# Two-Phase Flow Modeling for Bed Erosion by a Plane Jet Impingement

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

- The impinging region. It is characterized by a stagnation point on the jet axis at the wall with a deflexion of flow streamlines [21].

## 2. Laboratory Experiments

#### 2.1. Experimental Setup

#### 2.2. Measure of the Crater Dimensions

#### 2.3. Experimental Results

## 3. Mathematical Model

#### 3.1. Governing Equations

#### 3.2. A Unified Momentum Equation for the Solid Phase

#### Solid-Like Model

#### 3.3. Solid-Liquid Transition of the Granular Phase

#### 3.4. Numerical Technique

## 4. Numerical Results

#### 4.1. Original and Unified Model

#### 4.2. Crater Size Predictions

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

2D | Two Dimensional |

3D | Three Dimensional |

JET | Jet Erosion Test |

WID | Water Injection Dredging |

CFD | Computational Fluid Dynamics |

MPI | Message Passing Interface |

SOR | Successive Over Relaxation |

GPU | Graphics Processing Unit |

TVD | Total Variation Diminishing |

PMMA | Polymethyl Methacrylate |

## References

- Metzger, P.T.; Latta, R.C.; Schuler, J.M.; Immer, C.D. Craters formed in granular beds by impinging jets of gas. AIP Conf. Proc.
**2009**, 1145, 767–770. [Google Scholar] - Metzger, P.T.; Immer, C.D.; Donahue, C.M.; Vu, B.T.; Latta, R.C.; Deyo-Svendsen, M. Jet-induced cratering of a granular surface with application to lunar spaceports. J. Aerosp. Eng.
**2009**, 22, 24–32. [Google Scholar] [CrossRef] [Green Version] - Rouse, H. Criteria for similarity in the transportation of sediment. In Proceedings of the 1st Hydraulics Conference, Bulletin 20, State University of Iowa, Iowa City, IA, USA, 12–15 June 1939; pp. 33–49. [Google Scholar]
- Hanson, G.J.; Cook, K.R. Apparatus, test procedures, and analytical methods to measure soil erodibility in situ. Appl. Eng. Agric.
**2004**, 20, 455–462. [Google Scholar] [CrossRef] - Hanson, G.J.; Hunt, S.L. Lessons learned using laboratory JET method to measure soil erobibility of compacted soils. Appl. Eng. Agric.
**2007**, 23, 305–312. [Google Scholar] [CrossRef] - Perng, A.T.H.; Capart, H. Underwater sand bed erosion and internal jump formation by a travelling plane jets. J. Fluid. Mech.
**2008**, 595, 1–43. [Google Scholar] [CrossRef] [Green Version] - Kobus, H.; Leister, P.; Westrich, B. Flow field and scouring effects of steady and pulsating jets impinging on a movable bed. J. Hydr. Res.
**1979**, 17, 175–192. [Google Scholar] [CrossRef] - Rajaratnam, N.; Beltaos, S. Erosion by impinging circular turbulent jets. Procs. ASCE J. Hydr. Div.
**1977**, 103, 1191–1205. [Google Scholar] [CrossRef] - Aderibigbe, O.O.; Rajaratnam, N. Erosion of loose beds by submerged circular impinging vertical turbulent jets. J. Hydraulic Res.
**1996**, 34, 19–33. [Google Scholar] [CrossRef] - Badr, S.; Gauthier, G.; Gondret, P. Crater jet morphology. Phys. Fluids
**2016**, 28, 033305. [Google Scholar] [CrossRef] - Rajaratnam, N.; Mazurek, K.A. Erosion of Sand by Circular Impinging Water Jets with Small Tailwater. J. Hydr. Eng.
**2003**, 129, 225–229. [Google Scholar] [CrossRef] - Rajaratnam, N.; Mazurek, K.A. Impingement of circular turbulent jets on rough boundaries Impact des jets turbulents sur des parois rugueuses. J. Hydr. Res.
**2005**, 43, 688–694. [Google Scholar] [CrossRef] - Mazurek, K.A.; Gheisi, A.R. Assessment of the Erodibility of a Cohesive Soil using a Submerged Circular Turbulent Impinging Jet. In Proceedings of the 33rd IAHR Congress, Vancouver, BC, Canada, 9–14 August 2009. [Google Scholar]
- Sutherland, B.R.; Dalziel, S. Bedload transport by a vertical jet impinging upon sediments. Phys. Fluids
**2014**, 26, 035103. [Google Scholar] [CrossRef] - Badr, S.; Gauthier, G.; Gondret, P. Erosion threshold of a liquid immersed granular bed by an impinging plane liquid jet. Phys. Fluids
**2014**, 26, 023302. [Google Scholar] [CrossRef] [Green Version] - Beltaos, S.; Rajaratnam, N. Plane turbulent impinging jets. J. Hydr. Res.
**1973**, 11, 29–59. [Google Scholar] [CrossRef] - Rajaratnam, N. Erosion by plane turbulent jets. J. Hydr. Res.
**1981**, 19, 339–358. [Google Scholar] [CrossRef] - Phares, D.; Smedley, G.; Flagan, R. The wall shear stress produced by a normal impingement of a jet on a flat surface. J. Fluid Mech.
**2000**, 418, 351–375. [Google Scholar] [CrossRef] [Green Version] - Tritton, D.J. Physical Fluid Dynamics, 2nd ed.; Springer: Oxford, UK, 1988. [Google Scholar]
- Pope, S.B. Turbulent Flows, 1st ed.; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Donalson, C.D.; Snedeker, R.S. A study of free jet impingement. Part 1. Mean properties of free and impinging jets. J. Fluid Mech.
**1971**, 45, 281–319. [Google Scholar] [CrossRef] - Glauert, M.B. The wall jet. J. Fluid Mech.
**1956**, 1, 625–643. [Google Scholar] [CrossRef] - Launder, B.E.; Rodi, W. The turbulent wall: Measurements and modeling. Ann. Rev. Fluid Mech.
**1983**, 15, 429–459. [Google Scholar] [CrossRef] - Magyari, E.; Keller, B. The algebraically decaying wall jet. Eur. J. Mech. B Fluids
**2004**, 23, 601–605. [Google Scholar] [CrossRef] - Weidner, K.; Petrie, J.; Diplas, P.; Nam, S.; Gutierrez, M.; Ellenberg, M. Numerical simulation of jet test and associated soil erosion. In Proceedings of the 6th International Conference on Scour and Erosion, ICSE6, Paris, France, 27–31 August 2012. [Google Scholar]
- Mercier, F.; Bonelli, S.; Anselmet, F.; Pinettes, P.; Courivaud, J.R.; Fry, J.J. On the numerical modelling of the Jet Erosion Test. In Proceedings of the 6th International Conference on Scour and Erosion, ICSE6, Paris, France, 27–31 August 2012. [Google Scholar]
- Wang, T.; Song, B. Study on deepwater conductor jet excavation mechanism in cohesive soil. Appl. Ocean Res.
**2019**, 82, 225–235. [Google Scholar] [CrossRef] - Kuang, S.B.; LaMarche, C.Q.; Curtis, J.S.; Yu, A.B. Discrete particle simulation of jet-induced cratering of a granular bed. Powder Technol.
**2013**, 239, 319–336. [Google Scholar] [CrossRef] - Benseghier, Z.; Cuéllar, P.; Luu, L.H.; Delenne, J.Y.; Bonelli, S.; Philippe, P. Relevance of free jet model for soil erosion by impinging jets. J. Hydraul. Eng.
**2020**, 146, 04019047. [Google Scholar] [CrossRef] - Benseghier, Z.; Cuéllar, P.; Luu, L.H.; Bonelli, S.; Philippe, P. A parallel GPU-based computational framework for the micromechanical analysis of geotechnical and erosion problems. Comput. Geotech.
**2020**, 120, 103404. [Google Scholar] [CrossRef] - Boyaval, S.; Caboussat, A.; Mrad, A.; Picasso, M.; Steiner, G. A semi-Lagrangian splitting method for the numerical simulation of sediment transport with free surface flows. Comput. Fluids
**2018**, 172, 384–396. [Google Scholar] [CrossRef] - Qian, Z.D.; Hu, X.Q.; Huai, W.X.; Xue, W.Y. Numerical simulation of sediment erosion by submerged jets using an Eulerian model. Sci. China Technol. Sci.
**2010**, 53, 3324–3330. [Google Scholar] [CrossRef] - Yuan, Q.; Zhao, M.; Wang, C.; Ge, T. Numerical study of sand scour with a modified Eulerian model based on incipient motion theory. Mar. Georesour. Geotechnol.
**2018**, 36, 818–826. [Google Scholar] [CrossRef] - Uh Zapata, M.; Pham Van Bang, D.; Nguyen, K.D. Parallel simulations for a 2D x/z two-phase flow fluid-solid particle model. Comput. Fluids
**2018**, 173, 103–110. [Google Scholar] [CrossRef] - Wang, B.; van Rhee, C.; Nobel, A.; Keetels, G. Modeling the hydraulic excavation of cohesive soil by a moving vertical jet. Ocean Eng.
**2021**, 227, 108796. [Google Scholar] [CrossRef] - Greenshields, C.J.; Weller, H.G. A unified formulation for continuum mechanics applied to fluid-structure interaction in flexible tubes. Int. J. Numer. Meth. Engng.
**2005**, 64, 1575–1593. [Google Scholar] [CrossRef] - Chua, L.P.; Lua, A.C. Measurements of a confined jet. Phys. Fluids
**1998**, 10, 3137–3144. [Google Scholar] [CrossRef] - Midi, G.D.R. On dense granular flows. Eur. Phys. J. E
**2004**, 14, 341–365. [Google Scholar] [CrossRef] [Green Version] - Loiseleux, T.; Gondret, P.; Rabaud, M.; Doppler, D. Onset of erosion and avalanche for an inclined granular bed sheared by a continuous laminar flow. Phys. Fluids
**2005**, 17, 103304. [Google Scholar] [CrossRef] - Drew, D.A.; Lahey, R.T. Analytical Modelling of Multiphase Flow, in Particulate Two-Phase Flow; Roco, M.C., Ed.; Butterworth-Heinemann: Boston, MA, USA, 1993. [Google Scholar]
- Barbry, N.; Guillou, S.; Nguyen, K.D. Une approche diphasique pour le calcul du transport sédimentaire en milieux estuariens. C.R. Acad. Sci. IIb
**2000**, 328, 793–799. [Google Scholar] [CrossRef] - Nguyen, K.D.; Guillou, S.; Chauchat, J.; Barbry, N. A two-phase numerical model for suspended-sediment transport in estuaries. Adv. Water Resour.
**2009**, 32, 1187–1196. [Google Scholar] [CrossRef] - Nguyen, D.H.; Levy, F.; Van Bang, D.P.; Nguyen, K.D.; Guillou, S.; Chauchat, J. Simulation of dredged sediment releases into homogeneous water using a two-phase model. Adv. Water Resour.
**2012**, 48, 102–112. [Google Scholar] [CrossRef] - Lundgren, T. Slow flow through stationary random beds and suspensions of spheres. J. Fluid. Mech.
**1972**, 51, 273–299. [Google Scholar] [CrossRef] - Graham, A.L. On the viscosity of suspensions of solid spheres. Appl. Sci. Res.
**1981**, 37, 275–286. [Google Scholar] [CrossRef] - Chauchat, J.; Guillou, S. On turbulence closures for two-phase sediment-laden flow models. J. Geophys. Res.
**2008**, 113, C11017. [Google Scholar] [CrossRef] [Green Version] - Haider, A.; Levenspiel, O. Drag coefficient and terminal velocity of spherical and non-spherical particles. Powder Technol.
**1989**, 58, 63–70. [Google Scholar] [CrossRef] - Guillou, S.; Barbry, N.; Nguyen, K.D. Calcul numérique des ondes de surface par une méthode de projection et un maillage eulérien adaptatif. C.R. Acad. Sci. IIb
**2000**, 328, 875–881. [Google Scholar] - Uh Zapata, M.; Zhang, W.; Marois, L.; Hammouti, A.; Pham Van Bang, D.; Nguyen, K.D. Two-phase experimental and numerical studies on scouring at the toe of vertical seawall. Eur. J. Mech. B Fluids
**2022**, 93, 13–28. [Google Scholar] [CrossRef] - Chorin, A.J. Numerical solution of the Navier–Stokes equations. Math. Comp.
**1968**, 22, 745–762. [Google Scholar] [CrossRef] - Guillou, S.; Nguyen, K.D. An improved technique for solving two-dimensional shallow water problems. Int. J. Num. Meth. Fluids
**1999**, 29, 465–483. [Google Scholar] [CrossRef]

**Figure 1.**Geometrical parameters of the eroded sediment bed for a flow (

**a**) weakly and (

**b**) strongly deflected. Here, H and D are the crater depth and width, respectively, and L is the distance above the granular bed.

**Figure 2.**Sketch of the device to study the erosion of a granular bed by a water jet: (

**a**) hydraulic circuit, and (

**b**) water jet injector dimensions.

**Figure 3.**Experimental results of the crater (

**a**) depth and (

**b**) width normalized by the jet width b as function of the normalized jet-bed distance $L/b$ for different jet Reynolds numbers.

**Figure 4.**Experimental results of the crater (

**a**) depth and (

**b**) width normalized by the reduced jet-bed distance $L-\lambda $ as a function of E for different jet Reynolds numbers. The solid line corresponds to the linear fit model through the data.

**Figure 5.**Numerical simulations at different stages for the standard two-phase flow model: the solid concentration with fluid field (

**top**), and velocity magnitude of the fluid (

**bottom**).

**Figure 6.**Solid concentration (

**top**) and fluid velocity magnitude (

**bottom**) for the proposed two-phase flow model using a solid–liquid transition model of the sediment phase at different stages.

**Figure 7.**Numerical results of the liquid–solid transition function F at different stages. The fluid velocity field is also plotted as a reference.

**Figure 8.**Numerical results of the fluid velocity (

**a**) field, and (

**b**) magnitude with streamlines at $t=1.5$ s using $b=8$ mm, ${U}_{J}=0.323$ m/s, and $L=4.6$ cm for different grid resolutions.

**Figure 9.**Solid volume fraction at $t=1.5$ s using $b=8$ mm, ${U}_{J}=0.323$ m/s, and $L=4.6$ cm for different grid resolutions.

**Figure 10.**Velocity field and crater shapes for different jet configurations using $b=4$ mm and fine mesh ($\Delta x=\Delta z=0.5$ mm).

**Table 1.**Experimental results using different experimental parameters of ${U}_{J}$, L and jet thickness $b=0.004$ m.

Mean Velocity | Reynolds Number | Impingement Distance | Crater Width | Crater Depth | Mean Velocity | Reynolds Number | Impingement Distance | Crater Width | Crater Depth |
---|---|---|---|---|---|---|---|---|---|

${\mathit{U}}_{\mathit{J}}$ (m/s) | Re | L (m) | D (m) | H (m) | ${\mathit{U}}_{\mathit{J}}$ (m/s) | Re | L (m) | D (m) | H (m) |

0.471 | 1885 | 0.145 | 0.015 | 0.091 | 0.314 | 1256 | 0.111 | 0.007 | 0.044 |

0.471 | 1885 | 0.130 | 0.020 | 0.095 | 0.314 | 1256 | 0.103 | 0.008 | 0.042 |

0.471 | 1885 | 0.084 | 0.022 | 0.067 | 0.314 | 1256 | 0.088 | 0.011 | 0.040 |

0.471 | 1885 | 0.059 | 0.025 | 0.057 | 0.314 | 1256 | 0.075 | 0.013 | 0.043 |

0.471 | 1885 | 0.036 | 0.025 | 0.034 | 0.314 | 1256 | 0.062 | 0.017 | 0.040 |

0.471 | 1885 | 0.108 | 0.025 | 0.090 | 0.314 | 1256 | 0.051 | 0.015 | 0.045 |

0.431 | 1726 | 0.159 | 0.005 | 0.087 | 0.314 | 1256 | 0.041 | 0.021 | 0.050 |

0.431 | 1726 | 0.125 | 0.012 | 0.086 | 0.235 | 942 | 0.068 | 0.002 | 0.025 |

0.431 | 1726 | 0.079 | 0.020 | 0.069 | 0.235 | 942 | 0.064 | 0.005 | 0.028 |

0.431 | 1726 | 0.063 | 0.020 | 0.047 | 0.235 | 942 | 0.052 | 0.008 | 0.027 |

0.431 | 1726 | 0.042 | 0.025 | 0.050 | 0.235 | 942 | 0.039 | 0.011 | 0.024 |

0.392 | 1570 | 0.106 | 0.014 | 0.078 | 0.235 | 942 | 0.032 | 0.011 | 0.019 |

0.392 | 1570 | 0.101 | 0.023 | 0.071 | 0.235 | 942 | 0.020 | 0.012 | 0.012 |

0.392 | 1570 | 0.076 | 0.021 | 0.061 | 0.235 | 942 | 0.009 | 0.020 | 0.019 |

0.392 | 1570 | 0.056 | 0.021 | 0.049 | 0.157 | 628 | 0.053 | 0.004 | 0.018 |

0.392 | 1570 | 0.034 | 0.022 | 0.032 | 0.157 | 628 | 0.041 | 0.006 | 0.016 |

0.392 | 1570 | 0.020 | 0.024 | 0.035 | 0.157 | 628 | 0.029 | 0.007 | 0.016 |

**Table 2.**Numerical results of the crater dimensions using different parameters (${U}_{J}$, L, b) and $\lambda =8.75$.

Mean Velocity | Impingement Distance | Jet Width | Jet-Bed Distance | Reynolds Number | Erosion Parameter | Crater Width | Crater Depth |
---|---|---|---|---|---|---|---|

${\mathit{U}}_{\mathit{J}}$ (m/s) | L (mm) | b (mm) | $\mathit{L}/\mathit{b}$ | Re | E | D (mm) | H (mm) |

0.18 | 40 | 4 | 10 | 720 | 2.24 | 12 | 3 |

0.18 | 50 | 4 | 12.5 | 720 | 1.29 | 15 | 5 |

0.28 | 50 | 4 | 12.5 | 1120 | 2.01 | 32 | 10 |

0.28 | 60 | 4 | 15 | 1120 | 1.56 | 35 | 6 |

0.30 | 40 | 4 | 10 | 1200 | 3.74 | 23 | 10 |

0.30 | 60 | 4 | 15 | 1200 | 1.67 | 40 | 10 |

0.37 | 46 | 4 | 11.5 | 1480 | 3.11 | 40 | 15 |

0.37 | 80 | 4 | 20 | 1480 | 1.54 | 50 | 10 |

0.28 | 70 | 6 | 11.66 | 1680 | 2.28 | 38 | 15 |

0.28 | 80 | 6 | 13.33 | 1680 | 1.82 | 42 | 13 |

0.18 | 80 | 8 | 10 | 1440 | 2.24 | 25 | 10 |

0.18 | 90 | 8 | 11.25 | 1440 | 1.58 | 30 | 10 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pham Van Bang, D.; Uh Zapata, M.; Gauthier, G.; Gondret, P.; Zhang, W.; Nguyen, K.D.
Two-Phase Flow Modeling for Bed Erosion by a Plane Jet Impingement. *Water* **2022**, *14*, 3290.
https://doi.org/10.3390/w14203290

**AMA Style**

Pham Van Bang D, Uh Zapata M, Gauthier G, Gondret P, Zhang W, Nguyen KD.
Two-Phase Flow Modeling for Bed Erosion by a Plane Jet Impingement. *Water*. 2022; 14(20):3290.
https://doi.org/10.3390/w14203290

**Chicago/Turabian Style**

Pham Van Bang, Damien, Miguel Uh Zapata, Georges Gauthier, Philippe Gondret, Wei Zhang, and Kim Dan Nguyen.
2022. "Two-Phase Flow Modeling for Bed Erosion by a Plane Jet Impingement" *Water* 14, no. 20: 3290.
https://doi.org/10.3390/w14203290