# 2D Numerical Modeling on the Transformation Mechanism of the Braided Channel

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. Model Description

#### 2.2. Consideration of the Riparian Vegetation Influence

_{r}is the vegetation stress term (Pa); ${v}^{\prime}$, ${u}^{\prime}$ are the fluctuating velocity in the longitudinal and transverse direction (m/s), respectively; S is the slope, H is the averaged water depth (m) and $\theta $ is the inclination of the location, often $\theta \approx 0$, Equation (1) can be reduced to:

^{−1}), defined as $a=d/\left({l}_{x}{l}_{y}\right)$, d is the radius of the vegetation (m) and l

_{x}and l

_{y}are the distance of vegetation in the longitudinal and transverse directions (m).

_{D}is the drag coefficient of vegetation. Consider the influence range of the vegetation coefficient, let C

_{D}= 1.5 when the vegetation zones near the river bank [39]; if the zones of vegetation are in the river channel, we assumed the influence of vegetation was proportionate to the distance from the channel center in the form:

_{1}and h

_{2}are the lame coefficients in the $\xi $ and $\eta $ direction, respectively; U and V are the depth-averaged flow velocity components in the $\xi $ and $\eta $ direction; the unit discharge vector is $\overline{q}=(q,p)=(UH,VH)$; z is the water level relative to the reference plane; $\beta $ is the correction factor for non-uniformity of the vertical velocity profile; f is the Coriolis parameter, which was neglected in this study; g is the gravitational acceleration; C is the Chezy coefficient; ${\nu}_{e}$ is the depth mean effective vortex viscosity, ${z}_{s}$ and ${z}_{b}$ are the dependent water levels at the water surface and channel bed, respectively.

#### 2.3. Verification

_{a}= 15 m. A real time period of two years was simulated, and the calculated results of flow velocity, water stage and morphological changes were compared with the measured data.

## 3. Numerical Modeling on the Transformation of Braided and Meandering Channel

#### 3.1. Formation of the Braided Channel

#### 3.2. The Transformation of the Braided Channel under Control Variables

## 4. Discussion

#### 4.1. The Cross Section Change

#### 4.2. The Channel Planform Change

_{ctot}is the sum of the mid-channel lengths of all the segments of primary channels in a reach and L

_{cmax}is the mid-channel length of the same channel.

#### 4.3. Comparison with the Empirical Dimensionless Braiding Criterion

^{*}:

^{3}/s). Channel pattern is determined at least in part by both the rate and mode of sediment transport, an obvious shortcoming of Equation (9) is the absence of bed material size. Henderson [43] reanalyzed the Leopold and Wolman data and derived an equivalent expression:

_{50}is the median bed surface grain size (m). Equation (10) can be expressed using the dimensionless discharge defined by Parker [44]. The dimensionless discharge, Q

^{*}, is given by:

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

${\tau}_{ij}$ | The shear-stress tensor |

$\tau $ | The total shear stress near the river bank |

S | The slope of the water surface |

u, v | The time-averaged flow velocity components in the Cartesian coordinate system |

a | The vegetation density |

l_{x}, l_{y} | The distance of vegetation in the longitudinal and transverse direction |

ξ, η | The orthogonal curvilinear coordinates |

h_{1}, h_{2} | The Lamé coefficients |

J | The Jacobian of the transformation J = h_{1}h_{2} |

Z | The water level relative to the reference plane |

H | The averaged water depth |

U, V | The depth-averaged velocity components in the ξ and η directions |

β | The correction factor for the non-uniformity of the vertical velocity |

f | The Coriolis parameter |

g | The gravitational acceleration |

C | The Chezy coefficient |

${\upsilon}_{\mathrm{e}}$ | The depth mean effective vortex viscosity |

D_{11}, D_{12}, D_{21}, D_{22} | The depth-averaged dispersion stress terms |

z_{s}, z_{b} | The dependent water levels for the water surface and channel bed |

$\theta $ | The inclination of the location |

${D}_{r}$ | The vegetation stress term |

k | von Karman constant |

Δt | The time increment |

B | The “braided-channel ratio” |

L_{ctot} | The sum of the mid-channel lengths of all the segments of primary channels in a reach |

L_{cmax} | The mid-channel length of the same channel |

S^{*} | The meandering-braiding threshold slope |

Q | The bankfull discharge |

Q^{*} | The dimensionless discharge |

D_{50} | The median grain size |

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**Figure 1.**Layout of the field study reach section and its computational mesh. (

**a**) layout of the study river section; (

**b**) computational mesh.

**Figure 2.**Measured and calculated cross-sectional profiles of depth-averaged velocity. (

**a**) cross section S1; (

**b**) cross section S2.

**Figure 3.**Comparison of the water stages at two control stations. (

**a**) Shashi station; (

**b**) Xinchang station.

**Figure 4.**Calculated and measured scour or deposition depths of reach section (x: the distance from the x-direction/m; y: the distance from the y-direction/m; dz: the bed level changes/m). (

**a**) Measured; (

**b**) calculated; (

**c**) measured; (

**d**) calculated; (

**e**) measured and (

**f**) calculated.

**Figure 5.**Measured and calculated bed deformation at various typical cross sections. (

**a**) cross section S3 and (

**b**) cross section S4.

**Figure 7.**Layout of the experimental channel after 600 days (HCEN: Bed level/m). (

**a**) Run No. 1; (

**b**) Run No. 2; (

**c**) Run No. 3; (

**d**) Run No. 4.

**Figure 9.**Sketch of the braided reach for initial and run 1–4. (

**a**) Initial reach; (

**b**) run 1; (

**c**) run 2 and (

**d**) run 4.

**Figure 10.**Temporal changes in flow field including velocity and bed elevation of Run No. 4. (

**a**) T = 300 d and (

**b**) T = 300 d.

No. | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|

Size (mm) | 0.004 | 0.008 | 0.016 | 0.031 | 0.062 | 0.125 | 0.25 | 0.5 |

Proportion | 30 | 12.7 | 13.4 | 14.6 | 13.1 | 8.2 | 6.5 | 1.5 |

No. | Group Percentage of Bed Materials | D_{50} (mm) | Year | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

0.004 | 0.008 | 0.016 | 0.03 | 0.062 | 0.125 | 0.25 | 0.5 | 1 | |||

% | 0 | 0 | 0 | 0.1 | 1.1 | 13.2 | 55.3 | 30 | 0.3 | 0.193 | 2002 |

River Section | Total Distance (km) | Section Length (km) | Measured (10^{6} m^{3}) | Calculated (10^{6} m^{3}) |
---|---|---|---|---|

Taipingkou-Shashi | 8.47 | 8.47 | −827.26 | −1185.91 |

Shashi-Haoxue | 58.65 | 50.19 | −1705.39 | −1730.82 |

Haoxue-Xinchang | 73.62 | 14.96 | −1353.62 | −924.21 |

Xinchang-Shishou | 93.38 | 19.76 | −1508.87 | −1719.86 |

Time Period | Time (d) | Discharge (m^{3}/s) | The Medium Grain Size (mm) | Sediment Supply (kg/m^{3}) |
---|---|---|---|---|

1 | 360 | 150 | 0.1 | 1 |

2 | 360 | 300 | 0.1 | 5 |

No. | Flow Discharge (m^{3}/s) | Sediment Supply (kg/m^{3}) | Bank Vegetation | Time (d) |
---|---|---|---|---|

1 | 150 | 5 | Yes | 600 |

2 | 300 | 1 | No | 600 |

3 | 300 | 5 | No | 600 |

4 | 150 | 1 | Yes | 600 |

No. | Number of Breaches | Braided-Channel Ratio (B) | Sinuosity (P) |
---|---|---|---|

Run No. 1 | 6 | 2.11 | 1.06 |

Run No. 2 | 5 | 1.9 | 1.00 |

Run No. 3 | 4 | 1.97 | 1.01 |

Run No. 4 | 2 | 1.22 | 1.35 |

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**MDPI and ACS Style**

Yang, S.; Xiao, Y. 2D Numerical Modeling on the Transformation Mechanism of the Braided Channel. *Water* **2019**, *11*, 2030.
https://doi.org/10.3390/w11102030

**AMA Style**

Yang S, Xiao Y. 2D Numerical Modeling on the Transformation Mechanism of the Braided Channel. *Water*. 2019; 11(10):2030.
https://doi.org/10.3390/w11102030

**Chicago/Turabian Style**

Yang, Shengfa, and Yi Xiao. 2019. "2D Numerical Modeling on the Transformation Mechanism of the Braided Channel" *Water* 11, no. 10: 2030.
https://doi.org/10.3390/w11102030