# Dynamic Performance of Suspended Pipelines with Permeable Wrappers under Solitary Waves

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Methods

^{®}(version 11.1.0; 2014; https://www.flow3d.com (accessed on 10 December 2022); Flow Science, Inc., Santa Fe, NM, USA). Flow-3D aims to solve the transient response of fluids under interactions with structures, internal and external loads and multi-physical processes. It features some advantages in terms of a high level of accuracy in solving the Navier-Stokes equation with the volume of fluid (VOF) method, efficient meshing techniques for complex geometries, and high efficiency level for large-scale problems. Also, Flow-3D provides the flexibility and utility for flowing through porous media. A two-dimensional numerical wave tank was constructed by using the immersed boundary (IB) method and an in-house subroutine termed as IFS_IB. A submarine pipeline and porous medium were two-way coupled at the interface described by the individual volume fractions [29]. The pipeline was wrapped with a layer of a porous medium. A solitary wave was generated at the inlet boundary of the tank to simulate an approaching tsunami. Non-slip wall conditions were assigned at the bottom of the tank and the pipe surface, which was also specified with a roughness coefficient. The top boundary was defined as a free boundary and configured with the atmospheric pressure. A Neumann-type absorbing boundary condition, a stable, local, and absorbing numerical boundary condition for discretized transport equations [30], was imposed on the outlet boundary to attenuate the reflections of the outgoing waves. A transition zone is set within a certain range from the boundary to reduce the horizontal gradient force of the elements near the boundary and suppress the calculation wave caused by this boundary condition. Through the relaxation coefficient, the predicted value on the inner boundary of the transition zone and the initial value on the outer boundary are continuously transitioned to achieve the purpose of reducing the reflection of propagating waves. The CUSTOMIZATION function of the software FLOW-3D was utilised to impose the Neumann-type absorbing boundary condition. The FLOW-3D distribution includes a variety of FORTRAN source subroutines that allow the user to customize FLOW-3D to meet their requirements. The FORTRAN subroutines provided allow the user to customize boundary conditions, include their own material property correlations, specify custom fluid forces (i.e., electromagnetic forces), add physical models to FLOW-3D, and have additional benefits. Several “dummy” variables have been provided in the input file namelists that users may use for custom options. A user definable namelist has also been provided for customization. Makefiles are provided for Linux and Windows distributions and Visual Studio solution files are provided for Windows distributions to allow users to recompile the FLOW-3D code with their customizations.

#### 2.1. Governing Equations

**U**is the velocity vector,

**X**is the Cartesian position vector,

**g**denotes the gravitational acceleration vector, and ρ represents the weighted averaged density. The term μ is the viscosity. σκ∇α identifies the surface tension effects with σ as the surface tension and α as the fluid volume fraction. Each cell in the fluid domain has a water volume fraction (α) ranging between 0 and 1, where 1 represents cells that are fully occupied with water, while 0 represents cells that fully occupied with air. Values between 1 and 0 represent free surface between air and water. The free surface elevation is defined by using the volume of fluid (VOF) function:

_{F}is the volume of fluid fraction, F

_{SOR}is the source function, F

_{DIF}is the diffusion function; A

_{x}, A

_{y}, and A

_{z}represent the fractional areas; and u, v, and w are the velocity components in the x, y, and z directions.

#### 2.2. Porous Media Module

_{d}u

_{i}is added to the righthand-side of Equation (2):

**U**| is the norm of the velocity vector, n the porosity, and a and b are the factors.

#### 2.3. Solitary Wave Boundary

_{0}t; ${c}_{0}=g\sqrt{H+h}$; H is the wave height; and t is the elapsed time.

## 3. Validation

#### 3.1. Propagation over a Porous Breakwater

_{1}) and behind (WG

_{2}) the breakwater, respectively. The initial still water depth h was assumed to be 10.6 cm. Height of the solitary wave H was considered to be 4.77 cm. In the numerical model, the calculation zone had dimensions of 5 m in length and 0.25 m in height. The second order quadrilateral mesh elements were adopted. The grid around the breakwater was the finest of 0.001 m. The adopted time step size was 0.05 s. The numerical predictions of the water elevations at the locations WG

_{1}and WG

_{2}by the adopted numerical tool FLOW-3D are close to both the experimental measurements and the numerical predictions from another CFD FLUENT version 14.0.1 [33] (Figure 1). Figure 1b,c show the comparison of monitored water levels at the two water level monitoring points in Figure 1a. It can be seen that the experimental results of the two monitoring points are consistent with the numerical simulation results, indicating that the propagating solitary wave energy is basically completely dissipated and then flows out. If the propagating wave energy is not dissipated, the phenomenon of wave reflection will occur. The waves monitored at the two monitoring points will appear superposition of propagating waves and reflected waves. The numerical simulation results do not agree with the physical experiment results. The fluctuations of the water surface elevation after the bypass of the incoming wave are due to its residual reflection at the right absorbing boundary condition, which arrives at WG

_{2}at an earlier time than WG

_{1}. Evolution of the wave surfaces was also compared between the experimental and the numerical models (Figure 2), which demonstrates that the numerical tool is sufficiently reliable. The velocity of the wave is reduced by the porous medium as it partially infiltrates into the breakwater, which is shown as in Figure 3 by comparing the horizontal velocity distributions between the experimental and numerical results at times of 1.5 s and 2 s. The numerical predictions of the flow velocities have slight discrepancies with the experimental measurements, which are attributed to the material assumptions made in the numerical model for the glass beads in the experimental setup.

#### 3.2. Forces on Pipeline

^{2}/4) with L as the unit length of 1 m, is compared between the experimental and numerical results (Figure 4). Both the peak values and the transient variations of the forces predicted by the numerical analysis converge to the measured values in the experimental test. The slight discrepancy between the numerical and experimental results at 2.5 s and 3.1 s, which may be induced by the error of the numerical model simulating the complicated turbulence behaviour, is acceptable in relation to the requirements of this study as our concern is mainly the peak values of forces.

## 4. Results and Discussion

#### 4.1. Effect of Porous Wrapper

#### 4.1.1. Wrapper Porosity

^{2}/4) (Figure 10). With a fully solid (i.e., n = 0.0) wrapper, the pipeline tends to be unaffected by the external flow. Hence, the hydrodynamic forces are zero while the forces on the wrapper reach their maximum. With porous wrappers, water seeps into the wrapper, buffering the impact of the incoming waves on the pipe. As the porosity coefficient increases, the induced forces on the pipeline increase while those on the wrapper decrease. When the porosity coefficient is 0.4, the forces on the external wrapper become higher than that on the internal pipeline. Therefore, the porous wrapper is capable of protecting the pipeline. The smaller the porosity coefficient the better protection the wrapper provides to the pipeline. The pressure gradient and shear stress forces are also shown in Figure 11.

#### 4.1.2. Thickness of Wrapper

#### 4.2. Effect of Pipeline Structure

#### 4.2.1. Suspended Pipelines

#### 4.2.2. Pipelines in Tandem

^{n}= (f

_{f,max}−f

_{r,max})/f

_{f,max}, where f

_{f,max}and f

_{r,max}are the maximum forces on the pipeline or wrapper. It is found that the horizontal loads on the rear pipe and wrapper tend to be always higher than their counterparts on the front pipe. This means that a turbulent flow in the horizontal direction on rear pipe is more intense than that on the front pipe. For different distances, deviations for the forces on the pipelines and wrappers are also different. The deviation is found to be maximized at a distance of 1 m and this indicates that the pipeline is not well protected and needs to be avoided in engineering practice.

#### 4.3. Effect of Wave Height

_{f}

_{,max}− H

_{r}

_{,max})/H

_{f}

_{,max}. The wave height attenuation becomes more significant as the wave height increases. This means that waves with larger heights are more easily affected by the pipelines.

## 5. Conclusions

- (1)
- When a pipe is wrapped by a porous medium, the velocity in the wrapper is relatively small because the porous medium can consume the water energy and weaken the flow. With an increase in the porosity, the range of the low-speed flow at the bottom of the pipeline expands. This indicates that the porous wrapper can slow down the flow and affect a wider region of the surrounding water. After the bypass of the wave through the pipe, the number and volume of the vortices behind the porous wrapper are larger than those for a pipeline with a solid wrapper or without a wrapper. As the porosity coefficient increases, the impact forces on the pipe increase, while those on the wrapper decrease. This implies that the porous wrapper is capable of protecting the pipeline.

- (2)
- For a wave bypassing a pipe with different heights, a symmetric speed change similar to a fisheye appears behind the pipeline, along with two antisymmetric vortices shedding off from the wrapper.

- (3)
- When the waves with different heights pass over the pipeline, the height of the wave is reduced because of the blockage function from the pipeline and the dissipation characteristic of the flow energy. When the wave height is increased, the velocity around the pipeline increases, inducing an increase in the TKE. As the wave height increases, all the maximum forces on the pipeline and wrapper also increase. Note that an increase in the vertical forces on the pipeline is the most significant change because the weight of the water above the pipeline increases, which implies that the protection function of the wrapper is enhanced by the reduction in the wave height.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The diagrammatic sketch of the numerical setup (non–scaled) (

**a**) and the temporal evolution comparison of water surface between experimental and numerical results (

**b**,

**c**).

**Figure 3.**Comparison of horizontal velocity distribution between experimental and numerical results.

**Figure 6.**The velocity contours of the flow fields under different porosities; (

**a**) n = 0.0; (

**b**) n = 0.4; (

**c**) n = 0.6; (

**d**) n = 1.0; left to right: arrival, departure.

**Figure 7.**The vorticity contours of the flow fields under different porosities; (

**a**) n = 0.0; (

**b**) n = 0.4; (

**c**) n = 0.6; (

**d**) n = 1.0; left to right: arrival, departure.

**Figure 8.**Comparison of horizontal and vertical velocities at front and rear of pipeline under different porosities.

**Figure 9.**Comparison of turbulent kinetic energy at front and rear of wrapper under different porosities.

**Figure 12.**Temporal evolutions of vorticity contours around pipeline with wrapper thickness of 0.25 m (

**a**) 6.0 s (

**b**) 6.6 s (

**c**) 7.2 s (

**d**) 7.8 s (

**e**) 8.1 s (

**f**) 8.7 s (

**g**) 9.0 s (

**h**) 10.2 s (

**i**) 12.6 s.

**Figure 13.**Comparisons of flow field streamtraces and velocity contours under different wrapper thicknesses; (

**a**) T = 0.2 m; (

**b**) T = 0.3 m; (

**c**) T = 0.4 m; (

**d**) T = 0.5 m.

**Figure 14.**Comparisons of the maximum elevations and velocities in front and rear of the pipelines with different wrapper thicknesses; (

**a**) free surface elevation (note: original water depth is 6 m); (

**b**) velocity.

**Figure 16.**The velocity contours of the flow fields under different gaps; (

**a**) G = 0.2 m; (

**b**) G = 0.6 m; (

**c**) G = 1.0 m. Left to right: 6.3 s, 7.2 s, and 10.2 s. Left to right: arrival, stay, departure.

**Figure 17.**The vorticity contours of the flow fields under different gaps; (

**a**) G = 0.2 m; (

**b**) G = 0.6 m; (

**c**) G = 1.0 m. Left to right: 6.3 s, 7.2 s and 10.2 s.

**Figure 18.**Comparisons of the maximum horizontal and vertical hydrodynamic forces on the pipeline and wrapper under different gaps.

**Figure 19.**The velocity contours of the flow fields under different spacings; (

**a**) S = 2.5 m; (

**b**) S = 3.5 m; (

**c**) S = 4.5 m. Left to right: 6.3 s, 7.2 s and 10.2 s.

**Figure 20.**The vorticity contours of the flow fields under different porosities; (

**a**) S = 2.5 m; (

**b**) S = 3.5 m; (

**c**) S = 4.5 m. Left to right: 6.3 s, 7.2 s and 10.2 s.

**Figure 22.**The deviation of the forces on the front and rear pipelines and wrappers under different distances.

**Figure 24.**The temporal evolutions of forces on the pipeline and wrapper; (

**a**) Horizontal maximum force on pipeline; (

**b**) Vertical maximum force on pipeline; (

**c**) Horizontal maximum force on wrapper; (

**d**) Vertical maximum force on wrapper.

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## Share and Cite

**MDPI and ACS Style**

Dong, Y.; Zhao, E.; Cui, L.; Li, Y.; Wang, Y.
Dynamic Performance of Suspended Pipelines with Permeable Wrappers under Solitary Waves. *J. Mar. Sci. Eng.* **2023**, *11*, 1872.
https://doi.org/10.3390/jmse11101872

**AMA Style**

Dong Y, Zhao E, Cui L, Li Y, Wang Y.
Dynamic Performance of Suspended Pipelines with Permeable Wrappers under Solitary Waves. *Journal of Marine Science and Engineering*. 2023; 11(10):1872.
https://doi.org/10.3390/jmse11101872

**Chicago/Turabian Style**

Dong, Youkou, Enjin Zhao, Lan Cui, Yizhe Li, and Yang Wang.
2023. "Dynamic Performance of Suspended Pipelines with Permeable Wrappers under Solitary Waves" *Journal of Marine Science and Engineering* 11, no. 10: 1872.
https://doi.org/10.3390/jmse11101872