1. Introduction
When subsea structures are placed on a sea floor, vortices form due to currents and waves occur around foundations and subsea pipelines. The formation of a vortex structure is determined by the geometry and flow [
1]. The vortices are related scour, which results in the loss of the soil and erodes the seabed. The scour process around a subsea pipeline is expressed by the interaction between the fluid flow and the seabed [
2]. The fluid flow around the subsea pipeline may cause a free span. Excessive bending moment and fatigue load, due to the free span, can reduce the stability of the pipeline, leading to a structural failure [
3]. Pipelines installed on the sea bottom can be classified into two main purposes: submarine cables for power supply and communication systems and subsea pipelines that transport oil or gas to offshore or onshore processing facilities. When the structure is damaged, it causes economic damage, due to the interruption of continuous resource production work. In addition, it takes enormous cost and time to recover the polluted marine environment caused by the oil spill. For this reason, it is important to accurately predict scour around the subsea pipeline, which requires an understanding of the interaction between the fluid flow and seabed.
Many researchers have done a series of experiments to study the behavior of scour around the subsea pipeline [
3,
4,
5,
6]. Mao [
3] concluded that the difference in pressure between the upstream and downstream of the pipeline caused the erosion of sand particles beneath the pipeline. Chiew [
7] used the term piping to indicate a dominant factor in the early stage of scour. For factors influencing the scour process, stream velocity, grain size, pipe diameter, initial gap between a pipeline and a seabed, and flow depth were experimentally defined, with various environmental conditions [
3,
5,
7]. Empirical equations for the depth of scour beneath the subsea pipeline was presented [
5,
8,
9].
Various numerical models have been proposed, such as single- and two-phase models, to simulate a scour process. Single-phase models could be subdivided into a rigid seabed approaches and deformable seabed approaches. In the rigid seabed approaches, scour was predicted by the shear stress magnitude at the seabed boundary [
10,
11]. On the other hand, in the deformable seabed approaches, the elevation of the seabed was changed by a morphological model, which is considered a sediment transport model [
12,
13,
14]. To consider the deformable seabed, the periodic updates of the deformed mesh were required [
15,
16]. Zhao et al. [
17] investigated a local scour around various pipelines with combined wave and current conditions by a fully coupled numerical model. Liang et al. [
18] established a three-dimensional model to investigate scour around underwater structures and found that the equilibrium scour depth increased with the pipeline length until the pipeline length exceeded four times the pipe diameter. Above all, the single-phase models were unable to consider interactions between particles.
The two-phase models could be subdivided into two approaches depending on the fluid and seabed modeling method: Euler–Euler approach and Euler–Lagrangian approach. The Euler–Euler approach was based on an interpenetrating continuum for the fluid and particles. The movement of the particles that made up the seabed was explained by the sediment concentration. Yeganeh-Bakhtiary et al. [
19] employed the Euler–Euler two-phase model to simulate the tunnel erosion beneath a pipeline and successfully predicted the bed profile and flow behavior. Fraga et al. [
20] investigated scour beneath piggyback pipelines with current flow by a two-phase Euler–Euler model. The Euler–Euler two-phase model had limitations, in that it was difficult to provide information about particles, such as particle and particle interactions, as well as particle position and velocity [
21]. On the other hand, the Euler–Lagrangian approach treated the soil as a discrete phase that allowed tracking of individual particles. The Euler–Lagrangian approach has been applied in various engineering problems to simulate fluid and particle flow.
Computational fluid dynamics (CFD) and discrete element method (DEM) coupled models were commonly used in the Euler–Lagrangian approach. DEM had the advantage of being able to predict the exact behavior of particles and handle a large amount of particles [
22,
23]. CFD and DEM coupled models could be divided into resolved or unresolved approaches, depending on how fluid and particle interactions were considered. The resolved CFD and DEM models were fully resolving the flow around each particle and directly calculating the drag force. A grid that was at least 8–10 times smaller than a particle diameter was required, indicating huge computational requirements [
24]. The resolved methods were adequate for a dilute flow system or small system. In the resolved CFD and DEM model, a dense grid system was needed to compute the fluid force acting on a particle; whereas, in the unresolved CFD and DEM coupled model, a dense grid was not required to obtain an accurate drag [
23,
25,
26]. The unresolved CFD and DEM coupled model presented high computational efficiency for flows with bulk particles [
23,
27]. However, there was the grid dependency that yields valid computational results for fluid and particle interactions [
28]. Various numerical methods have been proposed to increase the reliability of the results, regardless of the grid size ratio [
29,
30,
31,
32,
33]. Song and Park [
33] used a kernel function-based averaging method to reduce the grid dependency.
The CFD and DEM coupled models have been applied to geotechnical engineering problems, e.g., the seepage flow in soils, sediment transport, and sand pile formation [
34,
35,
36]. Simulations for the scour process around a subsea pipeline were recently carried out, and the applicability of the CFD and DEM coupled model was discussed [
37,
38]. Zhang et al. [
37] used the CFD and DEM coupled model to investigate the onset of scour, and the motion of each particle was considered in detail. Yang et al. [
38] applied the CFD and DEM coupled model to simulate the scour process and provided detailed information to better understand the scour mechanism below a pipeline. Hu et al. [
39] adopted the CFD and DEM coupled model to study the effect of gap ratio and incipient velocity on a local scour around two pipelines in tandem and revealed that a scour development was closely related to the individual particle behavior. Liu and Tau [
21] employed the CFD-DEM coupled model to simulate a local scour behavior around a bridge pier with clear-water conditions. Yazdanfar et al. [
40] proposed a novel upscaling methodology, by a microscale CFD and DEM coupled model, to predict a live-bed scour. However, studies on local scour around the pipeline are still insufficient. More research on improved CFD and DEM coupled models is needed to overcome existing constraints, such as high computational cost, inaccuracies, due to the use of spherical particles, and grid dependency [
21,
28,
37,
38]. Recently, Song and Park [
33] presented an improved unresolved CFD and DEM coupled solver for particulate flow. The solver showed that it can reduce grid dependency and predict particle sedimentation for the single-particle settlement.
The purpose of this paper is to investigate the applicability of the CFD and DEM coupled solver, developed by Song and Park [
33], to simulate fluid flow and seabed interactions around a subsea pipeline. For the simulations, the unresolved CFD and DEM coupled solver using the kernel-based averaging method was used [
33]. The novelty and purpose were to suggest a proper selection procedure for the numerical model parameter and apply on scour around the pipeline. From the selected rolling friction coefficient, scour with non-spherical particles could be predicted by spherical particles. The selected solver was developed based on the open source CFD (OpenFOAM) and DEM (LIGGGHTS) libraries. To verify and validate the numerical methods, angles of repose and an incipient motion were simulated.
The present paper is organized as follows.
Section 2 describes computational methods including governing equations for the unresolved CFD and DEM coupled solver and numerical methods.
Section 3 presents the problem description.
Section 4 shows numerical model parameters.
Section 5 shows the results and discussion for the angle of repose, incipient motion of particles, and scour process around a pipeline. Finally, in
Section 6, some concluding remarks are provided.
3. Problem Description
The computational domain for the CFD and DEM simulation is shown in
Figure 1. Here,
D is the pipeline diameter. The lengths from the center of the pipeline to the inlet and outlet boundaries were 10
D and 12
D, respectively. The water depth (
H) and seabed depth (
h) were 5
D and 1
D, respectively. The soil particles for the DEM simulation were randomly distributed and then settled under gravity to form the seabed. The pipeline was initially laid just above the seabed without an initial gap and fixed during the scour process [
3]. The length of the seabed was set to 20
D, and the lengths of the inlet and outlet directions from the pipeline center were 9
D and 11
D, respectively. The width of the seabed was set to 2.16 mm in the out-of-plane direction. Due to the high computing cost, the length of the seabed was shorter than that of Mao’s experiment; however, Yang et al. [
38] confirmed that the development of scour could be sufficiently considered.
The simulation conditions are presented in
Table 1. The pipeline diameter (
D) was 0.05 m. The shape of soil particles in the simulation was a uniform spherical particle with a diameter (
) of 1 mm. The Shields parameter (
) was 0.33 [
3], which meant that live-bed scour occurred when the mean bed shear stress in the upstream was larger than the threshold value required to move the soil particles. The large Shields parameter (
) was chosen to accelerate scour development. The particle diameter was larger than that of Mao’s experiment. To keep the Shields parameter (
to be the same as that in Mao’s experiment, the mean flow velocity (
) was increased to be 1.21 m/s. Based on the diameter of the pipeline, the Reynolds (
) and Froude (
numbers were
and 1.71, respectively.
On the inlet boundary, the Dirichlet condition was applied for the velocity, turbulence properties, and volume fraction, while the Neumann condition was applied for the pressure. On the outlet boundary, the Neumann condition was applied for the velocity, turbulence properties, and volume fraction, while the Dirichlet condition was applied for the pressure. A zero normal gradient was applied for the Neumann conditions of all variables. The no-slip condition was applied to the pipeline and bottom surfaces. At the free surface, the symmetry boundary conditions were applied. The fluid inlet boundary was specified with a logarithmic velocity profile, based on the friction velocity and bed roughness [
19]. The mean velocity was calculated from the integral of the flow velocity over the depth at the inlet boundary. The mean flow velocity specified at the inlet was increased linearly from zero to 0.87 m/s over the first 6.4 s, in order to obtain the numerical stability [
37,
46].
The flow velocity profile was reproduced when there were the particles in the seabed as shown in
Figure 1. The flow velocity profile by the selected unresolved CFD and DEM coupled solver was compared with that of a CFD solver. In the coupled solver simulation, the seabed particles were considered, while, in the simulation of the CFD solver, the seabed particles were not considered.
Figure 2 shows the flow velocity profiles simulated by both solvers at a central location in a domain without the pipeline. The height of 0 m means the top of the sand surface. It was confirmed that the desired flow velocity profile could be obtained, even with the particles.
Table 2 shows the simulation parameters. A total number of 134,285 particles was used to form the seabed. The density of the particles was 2600 kg/m3. Young’s modulus and Poisson ratio of the particles were
Pa and 0.45, respectively. The friction and rolling friction coefficients were 0.6 and 0.125, respectively. The time step for the CFD and DEM solvers were
s and
s, respectively. Every forty steps, both CFD and DEM results were coupled. The computational time for a single test took 10 days on a workstation equipped with Intel Xeon(R) Gold 6226R CPU
2.9 GHz processors.
5. Results and Discussion
Simulations for scour around a subsea pipeline were carried out using the numerical methods used in the simulations of the angles of repose and incipient motion of the particles. A grid sensitivity study was performed using three different cases in
Table 6. Here, 1, 2, and 3 represent coarse, medium, and fine grids, respectively. The total number of the grid was increased by 2 times. A representative grid size (
) was used [
54]. The refinement factor for the representative grid size was
. The particle size (
) was set consistently to 1 mm for the three cases. As the grid system was fined, the scour depth and drag force acting on the pipeline were converged. The drag coefficient showed the difference less than 5% [
55]. The scour depth was converged in the case 2.
The evolution of scour depth with time is shown in
Figure 10. The depth of the scour hole (
) was normalized by the pipeline diameter (
). The simulation results in the early stage of scour slightly underestimated the scour depth, in which the inlet velocity was gradually increased [
38]. In other words, the onset of scour was postponed, due to a low velocity, which corresponded to about
s. The scour depth increased sharply after the onset of scour, which corresponded to the tunnel erosion (about
s). After the tunnel erosion, the scour depth increased at a relatively lower rate, which corresponded to the lee-wake erosion (about
s). Additionally, the scour depth reached the equilibrium state. This trend was consistent with the experimental results [
3]. In Yang et al.’ simulation [
38], the scour depth predicted lower than the experimental data in the onset of scour and tunnel erosion. While the scour depth was in good agreement with the experimental data in the lee-wake erosion, the selection of the coefficient of rolling friction by the validation test in this study more accurately influenced the prediction in the onset of scour and tunnel erosion.
Figure 11 shows the evolution of a bed profile with time. The onset of scour was occurred at about
s. The onset of scour was initiated by the seepage flow beneath the pipeline and pressure difference between the upstream and the downstream around the pipeline [
7]. In the single-phase models, and some two-phase models, it was challenging to reproduce the onset of scour without a very small gap between the seabed bottom and the pipeline [
13,
19]. After the onset of scour, the tunnel erosion proceeded to
< 0.3 (about
s), due to an increase in fluid flow entering the opening beneath the pipeline. The eroded particles beneath the pipeline formed a sand dune about half the diameter of the pipeline downstream. During the scour process, a large amount of particle movement at the rear side of the pipeline occurred at the lee-wake erosion (about
s). In addition, the scour hole was deepened and expanded downstream, while the sand dune moved farther and flattened in the direction of flow. These phenomena were consistent with the experimental observations [
3,
7].
Figure 12,
Figure 13,
Figure 14,
Figure 15 and
Figure 16 show the results at the three stages of scour (
s).
Figure 12 shows the contours of the fluid velocity normalized by the inlet velocity. In the onset of scour, the small velocity of the flow in the opening beneath the pipeline appeared. In the tunnel erosion, the small opening beneath the pipeline expanded into the tunnel, due to the increased flow. In the lee-wake erosion, the flow was affected by the seabed, and the deformed seabed region could be observed.
Figure 13 shows the velocity of the individual particles. The particles beneath the pipeline started to move at the onset of scour (
s). In the tunnel erosion (
s), the movement of the particles and number of moving particles around the pipeline were increased, indicating that the seabed under the pipeline was exposed to a strong erosion. In the lee-wake erosion (
s), the particles in the scour hole showed small movement, while the particles in the downstream of the scour hole moved in the flow direction.
The drag force is the dominant force in the fluid and particle interaction.
Figure 14 shows the drag force acting on the individual particles in the flow direction. Until the tunnel erosion, the drag force increased mainly beneath the pipeline, causing the high scour depth rate. In the lee-wake erosion, the drag force applied to the seabed led to the expansion of the scour hole and the movement of many particles. The particle transport capacity in the scour process increased with the increase of the drag force.
Figure 15 shows the drag force acting on individual particles in the vertical direction. The drag force acting in the vertical direction on the front and rear sides of the scour hole increased at the early stages and drove the motion of particles on the inclined bed inside the scour hole. From the results, the drag force acting on the particles played an important role in the transport of the sand particles.
Figure 16 shows the total force of the seabed particles. As the particles moved during the scour process, the interaction between the particles increased. In the region where the particle transport was most active, the interaction force between the particles was also the most intense. In
Figure 13,
Figure 14 and
Figure 15, it was shown that the particle transport was most affected by drag force acting on the particles. However, it should be noted that the particle and particle interaction was also active in the rear side of the sand dune, where the drag force was not dominant. It concluded that the interaction between the particles should be considered in the development process of the scour hole and sand dune in the tunnel erosion. These results show why the CFD and DEM coupled model is more suitable than the single-phase model for scour simulation.
6. Conclusions
In this study, the simulations were carried out on scour around the seabed pipeline, in order to investigate the applicability of the CFD and DEM coupled solver [
33]. The unsolved CFD and DEM coupled solver was used to track the particle motion caused by the fluid flow. For the verification of the numerical methods, the angles of repose were simulated and compared with the experimental data. The predicted angles of repose were in good agreement with the experimental data, and the rolling friction coefficient of the sand particles was decided for the scour simulation. The rolling friction coefficient should be used to compensate for non-spherical sand particles and determine the proper value for application to scour simulation. To validate the selected numerical model parameters, the incipient motion was simulated and discussed the critical state to validate the interaction between the fluid flow and seabed particles. The incipient motion of the sand particles was predicted close to the critical velocity in the Hjulstrom diagram.
The scour simulations around the subsea pipeline were carried out and compared with the experimental data [
3]. From the grid dependency test, the CFD and DEM coupled solver could be applied without grid dependency constraints. The flow fields and interactions between the flow and particle were presented in the three scour stages. At the onset of the scour stage, the sand particles beneath the pipeline started to move, due to the seepage flow and pressure difference between the upstream and downstream around the pipeline. During the tunnel erosion stage, the sediment transport was significantly increased with the increasing drag force, indicating that the seabed beneath the pipeline was exposed to the strong erosion. The strong erosion by the tunnel flow also expanded the scour hole and formed the sand dune. It was confirmed that the interaction between the particles was most active beneath the pipeline and in the sand dune during the tunnel erosion stage. With entering the lee-wake erosion stage, the particle movement was very small in the scour hole but very active when moving downstream. The main particle motions around the pipeline were generated by the drag force, due to the high flow velocity and the interaction between the particles.
The present CFD and DEM coupled solver required a lot of computational time, but all of the above findings can be explained from the information of each individual particle without using empirical formulas or solving constitutive equations, including sand concentration. The CFD and DEM coupled model showed great potential to simulate particle and particle interactions, as well as the fluid and particle interactions in previous research and the current study.