# Numerical Simulation of Boulder Fluid–Solid Coupling in Debris Flow: A Case Study in Zhouqu County, Gansu Province, China

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## Abstract

**:**

## 1. Introduction

## 2. Background

^{2}. The gully is generally distributed from south to north, with high terrain from north to south and low terrain from south to north, which is typical alpine canyon topography. The highest point of the basin is 3828 m above sea level and the lowest point is only 1340 m above sea level at the estuary, with a relative elevation difference of 2488 m. According to statistics, there have been many large-scale debris flows in this area and the “8.7” Zhouqu debris flow is the most serious debris flow disaster in the history of this area [46,47].

## 3. Numerical Model of the Zhouqu Debris Flow

_{T}is the turbulent kinetic energy, V

_{F}is the fractional volume open to flow, and A

_{x}, A

_{y}, and A

_{z}are the fractional area open to flow in the x-, y-, and z-directions, respectively. P

_{T}is the turbulent kinetic energy production term, G

_{T}is the buoyancy production term, Diff is the diffusion term, and ε

_{T}is the turbulence dissipation term. CDIS1 = 1.42, CDIS2, and CDIS3 = 0.2 are dimensionless parameters. CDIS2 is computed from the turbulent kinetic energy (k

_{T}) and turbulent production (P

_{T}) terms [48].

^{3}according to the field survey data. Owing to the large mass of the boulders, the restitution coefficients (e) and the friction coefficient (μ) used in the simulation are 0.10 and 0.35, respectively. The restitution coefficients (e) are the square root of the ratio of h

_{2}to h

_{1}(h

_{1}is the height before the rebound, h

_{2}is the height after the rebound), which is between 0 and 1. Due to the large mass of the boulders and the soft ground with water on its surface, we believe that the boulders basically have no rebound; therefore, this value is empirically obtained as 0.10. The friction coefficient (μ) is the ratio of the tangential force to the normal stress on the contact surface. Because there is no relevant research on this parameter, we used the friction coefficient of hard clay against the base of the retaining wall to approximate this coefficient. This value can be found in Table 6.6.5-2 of P44 of the National standard of the People’s Republic of China, Code for Design of Building Foundation [49]. The specific information of these boulders is shown in Figure 3.

_{y}is the yield stress, η is the viscosity coefficient, and du/dy is the flow velocity gradient. According to the quantitative relationship between the yield stress and density for Jiangjia gully debris flows (Equation (4)) [52], as well as that between the stiffness coefficient and the yield stress for a debris flow density larger than 1300 kg/m

^{3}(Equation (5)) [53], the following equations were used:

_{y}is the yield stress, ρ is the debris flow density, and η is the viscosity coefficient. According to Equations (4) and (5), the simulation parameters of debris flow densities of 1200 kg/m

^{3}, 1500 kg/m

^{3}, 1800 kg/m

^{3}, and 2100 kg/m

^{3}can be calculated, as shown in Table 1.

## 4. Results of the Simulation

#### 4.1. Movement Process of the Boulders

^{3}. In the simulation, the boulders are mainly arranged in the Dayanyu gully, the movement progress of which can be divided into two stages according to the obvious change in the terrain slope. In the first stage, the boulder moves in the circulation area where the Dayanyu gully and Xiaoyanyu gully intersect. In this stage, the terrain slope is relatively large, and the average gradient of the gully reaches 12%. In the second stage, the movement is in the accumulation area from the Sanyanyu gully to the Bailongjiang River and the terrain slope is relatively gentle with an average gradient of 9%. The obvious change in terrain slope in these two stages will have a great impact on the movement of boulders. In the simulation (Figure 6a), the debris flows in the Dayanyu gully and Xiaoyanyu gully began to converge after 30 s. When the simulation time was 90 s, the debris flow moved in the Sanyanyu gully with a gentle slope. When the simulation time was 180 s, the debris flow moved into the Bailongjiang River. According to the simulation, the surface velocity of the debris flow reached its maximum of approximately 30 m/s downstream of the intersection. After that, the flow velocity gradually decreased to 18 m/s. The boulders moved along the gully under the action of debris flow. At 30 s, the direction of the boulder movement began to turn due to the influence of terrain and moved downstream along the Sanyanyu gully and some boulders rolled into the Bailong River. By observing boulder movement under other simulation conditions, it can be seen that boulder movement is not only affected by the size, shape, and mass of the boulder, but also by topographic factors, the coupling effect of debris flow, and the collision of boulders in the movement process. Therefore, it is necessary to analyze the laws of the movement and accumulation of boulders from the changes in their motion parameters. Figure 6b show the snapshots of cube and spherical boulders flowing into the river for a debris flow of 2100 kg/m

^{3}.

#### 4.2. Velocity Variation Characteristics of Boulders

#### 4.2.1. Influence of Density

^{3}, 1500 kg/m

^{3}, 1800 kg/m

^{3}, and 2100 kg/m

^{3}. It can be seen from Figure 7 that the velocity of the four shapes of boulders increases first and then decreases gradually, which is consistent with the change in terrain slope. At T = 75 s, the slope of the terrain gradually decreased, and the average velocity of the boulders gradually slowed to approximately 13 m/s (Figure 7a). In this stage, the motion velocity of the spherical boulder is essentially positively correlated with the increase in the debris flow density and the variation amplitude of the velocity is relatively small, whereas the motion velocity of other shaped boulders has little correlation with the debris flow density. Similar conclusions can be preliminarily seen from our previous research, that is, the motion velocity of spherical boulders is positively correlated with the density of debris flow [55]. In this study, the velocity of the spherical boulders is not only affected by debris flow density, but also by the irregular slope of natural terrain. Therefore, this correlation was not always valid in this study and we cannot give a specific correlation formula between them.

#### 4.2.2. Influence of Shape

^{3}and 2100 kg/m

^{3}, the velocity of cuboid and rhombohedron boulders reaches zero and stops first compared with the spherical and cube boulders (Figure 8a,b). This may be related to the inconsistency of the edge length of cuboid and rhombohedron boulders, which makes it difficult for them to move along a straight line. The spherical and cube boulders have a symmetrical appearance and can move longer distances under the same conditions.

#### 4.3. Angular Velocity Variation Characteristics of Boulders

#### 4.3.1. Influence of Density

#### 4.3.2. Influence of Shape

^{3}, 1500 kg/m

^{3}, 1800 kg/m

^{3}, and 2100 kg/m

^{3}. It can be seen from the figures that the angular velocity of spherical and cube boulders is greater than that of boulders of other shapes most of the time under four kinds of flow densities, with the angular velocity of the cuboid boulder being the smallest. This is because the shape of the boulder with spherical symmetry and the smoothness of its surface has an important influence on its angular velocity. The spherical boulder not only has a spherically symmetrical shape but also has a smooth surface; therefore, the angular velocity is the largest. Although the axis of symmetry of the cube boulder is the same in length and approximately spherically symmetric, the surface is not smooth enough. The length of the symmetry axis of rhombohedron boulder is different but the surface is smooth. However, the length of the symmetry axis of a cuboid boulder is not the same and the surface is not smooth. Therefore, most of the time, the angular velocity of the cuboid boulder is the least.

#### 4.4. Kinetic Energy Variation Characteristics of Boulders

_{c}is the rotational inertia of the rigid body to the centroid axis, and ω is the angular velocity of the rigid boulders.

#### 4.4.1. Influence of Density

#### 4.4.2. Influence of Shape

^{3}, 1500 kg/m

^{3}, 1800 kg/m

^{3}, and 2100 kg/m

^{3}. It can be seen from the figures that the kinetic energy of the sphere is far greater than that of other shapes of boulders, which is related to its larger mass. In addition, the kinetic energy also decreases after the slope becomes gentle, which indicates that the terrain slope has a great influence on the velocity and kinetic energy of boulders. Therefore, corresponding engineering measures can be adopted to reduce the energy of boulders in the debris flow, such as the use of check dams to block the boulders in the channel or the installation of concrete retaining walls and deceleration baffles in front of buildings to absorb the energy [56].

## 5. Discussion

#### 5.1. Boulder Movement Range

#### 5.2. Boulder Movement Distance

#### 5.2.1. Influence of Density

_{n}is the distance between the nth point and the n-1th point, X

_{n}, Y

_{n}, and Z

_{n}are the spatial coordinates of the nth point, and S

_{n}is the total distance from the nth point to the initial position.

#### 5.2.2. Influence of Shape

^{3}, 1500 kg/m

^{3}, 1800 kg/m

^{3}, and 2100 kg/m

^{3}. From Figure 15, it can be seen that the spherical and cube boulders have a longer movement distance, whereas the cuboid and rhombohedron boulders have lesser movement distance and some of them will stay in the accumulation area. This is because the closer the boulder shape is to the spherical symmetry, the smoother the surface is, and the greater the cumulative movement distance is. In the movement process, a boulder with a non-spherical shape may not be able to move in a straight line due to its appearance. Instead, it may move to one side of the gully, out of the range of the debris flow, and may stop due to lack of dynamics. On the other hand, the gravity center of the cuboid and rhombohedron boulders will fluctuate up and down during the rolling movement, while the height of the gravity center of the spherical boulder will remain unchanged during the movement. Therefore, a part of the kinetic obtained from the debris flow of the cuboid and rhombohedron boulders used to overcome the acting of its gravity, thus reducing the kinetic energy obtained in the horizontal direction. This is also an important reason for their small moving distance [31,57]. These simulation results are essentially consistent with the field research of Hu et al. [19] that most of the boulders in the accumulation area are cuboid and rhombohedron boulders.

#### 5.3. Dynamic Impact Force of the Boulders

_{2}− mv

_{1}, the collision force between a boulder and a structure is inversely proportional to its collision time. We assume that the velocity of the boulder is zero after the collision, and compare the impact force of the rhombohedron boulder when the three collision durations are 0.1 s, 0.3 s, and 0.6 s. The simulation results show that the velocity of rhombohedron boulder is 14.64 m/s when t = 150 s under the action of debris flow with a density of 1800 kg/m

^{3}. It can be calculated that the impact force of the rhombohedron boulder on a structure is 4.94 × 10

^{7}N, 1.65 × 10

^{7}N, and 0.82 × 10

^{7}N when the interaction time is 0.1 s, 0.3 s, and 0.6 s, respectively. The impact force of the rhombohedron boulder can reach six times when the action time is reduced from 0.6 s to 0.1 s. Therefore, reducing the rigid collision between boulders and buildings can effectively reduce the impact force on infrastructures from the perspective of engineering protection. In engineering practice, decelerating obstacles such as waste tires, mounds, or concrete piers can be arranged in front of the protected objects to absorb the impact of boulders.

#### 5.4. Limitation of the Study

## 6. Conclusions

- (1)
- The motion characteristics of boulders in the 8.7 Zhouqu debris flow were simulated using FLOW-3D software with the GMO and RNG coupled models. The simulation results can be used to obtain various motion parameters of the boulders that interact with the debris flow slurry, such as the centroid velocity, angular velocity, kinetic energy, and motion trajectory. This method has a certain reference for studying the movement mechanism of debris flow.
- (2)
- The simulation results show that the movement velocity of boulders is affected by many factors such as terrain conditions, debris flow densities, and shapes of the boulders. When the terrain slope is relatively large, the boulders have greater potential energy, and the influence of terrain on the movement velocity of boulders is greater than that of debris flow density and boulder shape. When the slope is less, the appearance of boulders has a great influence on its movement. The velocity of a spherical boulder increases with debris flow density. However, the movement of cuboid, rhombohedron, and cube boulders presents a complex motion state, and it is difficult to predict the change law of their velocity.
- (3)
- The simulation results show that the angular velocity of the spherical boulder fluctuates slightly with time during the whole movement progress, whereas the angular velocity of the cuboid, rhombohedron, and cube boulders fluctuates continuously during the movement, presenting a peak form of “increasing-decreasing.” In addition, in most cases, the angular velocity of the spherical and cube boulders is larger than that of the cuboid and rhombohedron boulders. This indicates that the angular velocity variation law of boulder motion is related to whether the shape of the boulder is spherically symmetric, as well as the smoothness of the surface.
- (4)
- The simulation results show that motion distance is greatly influenced by the terrain slope and shape of the boulders. When the topographic slope is relatively large, the debris flow density and appearance characteristics of boulders have little influence. As the slope decreases, the boulders will gradually stop owing to insufficient dynamics. Moreover, the movement distance of the spherical and the approximately spherically symmetric cube boulders are greater than those of the cuboid and rhombohedron boulders. The accumulation characteristics of boulders in the simulation are generally consistent with the field investigation results.
- (5)
- Although the results of this study have simulated the coupled motion of debris flow and boulders under the influence of multiple factors, there are still some deficiencies in this study, such as the inaccuracy of the topographic map, limited computational resources, and boulder modeling. Further improvements can be made in subsequent studies.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Picture of the Zhouqu debris flow: (

**a**) boulders left in the accumulation area after the Zhouqu debris flow, (

**b**,

**c**) boulders in the gully, and (

**d**) a boulder approximately 18 m in diameter upstream of a check dam.

**Figure 3.**Parameters of boulders (the origin of x and y coordinates is the left and lower side of the DEM model, and z is the elevation).

**Figure 5.**(

**a**) Meshing range of the simulation; (

**b**) schematic diagram of debris flow inlet, as shown in the red area.

**Figure 6.**Simulation results. (

**a**) instantaneous images of debris flow and boulder movement at different times for a debris flow of 2100 kg/m

^{3}; (

**b**) snapshots of cube and spherical boulders flowing into the river for a debris flow of 2100 kg/m

^{3}.

**Figure 7.**Influence of debris flow density on movement velocity of boulders with different shapes of boulders: (

**a**) spherical, (

**b**) cuboid, (

**c**) rhombohedron, (

**d**) cube.

**Figure 8.**Influence of boulder shape on movement velocity under different density conditions: (

**a**) ρ = 1200 kg/m

^{3}, (

**b**) ρ = 1500 kg/m

^{3}, (

**c**) ρ = 1800 kg/m

^{3}, and (

**d**) ρ = 2100 kg/m

^{3}.

**Figure 9.**Influence of debris flow density on angular velocity of boulders with different shapes: (

**a**) spherical, (

**b**) cuboid, (

**c**) rhombohedron, and (

**d**) cube.

**Figure 10.**Influence of boulder shape on angular velocity under different density conditions, (

**a**) ρ = 1200 kg/m

^{3}, (

**b**) ρ = 1500 kg/m

^{3}, (

**c**) ρ = 1800 kg/m

^{3}, and (

**d**) ρ = 2100 kg/m

^{3}.

**Figure 11.**Influence of debris flow density on kinetic energy of boulders with different shapes, (

**a**) spherical, (

**b**) cuboid, (

**c**) rhombohedron, (

**d**) cube.

**Figure 12.**Influence of boulder shape on kinetic energy under different density conditions: (

**a**) ρ = 1200 kg/m

^{3}, (

**b**) ρ = 1500 kg/m

^{3}, (

**c**) ρ = 1800 kg/m

^{3}, and (

**d**) ρ = 2100 kg/m

^{3}.

**Figure 14.**Influence of debris flow density on movement distance of boulders with different shapes: (

**a**) spherical, (

**b**) cuboid, (

**c**) rhombohedron, and (

**d**) cube.

**Figure 15.**Influence of boulder shape on movement distance under different density conditions, (

**a**) ρ = 1200 kg/m

^{3}, (

**b**) ρ = 1500 kg/m

^{3}, (

**c**) ρ = 1800 kg/m

^{3}, and (

**d**) ρ = 2100 kg/m

^{3}.

Condition | Debris Flow Density (kg/m^{3}) | Yield Stress η (Pa) | Viscosity Coefficient (Pa·s) |
---|---|---|---|

1 | 1200 | 4.706 | 0.023 |

2 | 1500 | 18.900 | 0.091 |

3 | 1800 | 75.900 | 0.364 |

4 | 2100 | 304.770 | 1.463 |

Debris Flow Inlet | Extent of the Rectangular Inlet (m) | Inlet Width (m) | Depth of Debris Flow (m) | Debris Flow Velocity (m/s) |
---|---|---|---|---|

A | x direction: 2062–2200 z direction: 1724–1732 | 138 | 8.0 | 7.10 |

B | X direction: 2734–2832 z direction: 1730–1736 | 98 | 6.0 | 5.80 |

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## Share and Cite

**MDPI and ACS Style**

Wang, F.; Wang, J.; Chen, X.; Zhang, S.; Qiu, H.; Lou, C. Numerical Simulation of Boulder Fluid–Solid Coupling in Debris Flow: A Case Study in Zhouqu County, Gansu Province, China. *Water* **2022**, *14*, 3884.
https://doi.org/10.3390/w14233884

**AMA Style**

Wang F, Wang J, Chen X, Zhang S, Qiu H, Lou C. Numerical Simulation of Boulder Fluid–Solid Coupling in Debris Flow: A Case Study in Zhouqu County, Gansu Province, China. *Water*. 2022; 14(23):3884.
https://doi.org/10.3390/w14233884

**Chicago/Turabian Style**

Wang, Fei, Jiading Wang, Xiaoqing Chen, Shaoxiong Zhang, Haijun Qiu, and Canyun Lou. 2022. "Numerical Simulation of Boulder Fluid–Solid Coupling in Debris Flow: A Case Study in Zhouqu County, Gansu Province, China" *Water* 14, no. 23: 3884.
https://doi.org/10.3390/w14233884