# A Two-Dimensional Depth-Averaged Sediment Transport Mobile-Bed Model with Polygonal Meshes

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

#### 2.1. Flow Equations

#### 2.2. Sediment Transport Equations

_{x}and D

_{y}are the sediment mixing coefficients in the x- and y-directions, respectively, and ${S}_{ek}$ is the sediment exchange rate between sediments in the water column and those in the active layer or on the bed. The above equation was derived from the mass conservation law by Greimann et al. [30] and took the non-equilibrium nature of a sediment load into account.

_{x}and D

_{y}, include contributions from both turbulence as well as dispersion. For many cases, zero mixing coefficients may be used. Otherwise, the coefficients are set to equal to the turbulent viscosity with the Schmidt number specified by a user.

#### 2.3. Mobile-Bed Equations

## 3. Numerical Methods

_{1}P

_{2}in Figure 1) and evaluated at the side center C, $\overrightarrow{n}$ is polygon side unit normal vector, $\overrightarrow{s}$ is the polygon side distance vector (e.g., from P

_{1}to P

_{2}in Figure 1

**),**and ${S}_{\mathsf{\Phi}}={S}_{\mathsf{\Phi}}^{*}A$.

_{1}P

_{2.}A second-order interpolation for point I may be derived to be:

## 4. Model Verification and Validation

#### 4.1. Aggradation in a Straight Channel

_{50}) of 0.32 mm, sand depth of 0.15 m, and specific gravity of 2.65. The case has a constant flow unit discharge of 0.0355 m

^{2}/s, average velocity of 0.493 m/s, and average water depth of 0.072 m. Equilibrium of flow and sediment transport is established first by simulating long enough in time so that the upstream equilibrium sediment rate is balanced by the transport capability. Excessive sediment is then added suddenly from the upstream. The Manning’s roughness coefficient is 0.02294 in order to establish the flow equilibrium given the bed slope and flow velocity. Aggradation process is initiated immediately after a sudden increase of the sediment supply rate above the capacity. The excess sediment supply is $0.9{q}_{seq}$ with ${q}_{seq}$ the sediment transport capacity. Bedload transport mode was observed in the flume experiment and is used for the simulation.

#### 4.2. Erosion in a Straight Channel

^{3}/s, an average velocity of 0.654 m/s, and water depth of 6.0 mm at the downstream boundary.

#### 4.3. Erosion and Depostion in Bends

^{3}/s is imposed, and the sediment supply rate is estimated with the sediment capacity equation. At the exit, the water surface elevation of 0.1 m is maintained. The Engelund-Hansen sediment capacity equation is used and the beadload transport mode is adopted based on the laboratory observation. The equilibrium bedform is reached after a sufficient time (about 10 hours) and the computed bed elevation is shown in Figure 6. In addition, a comparison of the computed and measured water depths along two profiles, 0.375 m from the inner and the outer banks, is shown in Figure 7.

^{o}turn and 41.5 m in length. Both the entrance and exit of the bend have a section of the straight channel attached. The radius of the arc from the centerline was 22 m. Initial bed had a slope of 0.128% longitudinally but flat laterally. The bed was covered with non-uniform mixtures having ${d}_{50}=0.6$ mm. Some characteristic parameters of the case are listed in Table 3.

^{3}/s is imposed and the sediment supply is calculated with the sediment capacity equation. At the exit, the water surface elevation is maintained at 0.12 m.

#### 4.4. Alternating Bar Formation Downstream of an Inserted Dike

_{50}) of 0.216 mm. When the bed reached equilibrium, bed topography was measured which is available for numerical model verification. The experimental conditions are shown in Table 4.

^{3}/s is imposed and the sediment supply rate is based on the sediment transport capacity as equilibrium solution is sought. At the downstream (x = 25 m), a water depth of 0.044 m is specified. The Parker capacity equation is used.

#### 4.5. Erosion and Deposition on a Section of the Middle Rio Grande

## 5. Conclusions

## Funding

## Conflicts of Interest

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**Figure 2.**Comparison of bed elevation changes in time between model prediction and flume data for the aggradation case of Soni [52]). (

**a**) upstream, sediment rate is 1.9 the capacity; (

**b**) upstream sediment supply is 15% more than (

**a**).

**Figure 3.**Comparison of predicted and measured bed gradation at a location 10-m from the downstream model boundary; measured data are from Ashida-Michiue [53].

**Figure 4.**Comparison of predicted and measured scour depth variation in time at three locations: 13 m (red), 10 m (blue), and 7 m (black) from the downstream model boundary; “Measured” = Ashida-Michiue [53]; “Wu Grain Stress” = the results obtained with the modified grain shear stress calculation.

**Figure 6.**Computed equilibrium bed elevation for case T2 of Struiksma et al. [54].

**Figure 7.**Comparison of the computed and measured water depths along lines 0.375 m from inner and outer banks for case T2 of Struiksma et al. [54].

**Figure 8.**Computed equilibrium bed elevation for case T4 of Struiksma et al. [54].

**Figure 9.**Comparison of computed and measured water depth (

**left**) and ${d}_{50}$ (

**right**) along the lines 0.11B from the inner and outer banks for case T4 of Struiksma et al. [54].

**Figure 10.**Channel geometry, dimension, and 2D mesh for the Struiksma and Crosato [35] case.

**Figure 12.**Comparison of predicted and measured erosion and deposition depths along a straight line of y = 0.1 m; depth is normalized with the average water depth of 0.044 m.

**Figure 13.**(

**a**) Cross sections (XS) of the field measurement. (

**b**) The 2D mesh for the numerical modeling (flow is from top to bottom, or North to South).

DiameterRange (mm) | 0.2– 0.3 | 0.3– 0.4 | 0.4– 0.6 | 0.6– 0.8 | 0.8– 1.0 | 1.0– 1.5 | 1.5– 2.0 | 2.0– 3.0 | 3.0– 4.0 | 4.0– 6.0 | 6.0– 8.0 | 8.0– 10. |

Content (%) | 7.45 | 12.4 | 15.9 | 4.4 | 3.6 | 6.79 | 4.0 | 9.18 | 10.2 | 18.1 | 6.0 | 2.0 |

**Table 2.**Flume geometry and experiment conditions for case T2 of Struiksma et al. [54].

Flume Width (m) | Discharge (m ^{3}/s) | Water Depth (m) | Velocity (m/s) | Bed Slope | Froude Number | d_{50}(mm) | Manning’s Coefficient |
---|---|---|---|---|---|---|---|

1.5 | 0.062 | 0.10 | 0.41 | 0.203% | 0.41 | 0.45 | 0.023 |

**Table 3.**Flume geometry and experiment conditions for case T4 of Struiksma et al. [54].

Flume Width (m) | Discharge (m ^{3}/s) | Water Depth (m) | Velocity (m/s) | Bed Slope | Froude Number | d_{50}(mm) | Manning’s Coefficient |
---|---|---|---|---|---|---|---|

2.3 | 0.121 | 0.12 | 0.44 | 0.128% | 0.41 | 0.60 | 0.02 |

**Table 4.**Key flume geometry and experiment conditions with the Struiksma and Crosato [35] case.

Flume Width (m) | Discharge (m ^{3}/s) | Water Depth (m) | Velocity (m/s) | Bed Slope | Froude Number | Manning’s Coefficient |
---|---|---|---|---|---|---|

0.60 | 0.00685 | 0.044 | 0.26 | 0.3% | 0.39 | 0.0263 |

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**MDPI and ACS Style**

Lai, Y.G. A Two-Dimensional Depth-Averaged Sediment Transport Mobile-Bed Model with Polygonal Meshes. *Water* **2020**, *12*, 1032.
https://doi.org/10.3390/w12041032

**AMA Style**

Lai YG. A Two-Dimensional Depth-Averaged Sediment Transport Mobile-Bed Model with Polygonal Meshes. *Water*. 2020; 12(4):1032.
https://doi.org/10.3390/w12041032

**Chicago/Turabian Style**

Lai, Yong G. 2020. "A Two-Dimensional Depth-Averaged Sediment Transport Mobile-Bed Model with Polygonal Meshes" *Water* 12, no. 4: 1032.
https://doi.org/10.3390/w12041032