Open AccessArticle

Vortex-Induced Vibration of Deep-Sea Mining Pipes: Analysis Using the Slicing Method

Xiangzhao Wu

¹,

Song Sang

^1,*,

Youwei Du

²,

Fugang Liu

³ and

Jintao Zhang

College of Engineering, Ocean University of China, Qingdao 266404, China

College of Intelligent Manufacturing, Huanghai College of Qingdao, Qingdao 266427, China

Binzhou Polytechnic, Institute of Oceanography, Binzhou 255603, China

Author to whom correspondence should be addressed.

Appl. Sci. 2024, 14(24), 11938; https://doi.org/10.3390/app142411938

Submission received: 14 October 2024 / Revised: 30 November 2024 / Accepted: 18 December 2024 / Published: 20 December 2024

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Figure 1
Model sketch (left: traditional riser; right: deep-sea mining pipe). "> Figure 2
Schematic diagram of 2D flow-field slice division. "> Figure 3
The field of prospective computing and a grid division diagram. "> Figure 4
Field of background computing and grid-partitioning diagram. "> Figure 5
The numerical solution process of slicing method. "> Figure 6
Test device diagram. "> Figure 7
The response-time curve and the moving-track diagram of the down-flow and cross-flow displacement of each section of the deep-sea mining pipe. "> Figure 7 Cont.
The response-time curve and the moving-track diagram of the down-flow and cross-flow displacement of each section of the deep-sea mining pipe. "> Figure 8
Envelope diagram of instantaneous dimensionless displacement in the cross-flow direction of the riser. "> Figure 9
Schematic diagram of oscillation period of oscillating flow. "> Figure 10
Displacement diagram of mining pipe movement trajectory under different superimposed frequencies. "> Figure 11
Envelope diagram of transverse vibration of deep-sea mining pipes under different superimposed frequencies. "> Figure 12
Frequency amplitude plots of the transverse flow direction of deep-sea mining tubes at different stacking frequencies. "> Figure 13
Envelope plots of instantaneous dimensionless displacements in the transverse flow direction of deep-sea mining tubes with different weight intermediate warehouses. ">

Versions Notes

Abstract

Deep-sea mining pipes are different from traditional ocean risers articulated at both ends: they are free-suspended, weakly constrained at the bottom, and have an intermediate silo at the end, compared to which relatively little research has been carried out on vortex-induced vibration in mining pipes. In this study, a sophisticated quasi-3D numerical model with two degrees of freedom for the flow field domain and structural dynamics of a deep-sea mining pipe is developed through a novel slicing method. The investigation explores how the vortex-induced vibrations of the mining pipe behave in various scenarios, including uniform and oscillating flows, as well as changes in the mass of the relay bin. The findings indicate that the displacement of the deep-sea mining pipe increases continuously as it moves from top to bottom along its axial direction. The upper motion track appears chaotic, while the middle and lower tracks exhibit a stable “8” shape capture, with the tail capturing a “C” shape track. Furthermore, with an increase in flow velocity, both transverse vibration frequency and vibration modes of the mining pipe progressively rise. Under oscillating flow conditions, there exists a “delay effect” between vibration amplitude and velocity. Additionally, an increase in oscillation frequency leads to gradual sparsity in the vibration envelope of the mining pipe in transverse flow direction without affecting its overall vibration frequency. Under the same flow velocity and different bottom effects, the main control frequency of the deep-sea mining pipe is basically unchanged, but the vibration mode of the mining pipe is changed.

Keywords:

deep-sea mining; cantilever riser; vortex-induced vibration; fluid–solid coupling; slicing method

1. Introduction

The deep seabed is rich in high-quality mineral resources, including essential minerals such as nickel, cobalt, copper, and manganese and other metals, which are indispensable. These resources are critical for various emerging industries, such as the development of sectors like aerospace and the production of new energy vehicles, as cited in recent studies [1,2]. Therefore, the development of deep-sea mineral resources has become an inevitable choice for a country’s future development [3,4]. Researchers around the world have conducted a host of experiments on a variety of deep-sea mining tools and systems and have proposed a technical model with continuous and efficient mining capability, the vertical transport system (VTS) [5]. The connection between the upper hull structure and the lower underwater mining system is necessary for deep-sea mining missions, and, as such, it is often subject to a variety of factors, such as marine environmental loads, hull vibrations, and weight variations in the lower relay silo. Compared with the conventional riser fixed by hinges at both ends, the vortex-excited vibration response characteristics of the deep-sea mining pipe are more complex and are characterized by free suspension, large aspect ratios, and weak bottom constraints. Therefore, this paper investigates the vortex-excited vibration response characteristics of deep-sea mining tubes under uniform and oscillating flow variations to facilitate the understanding of the mechanism of vortex-excited vibration in deep-sea mining tubes [6,7]. The mining system has elongated mining pipes, and the vortex-induced vibration caused by ocean currents and waves can seriously affect the safety and service life of these mining pipes during operation [8,9,10].

The investigation presented in this document employs a slicing method to analyze vortex-induced vibrations in deep-sea mining pipes. However, since there are fewer studies on cantilever risers, such as deep-sea mining pipes; since most of the studies are based on traditional risers with articulated restraints at both ends; and since deep-sea mining pipes have many similarities with traditional risers in terms of the appearance and materials of their physical models, the study in this paper can refer to some of the traditional risers in vortex-induced vibration. Chaplin et al. [11] and Ge et al. [12] found that the change in flow velocity leads to the change in vibration modes induced by vortex-induced vibration in marine risers. Yuan et al. [13] developed a sophisticated time-domain force-decomposition model designed to accurately forecast the nonlinear dynamic behaviors of risers subjected to complex, non-uniform, and non-constant flow conditions. Zhou et al. [14] conducted a detailed investigation into the vortex-induced vibrations of deep-sea risers subjected to bidirectional shear flows, detailing the specific amplitude and frequency responses observed under such dynamic conditions. Han Xiangxi et al. [15] devised a new model that incorporates the variability of tensile forces in risers that effectively delineates the dual-peaked nature of amplitude ratio curves and the patterns of vortex shedding observed. Utilizing the principles of Euler–Bernoulli and the Van der Pol wake oscillation theory, Zhang et al. [16] established a comprehensive governing equation for TTR VIV and explored the effects of top tension alterations on the natural vibrations within the TTR system. Wang et al. [17] crafted a detailed model test scheme for assessing vortex-induced vibrations in submarine patch cords that encompasses the principles of model testing and extends to the intricate design and manufacturing processes of the patch cords, culminating in a discussion of high-level testing outcomes. In a significant contribution to the field, Liu et al. [18] investigated how variations in oscillation frequency impact the vortex-induced vibration characteristics of risers under oscillatory flows, demonstrating how such changes influence the structure of the cross-flow vibration envelope. Guoqing Jin et al. [19] performed a comparative analysis of the vibrational responses of end-hinged and cantilever risers in deep-sea mining applications, revealing a consistent pattern in the evolution of vibration amplitude and the primary vibration frequencies. Zhen Wang et al. [20] considered natural frequency influence and special boundary conditions, as well as tension variation along pipes, proposing five vibration modes. Guoqing Jin et al. [19] calculated cantilever riser’s vortex-induced vibrations using a quasi-three-dimensional coupling algorithm based on DVM and FEM, observing an amplitude hopping phenomenon during continuous principal state transitions.

Conclusively, the study of fluid–solid interactions emerges as a pivotal component in the field of vortex-induced vibration analysis. This approach not only forecasts the dynamics of vortex-induced vibrations in deep-sea mining pipes with high precision but also enhances theoretical frameworks that support the design and systematic maintenance of these infrastructures. This document delineates new investigative efforts aimed at exploring the unique characteristics of vortex-induced vibrations affecting deep-sea mining tubes, thereby broadening the understanding of their operational dynamics: (1) The computational model of viscous flow field is constructed by slicing and overlapping mesh methods in the fluid domain, while the finite element method is applied to construct a 3D computational model of the tube in the deep-sea mining tube structure domain. The two-way, sequential coupling between the fluid domain and the deep-sea mining pipe structure is achieved through UDF programming, so that a numerical model of the coupled flow–solidity of the vortex-excited vibration of the mining pipe is established and verified for its accuracy. (2) In this paper, the mechanism of vortex-induced vibration of a deep-sea mining pipe under the action of oscillating flow is revealed.

2. Theoretical Analysis

CFD (Computational Fluid Dynamics) is a method that uses computer technology to calculate and analyze physical phenomena such as heat conduction and fluid flow and data processing. This method, together with experimental measurement methods and theoretical analysis methods, constitutes the research system of fluid flow problems. The CFD method is not limited by experimental models, has a high degree of flexibility and a wide range of applications, and can provide complete and detailed calculation data.

2.1. Introduction of Relevant Parameters

(1): Reduction velocity, u_r

In the study of vortex-excited vibration, due to the differences in structural parameters such as riser diameter, the actual velocity is usually dimensionless to generate an important dimensionless parameter called reduced velocity, u_r, which is defined as follows:

u_{r} = \frac{u}{f_{n} d},

(1)

where u is for incoming flow velocity, m/s; and

f_{n}

is the natural frequency of the riser structure, Hz.

(2): Displacement root mean square value, $y_{RMS}$

Subtract the average value of each instantaneous displacement of the riser, sum it after square, divide it by the quantity, n, and take square root to calculate the root mean square of the transverse flow displacement, which is defined as follows:

y_{RMS} = \sqrt{\frac{1}{n} {\sum_{i = 1}^{n} [y_{i} (t) - \bar{y}]}^{2}},

(2)

where

y_{RMS}

represents the root-mean-square value of the displacement of the riser in the y direction,

\bar{y}

represents the average displacement of the riser in the y direction, and

y_{i} (t)

indicates that the riser is shifted in the y direction at every moment.

2.2. Governing Equations and Boundary Conditions

Derived from fundamental principles such as the conservation of mass, Newton’s second law of motion, and the conservation of energy, the foundational equations of fluid mechanics encompass the equations for mass conservation, momentum, and energy. In scenarios where thermal gradients do not play a significant role, such as with the flow of incompressible viscous fluids, temperature variations are generally negligible. Consequently, for the analysis within the fluid dynamics domain under these conditions, the equations primarily considered are those pertaining to the conservation of momentum and mass.

(1): Mass conservation equation

When the mass of a particle remains the same over a certain period of time, and the mass of the fluid in it remains stable, then the motion of the particle follows the principle of mass conservation. The specific mass conservation formula can be expressed as Formula (3):

\frac{\partial ρ}{\partial t} + \nabla \cdot (ρ \vec{U}) = 0,

(3)

where t represents time,

\nabla

is the Hamiltonian operator, and

\vec{U}

represents the velocity vector of the fluid.

For incompressible fluid, the density is constant, and we can simplify the above formula:

\nabla \cdot \vec{U} = \frac{\partial u}{\partial x} + \frac{\partial ν}{\partial y} + \frac{\partial w}{\partial z} = 0,

(4)

where the components of

\vec{U}

in the x, y, and z directions are represented by u, v, and w, respectively.

(2): Momentum conservation equation

The resultant force acting on a particle is equal to the increase in momentum on the particle, expressed as follows:

\frac{D \vec{U}}{D t} = \vec{F} - \frac{1}{ρ} \nabla \vec{P} + ν \nabla^{2} \vec{U},

(5)

where

\frac{D \vec{U}}{D t}

is the random derivative operator,

\vec{F}

is the mass force,

\nabla

represents the Hamiltonian operator, and ν is the viscosity coefficient of fluid motion.

(3): Turbulence model and solution method

Menter [21] proposed the SST k-ω model, which is particularly suitable for calculating fluids with reverse pressure differences [12], and which consists of two components, the dissipation rate (ω) and the turbulent kinetic energy (k):

ρ \frac{\partial k}{\partial t} + k ρ \frac{\partial u_{i}}{\partial x_{i}} = \frac{\partial}{\partial x_{i}} [Γ_{k} \frac{\partial k}{\partial x_{j}}] + {\tilde{G}}_{k} - Y_{k} + S_{k},

(6)

ρ \frac{\partial ω}{\partial t} + ρ ω \frac{\partial u_{i}}{\partial x_{i}} = \frac{\partial}{\partial x_{j}} [Γ_{ω} \frac{\partial ω}{\partial x_{i}}] + G_{ω} - Y_{ω} + D_{ω} + S_{ω},

(7)

The turbulence simulation method for numerical simulation analysis in this paper is based on the SST k-ω model of the Reynolds mean method, coupled with the application of overlapping mesh and dynamic mesh.

2.3. Dynamic Analysis of Riser Structure

The simplified stress model of the traditional riser (hinged at both ends) is shown in Figure 1 (left). The main subject of this paper is the deep-sea mining pipe. The deep-sea mining pipe model can be simplified into the Euler–Bernoulli beam model with one end hinged and one end free (z-direction constraint), and the top end subject to preload, as shown in Figure 1 (right).

By discretizing the deep-sea mining pipe into a finite number of three-dimensional units, the structural motion control equation can be discretely expressed as follows:

[M] \{\ddot{x} (t)\} + [C] \{\dot{x} (t)\} + [K] \{x (t)\} = \{F (t)\},

(8)

In this formulation, the matrix,

[M]

, represents the total mass matrix, while

[C]

denotes the comprehensive damping matrix, where Rayleigh damping is implemented;

[K]

signifies the complete stiffness matrix; the vector

\{F (t)\}

encapsulates the total external forces acting on the structure as a column vector; and the displacement of all finite element nodes within the tube is captured by the vector

x (t)

Natural frequency usually affects structural damping by using Rayleigh damping matrix,

[C]

. The calculation formula is as follows:

[C] = α [M] + γ [K],

(9)

In this context, α and γ represent the coefficients for Rayleigh damping and are derived based on the tube’s natural frequency and damping ratio. These coefficients are computed as per the formula presented in Equation (10):

[\begin{matrix} α \\ γ \end{matrix}] = \frac{2 ζ}{ω_{n 1} + ω_{n 2}} [\begin{matrix} ω_{n 1} ω_{n 2} \\ 1 \end{matrix}],

(10)

where ζ = 0.03,

ω_{n 1}

takes the first order natural frequency, and

ω_{n 2}

takes the second order natural frequency.

In this paper, the Newmark-beta method is used to solve the control equation of a deep-sea mining pipe structure by self-compiling UDF.

3. Construction and Verification of Numerical Model

3.1. Establishment of Viscous Flow-Field Calculation Model

3.1.1. Slice Two-Dimensional Flow Field

The research of Willden and Graham [22] indicates that the slicing method can convert the intricate three-dimensional flow issue into multiple fundamental two-dimensional flow problems. In structural dynamics, the storage and computational demands are relatively minimal, whereas CFD requires extensive computational resources. Consequently, the number of slice points is generally fewer than the number of nodes on the riser. By enhancing the granularity of the finite element node distribution, the precision in simulating the riser’s structural motion is improved. Simultaneously, reducing the number of CFD slices contributes to computational efficiency. Typically, the count of these slices is an integer multiple of the structural units. For the purposes of this study, twelve two-dimensional flow-field slices are strategically positioned along the riser to ensure coverage of at least three times the highest mode number depicted in Figure 2. The orientation is defined by the flow direction aligned positively along the x-axis and the axial direction of the riser extending from the bottom upward along the z-axis, sequentially setting the twelve slices along the tube’s length.

In this paper, the regulations for setting two-dimensional flow-field slices and three-dimensional finite element nodes are as follows: The corresponding positions of two-dimensional flow-field slices and risers are defined as type 1 of finite element nodes, and there are 12 finite element nodes in total. Subsequently, the riser undergoes segmentation using the second implementation of the finite element method. This process involves creating additional finite element nodes, termed “type 2 finite element nodes”, positioned at midpoints between consecutive original nodes based on the distribution from the initial method. In total, there are thirty-eight type 2 finite element nodes established.

3.1.2. Overlapping Grid and Grid Nesting Methods

In 1983, Steger et al. [23] for the first time, proposed a technique called superimposed mesh, which is able to finely mesh the fluid domain of two-dimensional flow-field slices. At present, the overlapping grid method has been widely used in numerical simulation.

Firstly, the size of the selected computing domain is 30D × 60D, and the region is properly encrypted during grid division. Finally, the final two-dimensional flow-field grid is created through grid-nesting technology. In addition, during the meshing process, we need to pay special attention to the mesh size of the outermost mesh of the foreground grid, as it should be slightly larger than the mesh size of the encrypted area of the background grid, so as to ensure that the overlap of the foreground grid can find interpolation points on the background grid at any time. The entire computing domain is divided into two main computing domains, namely the foreground computing domain (Figure 3) and the background computing domain (Figure 4).

3.2. Numerical Solution

In this study, we use the Fluent separation method to calculate the viscous flow field and set the initial conditions. Then, the fluid force on the surface of the tube is obtained by numerical calculation, and it is applied to the motion equation of the tube structure and solved numerically to calculate the instantaneous motion velocity and instantaneous motion displacement of the cylinder in the current time step. In this way, the movement of the tube and the deformation of the mesh are controlled. By using the UDF program compiled in the C language environment in Fluent, we can calculate the coupling between the viscous flow field and the pipe structure. The numerical solution process of the slice method is shown in Figure 5.

3.3. Grid Convergence Test

A total of three grid models were constructed in order to evaluate the convergence of the grids. The vortex-excited vibration fluid–solid coupling numerical model was established using the previously mentioned method, and numerical calculations were carried out. The incoming flow is a uniform flow with a velocity magnitude of 0.15 m/s and a Reynolds number of 3000. The numerical simulation results are shown in Table 1 below. The change rate of the main control frequency should be kept below 5%. It can be seen that the three sets of meshes have a negligible effect on the numerical simulation. Therefore, in order to combine both computational accuracy and computational time, Grid II was chosen as the grid used in this paper for subsequent studies.

3.4. Physical Model Selection and Model Verification

Since there are few research studies on physical experiments of deep-sea mining pipes, and deep-sea mining pipes and traditional risers have many similarities, the methods adopted in this paper are basically the same for the model establishment and fluid–structure coupling solution of deep-sea mining pipes and traditional risers. In this section, model tests with fixed boundary conditions at both the top and bottom are used for numerical verification.

To validate the fidelity of the numerical model in this research, the experimental setup designed by Kang Zhuang [24] is utilized as a benchmark for conventional riser simulation analyses discussed in this section. The experimental configuration features the riser’s bottom end coupled to an adjustment module via precision universal joints, while its top end is linked to the upper motion module, integrating tension sensors and tensioning devices. A schematic representation of this experimental arrangement is depicted in Figure 6. For the purpose of assessing the vibration response characteristics of the riser, a total of 12 detection points are strategically positioned along its axial direction. The position of the 12 two-dimensional flow-field slices divided in this paper is the same as that of the above experimental monitoring points. The numerical simulation method used is the same as that described above. Other experimental parameters are shown in Table 2.

Five groups of homogeneous flow conditions were taken to conduct our numerical simulation of the vortex-induced vibration of the riser, and the

y_{RMS}

results at slice 06 of the riser were compared with the experimental results of Kang Zhuang et al., as shown in Table 3.

According to the above data, we found that in the five working conditions we selected, the dimensionless value of the transverse flow displacement of the riser obtained by the numerical simulation method in this study was slightly higher than the result of the reference test, but these deviations were all lower than 8%, which was within a certain range, so we could confirm that the numerical model we established had high accuracy.

4. Numerical Simulation and Results

4.1. Vortex-Induced Vibration Response of Deep-Sea Mining Pipe

In the realm of practical marine engineering, cylindrical structures often align and move with seawater currents. These movements are susceptible to vortex-induced vibrations triggered by incoming flows, potentially leading to fatigue damage or even catastrophic structural failures. Over the last century, extensive research involving numerous physical and numerical simulations has been conducted by both domestic and international scholars. This substantial body of work has established a robust foundation for ongoing research in this field. This section employs Fluent2020R2 software, alongside a C language-compiled UDF program and overlapping grid technology, to systematically explore the vortex-induced vibration behaviors of a quasi-three-dimensional deep-sea mining pipe across various flow rates. Additionally, this research delves into how oscillating flows impact the vortex-induced vibrations of such structures. Also, when we select the 5 m riser, other properties do not change; only the lower-end constraint changes (fixed end becomes free end). And a downward vertical tension is applied to the free end of the deep-sea mining pipe, which is the same as the top tension value of the traditional riser in the model verification, and other physical parameters are unchanged. This ensures that the natural frequency of the pipe is consistent.

4.1.1. Analysis of Vibration Track Characteristics of Deep-Sea Mining Pipe

Figure 7 shows the time-history curve of cross-flow direction and parallel-flow direction displacement response of 24 nodes along the length direction of mining pipe from top to bottom when u_r = 20. It can be found that the down-flow displacement at each node vibrates slightly relative to the equilibrium position of each node. However, unlike the phenomenon that the closer the riser is to the middle of the riser, the greater the downstream displacement, because the mining pipe has the characteristics of free movement and no constraint at the bottom, its maximum displacement is located at the bottom, about 3.198 d. In the context of the cross-flow dynamics of the mining pipe, the displacement exhibits a reciprocating motion at y/d = 0, although the amplitude of the vibration remains inconsistent, reminiscent of the dynamic behavior observed in conventional risers. The nature of the vortex-induced vibrations, whether transverse or longitudinal, manifests greater complexity. The movement pattern of the mining pipe more closely resembles the irregular paths observed during “beating” phenomena in vortex-induced vibrations of a two-dimensional cylinder, devoid of the predictable sinusoidal or co-sinusoidal trajectories. Studies indicate that the amplitude of vibrations in the direction perpendicular to the flow is significantly larger than those aligned with the flow. Correspondingly, the frequency of vibrations in the direction of the flow is higher than in the perpendicular direction, revealing a binary relationship between these variables.

It can be found that the movement of the upper end of the mining pipe is relatively chaotic, and its shape is like a “cashew nut”; this is due to the locking of the downstream vibrations occurring, which affects the cross-flow vibration frequency and leads to chaotic vibration trajectories. From slice 12 to slice 10, the outline of the motion track starts to become confused gradually when slice 9 begins, but when it is near the bottom, it can be observed that the moving track of the mining pipe has an obvious “8” shape stage, which is caused by the binding force generated by the bottom of the mining pipe under the effect of the middle bin at the bottom, resulting in a change in the phase of the cross-flow and down-flow displacements, also making the bottom movement mixed with stability in the chaos. The randomness and complexity of mining-pipe movement are more obvious.

4.1.2. Modal Characteristics Analysis of Vortex-Induced Vibration of Deep-Sea Mining Pipe

Figure 8 illustrates the envelope of the instantaneous dimensionless displacement in the cross-flow direction of a deep-sea mining pipe at different reduced velocity values: ur = 8, 18, 36, 10, 21, and 38.

Our analysis reveals distinct vibration states: At ur = 8, the mining pipe exhibits a primary mode with the maximum displacement at the pipe’s bottom reaching approximately 0.72 d. At ur = 18, it transitions to a second-order mode, with displacement increasing to about 0.81 d. Further elevation to ur = 36 leads to a third-order mode, where the bottom displacement peaks at around 0.86 d. These modal transitions are vividly captured in the figure, demonstrating the envelope of dimensionless displacements across these states. As ur values escalate, not only do the number of vibration modes in the transverse flow direction multiply, but the manifestation of multimodal vibrations becomes increasingly pronounced.

At reduced velocities of ur = 10, ur = 21, and ur = 38, a progressive transition in the vibration modes of the deep-sea mining pipe can be observed in the cross-flow direction. Specifically, the mining pipe transitions from a first-order mode to a second-order mode, then from a second-order mode to a third-order mode, and finally to a fourth-order mode as the reduced velocity increases. Concurrently, a noticeable reduction in the maximum displacement at the bottom of the mining pipe accompanies each modal transition. For instance, at ur = 10, the maximum displacement, y/d, at the pipe’s bottom is approximately 0.31 d. At a reduced velocity of ur = 21, this displacement increases to about 0.74 d, indicating a larger amplitude of vibration. At ur = 38, while the mining pipe exhibits characteristics of a first-order master mode, the maximum displacement, y/d, at the bottom reaches about 0.79 d, further illustrating the dynamic behavior of the mining pipe under varying flow conditions.

4.2. Impact of Oscillatory Flow on Vortex-Induced Vibration of Deep-Sea Mining Pipes

This section will study the vortex-induced vibration characteristics of deep-sea mining pipe under the action of oscillating flow and reveal the vortex-induced vibration characteristics of deep-sea mining pipe under the special constraint condition that the upper end is hinged and the lower end is free (z-direction constraint) through numerical simulation.

Four oscillating flow frequencies, f = 0.1f₁, f = 0.2f₁, f = 0.3f₁, and f = 0.4f₁, were selected for the oscillating flow conditions selected in the numerical simulation in this section, as shown in Figure 9 for details.

4.2.1. Analysis of Vibration Displacement Trajectory Characteristics of Mining Pipe by Oscillation Frequency

The displacement trajectory of the mining pipe under different oscillating flow frequencies was drawn. The movement trajectory at the centroid of the mining pipe at 13 places was intercepted from top to bottom, and the drawing software Origin2020 was used to draw the following figure.

The change in vibration displacement at different positions of the deep-sea mining pipe with the oscillation frequency is analyzed by observing Figure 10. It is found that the moving distance of the mining pipe increases continuously along the current, which accords with the feature of free bottom end of the mining pipe. Initially moving in the negative direction of the x-axis, the riser soon reverses its course to the positive x-axis due to the phenomenon known as “Hysteresis”. The maximum vibrational displacement of the riser in the cross-flow direction exhibits a unique pattern: it in-creases from the top downwards, diminishes, and then rises again around 0.3 z/L. This pattern is likely induced by the second-order vibration mode of the mining pipe in the cross-flow direction. It is found that the instantaneous amplitude in the downstream direction is much larger than the instantaneous amplitude in the transverse direction, and the vibration track of the mining pipe has no fixed shape when the transverse direction displacement increases gradually. With the increase in down-flow displacement, the instantaneous amplitude of mining pipe in the down-flow direction gradually decreases and gradually changes into down-flow instantaneous amplitude less than cross-flow instantaneous amplitude. When the forward displacement reaches the maximum value, the mining pipe appears to have an “8” shape vibration track, which is similar to that when the oscillating flow acts on the traditional riser, because the two have the same physical appearance and belong to the category of cylindrical flow.

4.2.2. Analysis of Vibration Modal Characteristics of Deep-Sea Mining Pipe by Oscillation Frequency

Figure 11 below is the transverse flow vibration envelope diagram of the deep-sea mining pipe under different superimposed frequencies. Four oscillating flows with an average flow velocity of 0.15 m/s and an oscillation frequency of 0.1f₁, 0.2f₁, 0.3f₁, and 0.4f₁ were selected.

It can be seen from the above figure that when the oscillation frequency is 0.1f₁~0.4f₁, the transverse flow vibration envelope is symmetrical, consisting of the first- and second-order formation, and the whole is the second order master mode. When the vibration frequency is 0.1f₁, the upper and lower parts of the mining pipe are more uniform, and the second-order modes coexist with the first-order modes. When the vibration frequency is 0.2f₁, the risers are also more symmetrical above and below the cross-flow upward vibration envelope, but the proportion of the excitations of the second-order modes increases, and this change is particularly obvious when the vibration frequency is 0.3f₁. When the risers are at the cross-flow upward vibration envelope, most of the time is in the first-order vibration state during a fixed period. At the same time, the proportion of second-order vibration states decreased significantly, and the envelope became sparse. When the vibration frequency of 0.4f₁ risers in the cross flow upward vibration envelope range begins to shrink, the second order state becomes no longer obvious. This phenomenon should also be related to the pre-tension of the riser and the self-weight of the riser. At the same time, with the increase in oscillation frequency, the time for the flow rate to continuously reach 0.237 m/s becomes shorter, and the frequency of the leakage vortex generated by this flow rate is consistent with the second-order natural frequency of the tube, so this may also be one of the important reasons for the above phenomenon.

4.2.3. Vibration Frequency Analysis of Deep-Sea Mining Pipe Vibration Frequency Characteristics

Figure 12 vividly illustrates that regardless of the variations in oscillation frequencies applied, the frequency characteristics of the cross-flow vibrations of the deep-sea mining pipe remain consistent. This observation underscores that the alterations in the frequency of superimposed oscillatory flows do not significantly impact the inherent vibration frequencies of the mining pipe. Notably, across five specified axial positions along the pipe, the amplitude frequencies are remarkably uniform, lying within a band from 0.5 Hz to 4.0 Hz, where the predominant frequency consistently hovers around 2.7 Hz. The stability of this frequency band can be attributed to the adjustments in the vibration frequency within the cross-flow direction, which correlate directly with changes in the velocity of the oscillating flow. Furthermore, an examination of the amplitude reveals that the axial vibration amplitude of the mining pipe demonstrates a gradual enhancement from the top towards the bottom, illustrating a clear gradient in vibrational intensity along the pipe’s length.

4.3. Influence of Relay Bin Weight Variation on Vortex-Induced Vibration Characteristics of Deep-Sea Mining Tubes

In this section, the vortex-induced vibration characteristics of deep-sea mining tubes under different bottom-end forces W_b will be investigated. In this section, the properties of the mining tubes are selected to be the same as those of the mining tubes under the uniform flow in the previous section, and the properties of the intrinsic frequencies of the mining tubes are shown in Table 4.

The instantaneous dimensionless displacement envelopes for u = 0.2 m/s, Wb = 1000 N, Wb = 1500 N, and Wb = 2000 N in the transverse flow direction of the deep-sea mining tube are given in Figure 13, respectively.

At a top tension of Wb = 1000 N, the envelope of vibration displacement in the cross-flow direction of the deep-sea mining pipe predominantly exhibits the second-order mode, with the first and third-order modes appearing less distinctly. When the top tension increases to Wb = 1500 N, the displacement envelope still primarily shows the second-order mode, but it now includes a more prominent third-order mode, resulting in a more complex overall displacement pattern. At Wb = 2000 N, the primary mode of vibration shifts to the first order, with a less pronounced second-order mode also observable. With incremental increases in top tension, T, the dominant vibration mode excited by the cross-flow diminishes in complexity at a constant flow velocity, transitioning from a second-order mode to a more simplified first-order mode as the top tension ranges from Wb = 1000 N to Wb = 2000 N at a flow velocity of u = 0.2 m/s. Concurrently, the maximum dis-placement observed at the bottom end of the pipe decreases progressively as the intermediate warehouse weight increases.

5. Summary and Outlook

5.1. Summary of Work

(1): Along with its axial direction from top to bottom, the moving displacement of the mining pipe is constantly increasing, the upper motion trajectory is relatively chaotic, the middle and lower trajectory is more stable to capture the “8” font, and the tail can capture the “C” font trajectory.
(2): With the increase in flow velocity, the transverse vibration mode of the mining pipe increases step by step, and the vibration frequency gradually increases. With the increase in reduction velocity, the maximum displacement at the bottom of mining pipe first increases and then decreases, rather than simply increasing.
(3): In terms of vibration frequency, the frequency of the mining pipe is relatively stable at the axial 2/5 L, which is not easy to stimulate the phenomenon of multi-mode competition. As the mining pipe is axial from top to bottom, the amplitude stimulated by its main mode continues to increase.
(4): When the oscillating flow acts, the vibration amplitude of the mining pipe has a “delay effect” relative to the velocity change. The mining pipe will cross the position of the primary axis in the process of moving in the forward direction of the flow, and it will move in the negative direction of the x-axis for a short period of time before returning to the positive direction of the x-axis.
(5): With the increase in oscillation frequency, the vibration envelope of the deep-sea mining pipe in the direction of cross-flow is gradually sparse.
(6): At the same flow rate, as the force at the bottom end increases, the mining pipe cross-flow direction vibration mode gradually decreases, and the maximum displacement at the bottom end also gradually decreases.

5.2. Prospect of Work

In this paper, the study of vortex-induced vibration characteristics of deep-sea mining pipe under different oscillating flow frequencies and oscillating flow velocities is carried out by numerical simulation methods, providing a reference example for solving other similar fluid–structure coupling problems.

However, considering the limitation of time and computational resources, there are still some deficiencies in our study, so some corresponding future research directions are proposed here:

(1) In this paper, the impact of three-dimensional flow characteristics on the study of the fluid domain using the slice method is not fully considered. Since the slicing method focuses on two-dimensional analysis, it is difficult to fully capture the complex flow changes in three-dimensional space, such as the anisotropy of velocity gradient and the three-dimensional morphology of vortex structure, resulting in a certain degree of lack of accurate description of the real state of the three-dimensional flow field in the related research.

In view of the continuous progress of computer technology, the “thick slice method” can be introduced to construct 3D slice structure models for subsequent numerical simulation and analysis, which can be applied in the field of flow computation. By increasing the thickness of the slices in a certain dimension, this method can take into account the computational efficiency, better restore the three-dimensional flow field characteristics, and then reach a more accurate fluid–solid coupling between the three-dimensional fluid region and the three-dimensional riser structure, which will strongly improve the accuracy of the description of the riser’s kinematic characteristics and provide more reliable data support and a more reliable theoretical basis for the relevant marine engineering applications.

(2) In actual engineering, during the transport process of a deep-sea mining pipe, the internal mineral flow also has the coupling vibration phenomenon with the pipe itself, which can be taken into consideration in the future in order to obtain more accurate vortex-induced vibration characteristics of the deep-sea mining pipe.

The numerical analysis of riser vortex vibration has important implications for future research in many ways: in the field of ocean engineering, it helps to optimize riser design. By accurately simulating vortex-excited vibration, more reasonable riser structural parameters, such as diameter, wall thickness, and material selection, can be determined, improving the stability and durability of risers in complex ocean flow fields and reducing construction and maintenance costs. It can provide a theoretical basis for the development of new riser concepts and promote the expansion of ocean engineering to deeper waters and harsher environments.

Author Contributions

Conceptualization, X.W.; Methodology, X.W.; Software, X.W.; Validation, X.W.; Formal analysis, X.W.; Investigation, S.S.; Resources, S.S.; Data curation, S.S.; Writing—original draft, X.W.; Writing—review & editing, Y.D.; Visualization, F.L. and J.Z.; Supervision, F.L. and J.Z.; Project administration, S.S.; Funding acquisition, S.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research project is supported by multiple funding sources, including the NSFC-Shandong Joint Fund (U2106223), the Shandong Provincial Major Scientific and Technological Innovation Project (2021CXGC010707), and the Shandong Provincial Natural Science Foundation (ZR2022ME092).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare that they have no conflict of interest.

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Figure 1. Model sketch (left: traditional riser; right: deep-sea mining pipe).

Figure 2. Schematic diagram of 2D flow-field slice division.

Figure 3. The field of prospective computing and a grid division diagram.

Figure 4. Field of background computing and grid-partitioning diagram.

Figure 5. The numerical solution process of slicing method.

Figure 6. Test device diagram.

Figure 7. The response-time curve and the moving-track diagram of the down-flow and cross-flow displacement of each section of the deep-sea mining pipe.

Figure 8. Envelope diagram of instantaneous dimensionless displacement in the cross-flow direction of the riser.

Figure 9. Schematic diagram of oscillation period of oscillating flow.

Figure 10. Displacement diagram of mining pipe movement trajectory under different superimposed frequencies.

Figure 11. Envelope diagram of transverse vibration of deep-sea mining pipes under different superimposed frequencies.

Figure 12. Frequency amplitude plots of the transverse flow direction of deep-sea mining tubes at different stacking frequencies.

Figure 13. Envelope plots of instantaneous dimensionless displacements in the transverse flow direction of deep-sea mining tubes with different weight intermediate warehouses.

Table 1. Grid convergence test (Re = 3000).

Grid No.	Total Foreground Grid Cells	Outermost Thickness of Foreground Grid	Near-Wall Grid Thickness	Total Background Grid Cells	C_LRMS	C_Dmean
I	30,820	0.01D	0.001D	251,835	1.581	1.039
II	43,152	0.008D	0.001D	359,676	1.584	1.042
III	58,310	0.006D	0.001D	486,822	1.577	1.046

Table 2. Model parameters of the riser.

Correlation Parameter	Numerical Value	Unit
External diameter (d)	0.02	m
Length (L)	5	m
Top tension (T)	80	N
Bending stiffness (EI)	42.62	Nm²
Tensile stiffness (EA)	1.47 × 10⁶	N
Mass ratio (m*)	1.96	-
First-order inherent frequency (f₁)	0.99	Hz
Second-order inherent frequency (f₂)	2.34	Hz
Third-order inherent frequency (f₃)	4.64	Hz
Fourth-order inherent frequency (f₄)	7.67	Hz

Table 3. Comparison of numerical simulation and experimental results.

$u$ (m/s)	$u_{r}$	$\frac{y_{RMS}}{d}$ (Primary Experiment)	$\frac{y_{RMS}}{d}$ (Numerical Simulation of Paper)
0.10	5.04	0.26	0.28
0.15	7.55	0.28	0.30
0.20	10.07	0.27	0.29
0.25	12.59	0.23	0.24
0.30	15.11	0.46	0.49

Table 4. Intrinsic frequency of deep-sea mining tubes in Intermediate warehouses of different weights.

T	f1 (Hz)	f2 (Hz)	f3 (Hz)	f4 (Hz)
Wb = 2000 N	0.47	1.42	2.06	3.04
Wb = 1500 N	0.39	0.01	1.52	2.15
Wb = 1000 N	0.30	0.72	0.07	1.50

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Wu, X.; Sang, S.; Du, Y.; Liu, F.; Zhang, J. Vortex-Induced Vibration of Deep-Sea Mining Pipes: Analysis Using the Slicing Method. Appl. Sci. 2024, 14, 11938. https://doi.org/10.3390/app142411938

AMA Style

Wu X, Sang S, Du Y, Liu F, Zhang J. Vortex-Induced Vibration of Deep-Sea Mining Pipes: Analysis Using the Slicing Method. Applied Sciences. 2024; 14(24):11938. https://doi.org/10.3390/app142411938

Chicago/Turabian Style

Wu, Xiangzhao, Song Sang, Youwei Du, Fugang Liu, and Jintao Zhang. 2024. "Vortex-Induced Vibration of Deep-Sea Mining Pipes: Analysis Using the Slicing Method" Applied Sciences 14, no. 24: 11938. https://doi.org/10.3390/app142411938

APA Style

Wu, X., Sang, S., Du, Y., Liu, F., & Zhang, J. (2024). Vortex-Induced Vibration of Deep-Sea Mining Pipes: Analysis Using the Slicing Method. Applied Sciences, 14(24), 11938. https://doi.org/10.3390/app142411938

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