Dynamic Response Analysis of a Subsea Rigid M-Shaped Jumper under Combined Internal and External Flows

Abstract

To analyze the dynamic response of a rigid M-shaped jumper subjected to combined internal and external flows, a one-way coupled fluid–structure interaction process is applied. First, CFD simulations are conducted separately for the internal and external fluid domains. The pressure histories on the inner and outer walls are exported and loaded into the finite element model using inverse distance interpolation. Then, FEA is performed to determine the dynamic response, followed by a fatigue assessment based on the obtained stress data. The displacement, acceleration, and stress distribution along the M-shaped jumper are obtained. External flow velocity dominates the displacements, while internal flow velocity dominates the vibrations and stresses. The structural response to the combined effect of internal and external flows, plus the response to gravity alone, equals the sum of the structural responses to internal flow alone and external flow alone. Fatigue damage is calculated for the bend exhibiting the most intense vibration and higher stress levels, and the locations with significant damage correspond to areas with high maximum von Mises stress. This paper aims to evaluate multiple flow fields acting simultaneously on subsea pipelines and to identify the main factors that provide valuable information for their design, monitoring, and maintenance.

1. Introduction

Subsea rigid M-shaped jumpers play a crucial role as connectors in subsea oil and gas production systems, offering a pressure-resistant and watertight space for transporting a variety of objects, including gases, liquids, oils, and particles. These jumpers serve as crucial parts in connecting various subsea equipment, such as pipeline end terminals (PLETs) and pipeline end manifolds (PLEMs), as well as Christmas trees and subsea manifolds. Their arrangement is adaptable and customizable based on the two end units and specific seabed topography, such as horizontal or vertical arrangements. Different shapes, such as M-shaped and inverted U-shaped shapes, can be employed to meet varying connection requirements.

Subsea rigid jumpers have the characteristics of low difficulty in manufacturing and installation. They are usually assembled entirely on land and then placed on the seabed from the sea surface. Due to the high pressure of the deep sea, rigid jumpers have better pressure resistance and can be suitably used under large water depths. As a result, rigid jumpers are frequently chosen for deep-water subsea connections, consistent with the trend of oil and gas production systems moving deeper into the ocean.

The focus of this paper is on a commonly used rigid M-shaped jumper, characterized by its M-shaped arrangement consisting of multiple horizontal or vertical segments with in-plane bends. Despite its rigid structure, this jumper can reduce the impact of internal high-temperature and high-pressure flow through certain deformations [1]. This flexibility sets it apart from conventional straight pipes. However, the unique structural arrangement also introduces various challenges and issues that need to be addressed.

When an external current flows through the circular pipe cross-section, vortex shedding alternately occurs on both surfaces of the long span of the jumper. This phenomenon leads to fluctuating loads attributed to alternating vortex shedding, resulting in vortex-induced vibrations (VIVs). The potential for resonance increases when the frequency of vortex shedding approaches the natural frequency of the jumper. The recurrent shedding of vortices induces VIVs in subsea pipelines, representing a primary contributor to fatigue failure in subsea pipeline systems. Consequently, the analysis of fatigue damage and reliability in subsea pipelines often centers around the critical examination of VIVs.

Investigating VIVs in jumpers subjected to external flows is challenging due to the long total span of the actual jumpers commonly employed in subsea oil and gas production systems, often exceeding spans of 20 m [2]. Due to experimental limitations, it is necessary to use scaled-down models for VIV measurements in laboratory settings or to rely on computational fluid dynamics (CFD) for simulation studies. In practical measurements, the requirement for a large flow rate often demands more extensive and costly water pumps and circulation pipes. To achieve the necessary flow rate, two distinct methods are employed based on the relative motion of the jumper and the external flow. One method involves dragging the jumper at a constant speed. In the other method, a continuous flow of water is generated using a pump. Although the two methods differ, both are considered feasible and have consistent results when simulating the impact on the jumpers.

The primary experimental investigation of VIVs for M-shaped jumpers was a scaled-down model experiment conducted in a 200 m towing tank by ExxonMobil [3]. This study involved the use of two jumpers, with and without an external strake, towing within a speed range of approximately 0 to 1 m/s at three different angles (10°, 45°, and 90°). The scaled model of the M-shaped jumper was positioned upside down on the towing structure, forming a “W” shape. The vibration data were recorded through 13 accelerometers and three strain gauges placed within the jumpers.

The structural vibration data obtained from the above experiments serve as the basis for conducting structural analysis on the jumper model. The primary analytical objective is to determine the stresses and strains on a jumper through various methods, which are crucial for assessing the fatigue damage and fatigue life. Several researchers have evaluated the VIV response of jumpers using diverse methods. Zheng et al. [4] employed two distinct methods, the global response method based on modal scalar analysis and the spectral analysis method. Both methods proved suitable for a variety of towing speeds and directions. Igeh et al. [5] predicted fatigue damage resulting from VIVs using the global response model method from DNV-RP-F105 (2006) and VIVANA, incorporating torsional stress in the fatigue assessment. Kapoor et al. [6] conducted a fatigue assessment utilizing the energy-critical plane method instead of the classical method based on stress ranges and S–N curves. This approach highlighted the difference in the response frequency between two stresses and their impact on structure assessments. Liu et al. [7] evaluated the fatigue damage of a scaled-down rigid M-shaped jumper model considering torsional stresses. They compared two fatigue damage calculation methods, one accounting for bending and torsional stresses separately and the other based on the first principal stress. Qu et al. [8] utilized the linear Euler–Bernoulli beam theory to analyze the multi-frequency and multi-mode dynamic response. They employed a uniformly distributed wake oscillator model to simulate hydrodynamic forces in three flow directions (10°, 45°, and 90°) with different flow velocities (0.02 m/s to 0.90 m/s).

Other VIV experiments on jumpers under external flow conditions, such as that by Mobasheramini et al. [9], have simplified and scaled down the jumper to meet the experimental requirements. The experiment was carried out in a wave and current experimental tank with dimensions of 22 m × 1.4 m × 1 m and a water depth of 0.5 m. The flow velocity ranged from 0.04 m/s to 0.44 m/s, with a model scale factor of 1:25. Gross et al. [10] conducted VIV experiments and CFD simulations of a jumper, analyzing modes, natural frequencies, and deformations. The experimental tank had dimensions of 16 m in width and 40 m in length, with an external flow velocity of 0.3 m/s.

The utilization of CFD simulations offers the advantage of directly employing full-size jumper models without the necessity for scaling down. This approach allows for the flexible arrangement of monitoring positions, providing comprehensive information. However, a limited number of CFD simulations of jumpers under external flows have been conducted, primarily due to the large span of full-size models. Achieving sufficient simulation accuracy often requires a significant increase in the number of grids, leading to computational inefficiency. Holmes et al. [11] conducted CFD simulations on both a bare jumper and a jumper with a strake. The observations included vortex shedding and deformations of the jumper, which had a total span of 30 m and was subjected to external flow velocities ranging from 0.1 m/s to 0.5 m/s.

During various stages of subsea production system operations, monitoring the oil or gas content in the internal flow through the pipeline is crucial for exploitation efficiency. However, the complex geometry of the jumper results in recurrent changes in the flow direction, leading to complex alterations in the multiphase mixed-flow pattern. Dramatic changes in multiphase volume fractions can result in significant fluctuations in the inner wall pressure, potentially causing flow-induced vibrations (FIVs).

Observing variations in flow patterns within a jumper is challenging in practical applications. Hence, monitoring the internal flow pattern variations of scaled-down models in the laboratory is a practical approach. The installation of monitoring windows allows the observation of flow patterns in each pipe segment. However, adding monitoring windows compromises the integrity of the walls. To obtain accurate structural response results, an additional integrated jumper needs to be constructed or analyzed via fluid–structural interaction (FSI) simulation. Li et al. [12] carried out experiments and one-way coupled FSI simulations to visualize the flow pattern of an air–water mixed flow in a rigid M-shaped jumper. The dynamic responses were analyzed with a focus on bubble generation in the first rising segment. Li et al. [13] experimentally investigated the slug characteristics and FIV of a rigid M-shaped jumper and discovered that resonance may occur when the slug frequency is close to the natural frequency of the jumper. And electrical capacitance tomography (ECT) was utilized to capture four typical flow patterns in an M-shaped jumper conveying a mixed air–water two-phase flow: stratified flow, bubble flow, annular flow, and slug flow [14]. Li et al. [15] carried out experiments on a mixed air–water two-phase flow to investigate the flow characteristics of a Z-shaped rigid jumper. The flow patterns, including stratified flow, wavy flow, bubble flow, annular flow, churn flow, and slug flow, were recorded using a high-speed camera.

Assessing the structural impact of changes in flow patterns within the jumper necessitates an evaluation based on the structural response results. Determining the safety of the flow pattern inside each segment requires a thorough structural assessment. FSI simulation provides a means to perform structural response calculations on a full-size jumper. In a study by Zhu et al. [16], two-way FSI simulations were employed to investigate the limited dynamic responses and flow pattern variations of a mixed air–water two-phase flow within a reversed U-shaped jumper. Zhu et al. [17] utilized two-way coupled FSI simulations to numerically examine the multiphase FIV of an M-shaped jumper. Following the input slug flow, variations in flow patterns inside each segment were observed, revealing that the most severe oscillations occurred in the middle of the long span of the jumper.

Although the focus of a substantial amount of research is on either the internal or external flows of jumpers, a common research objective is to extend the investigations to comprehend the effects of combined internal and external flows. Despite this, there are a limited number of studies specifically addressing jumpers under the influence of both internal and external flows. The scarcity of such studies is primarily attributed to the inherent complexity of the multi-field problem, necessitating the simultaneous consideration of both flow fields within the intricate model of the jumper.

Semi-empirical models are commonly employed in the analysis of underwater pipelines such as risers under the combined influence of internal and external flows. When exploring risers subject to both flows, Euler–Bernoulli beam models are often utilized, and the assessment of VIVs is based on wake oscillator models. In these studies, vibrational characteristics and modal changes have been predominantly focused on, and multiphase internal flows and unique states, such as rotating and slug flows, have been considered. Xie et al. [18] examined the nonlinear dynamic response of a flexible riser conveying variable-density flow by modeling VIVs using the van der Pol wake oscillator. Gu et al. [19] developed a dynamic model for both internal and external flows based on the Euler–Bernoulli beam model, slip ratio coefficient model, and wake oscillator model. They investigated variations in the frequency, amplitude, and mode of riser vibrations for different internal flow rates, void fractions, aspect ratios, and external flow rates. Leng et al. [20] explored the effects of varying internal flow velocities and densities on VIV responses under different support conditions utilizing the finite element (FE) method and Newmark method for numerical modeling. Zhang and Chen [21] conducted a VIV assessment for a high-velocity helical flow inside a flexible riser, employing the Lagrangian slug tracking model and classical van der Pol oscillator model for the VIV response. Duan et al. [22] studied the cross-flow VIVs of a flexible riser conveying flows with different density ratios and velocities via the FE method for a semi-empirical hydrodynamic force model. Xu et al. [23] studied the VIV response characteristics of a curved riser conveying internal flows of varying velocities under steady or oscillatory flow conditions. The riser was modeled as a simply supported Euler–Bernoulli beam, and the external flow was calculated using the van der Pol equation.

A subsea jumper, characterized by multiple bends, possesses a complex geometry different from that of risers. In the analysis of subsea jumpers, a simplified approach, in which the structure of the jumper is represented as multiple straight pipes, is often implemented in semi-empirical models. However, this simplification results in the complex multiphase flow conditions within the bends and the shedding of complex external vortices being overlooked. Additionally, the stress levels at the bends tend to increase. Consequently, a more comprehensive investigation is imperative to thoroughly assess the suitability and applicability of semi-empirical models in the specific context of jumpers.

Conducting research through experiments is constrained by the considerable cost of the experimental system. For internal flow analysis, an internal circulating multiphase pipeline system is necessary, while external flow requires water pumps or a constant velocity drag tank. A large total span and flow rate require a high pump power and large circulating pipe diameter, contributing to a relatively high implementation cost. Therefore, rigorous verification of the reliability and safety of these methods is needed. Consequently, a more practical approach involves utilizing CFD and finite element analysis (FEA) instead of semi-empirical models and experiments for analyzing the dynamic response of the jumper under both internal and external flows. This approach allows for the observation of specific variations in the internal and external flow fields, the consideration of multiple variables, and the provision of a variety of information.

In this paper, a one-way coupled fluid–structure interaction process is used to analyze the dynamic response of an M-shaped jumper under combined internal and external flows. First, CFD simulations are solved separately for the internal and external fluid domains of the M-shaped jumper. Once the flow conditions are stable, pressure histories are exported for the inner and outer wall grid surfaces. Then, FEA is conducted to obtain the dynamic response. The pressure histories on the structural element faces are interpolated using inverse distance weighting (IDW). Subsequently, the displacement, vibration, and stress are obtained through implicit dynamic analysis, and the fatigue damage is assessed based on the stress results.

The application of this method to determine the dynamic response of a structural model under the influence of multiple flow fields offers the following advantages:

(1)

The internal and external flow fields are segregated for CFD simulation and subsequently loaded onto the FE model. This approach ensures that the influences of multiple flow fields are appropriately considered.

(2)

The algorithm selected for each fluid domain and structural model is characterized by a high degree of flexibility, allowing for additional modifications and supplementary calculations to be conducted based on specific requirements as needed.

(3)

The quantity of CFD simulations is minimized. Specifically, for combinations of internal and external flow conditions, only two corresponding individual flow conditions are chosen for pairing.

The structure of this paper is organized as follows: Section 2 introduces the model and specific working conditions, with the monitoring cross-sections and point locations used to describe the dynamic changes in the jumper. In Section 3, the governing equations of the CFD and FEA simulations and a detailed computational scheme are presented. Section 4 comprises the convergence analysis, with separate discussions on mesh and timestep considerations for each part of the simulation, given the presence of two fluid domains and one structural model. In Section 5, the results, including the displacement, vibration, stress, and fatigue damage of the rigid M-shaped jumper, are provided. Finally, the major conclusions drawn from the study are presented in Section 6.

2. Problem Description

2.1. Analysis Model

The M-shaped jumper analyzed in this paper is configured as a vertical overhang, connected to other equipment at both ends, without additional support elsewhere. Positioned along the X-axis, with the height corresponding to the positive Z-axis, the jumper is depicted in Figure 1. It has an outer diameter (D) of 60 mm, an inner diameter (d) of 48 mm, and a wall thickness of 6 mm. The structure is left–right symmetrical, comprising eight bends, three horizontal segments, and four vertical segments with dimensions specified in

Table 1. Specific parameters of the jumper.

Parameter	Value (mm)
Total length	3744
Inner diameter ($d$)	48
Outer diameter (D)	60
Wall thickness	6
Radius of bends	72
L1	800
L2	800
L3	1000
L4	2000

.

The flow field inside the jumper transfers an air–water two-phase flow. The fully developed stratified flow enters the jumper in the positive X-direction and exits after the bends change the flow direction multiple times. In analyzing the flow field inside the jumper, the effects of phase ratio and mixed-flow velocity are considered. The flow field outside the jumper is a constant-velocity field, flowing positively along the Y-axis towards the jumper. The internal and external flow field, as well as the FE model of the jumper, are analyzed at the same location and named consistently to facilitate pressure interpolation.

In Figure 2, eight monitoring points are located at the inlet, forming eight paths along the jumper from the inlet to the outlet using 74 cross-sections. The overall distribution of displacement, acceleration, and stress of the jumper is described based on these cross-sections. Given the absence of expansion or wall thickness changes, the overall displacement and acceleration are assessed using the path generated by “Out2” along the outer wall surface. In the analysis, cross-section 38 in the middle of the long-suspended span and cross-section 20 on the bend are marked in Figure 2.

2.2. Flow Conditions

The analysis process employed in this paper involves conducting CFD simulations for two fluid domains separately. For the inner fluid domain of the jumper, it is imperative to achieve a stable working status for each flow condition. This stability is reflected in the historical pressure at the inner wall monitoring points and the consistent variation in the flow pattern in each segment. The pressure history on the inner wall surface of the relevant segment is then exported from the inner wall surface grids. Similarly, for the external fluid domain, regular vortex shedding on the segments for each flow condition and a consistent pressure history of the outer wall monitoring points are necessary. The historical pressure on the outer wall surface of the segment of interest is exported based on the outer wall surface grids.

In the internal fluid domain, the mixed velocity and gas–liquid ratio, with a total of five designated flow conditions, are considered. The external fluid domain accounts for the flow velocity, with a total of three flow conditions. The external flow is directed transversely to the M-shaped jumper, perpendicular to its plane. This direction (90°) is chosen because it has the largest projected area across the M-shaped jumper and is considered the most critical direction. Consequently, only one direction (90°) is currently assigned for each flow condition. By summarizing the flow conditions for both internal and external flows, a total of 15 conditions are established.

The fluid domains are analyzed separately through the one-way coupled fluid–structure interaction process. Instead of conducting 15 CFD calculations, only 8 calculations are needed for all flow conditions. The detailed flow conditions are presented in

Table 2. Detailed parameters of the flow conditions.

Internal Flow Conditions
Case	Flow condition		Simulation time (s)
	Mixed velocity vi (m/s)	Volume fraction of water rw	Stable state (s)	Pressure export (s)
1	1.00	0.5	15	10
2	2.00	0.2	15	10
3	2.00	0.5	15	10
4	2.00	0.8	15	10
5	4.00	0.5	15	10
External Flow Conditions
Case	Flow condition		Simulation time (s)
	Velocity ve (m/s)	Flow direction (°)	Stable state (s)	Pressure export (s)
1	0.50	90	30	10
2	0.75	90	20	10
3	1.00	90	20	10
Combined Internal and External Flow Conditions
Case (total)	Flow condition		Simulation time (s)
15 (5 × 3)	According to the internal and external flow conditions, respectively		10

.

3. Method

3.1. Fluid Domain

In this paper, the internal and external fluid domains of a rigid M-shaped jumper are simulated using the commercial CFD software Fluent (2020 R1). The numerical model is based on the unsteady 3D Reynolds-averaged Navier–Stokes (RANS) equations, wherein velocities and other relevant solution variables represent time-averaged values. The conservation equations for mass and momentum can be expressed in the form of the Cartesian tensor as follows:

$${\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho u_i\right)=0$$

(1)

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho u_i\right)+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho u_i{u}_j\right)=-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left[\mu \left({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}-{\displaystyle \frac{2}{3}}{\delta }_{ij}{\displaystyle \frac{\partial u_l}{\partial x_l}}\right)\right]+{\displaystyle \frac{\partial }{\partial x_j}}\left(-\rho \overline{u_i^{\prime }{u}_j^{\prime }}\right)$$

(2)

where ρ represents the density, and in the case of multiphase flow, it represents the average density of the mixture. u_i and u_j are the mean velocity components. u_i′ and u_j′ are the fluctuating velocity components. μ represents the dynamic viscosity. p represents the mean static pressure. δ_ij represents the Kronecker delta. $-\rho \overline{u_i^{\prime }{u}_j^{\prime }}$ is the Reynolds stress. Subscripts i and j represent integer values ranging from 1 to 3.

By adopting the one-way coupled fluid–structure interaction process in this paper, the fluid domains inside and outside the jumper are initially simulated independently. Subsequently, the pressure data are loaded into the structural model. As a result, different turbulence models and calculation schemes can be employed for the internal and external fluid domains, which operate relatively independently and do not impact each other. This separate calculation and loading of the fluid simulation contribute to the substantial flexibility of the combined CFD-FEA method. This approach not only significantly reduces the computational burden but also enhances the efficiency of the analysis.

In the internal fluid domain, the realizable k-ε turbulence model is adopted in the CFD simulations for conveying a mixed gas–liquid two-phase flow. The standard wall function is applied to address near-wall areas. The mixed gas–liquid two-phase flow within the jumper consists of air and water, with densities of 1.225 kg/m³ and 998.2 kg/m³, respectively, and kinetic viscosities of 1.7894 × 10⁻⁵ Pa·s and 0.001003 Pa·s, respectively. A surface tension model is implemented between the two phases, with a constant surface tension coefficient of 0.072 N/m.

In the external fluid domain surrounding the M-shaped jumper, the SST k-Ω turbulence model is implemented in the CFD simulations for flow separation around the pipe wall. The external fluid domain is composed of a single homogeneous liquid, water, with a density of 1000 kg/m³ and a kinetic viscosity of 0.001 Pa·s.

To address different flow phenomena within the internal and external flow fields, specific turbulence models are employed. Regarding the internal flow field, a high y+ mesh division is utilized with the realizable k-ε turbulence model along with the volume of fluid (VOF) model to effectively handle multiphase problems. Conversely, for the external flow field, a low y+ mesh division is used with the SST k-Ω turbulence model to primarily observe the wall-induced flow separation. Further detailed complementary settings can be found in

Table 3. Detailed settings used in the CFD simulations.

Method		Internal Fluid Domain	External Fluid Domain
Turbulent model		Realizable k-ε	SST k-Ω
Pressure–velocity coupling scheme		PISO	PISO
Spatial discretization	Gradient	Least squares cell-based	Least squares cell-based
	Pressure	PRESTO!	PRESTO!
	Momentum	QUICK	QUICK
	Volume fraction	Geo-Reconstruct	-
	Turbulent kinetic energy	Second order upwind	Second order upwind
	Specific Dissipation Rate	Second order upwind	Second order upwind
Transient formulation		First order implicit	Bounded second order implicit

.

3.2. Structural Model

Abaqus (2022) is utilized to conduct the FEA of the structural model. The implicit dynamic analysis algorithm of Abaqus/Standard is employed to determine the dynamic response under both internal and external flows, only internal flow, and only external flow. The discrete governing equation for the transient motion of the structural model can be expressed as follows:

$$\left[\mathrm{M}\right]\ddot{X}+\left[\mathrm{C}\right]\dot{X}+\left[\mathrm{K}\right]X=F$$

(3)

[M], [C], and [K] in Equation (3) represent the mass, damping, and stiffness matrices, respectively. F represents the current load on the node of interest. $X$ represents the node displacement, $\dot{X}$ represents the node velocity, and $\ddot{X}$ represents the node acceleration.

The Newmark-β method is a widely used implicit time integration scheme. The Newmark-β method is employed for the M-shaped jumper to obtain the iterative solution of the dynamic response. The method is primarily controlled by the following three parameters, α, β, and γ, as shown in Equation (4). When α = 0, the Newmark-β method is employed.

$$\beta ={\displaystyle \frac{1}{4}}(1-{\alpha }^2), \gamma ={\displaystyle \frac{1}{2}}-\alpha , -{\displaystyle \frac{1}{2}}\le \alpha \le 0$$

(4)

At the new moment n + 1, with increasing time $\Delta t$, the transient structural motion is obtained through implicit dynamic analysis as follows:

$$\begin{array}{ll}({\displaystyle \frac{1}{\beta {{(}_{\Delta }t)}^2}}\left[\mathrm{M}\right]+{\displaystyle \frac{\gamma }{{\beta }_{\Delta }t}}\left[\mathrm{C}\right]+\left[\mathrm{K}\right]){\left.X\right|}_{n+1}=& {\left.F\right|}_{n+1}+\left[\mathrm{M}\right]({\displaystyle \frac{1}{\beta {{(}_{\Delta }t)}^2}}{\left.X\right|}_n+{\displaystyle \frac{1}{{\beta }_{\Delta }t}}{\left.\dot{X}\right|}_n+({\displaystyle \frac{1}{2\beta }}-1){\left.\ddot{X}\right|}_n)\\ & +\left[\mathrm{C}\right]({\displaystyle \frac{\gamma }{{\beta }_{\Delta }t}}{\left.X\right|}_n+({\displaystyle \frac{\gamma }{\beta }}-1){\left.\dot{X}\right|}_n+({\displaystyle \frac{\gamma }{2\beta }}-1{)}_{\Delta }t{\left.\ddot{X}\right|}_n)\end{array}$$

(5)

A frequency-dependent Rayleigh damping model is incorporated into the dynamic analysis of the FE model’s dynamic response. In Rayleigh damping, it is assumed that the damping matrix [C] can be expressed as a linear combination as follows:

$$\left[\mathrm{C}\right]=a\left[\mathrm{M}\right]+b\left[\mathrm{K}\right]$$

(6)

In Equation (6), a and b represent Rayleigh damping parameters. By defining a and b, the Rayleigh damping is determined. These parameters are related to the mass and stiffness matrices, and their relationship with the damping ratio (ξ) is as follows:

$$\xi ={\displaystyle \frac{a}{2{\omega }_n}}+{\displaystyle \frac{b{\omega }_n}{2}}$$

(7)

In Equation (7), the damping ratio (ξ) is set to 1.2% in this paper, in which subsea pipelines with similar materials and environmental conditions are assumed [24]. Here, n represents the nth vibration mode, and ω_n (rad/s) denotes the natural frequency of mode n. The parameters of the Rayleigh damping model are linked to the natural frequency of each mode of the M-shaped jumper. Consequently, the parameters a and b can be determined based on the lower and upper bounds i and j of the relevant modes:

$$a=\xi {\displaystyle \frac{2{\omega }_i{\omega }_j}{{\omega }_i{\omega }_j}},\hspace{1em}\hspace{1em}b=\xi {\displaystyle \frac{2}{{\omega }_i+{\omega }_j}}$$

(8)

For the rigid M-shaped jumper in this paper, the natural frequency range of the relevant modes is assumed to be within 100 Hz for the first 13 modes. The parameters for the Rayleigh damping model are determined to be a = 1.160 and b = 4.368 × 10⁻⁵.

And for the other fundamental material properties, the density of the solid element is 7850 kg/m³, with the Young’s modulus and Poisson’s ratio set at 2.1 × 10¹¹ Pa and 0.3, respectively.

3.3. Pressure Interpolation

In the respective CFD simulations, pressure histories are recorded at the grid face centers of both the inner and outer wall surfaces. Using the locations and pressure histories at the fluid grid face centers, IDW interpolation is applied to compute the pressure history at the face centers of the structural element faces.

To calculate the pressure at a specific point for interpolation at the current moment t, the IDW interpolation method is utilized by selecting the c data points closest to the point for interpolation. Subsequently, the distances from the point for interpolation to each of these data points are computed using Equation (9).

$$di{s}_n=\sqrt{{(x-x_n)}^2+{(y-y_n)}^2+{(z-z_n)}^2}$$

(9)

In Equation (9), n represents the nth point nearest to the point for interpolation, $1\le n\le c$. dis_n is the distance of the nth data point from the point for interpolation. x, y, and z are the locations of the points used for interpolation. x_n, y_n, and z_n are the coordinates of the nth data point. The weight W_n of each data point is calculated in Equation (10).

$$W_n={\displaystyle \frac{di{s}_n^{-e}}{{\displaystyle {\sum }_{i=1}^{c}di{s}_i^{-e}}}}$$

(10)

In Equation (10), the value of the distance is controlled through e to influence the interpolation. Using the weights W_n and the pressure p_n,t of each data point n at the current moment t, the pressure P_t at the point for interpolation can be computed by Equation (11). For the calculations in this paper, the values of c and e are 4 and 1, respectively.

$$P_t={\displaystyle {\sum }_{n=1}^c{W}_n{p}_{n,t}}$$

(11)

As shown in Figure 3, four data points are chosen for interpolation calculations, expressed in ascending order as 1 to 4, relative to their distances from the point for interpolation. The reciprocal of the distance is employed as the weighting factor. In the case of an exact match between the point for interpolation and one of the data points, the variable associated with that data point is directly selected as the variable for the point for interpolation. This method is deemed sufficiently accurate for the interpolation calculations in this paper.

The pressure of the point for interpolation is calculated at multiple timesteps in a loop to generate the pressure history. The acquired pressure history at the grid face center on the inner and outer wall surfaces is applied to the structural model using uniform pressure. For accurate pressure data, similar mesh sizes are necessary due to IDW interpolation between fluid grid face centers and structural element face centers.

3.4. Fatigue Damage

The focus of real-time monitoring for fatigue damage is on sensing long-term stress results. The current value of fatigue damage is obtained by adding the incremental damage of the concerned time to the previous damage [25]. Using the software Fe-safe (2022), stress results derived from the FEA are employed for additional fatigue damage analysis to identify the location of the maximum damage. Fatigue damage assessment is performed using the time history results of direct stress and shear stress obtained from FEA. The algorithm is assessed using the critical plane method, which takes into account the stress results on multiple planes where a point is located. Fatigue cycles and stress amplitudes are determined separately for each plane, followed by the application of a mean stress correction. The final fatigue damage at a specific location is the maximum fatigue damage among all planes.

The rainflow counting method, widely accepted as the most suitable signal analysis approach for interpreting fatigue life, is employed for the analysis of historical stresses. The extensive stress data over time are processed to extract the stress amplitude and the corresponding cycle count, disregarding temporal information. However, varying mean stresses also impact the fatigue life. For a given stress amplitude, a lower mean stress results in a longer fatigue life. Hence, mean stress correction is crucial in fatigue analysis. In this paper, a linear Goodman stress correction, as expressed in Equation (12), is utilized.

$${\displaystyle \frac{S_a}{S_i}}+{\displaystyle \frac{S_m}{S_{ult}}}=1$$

(12)

where S_a represents the current stress amplitude under consideration, S_m is the corresponding mean stress, and S_ult is the ultimate tensile strength of the material. Accordingly, S_i is the corrected stress amplitude at zero mean stress.

The corrected stress amplitude S_i and its corresponding number of cycles n_i are obtained, and the maximum number of cycles N_i given by the S–N curve of the material can be used for damage calculation, as shown in Equation (13).

$$D_i={\displaystyle \frac{n_i}{N_i}}$$

(13)

4. Convergence Analysis

4.1. Internal Fluid Domain

4.1.1. Geometry and Mesh

The specific arrangement of the internal fluid domain in this paper is illustrated in Figure 4. The rigid M-shaped jumper is positioned positively along the X-axis, with the height direction representing the positive Z-axis direction, situated in the XOZ plane. If the displacement of the jumper resulting from the load is confined to the XOZ plane, the vibration is considered in-plane. If the jumper undergoes substantial displacement in the Y-direction, the vibration type is categorized as out-of-plane.

The inlet and outlet boundary conditions are set as a velocity inlet and a pressure outlet, respectively. The remaining segments are defined as walls. The fluid domain is configured with inlet and outlet segments at both ends. Extended inlet and outlet segments are designed to minimize the impact of boundaries on the segment of interest. Starting from the inlet, the flow traverses a long development segment before reaching the M-shaped segment of interest. The mixed air–water flow induces stratified flow due to density variations, which are determined by the specified volume fraction. Eight pressure monitoring points are positioned along the jumper, from the inlet to the outlet, to observe changes in wall pressure.

A real jumper utilized in a subsea production system is linked to the subsea equipment through segment1 and segment7 at its ends. In contrast, the M-shaped segment in this study is connected to segment_inlet and segment_outlet at both terminations through bends. The fluid domain and the subsequent model configuration used in this paper are referenced to the multiphase flow circulation pipe system at Dalian Maritime University [14].

The unstructured mesh is created based on the geometric model of the M-shaped jumper. Given the absence of specific structures on the inner wall, the sweep method is utilized to generate a mesh. The detailed mesh configuration is illustrated in Figure 5. The first layer, situated within 5.5 × 10⁻⁴ m of the segment wall, features a high y+ mesh division. The gradient variation near the wall is addressed using the standard wall function. The grid dimensions remain consistent along the length of both the vertical and horizontal segments. Additional refinement is applied at the bends to capture the flow state with greater precision.

4.1.2. Mesh and Timestep Analysis

The mesh is generated from the cross-section grids for the entire internal fluid domain. A total of 64 parts are divided along the circumference, which is deemed sufficient for accurately capturing the air–water two-phase distribution within the cross-section. Three different meshes are created by setting distinct grid sizes along the length, measured at 0.075 m, 0.01 m, and 0.0125 m, and simulated with timesteps of 0.001 s. The total number of meshes for each configuration is 7.4 × 10⁵, 9.2 × 10⁵, and 1.2 × 10⁶, respectively. Simulations are then conducted using three different time steps (0.002 s, 0.001 s, and 0.0005 s), and the mesh with the intermediate number is selected.

The mid-level working condition is employed for the convergence analysis as a fundamental working condition (v_i = 2 m/s, r_w = 0.5). The average pressure of the eight monitoring points over the last 5 s of the steady state serves as the evaluation variable. As shown in Figure 6, the mesh with 9.2 × 10⁵ elements and a timestep of 0.001 s is chosen for the subsequent CFD simulations within the jumper. The locations of the pressure monitoring points are indicated in Figure 4.

4.2. External Fluid Domain

4.2.1. Flow around a Cylinder

If 3D CFD simulations of the entire jumper are conducted directly, the computational burden caused by the large total span and the number of meshes can significantly impact the efficiency. Therefore, in this paper, a mesh and timestep analysis is initially performed for 2D flow around a cylinder. Subsequently, the same meshing method is applied to the 3D simulation of the jumper.

The diameter of the cylinder equals the outer diameter of the jumper, D = 0.06 m. A 2D computational domain with only a single layer of grids in the thickness direction is employed. The center of the cylinder is positioned at 10D from the inlet, top, and bottom boundaries and 40D from the outlet boundary to minimize the effect of the boundaries on the fluid domain in the area around the cylinder. The boundary conditions for the inlet and outlet are the velocity inlet and pressure outlet, respectively. The surface of the cylinder is treated as a wall, and the rest of the domain boundary conditions are symmetrically set. The specific boundary locations and conditions are illustrated in Figure 7. The turbulence model and other setup parameters remain consistent with the external fluid domain of the jumper.

Since the meshing method of the flow around a cylinder is considered to be applied to the 3D fluid domain mesh of the jumper, it is necessary to decrease the number of grids as much as possible to improve the efficiency of 3D simulations while maintaining computational accuracy. Figure 8 shows the changes from the nearby area to the outer boundary through four grid size changes to capture the flow phenomena. The grid is refined at a certain distance behind the cylinder to capture the vortex shedding in the wake flow region.

The grid sizes in the proximity of the cylinder are partitioned into basic sizes of D/15, D/20, and D/25. The thickness (h) of the fluid domain is set to half the basic size, ensuring that only a single layer of mesh is retained throughout the entire fluid domain. The boundary layer comprises 12 layers with a growth rate of 1.5, and the thickness of the 12th layer is half of the basic size. This layering method is employed to achieve a wall y+ approaching 1. The chosen velocity values for the external fluid domain in this paper, 0.50 m/s and 1.00 m/s, are utilized for the convergence analysis, corresponding to Reynolds numbers of 30,000 and 60,000, respectively.

The convergence analysis begins with three different basic sizes with a timestep of 0.001 s. Subsequently, the intermediate basic size is selected, and simulations are conducted with three different timesteps: 0.0005 s, 0.001 s, and 0.002 s. Given that the primary focus of this paper is the pressure distribution on the wall, 16 pressure monitoring points are evenly distributed across the cylinder. The nondimensional average pressure for the last 10 s of the steady state is used in the convergence analysis. Upon comparison of the nondimensional average pressure at each point as shown in Figure 9, C_D (drag coefficient) and C_L (lift coefficient) as shown in Figure 10, a basic size of D/20 and a timestep of 0.001 s are chosen for further analysis.

4.2.2. Geometry and Mesh

The jumper is positively aligned along the X-axis, and the Z-axis denotes the positive height direction. The boundaries of the external fluid domain are shown in Figure 11. The fluid domain is designed considering the fluid domain of the flow around a cylinder, 10D from the inlet and 40D from the outlet. For the M-shaped segment of interest, a distance of 5D is retained from the top, bottom, left, and right boundaries. The inlet and outlet boundaries are assigned as the velocity inlet and pressure outlet, respectively. The outer wall surface of the jumper is designated a no-slip wall. Symmetry boundaries are applied to other boundaries. The external flow direction is along the positive Y-axis, and three velocity values (0.50 m/s, 0.75 m/s, and 1.00 m/s) are chosen, corresponding to Reynolds numbers of 30,000, 45,000, and 60,000, respectively.

For the external fluid domain, the mesh configuration is the same as the principles established for the flow around a cylinder. The basic grid size near the jumper is D/20. The grid size changes four times from the nearby area to the external domain boundaries. The boundary layer for the jumper is composed of 12 layers with a growth rate of 1.5, and the thickness of the 12th layer is half the basic size. Refined areas are positioned behind each segment to capture vortex shedding as the flow traverses through. The sphere-type refined areas are strategically placed behind bends to handle intricate vortex shedding near these features. The total number of grids in the external fluid domain is 9.7 × 10⁶. The specific division of the mesh is shown in Figure 12. This mesh setup ensures a comprehensive representation of the external fluid domain, incorporating refined regions to capture complex flow phenomena, particularly around bends and behind segments, with an overall focus on accuracy and computational efficiency.

4.3. Structural Model

4.3.1. Geometry and Mesh

The FE model for the jumper mirrors the configuration and positioning within the fluid domains. In Figure 13, the jumper is oriented positively along the X-axis, with the height being the same as that in the Z-axis positive direction. The FE model specifications include an inner diameter of 48 mm, an outer diameter of 60 mm, and a wall thickness of 6 mm. Each segment within the FE model corresponds spatially with its counterpart in both the internal and external fluid domains. This alignment ensures that the FE face centers can be seamlessly matched with nearby grids in the fluid domain during subsequent interpolation calculations. The name and length of each segment are consistent across the internal and external fluid domains.

The pressure history at the center of the element face is obtained through interpolation and subsequently applied to the faces using uniform pressure in this study. To ensure accurate loading with the correct pressure on the faces, it is imperative to maintain a grid size for the FE model similar to the mesh size of the fluid domains. However, due to the extended analysis duration, minimizing the number of elements becomes essential for alleviating the storage capacity requirements. The sweep method is employed for mesh generation in the FE model of the M-shaped jumper, as depicted in Figure 14. In the FE model, an eight-node brick element with reduced integration (C3D8R) is used. The specific meshing methods consider the sweep cross-section, the number of parts divided on the circumference, and the layers of the wall thickness. Furthermore, the sweep paths along the straight segments and bends incorporate a defined mesh size. A detailed division method is provided in

Table 4. Specific division method of the FE model.

Mesh	Cross-Section		Path
	Layers of the Wall Thickness	Parts Divided on the Circumference	Straight Segments (m)	Bends (m)
L3C16	3	16	0.02	0.01
L3C16R	3	16	0.01	0.01
L4C16	4	16	0.02	0.01
L5C16	5	16	0.02	0.01
L3C20	3	20	0.02	0.01
L3C24	3	24	0.02	0.01

.

4.3.2. Modal

Modal analysis is conducted on the FE model of the M-shaped jumper, employing various meshing methods for convergence analysis. The first 13 modes within the frequency range of 100 Hz, as illustrated in Figure 15, are focused on in the analysis. Minimal discrepancies are observed in each mode among the different meshing methods for the FE model, and the L3C16 method is suitable for subsequent calculations.

Table 5. Frequencies and vibration modes.

Mode	Frequency (Hz)	Type
1	8.520	Out-of-plane
2	13.825	In-plane
3	17.473	In-plane
4	20.628	Out-of-plane
5	21.707	Out-of-plane
6	25.497	In-plane
7	31.608	Out-of-plane
8	54.246	In-plane
9	59.186	In-plane
10	74.268	In-plane
11	74.622	Out-of-plane
12	77.364	Out-of-plane
13	78.933	Out-of-plane

provides a comprehensive list of frequencies and corresponding vibration types for these 13 modes.

The modal analysis results offer insights into the distinctions between the laboratory model and its full-scale counterpart. Subsea jumpers deployed under practical working conditions may exceed 20 m in size, exhibiting lower natural frequencies than their laboratory-scaled counterparts.

4.3.3. Time History Analysis

Convergence analysis involves applying historical pressure data to the element faces of the FE model. Given that the timestep for the historical load is 0.001 s, as determined from previous CFD simulations, a timestep of 0.001 s is used in the dynamic response calculations for the FE model. To balance storage considerations and computational efficiency, a 5 s simulation is conducted. Accounting for the initial moments that require a buffer, the results from the last 3 s are used for historical convergence assessment. The chosen flow conditions are v_i = 2 m/s and r_w = 0.5 for the internal air–water mixed flow and v_e = 1.00 m/s for the external flow in the 90° direction.

The evaluation of jumper displacement involves comparing different meshes. The average historical displacement of nodes distributed along the outer wall of the jumper is used for this comparison. The monitoring points’ locations, as illustrated in Figure 2, are generated by 74 cross-sections. The historical averages provide insights into the specific form of displacement. As depicted in Figure 16, with the bends shown in gray, minimal differences are observed among the various mesh divisions in all three directions, including the total displacement.

The assessment of stresses applied to the FE model involves using the maximum von Mises stress at each monitoring point of the concerned simulation time. As depicted in Figure 17, there is only a slight difference in the maximum von Mises stress at each monitoring point. Considering both the evaluation results of displacement and stress, the L3C16 mesh is chosen for the subsequent historical dynamic calculations.

5. Results

5.1. Displacement

The analysis of the overall displacement of the jumper involves evaluating the average historical displacement over the last 5 s of the 10 s FEA results. The same analysis time is used in the subsequent analyses. This period excludes the initial 5 s, which are deemed unstable due to the loads acting just on the FE model. For the analysis of the displacements, the average of the historical displacements is used to determine the deformation trend of the jumper, and its steady-state deformation under environmental loads.

A comprehensive analysis of the overall displacement is conducted under basic working conditions (v_i = 2 m/s, r_w = 0.5, and v_e = 1.00 m/s). Figure 18a shows the distribution of the displacement along the jumper for this working condition with the bends highlighted in gray. In the X-direction, both segment2 and segment6 exhibit displacement toward the center. In the Y-direction, influenced by the positive Y-direction external flow, the maximum displacement occurs at the midpoint of the long span. In the Z-direction, the maximum displacement is observed at the middle of the long overhang span due to the gravitational effects. Moreover, the peak total displacement is centralized in the middle of the long span.

Focusing on the historical displacement at P38 (monitoring point 38 in the middle of the long span), as shown in Figure 18b, the displacement in the X-direction oscillates around zero, displacement in the Y-direction occurs due to external flow, and downward displacement due to gravity is consistently present in the Z-direction.

To consider the effect of external flow velocity on displacement, three working conditions with only external flow and three with combined internal and external flow are compared at different external flow velocities (0.50 m/s, 0.75 m/s, and 1.00 m/s). As shown in Figure 19a, the displacement in the X-direction is minimally affected by the external flow velocity. However, compared to only external flow, the combined internal and external flows increase the displacement of segment2 and segment6, causing them to deform towards the middle of the jumper. As shown in Figure 19b, the displacement in the Y-direction is primarily influenced by the external flow. As the velocity increases, the displacement at the monitoring points along the jumper also increases, with the maximum displacement occurring in the middle of the long overhanging span due to the lack of support except at the two ends. In Figure 19c, the displacement in the Z-direction is minimally affected by the external flow velocity. The overall downward displacement increases when both internal and external flows act together compared to only external flow, potentially causing the long overhanging span to contact the seafloor soil more easily. In Figure 19d, increasing the external flow velocity significantly increases the total displacement of the long overhanging span. The displacement is greater under combined internal and external flows compared to only external flow.

To analyze the effect of internal flow velocity on the overall displacement of the jumper, three working conditions with the same water volume fraction (r_w) but different internal flow velocities (v_i) are selected under conditions of only internal flow and combined internal and external flows. As shown in Figure 20a, the displacement in the X-direction shows no significant variation with different internal flow velocities. In Figure 20b, the Y-direction displacement is primarily influenced by the external flow, with no significant displacement observed at any monitoring point under only internal flow. In Figure 20c, the internal flow velocity has a minimal effect on the Z-direction displacement, with an increase only in the middle of the long overhanging span as the internal flow velocity increases. Finally, as shown in Figure 20d, changing the internal flow velocity has a minimal impact on the overall displacement of the jumper.

Using three different volume fractions of water (r_w) with the same internal flow velocity (v_i), six different working conditions are comparatively analyzed for only internal flow and combined internal and external flows. As shown in Figure 21a, for the displacement in the X-direction, increasing the volume fraction of water mainly affects segment2 and segment6, which are displaced more toward the middle of the jumper as the volume fraction of water increases. In Figure 21b, the displacement in the Y-direction is primarily dominated by the external flow, with the internal flow having no significant effect. In Figure 21c, increasing the volume fraction of water significantly decreases the height of the long overhanging span. Finally, as shown in Figure 21d, changing the volume fraction of water significantly increases the displacement of the long-suspended span.

For the overall displacement of the jumper, the external flow is the primary influencing factor. The constant external flow leads to significant displacement in the direction of the external flow velocity, making the jumper’s displacement most affected by the external flow velocity. Regarding the internal flow, the volume fraction of water has a significant impact, but the influence of the internal flow is limited to the plane where it is located. The overall deformation trend of the jumper shows that both sides deform towards the middle, especially segment2 and segment6. Additionally, the height of the long overhanging segment decreases, which may reduce the distance between the long overhanging span and the seafloor soil. This reduction could result in the possibility of the long overhanging span touching the bottom, influenced by soil particles moved by the external sea current.

The historical displacement at one point is used to compare the one-way coupled FSI simulation process for handling combined internal and external flows as well as single flow. Three different operational conditions are considered for P38: internal flow only, external flow only, and both internal and external flows. As the pressures applied in this study remain consistent under the same working conditions, the dynamic responses of the three distinct flow conditions can be assessed by comparing specific values at each timestep. Given that gravity is a factor in all flow conditions, it is essential to incorporate the results of gravity alone into the results of both internal and external flows before making comparisons with the results of internal flow only and external flow only.

Figure 22a–c show the historical displacement of P38 in three different directions. “In” represents internal flow only; “Ex” represents external flow only; “g” represents gravity only; “C” represents the result caused by both internal and external flows; and “In + Ex” represents the result caused by internal flow only plus the result caused by external flow only. When employing the one-way coupled fluid–structure interaction method, if the gravity of the jumper is not considered in the FEA, the impact on the displacement under both internal and external flows is equivalent to the direct summation of the internal flow only and the external flow only. Comparing the historical averages of the total displacement for all working conditions, as shown in Figure 22d, the historical average displacement caused by both internal and external flows plus the result caused by gravity only is equal to the sum of the result caused by internal flow only and the result caused by external flow only.

5.2. Vibration

The root mean square (RMS) of the historical acceleration at the monitoring points along the jumper under basic working conditions (v_i = 2 m/s, r_w = 0.5, and v_e = 1.00 m/s) is utilized to evaluate the overall vibration, as shown in Figure 23a. In the X-direction, segment2, segment4, and segment6 exhibit more violent vibrations, as these three segments are horizontal. In the Y-direction, there is a significant increase in the vibration of the bends. In the Z-direction, segment3, segment4, and segment5, along with the four connected bends, experience larger vibrations. For the total acceleration, the vibration is maximum at bend3 and bend6.

Focusing on P20, which is located in the middle of bend3, the historical accelerations are illustrated in Figure 23b. The Z-acceleration experiences more intense vibrations with the largest amplitude. The acceleration in all three directions oscillates around 0. From the RMS results, it can be concluded that the acceleration at this point vibrates the most in the Z-direction and the least in the Y-direction.

The RMS distribution of acceleration from the inlet to the outlet of the jumper is analyzed under six working conditions, to investigate the effect of the external flow velocity (v_e) on the vibration of the jumper. As shown in Figure 24a, the effect of the external flow velocity on the vibration in the X-direction is insignificant. Segment2 and segment6 vibrate more in the X-direction, which aligns with the overall tendency of the jumper to deform towards the middle. In Figure 24b, in the three cases where only the external flow acts, the RMS of the acceleration in the middle of the long overhanging span increases with the increase in the external flow velocity, indicating that the amplitude in the middle increases and the vibration becomes more intense with a higher external flow velocity. In Figure 24c, changes in the external flow velocity have no significant effect on the RMS of the acceleration in the Z-direction. Overall, as shown in Figure 24d, the effect of the external flow velocity on the RMS of total acceleration is minimal.

The RMS distribution of acceleration is analyzed by comparing different internal flow velocities (v_i) under only internal flow and combined internal and external flows for the same water volume fraction (r_w). The RMS of acceleration in all three directions and in total increases significantly with the increase in internal flow velocity. In Figure 25a, the amplitudes of segment2 and segment6 are larger in the X-direction. Figure 25b shows that the amplitude of the bend is larger in the Y-direction. In Figure 25c, segment3 and segment5 have larger amplitudes in the Z-direction. And Figure 25d shows that bend3 and bend6 have the largest amplitudes in terms of total acceleration.

Comparative analysis is conducted at the same internal flow velocity (v_i) with three different water volume fractions (r_w) under conditions of only internal flow and combined internal and external flows. As shown in Figure 26a, in the X-direction, the RMS is maximum when the water volume fraction is 0.5. Increasing or decreasing the water volume fraction reduces the RMS. In Figure 26b, in the Y-direction, at bend2, bend4, bend5, and bend7, the RMS is maximum when the water volume fraction is 0.5, while the RMS of the remaining segments increases with the increase in the water volume fraction. Figure 26c shows that in the Z-direction, the RMS is maximum at the middle position of the long overhanging span for a water volume fraction of 0.5, while the RMS of the remaining segments decreases with the increase in water volume fraction. Finally, Figure 26d shows that the total RMS at bend3, bend4, bend5, bend6, segment3, and segment5 decreases with the increase in the water volume fraction, with the RMS being maximum at the remaining positions when the water volume fraction is 0.5.

The RMS of the acceleration represents the amplitude of vibration. Comparing the three variables of external flow velocity, internal flow velocity, and water volume fraction, the effect of external flow velocity on the vibration is insignificant, likely due to the use of a constant flow velocity. Internal flow has a greater impact on vibration, with the internal flow velocity having a more pronounced effect than the water volume fraction. The effect of the internal flow velocity on vibration follows a consistent trend along the jumper from inlet to outlet, while the water volume fraction causes more complex variations in different segments.

A comparison of the historical acceleration at a monitoring point is used to analyze the difference between the dynamic responses of the structure under the influence of single flow and combined internal and external flows. Based on the previous analysis, monitoring point 20 on bend3, which exhibits significant vibration, is selected. As shown in Figure 27a–c, the historical acceleration at this point is small under the influence of only external flow, and it is mainly dominated by the internal flow. Under different conditions, comparing the effects of internal and external flow acting alone and together, and considering the gravity factor, the sum of the squares of the internal and external flow acting alone is equal to the sum of the squares of the internal and external flow acting together and gravity acting alone.

5.3. Stress

The stress variations are analyzed along the eight paths of the jumper, with von Mises stresses used to unify the multiaxial stresses for analysis, as illustrated in Figure 28a–b. Under basic working conditions (v_i = 2 m/s, r_w = 0.5, and v_e = 1.00 m/s), higher stresses are observed due to the fixed supports at both ends. Notably, significant stress increases are observed in bend 2, bend 3, bend 6, and bend 7. Focusing on the cross-section where P20 experiences greater stress, the historical stress is shown in Figure 28b. “Out2”, located below the outer wall of the cross-section, is under higher stress.

As depicted in Figure 29a, different external flow velocities have a minor effect on the overall distribution of the maximum historical stress along the jumper. While there is a slight increase in the maximum stress recorded at each monitoring point with a higher external flow velocity, this factor does not provide the primary influence. In contrast, Figure 29b illustrates a significant increase in the maximum historical stress at each monitoring point of the jumper with a higher internal flow velocity, highlighting the predominant influence of the internal flow velocity on the stress distribution. Additionally, Figure 29c demonstrates that under different water volume fractions, the maximum historical stress is highest when the water volume fraction is 0.5 in segments experiencing a higher stress level. Increasing or decreasing the water volume fraction will reduce the stress level.

Comparing the effects of different internal and external flow elements on the overall stress distribution of the jumper, it can be concluded that the primary influencing factor is the internal flow velocity. Reducing the internal flow velocity can significantly lower the stress levels of the jumper, thereby enhancing the safety of the system. A significant stress concentration is observed at the bends. In addition to the higher stresses at both ends due to fixed constraints, bend3 and bend6 exhibit higher stress levels and require intensive monitoring.

Monitoring point 20 on bend3, characterized by higher stress levels, is analyzed to compare stress time histories under different conditions involving single flow and combined internal and external flows. Figure 30a shows a comparison of three different working conditions: internal flow only, external flow only, and both internal and external flows. The stress results under both internal and external flows, combined with the results caused by gravity only, are equal to the sum of the results caused by internal flow only and external flow only. A similar observation can be made for comparisons using maximum von Mises stresses under different working conditions, as shown in Figure 30b.

5.4. Fatigue Assessment

The software Fesafe is utilized for fatigue assessment. It is a fatigue analysis software for FE models that can directly import stress results calculated by Abaqus. Both the shear and direct stresses are employed in the fatigue calculation. The stress amplitude is corrected using the Goodman curve. Historical stress is quantified using the rainflow counting method to determine the number of cycles and their corresponding stress amplitudes. Subsequently, the S–N curve is applied, and fatigue damage is calculated as a linear sum using Miner’s rule.

The working conditions examined in this paper are based on safe laboratory conditions. The most dangerous working conditions (v_i = 4.00 m/s, r_w = 0.5, and v_e = 1.00 m/s) are chosen for damage calculations while ensuring safety during analysis. To observe the maximum fatigue damage and its location over one day of stable operation, the stresses for this working condition are magnified by a factor of 20.

No significant fatigue damage is observed at all locations except for bend3, bend6, and the ends with fixed support. Due to the structural symmetry of the M-shaped jumper, only bend3 is analyzed for fatigue damage to assess the effect of fluid. Fatigue damage analysis is conducted on bend3, which was identified as the critical component with the most intense vibration and highest stress. Seventeen paths are established along the pipe segment based on the inner and outer walls, each containing 16 nodes per path. The specific monitoring nodes are illustrated in Figure 31.

The inner and outer wall surfaces are unfolded, and a comparison of the maximum historical stresses and damages on these surfaces is presented in Figure 32a–d. The damage value represents the specific fatigue damage value accumulated after one day of operation. The reciprocal of the damage value indicates the number of days the system can operate stably. It is worth noting that a scaling factor is used in this paper, so the value is intended only to show where the maximum damage occurs. On the inner wall, the maximum damage is observed in the negative Y-direction, indicating the highest fatigue damage. This is primarily because the direction of the external flow is negative along the Y-direction. Conversely, on the outer wall, the maximum damage occurs in the negative Z-direction. This is due to the weight of the long unsupported overhanging span. The significant maximum historical stresses coincide with the locations of large damages, implying that the areas of substantial damage can be promptly identified through stress analysis.

In this analysis, the locations where fatigue failure is most likely to occur on the most critical segment of the M-shaped jumper during operation are identified as having higher maximum stress levels over a long period of time, leading to larger fatigue damage values and reduced fatigue life. This information provides vital guidance for the safety monitoring of jumpers during their service life.

6. Conclusions

A one-way coupled fluid–structure interaction process is employed in this study to analyze the dynamic response of a rigid M-shaped jumper under combined internal and external flows. The process starts with separate CFD simulations of the inner and outer fluid domains. Historical pressure data on the inner and outer walls of interest are subsequently exported separately. These pressure data are then loaded onto the FE model using IDW interpolation. Then, implicit dynamic simulations are conducted, followed by fatigue damage assessments based on the obtained stress results. A comparison of the historical displacement, acceleration, and stress results reveals that the method for addressing both internal and external flows produces results equivalent to separately analyzing internal and external flows and then summing them. The combined action of internal and external flow fields is more dangerous compared to the action of a single flow field. The fatigue assessment identifies the maximum values and corresponding locations of damage on the inner and outer wall surfaces of the segment of interest.

This one-way coupled fluid–structure interaction process facilitates the simultaneous consideration of both internal and external flows on the structural model, providing flexibility in adapting simulation schemes. The cases for both internal and external flows are constructed by combining individual flow cases, resulting in a reduction in the number of required CFD simulations. Given the challenges associated with experimental methods and semi-empirical models, particularly for the M-shaped jumper with complex geometry, CFD simulations are employed for both internal and external flows, enabling the comprehensive evaluation of the jumper under these conditions.

In this paper, a methodology for addressing the concurrent influence of multiple flow fields on subsea pipelines is presented. Moreover, an analysis of the historical dynamic response of the jumper is conducted, and the fatigue damage on the relevant segment is assessed, offering valuable insights for the design, monitoring, and maintenance of an M-shaped jumper. The primary conclusions are as follows:

(1)

The overall displacement of the jumper is assessed by averaging point displacements. Analysis of the total displacement and the displacements along the jumper in all three directions from inlet to outlet shows that the long span experiences maximum displacement at the middle, with the external flow velocity dominating the displacement. It is crucial to note that in the FEA, when considering gravity, the displacement results influenced by both internal and external flows, in addition to the results influenced solely by gravity, are equal to the sum of the results influenced solely by internal flow and solely by external flow.

(2)

The vibration of each monitoring point distributed along the jumper is assessed based on the RMS of the historical accelerations. The positions with the most intense vibrations are observed at bend3 and bend6 among the total of eight bends. The effect of external flow velocity is not significant. The variation in internal flow velocity has the greatest impact on vibration, with the amplitude increasing rapidly as the internal flow velocity increased. The variation in amplitude with water volume fraction is more complex, displaying different patterns of change on different segments. The sum of the squares of the internal and external flow acting alone is equal to the sum of the squares of the internal and external flow acting together and gravity acting alone.

(3)

This analysis focuses on the distribution of maximum von Mises stress on the inner and outer walls of the jumper. The highest stresses are observed at both ends, attributable to the fixed supports. Significant stress concentrations are observed at the bends, particularly at bend3 and bend6. The stress increased significantly with a higher internal flow velocity, identified as the primary influencing factor. According to the historical stress results, the impact on the structure from both internal and external flows, combined with the effects induced by gravity only, is equivalent to the sum of the effects when subjected only to internal flow and when subjected only to external flow.

(4)

Fatigue damage assessments are conducted on the inner and outer walls of bend3, where significant stresses and vibrations occur. The location of maximum damage on the inner and outer walls corresponds to the position with a high maximum von Mises stress. This correlation facilitates the quick identification of the location with the maximum damage through the analysis of the von Mises stress.

In the method employed in this study, the displacement of the structural model is assumed not to influence the fluid domains both inside and outside. While the M-shaped jumper is considered rigid and experiences limited displacement, a more comprehensive evaluation using two-way FSI simulations is necessary, particularly under more challenging working conditions. Due to computational resource constraints, the CFD simulations in this work rely on RANS, and the consideration of other turbulence models, such as large eddy simulation (LES), could provide insights into more detailed flow conditions. In subsequent investigations, semi-empirical models for M-shaped jumpers under both internal and external flows may be explored to provide a comparative analysis against the FEA results.