	0.412	0.184
4	4.2	0.417	2.25	0.1658	8.37 × 10−3	8.43 × 10−2	3.77 × 105	1.65 × 105	0.572	0.255
5	4.2	0.16	2.51	0.0516	2.59 × 10−3	6.80 × 10−2	1.42 × 105	7.42 × 104	0.261	0.139
6	4.2	0.27	2.51	0.0871	4.37 × 10−3	6.80 × 10−2	2.40 × 105	1.25 × 105	0.441	0.235
7	4.2	0.37	2.51	0.1193	5.99 × 10−3	6.80 × 10−2	3.29 × 105	1.72 × 105	0.604	0.322
8	4.2	0.528	2.51	0.1703	8.55 × 10−3	6.80 × 10−2	4.70 × 105	2.45 × 105	0.861	0.459
9	4.2	0.174	2.77	0.0466	2.32 × 10−3	5.59 × 10−2	1.52 × 105	8.66 × 104	0.325	0.198
10	4.2	0.313	2.77	0.0839	4.17 × 10−3	5.59 × 10−2	2.73 × 105	1.69 × 105	0.585	0.356
11	4.2	0.499	2.77	0.1337	6.64 × 10−3	5.59 × 10−2	4.35 × 105	2.68 × 105	0.933	0.568
12	4.2	0.65	2.77	0.1742	8.65 × 10−3	5.59 × 10−2	5.67 × 105	3.30 × 105	1.215	0.740
13	4.2	0.202	3.11	0.0442	2.13 × 10−03	4.43 × 10−2	1.71 × 105	1.15 × 105	0.437	0.301
14	4.2	0.432	3.11	0.0946	4.55 × 10−3	4.43 × 10−2	3.66 × 105	2.45 × 105	0.935	0.644
15	4.2	0.64	3.11	0.1402	6.75 × 10−3	4.43 × 10−2	5.43 × 105	3.63 × 105	1.386	0.953
16	4.2	0.237	3.45	0.0437	2.03 × 10−3	3.59 × 10−2	1.97 × 105	1.42 × 105	0.579	0.434
17	4.2	0.503	3.45	0.0928	4.30 × 10−3	3.59 × 10−2	4.18 × 105	3.19 × 105	1.229	0.921
18	4.2	0.738	3.45	0.1362	6.31 × 10−3	3.59 × 10−2	6.13 × 105	4.57 × 105	1.804	1.352
19	4.2	0.296	3.98	0.0441	1.91 × 10−3	2.70 × 10−2	2.41 × 105	1.91 × 105	0.850	0.693
20	4.2	0.49	3.98	0.0730	3.15 × 10−3	2.70 × 10−2	3.98 × 105	3.16 × 105	1.407	1.148
21	4.2	0.699	3.98	0.1042	4.50 × 10−3	2.70 × 10−2	5.68 × 105	4.51 × 105	2.008	1.637
22	4.2	0.351	4.51	0.0442	1.76 × 10−3	2.11 × 10−2	2.81 × 105	2.30 × 105	1.158	0.994
23	4.2	0.745	4.51	0.0939	3.74 × 10−3	2.11 × 10−2	5.96 × 105	4.78 × 105	2.459	2.111

, and all wave elements of the groups are within the range of second-order Stokes waves.

3. Test Results and Analysis

3.1. Dynamic Pressure on the Cylinder Surface

When the wave period is small, the pressure on the surface of the cylinder shows that the dynamic pressure on the upper part of the cylinder is higher, and the dynamic pressure on the lower part is lower. Figure 4 shows the maximum dynamic pressure on the cylindrical surface when the relative submergence depth d_s/d is 0.25 and the wave period is 2.25 s. It can be observed that different wave heights exhibit the same conclusion. As the wave period increases and the wave height increases, the reflection effect of the cylinder on the wave field increases, and the maximum pressure position of the cylinder changes to the area of 45° from the vertical axis of the cylinder on the up-wave and back-wave sides. There are two reasons: (1) the trajectory of water quality points in short-period waves is circular, and the radius of motion decays quickly with an increase in water depth. However, the trajectory of water quality points is elliptical when long-period waves act on the cylinder, and the horizontal motion amplitude of water quality points is greater than the vertical motion amplitude. Therefore, the maximum pressure position of the cylinder is not at the top of the cylinder, and the movement of water quality points decreases relatively slowly with water depth; (2) the reason for the large wave front measurement is the reflection effect of the cylinder on the waves, while the reason for the large wave back side is that the waves break through the cylinder, causing bubbles to impact the surface of the cylinder. Figure 5 shows the maximum dynamic pressure on the cylindrical surface when the same relative submergence and the wave period is 3.11 s. It was found that as the wave height increased, the maximum pressure position shifted to both sides. Figure 6 shows the maximum dynamic pressure on the cylindrical surface when the wave period is 3.98 s. Due to the longer wave period and stronger reflection, the above phenomenon is more obvious.

When the depth of cylinder submergence increases to d_s/d = 0.50, the pressure distribution trend remains the same. Figure 7 shows the maximum dynamic pressure on the surface of a cylinder when the wave period is 2.77 s. Even if the wave height increases, it still shows the maximum dynamic pressure on the upper part and the minimum dynamic pressure on the lower part of the cylinder. Compared with the case of d_s/d = 0.25, when the wave height is large, due to the reflection on the wave field, the pressure at 45° is already higher than that in the middle, as shown in Figure 8. Figure 9 shows the dynamic pressure distribution under the action of long-period waves. It was found that under the action of long-period waves, due to the influence of a cylinder on the wave field being obvious, the position of the dynamic maximum pressure shifted to both sides in large wave heights. Figure 10 shows the experimental photos of interaction between the waves and the submerged cylinder.

3.2. Wave Force Analysis of Submerged Cylinder

Wave forces in horizontal and vertical directions are calculated firstly and the spectrum results are obtained by the FFT method. Figure 11 and Figure 12 show the wave force measurement and analysis results when H = 0.78 m, T = 4.51 s, d_s/d = 0.50. The first- and second-order components can be seen in the spectrum analysis results obviously, corresponding to the wave frequency and twice wave frequency, which are defined as first-order and second-order wave force. These two parts have an important role on the ocean engineering. The first-order wave force dominates the force situation of the structure, while the second-order wave force, due to its high frequency, may approach the natural frequency of the ocean engineering and cause the resonance failure of the structure.

Figure 13 shows the variation of the first- and second-order wave force with wave height, taking the relative submergence depth d_s/d = 0.25 and the wave periods of 2.51 s and 2.77 s as examples. From the experimental results, the horizontal and vertical forces acting on the submerged cylinder per meter are similar, with the horizontal force being slightly greater than the vertical force. The first-order force on the structure is directly proportional to the wave height, while the second-order force is directly proportional to the square of the wave height. Figure 14 shows the experimental results of all groups at a submerged depth of d_s/d = 0.25, indicating that the wave force increases with an increase in wave height. However, as the wave height continues to increase, the impact of the cylinder on the wave field becomes significant, and the waves will break after passing through the submerged cylinder, resulting in a smaller increase in structural force.

Figure 15 and Figure 16 demonstrate that the dimensionless first- and second-order wave force varies with the relative cylinder diameter, respectively. The first-order dimensionless wave force increases with an increase in D/L, indicating that the submerged cylinder has a smaller impact on short waves. However, for long waves, due to the larger size of the structure, it can cause wave breaking, and some energy is dissipated due to wave breaking. Therefore, the structure is subjected to less wave force, and the higher the wave height, the smaller the dimensionless wave force. The second-order wave force first increases and then decreases with an increase in D/L, and the reason for the increase part is the same as the first-order wave force. The main reason for the decrease is that the wave wavelength is small at this time, and the structure is less affected by surface waves, which in turn affects the results of the second-order wave force. Overall, when D/L is approximately 0.1, the structure is most significantly affected by the second-order wave force.

Figure 17 shows the force results of the cylinder with relative submergence depth d_s/d = 0.50 and wave period T = 2.25 s and 2.77 s. From the calculation results, the horizontal and vertical forces acting on the submerged cylinder per linear meter are similar, with the horizontal force slightly greater than the vertical force. Due to the deep submergence depth of the cylinder at this time, the structure is subjected to relatively small second-order wave forces, which differ by two orders of magnitude from the first-order wave forces and can be disregarded.

The results of all experimental groups with a relative submergence depth of d_s/d = 0.50 for the cylinder are shown in Figure 18. The wave force increases with an increase in wave height. Due to the deep submergence depth of the cylinder, the wave field propagation effect of the submerged cylinder is very small, and even in the case of long-period large waves, there is still no wave breaking phenomenon. Therefore, the force on the cylinder is almost proportional to the wave height.

Figure 19 and Figure 20 show the variation of first- and second-order dimensionless wave forces with d_s/d = 0.50 as a function of D/L, respectively. For first-order wave forces, there is little change in the dimensionless wave force of the submerged cylinder with an increase in D/L. At the same time, the influence of wave height on the wave force is not significant, indicating that the influence of the submerged cylinder on surface waves is already very weak. Through the comparison of first- and second-order wave forces, the deeper the submerged depth, the smaller the second-order wave force, which can be disregarded in the calculation.

3.3. Morison Formula Inertia Value

Other scholars have conducted research to obtain the inertial force coefficient values of small-scale submerged cylinders under different conditions, but there are certain differences in the values. For example, Li [25], Chen, and Zhu [26] calculated the inertial force coefficient of a horizontal cylinder between 1.2 and 2.0, while Hu et al. [27] and Mao et al. [28] calculated that in extreme cases, the inertial force coefficient of a horizontal cylinder that may be exposed to the water surface may be greater than 2.0. The submerged cylindrical structure developed in this article has a large scale and is mainly controlled by inertial force. The C_M values at different Reynolds numbers are obtained by incorporating the Morison equation. The results are shown in Figure 21. It is found that the inertial force coefficients of the cylinder are between 1.2 and 2.0 at each Reynolds number, indicating that the Morison formula is also applicable when calculating large-scale structures. Relevant calculation coefficients provided by Chinese regulations can be referred to in the calculation.

3.4. Influence on the Wave Field of the Cylinder

Based on the data of three wave probes in front of the cylinder, the three-point method of Mansard and Funke [29] is used to calculate the reflected wave surface curves and spectrum.

Assuming that the distances between the three wave probes are X12 and X13, respectively, the incident wave and reflected wave can be derived as follows:

$$Z_{I,k}=\frac{1}{D_k}\cdot \left[B_{1,k}\cdot \left(R1+\mathrm{i}\cdot Q1\right)+B_{2,k}\cdot \left(R2+\mathrm{i}\cdot Q2\right)+B_{3,k}\cdot \left(R3+\mathrm{i}\cdot Q3\right)\right]$$

(3)

$$Z_{R,k}=\frac{1}{D_k}\cdot \left[B_{1,k}\cdot \left(R1-\mathrm{i}\cdot Q1\right)+B_{2,k}\cdot \left(R2-\mathrm{i}\cdot Q2\right)+B_{3,k}\cdot \left(R3-\mathrm{i}\cdot Q3\right)\right]$$

(4)

where B_{1, k} is the Fourier transform results of wave elevation, and the expressions of other symbols are as below

$$\begin{array}{c}{D}_k=2·\left({sin}^2{\beta }_k\right.\left.+{sin}^2{\gamma }_k+{sin}^2\left({\gamma }_k-{\beta }_k\right)\right)\hfill \\ R{1}_k={sin}^2{\beta }_k+{sin}^2{\gamma }_k\hfill \\ Q{1}_k=sin{\beta }_k·cos{\beta }_k+sin{\gamma }_k·cos{\gamma }_k\hfill \\ R{2}_k=sin{\gamma }_{k}sin\left({\gamma }_k-{\beta }_k\right)\hfill \\ Q{2}_k=sin{\gamma }_k·cos\left({\gamma }_k-{\beta }_k\right)+2·sin{\beta }_k\hfill \\ R{3}_k=-sin{\beta }_{k}sin\left({\gamma }_k-{\beta }_k\right)\hfill \\ Q{3}_k=sin{\beta }_k·cos\left({\gamma }_k-{\beta }_k\right)-2·sin{\gamma }_k\hfill \end{array}$$

(5)

β_k and γ_k of the above expressions are shown as follows

$$\begin{gathered} {\beta }_k=\frac{2\pi \cdot \mathrm{X}12}{L} \\ {\gamma }_k=\frac{2\pi \cdot \mathrm{X}13}{L} \end{gathered}$$

(6)

Figure 22 and Figure 23 show the reflected wave surface curves and amplitude spectra under two different wave heights, with a submergence depth d_s/d = 0.25 and a wave period of 2.25 s. The results show that the reflected waves are mainly composed of two types of frequency waves, corresponding to the wave frequency and the lateral oscillation frequency of the water tank. We analyzed the amplitude corresponding to the reflected wave frequency and found that under the action of waves in this period, the reflection coefficient of the submerged cylinder is very small, ranging from approximately 0.070 to 0.080, indicating that the submerged cylinder has little impact on the waves of this period.

Subsequently, the wave field under different wave heights with a larger wave period T = 3.45 s was selected for analysis are shown in Figure 24 and Figure 25. From the figures, the wavelength also increases significantly with an increase in wave period, resulting in a significant increase in the depth of water quality point influence. Under the wave action of this period, the reflected waves are more pronounced compared to the short-period situation, with a reflection coefficient of 0.082~0.098.

The same analysis is applied in the cases of the submergence depth d_s/d = 0.50. After statistical analysis, the reflection coefficients under different D/L conditions for two different submergence depths are obtained, as shown in Figure 26. From the results, the deeper the submergence depth of the cylinder, the less obvious the reflection effect. When the submergence depth is constant, the submerged cylinder has a certain reflection effect on long-period waves. The main reason is that the action of long-period waves is deeper, which means that the water quality points still exhibit obvious motion characteristics in deeper situations. Therefore, the submerged cylinder has a certain reflection effect on long-period waves.

4. Conclusions

This paper conducted a series of physical model tests under high Reynolds numbers for waves and underwater submerged cylinders in the large-scale wave tank in TIWTE. By analyzing the pressure distribution, wave force, and reflection of waves by the submerged cylinder, it provides validation support for the current high-precision mathematical model as much as possible. The conclusions are as follows:

(1)

When the wave period is small, the dynamic pressure on the surface of the submerged cylinder shows a higher dynamic pressure in the upper part and a lower dynamic pressure in the lower part. The reflection effect of the cylinder on the wave field is enhanced with increases in wave periods and wave heights. Waves would break when passing through the cylinder, and the maximum stress position of the cylinder changed to an area 45° from the vertical axis of the cylinder on the up-wave and back-wave sides.

(2)

The horizontal and vertical forces acting on the submerged cylinder per linear meter are similar, with the horizontal force slightly greater than the vertical force. The first-order force on the structure is directly proportional to the wave height, while the second-order force is directly proportional to the square of the wave height; the overall force on the structure increases with an increase in wave height. However, as the wave height continues to increase, the cylinder has a significant impact on the wave field. After the wave passes through the submerged cylinder, it will break. Through calculation, the inertia force coefficient of the cylinder is between 1.2 and 2.0, indicating that the Morison formula can also be used for calculating the force on large-scale cylinders. The coefficient can be selected according to these specifications.

(3)

The deeper the submergence depth of the submerged cylinder, the less obvious the reflection effect. When the submergence depth is constant, the submerged cylinder has a certain reflection effect on long-period waves. The main reason is that the long-period wave action is deeper in water depth, which means that water quality points still exhibit obvious motion characteristics in deeper situations. Therefore, the submerged cylinder has a certain reflection effect on long-period waves.