# Two-Phase Flow Simulation of Tunnel and Lee-Wake Erosion of Scour below a Submarine Pipeline

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Governing Equations

#### 2.2. Granular Stress Models

#### 2.2.1. $\mu \left(I\right)$ Rheology

#### 2.2.2. Kinetic Theory for Granular Flows

#### 2.3. Turbulence Models

#### 2.3.1. $k\u2013\epsilon $ Model

#### 2.3.2. $k\u2013\omega $2006 Model

#### 2.3.3. Modified $k\u2013\epsilon $ Model

#### 2.4. Numerical Setup

#### 2.4.1. General Setup

#### 2.4.2. Simulations with the $k\u2013\epsilon $ Turbulence Model

#### 2.4.3. Simulations with the $k\u2013\omega $2006 and the Modified $k\u2013\epsilon $ Turbulence Models

#### 2.5. Brier Skill Score

## 3. Results

#### 3.1. Solid Phase Stress Model Sensitivity

#### 3.2. Turbulence Model Sensitivity

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Hydrodynamic Simulations

#### Appendix A.1. Numerical Setup

Turbulence Kinetic Energy | Dissipation | |
---|---|---|

$k\u2013\epsilon $ | ${k}_{inlet}={\displaystyle \frac{3}{2}}{\left(UI\right)}^{2}$ | ${\epsilon}_{inlet}={C}_{\mu}{\displaystyle \frac{{k}^{1.5}}{l}}$ |

$k\u2013\omega 2006$ | ${k}_{inlet}={\displaystyle \frac{3}{2}}{\left(UI\right)}^{2}$ | ${\omega}_{inlet}={\displaystyle \frac{\sqrt{k}}{l}}$ |

#### Appendix A.2. Results

## References

- Sumer, B.M.; Fredsoe, J. The Mechanics of Scour in the Marine Environment; World Scientific Publishing Company: Singapore, 2002; Volume 17. [Google Scholar]
- Mao, Y. The Interaction Between a Pipeline and an Erodible Bed; Institute of Hydrodynamics and Hydraulic Engineering København: Series Paper; Institute of Hydrodynamics and Hydraulic Engineering; Technical University of Denmark: Lyngby, Denmark, 1986. [Google Scholar]
- Chiew, Y. Prediction of Maximum Scour Depth at Submarine Pipelines. J. Hydraul. Eng.
**1991**, 117, 452–466. [Google Scholar] [CrossRef] - Li, F.; Cheng, L. Numerical Model for Local Scour under Offshore Pipelines. J. Hydraul. Eng.
**1999**, 125, 400–406. [Google Scholar] [CrossRef] - Leeuwenstein, W.; Wind, H.G. The Computation of Bed Shear in a Numerical Model. In Proceedings of the 19th International Conference on Coastal Engineering, Houston, TX, USA, 3–7 September 1985. [Google Scholar]
- Liang, D.; Cheng, L.; Li, F. Numerical modeling of flow and scour below a pipeline in currents: Part II. Scour simulation. Coast. Eng.
**2005**, 52, 43–62. [Google Scholar] [CrossRef] - Jenkins, J.T.; Hanes, D.M. Collisional sheet flows of sediment driven by a turbulent fluid. J. Fluid Mech.
**1998**, 370, 29–52. [Google Scholar] [CrossRef] - Revil-Baudard, T.; Chauchat, J. A two-phase model for sheet flow regime based on dense granular flow rheology. J. Geophys. Res. Ocean.
**2013**, 118, 619–634. [Google Scholar] [CrossRef][Green Version] - Hsu, T.J.; Liu, P.L.F. Toward modeling turbulent suspension of sand in the nearshore. J. Geophys. Res. Ocean.
**2004**, 109. [Google Scholar] [CrossRef][Green Version] - Zhao, Z.; Fernando, H.J.S. Numerical simulation of scour around pipelines using an Euler-Euler coupled two-phase model. Environ. Fluid Mech.
**2007**, 7, 121–142. [Google Scholar] [CrossRef] - Yeganeh-Bakhtiary, A.; Kazeminezhad, M.H.; Etemad-Shahidi, A.; Baas, J.H.; Cheng, L. Euler-Euler two-phase flow simulation of tunnel erosion beneath marine pipelines. Appl. Ocean Res.
**2011**, 33, 137–146. [Google Scholar] [CrossRef] - Lee, C.H.; Low, Y.M.; Chiew, Y.M. Multi-dimensional rheology-based two-phase model for sediment transport and applications to sheet flow and pipeline scour. Phys. Fluids
**2016**, 28, 053305. [Google Scholar] [CrossRef] - MiDi, G.D.R. On dense granular flows. Eur. Phys. J. E
**2004**, 14, 341–365. [Google Scholar] [CrossRef] - Wilcox, D.C. Turbulence Modeling for CFD; DCW Industries: La Canada, CA, USA, 2006. [Google Scholar]
- Nagel, T. Three-Dimensional Scour Simulations with a Two-Phase Flow Model. 2019; under review. [Google Scholar]
- Cheng, Z.; Hsu, T.J.; Calantoni, J. SedFoam: A multi-dimensional Eulerian two-phase model for sediment transport and its application to momentary bed failure. Coast. Eng.
**2017**, 119, 32–50. [Google Scholar] [CrossRef][Green Version] - Chauchat, J.; Cheng, Z.; Nagel, T.; Bonamy, C.; Hsu, T.J. SedFoam-2.0: A 3D two-phase flow numerical model for sediment transport. Geosci. Model Dev. Discuss.
**2017**, 10, 4367–4392. [Google Scholar] [CrossRef] - Boyer, F.; Guazzelli, E.; Pouliquen, O. Unifying Suspension and Granular Rheology. Phys. Rev. Lett.
**2011**, 107, 188301. [Google Scholar] [CrossRef] - Richardson, J.F.; Zaki, W. Sedimentation and Fluidization: Part I. Chem. Eng. Res. Des.
**1954**, 32, 35–53. [Google Scholar] - Schiller, L.; Naumann, A.Z. Über die grundlegenden Berechnungen bei der Schwerkraftaufbereitung. Ver. Deut. Ing.
**1933**, 77, 318–320. [Google Scholar] - Chauchat, J. A comprehensive two-phase flow model for unidirectional sheet-flows. J. Hydraul. Res.
**2017**, 2017, 1289260. [Google Scholar] [CrossRef] - Maurin, R.; Chauchat, J.; Frey, P. Dense granular flow rheology in turbulent bedload transport. J. Fluid Mech.
**2016**, 804, 490–512. [Google Scholar] [CrossRef][Green Version] - Jop, P.; Forterre, Y.; Pouliquen, O. A constitutive law for dense granular flows. Nature
**2006**, 441, 727–730. [Google Scholar] [CrossRef][Green Version] - Chauchat, J.; Marc, M. A three-dimensional numerical model for incompressible two-phase flow of a granular bed submitted to a laminar shearing flow. Comput. Methods Appl. Mech. Eng.
**2009**, 199, 439–449. [Google Scholar] [CrossRef] - Ding, J.; Gidaspow, D. A bubbling fluidization model using kinetic theory of granular flow. AIChE J.
**1990**, 36, 523–538. [Google Scholar] [CrossRef] - Carnahan, N.F.; Starling, K.E. Equation of State for Nonattracting Rigid Spheres. J. Chem. Phys.
**1969**, 51, 635–636. [Google Scholar] [CrossRef] - Van Rijn, L. Sediment Transport, Part II: Suspended Load Transport. J. Hydraul. Eng.
**1984**, 110, 1613–1641. [Google Scholar] [CrossRef] - Ferziger, J.H.; Peric, M. Computational Methods for Fluid Dynamics; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2002. [Google Scholar]
- Sutherland, J.; Hall, L.; Chesher, T. Evaluation of the Coastal Area Model PISCES at Teignmouth (UK); HR Wallingford Report TR125; 2001. [Google Scholar] [CrossRef]
- Brady, A.; Sutherland, J. COSMOS Modelling of COAST3D Egmond Main Experiment; HR Wallingford Report TR115; 2001. [Google Scholar] [CrossRef]
- Van Rijn, L.; Ruessink, B.; Mulder, J. Coast3D-Egmond: The Behaviour of a Straight Sandy Coast on the Time Scale of Storms and Seasons; Aqua Publications: Locust Valley, NY, USA, 2002. [Google Scholar]
- Sumer, B.; Fredsoe, J. Hydrodynamics Around Cylindrical Structures; World Scientific: Singapore, 2006; Volume 12. [Google Scholar]

**Figure 2.**Time evolution of the maximum scour depth for simulations with the $k\u2013\epsilon $ turbulence model using $\mu \left(I\right)$ rheology (orange line) and kinetic theory (green dotted line) compared with the experimental data from Mao (1986) [2] (red dots) and numerical data from Lee et al. (2016) [12] (purple dashed line).

**Figure 3.**Bed profiles from simulations with the $k\u2013\epsilon $ turbulence model using $\mu \left(I\right)$ rheology (orange line) and kinetic theory (green dotted line) at 11 s (top), 18 s (middle), and 25 s (bottom) compared with experimental data from Mao (1986) [2] (red dots) and numerical data from Lee et al. (2016) [12] (purple dashed line).

**Figure 4.**Time evolution of the maximum scour depth from simulations with the $\mu \left(I\right)$ rheology using the $k\u2013\epsilon $ (orange line) and $k\u2013\omega $2006 (red dotted line) turbulence models compared with the experimental data from Mao (1986) [2] (red dots).

**Figure 5.**Bed profiles at 25 s from simulations with the $\mu \left(I\right)$ rheology using the $k\u2013\epsilon $ (orange line) and $k\u2013\omega $2006 (red dotted line) turbulence models compared with the experimental data from Mao (1986) [2] (red dots).

**Figure 6.**Streamlines and sediment volumetric flux at 25 s for the simulation using the $k\u2013\omega /2006$ turbulence model.

**Figure 7.**Time evolution of the maximum scour depth from simulations with $\mu \left(I\right)$ rheology using the $k\u2013\omega $2006 (red dotted line) and modified $k\u2013\u03f5$ turbulence model with (brown line) and without (brown dashed dotted line) the negative contribution of the cross-diffusion term compared with the experimental data from Mao (1986) [2] (red dots).

**Figure 8.**Typical turbulent kinetic energy (k) and specific dissipation rate ($\omega $) profiles for free shear flows, boundary flows, and sediment transport configurations.

**Table 1.**Empirical coefficients for the $k\u2013\epsilon $ turbulence model from Chauchat et al. (2017) [17].

${\mathit{\sigma}}_{\mathit{k}}$ | ${\mathit{\sigma}}_{\mathit{\epsilon}}$ | ${\mathit{C}}_{1\mathit{\epsilon}}$ | ${\mathit{C}}_{2\mathit{\epsilon}}$ | ${\mathit{C}}_{3\mathit{\epsilon}}$ | ${\mathit{C}}_{4\mathit{\epsilon}}$ | ${\mathit{C}}_{\mathit{\mu}}$ |
---|---|---|---|---|---|---|

1.0 | 0.77 | 1.44 | 1.92 | 1.2 | 1.0 | 0.09 |

${\mathit{\sigma}}_{\mathit{k}}$ | ${\mathit{\sigma}}_{\mathit{\omega}}$ | ${\mathit{C}}_{1\mathit{\omega}}$ | ${\mathit{C}}_{2\mathit{\omega}}$ | ${\mathit{C}}_{3\mathit{\omega}}$ | ${\mathit{C}}_{4\mathit{\epsilon}}$ | ${\mathit{C}}_{\mathit{\mu}}$ | C${}_{\mathit{lim}}$ |
---|---|---|---|---|---|---|---|

0.6 | 0.5 | 0.52 | 0.0708 | 0.35 | 1.0 | 0.09 | 0.875 |

${\mathit{\sigma}}_{\mathit{k}}$ | ${\mathit{\sigma}}_{\mathit{\omega}}$ | ${\mathit{C}}_{1\mathit{\omega}}$ | ${\mathit{C}}_{2\mathit{\omega}}$ | ${\mathit{C}}_{3\mathit{\omega}}$ | ${\mathit{C}}_{4\mathit{\epsilon}}$ | ${\mathit{\sigma}}_{\mathit{d}}$ |
---|---|---|---|---|---|---|

1.0 | 0.856 | 0.44 | 0.0828 | 0.35 | 1.0 | 1.712 or Equation (32) |

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**MDPI and ACS Style**

Mathieu, A.; Chauchat, J.; Bonamy, C.; Nagel, T. Two-Phase Flow Simulation of Tunnel and Lee-Wake Erosion of Scour below a Submarine Pipeline. *Water* **2019**, *11*, 1727.
https://doi.org/10.3390/w11081727

**AMA Style**

Mathieu A, Chauchat J, Bonamy C, Nagel T. Two-Phase Flow Simulation of Tunnel and Lee-Wake Erosion of Scour below a Submarine Pipeline. *Water*. 2019; 11(8):1727.
https://doi.org/10.3390/w11081727

**Chicago/Turabian Style**

Mathieu, Antoine, Julien Chauchat, Cyrille Bonamy, and Tim Nagel. 2019. "Two-Phase Flow Simulation of Tunnel and Lee-Wake Erosion of Scour below a Submarine Pipeline" *Water* 11, no. 8: 1727.
https://doi.org/10.3390/w11081727