Optimization Analysis of the Arrangement of the Submerged Floating Tunnel Subjected to Waves

Abstract

The motion responses, mooring tensions, and submergence depth are the dominant factors for the arrangement of the Submerged Floating Tunnel (SFT) subjected to waves. Generally, the maximum values of motion responses, mooring tensions, and absolute submergence depth are mainly focused on. In the present study, experiments are implemented to measure the motion responses and mooring tensions of the SFT with different mooring patterns and submergence depths under waves with different characteristic wave heights and periods. In order to evaluate the arrangement of the SFT more effectively and comprehensively, besides the maximum values, several new characteristic parameters are introduced. Such parameters account for the motion responses in the frequency domain, the uniformity of the tension distribution, the length of time during which the cable reaches a relaxed condition during wave action, the KC number, the dimensionless period, the wave height, and the submergence depth. The results from the optimization analysis show the following: according to the characteristic values of motion responses and mooring tensions, the pattern of diagonal cables is better than that of diagonal cables + vertical cables; and within the range of the present experiments, there are optimal dimensionless parameters—the dimensionless submergence depth d₀/L_P ≥ 0.15, the KC number ≤ 0.8, or the dimensionless wave height Hs/d₀ ≤ 0.10—for the condition of which the dynamic responses and mooring tensions vary slightly.

1. Introduction

The Submerged Floating Tunnel (SFT) is a new type of cross-water transportation structure that mainly consists of tunnel tubes, constrained along the tunnel and at the ends. Equilibrium is established by the gravity and buoyancy of the tunnel tube and mooring tension of the restraints [1,2,3]. Based on the different restraints along the tunnel, the SFT can be classified as a fixed-support tunnel, floating-tube tunnel, and anchor-cable tunnel. The anchor-cable tunnel has shown a better overall performance as compared with the fixed-support tunnel and floating-tube tunnel. This type of SFT has been the primary focus of researchers and is the most commonly produced structural form.

With respect to the mooring pattern of the SFT, the mooring system plays an important role in the motion responses of the floating structures subjected to waves, while the majority of the tension in the mooring is connected to the wave actions [4]. Seo et al. [5] experimentally studied the dynamic response of an SFT under regular waves and discussed the effect of the mooring pattern. The test results show that the single vertical moored SFT oscillates severely, while the w-type double-moored SFT oscillates slightly. Cifuentes et al. [6] numerically simulated the dynamic response of an SFT under regular waves. The vertical and inclined mooring systems were compared to find a better design to limit SFT movement and minimize mooring tension. The results show that the inclined mooring system is more effective in restricting SFT movement compared to the vertical mooring system. Jeong et al. [7] analyzed the feasibility of the SFT with a mooring system combining vertical and inclined tethers by using the finite element program ABAQUS-AQUA and pointed out that it could be effectively applied when the spacing is less than 40.0 m and the inclination angle is 45.0 degrees. Lee et al. [8] used numerical software OrcaFlex and CHARM3D to simulate the dynamic responses of two types of SFTs under the waves and seismic excitations in the time domain. The dynamic responses of the short-rigid-free-end SFT with vertical and inclined mooring systems were investigated. The results show that mooring tension increases in the inclined mooring system under horizontal earthquakes, while for vertical earthquakes, there is an increase in mooring tension in both the inclined and vertical mooring systems. Especially, when the seismic frequency is close to the natural frequency of the SFT, it is much more dangerous. Muhammad et al. [9] numerically investigated the motion responses and internal forces of the SFT under hydrodynamic and three-dimensional seismic excitations to test the overall performance of the SFT. The dynamic responses of the SFTs with three different mooring systems were evaluated to compare the stability of different mooring configurations. The results show that the moored SFT with a combination of tension legs and single inclined mooring cables is effective under moderate environmental conditions. Under harsh environmental conditions, the SFT should be moored by a combination of tension legs and double inclined mooring cables or only by double inclined mooring cables. Jin et al. [10] used the time-domain numerical software OrcaFlex to simulate the dynamic responses of a 1000 m long circular SFT with both ends fixed under irregular waves. The vertical mooring system (VM) and inclined mooring system (IM) were compared to investigate their overall performances. The results show that the horizontal response of VM is larger than that of IM due to the smaller horizontal stiffness and its lowest natural frequency closer to the incident wave frequency. The vertical response of VM is mainly downward and is also affected by set-down. The bouncing tension of the mooring line increases sharply when the SFT bounces back from the slack mooring condition caused by large vertical downward motion. Won et al. [11] numerically simulated the dynamic responses of the SFT with a dual section. Two types of dual-SFTs were proposed according to the mooring system, and the analysis results of hydrodynamic characteristics show that as the wave steepness increases, the sway of the SFT increases significantly, while the increase in heave is smaller, and that for inclined mooring system, a relaxation phenomenon lowering from the initial position occurs when the steepness of the wave is 0.028 and the motion increases rapidly. Wu et al. [12] proposed a new SFT consisting of three circular section tubes and a fiber-reinforced plastic (FRP) rigid truss structure. In addition, a series of model tests were carried out under wave-flow conditions to investigate the anti-vibration performance of the SFT. The results show that compared with other models, the free vibration frequency of each degree of freedom for the proposed SFT is much lower, which can avoid the resonance with the wave frequency and ensure the safety of the structure. The proposed SFT has better anti-vibration performance under different external loads, especially in the roll motion component. The mooring system of the proposed SFT has a smaller mooring tension which is below the limits' value.

For the submergence depth of the SFT, Cifuentes et al. [6] numerically simulated the dynamic response of the SFT under regular waves. The results show that both the motion response and mooring tension increase with the wave height and period and decrease with submergence depth due to the reduced wave action. The inclined mooring system restricts the motion response more effectively than the vertical mooring system with the same submergence depth. Won et al. [13] used the nonlinear finite element code ABAQUS–AQUA to investigate the dynamic response of a new type of SFT under the irregular wave. The dynamic responses, including the motion response, internal force, and fatigue damage, under severe waves were systematically investigated. The results show that the dynamic response of the SFT can be effectively controlled by designing a “suitable” submergence depth. It results from the wave force reduction caused by the deep draft design. Chen et al. [14] proposed a theoretical method to investigate the nonlinear dynamic response of the SFT mooring system under the combined action of parametric excitation and hydrodynamic excitations (i.e., wave- and vortex-induced loading). Based on the Hamilton principle, the governing equations of the SFT system considering the coupled degrees of freedom and mooring cables are established and solved. The results show that the motion amplitude of the mooring cables and the tube decreases with the increase in the submergence depth. When the submergence depth is within the range of 5~40 m, the motion amplitude reduction rate of the mooring cable is greater than that of the tube. When the submergence depth is more than 40 m, the motion response of the tube will decrease more significantly. Deng et al. [15] experimentally investigated the drag force on a double-tube SFT with different Reynolds numbers and submergence depths using a simplified rigidly connected tandem twin-cylinder model under strong current. The results show that both the vortex-induced vibration response and mean drag coefficient decrease with the decrease in the submerged depth, especially when the depth ratio h* < 3.7 (h* = h/D, where h is the distance from the bottom of the SFT to the initial free surface and D is the outer diameter of the SFT). Yang et al. [16] experimentally investigated the 2D motion characteristics of the SFT under regular waves. The experimental phenomena of SFT motion during the interaction between the SFT and waves are described in detail. The effect of experimental parameters, including the wave height, wave period, submergence depth, buoyancy-to-weight ratio (BWR), and mooring line angle, are thoroughly analyzed. The results indicate that as the submergence depth increases, the amplitude of the motion characteristics decreases significantly. For the SFT moored under the water surface, the motion response of sway, heave, and roll is much smaller than those of the SFT located on the water surface. Chen et al. [17] numerically investigated the effects of wave height, wave period, submergence depth, and BWR on the nonlinear hydrodynamic characteristics of the wave–SFT interaction based on 49 designed cases. The results show that the maximum motion response and mooring tension increase with wave height and wave period and decrease with submergence depth. Won et al. [18] investigated the hydrodynamic characteristics of the SFT under regular and irregular waves by time-domain hydrodynamic analysis. The analysis results show that the dynamic response of the SFT can be effectively reduced by deep draft design.

The above studies show that the effects of the mooring pattern and submergence depth on the dynamic response of the SFT are critical. Previous studies on optimal mooring patterns for the SFT mainly refer to regular wave loads, the maximum motion response, and the mooring tension in the time domain. In addition, previous studies on the submergence depth of the SFT mostly focus on the absolute submergence depth of a floater. In engineering, submerged floating structures, such as the SFT, are subject to both “absolute” and “relative” submergence depth. The absolute submergence depth is established to guarantee a specific navigable depth, while the relative submergence depth is determined to reduce the surface-wave-induced dynamic response. Theoretically, the relative submergence depth can be defined by the ratio of the absolute submergence depth d₀ to the wavelength L_P, represented as d₀/L_P. The wavelength is influenced by two factors: the water depth and wave period. This implies that the impact of relative submergence depth on dynamic response is essential to the effect of the wave period. Moreover, the relationship between the significant wave height and absolute submergence depth (represented as H_S/d₀, which can be defined as the relative wave height), as well as the relationship between the natural period of the mooring system and wave period, are also important factors influencing the dynamic response of the SFT.

In the present study, it is experimentally investigated that the mooring pattern and submergence depth influence the dynamic responses of the SFT. Based on the motion response and mooring tension derived from the tests, the mooring pattern was examined comprehensively in terms of three factors: the maximum value and spectral area of the motion response, the maximum value and uniformity of the mooring tension, and the length of time during which the cable reached a relaxed condition during wave action; moreover, the effects of submergence depth on the dynamic response of the SFT were investigated comprehensively in terms of four factors: the relative submergence depth, relative period, relative wave height, and KC number. Finally, the optimal mooring pattern and submergence depth for the SFT were determined.

2. Experiments

2.1. Experimental Equipment and Instruments

The model tests were carried out in a test tank at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The test tank was 60.0 m long, 2.0 m wide, and 1.8 m deep. The wave generation system was a Hydro-servo random wave maker system, which can generate waves with periods ranging from 0.5 s to 5.0 s. At the end of the test tank, multiple layers of energy dissipation equipment were installed for effective wave dissipation.

The motion responses of the SFT were measured by a non-contact 6-degree-of-freedom measurement system, consisting of dual-CCD (charge-coupled device) cameras and a data acquisition system. Three light markers were arranged on a plane and fixed on top of the SFT model to record the six-degrees-of-freedom motion components by the dual-CCD cameras. The motions of the light markers were continuously measured by the dual-CCD cameras at 30 frames per second and the signals were processed to recover the instantaneous position of each marker in a calibrated coordinate system. The mooring tension was measured by the tension sensor with an accuracy of 0.1 N. The wave heights were measured by the DS30 wave measuring system, which controlled 64 wave gauges synchronously. The wave gauges were calibrated before tests with an accuracy of 0.1 mm.

2.2. Model Parameters and Layout

The engineering background of the SFT in the test was the conceptual design of the SFT by the China Communications Construction SFT Technical Joint Research Team. The outer diameter of the SFT was 12 m and the interval between the mooring cables along the longitudinal direction of the tunnel was 120 m. It should be noted that the SFT was considerably long. In this study, a finite-length section (120 m) was intercepted and simplified into a rigid submerged horizontal cylinder for the two-dimensional experiment. The SFT model was made of organic glass. The model was designed in accordance with the Froude scaling law, and the geometric scale was determined as λ = 60 according to the test equipment conditions, the geometrical dimension of the model, and the boundary effect. The clump weights were placed to adjust the center of gravity (COG) and buoyancy weight ratio (BWR) of the SFT model. The length of the SFT model was designed to be 1.96 m according to geometric scale (the test tank width is 2.0 m) and six smooth universal balls were arranged on both ends of the SFT model to prevent the tunnel section from colliding with the flume side wall during the tests. The mechanical parameters of the prototype and model SFT are presented in

Table 1. Geometric and hydrodynamic parameters of the SFT (λ = 60).

Parameters	Symbols	Prototype	Unit	Model	Unit
Length	B	120.0	m	200.0	cm
Diameter	D	12.0	m	20.0	cm
Mass	G	11,232	t	52.0	kg
Centre of gravity	b	9.6	m	16.0	cm
Centre of buoyancy	b0	12.0	m	10.0	cm
Buoyancy weight ratio	BWR	1.20	-	1.20	-
Absolute submergence depth	d0	25.2	m	42.0	cm

.

Three mooring patterns of the SFT were selected, which are hereafter denoted as C_M1, C_M2, and C_M3 for convenience. The lower ends of the mooring lines were anchored to the bottom of the wave flume. The top ends together with tension sensors were connected to the circular cross-section, and each connecting point was 20 cm measured from the edge of the horizontal cylinders. The plan views of the three mooring patterns are the same as those shown in Figure 1a. The number of cables as well as the angle and position of the mooring lines of each mooring pattern were different, as shown in Figure 1b–d. The specific setups of the three mooring patterns are briefly described as follows:

Mooring pattern C_M1: There are three cables at either end of the horizontal cylinder, two cables are diagonal and one cable is vertical, where the diagonal cables are at 45° to the vertical, as shown in Figure 1b. The system is in equilibrium when the initial tension of each cable is F₀ = 17 N.

Mooring pattern C_M2: There are four cables at either end of the horizontal cylinder, two cables are diagonal and two cables are vertical, where the diagonal cables are at 45° to the vertical, as shown in Figure 1c. The system is in equilibrium when the initial tension of each cable is F₀ = 11 N.

Mooring pattern C_M3: There are four cables at either end of the horizontal cylinder, all the cables are diagonal, where the diagonal cables are at 45° to the vertical, as shown in Figure 1d. The system is in equilibrium when the initial tension of each cable is F₀ = 13 N.

The prototype mooring cable was a steel cable with a diameter of 174 mm, and the model mooring cable was simulated with a combination of wire rope, a fixed-length spring, and a unit counterweight. The fixed-length spring was configured to simulate the elongation of the mooring cable and the spring constant was determined from Equation (1) [19]. To simulate the model mooring cable, not only were the length and weight scaled, but the curve of mooring tension (T_m)–relative elongation (∆s) should also be matched. In order to accurately simulate the elasticity of mooring cables under the Froude similarity rule, the elastic characteristics of the prototype and model cables need to satisfy the following equation:

$$T_m=\frac{{C_P{d}_P}^2{(\Delta S/S)}^n}{{\lambda }^3}$$

(1)

where T_m is the mooring tension of the model cable (N), C_P is the elasticity coefficient of the prototype cable (for steel cable, C_P = 26.97 × 10⁴ MPa), d_P is the diameter of the prototype cable (m), ∆S/S is the relative elongation of the cable, and n is the index with the steel cable adopting n = 1.5. An example of a theoretical mooring tension–relative elongation curve and measured scatters for the case of C_M3 are presented in Figure 2. It shows that excellent agreement was achieved. Figure 3 shows the layout of the model SFT moored in the test tank.

The experimental coordinate system is shown in Figure 4. The origin of the coordinate system is located at the center of the model. Positive x is along the wave propagation direction, positive z is vertical up along the water depth, and positive y is determined by the right-hand rule. For the two-dimensional test, three motion components were considered: surge, heave, and pitch. Surge is the longitudinal motion component along the x-axis (wave propagation direction is +x); heave is the vertical motion component along the z-axis (vertical up is +z); and pitch is the rotation around the y-axis (clockwise direction is +y).

2.3. Experimental Contents and Parameters

The experimental contents included the free decay tests aiming to determine the natural period of the mooring system, RAO tests (under regular wave), and dynamic response tests under irregular waves. Free decay tests were first conducted on the SFT with three degrees of freedom (surge, heave, and pitch) to obtain the natural periods and damping coefficients. The RAOs of the SFT were obtained through regular wave tests under the same test conditions as the free decay tests (in terms of water depth, submergence depth, mooring pattern, mooring stiffness, and initial tension of the mooring cable). For the tests, the water depth d and submergence depth d₀ were 120 cm and 42 cm, respectively. Correspondingly, the dimensionless submergence depth d₀/d was 0.35. The RAOs of the SFT were obtained through regular waves with an average wave height of H = 4.0 cm and an average period of T = 0.8 s~2.0 s. Three types of mooring patterns (C_M1, C_M2, and C_M3) were considered. The experimental parameters are summarized in

Table 2. Experimental parameters for the RAO tests (under regular waves).

Mooring Pattern CMi	Initial Tension F0 (N)	Average Period T (s)	Wavelength L (m)	Relative Submergence Depth d0/L
CM1	17	0.8	0.99	0.360
		1.0	1.56	0.231
		1.2	2.24	0.161
		1.4	3.02	0.119
		1.6	3.84	0.094
		1.8	4.67	0.077
		2.0	5.49	0.066
CM2	11	0.8	0.99	0.360
		1.0	1.56	0.231
		1.2	2.24	0.161
		1.4	3.02	0.119
		1.6	3.84	0.094
		1.8	4.67	0.077
		2.0	5.49	0.066
CM3	13	0.8	0.99	0.360
		1.0	1.56	0.231
		1.2	2.24	0.161
		1.4	3.02	0.119
		1.6	3.84	0.094
		1.8	4.67	0.077
		2.0	5.49	0.066

. Based on the results of free decay tests and RAO tests, the dynamic responses under irregular waves were tested to further compare the optimal mooring pattern and submergence depth. The experimental parameters are summarized in

Table 3. Experimental parameters for mooring optimization (under irregular waves).

Mooring Pattern CMi	Initial Tension F0 (N)	Significant Wave Height HS (cm)	Spectrum Peak Period TP (s)	Relative Wave Height		Relative Period			Relative Submergence Depth d0/LP
				HS/d	HS/d0	TP/T0S	TP/T0H	TP/T0P
CM1	17	4.0	0.8	0.033	0.095	0.39	0.40	0.39	0.420
			1.0			0.49	0.50	0.49	0.269
			1.2			0.58	0.60	0.59	0.188
			1.4			0.68	0.70	0.68	0.139
			1.6			0.78	0.80	0.78	0.109
			1.8			0.87	0.90	0.88	0.090
			2.0			0.98	1.00	0.98	0.077
		1.0	1.6	0.008	0.024	0.78	0.80	0.78	0.109
		2.0		0.017	0.048
		3.0		0.025	0.071
		4.0		0.033	0.095
CM2	11	4.0	0.8	0.033	0.095	0.49	0.52	0.45	0.420
			1.0			0.62	0.65	0.56	0.269
			1.2			0.74	0.77	0.67	0.188
			1.4			0.86	0.90	0.79	0.139
			1.6			0.99	1.03	0.90	0.109
			1.8			1.11	1.16	1.01	0.090
			2.0			1.23	1.29	1.12	0.077
		1.0	1.6	0.008	0.024	0.99	1.03	0.90	0.109
		2.0		0.017	0.048
		3.0		0.025	0.071
		4.0		0.033	0.095
		5.0		0.042	0.120
CM3	13	4.0	0.8	0.033	0.095	0.67	0.67	1.60	0.420
			1.0			0.83	0.83	2.00	0.269
			1.2			1.00	1.00	2.40	0.188
			1.4			1.17	1.17	2.80	0.139
			1.6			1.33	1.33	3.20	0.109
			1.8			1.50	1.50	3.60	0.090
			2.0			1.67	1.67	4.00	0.077
		2.0	1.6	0.017	0.048	1.25	1.23	2.96	0.109
		3.0		0.025	0.071
		4.0		0.033	0.095
		5.0		0.042	0.119
		6.0		0.050	0.143
		7.0		0.058	0.167
		8.0		0.067	0.190

and

Table 4. Experimental parameters for relative submergence depth (under irregular waves).

Significant Wave Height HS (cm)	Spectrum Peak Period TP (s)	Relative Wave Height		Relative Period			Relative Submergence Depth d0/LP
		HS/d	HS/d0	TP/T0S	TP/T0H	TP/T0P
8.0	0.8	0.066	0.190	0.63	0.62	1.60	0.420
	1.0			0.78	0.77	2.00	0.269
	1.2			0.94	0.92	2.40	0.188
	1.4			1.09	1.08	2.80	0.139
	1.6			1.25	1.23	3.20	0.109
	1.8			1.41	1.38	3.60	0.090
	2.0			1.56	1.54	4.00	0.077
	2.2			1.72	1.69	4.40	0.067
	2.4			1.88	1.85	4.80	0.059
	2.6			2.03	2.00	5.20	0.054
2.0	1.6	0.017	0.048	1.25	1.23	3.20	0.109
3.0		0.025	0.071
4.0		0.033	0.095
5.0		0.042	0.119
6.0		0.050	0.143
7.0		0.058	0.167
8.0		0.067	0.190
9.0		0.075	0.214
10.0		0.083	0.238

. For submergence depth tests, the initial tension F₀ of each cable was 13 N. T_0s, T_0H, and T_0p are the natural periods of surge, heave, and pitch, respectively.

3. Results and Discussion

3.1. Free Decay Test and RAO Test Results

With the three mooring patterns (CM₁, CM₂, CM₃) as shown in Figure 1, the natural periods, damping coefficients, and RAOs of the SFT could be obtained and analyzed based on the free decay tests and RAO tests of the SFT. It should be noted that the hydrostatic equilibrium of the SFT was upheld by counterbalancing the vertical components of the mooring tension against the net buoyancy force of the SFT.

3.1.1. Natural Period and Damping Coefficient

The decay test was performed by applying an artificial displacement/rotation on the floater and releasing it in still water. The decay curve of the motion process had been completely collected until the moored SFT resumed to the static state. The natural periods and damping coefficients of surge, heave, and pitch were analyzed and are summarized in

Table 5. Natural period and damping coefficient of the SFT with different mooring patterns.

Mooring Pattern CMi	Surge		Heave		Pitch
	Natural Period (s)	Damping Coefficient	Natural Period (s)	Damping Coefficient	Natural Period (s)	Damping Coefficient
CM1	2.06	0.09	2.01	0.10	2.05	0.15
CM2	1.62	0.08	1.55	0.11	1.78	0.16
CM3	1.28	0.11	1.30	0.09	0.54	0.22

. Theoretically, the larger the stiffness of the mooring system, the smaller its natural period is. The natural periods of the three mooring systems were in the order of CM₁ > CM₂ > CM₃. The natural periods of surge and heave for mooring pattern CM₃ were approximately 50% of that for mooring pattern CM₁, and the natural period of pitch was only approximately 25% of that for mooring pattern CM₁. In addition, the natural period of the pitch was significantly shorter than those of the surge and heave, indicating that mooring system CM₃ had a strong constraint effect on the pitch. Table 5 shows that no considerable differences were observed in terms of the damping coefficients of surge and heave for three mooring patterns. However, considerable differences were observed in the damping coefficient of pitch. The damping coefficient of pitch for mooring pattern CM₃ was larger than those of mooring patterns CM₁ and CM₂, with a difference of approximately 30%.

3.1.2. Response Amplitude Operators (RAOs)

The RAOs of the SFT under different mooring patterns were obtained through regular wave tests. The water depth was fixed at d = 1.2 m and the submergence depth d₀ = 42 cm (d₀/d = 0.35, d₀/D = 2.1, where D is the diameter of the horizontal cylinder). The wave height in the test was selected as H = 4.0 cm, with average period T = 0.8~2.0 s. The period interval for each test was 0.2 s. Due to the limitation of the wave maker, the test period cannot cover the natural period of pitch under mooring pattern C_M3. Figure 5 shows the RAO test results of the SFT under three mooring patterns. The abscissas of the RAO curves for surge, heave, and pitch were dimensionless with their respective natural periods, and the abscissas of the RAO curves for mooring tension were dimensionless with the natural period of surge. The ordinate represents the dynamic response values under unit height waves. Max and Min RAO represent the maximum positive value and minimum negative value of the corresponding motion response in the RAO tests, respectively.

As shown in Figure 5, in terms of mooring patterns C_M1 and C_M2, the maximum surge, heave, and pitch of the SFT occur at their natural periods and the RAO variations of the three motion components were similar. In terms of mooring pattern C_M3, the maximum surge and heave occurred at their natural periods and the RAO variations of the three motion components were similar. However, within the range of the wave period in the experiment, the maximum pitch occurred at T_p/T_0P ≈ 2.0, twice the natural period of pitch. The maximum pitch occurred almost synchronously with the maximum surge and heave (T_p/T_0S ≈ 1.0, T_p/T_0H ≈ 1.0, T_p/T_0P ≈ 2.0), indicating a significant coupling effect among the natural periods of the three motion components. Simultaneously, the RAO variations in the mooring tension were nearly identical to those of the surge, with the maximum mooring tension appearing near T_p/T_0S ≈ 1.0. The RAOs of all tested dynamic response parameters under mooring pattern C_M3 were significantly shorter than those under C_M1 and C_M2. In other words, mooring pattern C_M3 effectively restricted the motion response of the SFT, while the mooring tension was also less than those of C_M1 and C_M2. This shows that mooring pattern C_M3 was optimal in terms of RAOs.

3.2. Optimization Analysis of the Mooring Pattern

Section 3.1 describes the RAO tests and the results showed that of the mooring patterns, C_M3 was the most optimal. Next, the advantages and disadvantages of the three mooring patterns were further comprehensively investigated through irregular wave tests in terms of three factors: the maximum value and spectral area of the motion response, the maximum value and uniformity of the mooring tension, and the length of time during which the cable reached a relaxed condition during wave action.

3.2.1. Motion Response of the SFT under Three Mooring Patterns

(1)

The maximum motion response

For the tests, the water depth was fixed at d = 1.2 m and the submergence depth d₀ = 42 cm (d₀/d = 0.35, d₀/D = 2.1). Two irregular wave series were generated, one wave series with the characteristics of T_P = 1.6 s and Hs/d₀ = 0.02~0.19, and the second wave series with the characteristics of Hs = 4.0 cm (Hs/d₀ = 0.10) and T_P = 0.8~2.0 s. The detailed wave parameters in the experiments are summarized in

Table 6. The 0th-order moments (spectral areas) of the motion response varying with the relative wave height.

Significant Wave Height HS (cm)	Relative Wave Height HS/d	Spectrum Peak Period TP (s)	Surge m0 (cm2)			Heave m0 (cm2)			Pitch m0 (deg2)
			CM1	CM2	CM3	CM1	CM2	CM3	CM1	CM2	CM3
2.0	0.017	1.6 s	191.84	0.46	0.06	9.83	0.07	0.00	138.68	0.74	0.02
3.0	0.025		347.69	15.98	0.06	20.11	1.09	0.01	252.20	18.51	0.03
4.0	0.033		453.88	53.41	0.13	28.21	3.34	0.02	325.42	52.69	0.05

.

Figure 6 shows the statistical characteristic values of the surge, heave, and pitch of the SFT induced by the irregular wave versus the relative wave height under three mooring patterns. Figure 7 shows the statistical characteristic values of the surge, heave, and pitch of the SFT induced by the irregular wave versus the relative period under three mooring patterns. In Figure 6, the surge, heave, and pitch of the SFT are all significantly correlated with the relative wave height. The positive and negative maximum values of each motion component increase nonlinearly as the relative wave height increases under three mooring patterns. Moreover, the maximum motion responses under three mooring patterns remain in the order of C_M1 > C_M2 > C_M3. In addition, Figure 6 shows that mooring pattern C_M3 constrains the heave of the 2D SFT relatively well, even though all the cables of the mooring system are diagonal, where the diagonal cables are at 45° to the vertical, as shown in Figure 1d. Mooring pattern C_M3 not only has an overwhelming advantage over the other two mooring patterns in limiting the surge and pitch of the 2D SFT, but it also has a better constraint on heave than the C_M1 and C_M2 mooring patterns. It should be noted that regardless of the presence or absence of vertical cables, all the three mooring patterns show a tendency in which the heave of the 2D SFT increases significantly with an increase in H_S/d₀ after a certain relative wave height is reached.

In Figure 7, a comparison of the statistical characteristic values of the motion response versus the relative period under three mooring patterns revealed that C_M3, for which the range of test periods was approximately 1.5~4.0 times the natural period of the motion components, had an overwhelming advantage over the other two mooring patterns in terms of restraining the motion response.

(2)

Spectral area of the motion response

The statistical characteristic values of the motion response of the 2D SFT versus relative wave height and relative period under three mooring patterns were previously discussed. In this section, based on the measured time histories, spectral analyses were performed to obtain the frequency-domain characteristics of the motions of the SFT. The spectral areas for the three types of mooring patterns were compared for the global analysis of motion responses.

Table 6 and

Table 7. The 0th-order moments (spectral areas) of the motion response varying with the relative period.

Significant Wave Height HS (cm)	Relative Wave Height HS/d	Spectrum Peak Period TP (s)	Surge m0 (cm2)			Heave m0 (cm2)			Pitch m0 (deg2)
			CM1	CM2	CM3	CM1	CM2	CM3	CM1	CM2	CM3
4.0	0.033	0.8	4.49	5.30	0.04	0.21	0.21	0.01	3.01	4.13	0.01
		1.0	13.83	12.71	0.13	0.65	0.53	0.04	9.25	12.61	0.07
		1.2	55.54	31.02	0.14	2.64	1.28	0.03	37.77	30.96	0.06
		1.4	231.05	48.01	0.14	12.86	1.79	0.02	161.73	46.12	0.05
		1.6	478.50	56.70	0.13	29.72	2.93	0.02	341.80	56.09	0.05
		1.8	672.05	57.10	0.12	44.77	3.01	0.01	493.60	57.24	0.03
		2.0	801.98	28.25	0.09	60.93	1.82	0.01	626.99	30.75	0.03

list the 0th-order moments (spectral areas) of each motion response component of the 2D SFT under three mooring patterns varying with the relative wave height (spectral peak period is fixed) and relative periods (significant wave height is fixed), respectively.

From Table 6 and Table 7, it can be seen that the 0th-order moments (spectral areas) of the motion response are in the order of C_M1 > C_M2 > C_M3 from largest to smallest. For mooring pattern C_M3, the 0th-order moments (spectral areas) for all three motion components were significantly smaller than those of the other two mooring patterns and the difference was particularly significant for large wave actions.

Combining the test results in Figure 6 and Figure 7 and Table 6 and Table 7, it can be concluded that C_M3 was the optimal pattern of the three mooring patterns in terms of restraining the motion response of the 2D SFT.

3.2.2. Mooring Tension and Its Uniformity of Distribution under Three Mooring Patterns

In this section, the mooring tension of the upstream (1#) and downstream (3# and 4#) cables of the SFT were selected as representatives for discussion. The mooring tensions of the other cables were described in the uniformity of the mooring tension. The test conditions were the same as those described in Section 3.2.1. Figure 8 shows the statistical characteristic values of the upstream (#1) and downstream (#4) mooring tension of the SFT induced by the irregular wave versus the relative wave height under three mooring patterns. Figure 9 shows the statistical characteristic values of the upstream (#1) and downstream (#4) mooring tension of the SFT induced by the irregular wave versus the relative period under three mooring patterns.

Figure 8 and Figure 9 show that the maximum mooring tension of mooring patterns C_M1 and C_M2 were generally the same and the maximum mooring tension transferred from each other alternatively with relative wave height and relative period. However, both mooring patterns yielded a larger mooring tension as compared with mooring pattern C_M3.

Table 8. The 0th-order moments (spectral areas) of upstream (#1) mooring tension varying with the relative wave height.

Significant Wave Height HS (cm)	Relative Wave Height HS/d	Spectrum Peak Period TP (s)	Mooring Tension m0 (N2)
			CM1	CM2	CM3
2.0	0.017	1.6	56.70	68.03	13.00
3.0	0.025		107.45	192.88	29.71
4.0	0.033		156.73	295.10	54.25

and

Table 9. The 0th-order moments (spectral areas) of upstream (#1) mooring tension varying with the relative periods.

Significant Wave Height HS (cm)	Relative Wave Height HS/d	Spectrum Peak Period TP (s)	Mooring Tension m0 (N2)
			CM1	CM2	CM3
4.0	0.033	0.8	8.97	7.22	8.29
		1.0	17.13	136.02	50.65
		1.2	28.30	229.30	56.72
		1.4	75.54	269.19	56.73
		1.6	161.53	312.87	53.97
		1.8	255.70	314.55	47.12
		2.0	407.27	269.14	41.49

list the 0th-order moments (spectral areas) of upstream (1#) mooring tension of the SFT under three mooring patterns varying with the relative wave height (spectral peak period is fixed) and relative periods (significant wave height is fixed), respectively. Table 8 and Table 9 show that the 0th-order moments (spectral areas) of upstream (1#) mooring tension of mooring pattern C_M3 were significantly smaller than those of the other two mooring patterns.

The uniformity of the mooring tension is a major factor in determining the optimal mooring pattern. For the two-dimensional case, the cables at either end of the horizontal cylinder were basically the same. Therefore, the mooring tension of all cables at one end was selected for discussion. The dimensionless mooring tensions were used to account for the uniformity of the mooring tension, expressed as F_r#/F_i#, where i# denotes the cable with the minimum mooring tension for most cases at one end, and r# denotes the other cables. Figure 10 and Figure 11 show the dimensionless mooring tensions versus the relative wave heights and relative periods under three mooring patterns, respectively.

From Figure 10 and Figure 11, the dimensionless mooring tensions (F_r#/F_i#) of all the cables under three mooring patterns were C_M1: 1.0~1.36, C_M2: 0.8~1.37, and C_M3: 1.0~1.12. The results reveal that the uniformity of the mooring tension for mooring pattern C_M3 was better than those for the other two mooring patterns.

3.2.3. The Relaxed Time of the Cables under the Three Mooring Patterns

Figure 12, Figure 13 and Figure 14 show the time history of the upstream (1#) mooring tension under C_M1, C_M2, and C_M3; the vertical (2#) mooring tension under C_M1 and C_M2; and the downstream (C_M1: 3#, C_M2 and C_M3: 4#) mooring tension under C_M1, C_M2, and C_M3. From Figure 12, Figure 13 and Figure 14, the cables under mooring pattern C_M2 exhibited the longest relaxed time, while the relaxed time for mooring pattern C_M3 was shortest. The relaxed time for mooring pattern C_M1 was centered among three types of mooring patterns. The relaxed time means the length of time during which the cable reached a relaxed condition during wave action. It should be noted that when the 2D SFT moved vertically downward, the cable reached a relaxed state; therefore, compared with mooring pattern C_M1, an additional vertical cable was not more effective in constraining the vertically downward movement of the 2D SFT. Furthermore, the vertical combined initial tension was equal to the net buoyancy of the 2D SFT, and the initial tension of the cables under C_M2 (F₀ = 11 N) was less than that of the cables under C_M1 (F₀ = 17 N), which makes the cables more susceptible to a relaxed state.

In summary, according to the comparison of three types of mooring patterns in terms of the maximum value and spectral area of the motion response, the maximum value and uniformity of the mooring tension, and the relaxed time of the cables, mooring pattern C_M3 was determined to be the optimal mooring pattern.

3.3. Optimization Analysis of the Submergence Depth

As mooring pattern C_M3 was determined to be the best of the three mooring patterns, in this section, the correlation between the dynamic responses of the SFT and the dimensionless parameters (the dimensionless submergence depth, period and wave height, KC number) for mooring pattern C_M3 is further investigated and an effective method is subsequently devised for determining the optimal submergence depth in the engineering context.

3.3.1. Variation in the Dynamic Response with Relative Submergence Depth d₀/L_P

For the tests, the water depth was fixed at d = 1.2 m and the submergence depth d₀ = 42 cm (d₀/d = 0.35, d₀/D = 2.1). The irregular wave series were generated with the characteristics of the spectrum peak periods T_P = 1.0~2.6 s (the relative periods T_P/T_0S = 0.63~2.03, T_P/T_0H = 0.62~2.00, T_P/T_0P = 1.60~5.20; the relative submergence depth d₀/L_P = 0.05~0.27) and the significant wave heights Hs = 4.0 cm and 8.0 cm (the relative wave height Hs/d₀ = 0.10 and 0.2). The detailed wave parameters in the experiments are summarized in Table 4.

Figure 15 shows the statistical characteristic values of the surge, heave, pitch, and upstream (1#) mooring tension of the SFT induced by the irregular wave versus the relative submergence depth d₀/L_P. According to the analytical solution, increasing draft can sufficiently restrain the motion response and tether stress of the SFT [20]. Consistently, the present experimental results suggest that when the relative wave height is smaller (Hs/d₀ = 0.10), the motion response of the SFT remains basically unchanged and maintains a stable and small value within the range of the relative submergence depths in the experiment, where the surge is less than 0.015D (D denotes the diameter of the horizontal cylinder), the heave is less than 0.01D, and the pitch is less than 0.3°. The mooring tension of the cable oscillates slightly in the range of d₀/L_P ≤ 0.15, with the oscillation amplitude ranging from 1.7 to 2.0F₀ (F₀ denotes the initial tension F₀ = 13 N), and then gradually decreases to a smaller value (1.7F₀) as the relative submergence depth increases.

As the relative wave height increases to Hs/d₀ = 0.20, the dynamic response characteristic can be distinguished into two parts bounded by d₀/L_P ≈ 0.15 within the range of the relative submergence depth in the experiment. Within the interval of d₀/L_P ≤ 0.15, all the dynamic response components oscillated considerably, with relatively high peak values occurring in the interval of d₀/L_P = 0.07~0.10 (where the surge is 0.25D, heave is 0.10D, pitch is 2.8°, and mooring tension is 2.7F₀). Within the interval of d₀/L_P > 0.15, the dynamic response gradually decreases to a relatively stable value with an increase in relative submergence depth (where the surge is basically stable at 0.07D, heave is basically stable at 0.03D, pitch is basically stable at 0.9°, and mooring tension is basically stable at 2.0F₀). The present experimental results suggest that there exists an “appropriate” relative submergence depth where the dynamic responses of the SFT are minimal for a certain Hs/d₀. Notably, under this submergence depth, the dynamic responses remain unaffected by the variation in the wave period. In the experiment, when Hs/d₀ = 0.2, the “appropriate” relative submergence depth is d₀/L_P ≥ 0.15.

3.3.2. Variation in the Dynamic Response with Relative Period

The relative submergence depth can be defined by the ratio of the absolute submergence depth to the wavelength, represented as d₀/L_P. The wavelength is influenced by the water depth and wave period. This implies that the impact of relative submergence depth on dynamic response is essentially the wave period.

In this section, the ratio of spectrum peak period T_P and the natural period of surge T_0S (which is also the natural period of the heave T_0H) is taken as the abscissa, and the dynamic response of the SFT induced by the irregular wave versus the relative period under the experimental conditions consistent with Section 3.3.1 are graphed in Figure 16.

As shown in Figure 16, when the relative wave height is smaller (Hs/d₀ = 0.10), the motion response of the SFT remains basically unchanged and maintains a stable and small value within the range of the relative period in the experiment, where the surge is less than 0.015D, the heave is less than 0.01D, and the pitch is less than 0.3°. The mooring tension of the cable oscillates slightly in the range of the relative period in the experiment, with the oscillation amplitude ranging from 1.7 to 2.0F₀ (F₀ denotes the initial tension F₀ = 13 N).

With the relative wave height increasing to Hs/d₀ = 0.20, both the motion response and mooring tension exhibit a notable peak at T_P/T_0S ≈ 1.4. The dynamic response exhibits significant fluctuations within the relative period range in the experiment. Notably, the maximum motion response and mooring tension did not appear at the natural frequency of the surge, heave, and pitch (T_P/T_0S = 1.41, T_P/T_0H = 1.38, T_P/T_0P = 3.60). This discrepancy could be due to the combined influences of the system’s natural frequency and the vortex release frequency.

3.3.3. Variation in the Dynamic Response with Relative Wave Height H_S/d₀

The relationship between wave height and absolute submergence depth, represented as H_S/d₀, is also an important influencing factor for the dynamic response of the SFT. In this section, the statistical characteristic of the dynamic response of the SFT varying with the relative wave height H_S/d₀ under irregular waves was investigated.

For the tests, the water depth was fixed at d = 1.2 m and the submergence depth was d₀ = 42 cm (d₀/d = 0.35, d₀/D = 2.1). The irregular wave series were generated with the characteristics of T_P = 1.6 s and Hs/d₀ = 0.02~0.19. Figure 17 shows the statistical characteristic of the surge, heave, pitch, and mooring tension of the SFT varying with the relative wave height H_S/d₀ under irregular waves. In Figure 17, the surge, heave, and pitch of the SFT are all significantly correlated with relative wave height H_S/d₀. The positive and negative maximum values of each motion component increase nonlinearly as the relative wave height increases. As shown in Figure 17, when the relative wave height H_S/d₀ is smaller (Hs/d₀ ≤ 0.10), the motion response of the SFT remains basically unchanged and maintains a stable and small value within the range of relative wave height H_S/d₀, where the surge is less than 0.06D (D denotes the diameter of the horizontal cylinder), the heave is less than 0.02D, and the pitch is less than 0.6°. With the relative wave height Hs/d₀ > 0.10, the motion response increases nonlinearly as the relative wave height increases, which indicates that the relative wave height H_S/d₀ has significant effects on the motion responses. In addition, Figure 17 shows that the mooring system constrains the heave of the SFT well, even though all the cables are diagonal, where the diagonal cables are at 45° to the vertical, as shown in Figure 1d. Moreover, both the upstream and downstream mooring tensions of the SFT increase almost linearly with the increase in relative wave height H_S/d₀. Meanwhile, the upstream mooring tension is slightly larger than that of the downstream.

3.3.4. Variation in the Dynamic Response with KC Number

Considering that the KC number can account for the wave height, wave period, and absolute submergence depth, it is introduced into this study to investigate the relationship between the dynamic response of the 2D SFT and KC number. The KC number is a key parameter for analyzing the hydrodynamic problems of the cylinder [21]. The KC number is defined as [22]

$$KC=\frac{u_{max}T}{D}$$

(2)

$$u\left(t\right)=\frac{\pi H}{T}\frac{coshkz}{sinhkd}\mathrm{c}\mathrm{o}\mathrm{s}(kx-wt)$$

(3)

where u_max indicates the maximum horizontal velocity at the water level calculated by linear wave theory. H, T, k, and d denote the wave height, wave period, wave number, and water depth, respectively. z is the vertical coordinate with the bottom of the water z = 0 and the still water surface z = d. In the present study, the relevant parameters are specified as follows: the wave height is considered as the maximum wave height in the wave sequence; the period is considered as the spectrum peak period; and the location z is considered as the center of the SFT (z = d − d₀ − D/2, where d₀ represents the absolute submergence depth, d represents the water depth, and D represents the diameter of the horizontal cylinder).

Figure 18 shows the statistical characteristic of the surge, heave, pitch, and upstream (1#) mooring tension of the SFT induced by the irregular wave versus the KC number.

As shown in Figure 18, the variation in the surge, heave, pitch, and upstream (1#) mooring tension of the SFT with KC number is basically opposite to the variation in the dynamic response with relative submergence depth d₀/L_P. This is due to the fact that as the relative submergence depth d₀/L_P increases, the KC number decreases. Hence, opposite variations occur in the curves of the dynamic response with d₀/L_P and KC number as the abscissas.

Thus, a similar conclusion can be drawn: for a certain Hs/d₀, there exists an “appropriate” KC number, with which the dynamic response of the SFT varies significantly. In the experiment, the “appropriate” KC number is KC ≤ 0.8 for Hs/d₀ = 0.2.

4. Conclusions

The present study focused on the anchor-cable SFT, and a series of experiments were conducted under regular and irregular waves. Based on the motion response and mooring tension derived from the tests, the effects of mooring pattern and submergence depth were examined comprehensively. Subsequently, the optimal mooring pattern and the submergence depth for the SFT were then determined. The main conclusions are summarized as follows:

(1)

For the mooring system along the SFT, the pattern of diagonal cables is better than that of the diagonal cables + vertical cables. The 0th-order moments of the dynamic response are in the order of C_M1 > C_M2 > C_M3 from largest to smallest. The uniformity of the mooring tension of all the cables under mooring pattern C_M3 is better than those of the other two mooring patterns. The length of the relaxed time for mooring pattern C_M2 is longest, while that for mooring pattern C_M3 is shortest.

(2)

For the SFT subjected to waves, there is an “appropriate” relative submergence depth under which the dynamic responses remain relatively stable and are unaffected by the variation in the wave period. When the relative wave height is smaller (Hs/d₀ = 0.10), the motion response of the SFT remains basically unchanged and maintains a stable and small value within the range of relative submergence depths. With the relative wave height Hs/d₀ = 0.20, the “appropriate” submergence depth is d₀/L_P > 0.15. In the range of d₀/L_P > 0.15, the dynamic response gradually decreases to a relatively stable value with an increase in relative submergence depth. Under the same condition, if the KC number is used to determine the “appropriate” submergence depth, the KC number corresponds to KC ≤ 0.8.

(3)

With the relative wave height Hs/d₀ = 0.20, both the motion response and mooring tension exhibit a notable peak at T_P/T_0S ≈ 1.4. The dynamic response exhibits significant fluctuations within the relative period range. Notably, the maximum motion response and mooring tension do not appear at the natural frequency of the surge, heave, and pitch (T_P/T_0S = 1.41, T_P/T_0H = 1.38, T_P/T_0P = 3.60). This discrepancy could be due to the combined influences of the system’s natural frequency and the vortex release frequency.