# Effect of Density and Total Weight on Flow Depth, Velocity, and Stresses in Loess Debris Flows

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Materials and Sample Preparation

#### 2.1. Similarity Principle

_{0}); characteristic depth of the fluid (h); characteristic diameter of the particles (d); channel length (L); channel width (W); slope angle (α); fluid density (ρ

_{f}); density of solid material (ρ

_{s}); dynamic viscosity coefficient (μ); gravitational acceleration (g); elastic modulus of grain (E

_{s}); coefficient of friction (c); initial flow velocity (v

_{0}). The distribution of the flow velocity can be expressed as follows:

_{f}, v

_{0}, and L as a unit system:

_{e}is the Reynolds number, which represents the ratio of inertial forces to viscosity; F

_{r}is the Froude number, which represents the ratio of inertial forces to gravitational forces; and ε represents the ratio of the fluid depth to channel length. The dimensionless function format of flow velocity can then be obtained by using similarity theory:

_{e}and F

_{r}appear in the distribution function for the dynamic characteristics, and must be equal in the model being developed and the actual process it represents. To achieve this goal, it’s necessary to build a model of the study system that has the same scale as the system, and this is unrealistic in practice [58]. To achieve accurate experimental results, similarity methods are used [58]. To meet this goal for the geometric similarity and kinematic similarity, R

_{e}and F

_{r}are often used [25,58]. Because the gravitational force is one of the most important driving forces in flume experiments [58], we adopted F

_{r}as the research object.

#### 2.1.1. Geometric Similarity

_{l}is the length ratio, l

_{a}is the length of the actual, and l

_{f}is the length of the flume.

#### 2.1.2. Material Similarity

#### 2.1.3. Velocity Similarity

_{0}similarity ratio.

#### 2.2. Experimental Apparatus

_{l}(m) is the length of the flume, and Δ

_{t}is the elapsed time (s).

#### 2.3. Sample Preparation

_{5}) was 0.36 mm. Prior to conducting the experiment, the mixtures were prepared according to the required densities (including 1500 kg/m

^{3}, 1650 kg/m

^{3}, 1800 kg/m

^{3}, 1900 kg/m

^{3}, 2050 kg/m

^{3}, 2100 kg/m

^{3}, and 2300 kg/m

^{3}) and total weights (including 200 kg, 300 kg, 400 kg, and 500 kg).

## 3. Results and Discussion

#### 3.1. General Flow Patterns

#### 3.2. Flow Depth

^{3}to 2.3 g/cm

^{3}. The flow depths increased to approximately 1.3 times and 1.5 times the minimum depth when the density increased from 1.5 g/cm

^{3}to 1.8 g/cm

^{3}and from 1.8 to 2.3 g/cm

^{3}, respectively.

#### 3.3. Flow Velocity

^{3}to 2.3 g/cm

^{3}. The flow velocity also decreased by approximately 1.2 times and 1.5 times as the density increased from 1.5 g/cm

^{3}to 1.8 g/cm

^{3}and from 1.8 g/cm

^{3}to 2.3 g/cm

^{3}, respectively. This indicated that low-density debris flow fastest, and that the high-density debris flowed slowly. When the mixture weight decreased from 500 kg to 400 kg, the flow velocity decreased by about 8%. Because of the inertial force increased with increasing mixture weight, it gradually increased for a given mixture density [67]. Our results therefore confirm the results of previous studies, which reported that debris flow density and mixture weight both strongly affected the debris flow velocity [5,38,72]. Combined with the flow depth analysis, these results let us calculate the impact force of the flowing debris.

#### 3.4. Stress and Pressure Measurements

#### 3.4.1. Total Normal Stress

^{3}, and at position S1 when the density was low (i.e., 1.5 or 1.65 g/cm

^{3}). These results illustrate that the maximum values moved upstream as the mixture density decreased (Figure 7a). For a given density and monitoring point, the peak total normal stress increased with increasing weight, and the peak position gradually occurred later (Figure 7b). The magnitude of this change increased as the weight of the mixture increased from 200 kg to 500 kg. An especially large difference in the stress was observed when the weight increased from 300 kg to 400 kg.

^{3}to 1.8 g/cm

^{3}. Therefore, there were obvious changes in the total normal stress as the mixture density increased (Figure 7c). The changes with respect to density result from the increasing proportion of the weight accounted for by water as the mixture density decreased, since the water decreased the resistance to flow, resulting in a faster flow velocity and reduced total normal stress [75]. However, the results differed for the high-density debris flow; although the weight of the mixture (thus, the gravitational force) was higher due to the higher content of particles, the resistance to flow also increased due to the decreased water content.

^{3}to 2.3 g/cm

^{3}. The stress also increased to between 1.5 and 3 times the initial value when the density increased from 1.5 g/cm

^{3}to 1.8 g/cm

^{3}, versus approximately 1.8 times as the density increased from 1.8 g/cm

^{3}to 2.3 g/cm

^{3}. Thus, the total normal stress was strongly influenced by the mixture density. In addition, as the mixture weight increased from 200 kg to 500 kg at a given density, the stress increased to approximately 6 times the initial value. However, the change was largest when the weight increased from 300 kg to 400 kg.

#### 3.4.2. Pore Fluid Pressure

^{3}or 1.9 g/cm

^{3}. The maximum peak was upstream when the mixture had a density <1.8 g/cm

^{3}. These results illustrate that the maximum values shift from the top of the flume to the bottom as the mixture density decreases (Figure 8a). The peak became increasingly large as the mixture weight increased at a given density and measurement position, and the position of the peak occurred later in the flow (Figure 8b). The magnitude of change changed greatly as the weight of the mixture increased from 200 kg to 500 kg. The magnitude of increase was largest when the mixture weight increased from 200 kg to 300 kg. For a given mixture weight and monitoring position, the peak occurred later as the mixture density decreased. The magnitude of the decrease was largest when the density increased from 1.65 g/cm

^{3}to 1.8 g/cm

^{3}. These results indicate that the pore fluid pressure changed greatly as the density changed (Figure 8c). In addition, a low-density debris flow sustains high pore fluid pressure during the flow, thus increasing the mobility of the debris and increasing the likelihood that the debris will be liquefied [76], and our results agree with previous experimental results [6]. However, for the high-density debris flow, we found reduced mobility of the debris and a reduced likelihood that the debris will liquefy.

^{3}to 1.4 g/cm

^{3}. The magnitude of the pore fluid pressure increased by 2 to 4 times with the density was high. However, when the density ranged from 1.8 g/cm

^{3}to 1.5 g/cm

^{3}, the pore fluid pressure increased by only approximately 1.5 times. For a given mixture density, the increase in pore fluid pressure at 500 kg was to approximately 6 times the value with a mixture weight of 200 kg.

#### 3.5. Development of Predictive Functions Based on the Study Data

#### 3.5.1. Parameterization of the Fitting Functions

^{dγ})

_{g}m + s

_{h}) × sS

_{i}× lnγ + s

_{j})

_{l}m + p

_{n}) × (p

_{0}× lnγ + p

_{q}),

#### 3.5.2. Parameterized Functions

^{2}) to evaluate the fitting accuracy [77,78]. The results were all excellent, with an adjusted R

^{2}value >0.93. Thus, our functions did a good job of explaining the variation in flow depth, flow velocity, total normal stress, and pore fluid pressure as a function of the mixture density and weight, which agrees with previous results [58,67,75].

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the experimental apparatus. Total normal stress sensors (labeled “S”) and pore fluid-pressure sensors (labeled “P”) are installed at upstream (S1 and P1), midstream (S2 and P2), and downstream (S3 and P3) positions.

**Figure 3.**Video frames from the debris flow experiments with densities of (

**a**) 1650 kg/m

^{3}and (

**b**) 2050 kg/m

^{3}.

**Figure 4.**Example of a representative set of experimental results for the normal stress, pore pressure, and flow depth for a sample with a density of γ = 2200 kg/m

^{3}and a weight of w = 500 kg. Data are for sampling positions S2 and P2 in Figure 1.

**Figure 5.**Effect on flow depth of (

**a1**,

**a2**) mixture density (γ) for two representative weights and (

**b1**,

**b2**) total mixture weight (w) for two representative densities.

**Figure 7.**Effect of mixture density (γ) and total weight (w) on the total normal stress at monitoring points S1 to S3 in Figure 1: (

**a1**,

**a2**) hydrographs for mixtures with constant weight and density at the three monitoring points (whose positions are shown in Figure 1); (

**b1**,

**b2**) hydrographs for mixtures with different weights but constant density at the three monitoring points; and (

**c1**,

**c2**) hydrographs for mixtures with different densities but constant weight at the three monitoring points.

**Figure 8.**Changes in the pore fluid pressure as a function of mixture density and weight: (

**a1**,

**a2**) hydrographs for mixtures at the three monitoring points P1 to P3 (Figure 1) but with constant weight and density; (

**b1**,

**b2**) hydrographs for mixtures with different weights but with constant density at a given monitoring point; and (

**c1**,

**c2**) hydrographs for mixtures with different densities but with constant weight at a given monitoring point.

**Figure 9.**Fitting functions for the maximum flow depth as a function of the mixture weight and density.

**Figure 10.**Fitting functions for the maximum flow velocity as a function of the mixture weight and density.

**Figure 11.**Fitting function for the total normal stress: (

**a**) the mixture weight and total normal stress; (

**b**) the mixture density and total normal stress. The locations of positions S1 to S3 are shown in Figure 1.

**Figure 12.**Fitting function for the pore fluid pressure as a function of: (

**a**) the mixture weight and pore fluid; (

**b**) the mixture density and pore fluid pressure. The locations of positions P1 to P3 are shown in Figure 1.

**Figure 13.**Response surfaces for the (

**a**) maximum flow depth, and (

**b**) mean flow velocity as a function of mixture weight and density.

Index | a | B | c | d | Adj. R^{2} |
---|---|---|---|---|---|

Coefficient value | 0.0524 | 27.2243 | 0.1044 | 0.8405 | 0.9729 |

Index | u | K | y | z | Adj. R^{2} |
---|---|---|---|---|---|

Coefficient values | 0.0009 | 0.6280 | −1.5142 | 4.9659 | 0.9834 |

**Table 3.**Coefficients in Equation (19) for maximum total normal stress at monitoring positions S1 to S3 (Figure 1).

Position | s_{g} | s_{h} | S_{i} | s_{i} | Adj. R^{2} |
---|---|---|---|---|---|

S1 | 0.0022 | 0.5259 | 3.2798 | −0.4498 | 0.9603 |

S2 | 0.0019 | 0.5764 | 4.3714 | −0.9746 | 0.9370 |

S3 | 0.0016 | 0.6155 | 5.8129 | −1.8288 | 0.9412 |

**Table 4.**Coefficients in Equation (20) for the maximum pore fluid pressure at monitoring positions P1 to P3 (Figure 1).

Position | p_{l} | p_{n} | p_{o} | p_{q} | Adj. R^{2} |
---|---|---|---|---|---|

P1 | 0.0017 | 0.4085 | −6.0102 | 5.5407 | 0.9548 |

P2 | 0.0019 | 0.3766 | −4.6781 | 4.7382 | 0.9586 |

P3 | 0.0021 | 0.2782 | −3.3683 | 3.8258 | 0.9423 |

© 2018 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**

Shu, H.; Ma, J.; Yu, H.; Hürlimann, M.; Zhang, P.; Liu, F.; Qi, S.
Effect of Density and Total Weight on Flow Depth, Velocity, and Stresses in Loess Debris Flows. *Water* **2018**, *10*, 1784.
https://doi.org/10.3390/w10121784

**AMA Style**

Shu H, Ma J, Yu H, Hürlimann M, Zhang P, Liu F, Qi S.
Effect of Density and Total Weight on Flow Depth, Velocity, and Stresses in Loess Debris Flows. *Water*. 2018; 10(12):1784.
https://doi.org/10.3390/w10121784

**Chicago/Turabian Style**

Shu, Heping, Jinzhu Ma, Haichao Yu, Marcel Hürlimann, Peng Zhang, Fei Liu, and Shi Qi.
2018. "Effect of Density and Total Weight on Flow Depth, Velocity, and Stresses in Loess Debris Flows" *Water* 10, no. 12: 1784.
https://doi.org/10.3390/w10121784