# Numerical Simulation of Non-Homogeneous Viscous Debris-Flows Based on the Smoothed Particle Hydrodynamics (SPH) Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fundamental Theories and Numerical Modeling

#### 2.1. The SPH Method

#### 2.1.1. SPH Interpolation

#### 2.1.2. Gradient and Divergence

_{a}and

_{b}represent the target particles and the particles in the influence domain, respectively, affecting the position of particle i. This choice of discretization operators ensures that an exact projection algorithm is produced. To date, there are various options to represent these operators, but only certain specific ones [34,35] have proven to be more convenient in terms of the accuracy and robustness of the method.

#### 2.2. Governing Equations

**g**is the gravitational acceleration, $P$ stands for pressure, and D/Dt refers to the material derivative. The density ρ has been intentionally kept in the equations to be able to enforce the incompressibility of the fluid. Using an appropriate constitutive equation to model the shear stress tensor $\tau $, Equations (6) and (7) can be used to solve both Newtonian and non-Newtonian flows.

#### 2.2.1. Equation of State

#### 2.2.2. Viscous Terms

## 3. Experimental Setup and Boundary Conditions

^{3}; ρ = 1500 kg/m

^{3}and ρ = 1600 kg/m

^{3}. Different layers patterns were selected according to the different configurations displayed in Figure 2. By adding water to the flume, when the water level reached the height of the mixing fluid, the front-end steel gate of the mixing area was released at a speed of 3 m/s. In order to maintain the driving force of the mixtures, the water level behind the mixtures was kept at h = 0.2 m during the experiment.

^{3}, (ii) ρ = 1500 kg/m

^{3}, and (iii) ρ = 1600 kg/m

^{3}. The slurry rheology coefficient was measured by the MCR301 advanced rotary rheometer manufactured by Anton Paar, Austria [39]. Values of viscosity μ for the fluids simulated are displayed in Table 1.

## 4. Simulation and Analysis

_{p}= 0.0025 m, solid particle density ρ

_{s}= 2200 kg/m

^{3}, thickness of solid phase h

_{s}= 0.1 m and thickness of liquid phase h

_{l}= 0.1 m for configuration a and c in Figure 2. Similar to the tests conducted on the experimental facility, three different viscosity coefficients for the liquid phases (as shown in Table 1) were selected in the numerical simulation. The inflow conditions were the same as those applied experimentally, and the water level as driving force of the debris flow was kept at 0.2 m for each entire simulation.

#### Characterization of Intermittent Debris Flow

_{k}and the height of the free surface $H$ (related to the potential energy of particles E

_{p}) were calculated for the tests conducted for configuration b. As shown in Figure 7a, the height of the free surface $H$ decreases as the debris flow develops. There is a noticeable correlation between the fluctuation of the free surface associated with the fluctuation of the moisture content. In the regions of L = 1.00–1.38 m and L = 1.82−2.04 m, the height of free surface decreases linearly, and in these two regions, the water content remains in the range 0.2–0.65. The points that obviously exceed this threshold are L = 0.90, L = 1.52, L = 1.6, L = 1.74, and L = 1.80−1.84, and the height of free surface is different from that of linear decline in these areas or vicinity. When the moisture content is within the range 0.20–0.65, the free surface of the debris flow is characterized by a linear change, but when the moisture content exceeds this range, it generates an impact on the free surface.

_{k}and the moisture content ϕ. To almost every peak of the kinetic energy E

_{k}(highlighted as green circles in Figure 7b) calculated corresponds a peak of the moisture content ϕ (highlighted as blue circles in Figure 7b), which indicates that kinetic energy E

_{k}and moisture content ϕ interact directly. However, this effect can only be assigned to small-scale portions of the particle kinetic energy fluctuations. Looking at Figure 7b, at the location of L = 1.74 m, the moisture content value corresponds to ϕ = 0.7619 and it is the maximum value measured in this region, and the corresponding kinetic energy E

_{k}records its minimum value. But because of the large kinetic energy of the particles recorded in this region, the corresponding kinetic energy E

_{k}= 5.4848 J is still higher than that recorded at the position of L = 1.64 m in the adjacent one E

_{k}= 0.30263 J.

_{k}and the height of the free surface H on a large scale, for configurations a and c. For both configurations, where the solid phase is located at the top and the bottom, the free surface is greatly affected by the magnitude of the moisture content ϕ, while the kinetic energy E

_{k}is greatly affected by the derivative of moisture content along the length of the slope $\frac{d\phi}{dL}$. However, the fluctuation of the moisture content ∆ϕ along the length L, especially for configuration a where the solid phase is displayed at the top of the debris flow, is relatively small. So the kinetic energy E

_{k}and potential energy E

_{p}curves show relatively large-scale area fluctuations and linear characteristics in comparison to the mixed distribution fluid conditions typical of configuration b.

_{p}= mgH) and the total energy (E

_{0}= E

_{k}+ E

_{p}) of fluids decreases at a similar rate. The difference between three fluids is mainly reflected on kinetic energies. When comparing the set of fluids with the smallest density (ρ = 1400 kg/m

^{3}, ${\mu}_{0}$ = 0.00004, ${\mu}_{1}$ = 0.0048, and ${\mu}_{2}$ = 0.0197), results shows that velocity values increase from t = 0.0 s up to t = 0.6 s, reaching almost the highest values, and then the kinetic energy of the three fluids tends to be equal. As the time progresses, the same order appears again in the kinetic energy magnitude arrangement, which is E

_{k,ρ}= 1400 kg/m

^{3}> E

_{k,ρ}= 1500 kg/m

^{3}> E

_{k,ρ}= 1600 kg/m

^{3}. This phenomenon is also due to the stronger fluctuation of the less dense fluids and these effects caused by different viscous fluids on debris flow array and collision, and friction forces on debris flow movement, will require further investigation in the future.

## 5. Summary and Conclusions

^{3}has the largest values. The results obtained can be summarized as follows:

- By comparing the shape and velocity of debris flows under different configurations, it was found that the vertical distribution of particles played a very important role in debris flow fluctuation, with a greater influence than on the viscous coefficient. The third configuration with mixed fine and coarse particles showed to fluctuate more violently, and this outcome confirmed one of the main assumptions for intermittent debris flows.
- By analyzing the characteristics of the fluid movement processes, it was found that when the two layers (fine and coarse particles) are mixed with the water, liquid particles tended to gather towards the bottom side of the debris flow causing a correspondent decrease of height. However, this effect could only be observed at small-scale areas. The potential energy was greatly affected by the magnitude of the moisture content, while the kinetic energy was significantly affected by the derivative of moisture content in the L direction.
- The differences of the energy conversion curves associated to different viscous coefficients were mainly noticed in kinetic energies. Fluids with smaller densities exhibited higher initiation velocities and higher fluctuations values.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Kim, M.I.; Kwak, J.H.; Kim, B.S. Assessment of dynamic impact force of debris flow in mountain torrent based on characteristics of debris flow. Environ. Earth Sci.
**2018**, 77, 538. [Google Scholar] [CrossRef] - Zhang, S.; Zhang, L.M. Impact of the 2008 Wenchuan earthquake in China on subsequent long-term debris flow activities in the epicentral area. Geomorphology
**2017**, 276, 86–103. [Google Scholar] [CrossRef] - Silhan, K.; Tichavsky, R. Recent increase in debris flow activity in the Tatras Mountains: Results of a regional dendogeomorphic reconstruction. CATENA
**2016**, 143, 221–231. [Google Scholar] [CrossRef] - Brandinoni, F.; Mao, L.; Recking, A.; Rickenmann, D.; Turowski, J.M. Morphodynamics of steep mountain channels. Earth Surf. Process. Landf.
**2015**, 40, 1560–1562. [Google Scholar] [CrossRef] - Takahashi, T.; Das, D.K. Debris Flow: Mechanics, Prediction and Countermeasures; CRC Press: London, UK, 2014. [Google Scholar]
- Shu, A.P.; Tian, L.; Wang, S.; Rubinato, M.; Zhu, F.Y.; Wang, M.Y.; Sun, J.T. Hydrodynamic characteristics of the formation processes for non-homogeneous debris-flow. Water
**2018**, 10, 452. [Google Scholar] [CrossRef] - Liu, J.J.; Li, Y. A review of study on drag reduction of viscous debris flow residual layer. J. Sedim. Res.
**2016**, 3, 72–80. (In Chinese) [Google Scholar] - Johnson, A.M. Physical Processes in Geology; Freeman Cooper & Company: San Francisco, CA, USA, 1970. [Google Scholar]
- Bagnold, R.A. Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc. R. Soc. Lond. Ser. A
**1954**, 22, 49–63. [Google Scholar] - Takahashi, T. Mechanical characteristics of debris flow. J. Hydraul. Div. Am. Soc. Civ. Eng.
**1978**, 104, 1153–1169. [Google Scholar] - Chen, C.L. Generalized viscoplastic modelling of debris flow. J. Hydraul. Div. Am. Soc. Civ. Eng.
**1988**, 114, 237–258. [Google Scholar] [CrossRef] - Chen, C.L. General solutions for viscoplastic debris flow. J. Hydraul. Div. Am. Soc. Civ. Eng.
**1988**, 114, 259–282. [Google Scholar] [CrossRef] - O’Brien, J.S.; Julien, P.Y. Laboratory analysis of mudflow properties. J. Hydraul. Div. Am. Soc. Civ. Eng.
**1988**, 114, 877–887. [Google Scholar] [CrossRef] - O’Brien, J.S.; Julien, P.Y.; Fullerton, W.T. Two-dimensional water flood and mudflow simulation. J. Hydraul. Div. Am. Soc. Civ. Eng.
**1993**, 119, 244–261. [Google Scholar] [CrossRef] - Dai, Z.; Huang, Y.; Cheng, H.; Xu, Q. 3D Numerical modeling using smoothed particle hydrodynamics of flow-like landslide propagation triggered by the 2008 Wenchuan earthquake. Eng. Geol.
**2014**, 180, 21–33. [Google Scholar] [CrossRef] - Hosseini, S.M.; Manzari, M.T.; Hannani, S.K. A fully explicit three-step SPH algorithm for simulation of non-Newtonian fluid flow. Int. J. Numer. Methods Heat Fluid Flow
**2007**, 17, 715–735. [Google Scholar] [CrossRef] - Rodriguez-Paz, M.X.; Bonet, J. A corrected smooth particle hydrodynamics method for the simulation of debris flows. Numer. Methods Part Differ. Equ.
**2004**, 20, 140–163. [Google Scholar] [CrossRef] - Savage, S.B.; Hutter, K. The dynamics of avalanches of granular materials from initiation to run out. Acta Mech. Sin.
**1991**, 86, 201–223. [Google Scholar] [CrossRef] - Trunk, F.J.; Dent, J.D.; Lang, T.E. Computer modeling of large rock slides. J. Geotech. Eng.
**1986**, 112, 348–360. [Google Scholar] [CrossRef] - Iverson, R.M. The physics of debris flows. Rev. Geophys.
**1997**, 35, 245–296. [Google Scholar] [CrossRef] [Green Version] - Xenakis, A.M.; Lind, S.J.; Stansby, P.K.; Rogersb, B.D. An incompressible SPH scheme with improved pressure predictions for free-surface generalised Newtonian flows. J. Non-Newton. Fluid Mech.
**2015**, 218, 1–15. [Google Scholar] [CrossRef] - Springel, V.; Hernquist, L. Cosmological smoothed particle hydrodynamics simulations: A hybrid multiphase model for star formation. Mon. Not. R. Astron. Soc.
**2003**, 339, 289–311. [Google Scholar] [CrossRef] - Gómez-Gesteira, M.; Dalrymple, R.A. Using a three-dimensional Smoothed Particle Hydrodynamics method for wave impact on a tall structure. J. Waterw Port Coast.
**2004**, 130, 63–69. [Google Scholar] [CrossRef] - Flebbe, O.; Muenzel, S.; Herold, H.; Riffert, H. Smoothed Particle Hydrodynamics: Physical viscosity and the simulation of accretion disks. Astrophys. J.
**1994**, 431, 754–760. [Google Scholar] [CrossRef] - Lucy, L.B. A numerical approach to the testing of the fission hypothesis. Astron. J.
**1977**, 82, 1013–1024. [Google Scholar] [CrossRef] - Liu, G.R.; Liu, M.B. Smoothed Particle Hydrodynamics: A Mesh-Free Particle Method; World Scientific Publishing Company: Singapore, 2003; pp. 27–33. [Google Scholar]
- Monaghan, J.J. Smoothed particle hydrodynamics. Annu. Rev. Astron. Astrophys.
**1992**, 30, 543–574. [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] - Gingold, R.A.; Monaghan, J.J. Kernel estimates as a basis for general particle methods in hydrodynamics. J. Comput. Phys.
**1982**, 46, 429–453. [Google Scholar] [CrossRef] - Monaghan, J.J. Particle methods for hydrodynamics. Comput. Phys. Rep.
**1985**, 3, 71–124. [Google Scholar] [CrossRef] - Fulk, D.A.; Quinn, D.W. An Analysis of 1-D Smoothed Particle Hydrodynamics kernels. J. Comput. Phys.
**1996**, 126, 165–180. [Google Scholar] [CrossRef] - Colagrossi, A.; Landrini, M. Numerical simulation of interfacial flows by Smoothed Particle Hydrodynamics. J. Comput. Phys.
**2003**, 191, 448–475. [Google Scholar] [CrossRef] - Bose, A.; Carey, G.F. Least-squares p-r finite element methods for incompressible non-Newtonian flows. Comput. Methods Appl. Mech.
**1999**, 180, 431–458. [Google Scholar] [CrossRef] - Wood, D. Collapse and fragmentation of isothermal gas clouds. Mon. Not. R. Astron. Soc.
**1981**, 194, 201–218. [Google Scholar] [CrossRef] [Green Version] - Bonet, J.; Lok, T.S.L. Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations. Comput. Method Appl. Mech.
**1999**, 180, 97–115. [Google Scholar] [CrossRef] - Loewenstein, M.; Mathews, W.G. Adiabatic particle hydrodynamics in three dimensions. J. Comput. Phys.
**1986**, 62, 414–428. [Google Scholar] [CrossRef] - Shao, S.; Lo, E.Y.M. Incompressible SPH method for simulating Newtonian and non-Newtonian flows with a free surface. Adv. Water Resour.
**2003**, 26, 787–800. [Google Scholar] [CrossRef] - Lo, E.Y.M.; Shao, S.D. Simulation of near-shore solitary wave mechanics by an incompressible SPH method. Appl. Ocean Res.
**2002**, 24, 275–286. [Google Scholar] - Yang, H.J.; Wei, F.Q.; Hu, K.H. Determination of the maximum packing fraction for calculating slurry viscosity of debris flow. J. Sediment. Res.
**2018**, 3, 382–390. [Google Scholar]

**Figure 1.**Experimental equipment for debris flow simulation [6].

**Figure 2.**Schematic representation (experimental and numerical) of configurations applied for this study [6].

**Figure 4.**Initial state of debris flow numerical simulation (mixed configuration). The water level behind the mixtures was kept at h = 0.2 m.

**Figure 5.**Debris flow free surface for tests 1, 2, 3, 4, 5, 6, 7, 8, and 9 (Table 1) and experimental results used for validation (dots), respectively.

**Figure 6.**Analysis of the debris flow behaviors at different locations. Solid particles and liquid particles are represented by red and blue dots, respectively.

**Figure 7.**Relationships between the moisture content, the kinetic energy, and the height of the free surface for configuration b.

**Figure 8.**Relationships between the moisture content ϕ, the kinetic energy E

_{k}, and the height of the free surface H for configuration a and c.

**Table 1.**Experimental conditions (viscosity values and vertical grading patterns) adopted for this study.

Test No. | Factors | ||||
---|---|---|---|---|---|

ρ (kg/m^{3}) | Solid Phase Level | μ_{0} (Pa) | μ_{1} (Pa·s) | μ_{2} (Pa·s^{2}) | |

1 | 1400 | Upper | 0.00004 | 0.0048 | 0.0197 |

2 | 1400 | Mixed | |||

3 | 1400 | Bottom | |||

4 | 1500 | Upper | 0.00006 | 0.0051 | 0.1654 |

5 | 1500 | Mixed | |||

6 | 1500 | Bottom | |||

7 | 1600 | Upper | 0.0001 | 0.0034 | 0.8242 |

8 | 1600 | Mixed | |||

9 | 1600 | Bottom |

© 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**

Wang, S.; Shu, A.; Rubinato, M.; Wang, M.; Qin, J.
Numerical Simulation of Non-Homogeneous Viscous Debris-Flows Based on the Smoothed Particle Hydrodynamics (SPH) Method. *Water* **2019**, *11*, 2314.
https://doi.org/10.3390/w11112314

**AMA Style**

Wang S, Shu A, Rubinato M, Wang M, Qin J.
Numerical Simulation of Non-Homogeneous Viscous Debris-Flows Based on the Smoothed Particle Hydrodynamics (SPH) Method. *Water*. 2019; 11(11):2314.
https://doi.org/10.3390/w11112314

**Chicago/Turabian Style**

Wang, Shu, Anping Shu, Matteo Rubinato, Mengyao Wang, and Jiping Qin.
2019. "Numerical Simulation of Non-Homogeneous Viscous Debris-Flows Based on the Smoothed Particle Hydrodynamics (SPH) Method" *Water* 11, no. 11: 2314.
https://doi.org/10.3390/w11112314