# A Semi-Analytical Model for the Hydraulic Resistance Due to Macro-Roughnesses of Varying Shapes and Densities

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## Abstract

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## 1. Introduction

## 2. Material and Method

#### 2.1. Bed and Model Description

#### 2.2. Emergent Roughness Elements

#### 2.3. Submerged Roughness Elements

## 3. Results and Discussion

#### 3.1. Calibration of Drag Coefficients

#### 3.2. Validation of Friction Coefficients for River Flows

#### 3.3. Log Law Parameters

#### 3.4. Implication for Engineering Practice

## 4. Conclusions

- The friction for emergent or submerged roughness elements to be obtained in a continuous sense, which can arise when modeling obstacles in a main channel that are steeply sloping or very cluttered.
- The inclusion of the shape factor $k/D$ for blocks. Indeed, not taking it into account can lead to serious errors when modeling emergent obstacles. In the proposed model, both rocky blocks and vegetation (large $k/D$) can be represented as, for example, plantations of trees in a floodplain or overland flows for hillslope hydrology.
- The inclusion of the concentration of obstacles that can be determined in the field or by remote sensing.
- A continuous trend towards “rough bed” formulae when there is a high degree of submergence.
- A calibration parameter ${C}_{d0}$ to be provided that depends only on the shape of the obstacle.

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Definition sketch of the input, output, and improvements of the model. S is the bed slope and ${k}_{s}$ the bed roughness heigth.

**Figure 2.**Turbulent length scale as a function of canopy aspect ratio (

**a**) and vertical profile of velocity and turbulent scale for submerged flow (

**b**).

**Figure 4.**Velocity profiles from model as a function of hydraulic and geometrical parameters for weak and high confinement cases (C = 0.2).

**Figure 6.**Friction coefficients from [18] for randomly spatialized obstacles as a function of the concentration, in comparison to the present model results ($h/k$ = 1.1, 1.5, 2, 3).

**Figure 7.**Comparison of roughness elements with the model assumption, i.e., vertically constant diameter D.

**Figure 8.**Comparison between the model and the experimental bulk velocity from [18] (${C}_{d0}$ = 0.3), ${R}^{2}$ = 0.59 (

**a**), from [22] (${C}_{d0}$ = 0.7) , ${R}^{2}=$ 0.78 and from [21] (${C}_{d0}$ = 1) (

**b**). The series correspond to various slopes, diameters, and concentrations tested in each study.

**Figure 9.**Friction factor as a function of Reynolds number compared with the experiments of [23] (gravels). The diameter and height are equal to the mean diameter D = k = 0.03 m for randomly spatialized obstacles with fixed concentration.

**Figure 10.**Friction factor as a function of Reynolds number compared with the experiments of [23] (cobbles). The diameter and height are equal to the mean diameter D = k = 0.2 m.

**Figure 11.**Comparison between the model and experimental correlations for rock ramps and steep slope. For the model $k/D$ = 0.5 and ${C}_{d0}$ = 0.7 deduced from analyzed experiments above [20].

**Figure 12.**Hydraulic resistance of real boulders for steep slope rivers. Data from [19] for sites with different configuration, comparison with the present model.

**Figure 13.**Displacement height as a function of the frontal density $Ck/D$ and compared to experimental correlation from [31].

**Figure 14.**Hydraulic roughness as a function of the confinement around a rough bed case C = 0.4, $k/d$ = 1, ${C}_{d0}$ = 0.3.

**Figure 16.**Ratio between shear velocity and velocity at the top of the canopy as a function of the spatial density.

**Figure 17.**Friction coefficient as a function of the geometrical model parameters (${C}_{d}$ = 0.3, $k/D$ = 0.5, C = 0.4 for the parameters which do not change).

**Table 1.**Dimension of macro-roughness. Data from [23] are corrected with the shape factor considering that the short axis is the dimension facing the flow D.

D (mm) | k (mm) | C | Shape | |
---|---|---|---|---|

[17] | 35 | 30–100 | 0.08–0.2 | cylinder |

[24] | 20 | 7 | 0.41 | truncated cone |

[1] | 38 | 19 | 0–0.38 | hemisphere |

[20] | 20 | 10 | 0.078/0.14/0.30 | hemisphere |

[18] | 38–76 | 26–57 | 0–0.9 | pebbles |

[21] | 3.2–8.3 | 100 | 0.05–0.15 | cylinder |

[22] | 46–108 | 23–54 | 0.06–0.82 | hemisphere |

[23] | (25.4–38.1) | (25.4–38.1) | 0.04–0.8 | gravels |

[23] | (127–250) | (127–250) | 0.09–0.83 | cobbles |

[19] | (620–830) | (220–410) | 0.05–0.37 | rocks |

h/k | S(%) | C | $\sqrt{\frac{8}{\mathit{f}}}$ | |
---|---|---|---|---|

[7] | 0.4–2 | 2.5–22 | rock ramp | $5.1log\left(\right)open="("\; close=")">h/k$ |

[5] | 0.5–6 | 0.5–3 | rock ramp | $5.62log\left(\right)open="("\; close=")">h/k$ |

[1] | 1–4 | 2.4–8.8 | 0–0.38 | $\left(\right)open="("\; close=")">-7.82S+3.04+log\left(\right)open="("\; close=")">h/k$ |

[25] | 0.1-30 | / | various | $6.4\times 2.5(h/k)/{\left(\right)}^{{6.5}^{2}}1/2$ |

**Table 3.**Models used to compute vertical velocity profile in the case of submerged roughness [14]: d is the displacement in the logarithmic law, ${z}_{0}$ is the hydraulic roughness, ${\alpha}_{t}$ is a turbulence length scale between the macro-roughnesses, and $\kappa $ the von Kármán constant (=0.41). The expression of these parameters results from the continuity of velocity and its gradient at the top of the roughness elements.

u (m s${}^{-1}$) | Parameters | |
---|---|---|

Upper layer | $u\left(z\right)=\frac{{u}_{*}}{\kappa}ln\left(\right)open="("\; close=")">\frac{z-d}{{z}_{0}}$ | $\frac{d}{k}=1-\frac{1}{\kappa}\frac{{\alpha}_{t}}{k}\frac{{u}_{k}}{{u}_{*}}=1-\frac{1}{\kappa}\frac{{l}_{0}}{k}$ |

$\frac{{z}_{0}}{k}=\left(\right)open="("\; close=")">1-\frac{d}{k}$ | ||

Lower layer | $u\left(z\right)={u}_{0}\sqrt{\beta \phantom{\rule{3.33333pt}{0ex}}\left(\right)open="("\; close=")">\frac{h}{k}-1\frac{sinh(\beta \phantom{\rule{3.33333pt}{0ex}}z/k)}{cosh\left(\beta \right)}+1}$ | ${\beta}^{2}=\frac{k}{{\alpha}_{t}}\frac{{C}_{d}Ck/D}{1-\sigma \phantom{\rule{3.33333pt}{0ex}}C}$ |

${C}_{d}$=${C}_{d0}{f}_{{h}_{*}}\left({h}_{*}\right)$ |

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**MDPI and ACS Style**

Cassan, L.; Roux, H.; Garambois, P.-A.
A Semi-Analytical Model for the Hydraulic Resistance Due to Macro-Roughnesses of Varying Shapes and Densities. *Water* **2017**, *9*, 637.
https://doi.org/10.3390/w9090637

**AMA Style**

Cassan L, Roux H, Garambois P-A.
A Semi-Analytical Model for the Hydraulic Resistance Due to Macro-Roughnesses of Varying Shapes and Densities. *Water*. 2017; 9(9):637.
https://doi.org/10.3390/w9090637

**Chicago/Turabian Style**

Cassan, Ludovic, Hélène Roux, and Pierre-André Garambois.
2017. "A Semi-Analytical Model for the Hydraulic Resistance Due to Macro-Roughnesses of Varying Shapes and Densities" *Water* 9, no. 9: 637.
https://doi.org/10.3390/w9090637