# Effects of Boulder Arrangement on Flow Resistance Due to Macro-Scale Bed Roughness

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{50}. According to some studies [4,5,6,7], if the hydraulic condition is defined by the ratio between h and the bed particle diameter d

_{84}(i.e., diameter for which 84% of the particles are finer), a macro-scale occurs for h/d

_{84}≤ 4. Macro-scale roughness condition is characterized by different dissipative mechanisms as compared to micro-scale one. Mendicino and Colosimo [7] stated that, for values higher than 100 of the depth/sediment ratio (i.e., the ratio between the hydraulic radius R or h and the particle diameter representing the characteristic roughness height), skin friction is due to drag effects related to the shape of each bed particle and viscous friction on its surfaces, and is also influenced by macro-scale bed-forms.

_{84}≤ 10, and in particular for h/d

_{84}~ 1), the resistance effects due to form drag and turbulent wakes caused by large roughness elements are high. When some elements protrude above the water surface, and the flow is locally supercritical, additional energy losses occur [8,9,10]. According to Bathurst [11], a channel can be defined with a “cobble and boulder bed” when d

_{50}> 64 mm, the flow resistance due to vegetation is negligible, and the occurring roughness condition is “transition” or “macro-scale”.

_{84}≤ 4), some authors [18,19] demonstrated that the velocity profile is S-shaped, with near-surface velocities higher than near-bed ones, and the logarithmic or power distribution can be assumed only for a bottom distance greater than the roughness size.

^{1/2}s

^{−1}), n is the Manning coefficient (m

^{−1/3}s), s is the channel slope, g is the gravitational acceleration, f is the Darcy–Weisbach friction factor, R is the hydraulic radius, and $\sqrt{gRs}={u}_{*}$, which is the shear velocity.

_{k}is the kinematic viscosity, Γ is a function estimated by experimental velocity measurements, and δ is an exponent calculated by the following relationship [24]:

_{k}is the flow Reynolds number.

_{v}of the function Γ [25]:

^{1/2}is the flow Froude number and a, b, and c are coefficients to be estimated experimentally.

## 2. Materials and Methods

#### 2.1. Experimental Data by CANOVARO et al. [38]

#### 2.2. Experimental Data by Ferro and Giordano [31]

## 3. Results

_{v}values obtained by Equations (5) and (6), with those calculated by Equations (8a) and (8b).

_{m}values and those calculated f

_{c}by Equations (9a) and (9b) and remarks that an accurate estimate of the Darcy–Weisbach friction factor can be obtained by the proposed approach. The friction factor values calculated by Equations (9) are characterized by errors in estimate E = (f

_{c}− f

_{m})/f

_{m}which are less than or equal to ±5% for 98.9% of cases and less than or equal to ±2.5% for 87.8% of cases.

_{m}. In other words, Equation (9a) systematically underestimates (−3%) the f values measured by Ferro and Giordano [31].

^{1.078}s

^{−0.58}, Γ) was investigated and a value of the a coefficient, equal to the slope coefficient of the best-fit straight line passing through the origin of the axes, was obtained. In particular, the a coefficient resulted equal to 0.3354 for the “Random” arrangement, 0.3372 for the “Transversal stripe” arrangement, and 0.335 for the “Longitudinal stripe” arrangement (Table 2). Figure 8 shows the comparison between the f measured values and those calculated coupling Equations (4) and (7) with b = 1.078, c = 0.58, and a varying with the arrangement. In this case, the errors E in the estimate of the Darcy–Weisbach friction factor are less than or equal to ±5% for 96.8% of cases and less than or equal to ±2.5% for 88.2% of cases.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Planimetric view of “Random” (

**a**), “Transversal stripe” (

**b**), and “Longitudinal stripe” (

**c**) arrangements investigated by Canovaro et al. [38].

**Figure 2.**Comparison between the Γ

_{v}values obtained by Equations (5) and (6), with those calculated by Equations (8a) and (8b) (

**a**), and between the measured f

_{m}values and those calculated f

_{c}by Equations (9a) and (9b) (

**b**) for the “Random” arrangement.

**Figure 4.**Frequency distribution of the errors E in the estimate of f applying Equation (9a) to the data by Ferro and Giordano [31].

**Figure 5.**Comparison between the measured f

_{m}values and those calculated f

_{c}by Equations (11) for the “Transversal stripe” arrangement.

**Figure 6.**Comparison between the measured f values and those calculated by Equation (13) for the “Longitudinal stripe” arrangement.

**Figure 7.**Trend of the Darcy–Weisbach friction factor for three selected values of concentration for low (Ch = 0, 18, and 42%) (

**a**) and high (Ch = 48, 71, and 89%) (

**b**) values of roughness concentration.

**Figure 8.**Comparison between the measured f values and those calculated coupling Equations (4) and (7) with b = 1.078, c = 0.58 and a varying with the arrangement.

**Figure 9.**Comparison between the measured f values and those calculated coupling Equations (4) and (7) with b = 1.1175, c = 0.587 and a varying with the arrangement.

Authors | Arrangement | Runs | s | Re | F | h/d |
---|---|---|---|---|---|---|

Canovaro et al. [38] | Random | 189 | 0.005–0.025 | 11,180–71,116 | 0.4–1.45 | 0.47–2.14 |

Canovaro et al. [38] | Transversal stripe | 160 | 0.002–0.06 | 12,960–64,251 | 0.42–2.06 | 0.46–2.32 |

Canovaro et al. [38] | Longitudinal stripe | 34 | 0.01–0.025 | 20,061–59,724 | 0.6–1.45 | 0.63–1.95 |

Ferro and Giordano [31] | Random | 416 | 0.007–0.094 | 2668–23,607 | 0.19–0.97 | 0.88–4.14 * |

_{84}.

Authors | Ch (%) | Runs | Arrangement | a | b | c |
---|---|---|---|---|---|---|

Canovaro et al. [38] | <48 | 84 | Random | 0.3398 | 1.0787 | 0.5772 |

Canovaro et al. [38] | ≥48 | 105 | Random | 0.3356 | 1.1177 | 0.5869 |

Canovaro et al. [38] | ≤35 | 140 | Transversal stripe | 0.3331 | 1.0762 | 0.5829 |

Canovaro et al. [38] | 100 | 20 | Transversal stripe | 0.3365 | 1.1172 | 0.5871 |

Canovaro et al. [38] | ≤35.8 | 30 | Longitudinal stripe | 0.331 | 1.0794 | 0.5829 |

Canovaro et al. [38] | <48 | 84 | Random | 0.3354 | 1.078 | 0.58 |

Canovaro et al. [38] | ≤35 | 140 | Transversal stripe | 0.3372 | 1.078 | 0.58 |

Canovaro et al. [38] | ≤35.8 | 30 | Longitudinal stripe | 0.335 | 1.078 | 0.58 |

Canovaro et al. [38] | ≥48 | 105 | Random | 0.3356 | 1.1175 | 0.587 |

Canovaro et al. [38] | 100 | 20 | Transversal stripe | 0.3366 | 1.1175 | 0.587 |

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**MDPI and ACS Style**

Nicosia, A.; Carollo, F.G.; Ferro, V.
Effects of Boulder Arrangement on Flow Resistance Due to Macro-Scale Bed Roughness. *Water* **2023**, *15*, 349.
https://doi.org/10.3390/w15020349

**AMA Style**

Nicosia A, Carollo FG, Ferro V.
Effects of Boulder Arrangement on Flow Resistance Due to Macro-Scale Bed Roughness. *Water*. 2023; 15(2):349.
https://doi.org/10.3390/w15020349

**Chicago/Turabian Style**

Nicosia, Alessio, Francesco Giuseppe Carollo, and Vito Ferro.
2023. "Effects of Boulder Arrangement on Flow Resistance Due to Macro-Scale Bed Roughness" *Water* 15, no. 2: 349.
https://doi.org/10.3390/w15020349