Effect of Stratified Flow on the Vibration of Anchor Cables in a Submerged Floating Tunnel

Abstract

This study investigates the vertical-type submerged floating tunnel with anchor cables. Based on the characteristics of the anchor cables, the anchor cables are simplified as a nonlinear beam model with hinged ends. Disregarding the axial displacement of the tunnel body, the loads will cause displacements in the x and z directions of the tunnel body. The vibrations of the anchor cables are decomposed into three directions, and the parameter excitation at the connection point between the anchor cables and the tunnel body is taken into account. The equations of motion for the three degrees of freedom of the anchor cables are established using Hamilton’s principle, and then the three equations are solved using the Galerkin method and the fourth-order Runge–Kutta method. The basic characteristics of an internal wave stratified flow acting on the anchor cables are considered, as well as the influence of the incident angle of the ocean currents on the three degrees of freedom of the anchor cables. The results indicate that (1) stratified flow weakens the first- and third-order vortex-induced vibrations of the anchor cables while enhancing the second-order vortex-induced vibrations. When considering the parameter excitation of the anchor cables, the first- and third-order vibrations are weakened, while the second-order vibration remains significant; (2) the first-order vibration of the anchor cables reaches its maximum value when the transverse oscillation frequency of the tunnel body is twice its natural frequency, and the second-order vibration of the anchor cables reaches its maximum value when the transverse oscillation frequency of the tunnel body is twice its natural frequency; (3) the downstream vibration of the anchor cables increases with the increase in the incident angle of the ocean currents, the cross-flow vibration of the anchor cables decreases with the increase in the incident angle of the ocean currents, and the axial vibration of the anchor cables reaches its maximum value when the incident angle of the ocean currents is 60 degrees; (4) stratified flow weakens the lock-in phenomenon of the anchor cables, and the influence of the 1/2 stratified flow on the vibrations of the anchor cables is greater than the influence of the 1/2 stratified flow.

1. Introduction

The concept of submerged floating tunnels (SFT) emerged as a solution for crossing lakes or oceans, particularly in areas with uneven straits or seabeds where traditional immersed tunnels are not suitable. Currently, the construction of SFT is still in the theoretical stage, with research primarily focused on the tunnel body, anchoring system, and pipe joints [1]. In the conceptual design of SFT, various cross-sectional types such as rectangular, circular, elliptical, polygonal, Bezier curves, and double circular shapes have been proposed. Circular sections are the most common due to their resistance and low drag coefficient [2]. However, other cross-sectional types have their own advantages and disadvantages [3]. Based on fluid dynamics optimization, a method using parameter Bezier curves to describe the pipe section has been proposed [4]. Inspired by tension leg platforms, the anchoring system of underwater floating tunnels can refer to tension legs. The anchoring system mainly consists of multiple steel wires twisted into steel ropes, which are then twisted into steel columns capable of bearing significant loads [5]. The arrangement of the anchoring system depends on the environmental conditions. A vertical arrangement is suitable for areas with minimal lateral forces, while mixed and inclined arrangements are considered for complex underwater environments [4,6]. Similar to the construction process of immersed tunnels, SFT are divided into different pipe segments connected by pipe joints. The reliability of pipe joints directly affects the usability of the tunnel. Compared to immersed tunnels, SFT present greater challenges due to differences in buoyancy between pipe segments and the complexity of marine environments. Therefore, construction and structural reliability must be rigorously considered [1]. After years of research, pipe joints can be classified as flexible or rigid. Rigid joints have simpler structures, while flexible joints offer a better deformation adaptability and waterproofing effects [7,8].

In recent decades, scientists have conducted extensive research on the response of SFT anchors. Some researchers have drawn inspiration from the action form of tension legs and hypothesized the anchor cables as elastic Euler beams [9,10,11,12,13,14]. Sun et al. [9] considered the effect of parametric excitation on SFT anchor cables, while Dong et al. [10] studied the effect of nonlinear excitation on the cables. Additionally, Jin et al. [11] established a two-dimensional SFT and conducted hydrodynamic analysis in the time domain, validating the related characteristics through experiments. Wu et al. [12,13] generated random seismic events in the time domain using the stochastic phase spectrum method and solved the anchor cable vibration equation using the Galerkin method. Luo et al. [14] investigated the effects of underwater impact and explosion loads on the anchor cables.

Some researchers have assumed the anchor cables to be nonlinear beams [15,16,17,18,19,20,21]. Xiang and Chen et al. [15,16,17,18] derived the coupled equations between the anchor cables and the pipe body, replacing the coupling vibration between the anchor cables and the pipe body with the Dirac function. They obtained the dynamic response equations for the pipe body coupled with multiple pairs of anchor cables. The coupling vibration was further analyzed by studying the effects of tidal current loads and anchor cable failure. Xiong et al. [19] applied seismic loads to the anchor cable vibration equation and analyzed the probability density function of the nonlinear seismic dynamic response of SFT based on the theoretical framework of nonlinear random dynamics and a random seismic model reflecting specific site response spectra characteristics. The seismic dynamic performance of SFT was evaluated based on dynamic reliability. Chen et al. [20] simulated the two-dimensional tunnel body subjected to concave internal solitary waves using the NS equation and the density transport equation. Won et al. [21] considered the time-series hydrodynamic loads caused by wave generation using the JONSWAP spectrum and studied the feasibility of SFT using finite-element software.

Another group of researchers further studied SFT through experimental methods. Won et al. [22] verified SFT experiments by studying different wave periods. Wang et al. [23] investigated the effects of wave height and wave period on SFT by changing the physical model experiment scale. Shekari et al. [24] compared the results of two-dimensional numerical simulations with physical model experiments to study the bi-directional effects of earthquakes on the flow-structure coupling of SFT. According to Xiang et al. [25], the vibration equation of an underwater anchor cable was derived based on its characteristics. Galerkin’s method was used for simplification, and the fourth-order Runge–Kutta method was used to solve the vibration equation. Han et al. [26] proposed a general modeling method and an improved Galerkin discretization method for underwater anchor cable systems. Yang et al. [27] analyzed the SFT (presumably referring to a specific system) based on modal superposition and Galerkin’s mixed method. The impact of traffic flow on SFT was considered using the finite-difference method.

The previous models proposed by the authors mainly focused on the vibration of a single degree of freedom or two degrees of freedom of the anchor, often neglecting the axial motion of the anchor. Building upon previous research, this study derives the vibration equation of a three-dimensional anchor based on the three-dimensional SFT vibration characteristics of the anchor. Regarding the motion of the pipe, the authors usually assume a unidirectional motion or neglect axial deformation in model experiments due to scaled-down models. In this paper, the authors consider the motion in two directions of the pipe as parameter excitation and derive the motion equation of the anchor. The derived equation is then simplified using the Galerkin method and solved using the fourth-order Runge–Kutta method to explore the vibration characteristics of the anchor. This research has a certain reference significance for the study of the multidimensional vibration of suspended tunnel anchors and provides a scientific basis for the construction of suspended tunnels.

2. Mathematical Modeling of Anchor Cables

2.1. Mathematical Model of Anchor Cables

Considering the impact of both longitudinal and transverse movements of the pipe on the anchor cable, it is assumed that the axial displacement of the pipe during the establishment process is negligible compared to the length of the pipe. Therefore, the axial deformation of the pipe is disregarded. Based on this assumption, the transverse displacement of the pipe is denoted as U, while the vertical displacement is denoted as V. As for the anchor cable, the vibration changes in the downstream and axial directions cannot be ignored, and the vibrations in all three directions of the anchor cable are considered. Therefore, the axial variation in the anchor cable is denoted as u, the transverse displacement as w, and the downstream displacement as v. The detailed model is shown in Figure 1. The flow velocity of the seawater is denoted as V_c.

The basic equation of Hamilton’s principle can be expressed as:

$$\delta {\displaystyle {\int }_{t_i}^{t_f}\left(KE-PE\right)dt}+\delta {\displaystyle {\int }_{t_i}^{t_f}Wdt=}0$$

(1)

From the equation, the potential energy, kinetic energy, and virtual work of the external force of the anchor cable can be obtained according to the characteristics of the vertical anchor cable as follows:

$$PE=\frac{1}{2}{\displaystyle {\int }_0^L\left[EA{(u\prime +\frac{1}{2}(v{\prime }^2+w{\prime }^2))}^2+EI({v^{″}}^2+{w^{″}}^2)\right]}dz$$

(2)

$$KE={\displaystyle {\int }_0^L\left\{\begin{array}{l}\frac{1}{2}\left[m({\dot{u}}^2+{\dot{v}}^2+{\dot{w}}^2)+{\rho }_{t}I({{\dot{v}}^{\prime }}^2+\dot{w}{\prime }^2)\right]\\ +C_s({\displaystyle {\int }_{v_{ti}}^{v_{tf}}vdv+{\displaystyle {\int }_{w_{ti}}^{w_{tf}}wdw})}\end{array}\right\}}dz$$

(3)

$$\delta {\displaystyle {\int }_{t_i}^{t_f}Wdt={\displaystyle {\int }_0^L\left[f_x(z,t)\delta u+f_y(z,t)\delta v+f_z(z,t)\delta w\right]}}dz$$

(4)

In these equations, L, E, I, m, C, ρ_t, and A correspond to the length, elastic modulus, moment of inertia, unit length mass, density, and cross-sectional area of the anchor cable, respectively. t_i and t_f represent the starting and ending times for the calculations, v_ti and v_tf represent the displacement values of v at the starting and ending times, and w_ti and w_tf represent the displacement values of w at the starting and ending times. The variables u, v, and w correspond to the displacements in the z-axis, x-axis, and y-axis directions of the anchor cable, respectively. u′ is the partial derivative of u with respect to the z-axis direction, and $\overline{u}$ is the partial derivative of u with respect to time t. Similar symbols are used for other variables. f_x(z,t), f_y(z,t), and f_z(z,t) respectively represent the external excitation terms of the anchor cable in the three coordinate axes.

The vibration control equation for the suspension bridge anchor cable can be derived from the above equation:

$$m\ddot{v}+C_s\dot{v}+(EI{v}^{″}{)}^{″}-{\rho }_{t}I{\ddot{v}}^{″}-[EA(u^{\prime }+\frac{1}{2}({v^{\prime }}^2+w{\prime }^2))v^{\prime }{]}^{\prime }=f_x(z,t)$$

(5)

$$m\ddot{w}+C_s\dot{w}+(EI{w}^{″}{)}^{″}-{\rho }_{t}I{\ddot{w}}^{″}-[EA(u^{\prime }+\frac{1}{2}({v^{\prime }}^2+w{\prime }^2))w\prime {]}^{\prime }=f_y(z,t)$$

(6)

$$m\ddot{u}-[EA(u^{\prime }+\frac{1}{2}({v^{\prime }}^2+w{\prime }^2)){]}^{\prime }=f_z(z,t)$$

(7)

Based on the physical conditions, the external excitations primarily affecting the x and y directions of the anchor cable are the hydrodynamic forces generated by the along-stream and cross-stream vortices caused by the action of ocean currents, as well as the hydrodynamic forces generated by the anchor cable itself. In the z-axis direction, the main external excitations on the anchor cable are buoyancy and gravity. When lock-in occurs, the fluctuating drag force per unit length can be approximately represented by a harmonic function with the shedding frequency of vortices. The water damping force and added mass force generated by the anchor cable vibration can be expressed using the Morrison equation. Thus, the external excitation force on the anchor cable can be determined as follows:

$$f_x(z,t)=\frac{1}{2}\rho D{V}_c^2{C^{\prime }}_D\cos 2{\omega }_{s}t-\rho A{C}_m\ddot{v}-\frac{1}{2}\rho D{C}_D\mathrm{sgn}(\dot{v}-V_c)\cdot {(\dot{v}-V_c)}^2$$

(8)

$$f_y(z,t)=\frac{1}{2}\rho D{V}_c^2{C}_L\cos {\omega }_{s}t-\rho A{C}_m\ddot{v}-\frac{1}{2}\rho D{C}_D\mathrm{sgn}(\dot{v})\cdot {\dot{v}}^2$$

(9)

$$f_z(z,t)=\rho Ag-mg$$

(10)

In these equations, ρ, g, D, C′_D, C_D, C_m, V_s, w_s, and sgn represent the density of seawater, diameter of the anchor cable, coefficient of pulsating drag force, coefficient of resistance, coefficient of lift, coefficient of mass, seawater velocity, shedding frequency of vortices, and the function of the positive and negative signs for the extraction parameters.

2.2. The Role of Internal Waves on Anchor Cables

In the actual environment of the ocean, the phenomenon of ocean fluctuations is very complex. The ocean currents change with a variety of objective factors such as depth, seawater density, seafloor environment, etc. We refer to the fluctuations occurring within the ocean as internal ocean waves. One of the simplest forms of internal waves is the fluctuation that occurs at the interface of two layers of seawater with different densities, which is called an interfacial internal wave [28]. The actual density of seawater is gradual, and a variety of objective factors such as salinity, currents, tides, etc., also affect internal waves. Zhao et al. [29] proposed a new method for estimating the phase velocity and vertical velocity of internal solitary waves (ISWs) based on dynamic control equations. By studying the general solution of three theoretical models of internal solitary waves, KdV, eKdV, and MCC, it can be concluded that the velocity changes rapidly at the velocity boundary layer. This conclusion can also be confirmed through CFD software calculations [30]. Therefore, in this study of internal waves, the waves are assumed to be stratified flows, taking into account the effect of stratified flows on anchor cables.

2.3. Determination of Boundary Conditions and Solution of Equations

In this paper, only the motion of the anchor cable is considered, so the tube part is simplified, and the ends of the anchor cable are assumed to be a nonlinear beam structure articulated at both ends. The boundary conditions of the anchor cable can be obtained as:

$$v(0,t)=0,v(L,t)=G(t)+v_{ti}$$

(11)

$$\omega (0,t)=0,(L,t)=H(t); \omega ″(0,t)=0,\omega ″(L,t)=0$$

(12)

$$u(0,t)=0,u(L,t)=S(t); u″(0,t)=0,u″(L,t)=0$$

(13)

In the equations, v_ti represents the displacement in the x direction at the starting time. This refers to the elongation generated by the difference between the buoyancy and gravity forces on the pipe, which is balanced by the mooring cables, and can be expressed using the principles of material mechanics:

$$v_{t0}=TL/EA$$

(14)

There are various theoretical methods for mooring the SFT, but there are no existing SFT that can serve as a reference to effectively determine the practical advantages and disadvantages of various mooring methods. Therefore, the mooring method chosen in this study is depicted in Figure 1a. Although the motion of the pipe is assumed to be that of a particle, the actual motion of the pipe can also affect the response of the anchor cable. In previous studies of the pipe, the axial excitation of the pipe was assumed to be a parameter excitation along the z-axis. Building upon this assumption, this paper considers the influence of the displacement along the x-axis of the pipe on the anchor cable. As shown in Figure 2, the angle between the ocean current and the pipe is α. Therefore, the lateral oscillation of the pipe can be decomposed into the downstream motion and the cross-flow motion. The motion of the tube body is assumed to be a simple harmonic function, while the motion of the anchor cable contact points connected to the tube body at this time can be expressed as follows, respectively:

$$G\left(t\right)=U\cos {\omega }_{u}t; H\left(t\right)=\sin \alpha \cdot V\sin {\omega }_{v}t; S\left(t\right)=\cos \alpha \cdot V\sin {\omega }_{v}t$$

(15)

where T, U, and V represent the initial static tension of the anchor cable, the amplitude of the pendulum motion of the tube, and the amplitude of the transverse motion of the tube, respectively, and w_u and w_v represent the frequencies of the heave and sway motions of the tube.

The vibration model of anchor cables can be represented using the Galerkin method:

$$v(x,t)=(G(t)+v_{t0})\frac{x}{L}+{\displaystyle \sum _{n=1}^R{v}_n(t)\sin \frac{n\pi x}{L}}$$

(16)

$$\omega (x,t)=H(t)\frac{x}{L}+{\displaystyle \sum _{n=1}^R{\omega }_n(t)\sin \frac{n\pi x}{L}}$$

(17)

$$u(x,t)=S(t)\frac{x}{L}+{\displaystyle \sum _{n=1}^R{u}_n(t)\sin \frac{n\pi x}{L}}$$

(18)

The modal vibration control equations for each order of the anchor cable can be simplified jointly with the above equation as:

$$\begin{array}{l}{\ddot{v}}_n(t)+{(-1)}^{n+1}\frac{2}{n\pi }\ddot{G}\left(t\right)+\frac{EA}{m}{\left(\frac{n\pi }{L}\right)}^2{v}_n(t)+\frac{EA}{m}{\left(\frac{n\pi }{L}\right)}^2\frac{H(t)}{L}{\omega }_n(t)\\ \begin{array}{cc}& \end{array}\hspace{1em}+\frac{EA}{m}{\left(\frac{n\pi }{L}\right)}^2\frac{S(t)}{L}{u}_n(t)=\left[1-{(-1)}^n\right]\frac{2}{mn\pi }(\rho Ag-mg)\end{array}$$

(19)

$$\begin{array}{l}{\ddot{\omega }}_n(t)+2u_n\zeta {\dot{\omega }}_n(t)+{(-1)}^{n+1}\frac{m+\rho A{C}_m}{\overline{m}}\frac{2}{n\pi }\ddot{H}(t)+{(-1)}^{n+1}\frac{4u_n\zeta }{n\pi }\dot{H}(t)+\\ \frac{EA}{\overline{m}}{\left(\frac{n\pi }{L}\right)}^2\frac{H(t)}{L}{v}_n(t)+\frac{3}{2}\frac{EA}{\overline{m}}{\left(\frac{H(t)}{L}\right)}^2{\left(\frac{n\pi }{L}\right)}^2{\omega }_n(t)+\frac{EA}{\overline{m}}\frac{H(t)}{L}\frac{S(t)}{L}{\left(\frac{n\pi }{L}\right)}^2{u}_n(t)\\ +\frac{EA}{\overline{m}}\frac{(G(t)+v_{t0})}{L}{\left(\frac{n\pi }{L}\right)}^2{\omega }_n(t)+\frac{1}{2}\frac{EA}{\overline{m}}{\left(\frac{S(t)}{L}\right)}^2{\left(\frac{n\pi }{L}\right)}^2{u}_n(t)+\frac{EI}{\overline{m}}{\left(\frac{n\pi }{L}\right)}^4{\omega }_n(t)\\ =-\frac{\rho D{C}_D}{\overline{m}L}KD(Q)+\left[1-{(-1)}^n\right]\frac{\rho D{V}_c^2{C^{\prime }}_D}{\overline{m}n\pi }\cos 2{\omega }_{s}t\end{array}$$

(20)

$$\begin{array}{l}{\ddot{u}}_n(t)+2u_n\zeta {\dot{u}}_n(t)+{(-1)}^{n+1}\frac{m+\rho A{C}_m}{\overline{m}}\frac{2}{n\pi }\ddot{S}(t)+{(-1)}^{n+1}\frac{4u_n\zeta }{n\pi }\dot{S}(t)\\ +\frac{EA}{\overline{m}}{\left(\frac{n\pi }{L}\right)}^2\frac{S(t)}{L}{v}_n(t)+\frac{3}{2}\frac{EA}{\overline{m}}{\left(\frac{S(t)}{L}\right)}^2{\left(\frac{n\pi }{L}\right)}^2{u}_n(t)+\\ \frac{EA}{\overline{m}}\frac{H(t)}{L}\frac{S(t)}{L}{\left(\frac{n\pi }{L}\right)}^2{\omega }_n(t)+\frac{EA}{\overline{m}}\frac{(G(t)+v_{t0})}{L}{\left(\frac{n\pi }{L}\right)}^2{u}_n(t)\\ +\frac{1}{2}\frac{EA}{\overline{m}}{\left(\frac{H(t)}{L}\right)}^2{\left(\frac{n\pi }{L}\right)}^2{\omega }_n(t)+\frac{EI}{\overline{m}}{\left(\frac{n\pi }{L}\right)}^4{u}_n(t)=-\frac{\rho D{C}_D}{\overline{m}L}KD(J)\\ +\left[1-{(-1)}^n\right]\frac{\rho D{V}_c^2{C}_L}{\overline{m}n\pi }\cos {\omega }_{s}t\end{array}$$

(21)

This equation introduces for convenience KD(P), Q, J, ${\omega }_n^2$, and $\overline{m}$, represented by the following parameters:

$$KD(P)={\displaystyle {\int }_0^L\mathrm{sgn}(P)\cdot {(P)}^2\sin \frac{n\pi x}{L}dz}\begin{array}{cc}& \left(P\right.=Q\left. \mathrm{or} \mathrm{J}\right)\end{array}$$

(22)

$$Q=\dot{H}(t)+{\displaystyle \sum _{n=1}^R{\dot{\omega }}_n(t)}\sin \frac{n\pi x}{L}-V_c; J=\dot{S}(t)+{\displaystyle \sum _{n=1}^R{\dot{u}}_n(t)}\sin \frac{n\pi x}{L}$$

(23)

$${\omega }_n^2=\frac{EI}{\overline{m}}{\left(\frac{n\pi }{L}\right)}^4+\frac{T}{\overline{m}}{\left(\frac{n\pi }{L}\right)}^2; \overline{m}=m+\rho A{C}_m+{\rho }_{t}I{\left(\frac{n\pi }{L}\right)}^2$$

(24)

3. Numerical Analysis

Calculation Parameters of Anchor Cable Structure for SFT

The internal wave assumption is made for stratified flow, with inflow directions set at 90°, 75°, 60°, 45°, and 30°. Three points of intersection are set for the stratified flow, namely the uniform flow as the control group, and the stratification at one-half and one-third positions. As for other parameters, due to the lack of completed global SFT projects for reference, the parameters used in this study are based on calculations from previous domestic and international cases. The specific parameters are as follows:

From the data in

Table 1. Parameters related to suspension tunnels.

Structure	Parameters	Symbol/Unit	Value
Suspension tunnel	Depth of flooding	h/m	30
	Section type	/	Round single tube
	Outer diameter of tube body	Dout/m	15
	Inside diameter of tube	Din/m	12
	density	ρtube/(kg·m−3)	7850
Suspension tunnel anchor	Length	L/m	140
	Caliber	D/m	0.5
	Density	ρcable/(kg·m−3)	7850
	Prestressing	T/N	2.7 × 107
	Modulus of elasticity	E/Pa	2.1 × 1011
	Damping ratio	ζ	0.0018
Other relevant parameters	Water depth	d/m	170
	Water density	ρ/(kg·m−3)	1025
	Quality factor	Cm	1
	Drag coefficient	CD	0.6
	Pulsating drag force coefficient	C′D	0.2
	Strouhal number	St	0.2
	Gravitational acceleration	g/(N·kg−1)	9.8

, it can be concluded that when the velocity of the current is 0.56 m·s⁻¹, the cross-current eddy-excitation vibration of the anchor cable can reach a frequency-locked state. When the velocity of the current is 1.117 m·s⁻¹, the cross-flow vibration of the anchor cable can reach the frequency-locked state. Since the anchor cable is in seawater, its eddy-excited vibration has more influence on it, so in order to explore more accurate dynamic characteristics of the anchor cable, the velocity of the sea current selected in this paper is 1.117 m·s⁻¹.

As the parameter excitation of the tube body is considered, the values of the motion values U and V of the tube body are also very critical. Since the tube body belongs to the elongated structure, its motion in the transverse and vertical oscillation directions is slowly deformed, and since its span is often several kilometers to start with in the actual construction, and in the sea by the repeated action of the currents, its buoyancy is also greater than the gravity. The displacement of the tube body is inevitable for suspension tunnels. Based on the study of the motion response of vertical fabric cables of different cross-section types, the maximum of the transverse and vertical oscillations generally does not exceed 5 m. It is assumed that the value of the tube body U is taken as 0 and 0.03 m for two cases, and the value of V is taken as 0, 0.3, 0.6, and 0.9 m for four cases to study the anchor cable, and all the cases are calculated for 200 s.

4. Discussion

4.1. Effect of Stratified Flow on Vortex-Excited Vibration

Firstly, the impact of stratified flow on anchor cable vortex-induced vibration is investigated, without considering the vibration of the pipe body, focusing on the vibration of the anchor cable. When not considering the vibration of the pipe body, U and V are both 0. According to the equation derived above, stratified flow has no effect on the axial vibration of the anchor cable. Therefore, the axial vibration of the anchor cable is not considered in this case. As shown in Figure 3a,b, the images illustrate the first-order vibration of the anchor cable. From Figure 3a, it can be observed that stratified flow reduces the transverse vibration of the anchor cable. The effect of stratified flow on the anchor cable is particularly evident at 1/2, indicating that stratified flow reduces the first-order transverse vortex-induced vibration of the anchor cable. Figure 3b also shows that stratified flow reduces the first-order inline vibration of the anchor cable. The second and third images clearly display the final stable position of the anchor cable. In the second image, due to stratification at 1/3, with the reverse flow of 2/3, the stable position is negative, while the uniform flow stabilizes at a positive value, which aligns with actual observations and demonstrates the reliability of the numerical results. Figure 3c,d illustrate the second-order vibration of the anchor cable, indicating that stratification increases the second-order vibration significantly, whereas under the influence of uniform flow, the second-order vibration of the anchor cable is almost negligible. Figure 3e,f represent the third-order vibration of the anchor cable, which is significantly lower compared to the first-order vibration. Stratified flow also has a noticeable weakening effect on the vibration of the anchor cable. The effect of uniform flow on the modal vibration of the first three sections of the anchor cable is the same as that concluded by Su et al. [10] with a maximum deviation of 0.1, which is due to the differences in the selected length of the anchor cable, the diameter of the anchor cable, and other related parameters, which proves the reliability of the program.

4.2. Influence of Internal Wave Stratified Flow on Transverse Flow Vibration of an Anchor Cable

The length of the anchor cable is 140 m. Under the condition where the angle between the ocean current and the anchor cable is 90°, the anchor cable vibration is compared longitudinally between a uniform flow and layering at the 1/2 and 1/3 positions. The root mean square of the anchor cable vibration for the first 200 s is calculated, as shown in Figure 4. From the first picture in Figure 4, it can be observed that the first-order transverse vibration of the anchor cable under the influence of a uniform flow is greater than the first-order transverse vibration of the anchor cable with layering at the 1/3 position, and the first-order transverse vibration of the anchor cable with layering at the 1/3 position is greater than the first-order transverse vibration of the anchor cable with layering at the 1/2 position. The theoretically calculated magnitude of the first-order mode vibration corresponds to the simulated results, demonstrating the reliability of the numerical simulation. The second picture in Figure 4 shows that the second-order transverse vibration of the anchor cable with layering at the 1/2 position is greater than the second-order transverse vibration of the anchor cable with layering at the 1/3 position, and it is also greater than the second-order transverse vibration of the anchor cable under the influence of a uniform flow. Considering the form of the ocean current and the vibration mode of the second order, the observed magnitude order of the second-order mode of vibration fully aligns with the theoretical expectations. The third picture in Figure 4 illustrates the vibration of the third-order anchor cable. Although the occurrence probability of the third-order mode in engineering practice is very low, the numerical simulation results validate the theoretical magnitude order, demonstrating the reliability and rigor of the numerical simulation. It can be concluded from the above that the stratified flow of internal waves has a significant impact on the vibration of the anchor cable, and changing the position of the layering in the stratified flow will affect the vibration of different mode orders.

Considering the parameters of the pipe excitation, previous research has found that when the excitation frequency of the pipe is equal to the natural frequency of the anchor cable, the anchor cable will experience a frequency locking phenomenon. When the excitation frequency of the pipe is twice the natural period of the anchor cable, the vibration of the anchor cable reaches its maximum value. Building upon this, the cross-flow frequency is set to the natural frequency of the anchor cable to further amplify the vibration. This is assuming that the transverse amplitude of the pipe is a fixed value of 0.03 m, obtained through the conversion of initial tension. Based on previous research, the transverse frequency of the anchor cable is set to one and two times the natural period. From the first and third picture in Figure 4, it can be observed that when the axial frequency of the pipe is twice the natural frequency of the anchor cable, the vibration response of the anchor cable significantly increases. However, for the second picture in Figure 4, the maximum value occurs when the transverse frequency of the pipe is equal to the natural period of the anchor cable, and the anchor cable vibration at 1/2 of the stratified flow layer is the highest. Therefore, it can be concluded that the first- and third-order vibrations of the anchor cable significantly increase when the frequency of the axial parameter excitation is twice the natural frequency of the anchor cable. However, this does not apply to the second-order modal vibration of the anchor cable. From the second picture in Figure 4, it can be seen that the effect of stratified flow has a significant impact on the second-order vibration of the anchor cable, with the maximum value occurring when the axial excitation frequency is equal to the natural frequency of the anchor cable.

4.3. The Effect of Internal Wave Stratified Flow on the Downstream Vibration of an Anchor Cable

Under different operating conditions, the first three upstream vibrations of the mooring cable are shown in Figure 5. When the effect of lateral motion of the pipe body is not considered, it can be observed that there is little change in the first- and third-order upstream vibration responses of the mooring cable. Except for the first mode, when the oscillation frequency is twice the natural frequency, the stratified flow has a significant impact on the upstream vibration of the mooring cable at the midpoint of the cable. For the second-order upstream vibration of the mooring cable, it can be clearly observed that the lines are very chaotic, which is significantly different from the first and third pictures in Figure 5. In the first and third pictures in Figure 5, the oscillation frequency of the pipe body is significantly increased compared to the other two types when the oscillation frequency is twice the natural frequency. This is also the conclusion of previous studies. In the second picture in Figure 5, the root mean square of the initial vibration amplitude of the uniform flow is the same, but considering the stratified flow, it is obvious that the second-order upstream vibration increases significantly. Moreover, when the oscillation frequency of the pipe body is one or two times the natural frequency, it can be seen that the increase in the second-order upstream vibration of the mooring cable is more pronounced at half of the fluid stratification. When considering the lateral motion of the pipe body, it can also be found that there is little change in the first- and third-order vibrations, but the influence on the second-order vibration of the mooring cable is significant. Under the same operating conditions, the second-order upstream vibration of the uniform flow is significantly smaller than the upstream vibration of the mooring cable at one-third stratification, and the upstream vibration of the mooring cable at one-third stratification is smaller than the upstream vibration of the mooring cable at one-half stratification, which conforms to theoretical principles. When compared with the lateral vibration, it is found that the upstream vibration is more affected by the variation in the lateral motion amplitude of the pipe body than the lateral vibration.

4.4. The Effect of Internal Wave Stratified Flow on the Axial Vibration of the Anchor Cable

The axial vibration of the anchor cable can be seen in Figure 6. From Figure 6, it can be observed that the starting points of all lines are almost the same. This is likely because the axial vibration itself is very small, and the impact of laminar flow on axial vibration can be ignored. However, when considering the lateral motion of the pipe, the curves begin to diverge significantly. This indicates that the lateral motion of the pipe exacerbates the influence of laminar flow on anchor cable vibration. By observing the first picture in Figure 6, it can be noted that there are only five distinct lines, while the rest of the lines overlap. When the axial vibration of the pipe is zero, the vibration of the anchor cable due to laminar flow and uniform flow overlap, meaning that laminar flow does not affect the axial vibration of the anchor cable when axial vibration is not considered. However, when axial vibration is taken into account, it can be observed that the first-order vibration of the anchor cable coincides with the axial vibration frequency being twice the natural frequency, and the third-order vibration being close to coincidence. When the axial vibration frequency is equal to the natural frequency, the first-order and third-order vibrations of the anchor cable are also very close. This indicates that the influence of laminar flow on the first- and third-order axial vibrations of the anchor cable is minimal. As for the second-order axial vibration of the anchor cable, when the lateral motion of the pipe is not considered, the vibration of the anchor cable still follows the previous research findings regarding the axial vibration frequency of the pipe. However, when considering the lateral motion of the pipe, the uniform flow will decrease when it is not affected by axial motion and when the axial frequency is twice the natural frequency of the anchor cable. On the other hand, under uniform flow, an increase in axial excitation frequency to twice the natural frequency of the anchor cable can be observed. Apart from this, it can be seen that the impact of laminar flow on anchor cable vibration is noticeable. When the laminar flow is at the midpoint of the anchor cable, the second-order axial vibration of the anchor cable increases significantly. A significant increase in the second-order axial vibration of the anchor cable can also be observed when laminar flow is at one-third of the anchor cable, although it is relatively smaller compared to the second-order axial vibration at the midpoint. This indicates that laminar flow has a significant impact on the axial second-order vibration.

4.5. Effect of Incidence Angle of Stratified Ocean Currents on the Vibration of Anchor Cables

Taking into consideration the influence of layered currents and the incident angle of the pipe on anchor cable vibration, the working condition at the 1/2 position with layered flow is adopted. The flow velocity of the current remains constant, while the incident angle varies from 90 degrees to 30 degrees, with an interval of 15 degrees. The axial vibration of the pipe is also considered. Due to the low probability of occurrence and the significantly smaller amplitude compared to the first- and second-order vibrations, the third-order vibration is neglected. The variation in anchor cable vibration is shown in Figure 7. Figure 7a,b depict the first two axial vibrations of the anchor cable. It can be observed that the vibration changes are relatively dense. This indicates that the incident angle of the current affects the anchor cable vibration, but it is not a determining factor. From the observation of the two figures, it can be seen that for the first-order vibration, the peak appears when the transverse frequency of the pipe is twice the natural frequency; for the second-order vibration, the peak appears when the transverse frequency of the pipe is equal to twice the natural frequency, and a maximum value occurs when the incident angle is 60 degrees. Figure 7c,d show the first two transverse vibrations of the anchor cable. Except for the third graph in Figure 7c, it can be seen that as the incident angle of the waves decreases, the transverse vibration of the anchor cable increases significantly, with a large variation in amplitude. From a theoretical perspective, when the axial excitation of the pipe is equal to the natural frequency, the vibration of the anchor cable reaches its maximum value; when the axial excitation is twice the natural frequency, the vibration of the anchor cable reaches its maximum value. Considering the effect of layered flow, it can be seen that when the axial excitation of the anchor cable is equal to the natural frequency, its value is not significantly different from when the axial excitation is not considered, indicating that layered flow affects the effect of the anchor cable when the axial excitation is equal to the natural frequency. For the third graph in Figure 7c, the maximum value occurs when the transverse vibration frequency of the pipe is equal to the natural frequency, but the minimum value occurs when the transverse vibration frequency of the pipe is equal to twice the natural frequency. If we exclude the possibility of calculation errors, the existence of this phenomenon may be due to the interaction between the anchor cable vibration excitations under the effect of layered currents, resulting in a decrease in anchor cable vibration. Further research can be conducted to understand the mechanism and reduce the damage to the anchor cable caused by vortex-induced vibration from a theoretical perspective. Figure 7e,f depict the first two downstream vibrations of the anchor cable. Overall, the downstream vibration of the anchor cable decreases as the incident angle decreases.

5. Conclusions

(1)

The effect of parametric excitation on the vibration of the mooring ropes is more pronounced, and the position of the stratified flow also affects the vibration of the mooring ropes;

(2)

When the axial vibration frequency of the pipe is twice the natural frequency of the mooring cable, the influence of stratified flow on the second-order transverse vibrations of the mooring cable is intensified. When the axial vibration frequency of the pipe is twice the natural frequency of the mooring cable, the influence of stratified flow on the second-order downstream and axial vibrations of the mooring cable is intensified. Furthermore, the effect generated at the half point of the mooring cable by the stratified flow is greater than the effect generated at the one-third point of the mooring cable;

(3)

The first-order vibration of the mooring cable reaches its maximum value when the lateral oscillation frequency of the pipe is twice the intrinsic frequency of the mooring cable. When the lateral oscillation frequency of the pipe is twice the intrinsic frequency of the mooring cable, the second-order vibration of the mooring cable reaches its maximum value. Laminar flow has a suppression effect on the frequency locking phenomenon;

(4)

As the incidence angle of the current increases, the downward vibration of the mooring cable increases, while the transverse vibration of the mooring cable decreases with the increase in the incidence angle of the current. The axial excitation of the mooring cable reaches its maximum value when the incidence angle of the current is 60 degrees.

This paper is based on theoretical research; there are some assumptions in the paper, and these assumptions have some theoretical errors. For example, it is inaccurate to set the damping as a fixed value, as the damping is affected by the frequency and other factors; the motion of the tube body is assumed to be a sinusoidal motion, which is also inaccurate, as the actual motion of the tube body is affected by the wave current force and frequency and other factors. The change in frequency on damping can be further explored in subsequent research to affect the vibration of the anchor cable, and the tube body motion law can also be further studied to further optimize the parameter excitation. For the calculation method used in this paper, there will be a certain error, the fourth-order Runge–Kutta method iteration process will have a certain error, and other numerical methods can also be used in the future to solve the problem.