# Application of an Inertia Dependent Flow Friction Model to Snow Avalanches: Exploration of the Model Using a Ping-Pong Ball Experiment

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## Abstract

**:**

## 1. Introduction

## 2. Model and Method

_{0}is a constant that is derived based on the experimental results. The value of I

_{0}for glass beads is approximately 0.3 according to [22], and we use this value even though our material is different. For the constant friction model, ${\mu}_{2}$ and ${\mu}_{S}$ are defined as

_{x}and V

_{y}are the flow velocities in the x and y directions, respectively. The first equation is a mass conservation equation, and the second and the third equations are momentum conservation equations in the x and y directions. The x-axis represents a downslope direction on a plane parallel to the slope in our simple terrain similar to a ski jump hill (Figure 1). The y-axis represents the width of the slope. C

_{0}is a constant showing a clear difference between the I-dependent model and Titan2D, being defined as ${C}_{0}=1/2$ in the I-dependent model, and as ${C}_{0}={k}_{ap}/2$ in Titan2D, where ${k}_{ap}$ denotes an earth pressure coefficient. Moreover, the source term of Equation (4) “$\overrightarrow{S}\left(\overrightarrow{U}\right)$” shows the features of each numerical model because this term includes friction effects. The source term of the I-dependent model is

_{int}and ϕ

_{bed}are the internal and bed frictions. As seen in these equations, the source term of Titan2D clearly consists of the gravity term, internal friction term, and basal friction term, while the source term of the I-dependent model consists of the gravity term and shear stress terms of T

_{x}and T

_{y}, neglecting the internal friction term (see [31] for another I-dependent model that retains internal friction). This includes $\mu $ of Equation (1). According to these equations, ${\delta}_{2}$ and ${\delta}_{s}$ of the I-dependent model are physically different from ϕ

_{int}and ϕ

_{bed}of Titan2D. We need to be careful of this difference when we discuss these two models. The stopping criterion is applied using the threshold value of the shear stress ${\sigma}_{c}$ defined as

_{s}= 22–26° and δ

_{2}= 28–32° were in agreement with the experimental results carried out using glass beads. Even though our experiments involved ping-pong balls or snow, and not glass beads, we used their values to define the input values of the friction angles δ

_{s}= 20–26° and δ

_{2}= 28–34°, because the best estimates of the ping-pong balls or snow material are not known.

## 3. Results

#### 3.1. Particle Diameter and Flow Characteristics

#### 3.2. Comparison with the Ping-Pong Ball Experiment

_{s}cases and seven δ

_{2}cases) as shown in Table 2. In our simulation, a cylindrical pile, which has a 5 m radius and a 5 m height, was released. In order to study the effect of particle diameter on our numerical model, we used five different sizes of particles—10 cm, 3.8 cm, 1 cm, 1 mm, and 0.1 mm—though the diameter of the real ping-pong ball used in the experiment was 3.8 cm. Therefore, we have carried out $4\times 7\times 5=$140 calculations for simulating the ping-pong ball experiment.

_{2}= 32° and δ

_{2}= 33° (Figure 10). The maximum velocity of the experimental results is especially similar to that in the simulated cases of {δ

_{s}= 26°, δ

_{2}= 32°} and {δ

_{s}= 20°, δ

_{2}= 33°}.

#### 3.3. Error Analysis

_{r}} = {10 cm, 20°, 32°, 0.1486}, {3.8 cm, 24°, 32°, 0.1665}, {1 cm, 26°, 32°, 0.1860}, {1 mm, 26°, 34°, 0.2442}, {0.1 mm, 26°, 34°, 0.4080}. According to these numbers, we could see that the larger the particle diameter is, the better the velocity of the experimental and simulated results fit. In order to clarify how the minimum relative error of the velocity varies with the particle diameter, these minimum error values were plotted against the particle diameter (Figure 12).

## 4. Discussion

#### 4.1. Shape Reproducibility

#### 4.2. Velocity Reproducibility

_{0}for ping-pong ball flows differs substantially from 0.3 and needs to be calibrated or extracted from laboratory experiments with ping-pong balls.

_{0}

_{.}Some of our simulations are in the unstable region, but no signs of numerical instability have been observed (perhaps due to a sufficient degree of code-inherent numerical viscosity). Therefore, it appears unlikely that ill-posedness of the model equations can explain the mismatch between the best-fit particle diameter and the real particle diameter.

_{0}. Unfortunately, our attempts at measuring the angle ${\delta}_{s}$, which can be viewed as a repose angle, for ping-pong balls, have failed so far.

## 5. Conclusions

_{0}should be chosen on the basis of experiments, rather than using the value 0.3 appropriate for glass beads.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Numerical simulation results of the ping-pong ball experiments. The flow direction is from left to right. Deposit shape predicted by (

**a**) the I-dependent model with ${\delta}_{2}=$30° and ${\delta}_{S}$= 20°; (

**b**) Titan2D with ${\varphi}_{int}$= 30° and ${\varphi}_{bed}$= 20°.

**Figure 3.**The particle diameter of snow avalanche deposit. The solid line shows the graph of a dry snow avalanche and the dotted line shows a wet avalanche. Both graphs show a peak in the 0–10 cm range [33].

**Figure 4.**Slopes used in the simulation for revealing the effect of particle diameter d. The broken line shows the particle release position. The steepness of the slope at the release position was 47° (

**red**), 37° (

**blue**), 26° (

**yellow**), and 15° (

**green**) and the release heights were 2190, 1640, 1090, and 550 m, respectively.

**Figure 5.**The final resting shape of the flows. Colored bars show the flow thickness in meters. The shape variation depending on the particle diameter was simulated with the I-dependent model on the 47° slope (at the release position). Cylindrical granular piles were set at the release positions initially and the flow started automatically. The particle diameters are (

**a**) 10 cm, (

**b**) 1 cm, (

**c**) 1 mm, and (

**d**) 0.1 mm.

**Figure 6.**A numerically simulated variation of the flow front positions over time. Simulations were implemented for a flow on the (

**a**) 47° slope (at the release position) and (

**b**) 26° slope (at the release position).

**Figure 7.**Schematic diagram of the Miyanomori ski jump hill modified from [7]. At the release position, ping-pong balls were released from a container.

**Figure 8.**The final deposit shape of simulated flows for δ

_{2}= 29° and δ

_{s}= 20° (case 05). The particle diameters are (

**a**) d = 10 cm, (

**b**) d = 3.8 cm, (

**c**) d = 1 cm, (

**d**) d = 1 mm, and (

**e**) d = 0.1 mm.

**Figure 9.**Comparison of the flow front velocity in the case of δ

_{2}= 29°. (

**a**) δ

_{s}= 20°, (

**b**) δ

_{s}= 22°, (

**c**) δ

_{s}= 24°, and (

**d**) δ

_{s}= 26°. In each graph, the simulation results of d = 10 cm (blue circle), d = 3.8 cm (sky blue star), d = 1 cm (green square), d = 1 mm (yellow triangle), and d = 0.1 mm (red diamond). Raw (black dots) and smoothed (green line) experimental results are included. (

**e**) The slope profile used to simulate the ping-pong ball simulations. A square in the slope figure shows the position of the ping-pong ball container. Raw and smoothed data of front velocity are from [8].

**Figure 10.**Comparison of the flow front velocity in the case of a particle diameter d = 3.8 cm. (

**a**) Simulations with δ

_{2}= 32°. (

**b**) Simulations with δ

_{2}= 33°. In each graph, the simulation results of δ

_{s}= 20° (blue circle), δ

_{s}= 22° (green square), δ

_{s}= 24° (yellow triangle), and δ

_{s}= 26° (red diamond). Ping-pong ball experimental results of raw (black dots) and smoothed data (green line) are included. (

**c**) The slope profile used to simulate the ping-pong ball simulations. A square in the slope figure shows the position of the ping-pong ball container. Raw and smoothed data of front velocity are from [8].

**Figure 11.**The relative error of velocity on the δ

_{s}and δ

_{2}plane for particle diameters (

**a**) d = 10 cm, (

**b**) d = 3.8 cm, (

**c**) d = 1 cm, (

**d**) d = 1 mm, and (

**e**) d = 0.1 mm. The white circles show the minimum error position in each plane. The number in each plane is the minimum relative error value.

**Figure 12.**Minimum relative errors of velocity between the simulated and experimental velocities against the particle diameter d. The simulations are implemented with ${\delta}_{s}=22-26$° and ${\delta}_{2}=28-32\xb0$.

**Figure 13.**Schematic graph showing (

**a**) the relationship between ${\mu}_{\mathrm{s}}$ and ${\mu}_{2}$ against the inertia number, and (

**b**) the relationship between ${\delta}_{2}$and ${\delta}_{s}$against ${h}_{stop}/d$. ${h}_{stop}$ is the minimum thickness, below which no flow occurs. Graph (

**a**) was drawn based on Figure 1 of [28], and graph (

**b**) was based on Figure 2c of [37].

**Table 1.**Width and tail length of the final resting shape of simulated granular flows. The width and tail length are measured using the simulation results shown in Figure 5.

Particle Diameter | ||||
---|---|---|---|---|

10 cm | 1 cm | 1 mm | 0.1 mm | |

Width (m) | 525 | 555 | 585 | 615 |

Tail length (m) | 155 | 345 | 525 | 580 |

δ_{2} = 28° | δ_{2} = 29° | δ_{2} = 30° | δ_{2} = 31° | δ_{2} = 32° | δ_{2} = 33° | δ_{2} = 34° | |
---|---|---|---|---|---|---|---|

δ_{s} = 20° | case 01 | case 05 | case 09 | case 13 | case 17 | case 21 | case 25 |

δ_{s} = 22° | case 02 | case 06 | case 10 | case 14 | case 18 | case 22 | case 26 |

δ_{s} = 24° | case 03 | case 07 | case 11 | case 15 | case 19 | case 23 | case 27 |

δ_{s} = 26° | case 04 | case 08 | case 12 | case 16 | case 20 | case 24 | case 28 |

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**MDPI and ACS Style**

Tsunematsu, K.; Maeno, F.; Nishimura, K. Application of an Inertia Dependent Flow Friction Model to Snow Avalanches: Exploration of the Model Using a Ping-Pong Ball Experiment. *Geosciences* **2020**, *10*, 436.
https://doi.org/10.3390/geosciences10110436

**AMA Style**

Tsunematsu K, Maeno F, Nishimura K. Application of an Inertia Dependent Flow Friction Model to Snow Avalanches: Exploration of the Model Using a Ping-Pong Ball Experiment. *Geosciences*. 2020; 10(11):436.
https://doi.org/10.3390/geosciences10110436

**Chicago/Turabian Style**

Tsunematsu, Kae, Fukashi Maeno, and Kouichi Nishimura. 2020. "Application of an Inertia Dependent Flow Friction Model to Snow Avalanches: Exploration of the Model Using a Ping-Pong Ball Experiment" *Geosciences* 10, no. 11: 436.
https://doi.org/10.3390/geosciences10110436