# Application of Smooth Particle Hydrodynamics to Particular Flow Cases Solved by Saint-Venant Equations

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background of the SPH-SWEs Formulations

**v**is the horizontal depth-averaged velocity vector, d is the water depth, b is the bottom elevation, $\mathit{g}$ is the acceleration due to gravity, and ${\mathit{S}}_{\mathit{f}}$ is the bed friction source term.

#### 2.1. Density Formulation

#### 2.2. Momentum Formulation

#### 2.3. Time Stepping

#### 2.4. Boundary Conditions

## 3. Materials and Methods

#### 3.1. Non-Uniform Steady State Profiles

#### 3.2. Wave Propagation

#### 3.2.1. Wave Attenuation

^{3}/s is applied, and the upstream boundary condition represents an inflow flood hydrograph with a flood peak of 500 m

^{3}/s over 2000 s as shown in Figure 1. A constant outflow discharge of 80 m

^{3}/s was set as the downstream boundary condition.

#### 3.2.2. Wave Translation

^{3}/s, and the inflow hydrograph is applied upstream with 800 m

^{3}/s peak discharge over a time base of 1500 s (Figure 1). A constant outflow discharge of 150 m

^{3}/s was set as the downstream boundary condition.

#### 3.3. Flooding

#### 3.3.1. Momentum Conservation over a Hump

^{3}/s and a time base of 30 s (Figure 2). The flow travels downhill with a steep slope of 1:200. The total volume of the inflow hydrograph is just sufficient to fill in the left depression, and some of the volume is expected to overtop the hump as a result of the flow inertia. The total simulation time is 15 min to allow the water to settle. The channel friction is represented by a uniform Manning coefficient value of 0.01.

#### 3.3.2. Filling of Floodplain Depressions

^{2}floodplain with 16 flattened egg-shape depressions of 0.5 m deep. The general slope of 1:1500 is applied from the north to south direction and the one of 1:3000 from the west to east direction, with a 2 m drop in elevation between the top left corner to the bottom right corner. The Digital elevation model (DEM) of this area is shown in Figure 3a. The Manning coefficient for bed friction is 0.03.

^{3}/s and a time base of 85 min, as shown in Figure 3b. Except for the inflow boundary, all other boundaries are closed boundaries. The total simulation time is 48 h, set to reach an inundation state over the whole domain. The water level in the middle center point of each depression is observed together with the final inundation extent.

^{3}for the time base of 85 min.

## 4. Results and Discussion

#### 4.1. Water Profile for Non-Uniform Flow

#### 4.2. Wave Propagation

#### 4.3. Flooding

#### 4.3.1. Momentum Conservation over a Hump

#### 4.3.2. Filling of Floodplain Depressions

## 5. Conclusions

^{−3}. These subcritical and supercritical flows were handled easily with no restrictions regarding bed complexity and steepness, which shows that the method can be used to solve such hydraulic problems. The momentum conservation property was verified, and the results were compared to other grid-based software results with an agreement of up to 0.5%. The overflow cases associated with low-momentum flow required high particle resolution to achieve better results.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Lucy, L.B. A numerical approach to the testing of the fission hypothesis. Astron. J.
**1977**, 1013–1024. [Google Scholar] [CrossRef] - Gingold, R.A.; Monaghan, J.J. Smoothed Particle Hydrodynamics: Theory and application to non-spherical stars. Mon. Not. R. Astron. Soc.
**1977**, 181, 375–389. [Google Scholar] [CrossRef] - Monaghan, J.J. An introduction to SPH. Comput. Phys. Commun.
**1988**, 48, 89–96. [Google Scholar] [CrossRef] - Liu, M.B.; Liu, G.R. Smoothed Particle Hydrodynamics (SPH): An Overview and Recent Developments. Arch. Comput. Methods Eng.
**2010**, 17, 25–76. [Google Scholar] [CrossRef][Green Version] - Monaghan, J.J. Smooth Particle Hydrodynamics. Rep. Prog. Phys.
**2005**, 68, 1703–1754. [Google Scholar] [CrossRef] - Mirauda, D.; Albano, R.; Sole, A.; Adamowski, J. Smoothed Particle Hydrodynamics Modeling with Advanced Boundary Conditions for Two-Dimensional Dam-Break Floods. Water
**2020**, 12, 1142. [Google Scholar] [CrossRef][Green Version] - Novak, G.; Tafuni, A.; Domínguez, J.M.; Cetina, M.; Žagar, D. A Numerical Study of Fluid Flow in a Vertical Slot Fishway with the Smoothed Particle Hydrodynamics Method. Water
**2019**, 11, 1928. [Google Scholar] [CrossRef][Green Version] - Kazemi, E.; Tait, S.; Shao, S.; Nichols, A. Potential Application of Mesh-Free SPH Method in Turbulent River Flows. In Hydrodynamic and Mass Transport at Freshwater Aquatic Interfaces; Rowiński, P., Marion, A., Eds.; GeoPlanet: Earth and Planetary Sciences; Springer: Cham, Switzerland, 2016. [Google Scholar] [CrossRef][Green Version]
- Tran-Duc, T.; Meylan, M.H.; Thamwattana, N.; Lamichhane, B.P. Wave Interaction and Overwash with a Flexible Plate by Smoothed Particle Hydrodynamics. Water
**2020**, 12, 3354. [Google Scholar] [CrossRef] - Gu, S.; Zheng, X.; Ren, L.; Xie, H.; Huang, Y.; Wei, J.; Shao, S. SWE-SPHysics Simulation of Dam Break Flows at South-Gate Gorges Reservoir. Water
**2017**, 9, 387. [Google Scholar] [CrossRef][Green Version] - Ata, R.; Soulaïmani, A. A stabilized SPH method for inviscid shallow water flows. Int. J. Numer. Methods Fluids
**2005**, 47, 139–159. [Google Scholar] [CrossRef] - Meister, M.; Rauch, W. Modelling aerated flows with smoothed particle hydrodynamics. J. Hydroinform.
**2015**, 17, 493–504. [Google Scholar] [CrossRef][Green Version] - Gan, B.S.; Nguyen, D.K.; Han, A.; Alisjahbana, S.W. Proposal for fast calculation of particle interactions in SPH simulations. J. Comput. Fluids
**2016**, 104, 20–29. [Google Scholar] [CrossRef] - Vacondio, R.; Rogers, B.; Stansby, P.; Mignosa, P. SPH Modeling of Shallow Flow with Open Boundaries for Practical Flood Simulation. J. Hydraul. Eng.
**2012**, 138, 530–541. [Google Scholar] [CrossRef] - Vacondio, R.; Rogers, B.D.; Stansby, P.K.; Mignosa, P. A correction for balancing discontinuous bed slopes in two-dimensional smoothed particle hydrodynamics shallow water modeling. Int. J. Numer. Methods Fluids
**2013**, 71, 850–872. [Google Scholar] [CrossRef] - Vacondio, R.; Rogers, B.D.; Stansby, P.K. Accurate particle splitting for smoothed particle hydrodynamics in shallow water with shock capturing. Int. J. Numer. Methods Fluids
**2012**, 69, 1337–1410. [Google Scholar] [CrossRef] - Rodriguez-Paz, M.; Bonet, J. A corrected smooth particle hydrodynamics formulation of the shallow-water equations. Comput. Struct.
**2005**, 83, 1396–1410. [Google Scholar] [CrossRef] - Ferrand, M.; Violeau, D.; Mayhoffer, A.; Mahmood, O. Correct boundary conditions for turbulent SPH. In Advances in Hydroinformatics; Gourbesville, P., Cunge, J., Caignaert, G., Eds.; Springer: Singapore, 2015; pp. 245–258. [Google Scholar]
- Sarfaraz, M.; Pak, A. SPH Numerical Simulation of Tsunami Wave Forces Impinged on Bridge Superstructures. Coast. Eng.
**2017**, 121, 145–157. [Google Scholar] [CrossRef] - SWE-SPHysics SWE-SPHysics Code v1.0. Available online: http://www.sphysics.org (accessed on 20 March 2021).
- Monaghan, J.J.; Kajtar, J.B. SPH particle boundary forces for arbitrary boundaries. Comput. Phys. Commun.
**2009**, 180, 1811–1820. [Google Scholar] [CrossRef] - Ferrari, A.; Dumbser, M.; Toro, E.F.; Armanini, A. A new 3D parallel SPH scheme for free surface flows. Comput. Fluids
**2009**, 38, 1203–1217. [Google Scholar] [CrossRef] - Napoli, E.; De Marchis, M.; Vitanza, E. Panormus-SPH. 2015, A new Smoothed Particle Hydrodynamics solver for incompressible flows. Comput. Fluids
**2015**, 106, 185–195. [Google Scholar] [CrossRef] - Néelz, S.; Pender, G. Benchmarking the Latest Generation of 2D Hydraulic Modelling Packages. 2013. Available online: https://publications.environment-agency.gov.uk/ms/BXoKPi (accessed on 1 March 2021).
- Hartanto, I.M.; Beevers, L.; Popescu, I.; Wright, N.G. Application of a coastal modelling code in fluvial environments. Environ. Model. Softw.
**2011**, 26, 1685–1695. [Google Scholar] [CrossRef][Green Version] - Beevers, L.; Popescu, I.; Pan, Q.; Pender, D. Applicability of a coastal morphodynamic model for fluvial environments. Environ. Model. Softw.
**2016**, 80, 83–99. [Google Scholar] [CrossRef]

**Figure 1.**Inflow hydrograph at the upstream boundary: for wave attenuation and wave translation cases.

**Figure 2.**Momentum conservation over a hump case: longitudinal profile and inflow hydrograph at the upstream boundary. Total inflow volume of the inflow hydrograph in Figure 2 is 1310 m

^{3}.

**Figure 3.**Filling of floodplain depressions case: (

**a**) DEM map showing the location of the upstream boundary condition (red line upper left corner) and ground elevation contour lines every 0.05 m; (

**b**) inflow hydrograph on the left upper corner of the domain.

**Figure 4.**Results of 1D simulation for M1 curve longitudinal profile: (

**a**) velocity and (

**b**) water level.

**Figure 5.**Results of 2D simulation for M1 curve longitudinal profile: (

**a**) velocity and (

**b**) water level.

**Figure 6.**Results of 2D simulation for M1 curve cross-sectional view: (

**a**) velocity and (

**b**) water level.

**Figure 7.**Results of 1D simulation for M2 curve longitudinal profile view: (

**a**) velocity and (

**b**) water level.

**Figure 8.**Results of 2D simulation for M2 curve longitudinal profile: (

**a**) velocity and (

**b**) water level.

**Figure 9.**Results of 2D simulation for M2 curve cross-sectional view: (

**a**) velocity and (

**b**) water level.

**Figure 16.**Filling of floodplain depressions: water level results for Depression 4 (closest to inflow boundary).

Model Set-Up | M1 Flow Curve | M2 Flow Curve | ||
---|---|---|---|---|

1D Schematization | 2D Schematization | 1D Schematization | 2D Schematization | |

Channel length (m) | 10,000 | 10,000 | 16,650 | 10,000 |

Channel width (m) | 1 | 400 | 1 | 400 |

Bed slope (-) | 0.001 | 0.001 | 0.001 | 0.001 |

Manning coefficient (m^{−1/3}s) | 0.01 | 0.02 | 0.02 | 0.02 |

Specific discharge (m^{3}/s/m) | 2.00 | 4.99 | 2.50 | 4.99 |

Normal depth (m) | 1.26 | 2.00 | 2.80 | 2.00 |

Critical depth (m) | 0.74 | 1.36 | 0.86 | 1.36 |

Upstream velocity (m/s) | 1.59 | 2.49 | 0.89 | 2.49 |

Downstream velocity (m/s) | 0.40 | 1.00 | 2.50 | 3.56 |

Upstream water level (m) | 1.25 | 2.00 | 2.80 | 2.00 |

Downstream water level (m) | 5.00 | 5.00 | 1.00 | 1.40 |

Particles spacing (m) | 100 | (a) 10 | 100 | (a) 10 |

(b) 100 | (b) 100 | |||

Initial number of particles | 100 | 400 | 100 | 400 |

40,000 | 40,000 |

Model Set-Up | Diffusive Wave | Kinematic Wave |
---|---|---|

(Attenuation) | (Translation) | |

Channel length (m) | 15,000 | 15,000 |

Channel width (m) | 60 | 60 |

Bed slope (-) | 0.001 | 0.01 |

Manning coefficient (m^{−1/3}s) | 0.03 | 0.02 |

Specific discharge (m^{3}/s/m) | 2.00 | 2.50 |

Normal depth (m) | 1.03 | 0.67 |

Critical depth (m) | 0.49 | 0.86 |

Upstream BC ^{1} velocity (m/s) | 2.30 | 7.22 |

Downstream BC water level (m) | 3.62 | 1.85 |

Particles spacing (m) | 10 | 10 |

Minimum depth for friction (m) | 0.05 | 0.05 |

Initial number of particles | 9000 | 9000 |

^{1}BC—Boundary conditions, for peak inflow conditions.

Model Set-Up | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 |
---|---|---|---|---|---|

BC ^{1} maximum velocity (m/s) | 3.275 | 3.275 | 3.275 | 3.275 | 2.0 |

BC maximum water depth (m) | 0.2 | 0.2 | 0.2 | 0.2 | 0.3275 |

Particles spacing (m) | 2.0 | 2.0 | 0.5 | 5 | 2 |

Split particles | No | No | No | Yes | No |

Min. depth for friction (m) | 0.05 | 0.005 | 0.05 | 0.05 | 0.05 |

^{1}BC—Boundary conditions.

Model Set-Up | Case 6 | Case 7 | Case 8 | Case 9 |
---|---|---|---|---|

BC ^{1} maximum velocity (m/s) | 0.80 | 2.67 | 0.50 | 0.80 |

B.C. maximum water depth (m) | 0.25 | 0.75 | 0.40 | 0.25 |

Particles spacing (m) | 20 | 20 | 20 | 10 |

Split particles | No | No | No | No |

Min. depth for friction (m) | 0.05 | 0.05 | 0.05 | 0.05 |

^{1}BC—Boundary conditions.

Non-Dimensional Error ^{1} | 1D SPH Model (Δx = 100 m) | 2D SPH Model (100 m × 100 m) | 2D SPH Model (10 m × 10 m) | |||
---|---|---|---|---|---|---|

M1 | M2 | M1 | M2 | M1 | M2 | |

Error d (water depth) | 1.42 × 10^{−2} | 5.80 × 10^{−2} | 7.91 × 10^{−2} | 6.1 × 10^{−2} | 4.86 ×10^{−2} | 2.43 × 10^{−2} |

Error v_{x} | 4.36 × 10^{−3} | 2.45 × 10^{−2} | 4.21 × 10^{−2} | 5.21 × 10^{−2} | 9.99 × 10^{−2} | 1.99 × 10^{−2} |

Error v_{y} | - | - | 1.19 × 10^{−2} | 4.15 × 10^{−3} | 5.01 × 10^{−2} | 3.16 × 10^{−2} |

^{1}See formulas in Equations (10)–(12).

Case | Loss in Inflow Water Volume (%) | Total No. of Particles at the End of Simulation | Computer Running Time (s) |
---|---|---|---|

Case1 | 2.6 | 1986 | 262 |

Case 2 | 2.6 | 1983 | 244 |

Case 3 | 0.5 | 31,735 | 67,956 |

Case 4 | 6.0 | 2112 | 2246 |

Case 5 | 6.5 | 1181 | 228 |

Case 6 | 1.2 | 1000 | 20,570 |

Case 7 | 13.7 | 2864 | 30,015 |

Case 8 | 0.3 | 314 | 23,287 |

Case 9 | 0.16 | 4350 | 54,728 |

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**MDPI and ACS Style**

Fadl-Elmola, S.A.M.; Ciocan, C.M.; Popescu, I. Application of Smooth Particle Hydrodynamics to Particular Flow Cases Solved by Saint-Venant Equations. *Water* **2021**, *13*, 1671.
https://doi.org/10.3390/w13121671

**AMA Style**

Fadl-Elmola SAM, Ciocan CM, Popescu I. Application of Smooth Particle Hydrodynamics to Particular Flow Cases Solved by Saint-Venant Equations. *Water*. 2021; 13(12):1671.
https://doi.org/10.3390/w13121671

**Chicago/Turabian Style**

Fadl-Elmola, Salman A. M., Cristian Moisescu Ciocan, and Ioana Popescu. 2021. "Application of Smooth Particle Hydrodynamics to Particular Flow Cases Solved by Saint-Venant Equations" *Water* 13, no. 12: 1671.
https://doi.org/10.3390/w13121671