# 3D–3D Computations on Submerged-Driftwood Motions in Water Flows with Large Wood Density around Driftwood Capture Facility

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Computational Model

#### 2.1.1. Flow Model

_{dr}

^{i}is the drag force that the driftwood exerts on the flow in a certain calculation cell, ρ is the density of water, C

_{D}is the drag coefficient, $\sqrt{g}$ is the Jacobian of the cell, A

_{k}is the flow direction projection area existing in the cell,

**u**is the flow velocity vector,

**u**is the driftwood motion velocity vector, U

_{p}^{i}is the flow velocity vector component in the i direction, U

_{p}

^{i}is the driftwood motion velocity component in the i direction, i is the generalized curvilinear coordinate system (ξ, η, ζ), and N

_{cell}is the number of spherical elements that constitute the driftwood existing in the calculation cell.

_{D}was considered as a function of the particle Reynolds number Re

_{d}as follows [26]:

#### 2.1.2. Driftwood Motion Model

- Outline of Driftwood Motion Model

- Bed Friction Terms

_{D}is the drag coefficient,

**u**is the water flow velocity vector,

**u**is the moving velocity vector of a sphere,

_{p}**g**is the gravity acceleration vector,

**F**is the interparticle collision force vector, t is the time, C

_{p}_{M}is the additional mass coefficient (C

_{M}= 0.5), A

_{2}and A

_{3}are the 2D and 3D shape coefficients (A

_{2}= π/4, A

_{3}= π/6), λ

_{A-sub}is the ratio of the submerged part of the projected area of the sphere in the flowing water direction, λ

_{V-sub}is the ratio of the submerged part of the sphere volume, and

**F**is the bottom friction force vector.

_{bed}**F**is expressed as

_{bed}_{p}is the height of the sphere center from the riverbed,

**F**is the bottom friction vector, and

_{b}**F**is the effect of gravity associated with the riverbed gradient.

_{s}**F**is expressed by the following equation:

_{b}_{p}is the friction coefficient between the sphere and riverbed and is described in detail later. In addition, N is the normal force at the contact point between the bottom and sphere. If the bottom surface gradient is small, it is approximately expressed by the following equation:

_{v-sub}in Equation (10) is a coefficient representing the volume ratio of the submerged part of the sphere, which is usually 1 for the driftwood near the bottom, but this value becomes less than 1 when the water depth is small.

_{p}, when the driftwood has a cylindrical shape, the friction coefficient of sliding in the driftwood axis direction and that of rolling in the transverse direction are considerably different. Thus, the friction coefficient was determined while considering the anisotropy of the frictional force based on the results of Kang and Kimura [18] and Kang et al. [19]. The profile of the friction force in such models is illustrated in Figure 2, which shows a 2D view of the presence of driftwood near the bottom. The distribution of friction coefficients in the driftwood axis direction (

**t**direction) and transverse direction (

**n**direction) has the maximum and minimum values, and the values in the other directions are assumed to be elliptical, as shown in the figure. Assuming that the friction coefficient of sliding in the driftwood axis (trunkwise) direction is μ

_{t}and the friction coefficient of rolling in the lateral direction is μ

_{n}, the friction coefficient μ

_{p}in the moving direction of the driftwood is expressed by the following equation:

_{t}is the angle formed by the driftwood movement direction and driftwood axis direction. In Equation (11), cos ψ

_{t}is obtained by the following equation:

_{t}, we considered the difference between the static friction coefficient μ

_{ts}and dynamic friction coefficient μ

_{tk}(μ

_{ts}≥ μ

_{tk}). Because it is impossible to theoretically determine the values of these coefficients, trial-and-error procedures were performed to more faithfully reproduce the driftwood behavior of the experiment. Table 1 lists the values of the friction coefficients used in the present study.

_{b}is the bed elevation. The components in the x and y directions of the gravity forces due to the slope effect

**F**in Equation (6) were modeled as follows:

_{s}- Drag Force

_{D}for a sphere can be evaluated using Equations (2) and (3). However, in the case of driftwood, the drag force changes depending on the directions of the flow and driftwood axis. The variation in the drag coefficient was recently studied by Persi et al. [28], who considered the relationship between the angle of attack and the drag in the case of a columnar object. On the other hand, Kang et al. [19] assumed that the drag coefficient C

_{D}was constant, but the projected area for each spherical element changes depending on the angle between the flow direction and driftwood axis direction. The difference between these two studies is whether the variable drag force was included in the drag coefficient C

_{D}or projected area $\overline{A}$. The latter approach was adopted in this study. Based on the study by Kang et al. [19], the variable projected area was described as follows:

^{2}), $\overline{A}$ is the modified projected area considering the angle between the driftwood and flow, ϕ is the angle between the driftwood axis direction and flow direction (Figure 3), and M is the number of constituent spheres of the driftwood. The direction of the driftwood axis is defined to be positive from the sphere with m = 1 to the sphere with m = M. Figure 4 shows the relationship between the projection area and flow direction. From this figure, cosϕ and |sinϕ| are calculated as follows (Figure 4):

#### 2.2. Computational Conditions

#### 2.2.1. Overview of Laboratory Experiment Performed by Kato et al.

_{10}= 469.7 m

^{3}/s) and 30-year return period flood (Q

_{30}= 680 m

^{3}/s). Two types of driftwood models with different lengths (6 and 12 m) and a common diameter (30 cm) were used. All the values were real-scale values. In each case, the specific gravity of driftwood was set to 1.1, considering the dominant species of trees (hardwood) at the site. The number of driftwood pieces was 3600 for 6 m long pieces and 1800 for 12 m long pieces.

#### 2.2.2. Computational Conditions

## 3. Results

#### 3.1. Cross-Sectional Flow Pattern

#### 3.2. Strength of Secondary Current

#### 3.3. Driftwood Behavior

#### 3.4. Driftwood Capture Ratio

## 4. Discussion

#### 4.1. Reproducibility of Basic Characteristics of Driftwood Behavior

#### 4.2. Effect of Manning Roughness Coefficient

#### 4.3. Effect of Turbulence Model

#### 4.4. Effect of Vertical Grid Division

#### 4.5. Effect of Wood Density

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**Modified projection area Ā considering the angle between flow direction and driftwood axis direction (=ϕ).

**Figure 5.**Planned site for driftwood capture facility in Omoto River, Iwate, Japan ((

**left**) aerial photograph, (

**right**) schematic diagram of planned driftwood facility). Reproduced with permission from Kato et al., Experimental study on design of driftwood capturing facility in Omoto River; published in Advances in river engineering. JSCE, 2018 [3].

**Figure 8.**Plan view of computational area and grid divisions (KP denotes the river kilometer (real scale)).

**Figure 9.**Cross-sectional velocity vectors and color contour of streamwise vorticity along A-A’ and B-B’ sections (upper: vertical grid division of 20 layers (Run 2), lower: vertical grid division of 10 layers (Run 1)).

**Figure 10.**Cross-sectional velocity vectors and color contour of streamwise vorticity along the A-A’ section with different Manning roughness coefficients.

**Figure 11.**Cross-sectional velocity vectors and color contour of streamwise vorticity along the A-A’ section with different turbulence models.

**Figure 12.**Profiles of Un/Us along the A-A’ section in the results with different Manning roughness coefficients (Run 1, 3, 4, and 5) (Us: flow in the channel direction near the bottom, Un: lateral flow toward the inner bank near the bed).

**Figure 13.**Profiles of Un/Us along the A-A’ section in the results of different turbulence models. “Nonlinear”: nonlinear k-ε model (Run 3); “linear”: linear standard k-ε model (Run 6); Us: flow in the channel direction near the bottom; Un: lateral flow toward the inner bank near the bed.

**Figure 14.**Simulated driftwood behavior together with streamlines in the vicinity of the riverbed (Run 1, bird-eye view, red bars: driftwood pieces, vertical grid division: 10 layers, t: elapsed time after driftwood input).

**Figure 15.**Plan view of the velocity vectors near the bottom and the driftwood distribution at t = 180 s (Run 1).

**Figure 16.**Driftwood deposition area in laboratory experiment. Reproduced with permission from Kato et al., Experimental study on design of driftwood capturing facility in Omoto River; published in Advances in river engineering. JSCE, 2018 [3].

**Figure 17.**Simulated driftwood distributions with different specific gravities (0.9 ((

**Left**) Run 8), 1.0 ((

**Middle**) Run 7), and 1.1 ((

**Right**) Run 1)) at t = 60 s. The velocity vectors at the bottom (for specific gravities of 1.0 and 1.1) and at the surface (specific gravity of 0.9) are shown together.

Type of Friction | Symbol | Value |
---|---|---|

Static Friction Coefficient | ${\mu}_{ts}$ | 2.0 |

Sliding Friction Coefficient | ${\mu}_{tk}$ | 1.0 |

Rolling Friction Coefficient | ${\mu}_{n}$ | 0.5 |

Run | Vertical Layer | Turbulence Model | Manning Roughness Coefficient n ^{1} | Specific Gravity of Wood ^{2} |
---|---|---|---|---|

1 | 10 | Nonlinear k-ε | 0.03 | 1.1 |

2 | 20 | Nonlinear k-ε | 0.03 | 1.1 |

3 | 10 | Nonlinear k-ε | 0.025 | 1.1 |

4 | 10 | Nonlinear k-ε | 0.02 | 1.1 |

5 | 10 | Nonlinear k-ε | 0.015 | 1.1 |

6 | 10 | Linear standard k-ε | 0.025 | 1.1 |

7 | 10 | Nonlinear k-ε | 0.03 | 1.0 |

8 | 10 | Nonlinear k-ε | 0.03 | 0.9 |

^{1}Averaged value of Manning roughness in the experiment is n = 0.03.

^{2}Specific gravity of wood used in the experiment is 1.1.

Run | Capture Ratio (%) |
---|---|

1 | 32 |

2 | 39 |

3 | 59 |

4 | 96 |

5 | 100 |

6 | 88 |

7 | 94 |

8 | 100 |

Laboratory experiment | 65~87.5 |

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**MDPI and ACS Style**

Kimura, I.; Kang, T.; Kato, K. 3D–3D Computations on Submerged-Driftwood Motions in Water Flows with Large Wood Density around Driftwood Capture Facility. *Water* **2021**, *13*, 1406.
https://doi.org/10.3390/w13101406

**AMA Style**

Kimura I, Kang T, Kato K. 3D–3D Computations on Submerged-Driftwood Motions in Water Flows with Large Wood Density around Driftwood Capture Facility. *Water*. 2021; 13(10):1406.
https://doi.org/10.3390/w13101406

**Chicago/Turabian Style**

Kimura, Ichiro, Taeun Kang, and Kazuo Kato. 2021. "3D–3D Computations on Submerged-Driftwood Motions in Water Flows with Large Wood Density around Driftwood Capture Facility" *Water* 13, no. 10: 1406.
https://doi.org/10.3390/w13101406