# Numerical Study of the Local Scouring Process and Influencing Factors of Semi-Exposed Submarine Cables

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Sediment Scouring Model

_{cr}of sediment is given as [23,24]

_{f}is the shearing velocity of bed surface, ρ

_{s}is the density of the sediment particle, ρ

_{f}is the fluid density, g is the acceleration of gravity, d

_{50}is the median size of sediment, and μ is the dynamic viscosity of sediment.

_{s}is the concentration of the sediment particle, $\overline{u}$ is the mean velocity vector of the fluid and the sediment particle, u

_{s}is the velocity of the sediment particle, f

_{s}is the volume fraction of the sediment particle, P is the pressure, F is the volumetric and viscous force, K is the drag force, and u

_{r}is the relative velocity.

## 3. Numerical Setup and Modeling

#### 3.1. Geometric Modeling and Mesh Division

#### 3.2. Physical Field Setup

#### 3.3. Mesh Independent Test

## 4. Results and Analysis

#### 4.1. Analysis of Local Scouring Process

^{−2}m/s to 3.98 × 10

^{−3}m/s and remained stable until the end of the sixth hour. This phenomenon explains why the upstream scouring hole developed rapidly in the first hour but remained stable for the following 5 h.

^{−2}m/s. It took approximately 5 h for the shear velocity to stabilize, and the stable shear velocity was 2.26 × 10

^{−3}m/s. Therefore, compared with the upstream scouring hole, the downstream scouring hole was deeper and required more time to reach stability.

^{−3}m/s, which almost does not change in the first hour. This leads to a very slow development of the scouring hole at the wake position in the early stages. The maximum shear velocity in this scouring hole gradually increased to 1.05 × 10

^{−2}m/s from the second to the fifth hour, and then decreased to 6.61 × 10

^{−3}m/s by the end of the eighth hour. This is why the scouring hole at the wake position grows fastest around the fifth hour. Consistent with the pattern of change in the scouring hole’s terrain, the location of the maximal shear velocity also shifted to the right with time.

#### 4.2. Influence Factors

#### 4.2.1. Sediment’s Critical Shields Number

_{cr}is set as 0.02, 0.03, 0.04, 0.05, 0.06, and 0.07, and the variations of scouring terrain over time under each θ

_{cr}are displayed in Figure 7.

_{cr}will affect the depth of the upstream scouring hole and the development speed of the scouring hole at the wake position, but it will have no significant impact on the expansion of the downstream scouring hole.

_{cr}, the upstream scouring hole will reach a temporary plateau within 1 h, at which time the stable depth will be about 0.04 m. When θ

_{cr}≤ 0.05, the upstream scouring hole will continue to expand after a few hours. The stable time is obviously affected by θ

_{cr}, which will gradually increase from 1 h to 11 h with the increase in θ

_{cr}. The terrain of the upstream scouring hole will gradually convert to deep on the left and to shallow on the right. Since the scouring hole at the wake position has not been stable, its state at the time of submarine cable spanning is studied emphatically. In the whole process of scouring, the scouring hole at the wake position continues to develop and does not reach a stable state. With the increase in θ

_{cr}, the development velocity of the scouring hole at the wake position will decrease considerably. Its average evolution velocity decreases from 3.88 cm/h to 1.62 cm/h, and its depth decreases from 21.9 cm to 18.8 cm. Under the condition of each θ

_{cr}, the downstream scouring hole will stabilize within 1 h, and the stable depth will be basically unchanged (all about 13.5 cm).

_{cr}increases, so does the sediment’s ability to withstand shearing forces, which will cause it to become increasingly difficult to be eroded or carried away by ocean currents. This effect has been directly reflected in the depth of scouring holes (upstream and wake position). Due to the blocking effect of semi-exposed submarine cables, the wake is elongated, which is why the downstream scouring hole develops before the scouring hole at the wake position and quickly reaches a stable state. However, due to the high wake intensity, this process is not significantly affected by the change of θ

_{cr}.

#### 4.2.2. Sediment Density

_{s}is set as 1550 kg/m

^{3}, 1600 kg/m

^{3}, 1650 kg/m

^{3}, 1700 kg/m

^{3}, 1750 kg/m

^{3}, and 1800 kg/m

^{3}, and the variation of scouring terrain over time under each ρ

_{s}are displayed in Figure 8.

_{s}will also affect the depth of the upstream scouring hole and the development speed of the scouring hole at the wake position. In addition, it can even have an impact on the downstream scouring hole depth.

_{s}conditions, the upstream scouring hole will always reach a temporary stable state in 1 h, at which time the stable depth will be 0.04 m. When ρ

_{s}≤ 1750 kg/m

^{3}, the upstream scouring hole will continue to expand after a few hours. The stabilization time of upstream scouring hole is more clearly affected by ρ

_{s}, which will gradually increase from 3 h to 13 h with the increase in ρ

_{s}. The terrain of the upstream scouring hole will gradually change to deep on the left and to shallow on the right. Since the scouring hole at the wake position has not been stable, its state at the time of the submarine cable spanning is studied emphatically, too. In the whole process of scouring, the scouring hole at the wake position continues to develop and does not reach a stable state. When ρ

_{s}is large, the development rate of scouring hole obviously decreased with time. With the increase in ρ

_{s}, the development velocity of the scouring hole at the wake position reduces from 3.38 cm/h to 1.14 cm/h, and the depth of this scouring hole declines from 20 cm to 15 cm. As ρ

_{s}increases, the stabilization time of the downstream scouring hole increases from less than 1 h to about 2 h, but the stabilization depth of the downstream scouring hole remains essentially the same (all around 13.5 cm).

_{s}will reduce the Shields number, thus weakening the shear action of the sediment by the ocean current, which explains the extension of the stability time of the upstream scouring hole. At the same time, with the increase in the depth of scouring hole at the wake position, its shear velocity will decreases. Therefore, under a larger ρ

_{s}value, the development speed of scouring hole at the wake position will decrease significantly with time. Possibly for the same reason, ρ

_{s}can affect the development rate of downstream scouring hole.

#### 4.2.3. Ocean Current Velocity

#### 4.3. Variation Rule of Spanning Time

_{cr}= 4.59 × 10

^{−2}. The relationship between spanning time t and sediment’s critical Shields number θ

_{cr}can be formulated by a cubic function as shown in Equation (6):

_{s}can be formulated by the first order function as shown in Equation (7):

## 5. Conclusions

- Under the steady-state ocean currents, scouring holes will be formed at the upstream, wake position and downstream of the semi-exposed submarine cable. The upstream and downstream scouring holes develop faster, which will reach a temporary stable state at about 1 h after the start of the scouring. The scouring hole at the wake position will continue to expand at a slower rate and eventually lead to the spanning of the submarine cable.
- There is a close relationship between the distribution of shear velocity and the scouring terrain. As the local scouring process occurs, the location of the maximum shear velocity within the scouring hole shifts and causes the bottom of the hole to move as well.
- When the sediment’s critical Shields number and density are significantly large and ocean current velocity is sufficiently low, the duration of the stable state of the upstream scouring hole will be prolonged, and the average development velocity of the scouring holes at the wake position and downstream will be reduced.
- The relationship between the spanning time and the critical Shields number θ
_{cr}can be formulated as a cubic function, in which the curve’s inflection point is θ_{cr}= 4.59 × 10^{−2}. The relationship between spanning time and sediment density can be formulated as a linear function. The relationship between spanning time and ocean current velocity can be formulated by exponential function.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 7.**Influence of sediment’s critical Shields number θ

_{cr}on local scouring around semi-exposed submarine cable: (

**a**) θ

_{cr}= 0.02; (

**b**) θ

_{cr}= 0.03; (

**c**) θ

_{cr}= 0.04; (

**d**) θ

_{cr}= 0.05; (

**e**) θ

_{cr}= 0.06; and (

**f**) θ

_{cr}= 0.07.

**Figure 8.**Influence of sediment density ρ

_{s}on local scouring around semi-exposed submarine cable: (

**a**) ρ

_{s}= 1550 kg/m

^{3}; (

**b**) ρ

_{s}= 1600 kg/m

^{3}; (

**c**) ρ

_{s}= 1650 kg/m

^{3}; (

**d**) ρ

_{s}= 1700 kg/m

^{3}; (

**e**) ρ

_{s}= 1750 kg/m

^{3}; and (

**f**) ρ

_{s}= 1800 kg/m

^{3}.

**Figure 9.**Influence of ocean current velocity v on local scouring around semi-exposed submarine cable: (

**a**) v = 0.35 m/s; (

**b**) v = 0.40 m/s; (

**c**) v = 0.45 m/s; (

**d**) v = 0.50 m/s; (

**e**) v = 0.55 m/s; and (

**f**) v = 0.60 m/s.

**Figure 10.**Influence of different parameters on spanning time of the semi-exposed submarine cable: (

**a**) Sediment critical Shields number; (

**b**) Sediment density; and (

**c**) Ocean current velocity.

Parameter | Value |
---|---|

Sediment Diameter | 8.5 × 10^{−5} m |

Sediment Density | 1650 kg/m^{3} |

Sediment Critical Shields Number | 0.05 |

Fluid Density | 1000 kg/m^{3} |

Fluid Viscosity | 0.01 kg/(m·s) |

Gravitational Acceleration | −9.81 m/s^{2} |

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**MDPI and ACS Style**

Li, Q.; Hao, Y.; Zhang, P.; Tan, H.; Tian, W.; Chen, L.; Yang, L.
Numerical Study of the Local Scouring Process and Influencing Factors of Semi-Exposed Submarine Cables. *J. Mar. Sci. Eng.* **2023**, *11*, 1349.
https://doi.org/10.3390/jmse11071349

**AMA Style**

Li Q, Hao Y, Zhang P, Tan H, Tian W, Chen L, Yang L.
Numerical Study of the Local Scouring Process and Influencing Factors of Semi-Exposed Submarine Cables. *Journal of Marine Science and Engineering*. 2023; 11(7):1349.
https://doi.org/10.3390/jmse11071349

**Chicago/Turabian Style**

Li, Qishun, Yanpeng Hao, Peng Zhang, Haotian Tan, Wanxing Tian, Linhao Chen, and Lin Yang.
2023. "Numerical Study of the Local Scouring Process and Influencing Factors of Semi-Exposed Submarine Cables" *Journal of Marine Science and Engineering* 11, no. 7: 1349.
https://doi.org/10.3390/jmse11071349