# Numerical Study on the Hydrodynamic Characteristics of Submarine Pipelines under the Impact of Real-World Tsunami-Like Waves

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}(*) wave profiles. The effects of prominent factors such as wave height, water depth, pipeline diameter, and gap-ratio on the hydrodynamic characteristics of submarine pipelines are discussed in detail. Hydrodynamic characteristics of pipelines in a tandem arrangement are also investigated. The paper is organized as follows: Section 2 describes governing equations and numerical methods; Section 3 describes the model validation; Section 4 presents the results and discussions; Section 5 summarizes the findings.

## 2. Numerical Model

#### 2.1. Governing Equations

**U**is the flow velocity vector, ρ is the mixture density of air–water, t is the time, μ

_{e}is the effective viscosity, ρ

_{r}is the reference density, p is the pressure, and

**g**is the gravity acceleration [24].

_{k}represents the generation of turbulence kinetic energy, G

_{ω}represents the generation of ω, Y

_{k}, and Y

_{ω}represent the dissipation of k and ω, S

_{k}and S

_{ω}are source terms. In the simulation, the k-ω model is used as the turbulence closure due to two reasons. Firstly, the simulation and calculation of the fluid nearby the wall are more stable using this turbulence model than other models. In order to capture the detailed flow around the wall of submarine pipeline, this turbulent closure is selected. Secondly, the calculation accuracy of the model can be ensured when the counter-pressure gradient flow is simulated. So, this turbulent closure is suitable for the solution of free shear turbulence flow, boundary layer turbulence flow, and moderate separation turbulence flow when the wave passes the submarine pipeline.

#### 2.2. Wave Generation Method

_{l}) is specified on moving boundary firstly. Based on the height, the layer of cells (layer j) adjacent to the moving boundary is split or merged with the layer of cells (layer i) next to it. The dynamic layering mesh adjacent to moving boundary is added or removed layers of cells. To produce the desired free surface profile, the position of the moving boundary changes with time [26,27]. Solitary and tsunami waves are generated using the piston-type wave maker depending on the following methods.

_{o}is the wave height, k

_{o}= (3H

_{o}/4h

^{3})

^{1/2}is the effective wave number, x

_{o}is the location of the wave crest at t = 0, C

_{o}= [g(H

_{o}+ h)]

^{1/2}represents the wave celerity, T

_{o}is the wave period of a solitary wave which is defined by the effective wave number.

^{2}(*) wave profiles is used to model the tsunami wave based on the concept of N-wave. The combination of three sech

^{2}(*) wave can be written as

_{o}is the wave frequency of a solitary wave with the same wave height as the tsunami-like wave.

## 3. Model Validation

^{2}/4), namely, F* = F/ρghA. In the rest of the paper, all hydrodynamic forces are presented using this method. Predicted horizontal and vertical forces are in good agreement with corresponding measurements in both peak force values and temporal evolution process. Forces for the medium and dense meshes are nearly identical whereas those computed on the coarse mesh are noticeably different from measurement. Hence, the resolution of dense mesh and medium mesh is sufficient. In order to guarantee the calculation accuracy, a similar resolution mesh setup as dense mesh will be applied in the following simulation. By carrying out the simulation work in this section, the computational capability of our model in predicting the hydrodynamic forces of submarine pipeline is well calibrated.

## 4. Results and Discussions

^{5}and the maximum iterations are 20. Depending on the numerical model, 2D simulations are conducted. The calculation was performed on the a Dell Precision 3630 Tower, which includes 12 central processing units (CUPs). The CPU type is Intel (R) Core (TM) i7-8700 CUP @ 3.20GHz which is producted by Intel Corporation, California, America. The storage of the internal memory and hardware are 16G and 2T, respectively. In order to save computation time, four cases were run simultaneously on one computer. The real duration for one case is about 115 h. When the simulation begins, the program reads the initial boundary conditions and mesh grids. Depending on the numerical methods, the results are generated and saved according to the storage time step. Because the moving-mesh method is used in the simulation, when the results are saved, the changed meshes are also saved.

#### 4.1. Single Pipeline

- (1)
- the leading-depression wave portion;
- (2)
- the preceding elevated wave portion;
- (3)
- the secondary elevated wave portion.

_{1}and L

_{2}under tsunami-like wave and solitary wave conditions are depicted in Figure 14. Since the diameter of the pipeline is relatively small compared to still water depth, for both solitary and tsunami-like waves, the wave profiles are almost unchanged. Although the blocking effect of the pipeline on wave propagation is very weak, the crests of two waves still fluctuate slightly. The wave heights decrease for both waves when they pass through the pipeline. The wave height of the solitary wave decreases more than that of the tsunami-like wave. The difference between the two wave shapes are independent of the difference in the wave peaks due to the different scattering. The maximum velocity of the tsunami-like wave measured at P is about 1.15 times of that of the solitary wave at the same wave height (Figure 15). Also, the duration of the high-velocity portion is much longer for the tsunami-like wave.

#### 4.1.1. Effect of Water Depth

_{1}at different water depths are plotted in Figure 17. It is seen that the wave profile of the tsunami-like wave gradually becomes fatter and its wave period becomes longer as water depth increases, thus increasing its energy carrying capacity.

#### 4.1.2. Effect of Wave Height

#### 4.1.3. Effect of Pipe Diameter

#### 4.1.4. Effect of Gap-Ratio

#### 4.2. Pipelines in Tandem Arrangement

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Fang, N.; Chen, G.M.; Zhu, H.W.; Meng, H.X. Statistical analysis of leakage accidents of submarine pipeline. Oil Gas Storage Transp.
**2014**, 33, 99–103. [Google Scholar] - Tong, F.F.; Cheng, L.; An, H.; Griffiths, T. The hydrodynamic forces on a circular cylinder in proximity to a wall with intermittent contact in steady current. Ocean Eng.
**2017**, 146, 424–433. [Google Scholar] [CrossRef] - Sumer, B.M.; Fredsoe, J.; Gravesen, H.; Bruschi, R. Response of Marine Pipelines in Scour Trenches. J. Waterway Port Coast. Ocean Eng.
**1989**, 115, 477–496. [Google Scholar] [CrossRef] - Zhao, E.J.; Shi, B.; Qu, K.; Dong, W.B.; Zhang, J. Experimental and Numerical Investigation of Local Scour around Submarine Piggyback Pipeline under Steady Currents. J. Ocean Univ. China
**2018**, 17, 244–256. [Google Scholar] [CrossRef] - Subbiah, K.; Cheong, H.F.; Shankar, N.J. Regular and random wave pressures around large diameter submarine pipeline near ocean bed. J. Hydraul. Res.
**1991**, 29, 49–66. [Google Scholar] [CrossRef] - Gao, F.P.; Gu, X.Y.; Jeng, D.S.; Teo, H.T. An experimental study for wave-induced instability of pipelines: The breakout of pipelines. Appl. Ocean Res.
**2002**, 24, 83–90. [Google Scholar] [CrossRef] - Haley, J.F.; Swan, C.; Gibson, R. An Experimental Investigation of Wave Impact Loads on a Slender Horizontal Cylinder. In Proceedings of the ASME 2014 33rd International Conference on Ocean, Offshore and Arctic Engineering, San Francisco, CA, USA, 8–13 June 2014; American Society of Mechanical Engineers: New York, NY, USA, 2014; p. V08BT06A041. [Google Scholar]
- Gao, N.; Yang, J.; Li, X.; Zhao, W. Wave forces on horizontal cylinder due to nonlinear focused wave groups. In Proceedings of the Twenty-fifth International Ocean and Polar Engineering Conference, Kona, HI, USA, 21–26 June 2015; International Society of Offshore and Polar Engineers: Houston, TX, USA, 2015. [Google Scholar]
- Chern, M.-J.; Odhiambo, E.A.; Horng, T.-L.; Borthwick, A.G.L. Numerical simulation of vibration of horizontal cylinder induced by progressive waves. Fluid Dyn. Res.
**2016**, 48, 015508. [Google Scholar] [CrossRef] [Green Version] - Ong, M.C.; Kamath, A.; Bihs, H.; Afzal, M.S. Numerical simulation of free-surface waves past two semi-submerged horizontal circular cylinders in tandem. Mar. Struct.
**2017**, 52, 1–14. [Google Scholar] [CrossRef] - Liang, D.F.; Gotoh, H.; Khayyer, A.; Chen, J.M. Boussinesq mdelling of solitary wave and N-wave runup on coast. Appl. Ocean Res.
**2013**, 42, 144–154. [Google Scholar] [CrossRef] - Hsiao, S.C.; Lin, T.C. Tsunami-like solitary waves impinging and overtopping an impermeable seawall: Experiment and RANS modeling. Coast. Eng.
**2010**, 57, 1–18. [Google Scholar] [CrossRef] - Limura, K.; Norio, T. Numerical simulation estimating effects of tree density distribution in coastal forest on tsunami mitigation. Ocean Eng.
**2012**, 54, 223–232. [Google Scholar] - Synolakis, C.E. The run-up of solitary waves. J. Fluid Mech.
**1987**, 185, 523–545. [Google Scholar] [CrossRef] - Gedik, N.; Irtem, E.; Kabdasli, S. Laboratory investigation on tsunami run-up. Ocean Eng.
**2005**, 32, 513–528. [Google Scholar] [CrossRef] - Goseberg, N.; Wurpts, A.; Schlurmann, T. Laboratory-scale generation of tsunami and long waves. Coast. Eng.
**2013**, 79, 57–74. [Google Scholar] [CrossRef] - Francesco, A.; Tripepi, G.; Meringolo, D.D.; Veltri, P. Solitary wave-induced forces on horizontal circular cylinders: Laboratory experiments and SPH simulations. Coast. Eng.
**2017**, 129, 17–35. [Google Scholar] - Madsen, A.; Schäffer, H.A. Analytical solutions for tsunami run-up on a plane beach: Single waves, N-waves and transient waves. J. Fluid Mech.
**2010**, 645, 27–57. [Google Scholar] [CrossRef] - Madsen, A.; Fuhrman, D.R.; Schäffer, H.A. On the solitary wave paradigm for tsunamis. J. Geophys. Res.
**2008**, 113, 286–292. [Google Scholar] [CrossRef] - Chan, I.C.; Liu, L.F. On the run-up of long waves on a plane beach. J. Geophys. Res.
**2012**, 117, 72–82. [Google Scholar] [CrossRef] - Qu, K.; Ren, X.Y.; Kraatz, S. Numerical investigation of tsunami-like wave hydrodynamic characteristics and its comparison with solitary wave. Appl. Ocean Res.
**2017**, 63, 36–48. [Google Scholar] [CrossRef] - Stefan, L.; Oumeraci, H. Solitary waves and bores passing three cylinders-effect of distance and arrangement. Coast. Eng. Proc.
**2014**, 1, 39. [Google Scholar] - Istrati, D.; Buckle, I.; Lomonaco, P.; Yim, S. Deciphering the Tsunami Wave Impact and Associated Connection Forces in Open-Girder Coastal Bridges. J. Mar. Sci. Eng.
**2018**, 6, 148. [Google Scholar] [CrossRef] - Zhao, E.J.; Mu, L.; Shi, B. Numerical Study of the Influence of Tidal Current on Submarine Pipeline Based on the SIFOM–FVCOM Coupling Model. Water
**2018**, 10, 1814. [Google Scholar] [CrossRef] - Rhie, T.M.; Chow, A. Numerical study of the turbulent flow past an isolated airfoil with trailing-edge separation. AIAA J.
**1983**, 21, 1525–1532. [Google Scholar] [CrossRef] - Gomes, M.N.; Olintoa, C.R.; Rochaa, L.A.O.; Souzaa, J.A.; Isoldi, L.A. Computational modeling of a regular wave tank. Therm. Eng.
**2009**, 8, 44–50. [Google Scholar] - Hafsia, Z.; Haj, M.B.; Lamloumi, H.; Maalel, K. Comparison between Moving Paddle and Mass Source Methods for Solitary Wave Generation and Propagation over A Steep Sloping Beach. Eng. Appl. Comput. Fluid Mech.
**2009**, 3, 355–368. [Google Scholar] [CrossRef] - Qu, K.; Ren, X.Y.; Kraatz, S.; Zhao, E.J. Numerical analysis of tsunami-like wave impact on horizontal cylinders. Ocean Eng.
**2017**, 145, 316–333. [Google Scholar] [CrossRef] - Sibley, P.O. The Solitary Wave and the Forces It Imposes on a Submerged Horizontal Circular Cylinder: An Analytical and Experimental Study. Ph.D. Thesis, City University London, London, UK, 1991. [Google Scholar]

**Figure 5.**Same as Figure 3 but for run 2.

**Figure 6.**Force comparison between measurement and simulation in run1; (

**a**) horizontal force F

_{x}*; (

**b**) vertical force F

_{z}*.

**Figure 7.**Force comparison between measurement and simulation in run 2; (

**a**) horizontal force F

_{x}*; (

**b**) vertical force F

_{z}*.

**Figure 12.**Snapshots of the velocity contour around the submarine pipeline at different times for the tsunami-like wave.

**Figure 13.**Snapshots of the velocity contour around submarine pipeline at different times for the solitary wave.

**Figure 14.**Time series of water elevations recorded at elevation sensors L1 and L2 for the solitary and tsunami-like wave.

**Figure 15.**Time series of velocity at velocity sensor P during the tsunami-like wave and solitary wave.

**Figure 16.**Time series of hydrodynamic forces at a pipeline under tsunami-like and solitary waves; (

**a**) horizontal force F

_{x}*; (

**b**) vertical force F

_{z}*.

**Figure 17.**Time series of water elevation recorded at elevation sensor L

_{1}for a tsunami-like wave at different water depths.

**Figure 18.**The variations of velocity at different water depths recorded at sensor P; (

**a**) time series of velocity; (

**b**) maximum velocity as function of water depth. α is the ratio of water depth h to the wave height H.

**Figure 19.**Plots of maximum forces as function of water depth; (

**a**) maximum horizontal force F

_{x,max}*; (

**b**) maximum vertical force F

_{z,max}*; (

**c**) minimum horizontal force F

_{x,min}*; (

**d**) minimum vertical force F

_{z,min}*.

**Figure 22.**Plots of maximum hydrodynamic forces at the pipeline as a function of wave height; (

**a**) maximum horizontal force F

_{x,max}*; (

**b**) maximum vertical force F

_{z,max}*; (

**c**) minimum horizontal force F

_{x,min}*; (

**d**) minimum vertical force F

_{z,min}*.

**Figure 24.**Plots of the maximum horizontal and vertical forces as function of pipeline diameter; (

**a**) maximum horizontal force F

_{x,max}*; (

**b**) maximum vertical force F

_{z,max}*; (

**c**) minimum horizontal force F

_{x,min}*; (

**d**) minimum vertical force F

_{z,min}*.

**Figure 25.**Snapshots of vortices contours for pipelines with different gap-ratios at t = 135 s under the tsunami-like wave.

**Figure 26.**Temporal evolution of hydrodynamic forces at a pipeline when the gap-ratio equals 0.25; (

**a**) horizontal force F

_{x}* (

**b**) vertical force F

_{z}*.

**Figure 27.**Plots of the maximum horizontal and vertical forces as a function of gap-ratio; (

**a**) maximum horizontal force F

_{x,max}*; (

**b**) maximum vertical force F

_{z,max}*; (

**c**) minimum horizontal force F

_{x,min}*; (

**d**) minimum vertical force F

_{z,min}*.

**Figure 28.**Snapshots of vortices contours around a tandem pipeline subjected to a tsunami-like wave at different pipeline spacing distances and t = 180 s. δ is the ratio of the distance between the two pipes to the pipeline diameter.

**Figure 29.**Same as Figure 28 but for a solitary wave at t = 22 s.

**Figure 30.**Temporal evolution of the hydrodynamic forces at pipelines with spacing distance 1 D under the impact of solitary and tsunami-like waves; (

**a**) horizontal force of upstream pipe F

_{x}*; (

**b**) vertical force of downstream pipe F

_{z}*; (

**c**) horizontal force of downstream pipe F

_{x}*; (

**d**) vertical force of downstream pipe F

_{z}*.

**Figure 31.**Horizontal and vertical forces on the upstream pipeline as function of spacing ratio; (

**a**) maximum horizontal force F

_{x,max}*; (

**b**) maximum vertical force F

_{z,max}*; (

**c**) minimum horizontal force F

_{x,min}*; (

**d**) minimum vertical force F

_{z,min}*.

**Figure 32.**Horizontal and vertical forces on the downstream pipeline as function of spacing ratio; (

**a**) maximum horizontal force F

_{x,max}*; (

**b**) maximum vertical force F

_{z,max}*; (

**c**) minimum horizontal force F

_{x,min}*; (

**d**) minimum vertical force F

_{z,min}*.

Run | h (m) | H (m) | D (m) | G (m) |
---|---|---|---|---|

1 | 0.17 | 0.034 | 0.034 | 0.0714 |

2 | 0.16875 | 0.03105 | 0.027 | 0.023625 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhao, E.; Qu, K.; Mu, L.; Kraatz, S.; Shi, B.
Numerical Study on the Hydrodynamic Characteristics of Submarine Pipelines under the Impact of Real-World Tsunami-Like Waves. *Water* **2019**, *11*, 221.
https://doi.org/10.3390/w11020221

**AMA Style**

Zhao E, Qu K, Mu L, Kraatz S, Shi B.
Numerical Study on the Hydrodynamic Characteristics of Submarine Pipelines under the Impact of Real-World Tsunami-Like Waves. *Water*. 2019; 11(2):221.
https://doi.org/10.3390/w11020221

**Chicago/Turabian Style**

Zhao, Enjin, Ke Qu, Lin Mu, Simon Kraatz, and Bing Shi.
2019. "Numerical Study on the Hydrodynamic Characteristics of Submarine Pipelines under the Impact of Real-World Tsunami-Like Waves" *Water* 11, no. 2: 221.
https://doi.org/10.3390/w11020221