Experimental and Theoretical Study on the Stability of Submarine Cable Covered by Articulated Concrete Mattresses on Flat Hard Seabed Under the Action of Currents

Abstract

The safe and stable operation of submarine cables is a critical issue in offshore wind power engineering. This study presents an experimental and theoretical study on the stability of submarine cable protected by a sleeve (SCPS) with Articulated Concrete Mattresses (ACMs) protection on a flat hard seabed under current conditions. The instability modes of the SCPS–ACMs were identified, and the effects of the number of spans, cover spacing, and ACMs length on the critical instability velocity were investigated. The experimental results indicate that the primary instability mode of the SCPS–ACMs is the overall slip mode. An increase in cover spacing enlarges the exposure scale of the SCPS in the flow environment, thereby reducing the critical velocity. Employing at least two spans effectively mitigates the boundary effect induced by the flow past the SCPS at its ends, thus ensuring the reliability of the experimental model. The critical velocity is fundamentally determined by the dimensionless parameter—the ACMs coverage ratio (incorporating both the ACMs length and cover spacing). Based on the experimental results and force analysis, a theoretical equation reflecting the intrinsic relationship between the ACMs’ cover spacing and critical velocity was established. Key parameters in the equation, such as the friction coefficient, hydrodynamic coefficients (including the lift coefficient and drag coefficient), and weight distribution coefficients, were determined. Finally, the theoretical results were validated against the experimental data, showing a good agreement and verifying the reliability of the theoretical formula. The findings of this research can provide crucial support for the optimal design of ACMs protection schemes for submarine cables on the hard seabed.

1. Introduction

Driven by the dual carbon goal, offshore wind power has become a core field in the global development of new energy [1,2]. As the only energy transmission carrier connecting offshore wind farms to onshore power grids, submarine cable has its service stability directly determining the power generation efficiency, grid-connection reliability, and full-life-cycle benefits of wind power projects [3,4]. Existing studies have shown that the core causes of submarine cable failures include the cable’s own instability driven by environmental loads [5], the instability of the protective structures [6], and local seabed liquefaction [7]. Particularly, in flat hard seabed environments, the back-and-forth friction between the submarine cable and seabed causes severe sheath wear, which seriously affects the safe operation of the cable [8]. Therefore, revealing the stability regulation mechanism of submarine cables and improving the design level of protective measures have become research hotspots in the field of marine engineering [9].

To enhance the lateral stability of submarine cables, the engineering community commonly adopts the following measures: installing cast iron sleeves to form local rigid protection while increasing the cable’s self-weight [10]; and using trench burial to place the cable in soil to resist the direct action of waves and currents [11]. In addition, other stabilization methods include anchoring, overweight clamps, ACMs, and rock dumping [12,13]. Existing studies have verified the effectiveness of trench burial through numerical models and laboratory tests: Sudhan et al. [14] found that, when a pipeline is laid with a specific burial depth ratio, the wave-induced pressure on the pipeline is significantly reduced; Sun et al. [15] further confirmed through tests that, when the trenching depth exceeds 1.5 times the cable diameter, the impact of the wave-current drag force on the submarine cable can be neglected. However, the compressive strength of rock masses in the flat hard seabed (such as reefs and weathered rocks) generally exceeds 30 MPa, making it difficult for traditional trenching equipment to excavate, and an effective burial depth cannot even be achieved. Moreover, protrusions on the rock surface easily form point-contact friction with the cable body, and the wear rate is 4–6 times higher than that of soft soil foundations [16]. Thus, there is an urgent need to develop targeted alternative protection technologies.

To address the challenges of the high strength and difficult trenching in the flat hard seabed, the engineering sector often adopts the basic scheme of “direct submarine cable laying with local cast iron sleeve protection”, where the sleeve resists direct wear from reef protrusions [17]. However, water flow loads cause the periodic oscillation of the exposed cable segments, leading to relative slip between the sleeve and the cable body and accelerating secondary wear of the sheath [5]. Meanwhile, the cable body lacking planar constraints is prone to lateral displacement under strong currents—an offshore wind power project in the East China Sea once suffered a 500 m deviation of the submarine cable from its designed path due to this issue, triggering a safety hazard of collision with anchor chains [8]. To make up for the shortcomings of the basic scheme, the ACMs technology has been introduced as a secondary stabilization measure: it is a flexible planar structure formed by hinging concrete blocks with steel cables, which can cover the surface of the submarine cable to form integral protection [18].

Existing studies have initially revealed the mechanical properties and failure laws of ACMs: Gaeta et al. [19] obtained the drag coefficient and lift coefficient of ACMs through 1:20 scale model tests, providing the basic parameters for the stability calculation; McLaren et al. [20] conducted flow field tests on ACMs with different block sizes and found that a 20% increase in the block side length can improve the anti-slip capacity by 35%; Godbold et al. [21] clarified that the core failure mechanism of ACMs is the rolling instability of the edge blocks caused by the water flow lift, which further leads to overall rolling. Due to the conservative design considerations, the full-length coverage of submarine cables with ACMs is commonly used in engineering practice, resulting in high protection costs. However, the feasibility of the spaced covering mode lacks systematic theoretical support [22]. At the same time, for the special boundary of the flat hard seabed, the synergistic mechanism of SCPS–ACMs and the influence law of the cover spacing on the stability of the SCPS–ACMs have not been clarified, which restricts the economical application of this technology. This indicates that the current ACMs protection design not only has room for cost optimization but also lacks targeted theoretical support.

This study mainly focuses on the stability of uncovered submarine cable on the flat hard seabed under the action of currents. Therefore, the local scour and the actions of the combined waves and currents [23] are not considered. Based on physical model experiment and theoretical analysis methods, the stability of SCPS covered by ACMs is investigated. The core contents include the following: revealing the synergistic force transfer and instability mechanism between ACMs and SCPS; and establishing a quantitative relationship between the cover spacing and the critical instability velocity of SCPS-ACMs. This study aims to fill the research gap in the stability of SCPS covered by ACMs at intervals on the flat hard seabed.

2. Flume Experiment

2.1. Experimental Setup

The physical model experiment was carried out in a large basin (30 m × 11 m × 1.5 m) at Hohai University. The flat hard seabed was set in the middle area of the basin, with a length of 2.5 m and a width of 4 m, and the overall layout is shown in Figure 1. The basin is equipped with an open-boundary multi-pump-controlled tidal current simulation system, which consists of multiple 32-channel open-boundary model multi-pump control cabinets, submersible pumps, water level and flow velocity measuring instruments, and other equipment, enabling the simulation of a uniform flow. In this study, a unidirectional uniform flow was adopted, with a water depth of h = 0.35 m.

During the physical test setup and data collection process, an Acoustic Doppler Velocimeter (ADV, HR 2MHz, manufactured by Nortek, Rud, Norway) and a high-speed camera (FASTCAMMiniWX50, manufactured by Photron, Tokyo, Japan) were used to simultaneously observe and statistically analyze the flow velocity and the motion response of the model.

2.2. Measuring Instrumentation

The probe of the ADV was installed 1.25 m upstream of the SCPS to monitor the far-field incoming flow velocity at a position 0.1h above the flat hard seabed. The height of this monitoring point is close to the top height of the ACMs, serving as the reference elevation for the critical velocity of the SCPS. The data was transmitted to the remote computer of the data acquisition system at a sampling frequency of 16 Hz. The high-speed camera was used to capture the movement patterns of the models during the experiment, facilitating a comprehensive analysis of the motion images.

2.3. Properties of Models

The prototype submarine cable is of 500 kV specification, with an outer diameter of 302 mm and a mass per unit length of 151.49 kg/m; it is externally equipped with a half-type spherical hinge ductile iron protective sleeve, which has a maximum outer diameter of 460 mm and a mass per unit length of 154 kg/m. A single piece of ACMs is 2700 mm long and 4350 mm wide, containing 40 concrete blocks with a mass of approximately 172 kg per block. The test model was designed based on Fr similarity (Froude Number Similarity), with a geometric similarity ratio $\lambda$ of 1:25. The solid density similarity ratio ${\lambda }_{\mathsf{\rho }}$ and gravitational acceleration similarity ratio ${\lambda }_{\mathrm{g}}$ between the prototype and the model were both set to 1. PC rigid pipes were used to make the integral model of SCPS. During the preliminary preparation of the experiments, the potential error introduced by the rigid assumption was one of the issues we considered. In practical engineering, small rotational movements between SCPS–ACMs units are indeed allowed. However, our investigation shows that the flexural stiffness of the SCPS sleeve is relatively high, and its joints are fixed with special locking parts, resulting in a very small actual rotation amplitude, which can be regarded as a “quasi-rigid” structure. Therefore, the interaction between the sleeve and the cable was ignored, and the weights of the two were superimposed and assigned to the integral model. To make the model reach the target mass, a homogeneous mixture of quartz sand and fine lead particles was poured into the pipe, and manual vibration was used to ensure the dense filling of the contents. Polypropylene ropes were used to connect the concrete blocks in series to make the ACMs model. Specific model parameters are shown in

Table 1. Main parameters of SCPS and ACMs.

Parameter		Value		Unit
		Prototype	Model
SCPS	External diameter, D	460	18.4	mm
	Mass per unit length (in air), ms	305.49	0.489	kg/m
	Mass ratio, 4ms/(πρD2)	1.92	1.92
ACMs	Concrete block length/width	500	20	mm
	Concrete block height	300	12	mm
	Spacing between blocks	50	2	mm
	Total length, Lm	2700	108	mm
	Total width, Bm	4350	174	mm
	Density, ρm	2291	2291	kg/m3

, and the actual scale model is shown in Figure 2.

2.4. Experimental Conditions and Procedures

Under the model flow velocities of 0.2–0.59 m/s, the Reynolds number of the SCPS ranges from 3.6 × 10³ to 1.06 × 10⁴, which falls within the subcritical flow regime. For the prototype SCPS under engineering flow velocities of 1.0–2.95 m/s, the Reynolds number can reach 4.4 × 10⁵ to 1.3 × 10⁶, corresponding to a supercritical flow regime [24].

To reduce the boundary effect caused by the flow past the ends of the SCPS in this study, we compared the influence of different numbers of spans of ACMs on the stability results. The experiment set the following working conditions: two pieces of ACMs with interval covering (-L-L-), three pieces of ACMs with interval covering (-L-L-L-), and six pieces of ACMs with interval covering (-L-L-L-L-L-L-). In practical engineering, continuous covering with two or more adjacent ACMs can improve the construction efficiency; therefore, this experiment additionally conducted a study on the combined covering configuration with two pieces of ACMs (-2L-2L-2L-). The specific layout is shown in Figure 3, and different covering spacings were set for each covering form; the arrangement of test conditions is shown in

Table 2. Test conditions.

Case	(ACMs) Cover Mode	ACMs Spacing, d (d = nLm, Lm = 10.8 cm)	SCPS Total Length, Lc (cm)
A1	-L-L-	3Lm	86.4
A2		4Lm	108
A3		5Lm	129.6
A4		6Lm	151.2
A5		7Lm	172.8
A6		8Lm	194.4
B1	-L-L-L-	3Lm	129.6
B2		4Lm	162
B3		5Lm	194.4
B4		6Lm	226.8
B5		7Lm	259.2
B6		8Lm	291.6
C1	-L-L-L-L-L-L-	3Lm	259.2
C2		4Lm	324
C3		5Lm	388.8
D1	-2L-2L-2L-	6Lm	259.2
D2		8Lm	324
D3		10Lm	388.8

, involving a total of 18 working conditions. Each working condition was tested at least three times, and the average value was finally taken as the result of that working condition.

Take the SCPS within half the covering spacing on both sides of each piece of ACMs as a “SCPS–ACMs unit” (as shown in Figure 3). The stability of a single SCPS–ACMs unit indicates that the overall structure of the SCPS is stable. Therefore, under different covering intervals, the overall length of the SCPS model will be adjusted accordingly.

The test processes are as follows: for flow parameters, the flow velocity at a depth of 0.1h from the flat hard seabed was selected as the monitoring flow velocity; each group of tests was conducted at three different positions in the test area to eliminate the error caused by the roughness variation; and each stability test started from a still water state, and the flow velocity was gradually and slowly increased until the model started to move. The flow velocity corresponding to this moment is the critical instability velocity, and the process and data were recorded simultaneously. Through the multi-pump control system, the flow velocity at the monitoring point can reach a maximum of 0.59 m/s. The quantitative criterion for instability is zero displacement, which meets the condition of absolute stability. It is specifically defined as follows: “when the model first exhibits measurable non-reciprocating net displacement from a static state (displacement = 0), the system is judged to have undergone critical instability, and the incoming flow velocity corresponding to this moment is the critical velocity”. Referring to the absolute stability criteria for submarine pipelines in DNV-RP-F109 (Edition 2021, amended 2025) [25], this study considers that, under the same environmental dynamic conditions, multiple SCPS–ACMs units are either stable or experience overall sliding failure, and the interactions between units are minimal, so the force transmission between units can be neglected. Therefore, it can be concluded that the startup velocity of the SCPS–ACMs units under the rigid assumption in this study is reasonably reliable.

3. Results

3.1. Instability Mode

We found that, on a flat hard seabed, the SCPS covered by ACMs at intervals exhibited a consistent instability mode of overall slip. Figure 4 presents the instability process of the SCPS covered by ACMs at intervals under currents, for the representative working condition (cover spacing d = 6L_m, and covering pattern -L-L-L-). At the initial stage of the experiment, as the incoming flow velocity increased slowly from 0, the overall structure remained stable throughout. However, when the flow velocity approached the critical velocity, the SCPS and ACMs slid forward slowly and synchronously—this indicates that the total constraint force exerted on the SCPS by the ACMs and hard seabed was greater than the frictional force between the ACMs and flat hard seabed. Shortly after that, however, the movement mode of the SCPS shifted from translation to a combination of translation and rotation. The occurrence of rotation significantly reduced the frictional resistance exerted by the flat hard seabed on the SCPS, causing its moving speed to accelerate and, subsequently, exceed the sliding velocity of the ACMs—in other words, the SCPS attempted to break away from the bottom of the ACMs. Nevertheless, after a brief force adjustment, the SCPS and ACMs resumed the mode of synchronous forward movement.

Figure 5 presents the instability process of the SCPS covered by ACMs at intervals under the action of currents, for the representative working condition (cover spacing d = 7L_m, and covering pattern -L-L-L-). The instability mode was similar to the above scenario, with the difference that the SCPS eventually broke away from the bottom of the ACMs. The reason lies in the fact that, as the exposed length of the SCPS increased, the fluid rotational moment grew, the rotation speed accelerated, and the SCPS could more easily break free from the constraint of the ACMs.

3.2. Number of Spans

Figure 6 presents the variation in the critical velocity of SCPS covered by ACMs at intervals with a cover spacing for different number of spans.

The results show that, when the number of spans ≥ 2, the critical velocity is not sensitive to the number of spans. As the number of spans increases, the critical velocity only decreases slightly. Overall, the results of the two-span configuration are relatively close to those of the five-span configuration, with a maximum deviation of 3.8. When the cover spacing d ≥ 4L_m, the results of the one-span configuration are almost consistent with those of the two-span configuration; however, in the case of a small spacing (d = 3L_m), the flow past the cable at both ends of the SCPS has a significant effect on reducing the total flow drag force, resulting in a relatively higher critical velocity.

3.3. Mattress Row Length

Figure 7 presents the variation in the critical velocity of SCPS covered by ACMs at intervals with a cover spacing for different ACMs lengths.

It can be observed from the figure that, if the mattress length and cover spacing are increased to twice their original sizes simultaneously, the critical velocity of the SCPS remains essentially unchanged. This phenomenon indicates that the mechanical properties of the SCPS covered by ACMs at intervals—more specifically, its hydrodynamic properties—are basically unaffected by the mattress length.

In addition, the magnitude of the cover spacing mainly depends on the ACMs coverage ratio (a parameter defined as ρ_c = L_m/(d + L_m), where L_m is the length of a piece of ACMs and d is the cover spacing). Under the condition of the same ACMs coverage ratio, the test results of the two covering patterns (-2L-2L-2L- and -L-L-L-) show little difference; within the entire test parameter range, the maximum deviation between the two does not exceed 4.8%.

The above conclusions provide a scientific basis for the rational formulation of mattress installation technology and the improvement of construction efficiency in practical engineering.

4. Theoretical Research

4.1. Establishment of Theoretical Model

According to the lateral stability design standard for submarine pipelines based on a static analysis in DNV-RP-F109 [25], the SCPS under current action must satisfy the following condition to maintain absolute static stability:

$$\gamma \cdot {\displaystyle \frac{F_{\mathrm{Dc}}+{\mu }_{\mathrm{c}}\cdot F_{\mathrm{Lc}}}{{\mu }_{\mathrm{c}}\cdot G_{\mathrm{c}}+F_{\mathrm{R}}}}\le 1.0$$

(1)

where γ is the safety factor; G_c is the underwater weight of the SCPS; F_R is the passive resistance due to lateral movement, which is zero for the flat hard seabed; and μ_c is the friction coefficient between the SCPS and the flat hard seabed, obtained from tests as μ_c = 0.334. F_Dc and F_Lc represent the maximum flow drag and lift on the SCPS, calculated as follows:

$$F_{\mathrm{Dc}}=c_{\mathrm{D}}{\displaystyle \frac{\rho }{2}}{u}_c^2{D}_{\mathrm{c}}{L}_{\mathrm{c}},F_{\mathrm{Lc}}=c_{\mathrm{L}}{\displaystyle \frac{\rho }{2}}{u}_c^2{D}_{\mathrm{c}}{L}_{\mathrm{c}}$$

(2)

Here, ρ is the water density; u_c is the mean perpendicular current velocity over the diameter of a SCPS, approximated at D_c/2; D_c and L_c are the SCPS diameter and length; and c_D and c_L are the drag and lift coefficients, with c_D = 1.0 and c_L = 0.9 per DNV-RP-F109 [25]. For γ = 1, Equation (1) is reduced to the force balance equation of the SCPS, which can be used to calculate the critical instability velocity theoretically:

$$F_{\mathrm{Dc}}+{\mu }_{\mathrm{c}}{F}_{\mathrm{Lc}}={\mu }_{\mathrm{c}}{G}_{\mathrm{c}}$$

(3)

The velocity at the reference point is determined from the velocity profile. According to DNVGL-RP-F109 [26], the profile is expressed as follows:

$$u(z)=u(z_{\mathrm{r}})\cdot {\displaystyle \frac{\ln (z+z_0)-\ln z_0}{\ln (z_{\mathrm{r}}+z_0)-\ln z_0}}\cdot \sin {\theta }_c$$

(4)

where z_r is the reference point, taken as 0.1h to match the experimental setup; z₀ is the flat hard seabed roughness, which is minimal for the concrete seabed; and θ_c is the angle between the flow and SCPS axis, taken as 90°. Figure 8 shows a good agreement between a typical measured profile (u(z_r) = 0.48 m/s) and a calculated profile, validating the reliability for estimating the velocity profile from Equation (4).

A stability experiment was conducted for SCPS (without ACMs), and the measured critical instability velocity was compared with the theoretical prediction, using u(z_r) as the unified reference velocity. As shown in Figure 9, the measured values are slightly lower than the theoretical results. This deviation is attributed to the instability mode observed in the experiment: the cable rapidly transitions from the initial minor translation to rolling. The rolling motion changes the contact form between the SCPS and the hard seabed from “line contact” to “surface contact”, resulting in a significant reduction in static friction resistance; at this time, the resistance that the SCPS needs to overcome to transition from the static stable state to the moving state decreases, and the corresponding hydrodynamic force threshold decreases, thereby reducing the critical velocity.

Based on the force analysis of the single SCPS, a further force-balance analysis was performed for the SCPS–ACM unit. Since the critical instability mode of this system is the overall slip mode, the concrete articulated mattress and SCPS can be regarded as a single unit, and the contact forces between them are internal forces that can be omitted in the equilibrium analysis. As illustrated in Figure 10, the unit is subjected to gravity (G_m, G_c), flow drag force (F_Dm, F_Dc), lift force (F_Lm, F_Lc), flat hard seabed friction resistance (F_Rm, F_Rc), and flat hard seabed support force (F_Sm, F_Sc).

As the flow velocity increases, the SCPS–ACM unit reaches the critical instability condition when the total hydrodynamic force equals the total frictional resistance. Based on the force balance, Equation (5) is obtained:

$$F_{\mathrm{Dc}}+{\mu }_{\mathrm{c}}{F}_{\mathrm{Lc}}+F_{\mathrm{Dm}}+({\mu }_{\mathrm{c}}{\beta }_{\mathrm{mc}}+{\mu }_{\mathrm{m}}{\beta }_{\mathrm{mb}})F_{\mathrm{Lm}}=F_{\mathrm{Rc}}+F_{\mathrm{Rm}}$$

(5)

$$F_{\mathrm{Rc}}={\mu }_{\mathrm{c}}{G}_{\mathrm{c}}+{\mu }_{\mathrm{c}}{\beta }_{\mathrm{mc}}{G}_{\mathrm{m}}$$

(6)

$$F_{\mathrm{Rm}}={\mu }_{\mathrm{m}}{\beta }_{\mathrm{mb}}{G}_{\mathrm{m}}$$

(7)

By substituting Equations (6) and (7) into Equation (5), the latter can be rewritten as follows:

$$F_{\mathrm{Dc}}+F_{\mathrm{Dm}}={\mu }_{\mathrm{c}}(G_{\mathrm{c}}-F_{\mathrm{Lc}})+({\mu }_{\mathrm{c}}{\beta }_{\mathrm{mc}}+{\mu }_{\mathrm{m}}{\beta }_{\mathrm{mb}})(G_{\mathrm{m}}-F_{\mathrm{Lm}})$$

(8)

where μ_m is the friction coefficient between the ACMs and the seabed, obtained from tests as μ_m = 0.481; and β_mc and β_mb are the weight distribution coefficients of the ACMs acting on the SCPS and on the flat hard seabed, respectively. The flow drag force F_Dm and lift force F_Lm acting on the ACMs are calculated as follows:

$$F_{\mathrm{Dm}}={\lambda }_{\mathrm{D}}{\displaystyle \frac{\rho }{2}}{u}_{\mathrm{m}}^2{L}_{\mathrm{m}}{H}_{\mathrm{m}},F_{\mathrm{Lm}}={\lambda }_{\mathrm{L}}{\displaystyle \frac{\rho }{2}}{u}_{\mathrm{m}}^2{B}_{\mathrm{m}}{L}_{\mathrm{m}}$$

(9)

where λ_D and λ_L are the drag and lift coefficients of the ACMs, and u_m is the characteristic velocity at half the arch height (H_m/2). The flow drag F_Dc and lift F_Lc acting on the SCPS are still computed using Equation (2). For a basic force unit, L_c = d, where d is the cover spacing between the ACMs. The hydrodynamic forces on the SCPS beneath the ACMs are neglected due to shielding.

4.2. Determination of Key Coefficients

The hydrodynamic forces acting on the ACMs were determined through a computational fluid dynamics simulation of the flow past the SCPS–ACMs unit. The computational fluid dynamics (CFD) model of the SCPS–ACMs unit was developed based on the unsteady Reynolds-averaged Navier–Stokes (URANS) equations. The Transition SST turbulence model was employed to close the URANS equations. For the solver configuration, the pressure–velocity coupling was handled using the Coupled algorithm, and the flux type was set to the Rie–Chow scheme based on the distance weighting. In terms of spatial discretization, the gradient was computed using the Least-Squares Cell-Based method; the pressure term was discretized with a second-order scheme, the momentum equations with a second-order upwind scheme, and both the turbulent kinetic energy and dissipation rate with first-order upwind schemes.

In the numerical model, a velocity inlet was applied at the upstream boundary, while a pressure outlet was set downstream. Symmetry boundary conditions were imposed on both lateral sides, and the bottom of the computational domain was treated as a no-slip wall. The scaled model domain measured 1.6 m × 1.0 m × 0.3 m, with the SCPS–ACMs unit positioned at the center of the bottom boundary. In the scaled model, the base mesh size was 0.01 m, while local mesh refinement was applied around the SCPS–ACMs unit, with a minimum cell size of 5 × 10⁻⁴ m to ensure the adequate resolution of near-wall flows and local pressure gradients. And mesh sensitivity analysis was performed using three mesh configurations with different refinement sizes (refinement regions centered on the SCPS–ACM unit). The influence of the mesh refinement size on the numerical results was assessed as shown in

Table 3. Parameters of mesh sensitivity analysis.

Grid	Refinement Mesh Size (m)	Number of Grids	λD	λL
Grid-1	1 × 10−3	1,134,632	0.822	0.243
Grid-2	5 × 10−4	1,659,478	0.829	0.246
Grid-3	2 × 10−4	2,214,502	0.827	0.245

.

Based on the three mesh configurations described above, Grid-2 and Grid-3 performed well overall, yielding drag (λ_D) and lift (λ_L) coefficients that are in close agreement, whereas Grid-1 exhibited relatively larger computational errors. Considering both the accuracy of the numerical results and computational efficiency, Grid-2 was selected as the computational mesh for the present mathematical model.

A uniform inlet velocity condition was applied, with the flow velocity ranging from 0.36 m/s to 1.0 m/s. Figure 11a presents the typical pressure and streamline distribution under an incoming velocity of 0.5 m/s. On the upstream face of the ACMs, the incoming flow converges rapidly, forming dense streamlines that closely follow the surfaces before diverging downstream. The lower upstream face forms the impingement zone, which exhibits the highest pressure in the flow field. The pressure distribution varies with the angle between the structural surface and the incoming flow, reaching a peak where direct impact occurs. As the flow passes over the ACMs crest, the adverse pressure gradient causes rapid kinetic energy loss, leading to boundary-layer separation. The separated region exhibits a gradual pressure reduction, forming a transition from upstream high pressure to downstream low pressure. A reverse flow forms on the downstream surface, creating a pronounced low-pressure zone that induces suction toward the downstream direction, which is unfavorable for the stability of the system.

Using u_m as the characteristic velocity, the calculated drag coefficient and lift coefficient of the ACMs are shown in Figure 11b. These coefficients decrease slightly with increasing flow velocity and gradually reach stable level. For theoretical analysis, the average values within the experimental velocity range are adopted: drag coefficient λ_D = 0.86 and lift coefficient λ_L = 0.26.

To determine the weight distribution coefficients, the interaction among the ACMs, submarine SCPS, and seabed was analyzed using the finite element method. Solid elements were used for the ACMs, SCPS, and seabed, and cable elements were used to model the lashing ropes. Contact algorithms were applied between the ACMs and the SCPS, ACMs and seabed, and SCPS and seabed, and between adjacent blocks, and the underwater weight of the ACMs was applied in the model. Under gravity, the ACMs adaptively adjusted their posture and reached an equilibrium configuration (Figure 12) consistent with the actual installation. Figure 13 shows the contact forces between a single ACMs and the SCPS (1.24 N) and between the ACMs and the seabed (1.19 N). Their sum equals the underwater weight of the ACMs (2.43 N). Thus, the weight distribution coefficients are β_mc = 0.51 and β_mb = 0.49.

4.3. Verification of Theoretical Equations

Based on Equation (8) and the determined key coefficients, the quantitative relationship between the critical instability velocity of the submarine SCPS and the cover spacing of the ACMs can be obtained. Since this study has systematically carried out stability tests on three core covering forms (-L-L-L-, -L-L-L-L-L-L-, and -2L-2L-2L-) in Section 3.2 and Section 3.3, it is confirmed that the variation trends of their critical velocities with a cover spacing are completely consistent, and the maximum deviation of the values under the same coverage ratio does not exceed 4.8% (much smaller than the test repeatability error), indicating that the stability control mechanism of different covering forms is uniformly dominated by the ACMs coverage ratio. Therefore, we select the representative -L-L-L- mode in engineering applications as the validation dataset of the theoretical model, which can not only accurately reflect the prediction reliability of the model but also avoid content redundancy caused by multi-form repeated validation. Figure 14 presents the theoretical results for the representative -L-L-L- condition, compared with the experimental measurements. The theoretical predictions show a good agreement with the experimental data, exhibiting similar trends with a maximum deviation below 8%. This comparison verifies the reliability of the theoretical model and provides technical support for extending the laboratory findings to engineering applications.

4.4. Applicability of Theoretical Model

The wave-induced effect and high-velocity tidal environment are not within the scope of this study. The experimental results are derived from idealized low-energy currents. However, the theoretical equations are derived based on the force equilibrium and adhere to objective physical principles, thus possessing a certain degree of universality. It not only is applicable to the experiments but can also be extended to high-velocity current conditions in actual engineering, where the drag and lift coefficients of the ACMs need to be re-determined through reliable CFD methods.

On the other hand, the theoretical model is applicable to the overall slip mode of the SCPS–ACMs, which occurs on a flat hard seabed with a relatively small friction coefficient. However, an actual hard seabed is usually composed of weathered rocks and reef limestone, with an irregular surface and a larger friction coefficient, so the instability mode of the SCPS–ACMs may change. Considering the site-specificity, finite element simulations are performed to capture the transformation of the instability mode under different friction coefficients. Firstly, the ACMs are laid on the SCPS under gravity and allowed to reach an equilibrium configuration. Then, a constant velocity is applied to the SCPS to make it slide forward slowly, and the ACMs are forced to move accordingly. The instability mode mainly depends on the constraint force (F_Co) of the ACMs on the SCPS and the friction force (F_Rm) between the ACMs and the seabed. Figure 15 presents the variations in F_Co with different friction coefficients μ_m. It can be calculated that, in the initial stage of movement, F_Co ≈ 0.73 N. If μ_m ≤ 0.6, we can see that F_Rm ≤ 0.72 N and F_Co > F_Rm; in this case, the SCPS–ACMs undergo overall slip. On the contrary, if μ_m ≥ 0.7, we can see that F_Rm ≥ 0.83 N and F_Co < F_Rm; in this case, the SCPS slides under the ACMs. Representative motion modes are shown in Figure 16.

Experimental and theoretical research on the slip instability of the SCPS under the ACMs will be further conducted in our future work.

5. Discussion

This study provides a quantitative tool for the design of the cover spacing, supporting the optimization of protection schemes for cables in offshore wind power and other marine engineering applications. Future work will further extend the model toward more complex ocean environments. Additional physical model experiments will be conducted to obtain wave-induced loading parameters, allowing wave–current phase differences and wave conditions to be incorporated into the force-balance formulation, thereby improving applicability under combined wave–current actions. The influence of different incident angles on the hydrodynamic coefficient will be systematically quantified, and an angular correction factor will be introduced to enhance the model performance under oblique incoming flows. Furthermore, the effects of the hard seabed roughness on the friction coefficient and passive resistance will be investigated, with boundary condition parameters supplemented to improve the adaptability to varying seabed types. These efforts will progressively extend the applicability of the theoretical formulation in real engineering scenarios, providing more reliable support for the design and implementation of submarine cable protection systems.

6. Conclusions

This study systematically reveals the instability mechanism of SCPS protected by spaced ACMs on a flat hard seabed and the variation of their critical instability velocity through physical model experiments and theoretical modeling. The main conclusions are as follows:

(1)

The dominant instability mode of the SCPS–ACMs system is the overall slip mode. On a flat hard seabed, SCPS and ACMs slip cooperatively. As the cover spacing increases, the hydrodynamic overturning moment acting on the exposed SCPS segment increases significantly, raising the risk of separation from the underside of the ACMs.

(2)

Cover spacing is the key parameter controlling system stability, and the critical instability velocity decreases notably with increasing spacing. Using ACMs with two or more spans effectively mitigates the boundary effect caused by end-flow separation, and thus improves the reliability of the experimental model.

(3)

The ACMs coverage ratio ρ_c is the dominant dimensionless parameter governing the instability velocity. When the ACMs length and spacing are scaled proportionally (e.g., simultaneous increases in L_m and d), the instability velocity remains nearly unchanged, indicating that the ACMs length itself has minimal influence on the overall stability.

(4)

A quantitative theoretical equation relating cover spacing to critical instability velocity is established based on a force-balance analysis, incorporating key parameters such as the friction coefficients, hydrodynamic coefficients, and weight distribution coefficients. The predicted results show an excellent agreement with the experimental data, confirming the model’s reliable engineering applicability.

This study provides a quantitative tool for the design of cover spacing and offers scientific support for optimizing submarine cable protection schemes in offshore wind power and other marine engineering applications.