Kinematic response of submerged structures under the action of internal solitary waves

Ocean Engineering 196 (2020) 106814

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Kinematic response of submerged structures under the action of internal solitary waves Junnan Cui a, Sheng Dong a, b, Zhifeng Wang a, b, *, Xinyu Han a, Peng Lv a a b

College of Engineering, Ocean University of China, Qingdao, 266100, China Shandong Province Key Laboratory of Ocean Engineering, Ocean University of China, Qingdao, 266100, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Kinematic response Submerged structure Internal solitary wave Particle image velocimetry

In this study, we investigate the kinematic responses of submerged bodies under the motion of internal solitary waves (ISWs) with laboratory experiment. A two-dimensional ISW was generated in an ISW tank of 15 � 0.4 � 0.6 m dimension with a water depth of 0.5 m. The velocity field of the ISW was measured by the particle image velocimetry and the kinematic responses were recorded using a charged coupled device camera. Different conditions, such as ISW amplitude, ratio of the thickness of the upper layer to that of the lower layer, and depth and shape of the submerged structure, were considered. The results show that the amplitude of ISWs significantly affects their heave motion, and the relative distances between a model and pycnocline significantly influences the surge motion. The amplitude of the heave motion of the model was lower than that of the ISW under the corresponding conditions. The model in fluid with a thin upper layer (which easily causes strong nonlinearity) undergoes a more violent motion response. An empirical expression of the relationship between the maximum displacement of the surge motion and the velocity of the ISWs was also proposed.

1. Introduction Internal solitary waves (ISWs) can be generated in stratified fluids with pycnocline. When a very sharp density change occurs along an interface, the smaller the difference in density, the lower the wave fre­ quency and the slower the propagation speed (Apel, 1987). ISWs travel hundreds of meters along the pycnocline, which causes unusually strong underwater currents that may create large loads that deep-sea drilling rigs or platforms are subject to (Cai et al., 2003). In addition, the internal waves on the continental margins have significant influences on biology, engineering, military activities, and large-scale ocean circulations (Wang et al., 2012; Garrett, 2003a; Garrett, 2003b; Lamb, 2014; MacKinnon et al., 2017). The exploration of natural resources on the continental shelf, where ISWs occur frequently, plays an important role in the development of ocean resources. However, numerous accidents have damaged floating structures. For example, the equipment on the Liuhua offshore oil platform was damaged by ISWs during tanker operations and platform installation in the South China Sea (Bole et al., 1994). The Discoverer 534 platform has provided the measurements of drillship responses to internal wave activities at depths ranging from 1900 to over 3400 ft, and

the maximum measured current velocity induced by an ISW reached 2.6 knots (Osborne et al., 1977). ISWs are important for various practical reasons. They are ubiquitous in areas where strong tides and stratification occur near irregular topography. A European Remote Sensing Satellite (ERS-1) synthetic aperture radar image showed four continental shelf ISW packets located northeast of the Hudson Canyon off New York (Apel, 2000). Such phe­ nomena have occurred in the Sulu Sea in the Philippines (Apel and Holbrook, 1983; Apel et al., 1985; Liu et al., 1985; Hsu and Liu, 2000), in the strait between the Luzon Strait and Taiwan (Hsu et al., 2000), in arcs of the Andaman and Nicobar Islands in the eastern Indian Ocean (Apel, 1979; Osborne and Burch, 1980; Alpers et al., 1997), in the semi-enclosed East China Sea in the western Pacific (Jackson and Apel, 2004), and in the South China Sea (Huang et al., 2016; Bai et al., 2017). Many studies on the effect of ISWs on marine structures have been conducted, and this effect is one of the factors that must be considered when designing marine structures, especially in areas where ISWs occur frequently. The shear forces and torques exerted by ISWs on a supposed rigid pile was calculated in a study by Si et al. (2012). Their results indicate that, the shear force and torque of an ISW on a rigid pile can reach up to 105 N and 109 Nm, respectively. The motion response of a

* Corresponding author. College of Engineering, Ocean University of China, Qingdao, 266100, China. E-mail address: [email protected] (Z. Wang). https://doi.org/10.1016/j.oceaneng.2019.106814 Received 17 July 2019; Received in revised form 30 November 2019; Accepted 2 December 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

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Spar platform subjected to ISWs was theoretically investigated in a study by Song et al. (2011). The maximum force induced by an ISW is smaller than that of surface waves. However, the surge motion induced by an ISW of the Spar platform is three times of that induced by the surface waves; thus, it should be considered. Furthermore, the motion response of a top-tensioned riser under excitation caused by ISWs, surface waves, and vessel motion was theoretically and numerically explored in research works by Guo et al. (2013) and Fan et al. (2017). The conclu­ sions of their works highlighted that an ISW will cause large displace­ ment of drilling risers, especially at the location where pycnocline is situated. Furthermore, the forces that ISWs create on fixed structures, i. e. cylindrical, semisubmersible platform, or slender-bodied structures, were experimentally, theoretically, and numerically studied in research works by Chen et al. (2017, 2018), Cai et al. (2003, 2006), and Wei et al. (2014), respectively. These researches demonstrated that an ISW has significant effect on fixed structures. Furthermore, the force and torque induced by an ISW vary with the layer thickness ratio and wave amplitude. Especially, in a study of Wei et al. (2014), they mentioned that characteristics of the internal wave forces depend on the wave amplitude and the model position in depth. Previous studies on the interactions between ISWs and floating or submerged structures mainly concentrated on the forces ISWs created on fixed structures, and very few studies have been conducted on the mo­ tion responses of structures under the action of ISWs. Du et al. (2017) used a flexible rope to suspend a submarine model for motion response experiments, and their results highlighted that the center of gravity (CG) tracks of the submarine under ISWs were irregular ellipticals, and the responses of the submarine to heaves, surges, and pitches were signifi­ cant. However, the motion response was affected by the restraints of the flexible rope. In our experiments, the motion response of free suspended models under the action of ISW was investigated. The particle image velocim­ etry (PIV) technology (Raffel et al., 2007) were used to detect the ve­ locity filed by the ISWs and the motion response of models. The experimental results were compared with the theoretical results. In this study, Experiment setup is shown in Section 2. In Section 3, we compared the experimental results of the ISW characteristics with the theoretical values. Furthermore, in Section 4, we described the motion response of the models in detail. We also considered the effects of amplitude of an ISW and the position of the model on motion response. In Section 5, we analyzed the dimensionless motion response results with the dimensionless ISW characteristics. Finally, the paper is sum­ marized in Section 6.

and the total depth of the fluid is 0.5 m. The fluid densities in the upper and lower layers are ρ1 ¼ 1030 kg=m3 and ρ2 ¼ 1051 kg=m3 , respec­ tively. The thickness ratios β between the upper and lower layers were set as 1:9, 2:8, and 3:7 under different conditions. A global coordinate system was set up as shown in Fig. 1. The ISW velocity, and heave and surge motions were described in this coordinate system. The positive direction of the pitch motion is anticlockwise. ISWs were generated by the Lock-release method (Kao et al., 1985; Sutherland et al., 2015). A lock-release area with a length of x0 ¼ 40 cm was set with a watertight, manually movable gate at the upstream end of the tank. Five height differences η0 (5–25 cm at 5 cm intervals) were set to generate the ISWs with different amplitudes. In a previous study, it was observed that the amplitudes of ISWs are linearly correlated with their height differences, as shown in Fig. 2; the results are from Cui et al. (2019). The shape of a submerged structure is an important factor that in­ fluences its motion response. Thus, two different models composed of nylon (spherical model with a diameter of D ¼ 8:8 cm, and a cylindrical model with a diameter of D ¼ 8:0 cm and length of L ¼ 29:8 cm) were tested. Both models were hollow in the middle to allow their specific gravities to be easily changed. By changing their specific gravities, the models can be suspended in the water at a specific depth. Fig. 3a shows images of the submerged models. The centers of gravity were set along the middle of the models, slightly lower than the center of the shape. Fig. 3b shows the spherical model suspended in the upper layer. The experimental area was located in the middle of the tank, which ranged from 6 to 9 m. The first half of this region is a velocity field tracing area, and the PIV method was followed to measure the velocity fields of the ISW. Two lasers with a maximum power of 6000 mW and wavelength of 450�2 nm were set 1.5 m above the tank and a CCD

2. Facility and experimental procedure The experiment was conducted in the State Key Laboratory of Physical Oceanography, Ocean University of China. The ISW tank shown in Fig. 1 is approximately 15 m long, has a cross-section of 0.4 � 0.6 m,

Fig. 2. Relationship between ISW amplitudes and height differences.

Fig. 1. Schematic diagram of the experimental system. 2

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� Grk xik ; zjk ¼

�

� Gk xik ; zjk � � 50 : Gk xik ; zjk > 50

0 255

(1)

All the pixels with a grayscale of 0 ðxwik ; zwik Þ are identified to form a set Wk ðxwik ; zwjk Þ ¼ 0; ​ i ¼ 1; ​ 2; …; ​ m; ​ j ¼ 1; ​ 2; …; ​ n. Further, the coordinates of the CG Pk ðXk ; Yk Þ of the model set can be expressed as follows: Xk ¼

m 1 X xwik þ dxk ; m i¼1

(2)

Zk ¼

n 1X zwjk þ dzk ; n j¼1

(3)

where Grk ðxik ; zik Þ is the grayscale of the reset post-point ðxik ;zik Þ, Gk ðxik ; zik Þ is the original grayscale of the point ðxik ;zik Þ, and dxk and dzk are the distance from the center of the form to the CGs parallel to the x and z axes. k represents the current frame number. The models were designed to be suspended in the upper layer, pyc­ nocline, and lower layer. Eighteen operating conditions (Table 1) with different layer thickness ratios, ISW amplitudes, and model shapes were tested. Case 1–3 for the cylindrical model (case 10–12 for the spherical model) were mainly considered of the influence of different ISW am­ plitudes on motion responses in the pycnocline. Case 4–6 and 7–9 for the cylindrical model (cases 13–15 and 16–18 for the spherical model) were focused on to discuss the different characteristics of motion response with different distances between the model and pycnocline with layer

Fig. 3. Images of the submerged models. (a) Cylindrical (left) and spherical models (right), and (b) spherical model suspended in the upper layer.

Table 1 Conditions of experiment.

camera, with a resolution of 1348 � 2560 and sampling frequency of 40 Hz, was set in front of the tank to record the particle motion. Further­ more, a polystyrene PIV particle material with a surface area of 54.093 μm2, volume of 60.627 μm3, and density of 1.04 kg/m3 was used in this experiment. The PIV images were analyzed with correlation analysis algorithm (Thielicke and Stamhuis, 2014). The interface displacement was determined by two different procedures (Grue et al., 1999): by measuring the vertical coordinate where there is a jump in the velocity, and by visual determination of the position of the pycnocline. The motion response of the model in the two-dimensional (2D) ISW tank was recorded as a 2D motion. Another CCD camera was set in front of the tank to record the motion responses of the models. An LED optical screen was placed behind the tank to increase the contrast ratio between the model and background. The motion was recorded as a binary file and was post-processed as a Bitmap image. Gray-level analysis was per­ formed to distinguish the model in the water. The CG and response of the model were traced by gray level analysis that provided the motion response results (Fig. 4). Furthermore, the bottom of the cylindrical model was linear fitted. The slope angle of the fitted line was considered as the pitch angle of the model. The grayscale of every pixel was reset by gray-level analysis. Pixels with grayscale less than 50, would be reset to 0, and pixels with the gray level more than 50 would be reset to 255.

Number

β

η0 (cm)

Model

d (cm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1:9 1:9 1:9 2:8 2:8 2:8 3:7 3:7 3:7 1:9 1:9 1:9 2:8 2:8 2:8 3:7 3:7 3:7

10 15 20 20 20 20 20 20 20 10 15 20 20 20 20 20 20 20

C C C C C C C C C S S S S S S S S S

0 0 0 þ5 0 5 þ5 0 5 0 0 0 þ5 0 5 þ5 0 5

C - cylindrical model, S - spherical model, β - layer thickness ratio, η0 - height difference, and d - the vertical distance between pycnocline and CG of the model; positive sign means model above pycnocline and negative sign means model below pycnocline.

Fig. 4. CCD image (left) and gray-level analysis results (right) of the cylindrical model. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article). 3

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thickness ratios of 2:8 and 3:7.

3.1.1. Theories of ISWs The KdV equation is used to describe the ISWs with a low amplitude in comparison to the total depth (Korteweg and De Vries, 1895; Miles, 1979; Koop and Butler, 1981; Kao et al., 1985). The interface displacement was expressed by the analytical solution of the KdV equation:

u1 ðXÞ ¼

(5)

and λKdV ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12c2 =ðζ0 c1 Þ

(6)

with 8 � 2 > > 1 h2 þ ρ2 h1 Þ < c0 ¼ gh1 h2 ðρ2 ρ1 Þ ðρ��� �� c1 ¼ 3c0 ρ1 h22 ρ2 h21 2 ρ1 h1 h22 þ ρ2 h21 h2 ; > �� > 2 2 : c ¼ c ρ h h þ ρ h h ½6ðρ h þ ρ h Þ� 2

0

2 1 2

1 1 2

1 2

2 1

the amplitude of an ISW from 0 to h, where h is the distance between the interface and critical level hc , was proposed: h ¼ h2 hc . An ISW that has its interface displacement governed by the m-KdV equation is expressed as follows: �� � ζðx; tÞ ¼ ζ0 sech2 ððx cmKdV tÞ = λmKdV Þ 1 μtanh2 ððx cmKdV tÞ = λmKdV Þ

� where μ ¼

4ðρ2

ρ1 Þhc ðh

(11) h

jh þ ζ0 j;

η22

0

2h2 η2

�2 ��

η32

;

(18)

ðWn Þzz þ μn N 2 Wn ¼ 0; Wn ð0Þ ¼ Wn ð HÞ ¼ 0;

(19)

Ψn ðx; z; tÞ ¼ cζðx; tÞWn ðzÞ;

(20)

w;

(21)

The experimental results of the thickness ratios of the upper and lower layers are as follows: 8 2:92; ​ case ​ 3 u1 < � 1:79; case ​ 4 ; (23) u2 : 1:27; case ​ 7

with coefficients on a two-layer model:

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12c2 =½ðc1 þ c3 ζ0 =2Þζ0 �;

�� 00 h 2 η2

The experimental results show that the ratios of maximum velocities of the upper and lower layers is similar to the ratios of their thickness. As shown in Table 2, the thickness ratios of each typical working condition (the height differences of case 3, 4, and 7 are 20 cm, in Table 1) are as follows: 8 η2 h2 ζ0 < 2:91; ​ case ​ 3 ¼ � 1:83; ​ case ​ 4 ; (22) η1 h1 þ ζ0 : 1:35; ​ case ​ 7

​ and ​ h ¼ h þ jh þ ζ0 j. The addition of a third-order nonlinear term to the KdV equation forms the e-KdV equation (Benney and Ko, 1978; Helfrich and Melville, 1986). An ISW interface displacement governed by the e-KdV equation is expressed as follows: �� � ζðx; tÞ ¼ ζ0 B þ ð1 BÞcosh2 ððx ceKdV tÞ = λeKdV Þ ; (12)

λeKdV ¼

6

ðh2 þ zÞ2 2

3.2. Comparison of theoretical and experimental results

’’

ceKdV ¼ c0 þ ζ0 ðc1 þ c3 ζ0 = 2Þ=3;

η22

in which, u and w are the horizontal and vertical velocities respectively.

(10)

��1=2 io1=2 � hc Þ ρ2 h2 ;

pffiffiffiffiffiffiffiffiffiffiffiffi h’’ =h’ ; h > 0 , hc ¼ h=½1 þ ρ1 =ρ2 �, h’ ¼ h’ =h’’ ; h < 0

η2

� þ

ðΨn Þz ¼ u; ðΨn Þx ¼

(9)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�pffiffiffiffiffiffiffiffiffiffiffiffiffi 3hh0 h00 ; hc Þ ðh hc Þ3 þ h3c

1

h2

where μn and Wn are eigenvalue and eigenfunction of the boundary €isa €la € frequency, value problem, n is the mode number, N is the Brunt-Va Ψn ðx; z; tÞ is the two-dimensional stream function. We have:

(8)

n . h c0m ¼ gh 2 1

(16)

:

where X � x ct, η’1 � η1X , η’2 � η2X , η’’1 � η1XX , η’’2 � η2XX , η1 ðXÞ ¼ h1 ζ, and η2 ðXÞ ¼ h2 þ ζ. By solving the eigenvalue function, the distribution of vertical and horizontal velocities of ISW can also be derived easily (Vlasenko, 1994; Vlasenko and Hutter, 2001; Buijsman et al., 2010; Zhao et al., 2010; Guo and Chen 2012):

where ζ is the interface displacement and ζ0 is the amplitude of the ISW. Another equation describing the ISW is the m-KdV equation (Kaku­ tani and Yamasaki, 1978; Mitsuaki and Masayuki, 1986), which predicts

λmKdV ¼ 2ðh

cζ

η2 ðXÞ

� u2 ðX; zÞ ¼ c 1

(7)

with coefficients on a two-layer model: n . o cmKdV ¼ c0m 1 1 2½ðh þ ζ0 Þ=ðh hc Þ�2 ;

(15)

;

Based on the theory of Camassa et al. (2006), which is widely used in the calculation of velocity of ISW, e.g. Xie et al. (2010) and Chen et al. (2017), the velocities in the upper and lower layers (u1 and u2 ) can be calculated by the following equations: � � 2 �� 0 �2 �� h1 η ðh1 zÞ2 h1 η001 2h1 η1 þ 1 ; (17) u1 ðX; zÞ ¼ c 1 3 2 η1 6 2 η1 η1

where the phase speed cKdV and characteristic length of the ISW λKdV of the two-layer model can be expressed as follows: cKdV ¼ c0 þ ζ0 c1 =3

cζ

η1 ðXÞ

u2 ðXÞ ¼

(4)

cKdV tÞ = λKdV Þ;

c3 ζ0 =ð2c1 þ c3 ζ0 Þ.

3.1.2. ISW velocity For travelling-wave solutions with speed c (where c could be any expression of the KdV, m-KdV, and e-KdV equations), the average ve­ locities in the upper and lower layers are as follows:

3.1. Theoretical description of ISWs

ζðx; tÞ ¼ ζ0 sech ððx

ρ2 h21 Þ=ðρ1 h2 þ ρ2 h1 Þ� =8 ðρ1 h32 þρ2 h31 Þ =ðρ1 h2 ­

þρ2 h1 Þg=ðh21 h22 Þ and B ¼

3. ISW characteristics

2

2

c3 ¼ 3c0 f7½ðρ1 h22

(13)

The horizontal mass transport is conserved; thus, it can be derived as follows:

(14)

u1 ¼ u2

where

η2 ; η1

(24)

where u1 is the average horizontal velocity in the upper layer, u2 is the 4

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velocities of the ISWs in the upper and lower layers are opposite and their magnitudes differ. There is a sudden change in velocity at the depth of the pycnocline. A nonlinear coefficient, α ¼ ζ0 =h1 , is proposed as a way to determine the degree of nonlinearity of an ISW; a larger value of α would indicate a stronger nonlinearity of ISW. The nonlinear coefficients shown in Fig. 5a, 5b, and 5c are 1.56, 0.83, and 0.45, respectively. In Fig. 5a, the velocity of the lower layer calculated by the KdV theory failed, and the bottom velocity changed from negative to positive. As both m-KdV and e-KdV have third order nonlinear term, the two theoretical models are more effective in the event of a strong nonlinearity. In Fig. 5b, the nonlinearity is lower, and the three theoretical results are reasonable; Whereas, the results of the KdV equation from the experimental results show large biases. However, the other two theoretical results agreed well with the experimental re­ sults. In Fig. 5c, the fitting effects of e-KdV and m-KdV are still in good consistency with the experimental results. The nonlinearity coefficient of the working conditions is small, and the fitting results of the KdV theory are much better than those of the first two working conditions. Generally, in the situation of our experimental setup, α ranges from ​ 0:45 ​ to ​ 1:56; β ​ ranges ​ from ​ 0:11 ​ to ​ 0:43; and m-KdV best fits the vertical distribution of horizontal velocity at the laboratory scale, and is suitable for various upper and lower layer thickness ratios along with strong nonlinearity. The e-KdV theory has not been invalidated by this experiment; however, the predicted results differ from the experimental results in the case of strong nonlinearity with low upper- and lower-layer thicknesses. The KdV theory can only be applied in the event of a high thickness ratio and weak nonlinearity. In these cases, the vertical dis­ tribution of horizontal velocity calculated by eigenfunction is larger than the results of other theories. However, only the results of eigen­ function shows the continuous change of velocity in the pycnocline.

Table 2 Experimental results of ISW measurements. Number

ζ0 (cm)

u1 (cm/s)

u2 (cm/s)

vu (cm/s)

vd (cm/s)

1 2 3

5.02 6.45 7.80

5.46 6.62 7.50

1.73 2.36 2.56

1.69 3.28 5.48

1.59 2.44 4.66

4 5 6

7.64 8.35 7.79

6.37 6.70 5.77

3.56 3.33 3.34

1.67 1.79 1.73

1.58 1.65 1.73

7 8 9

6.27 6.11 6.72

4.18 4.55 4.30

3.29 3.71 3.57

1.22 1.25 1.13

1.20 1.04 0.91

10 11 12

4.97 6.35 7.91

5.18 6.85 8.90

1.11 1.96 2.69

1.23 2.01 4.98

1.19 2.26 5.26

13 14 15

8.62 8.87 8.27

5.81 6.43 6.21

3.10 3.78 3.47

1.62 1.66 1.68

1.65 1.47 1.73

16 17 18

6.18 6.48 6.11

4.12 4.26 4.46

3.45 3.71 3.66

1.39 1.26 1.35

1.14 0.98 1.03

ζ0 - amplitude of the ISW, u1 - the averaged horizontal velocity in the upper layer, u2 - the averaged horizontal velocity in the lower layer, vu - the maximum upward vertical velocity, and vd - the maximum downward velocity.

average horizontal velocity in the lower layer, vu is the maximum up­ ward vertical velocity, and vd is the maximum downward vertical ve­ locity. The experimental and theoretical results are similar. Fig. 5 shows the vertical distribution of the horizontal velocity of an ISW. The velocity was observed at X ¼ 8 m in the tank. The horizontal

Fig. 5. Comparison of the theoretical and experimental vertical distribution of the horizontal velocity of an ISW. (a) β ¼ 1 : 9, ζ0 ¼ 7:8 cm, (b) β ¼ 2 : 8, ζ0 ¼ 8:3 cm, and (c) β ¼ 3 : 7, ζ0 ¼ 6:7 cm. The red, blue, green and black lines indicate the results of the KdV, m-KdV, e-KdV theories and eigenfunction; and the black circle represents the experimental results. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article). 5

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Fig. 6 shows a timeseries of the horizontal velocity in the upper and lower layers of an ISW. The observation points in the upper and lower layers are located at z ¼ 0.03 and 0.2 m, respectively. The horizontal velocities at the two observation points varied significantly with time. The velocity of the upper layer first increased in the positive direction and then restored to a small magnitude with time; while that of the lower layer was contrasting to this. In Fig. 6a, the e-KdV and m-KdV theories are in good agreement with the change trend in the velocity of the upper and lower layers. In Fig. 6b, the m-KdV theory fits well with the timeseries of the velocities in the upper and lower layer. Among the theoretical results of e-KdV and KdV, the duration of the velocity change is significantly shorter than that in the experiment. In Fig. 6c, the results of the e-KdV and m-KdV theories are in good agreement with the experimental results; however, the fitting results of the KdV theory are still unsatisfied. In these cases, the e-KdV theory is consistent with the experimental results for fitting the time series of the velocity of ISWs. In the KdV theory and eigenfunction, the duration of the velocity change is shorter than the experimental duration. However, in the m-KdV theory, the duration is longer. Considering both the vertical distribution of the horizontal velocity and the fitting results of the horizontal velocity time series, the e-KdV theory agreed with the experimental results better. The results from the theoretical calculation were essential for further research on motion response of submerged structures under the action of ISWs. Furthermore, a depth-time diagram of the velocity field of an ISW can be obtained from the PIV results, as shown in Fig. 7. The velocity magnitude and the interface displacement are clearly shown in this figure.

under 2-D ISWs include heave, surge, and pitch responses. In this sec­ tion, the experimental results are presented from two aspects, i.e., the variations in the kinematic response of the model with the amplitude of ISW, and the variations with the relative distances between model and pycnocline. The variations in the motion response of the submerged structure under the action of ISWs with these experimental variables are also further explored. 4.1. Influence of amplitude of ISW Oceanic ISWs are frequently observed to have vertical displacements ranging from tens to hundreds of meters in amplitude (Sutherland, 2010). The larger the amplitude, the higher the velocity induced by the ISW. Normally, ISWs with a larger amplitude pose a greater threat to submerged bodies (Wei et al., 2014). In this section, three different amplitudes of ISW (about 5, 6.4, and 7.8 cm) were set in conditions 1–3 and 10–12 to investigate the influence of different amplitudes on the motion response of the submerged bodies. The model did not travel particularly far in the x-direction in pres­ ence of ISW, as shown in Fig. 8a. The model was slightly positively displaced along the x-axis. Under the three conditions, the maximum displacement was almost 0.09 m. Fig. 8b shows the heave motion re­ sults. The model sank/rose with interface displacement. The heave motion response form is similar to the form of interface displacement. The amplitude of the heave motion is directly related to that of the ISW; however, it is 10.6%, 7.8%, and 13.6% lower for conditions 1–3, respectively. Fig. 8c shows the pitch motion of the model. The pitch motion began increasing in the negative direction after 7.5 s and sub­ sequently reached the maximum value. Further, it increased in the positive direction and reached the maximum value after 15 s. The pitch motion was also related to the amplitude of ISW. The pitch motion

4. Motion response of the submerged structure As mentioned above, the motion responses of submerged structures

Fig. 6. Comparison of the theoretical and experimental variations in velocity timeseries of an ISW. (a) β ¼ 1 : 9, ζ0 ¼ 7:8 cm, (b) β ¼ 2 : 8, ζ0 ¼ 8:3 cm, and (c) β ¼ 3 : 7, ζ0 ¼ 6:7 cm. The red, blue, green and black lines indicate the results of the KdV, m-KdV, e-KdV theories and eigenfunction; the black circle represents the experimental results; and the solid and dashed lines represent the velocities of the upper and lower layers. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article). 6

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Fig. 7. Depth-time diagram of velocity. (a) β ¼ 1 : 9, ζ0 ¼ 7:8 cm, (b) β ¼ 2 : 8, ζ0 ¼ 8:3 cm, and (c) β ¼ 3 : 7, ζ0 ¼ 6:7 cm. The dashed-dotted line represents the interface displacement.

Fig. 8. Motion response results of cylindrical model with β ¼ 1 : 9. (a) surge motion, (b) heave motion, (c) pitch motion, (d) track of CG. Solid line, ζ0 ¼ 5:02 cm; dash line, ζ0 ¼ 6:45 cm; dash dot line, ζ0 ¼ 7:80 cm.

became more violent with higher ISW amplitudes. The maximum pitch angle was almost 2.1� . The CG track of the model is shown in Fig. 8d, and the form of the CG track of the submerged structure is a rightward ‘V’ with a clockwise ellipse in the lower half of the ‘V’ shape. The range of the ellipse was positively correlated with the ISW amplitude. The results for the spherical model are shown in Fig. 9. The track of the model’s CG was a clockwise, irregular ellipse. During the action of the ISW, the surge motion of the model changed slightly in both positive and negative directions along the x axis, with a maximum displacement of less than 0.06 m. The heave motion exhibited the same trend as that

under conditions 1–3, and the maximum displacement was 7.5 cm in the negative direction along the z axis. With an increase in the amplitude of the ISW, the motion amplitude of the submerged structure increased gradually, which included the heave and pitch motions. When the amplitude of the ISW is too high, the pitching angle of the model will oscillate because of the wave packet following the leading wave. The amplitude of the heave motion of these six conditions were smaller than that of an ISW in different magnitudes (Fig. 8c).

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Fig. 9. Motion response results of spherical model with β ¼ 1 : 9. (a) Surge motion, (b) heave motion, and (c) CG track. Solid line, ζ0 ¼ 4:97 cm; dashed line, ζ0 ¼ 6:35 cm; and dashed-dotted line, ζ0 ¼ 7:91 cm.

Fig. 10. Motion response results of cylindrical model with β ¼ 2 : 8: ​ (a) Surge motion, (b) heave motion, (c) pitch motion, and (d) CG track. Solid line, d ¼ 5 cm; dashed line, d ¼ 0 cm; and dashed-dotted line, d ¼ 5 cm. 8

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4.2. Influence of relative distances between model and pycnocline

at the center of pycnocline. The amplitude of the surge motion in the upper layer was approximately 0.7 m in the positive direction along the x axis and that in the lower layer was approximately 0.35 m in the negative direction along the x axis. Under condition 8, d ¼ 0 cm, the surge motion amplitude was 0.1 m. Fig. 11b shows that the heave mo­ tion under conditions 7–9 follows the same trend, i.e., it first increases in the negative direction of z axis and then decreases. However, the am­ plitudes of the heave motion are not similar. The amplitude of heave motion of d ¼ 5 cm was maximum and that of d ¼ 5 cm as minimum in these three conditions. The same trend is also observed in conditions 16–18 in Fig. 12b. Fig. 11c shows that the pitch motion response of the model at the center of the pycnocline is much higher than that of the other two conditions, and it first increase in the negative direction and then increase in the positive direction. The maximum pitch angles in the negative and positive directions were 0.8� and 1.2� , respectively. The CG tracks of the model shown in Fig. 11d are similar to those shown in Fig. 10d. The variations in the motion response of the spherical model with different relative distances with pycnocline were also studied (Figs. 12 and 13). The overall trend of the results was similar to those of the re­ sults of conditions 4–9, thereby indicating that d greatly influences the surge motion; however, it has slight influence on the heave motion in these cases. Moreover, it is known that the amplitude of ISW will decrease as the increasing of the distance to pycnocline in the real ocean (Vlasenko and Hutter, 2001). It can be inferred that d has an effect on the amplitude of the heave motion when it is relatively large, and the amplitude of the heave motion will decrease with an increase in d. When the submerged structure is suspended near a free surface or at the bot­ tom, the amplitude of the heave motion should be extremely small. However, these are not the most dangerous situations we want to investigate. With a layer thickness ratio of 3:7 (Fig. 13), the variations in surge, heave, and CG track at different depths were very close to those shown

The seasonal pycnocline was situated at depths approximately be­ tween 50 and 150 m. The main pycnocline in the ocean, which extended to depths of 1000 m, was unaffected by the seasons (Sutherland, 2010). A submerged structure may be influenced by the upper, lower, or pyc­ nocline layer effects. This section explores the forms of motion response of the submerged structure at different relative distances to pycnocline d (see Fig. 1), with layer thickness ratios of 2:8 and 3:7, and the pycnocline was taken as the reference plane. The three distances were d ¼ 5, 0, and - 5 cm, which represent that the model was situated 5 cm above, at the center, and below the pycnocline, respectively. The ISW amplitude was approximately 8 and 6 cm at layer thickness ratios of 2:8 and 3:7, respectively. The accurate amplitudes are presented in Table 2. The surge motions significantly differed among conditions 4–6. When the model was in the upper layer (Fig. 10a, d ¼ 5 cm), it moved in the positive direction along the x axis by 0.7 m. The model at the center of the pycnocline (d ¼ 0 cm) also traveled in the positive direction along the x axis by 0.3 m. However, when the model was in the lower layer (d ¼ 5 cm), it traveled in the negative direction along the x axis by 0.15 m. As shown in Fig. 10b, the effect of d on the heave motion amplitude is negligible, because d values of these cases are relatively small. However, according to the ISW theory, the amplitude of the heave motion should decrease with the increase in d. In Fig. 10c, the pitch angle increased negatively first and then significantly increased to positive when d ¼ 0 cm. This was contrasting to the pitch motion results of the other two cases. Under the three conditions, the pitch amplitude was greater when d ¼ 0 cm, maximum positive angle was 1.3� , and maximum negative angle was 1.1� . In Fig. 10d, the model moves forward in the x direction by maximum 0.7 m and in the z direction by maximum 0.07 m. The amplitude of the motion response was high, especially when d ¼ 5 ​ cm. It can be seen from Fig. 11a, the surge motion responses of the submerged structure are higher in the upper and lower layers than those

Fig. 11. Motion response results of the cylindrical model with β ¼ 3 : 7. (a) Surge motion, (b) heave motion, (c) pitch motion, and (d) CG track. Solid line, d ¼ 5 cm; dashed line, d ¼ 0 cm; and dashed-dotted line, d ¼ 5 cm. 9

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Fig. 12. Motion response results of spherical model with β ¼ 2 : 8. (a) Surge motion, (b) heave motion, and (c) track of CG. Solid line, d ¼ 5 cm; dashed line, d ¼ 0 cm; and dashed-dotted line, d ¼ 5 cm.

Fig. 13. Motion response results of spherical model with β ¼ 3 : 7. (a) Surge motion, (b) heave motion, (c) pitch motion, and (d) CG track. Solid line, d ¼ 5 cm; dashed line, d ¼ 0 cm; and dashed-dotted line, d ¼ 5 cm. 10

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in Fig. 12. The amplitudes of the surge motion at d ¼ 5, 0, and 5 cm were 0.38, 0.07, and 0.27 m, respectively; and the amplitude of heave motion was 0.07 m. Shape is also a key factor that affects the motion response amplitude of the model. In most cases, the amplitudes of the surge motion of the spherical model were smaller than those of the cylindrical model. However, the heave motion response amplitude of these models did not exhibit a distinct difference. Comparing Figs. 10b and 12b, and Figs. 11b and 13b, the heave motion response trends of the cylindrical and spherical models with different values of d are similar. The surge motion amplitudes of the spherical model shown in Fig. 12a are 0.35, 0.08, and 0.20 m for d ¼ 5, 0, and 5 cm, respectively, which are significantly smaller than the results shown in Fig. 10a. The amplitude of the heave motion is approximately 0.078 m. That is to say, the shape has a great influence on the motion response of the model. Moreover, the size of the model, such as the radius and length of the cylinder model, and the radius of the spherical model, all have the potential to affect the motion response. We will study and discuss it in future research. Under these conditions, we experimentally studied the motion re­ sponses of submerged models with different relative distances with pycnocline and layer thickness ratios. The results showed that, when d varies, the direction of the surge motion will also change. When the submerged structure was located in the upper and lower layers, the submerged structures were affected by the flow velocity in a fixed di­ rection. Therefore, the amplitude of the surge motion was higher. When the submerged structure was located at the center of the pycnocline, the surge motion amplitude was not very distinct; however, the pitch angle changed greatly because of the influence of the shear effect of the ISW velocity field and the density difference of the upper and lower layers. For d ¼ 0 cm, the model was first affected by the density difference and the pitch angle increased in the negative direction. When the ISW propagated to the model, the pitch angle increased in the positive di­ rection because of the strong shear effect. For d ¼ 5 cm, the model was

situated in the upper layer, and only shear effect affected the pitch motion. The bottom of the model was close to the pycnocline and affected by a smooth velocity, and the top of the model was in the middle of the upper layer and affected by a strongly velocity. The conditions of d ¼ 5 cm were contrasting to this. As a result, the form of pitch motion in these conditions were different from that of conditions of d ¼ 0 cm. By comparing the corresponding results of the two different model shapes, it was observed that the amplitude of the surge motion of the spherical model was 29.8% lower on an average than that of the cylin­ drical model. Furthermore, the heave motion amplitude of the spherical model was 4.5% greater on an average than that of the cylindrical model. 5. Discussions To compare all the conditions tested in this experiment, a dimen­ sionless position parameter was defined as follows: PD ¼ ðh1 þ ζ0 þ dÞ=H:

(25)

By comparing three examples of each group of working conditions shown in Figs. 14a and 14b, we can intuitively determine a law for the action of ISWs on the motion of a submerged structure. When the layer thickness ratio is 1:9, the variable is the amplitude of the ISW, and d does not change. Therefore, the horizontal motion of the submerged structure is weak, and does not exceed 0.1 m. The vertical motion of the model increases with the ISW amplitude. The variations in amplitude mainly affect the vertical motion of the model; however, they have little effect on the horizontal motion when the submerged structure is located at the center of the pycnocline. For layer thickness ratios of 2:8 and 3:7, d is the only variable. Furthermore, the surge motion response is significantly influenced by d; thus, the maximum displacement can reach 15 times that of the minimum displacement. However, the heave motion is not affected greatly by d in these cases. Moreover, the amplitudes of the

Fig. 14. Variations in the amplitude of the motion response with a non-dimensional position parameter. (C - cylindrical model, S - spherical model, Xamp - amplitude of the surge motion, Zamp - amplitude of the heave motion, and αamp - amplitude of the pitch motion). 11

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motion responses of the cylindrical model are higher than those of the spherical model in most cases for a surge motion. In contrast, they are lower in the case of a heave motion. The response of pitch angle (Fig. 14c) shows a good relationship with the layer thickness ratio; the higher the layer thickness ratio, the smaller the amplitude of the pitch motion response. This is because the shear effect in water with a high layer thickness ratio is lower than that in water with a small layer thickness ratio, as shown in Fig. 15. The form of motion response of the submerged structure under an ISW was explored from several aspects. The experimental results clearly illustrated that ζ0 , β, d, and the shape of the submerged structure affect the motion response to varying degrees. Furthermore, the first three factors mainly affect the velocity and direction of the motion response. According to Eq. (15), the velocity of the upper layer is positively correlated with the amplitude and negatively correlated with the thickness ratio of the upper and lower layers, while the opposite is true for the velocity of the lower layer. Therefore, by comparing the average velocity at the position of the submerged structure to the amplitude of the surge motion under different working conditions, the velocity de­ termines the motion amplitude, and the direction of the velocity significantly affects that of the surge motion (Fig. 16). An empirical expression for the relationship between the nondimensional maximum displacement of the surge motion and nondimensional ISW velocity is proposed: X ¼ 6:94V

0:02;

Fig. 16. Non-dimensional amplitude of the surge motion with the nondimensional velocity. (U characteristic velocity, U ¼ 8 uup ; d ¼ 5cm < ðu þ udown Þ=2; d ¼ 0cm , linear fitting expression - X ¼ 6:94V 0:02). : up udown ; d ¼ 5cm

(26)

pffiffiffiffiffiffiffiffi where X ¼ Xdisp =H, V ¼ U= Hjgj, Xdisp is the maximum displacement of the surge motion, and U is the maximum characteristic velocity at the position of the submerged structure, 8 uup ; �� d ¼ 5cm < (27) U¼ u þ udown 2; d ¼ 0cm : : up d ¼ 5cm udown ; The horizontal velocity is the main factor that affect the displace­ ment of the surge motion, however, the amplitude of ISW plays an important role in the displacement of heave motion. The relationship between dimensionless amplitude of the heave motion and the dimen­ sionless amplitude of ISW is presented in Fig. 17. And this relationship is expressed as follows: Z ¼ 0:82A

0:01;

Fig. 17. Non-dimensional amplitude of the heave motion with the nondimensional amplitude of ISW. (linear fitting expression - Z ¼ 0:82A 0:01).

From Eqs. (26) and (28), if the dimensionless characteristics of an ISW were in the range of this study (α ranges ​ from 0:45 ​ to ​ 1:56 ​ and β ​ ranges ​ from ​ 0:11 ​ to ​ 0:43) the direction and distance of the surge and heave motion response of the submerged bodies can be approxi­ mated by using the theoretical calculation results of the characteristic velocity and amplitude of ISW.

(28)

where Z ¼ Zdisp =H, A ¼ ζ0 =H, Zdisp is the maximum displacement of the heave motion.

6. Conclusions In this study, we experimentally studied the kinematic responses of a submerged structure under 2D ISWs. The variations in the surge, heave, pitch, and track of CG with the ISW amplitudes, relative distances be­ tween the model and pycnocline, thickness ratio of the upper and lower layers, and the shape of a submerged structure are studied. The exper­ imental results were qualitatively and quantitatively analyzed, and a law of motion of a submerged structure under the action of ISWs is obtained in different conditions. This can guide the construction and operational safety of submerged structures in the future. The conclu­ sions of this study are listed as follows: 1. The amplitude of an ISW is the main factor that affects the amplitude of a submerged structure’s motion. The higher the amplitude of the ISW, the higher is the motion amplitude of the submerged structure. The influence of amplitude on the heave and pitch motions is more notable than that on the surge motion when the submerged structure

Fig. 15. Relationship between depth and shear (∂u=∂z) effect. (a) β ¼ 1 : 9, ζ0 ¼ 7:8 cm, (b) β ¼ 2 : 8, ζ0 ¼ 8:3 cm, and (c) β ¼ 3 : 7, ζ0 ¼ 6:7 cm. 12

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is located in the pycnocline. The amplitude of the heave motion is lower than the that of the ISW. 2. The position of the submerged structure in the stratified fluid affects the form of its motion. When the submerged structure is located in the upper layer, the direction of the surge motion depends on the direction of ISW propagation, and the surge motion distance is large; when the submerged structure is located in the pycnocline, the di­ rection of the surge motion depends on both the velocities of the upper and lower layers, and the shape of the submerged structure. However, the amplitude of the motion is low; when the submerged structure is in the lower layer, the direction of its surge motion is opposite to the direction of the ISW propagation, and the amplitude of motion is moderate. Changing the relative distance between the submerged structure and pycnocline had little effect on the heave motion when the structure is close to the pycnocline. The pitch motion is opposite to that in the upper and lower layers when the submerged structure is in the pycnocline. 3. The shape of the submerged structure also affects its motion response form. The motion response amplitude of the spherical submerged structure is notably lower than that of the cylindrical submerged structure. The surge motion of the spherical structure is 29.8% lower than that of the cylindrical submerged structure. The difference in the heave motions between the two is small, as the amplitude of the heave motion of the cylindrical submerged structure is 4.5% larger than that of the spherical submerged structure. Furthermore, when the models are in the pycnocline, the form of the track of the CG track of the cylindrical structure is similar to a ‘V’ shape and that of the spherical structure is similar to an ellipse. 4. The surge motion of the submerged structure mainly depends on the magnitude and direction of the flow velocity. The relationship be­ tween the nondimensional maximum displacement of the surge motion and nondimensional velocity of ISW can be concluded as X ¼ 6:94V 0:02. The velocity calculated from the theories could be used to estimate the maximum displacement of the submerged structures. The heave motion of the submerged structure mainly depends on the amplitude of ISW. The relationship between them is concluded as Z ¼ 0:82A 0:01. The displacement of heave motion could be estimated from amplitude of ISW by using this formula.

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The kinematic response of a submerged structure under the action of an ISW is investigated in this study. The strong flow in both the hori­ zontal and vertical directions induced by the ISW can greatly move as well as cause damage to a submerged structure. Author contributions section Junnan Cui: Conceptualization, Formal analysis, Investigation, Re­ sources, Writing-Original draft preparation, Writing-Reviewing and Editing. Sheng Dong: Data curation, Supervision, Project administration, Funding acquisition. Zhifeng Wang: Methodology, Investigation, Writing-Original draft preparation, Writing-Reviewing and Editing, Funding acquisition, Project administration. Xinyu Han: Investigation, Resources. Peng Lv: Investigation, Resources. Acknowledgements This study is financially supported by the NSFC-Shandong Joint Foundation (No. U1706226) and National Natural Science Foundation of China (Nos. 51779236 and 51509226). References Alpers, W., Heng, W.C., Lim, H., 1997. Observation of Internal Waves in the Andaman Sea by ERS SAR. Third ERS Symposium on Space at the Service of Our Environment, Florence, Italy, ESA, pp. 1287–1297.

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