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Comparative Analysis of HLLC- and Roe-Based Models for the Simulation of a Dam-Break Flow in an Erodible Channel with a 90^{∘} Bend

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

#### Capacity vs. Noncapacity Approach for Bedload Transport

## 3. Numerical Models and Simulations

#### 3.1. Roe-Based Solvers: R-Cap and R-NCap

- On the one hand, assuming the capacity hypothesis, the cell-centered value of the transport thickness at the next time step ${t}^{n+1}$ is directly computed using:$${\eta}_{i}^{n+1}=\frac{{k}_{E}}{{r}_{s}\phantom{\rule{0.166667em}{0ex}}{k}_{D}}\phantom{\rule{0.166667em}{0ex}}{(\Delta \theta )}_{i}^{n+1}\phantom{\rule{0.166667em}{0ex}}{d}_{s}$$
- On the other hand, the assumption of the noncapacity approach requires solving Equation (16) at each time step using the updating formula:$${\eta}_{i}^{n+1}={\eta}_{i}^{n}-\frac{\Delta t}{{A}_{i}}\sum _{k=1}^{NE}{F}_{k}^{\eta \downarrow}\phantom{\rule{0.166667em}{0ex}}{l}_{k}-\Delta t\phantom{\rule{0.166667em}{0ex}}{({\dot{\eta}}_{D}-{\dot{\eta}}_{E})}_{i}^{n}$$To compute the numerical flux ${\left({F}_{p}^{\eta}\right)}_{k}^{\downarrow}$ for the p-th sediment class at the k-th cell edge, the left-hand side of the transport Equation (16) is integrated along the normal direction ${\widehat{x}}_{n}$ to the edge, and the numerical flux at the intercell interface is approximated using the linearized homogeneous Riemann problem [32]:$$\begin{array}{c}{\displaystyle \frac{\partial \eta}{\partial t}}+{\tilde{\lambda}}_{\eta ,k}{\displaystyle \frac{\partial \eta}{\partial {\widehat{x}}_{n}}}=0\hfill \\ \\ \eta (\widehat{x},0)=\left\{\begin{array}{ccc}{\eta}_{i}^{n}& \mathrm{if}& {\widehat{x}}_{n}<0\\ {\eta}_{j}^{n}& \mathrm{if}& {\widehat{x}}_{n}>0\end{array}\right.\hfill \end{array}$$$${\tilde{\lambda}}_{\eta ,k}={\left(\frac{\delta \left({G\left|\mathbf{u}\right|}^{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{u}\xb7\widehat{\mathbf{n}}\right)}{(1-\xi )\phantom{\rule{0.166667em}{0ex}}\delta \eta}\right)}_{k}$$Therefore, the intercell numerical flux for the transport thickness is computed as:$${F}_{k}^{\eta \downarrow}=\left\{\begin{array}{ccc}\frac{1}{1-\xi}{\left({G\left|\mathbf{u}\right|}^{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{u}\xb7\widehat{\mathbf{n}}\right)}_{i}^{n}\hfill & \mathrm{if}\hfill & {\tilde{\lambda}}_{\eta ,k}>0\hfill \\ \frac{1}{1-\xi}{\left({G\left|\mathbf{u}\right|}^{2}\phantom{\rule{0.166667em}{0ex}}\mathbf{u}\xb7\widehat{\mathbf{n}}\right)}_{j}^{n}\hfill & \mathrm{if}\hfill & {\tilde{\lambda}}_{\eta ,k}<0\hfill \end{array}\right.$$

#### 3.2. HLLC-Based Solvers

#### 3.2.1. HLLC-CM

#### 3.2.2. HLLC-WCM

#### 3.3. Test Case

## 4. Results and Discussion

#### 4.1. Comparison between the Roe- and HLLC-Based Capacity Models

- First, the three models were able to predict the bed degradation close to the outlet boundary reasonably well. However, none of them were able to obtain the bed forms observed in the experimental measurements at the beginning of the inlet reach;
- Second, the R-Cap and the HLLC-WCM reproduced the main structures observed in the experiments for the final bed elevation well. However, the HLLC-CM led to a highly diffusive estimation of the bedload flux at the intercell edges, and the model was not able to reproduce the main features of the measured topography (see Figure 2). However, none of the schemes were able to accurately predict the absolute accumulation of bed material observed in the experiments downstream of the inner corner, nor the depth in the opposite eroded region;
- Third, the R-Cap and HLLC-WCM computed a marked eroded zone close to the inner corner. Although slight erosion was observed in this region in the laboratory, both numerical models overestimated the bed degradation. That was in reality due to the formation and development of a 3D vortex downstream of the inner corner. Such a vortex cannot correctly be reproduced by depth-averaged models, which would neglect some vertical recirculation, which would itself hamper the erosion near the inner corner. Relaxing the assumption of a hydrostatic pressure distribution along the flow column might help to improve the accuracy of the numerical model predictions, since the vertical accelerations within the flow play an important role in this region.

#### 4.2. Application of the R-NCap Model

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

HLLC | Harten–Lax–van Leer with Contact |

SWE | Shallow-Water Equations |

MPM | Meyer-Peter and Müller |

R-Cap | Roe-based Capacity Model |

R-NCap | Roe-based Noncapacity Model |

HLLC-CM | HLLC-based Coupled Model |

HLLC-WCM | HLLC-based Weakly Coupled Model |

wsl | water surface level |

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**Figure 1.**(

**a**) Plane view of the experimental flume and position of the gauges G1 to G5; (

**b**) vertical cut taken along the longitudinal axis of the first part of the flume.

**Figure 2.**Final topography obtained after the channel drainage using photogrammetry and averaged over the three experimental runs available.

**Figure 4.**Bed elevation ${z}_{b}$ 2D maps obtained with the R-Cap- and HLLC-based models at $t=180$ s.

**Figure 5.**Final bed level profiles along $x=6.34$ m (

**A**), $x=6.77$ m (

**B**), and $y=0.60$ m (

**C**) with the R-Cap, HLLC-CM, and HLLC-WCM. Experimental photogrammetric data are also plotted.

**Figure 6.**Flow structure with the R-Cap model: (

**left**) 2D map of the maximum $\Delta \theta $; (

**right**) zoom of the inner corner region. The velocity vectors are superimposed with the bed elevation.

**Figure 8.**Final bed level profiles along $x=6.34$ m (

**A**) and $x=6.77$ m (

**B**) with the R-NCap model. Experimental photogrammetric data and results from the R-Cap model are also plotted.

**Table 1.**Coefficients c, ${m}_{1}$, ${m}_{2}$, and ${\theta}_{c}$ for different capacity solid transport rate formulations (9).

Formulation | c | ${\mathit{m}}_{1}$ | ${\mathit{m}}_{2}$ | ${\mathit{\theta}}_{\mathit{c}}$ |
---|---|---|---|---|

MPM [26] | 8 | 0 | 3/2 | 0.047 |

Nielsen [27] | 12 | 1/2 | 1 | 0.047 |

Fernández-Luque [28] | 5.7 | 0 | 3/2 | 0.037 |

Wong [29] | 3.97 | 0 | 3/2 | 0.0495 |

Formulation | ${\mathit{\Gamma}}_{1}\left(\mathit{h}\right)$ | ${\mathit{\Gamma}}_{2}\left(\mathit{\theta}\right)$ | ${\mathit{\Gamma}}_{3}\left(\mathit{\eta}\right)$ | ${\mathit{\theta}}_{\mathit{c}}$ |
---|---|---|---|---|

MPM | $\frac{{n}_{b}^{3}\sqrt{g}}{({r}_{s}-1)\sqrt{h}}$ | $\frac{8\sqrt{\Delta \theta}}{{\theta}^{3/2}}$ | $\frac{{r}_{s}\phantom{\rule{0.166667em}{0ex}}{k}_{D}}{{k}_{E}}}{\displaystyle \frac{\eta}{{d}_{s}}$ | 0.047 |

Nielsen | $\frac{{n}_{b}^{3}\sqrt{g}}{({r}_{s}-1)\sqrt{h}}$ | $\frac{12}{\theta}$ | $\frac{{r}_{s}\phantom{\rule{0.166667em}{0ex}}{k}_{D}}{{k}_{E}}}{\displaystyle \frac{\eta}{{d}_{s}}$ | 0.047 |

Fernández-Luque | $\frac{{n}_{b}^{3}\sqrt{g}}{({r}_{s}-1)\sqrt{h}}$ | $\frac{5.7\sqrt{\Delta \theta}}{{\theta}^{3/2}}$ | $\frac{{r}_{s}\phantom{\rule{0.166667em}{0ex}}{k}_{D}}{{k}_{E}}}{\displaystyle \frac{\eta}{{d}_{s}}$ | 0.037 |

Wong | $\frac{{n}_{b}^{3}\sqrt{g}}{({r}_{s}-1)\sqrt{h}}$ | $\frac{3.97\sqrt{\Delta \theta}}{{\theta}^{3/2}}$ | $\frac{{r}_{s}\phantom{\rule{0.166667em}{0ex}}{k}_{D}}{{k}_{E}}}{\displaystyle \frac{\eta}{{d}_{s}}$ | 0.0495 |

Water density ${\rho}_{w}$ | 1000 kg/m^{3} | ||

Solid density ${\rho}_{s}$ | 2650 kg/m${}^{3}$ | ||

Solid particles’ diameter ${d}_{s}$ | 1.7 mm | ||

Manning’s roughness coeff. ${n}_{b}$ | 0.0165 sm${}^{-1/3}$ | ||

Bed porosity $\xi $ | $0.44$ | ||

Bedload formulation | Meyer-Peter and Müller [26] | ||

Critical Shields stress ${\theta}_{c}$ | $0.047$ |

Global RMSE for ${\mathit{z}}_{\mathit{b}}$ (cm) | ||
---|---|---|

HLLC-CM | HLLC-WCM | R-Cap |

1.07 | 0.96 | 1.02 |

**Table 5.**Bed-level ${z}_{b}$ RMSE for Profiles $x=6.34$ m, $x=6.77$ m, and $y=0.60$ m with the R-Cap, HLLC-CM and HLLC-WCM.

Profile | ${\mathit{z}}_{\mathit{b}}$ RMSE (cm) | ||
---|---|---|---|

HLLC-CM | HLLC-WCM | R-Cap | |

$x=6.34$ m | 1.53 | 1.47 | 0.95 |

$x=6.77$ m | 1.50 | 1.39 | 1.45 |

$y=0.60$ m | 0.50 | 0.48 | 0.45 |

Test | Model | ${\mathit{k}}_{\mathit{E}}/{\mathit{K}}_{\mathit{D}}$ | ${\mathit{K}}_{\mathit{E}}$ | ${\mathit{K}}_{\mathit{D}}$ | $\mathsf{\Delta}\mathit{\theta}$ | ${\mathit{\eta}}^{*}/{\mathit{d}}_{\mathit{s}}$ (15) | ${\mathit{L}}_{\mathit{b}}$ (46) |
---|---|---|---|---|---|---|---|

(-) | (-) | (-) | (-) | (-) | (cm) | ||

T0 | R-Cap | 20 | - | - | 1.4 | 9.4 | - |

T1 | R-NCap | 20 | 1.60 | 0.080 | 1.4 | 9.4 | 4.8 |

T2 | R-NCap | 20 | 0.20 | 0.010 | 1.4 | 9.4 | 38.1 |

T3 | R-NCap | 20 | 0.10 | 0.005 | 1.4 | 9.4 | 76.1 |

T4 | R-NCap | 20 | 0.05 | 0.0025 | 1.4 | 9.4 | 152.3 |

Test | Global RMSE for ${\mathit{z}}_{\mathit{b}}$ (cm) |
---|---|

T0: R-Cap | 1.02 |

T1: R-NCap ${k}_{E}=1.60$ | 1.01 |

T2: R-NCap ${k}_{E}=0.20$ | 0.98 |

T3: R-NCap ${k}_{E}=0.10$ | 1.02 |

T4: R-NCap ${k}_{E}=0.05$ | 1.15 |

**Table 8.**Bed level ${z}_{b}$ RMSE for the profiles $x=6.34$ m and $x=6.77$ m with the R-Cap and R-NCap models.

Test | ${\mathit{z}}_{\mathit{b}}$ RMSE (cm) | |
---|---|---|

$\mathit{x}=6.34$ m | $\mathit{x}=6.77$ m | |

T0: R-Cap | 0.95 | 1.45 |

T1: R-NCap ${k}_{E}=1.60$ | 1.70 | 1.32 |

T2: R-NCap ${k}_{E}=0.20$ | 1.61 | 1.05 |

T3: R-NCap ${k}_{E}=0.10$ | 1.51 | 1.13 |

T4: R-NCap ${k}_{E}=0.05$ | 1.69 | 1.34 |

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**MDPI and ACS Style**

Martínez-Aranda, S.; Meurice, R.; Soares-Frazão, S.; García-Navarro, P. Comparative Analysis of HLLC- and Roe-Based Models for the Simulation of a Dam-Break Flow in an Erodible Channel with a 90^{∘} Bend. *Water* **2021**, *13*, 1840.
https://doi.org/10.3390/w13131840

**AMA Style**

Martínez-Aranda S, Meurice R, Soares-Frazão S, García-Navarro P. Comparative Analysis of HLLC- and Roe-Based Models for the Simulation of a Dam-Break Flow in an Erodible Channel with a 90^{∘} Bend. *Water*. 2021; 13(13):1840.
https://doi.org/10.3390/w13131840

**Chicago/Turabian Style**

Martínez-Aranda, Sergio, Robin Meurice, Sandra Soares-Frazão, and Pilar García-Navarro. 2021. "Comparative Analysis of HLLC- and Roe-Based Models for the Simulation of a Dam-Break Flow in an Erodible Channel with a 90^{∘} Bend" *Water* 13, no. 13: 1840.
https://doi.org/10.3390/w13131840