A State-of-the-Art Review of the Hydrodynamics of Offshore Pipelines Under Submarine Gravity Flows and Their Interactions

Abstract

Submarine gravity flows, e.g., debris flows and turbidity currents, pose a significant threat to offshore pipeline integrity. This risk primarily manifests through the imposition of substantial dynamic loads on pipelines or their large displacement when impacted by such flows. To enhance our understanding of these threats and facilitate the development of more robust pipeline design and protection strategies, this work reviewed the interactions between submarine gravity flows and offshore pipelines. For an individual pipeline, critical focus lies in characterizing the influence of key parameters—including Reynolds number, span height, impact angle, pipe geometry, ambient temperature, and surface roughness—on both the resultant impact forces and the fluid-structure interaction dynamics. Then, investigations into the interactions between gravity flows and multiple pipes are summarized, where the in-line spacing distance between two pipes is a key factor in reducing the impact force. Further, flow-induced vibration responses of a single pipeline and two tandem pipelines under gravity flows are presented. Building upon a thorough review, we conducted overall evaluations. There are few experimental studies and most investigations ideally treat the seabed to be horizontal, which does not always occur in practical engineering. Choosing empirical formulas to evaluate hydrodynamic loads should carefully consider the specific working conditions. An appropriate non-Newtonian fluid model is significantly important to avoid uncertainties. Some practical risk reduction measures such as streamlined structures and reduction in roughness are recommended. Finally, suggestions for future study and practice are proposed, including the requirement for three-dimensional numerical investigations, assessment of fatigue damage by flow-induced vibrations, consideration of flexible pipeline, and more attention to multiple pipelines.

1. Introduction

Submarine gravity flows represent underwater currents carrying suspended sediments. Gravity is the primary energy source of submarine gravity flows, and it converts the potential energy of unstable slope sediments into kinetic energy, which facilitates the transport of eroded materials from continental slopes to abyssal plains. Gravity flows can not only generate impact forces on pipelines but also scour several meters deep, threatening even buried pipelines. Gravity flows form when sediment accumulation exceeds slope stability [1,2,3,4,5], often triggered by earthquakes, volcanic activity, and other internal or external mechanisms [6,7,8]. As shown in Figure 1, under the effect of gravity, the water content increases while density and strength decrease [9,10,11,12,13], resulting in various sliding movement forms, typically including plastic deformation, non-Newtonian fluid, transition fluid, and Newtonian fluid. Among the most significant types are debris flows and turbidity currents. Debris flows are cohesive, high-density mixtures of sediment (clay to boulders) and water, where particles are supported primarily by matrix strength rather than fluid turbulence. They exhibit poor sorting, chaotic deposition, and unstratified masses, and typically move as slow, viscous slurries confined to steep slopes or proximal basin settings. In contrast, turbidity currents are turbulent, dilute flows in which sediment is suspended by fluid turbulence, enabling long-distance transport. Their depositions are well-structured to form graded beds (Bouma sequences) with sharp basal contacts and sedimentary structures reflecting waning flow energy [14,15,16]. Key distinctions between debris flows and turbidity currents lie in their support mechanisms (matrix strength vs. turbulence), depositional fabrics (disorganized vs. stratified), and run-out distances (short vs. extensive). Debris flows may evolve into turbidity currents via dilution and turbulence incorporation, producing hybrid deposits with basal chaotic units overlain by graded turbidites [17]. Both processes are critical in shaping submarine fans, distributing organic-rich sediments, and triggering geohazards.

Gravity currents can transport large amounts of submarine sediment from the continental shelf to the deep ocean floor due to their general characteristics of massive volumes, large influence area, high velocity, and large run-out distance [19,20,21]. For example, the velocity of high-density turbidity flows can surprisingly reach up to 30 m/s [22], and move extensive run-out distance covering hundreds of kilometers [23], both of which have a significant influence on sediment behavior and a substantial geohazard to submarine installations. The unique phenomenon, hydroplaning, is the key mechanism that allows gravity currents to travel long distances, a phenomenon that is attributed to the reduction in frictional resistance at the interface between the flow and the seafloor [24,25,26]. Hydroplaning occurs when the hydrodynamic pressure at the flow head reaches sufficient magnitude to balance or exceed the dynamic pressure exerted by the sediment load, resulting in fluidization and subsequent water intrusion beneath the gravity flow. The hydroplaning phenomenon induces frontal acceleration in gravity flows, leading to progressive separation from the main flow body and ultimately surging behavior [27,28]. Hydroplaning can happen easily at a very small velocity of around 1 m/s for small-scale flows in laboratory experiments [27].

Clear classifications of submarine gravity currents are significantly important for analyses in flow dynamics and associated interactions with deep-sea installations. Dott [29] identified the elastic, plastic, and viscous flow behavior with the development of submarine landslide. This classification provides a simple framework to understand the sediment transportation and deposition processes by gravity currents. Based on the particle support methods, Middleton [30] classified gravity flows into four types including particle flow, debris flow, turbidity flow, and pure fluid flow. Further, researchers [31,32,33,34,35,36,37,38,39,40,41] found that gravity currents cannot be treated as one single fluid or adopt only one deposition mechanism in practical sediment and transportation process. Later, Lowe [31,32] proposed a new classification framework by considering both the rheological characteristics of fluids and the supporting mechanisms of sediments. Firstly, based on rheological features, gravity currents are divided into plastic debris flow and pure liquid flow. Then, according to different transport mechanisms, sediment gravity currents are classified into five types including turbidity flow, fluidized flow, liquefied flow, grain flow and debris flow [31,32]. This classification scheme, which not only analyzes rheological characteristics but also takes into consideration the influence of deposition mechanisms, has been widely used for a long time. Now, the most commonly adopted classification method is from the work of Shanmugam [33], as depicted in Figure 2. The concept of sandy debris flow encompasses the types of gravity flow that are unclear or controversial in traditional classifications, such as the transitional type between high-density and low-density turbidity current, turbidity current types with a higher sand content, sediment types with a mixture of mud and sand components, and types with the characteristics of a mixture of turbidity current and debris flow. This method has greatly improved the accuracy and practicality of classification and has been widely applied in the research of deep-water gravity flows. After complex water–sediment exchange and long-distance transportation, the submarine landslide can gradually evolve into debris flows with high density and large velocity, and, ultimately, develop into fluidized turbidity flows. Compared with fluidized turbidity flows (Newtonian fluid), the debris flows—which show higher shear stress and density—are generally considered non-Newtonian fluids [13,34,35,36], possibly causing significant damage to marine facilities [37]. Non-Newtonian fluid models such as Bingham plastic [37,38] and Herschel–Bulkley models [12,35,39,40,41] have been commonly adopted to model the yield stress behavior and viscous properties of debris flows.

The progressive development of offshore operations in deep water over 1000 m across continental slope regions increases submarine pipeline systems’ vulnerability to geohazard impacts from submarine gravity flows [37,42]. Historical evidence demonstrates their destructive potential. The most catastrophic recorded incident remains the 1929 Grand Banks turbidity current, which destroyed twelve transatlantic cables sequentially across 600 km, with estimated speeds of 60 km/h [43]. The Vancouver Island M7.3 earthquake in 1946 caused significant gravity flows, destroying coastal facilities and shearing submarine pipelines [44]. In 2006, submarine turbidity current triggered by the Pingtung earthquake caused damage to nine submarine optical cables in the nearby deep-water area [45]. Similarly, in Taiwan’s Gaoping Submarine Canyon, turbidity current velocities reaching 20 m/s have severed multiple international communication cables during typhoon events [46]. These cases underscore the urgent need to understand interactions between gravity flows and pipelines/cables to protect offshore infrastructure and marine environments.

In this work, we focus on the hydrodynamics of offshore pipelines under submarine gravity flows. By reviewing the published literature, key factors influencing interactions between gravity flows and a single pipe are first clarified, then multiple pipelines are considered, and, finally, the flow-induced vibrations of pipes are discussed. The main objectives of this work are to (1) provide evidence on how to choose the working conditions and design the pipelines under the impact of gravity currents, and (2) identify directions for future work that researchers can pursue.

2. Factors Influencing the Interactions Between Gravity Flows and a Single Pipe

2.1. Effect of Reynolds Number

Reynolds number is a significantly important non-dimensional parameter in fluid mechanics, particularly affecting the vortex shedding behavior and associated hydrodynamic forces. The definition of the Reynolds number for Newtonian fluids, e.g., flows around an airfoil where gravitational effects are ignored, is presented as follows:

$${\mathrm{Re}}_{\mathrm{Newtonian}}={\displaystyle \frac{\rho U_{\infty }D}{{\mu }_{\mathrm{abs}}}}.$$

(1)

where μ_abs is the absolute fluid viscosity, D is the characteristic length scale of the structure, ρ is fluid density, and U_∞ is the free stream velocity. However, for non-Newtonian fluids flowing around a typical circular-sectional body, Equation (1) is not applicable. The hydrodynamic characteristics of shear-thinning, non-Newtonian fluids was advanced by [34] through their proposal of a specialized Reynolds number formulation. This dimensionless parameter was specifically developed for fluids exhibiting power-law or Herschel–Bulkley rheological behavior. The Reynolds number for non-Newtonian fluids (Re_{non-Newtonian}) is defined as follows:

$${\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}={\displaystyle \frac{\rho {U_{\infty }}^2}{{\mu }_{\mathrm{app}}\dot{\gamma }}}={\displaystyle \frac{\rho {U_{\infty }}^2}{\tau }}$$

(2)

where μ_app is the apparent viscosity, defined as the ratio of shear stress to the rate of shear of a non-Newtonian fluid, and it is closely related to the changing rates of shear and ambient temperature; $\dot{\gamma }=U_{\infty }/D$ is the shear rate outside the boundary layer; $\tau$ is the fluid shear stress at the strain rate. To calculate Re_{non-Newtonian}, an appropriate μ_app should first be considered, defined as follows:

For power-law fluids:

$$\left\{\begin{array}{l}\tau =K\cdot {\dot{\gamma }}^n\\ {\mu }_{\mathrm{app}}=K\cdot {\dot{\gamma }}^{n-1}\end{array}\right.$$

(3)

For Herschel–Bulkley fluids:

$$\left\{\begin{array}{l}\tau ={\tau }_C+K\cdot {\dot{\gamma }}^n\\ {\mu }_{\mathrm{app}}={\displaystyle \frac{{\tau }_C}{\dot{\gamma }}}+K\cdot {\dot{\gamma }}^{n-1}\end{array}\right.$$

(4)

where K is the consistency index, n is the flow index, and ${\tau }_C$ is the yield stress. The rheological behavior for most clay-rich debris flows can be described by the Herschel–Bulkley model [47].

The Reynolds number of Newtonian fluids can vary from 10¹ to 10⁶. As Reynolds number increases, the fluid flow exhibits the subcritical, critical, and supercritical regimes [48,49]. In contrast, the Reynolds number of non-Newtonian fluids is much smaller. According to [34], in engineering practice, the submarine pipeline diameter is 0.1–1 m, a submarine landslide velocity is 1–10 m/s, and, consequently, the Re_{non-Newtonian} ranges from 0.6 to 240, as evidenced in

Table 1. Reynolds numbers in previously published studies on the effect of gravity flows on pipelines.

Fluids	Authors	Fluid Density (kg/m3)	Reynolds Number
Debris flows Non-Newtonian fluids	Zakeri et al. [34]	1681~1694	0.6~240
	Zakeri et al. [28]	1681~1694	2~320
	Perez-gruszkiewicz [52]	670	1.5~34
	Haza et al. [53]	2630	0.18~92.64
	Sahdi et al. [54]	/	0.00001~110
	Liu et al. [38]	1681~1694	2.25~353.47
	Dong et al. [12]	1500	10~100
	Fan et al. [55]	1681~1694	10~10,000
	Nian et al. [39]	1312~1468	0.24~410.12
	Sahdi et al. [50]	/	0.5~40
	Zhang et al. [56]	1681~1694	1.06~345.68
	Guo et al. [51]	1312~1468	0.2~130
	Guo et al. [57,58,59,60,61]	1312~1468	0.45, 3.4, 86.7, 346.47
	Qian et al. [62]	1681~1694	0.3~847.8
	Zhao et al. [63]	1687.7~1694	1000~15,000
	Fan et al. [64]	/	0.001~1241
	Fan et al. [65]	/	0.00001~100
	Fan et al. [41]	/	0.36~287
	Guo et al. [66]	1312~1468	1.81~723.42
	Guo et al. [67]	1312~1468	4~130
Turbidity flows Newtonian fluids	Ermanyuk and Gavrilov [68]	1008.081~1038.024	1000
	Gonznlez-Juze et al. [69]	/	2000, 6000
	Gonznlez-Juze et al. [70]	/	2000~45,000
	Jung and Yoon [71]	/	6000
	Xie et al. [72]	1008.081~1038.024	840, 23,500
	Guo et al. [73]	1466.3	95, 500
	Guo et al. [74]	1312	1112~333,559

. For flow past a circular pipe, the Reynolds number determines the vortex shedding mode behind the pipe. Figure 3 compares the typical vortex shedding behavior at low Reynolds numbers for Newtonian [50] and non-Newtonian fluids [51]. For both fluids, three distinct stages including symmetrical flow, vortex formation, and vortex shedding are clearly identified. At very low Reynolds numbers, the flow exhibits a symmetrical wake with fully attached streamlines, indicating a stable and predictable laminar flow pattern. Increasing Reynolds number leads to flow separation phenomenon, where two identical recirculation regions are observed while no vortex shedding occurs. When the Reynolds number exceeds the critical threshold, the flow undergoes a transition characterized by the emergence of periodic vortex shedding in the wake region downstream of the pipe. As evidenced in Figure 3, it is clearly seen that the boundaries of Reynolds numbers for non-Newtonian fluids in three stages are larger than those for Newtonian fluids, indicating the significant difference in fluid properties. For example, overcoming the yield stress of Herschel–Bulkley fluid requires additional energy, resulting in a delay in vortex formation and shedding.

For non-Newtonian fluids past a circular pipeline, the relationship between the drag and Reynolds number is attracting more and more attention, enabling engineers to estimate drag forces based on flow conditions and pipeline positioning. From the perspective of fluid framework, the drag force (F_D) can be expressed as follows:

$$F_{\mathrm{D}}={\displaystyle \frac{1}{2}}\rho {U_{\infty }}^{2}A\cdot C_{\mathrm{D}}$$

(5)

where A is the projected area and C_D is the drag coefficient. From Equation (5), we can find that the calculation of drag coefficient C_D is the key problem. In the experiments of Zakeri et al. [34], slurries were supplied by a head tank and flowed into a water compartment through a chute, and the fully developed debris flows finally interacted with the pipeline. According to the experimental results, Zakeri et al. [34] proposed the first C_D-Re_{non-Newtonian} relationship for the interactions between the submarine debris flows and pipelines.

For a suspended pipeline, the following is true:

$$C_{\mathrm{D}}=1.4+{\displaystyle \frac{17.5}{{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}^{1.25}}}$$

(6)

For a laid-on-seafloor pipeline, the following is true:

$$C_{\mathrm{D}}=1.25+{\displaystyle \frac{11}{{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}^{1.15}}}$$

(7)

This finding provides significant insights into interactions between a submarine pipeline and debris flows, highlighting the importance of drag force estimation and the necessity of integrating non-Newtonian rheological models into offshore pipeline designs to account for the complex shear-dependent behavior of debris flows. By using a lock-exchange system, Haza et al. [53] proposed a new equation to estimate the drag coefficient for a suspended pipeline:

$$C_{\mathrm{D}}=0.7+{\displaystyle \frac{29.1}{{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}^{1.3}}}$$

(8)

The distinct difference between the results of Zakeri et al. [34] and Haza et al. [53] indicates the significant effect of experimental conditions. According to the Morison equation, the total drag force applied on a suspended pipeline can be divided into two components, including velocity-related drag force and acceleration-induced inertia force. Qian et al. [62] suggested that the inertia force should be taken into consideration to provide accurate estimations of the total drag forces, which is calculated as follows:

$$F_{\mathrm{D}}={\displaystyle \frac{1}{2}}\rho {U_{\infty }}^{2}A\cdot C_{\mathrm{D}}+{\displaystyle \frac{\pi }{4}}\rho DA{\ddot{U}}_{\infty }\cdot C_{\mathrm{M}}$$

(9)

where the following is true:

$$C_{\mathrm{D}}=0.58+{\displaystyle \frac{18}{{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}^{0.9}}}$$

(10)

$$C_{\mathrm{M}}=\left\{\begin{array}{l}0,{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}<17.6\\ 0.2975\ln \left({\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}\right)-0.8533,{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}\ge 17.6\end{array}\right.$$

(11)

Apart from the abovementioned factors, the span height ratio [41], ambient temperature [39], impact angle [28,38,75], and surface roughness [60] also play crucial roles in estimating the drag forces, and they will be discussed later.

2.2. Effect of Span Height Ratio

Submarine free-spanning pipelines are important sections of offshore pipelines that are unsupported by the seabed over certain lengths due to uneven topography, scour erosion, or anthropogenic activities [76]. These suspended spans are subjected to complex hydrodynamic forces, including submarine landslides, vortex-induced vibrations (VIV), wave-current interactions, and potential fatigue damage, all of which may compromise structural integrity over time. The key parameter, span height ratio H* = H/D, is defined as the vertical distance between the seabed line and the lowest point of the pipeline. Investigations of the single-phase Newtonian fluids like pure water flowing past a submarine suspended pipeline have been extensively reported [77,78,79,80,81,82,83]. However, considerations of multi-phase Newtonian/non-Newtonian fluids, such as debris and turbidity flows past free-spanning pipelines, are insufficient, and the latest studies are summarized in

Table 2. Studies of debris and turbidity flows past a submarine suspended pipeline.

Flow Type	Authors	Slope Angle of the Seabed (°)	H*
Debris flow	Zakeri et al. [34]	3, 6	1
	Sahdi et al. [54]	/	6.2
	Zhang et al. [56]	0	0~3
	Guo et al. [51]	0	0~3
	Fan et al. [41,64,65]	0	0.08~10
	Guo et al. [59]	0	0~2.5
	Guo et al. [61]	0	0~1
Turbidity flow	Ermanyuk and Gavrilov [68]	0	0.03~4.5
	Gonznlez-Juze et al. [69]	0	0.067~1.33
	Gonznlez-Juze et al. [70]	0	0.15~1.5
	Jung and Yoon [71]	0	0.2~1.4
	Xie et al. [72]	0	0.05~1
	Ding et al. [84]	0	0.9~1.7
	Guo et al. [73]	0	1

.

For different span heights, the flow around the pipeline varies. Figure 4 plots the velocity field and the shedding vortices when gravity flows past a pipeline under three typical span heights. For a very low span height case, as shown in Figure 4c, the space between the pipe and seabed is partially or fully occupied by the submarine pipe. In this case, the pipeline creates a complete flow obstruction, resulting in zero underflow penetration, and the whole mass flux of the gravity-driven flow must be diverted over the pipeline structure. Consequently, two velocity deceleration regions on the lower front area and an acceleration region on the upper front area of the pipe are observed. This velocity distribution will lead to great pressure differences between the two surfaces of the pipe and consequently large lift forces [59], as well as an asymmetrical vortex shedding mode. When the span height increases to a moderate level, as sketched in Figure 4b, the flow field changes significantly. The approaching flow experiences significant deceleration upstream of the pipeline due to the obstruction effect, creating a stagnation zone. Simultaneously, the pressure potential energy of the gravity flow on the lower front surface of the pipeline undergoes conversion to kinetic energy, leading to a strong acceleration. The flow on the upper front surface forms a relatively weak acceleration. Vortices are observed to be alternately shed from both sides of the pipeline, while they are not identical due to the different flow channels. For a very large span height, the seabed effect can be ignored, and the flow field is considered to be infinite. In this case, as depicted in Figure 4a, two equal vortices with opposite signs are shed alternately, forming the classical Karman vortex street.

Since the span height has a significant effect on the flow field, the associated hydrodynamic loads should be carefully checked. For debris flows, Guo et al. [51] numerically studied the impact force coefficients of a suspended pipeline under a series of span heights, and the results are plotted in Figure 5. Three distinct modes are identified. At low Reynolds numbers less than 1 (mode I), the maximum drag and lift coefficients usually appear at H* = 2.5~3, and their minimum values often occur at H* = 0. The span height ratio has a significant effect on the hydrodynamic loads when Re_{non-Newtonian} < 1, especially for the lift. When 1 ≤ Re_{non-Newtonian} < 10 (mode II), the effect of span height is gradually weakening as the differences between the maximum and minimum hydrodynamic coefficients decrease as the Reynolds number increases. In mode III where Reynolds number is greater than 10, the span heights corresponding to the maximum and minimum forces have changed. The maximum and minimum drag appear at H* = 0~0.5 and H* = 2.5~3, respectively. The lift coefficient exhibits similar trends, with a critical transition threshold emerging at Re_{non-Newtonian} = 10. The finding that the lift coefficient decreases with increasing span height in mode III can be further confirmed by the results of Fan et al. [64].

For turbidity flows, Ermanyuk and Gavrilov [68] conducted flume experiments to study the pipeline loads with various span heights. The results revealed that hydrodynamic loads reach their maximum magnitude when the pipeline is in direct contact with the channel bed, while they decrease rapidly with increasing span height. Specifically, the rapid decrease in hydrodynamic loads occurs at H* = 1.5. Gonznlez-Juze et al. [69], Xie et al. [72], and Ding et al. [84] employed numerical methods to investigate the variations of peak drag and lift coefficients against span height ratio, and their results are compared in Figure 6. With increasing span height ratio, the peak drag coefficients generally decrease. However, clear fluctuations in C_D-P vs. H* can be found at a very high Reynolds number of 23,500, which belongs to the subcritical regime. These fluctuations reflect the significant Reynolds number effect. The results of Ding et al. [84] show that the peak lift coefficient of the pipeline at Re = 840 is very large when the pipeline is close to the seabed, while it rapidly decreases to a low level and does not show significant fluctuations with further increases in the span height ratio. In contrast, at a large Reynolds number of 2000, totally different variations are observed, where the lift coefficients show a nearly linear increasing trend with the span height ratio. The different trends of C_L-P vs. H* also reflect the Reynolds number effect.

2.3. Effect of Impact Angle

In practical engineering, submarine gravity flows may not always strike the pipeline vertically. Instead, there exists an impact angle (θ, defined as the angle between the oncoming flow and the pipe axis), varying from 0° to 90°. In that case, the normal and axial drag forces on the pipeline under gravity flows can be expressed as follows (Cohen and Kundu [85]):

$$F_{\mathrm{D}\text{-}90}=C_{\mathrm{D}\text{-}90}\cdot {\displaystyle \frac{1}{2}}\rho {U_{\infty }}^2\left(A\cdot \sin \theta \right)$$

(12)

$$F_{\mathrm{D}\text{-}0}=C_{\mathrm{D}\text{-}0}\cdot {\displaystyle \frac{1}{2}}\rho {U_{\infty }}^2\left(A\cdot \cos \theta \right)$$

(13)

Comparisons of the drag estimation methods under the effect of impact angle and span height ratio are listed in

Table 3. Conditions for drag estimation methods under the effects of impact angle as well as span height ratio.

Authors	Renon-Newtonian	Oncoming Flow Velocity (m/s)	θ (°)	H*
Zakeri [28]	1.5~317.1	0.35~3.4	0, 30, 45, 60, 90	1
Liu et al. [38]	2.25~353.47	0.35~3.4	0, 30, 45, 60, 90	1
Wang et al. [86]	1~350	Below 0.2 for the smallest velocity	0, 15, 30, 45, 60, 75, 90	1
Zhang et al. [56]	1.23~345.68	0.2~5	0, 15, 30, 45, 60, 75, 90	0~3

. Scientific researchers and engineers should note that Equations (14)–(21) are empirically obtained based on specific conditions, so it is important to choose an appropriate one according to practical applications.

Zakeri [28] employed computational fluid dynamics analysis to numerically simulate the impact of clay-rich submarine debris flows on a suspended pipeline at various impact angles. A primary method is presented for estimating the normal and axial drag forces of a pipeline, expressed as follows:

$$F_{\mathrm{D}\text{-}90}=\left(1.4+{\displaystyle \frac{17.5}{{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}^{1.25}}}\right)\cdot {\displaystyle \frac{1}{2}}\rho {U_{\infty }}^2\left(A\cdot \sin \theta \right)$$

(14)

$$F_{\mathrm{D}\text{-}0}=\left(0.08+{\displaystyle \frac{9.2}{{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}^{1.1}}}\right)\cdot {\displaystyle \frac{1}{2}}\rho {U_{\infty }}^2\left(A\cdot \cos \theta \right)$$

(15)

Based on the results of Zakeri [28], Liu et al. [38] examined more cases by expanding the flow velocity range and increasing impact angle conditions. The experimental findings demonstrated that while the debris flow impact angle significantly influences the normal drag coefficient, it exhibits a negligible effect on the axial drag coefficient. Therefore, the drag forces are written as follows:

$$F_{\mathrm{D}\text{-}90}=\left({\displaystyle \frac{0.46+\left(23.55/{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}\right)}{{\left(\sin \theta \right)}^{0.6}}}\right)\cdot {\displaystyle \frac{1}{2}}\rho {U_{\infty }}^2\left(A\cdot \sin \theta \right)$$

(16)

$$F_{\mathrm{D}\text{-}0}=\left({\displaystyle \frac{9.13}{{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}}}\right)\cdot {\displaystyle \frac{1}{2}}\rho {U_{\infty }}^2\left(A\cdot \cos \theta \right)$$

(17)

Later, Wang et al. [86] considered lower flow velocities, and further improved the estimation method of drag forces as follows:

$$F_{\mathrm{D}\text{-}90}=\left({\displaystyle \frac{0.65+\left(19.16/{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}\right)}{{\left(\sin \theta \right)}^{0.55}}}\right)\cdot {\displaystyle \frac{1}{2}}\rho {U_{\infty }}^2\left(A\cdot \sin \theta \right)$$

(18)

$$F_{\mathrm{D}\text{-}0}=\left({\displaystyle \frac{8.361}{{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}}}\cdot {\left(\cos \theta \right)}^{0.22}\right)\cdot {\displaystyle \frac{1}{2}}\rho {U_{\infty }}^2\left(A\cdot \cos \theta \right)$$

(19)

By considering the effect of both impact angle and span height ratio (H*), Zhang et al. [56] proposed an improved method to estimate the normal and axial drag force:

$$F_{\mathrm{D}\text{-}90}=\left({\displaystyle \frac{23.56-\frac{25}{4.6+e^{3H*}}}{{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}}}\right)\cdot {\displaystyle \frac{1}{2}}\rho {U_{\infty }}^{2}A{\left(\sin \theta \right)}^{0.45}$$

(20)

$$F_{\mathrm{D}\text{-}0}=\left({\displaystyle \frac{8.96-0.5e^{-2H*}}{{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}}}\right)\cdot {\displaystyle \frac{1}{2}}\rho {U_{\infty }}^{2}A{\left(\cos \theta \right)}^{1.22}$$

(21)

2.4. Effect of Pipe Geometry

To reduce the impact loads on offshore pipelines under submarine gravity flows, researchers have proposed several novel methods, including using streamlined profiles such as wedge, airfoil, arc-angle hexagon, and double ellipse (Perez-Gruszkiewicz [52]; Fan et al. [55]) and changing the pipe surface by arranging uniform honeycomb holes (Guo et al. [57,59]), as shown in Figure 7. Both experimental results by Perez-Gruszkiewicz [52] and numerical simulations by Fan et al. [55] demonstrated that the drag coefficients of all four streamlined pipelines are effectively reduced compared with a regular pipe. As evidenced in Figure 8, the drag reduction is also greatly affected by the Reynolds number of non-Newtonian fluids, where significant reductions are successfully achieved in the low Re_{non-Newtonian} range. Additionally, streamlined profiles are helpful for stabilizing lift force responses, which is attributed to the delay in the separation of boundary layers (Figure 9). Specifically, the wedge and double-ellipse pipelines exhibit the greatest reduction among the four streamlined pipelines. According to the C_D-Re_{non-Newtonian} relationship, Fan et al. [55] proposed estimation equations for the drag coefficients of different streamlined devices.

For wedge pipelines:

$$C_{\mathrm{D}}=0.75+{\displaystyle \frac{31.28}{{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}}}$$

(22)

For double-ellipse pipelines:

$$C_{\mathrm{D}}=0.7+{\displaystyle \frac{38.14}{{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}}}$$

(23)

For airfoil pipelines:

$$C_{\mathrm{D}}=0.71+{\displaystyle \frac{42.35}{{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}}}$$

(24)

For arc-angle hexagon pipelines:

$$C_{\mathrm{D}}=0.93+{\displaystyle \frac{38.7}{{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}}}$$

(25)

Focusing on the innovate pipelines, Guo and his team proposed pipelines with honeycomb holes [57] and dimples [59]. Both devices can achieve a significant reduction of about 20% in the peak drag forces of the pipeline under the impact of submarine debris flows. The drag reduction mechanism is mainly attributed to the reduced maximum pressure difference around the pipeline. In addition, the honeycomb holes and dimples are helpful for suppressing lift force fluctuations and effectively reducing the force amplitude, preventing potential pipeline damage from flow-induced vibrations.

2.5. Effect of Ambient Temperature

Wang et al. [86] conducted low-temperature rheology tests of debris flows, and they found the environmental temperature plays an important role in rheological properties of submarine debris flows. Based on this finding, Nian et al. [39] established a low-temperature rheological model to study the impact force on suspended pipelines under debris flows. Five ambient temperatures—0.5 °C, 4.5 °C, 8.5 °C, 12 °C, and 22 °C—were considered in their work. The results indicated that the drag and lift forces on the pipeline at 0.5 °C increase by 26.0% and 70.3%, respectively, compared with those at 22 °C. Moreover, calculations of the peak (C_D-P) and stable (C_D-S) drag coefficients were established as follows:

$$C_{\mathrm{D}\text{-}\mathrm{P}}=1.52+{\displaystyle \frac{17.21}{{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}}}$$

(26)

$$C_{\mathrm{D}\text{-}\mathrm{S}}=1.2+{\displaystyle \frac{16.06}{{\mathrm{Re}}_{\mathrm{non}\text{-}\mathrm{Newtonian}}}}$$

(27)

Later, Guo et al. [51] compared the impact of debris flows past a pipeline under low (0.5 °C) and normal (22 °C) ambient temperature. A similar conclusion was obtained: the loads on pipelines at 0.5 °C are significantly larger than those at 22 °C. Interestingly, the maximum rate of change of C_D-P and C_L-P (the peak lift coefficient) can reach 30% and 49%, respectively. For a laid-on-seafloor pipeline, a peak enhancement of 123% in the stable lift coefficient (C_L-S) is observed. Therefore, low temperature has a vital effect on the forces acting on the pipeline and must be carefully considered.

2.6. Effect of Roughness

In practice, the outer materials of submarine pipelines can be concrete, steel, rubber, asphalt, polyethylene, etc. These surfaces, with varying roughness, have a significant effect on the flow field around submarine pipelines. In addition, the attachment of microorganisms can alter the pipeline’s surface roughness, further affecting the flow and associated dynamic loads on the pipeline. Guo et al. [60] numerically investigated the characteristics of debris flow around circular pipelines with surface roughness using an equivalent sand-grain roughness model. The results demonstrated that increasing surface roughness leads to a significant increase in peak lift force, while peak drag force decreases. Specifically, for a suspended pipeline with surface roughness of 0.15 mm, the peak lift force increases by 62% and the peak drag force decreases by 17% compared to smooth pipelines. Based on these findings, the researchers established a chart methodology considering pipeline roughness effects for estimating impact forces, expressed as:

$$F_{\mathrm{D}}={\displaystyle \frac{1}{2}}\rho {U_{\infty }}^2(D+k_s)\cdot C_{\mathrm{D}}$$

(28)

where k_s is the roughness height.

3. Interactions Between Gravity Flows and Multiple Pipes

With the significant increase in the demand for submarine pipelines and cables, multiple pipelines in close proximity are increasingly common in order to meet the demands of the growth of offshore activities. For instance, the “Nord Stream 2” pipeline was constructed in parallel to the “Nord Stream 1” pipeline through the Baltic Sea to transport natural gas. Single-medium fluid flow with uniform incoming velocity past two circular pipes has been systematically reviewed, identifying several distinct flow regimes and interference modes [87]. In contrast, the hydrodynamic characteristics of multiple pipes under gravity flows involving at least two different fluids have been less studied. Guo et al. [73] employed the large-eddy simulation (LES) framework coupled with density transport equations to investigate the gravity-driven high-density turbidity current dynamics and their interactions with parallel suspended pipelines. The effect of the streamwise spacing distance between two pipes on the flow field and associated hydrodynamic loads is examined. The results indicated that the turbulent wake generated by high-density fluid flowing past the front pipe promotes velocity fluctuations, leading to increased impact forces on the rear pipe with increasing streamwise spacing distance up to eight times the pipe diameter. The results also suggested that the streamwise spacing distance should be less than two times the pipe diameter to minimize hydrodynamic loads on the rear pipe. Ding et al. [84] conducted LES simulations to study the interactions between gravity currents and two tandem pipes. Results showed that the distribution of the maximum dynamic pressure and vortices around two pipes is closely related to the spacing distance. The drag force of the front pipe is considerably smaller than that of an individual pipe when the spacing distance is small. Unfortunately, although three-dimensional simulations using the LES method were used in the studies of both Guo et al. [73] and Ding et al. [84], only two-dimensional results were analyzed, and the vorticity distribution and process in the spanwise direction were not discussed.

4. Flow-Induced Vibrations of Pipes Under Gravity Flows

Previous investigations mainly focused on gravity flows past a stationary pipeline. However, the pipeline may vibrate in both the in-line and transverse directions, commonly known as flow-induced vibrations (FIV). The studies about FIV responses of pipes under gravity flows are scarce. Zhao et al. [63] adopted an air-water-debris model coupled with the one-degree flow-induced vibrating pipe module to study the pipe dynamics under the debris flow. Specifically, the effects of debris flow (volume, density, and location) and pipe characteristics (stiffness coefficient, location, and number) on the hydrodynamic loads and vibrations are systematically investigated. As depicted in Figure 10, before the debris flow interacts with the pipe at t = 20.4 s, some vortices appear due to the turbulence of the two fluids, while flow fields are much similar for cases with and without a pipe. At t = 28 s when the debris flow passes the pipe, the pipe is pushed upward because of the large density difference between the debris flow and water. The head shape of the debris flow is similar for both cases, but the pipe-flow interactions lead to more vortices around the pipe, thereby enhancing turbulence intensity. At t = 28.8 s, the debris flow is located above the pipe, pushing the pipe downward. After the debris flow passes through the pipe, many pairs of fish-eye-like swirls are generated, strengthening the turbulence of both the water and the debris flow. At the same time, the pipe starts to rise again, exhibiting a reciprocating vibration behavior. As compared with perfectly periodic oscillations of the constant-density flow past a circular pipe [88], the displacements under debris flows show strong time-dependent features [63] (Zhao et al., 2021).

Compared with a single pipe, the mutual interference between two tandem pipes and their interactions with debris flows create a more complex flow field with strong turbulence, as shown in Figure 11. The displacement variations in Figure 11 show that (1) the vibration trajectory of the rear pipe closely mirrors that of the front pipe for a period of time; (2) the front pipe maintains its elevated position longer than the rear pipe, owing to the combined effects of direct debris flow impact and the shielding effect provided by the front pipe to its downstream counterpart; and (3) the vibration frequency of the rear pipe is significantly higher than that of the front pipe, a phenomenon that is mainly attributed to the wake interference.

5. Overall Evaluations

This work presents a review of interactions between submarine gravity flows and offshore pipelines. By synthesizing relevant experimental and computational findings, key factors (Reynolds number, span height, impact angle, pipe geometry, ambient temperature, and surface roughness) influencing the dynamics of a single pipeline and the resulting impact forces under gravity flow impact are first discussed. Further, flow fields and hydrodynamic characteristics of multiple pipelines under the impact of submarine gravity flows are presented. Finally, flow-induced vibrations of suspended pipelines are depicted. Some substantial advancements have been achieved in comprehending and mitigating submarine gravity flow effects on offshore pipelines, while there are still shortcomings in the studies referenced.

Numerous numerical studies have been conducted, while only a few flume experiments have been carried out, leading to a lack of physical experimental results for numerical validation. Moreover, in many studies, the seabed slope is commonly idealized and assumed to be horizontal (0°) for simplicity in modelling and analysis. However, in reality, natural seafloors often exhibit complex and varying degrees of slope, which can significantly influence hydrodynamic and geotechnical processes.

Based on experimental and numerical results, empirical formulas evaluating the gravity flow loads on the offshore pipelines have been proposed. These formulas are closely related to key parameters, such as Reynolds number, span height ratio, impact angle, pipe geometry, ambient temperature, and surface roughness. The establishment of empirical formulas provides important guidance for engineers to design offshore pipelines. Nevertheless, the prerequisite for using these formulas is to carefully consider the specific working conditions. For example, in Section 2.3, the varied Reynolds numbers, impact angles, and height ratios determine distinct methods to estimate the forces acting on the pipeline. Therefore, multi-parameter uncertainty evaluation and the sensitivity of empirical constants deserve more attention.

In most previous reports, clay-rich debris flows are described by the Herschel–Bulkley model. However, choosing an appropriate non-Newtonian fluid model is critically important. Modeling non-Newtonian flows is fraught with significant uncertainties, primarily stemming from the challenge of selecting an appropriate constitutive model that can accurately capture complex rheological behaviors like shear-thinning, yield stress, and viscoelasticity. This uncertainty is compounded by the inherent difficulty in precisely measuring the model’s parameters, which are highly sensitive and prone to experimental artifacts like wall slip. To mitigate these issues, a comprehensive rheological characterization is essential to guide model selection, while advanced uncertainty quantification techniques should be employed to calibrate parameters and assess their confidence intervals. Furthermore, the critical uncertainty arising from the imperfect coupling between a fluid’s evolving microstructure and its macro-scale flow behavior can be addressed by developing multi-scale modeling frameworks that dynamically link these scales. Finally, the numerical uncertainties from solving highly nonlinear equations are alleviated by using robust algorithms, adaptive mesh refinement, and validating predictions against detailed experimental data from advanced flow visualization techniques like Particle Image Velocimetry (PIV).

To mitigate the impact of gravity flows on offshore pipelines, some practical risk reduction measures can be directly used according to the results of previous reports. Choosing an appropriate span height ratio is helpful for reducing hydrodynamic loads on pipelines. Installing streamlined structures like double-ellipse and wedge around the pipelines can effectively recover base pressures and stabilize wake flows, resulting in reduced impact forces. Increasing ambient temperature and reducing roughness are good choices to weaken gravity flow effects.

6. Suggestions for Future Study and Practice

6.1. Three-Dimensional Numerical Studies

Transitioning from two-dimensional to three-dimensional numerical modeling approaches is essential for enhancing the understanding of submarine gravity flow dynamics and their complex interactions with offshore pipelines. Most existing numerical studies on gravity flow–pipeline interactions rely on simplified two-dimensional models, which suffer from inherent limitations. In two-dimensional simulations, out-of-plane effects, such as non-uniform sediment stresses, three-dimensional fluid flow patterns, and asymmetric pipeline deformations, are neglected or oversimplified. This can lead to inaccurate predictions of key behaviors, including localized scour around pipelines, uneven load distribution, and the true failure mechanisms of the pipeline-sediment system. Moreover, two-dimensional models often fail to account for the influence of seabed slope, transverse slope variations, and complex boundary conditions, which are critical in real-world scenarios. In contrast, three-dimensional numerical models can overcome these limitations by fully resolving spatial interactions. A three-dimensional framework enables the explicit representation of turbulent flow dynamics around pipelines, multi-directional sediment displacement, and the true geometry of the pipeline (e.g., joints and bends). It also allows for a more realistic simulation of phenomena such as vortex shedding, sediment transport heterogeneity, and the progressive development of failure zones in the surrounding sediment. Future research should prioritize the development of validated three-dimensional models, incorporating multi-physics couplings (e.g., fluid-structure-sediment interactions) and experimental data for calibration. Such efforts would bridge the gap between theoretical studies and practical engineering applications, ultimately improving the design and risk assessment of pipeline systems in gravity-driven flow environments.

6.2. Fatigue Damage by Flow-Vibration Responses

While previous studies on gravity flow-pipeline interactions have primarily focused on estimating impact forces under static pipeline conditions, the critical aspect of flow-induced vibration (FIV) remains largely unexplored. The assumption of fixed pipelines neglects important dynamic effects, such as vortex-induced vibration (VIV), periodic loading from turbulent fluctuations, and potential resonance phenomena that may lead to fatigue damage or structural failure over time. Future research should systematically investigate FIV mechanisms in gravity flow environments through coupled fluid-structure interaction (FSI) modeling, which can enable safer pipeline designs in gravity flow systems, particularly in scenarios involving unsteady flows.

6.3. Flexible Pipelines

Existing studies on interactions between the gravity flow and suspended pipeline have predominantly modeled pipelines as rigid, non-deformable structures. While this simplification facilitates computational analysis, it overlooks critical aspects of flexible pipeline behavior that are increasingly relevant in modern engineering applications. Future research should systematically investigate the dynamic response of flexible suspended pipelines under gravity flow conditions, with particular focus on (1) the coupled hydroelastic vibration mechanisms induced by unsteady flow separation and vortex shedding; (2) nonlinear deformation characteristics, including sagging effects and geometric nonlinearity, under combined hydrodynamic and self-weight loading; and (3) the fatigue performance under cyclic loading conditions. Advanced computational frameworks combining large-eddy simulation with nonlinear structural dynamics should be developed, incorporating both material nonlinearity (e.g., hyperelastic or viscoelastic behavior) and geometric nonlinearity (e.g., large deflection theory). These investigations will provide fundamental insights for optimizing the design and operation of suspended pipeline systems in applications such as deep-sea mining tailings disposal, subaqueous sediment transport, and other scenarios where pipeline flexibility significantly influences system performance and safety.

6.4. Multiple Pipelines

Real-world engineering applications often involve multiple pipelines arranged in complex configurations (e.g., parallel, staggered, or bundled systems). The hydrodynamic interference between adjacent pipelines and their collective impact on flow patterns, sediment transport, and local scour development remain poorly understood. Future studies should pay more attention to multiple pipeline systems under gravity flow conditions: (1) wake interference effects and their influence on hydrodynamic loading characteristics; (2) the amplification or sheltering effects on local scour depth and erosion patterns; and (3) coupled vibration mechanisms that may lead to collision risks or fatigue damage. Such research will provide essential insights for the optimal spacing design, risk assessment, and maintenance strategy of pipeline networks in gravity flow environments, particularly for offshore mining operations and subsea cable–pipeline crossings where multiple pipelines coexist.