# Determination of Skin Friction Factor in Gravel Bed Rivers: Considering the Effect of Large-Scale Topographic Forms in Non-Uniform Flows

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{max}is the maximum velocity of a particular velocity profile.${\delta}^{\ast}$ and $\theta $ can be calculated as [16]:

_{b}) to the median grain size (d

_{50}) [21].

_{s}is Nikuradse equivalent roughness size. Determination of k

_{s}is a controversial issue and there is a wide range of estimations from 1.23d

_{35}to 3d

_{90}or 6.6d

_{50}[23,24,25]. In another approach, the Shields parameter can be applied in the Keulegan Equation via a series of calculations. Critical shields parameter (τ*

_{cr}) can be defined as:

_{cr}can be written in association with ${S}_{f}$ as the following [14,26]:

_{s}= d

_{50}in Equation (5), and by comparing Equations (6) and (7),${f}^{\prime}$ can be calculated via logarithm rules. The final equation can be written as [14]:

## 3. Motivation and Objective

## 4. Methodology and Technical Approach

_{s}values). The equivalent skin friction of uniform flow was also calculated via the Einstein–Strickler Equation (Equations (8) and (9)) by applying the value of d

_{50}.

_{s}values. The results are presented in Table 2. RMSD should always be a non-negative value including 0, which represents the perfect fit, and generally, the lower value of RMSD is considered the better result.

## 5. Field Study and Data Collection

_{s}= D

_{50}. See Table 2). This method is also independent of the velocity profile. The results contain a higher level of errors in comparison with the two previous methods, although it is less scattered in comparison with the classic Einstein–Strickler Equation (Figure 7c).

## 6. Discussion

_{BLCM}− f’)/f’

_{BLCM}) was calculated for 100 velocimetry points (Figure 10). This was done for a wide range of energy slopes (0.001–0.1). The results showed that Equation (21) contains less than 5% of error in comparison to the boundary layer flow characteristics method for energy slopes between 0.0014 and 0.056, considering the normal range of friction factor in gravel-bed rivers (Figure 10a). For the high energy slope (S

_{f}> 0.06), the error increases linearly. The results also showed that for the conditions where the energy slope is not available, the modified Einstein’s Equation (based on the $\phi $ parameter’s concept) can reduce the error up to 50% on average (Figure 10b). In addition to the Marbor river, for other rivers with different energy slopes, the accuracy of results obtained from Equation (21) can be compared with the results obtained from the Einstein equation (Figure 11). According to the Figure, the Einstein equation is not appropriately accurate in low and high energy slopes. One reason can be attributed to the assumptions taken into account by Einstein. The normal or semi-normal flow regime is not prevalent in rivers with very steep energy slopes because of the tendency of flow to accelerate in such conditions. On the other hand, very low energy slopes are prevalent in the presence of flow blockage or decelerating flow. Despite the Einstein equation, Equation (21) takes energy slope into account. Consequently, the accuracy of the estimated skin friction factor increased on extreme slopes. However, it must be considered that the findings of this research are limited by the number of rivers where flow data are available. As presented in Figure 11a, the accuracy of results decreased sparsely in higher energy slopes. However, the accuracy of the results is still acceptable in comparison with classic methods. Despite the energy slope, relative errors do not comply with a predictable pattern for average velocity and relative roughness (Figure 12). The relative error remains below 6% in most cases. However, large relative errors have been observed in the presence of higher relative roughness. Consequently, it is recommended that future research studies focus on the accuracy of the proposed approach in rivers with higher relative roughness.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Comparison between values of modified momentum thickness of skin friction and their correlation with the Keulegan–Shields approach.

**Figure 5.**Topography of the bed, river plan, and the location of the sections in the selected reach (The lowest point has been set as the baseline of local elevation code)—Marbor River.

**Figure 6.**Water surface elevation (light blue dashed line), riverbed elevation (violet line), and velocity profiles (dark blue lines) along the center of mass’ line (centroid line)—selected reach of Marbor River.

**Figure 7.**Skin friction factor calculated using Equation (20) (

**a**), Equation (21) (

**b**), modified Einstein–Strickler Equation (

**c**), and classic Einstein–Strickler Equation (

**d**), in the Marbor River. The vertical axis shows the skin friction factor calculated with the flow characteristic method. The ranges of deviations are shown with dashed blue lines.

**Figure 9.**Surface waves and the contraction streamlines, which are visible near the wall of a contraction zone (Marbor River).

**Figure 10.**The relative errors of Equation (21) (

**a**) and Einstein Equation (

**b**) in comparison with BLCM for determining the skin friction factor. The horizontal axis represents the total energy slope.

**Figure 11.**Comparison between the values of skin friction factor calculated via BLCM and Equation (21) (

**a**) and Einstein equation (

**b**).

**Figure 12.**Relative error of skin friction factors calculated using Equation (24) and the BLCM method in comparison with average velocity (

**left**) and relative roughness (

**right**).

River | Location | No. of Profiles | Avg. Depth (cm) | Avg. U_{mean} (cm/s) | Avg. d_{50} (mm) | Avg. Fr | Avg. f | Average f’ (Keulegan) | Avg. S_{f} |
---|---|---|---|---|---|---|---|---|---|

Melodari | Italy | 14 | 13.6 | 64.9 | 20 | 0.554 | 1.204 | 0.101 | 0.038 |

Cerasia | Italy | 15 | 19.5 | 56 | 47 | 0.416 | 0.78 | 0.072 | 0.016 |

Valanidi | Italy | 8 | 15.8 | 61.4 | 35 | 0.502 | 1.089 | 0.086 | 0.026 |

Gallico | Italy | 13 | 24.5 | 72.2 | 52 | 0.470 | 1.182 | 0.088 | 0.028 |

Zayanderud | Iran | 5 | 73 | 77 | 10 | 0.291 | 0.092 | 0.044 | 0.001 |

Kaj | Iran | 8 | 27 | 63.9 | 10 | 0.396 | 0.153 | 0.042 | 0.004 |

Gamasyab | Iran | 24 | 30 | 84.2 | 19 | 0.287 | 0.107 | 0.043 | 0.003 |

Marbor | Iran | 13 | 22 | 92.6 | 17 | 0.433 | 0.116 | 0.050 | 0.006 |

**Table 2.**Calculated parameters to determine modified momentum thickness emerged by skin friction for different equivalent roughness in the Keulegan Equation.

${\theta}_{skin}=\phi \theta =\left(m*F{r}^{c1}*{\left(\frac{{d}_{50}}{d}\right)}^{c2}*{S}_{f}^{c3}*R{e}_{grain}^{c4}\right)*d$ | |||||||
---|---|---|---|---|---|---|---|

k_{s} | m | c1 | c2 | c3 | c4 | Pearson Correlation Coefficient | Equivalent Equation for Skin Friction Factor |

D_{50} | 0.013 | 0 | −0.8 | 0 | 0 | 0.89 | $0.017\ast {\left(\frac{{d}_{50}}{d}\right)}^{-0.47}$ |

1.5 D_{50} | 0.014 | 0 | −0.85 | 0 | 0 | 0.93 | $0.018\ast {\left(\frac{{d}_{50}}{d}\right)}^{-0.52}$ |

2 D_{50} | 0.015 | 0 | −0.9 | 0 | 0 | 0.94 | $0.019\ast {\left(\frac{{d}_{50}}{d}\right)}^{-0.57}$ |

Lamb–Shields Method (Equation (13)) | 0.225 | 0 | −0.33 | 0.33 | 0 | 0.80 | $0.3\ast {S}_{f}^{0.33}$ |

Station | IA | IC | IIA | IIB | IIC | IIIA | IIIB | IIIC | IVA | IVB | IVC | VA | VB |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

d_{50} (mm)
| 20 | 22 | 21 | 17 | 23 | 17 | 13 | 16 | 20 | 16 | 19 | 19 | 16 |

d (cm) | 21 | 23 | 21 | 21 | 17 | 23 | 21 | 17 | 29 | 23 | 11 | 33 | 25 |

U_{mean} (m/s)
| 1.26 | 1.22 | 0.94 | 1.11 | 0.58 | 1.16 | 1.08 | 0.94 | 0.94 | 0.76 | 0.46 | 0.88 | 0.71 |

u_{*} (m/s) | 0.16 | 0.17 | 0.14 | 0.162 | 0.092 | 0.098 | 0.086 | 0.074 | 0.095 | 0.094 | 0.036 | 0.14 | 0.075 |

Fr | 0.88 | 0.81 | 0.65 | 0.77 | 0.45 | 0.77 | 0.75 | 0.73 | 0.56 | 0.51 | 0.44 | 0.49 | 0.45 |

${\mathit{\delta}}^{\ast}$ (m) | 0.034 | 0.043 | 0.049 | 0.048 | 0.038 | 0.024 | 0.024 | 0.014 | 0.049 | 0.033 | 0.008 | 0.082 | 0.043 |

$\mathit{\theta}$ (m) | 0.019 | 0.023 | 0.025 | 0.027 | 0.018 | 0.016 | 0.017 | 0.01 | 0.028 | 0.022 | 0.006 | 0.043 | 0.028 |

$\mathit{f}$ (-) | 0.123 | 0.153 | 0.175 | 0.168 | 0.199 | 0.056 | 0.050 | 0.049 | 0.082 | 0.123 | 0.048 | 0.195 | 0.090 |

${\mathit{f}}^{\prime}$ (-) | 0.066 | 0.067 | 0.061 | 0.067 | 0.050 | 0.047 | 0.045 | 0.044 | 0.040 | 0.040 | 0.034 | 0.052 | 0.040 |

${\mathit{f}}^{\u2033}$ (-) | 0.057 | 0.086 | 0.114 | 0.101 | 0.148 | 0.009 | 0.005 | 0.005 | 0.042 | 0.083 | 0.014 | 0.143 | 0.049 |

${\mathit{f}}^{\prime}$_{Uniform} (-)
| 0.062 | 0.062 | 0.063 | 0.059 | 0.070 | 0.057 | 0.054 | 0.062 | 0.056 | 0.056 | 0.076 | 0.053 | 0.055 |

Section | I | II | III | IV | V | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Station | Left | Right | Left | Cent. | Right | Left | Cent. | Right | Left | Cent. | Right | Left | Cent. |

Calculated $\phi $ | 1.21 | 1.14 | 0.85 | 0.95 | 0.72 | 1.23 | 1.07 | 1.30 | 0.85 | 0.90 | 0.81 | 0.82 | 0.68 |

Measured $\phi $ | 1.18 | 1.11 | 0.83 | 0.93 | 0.71 | 1.22 | 1.07 | 1.30 | 0.78 | 0.77 | 0.87 | 0.81 | 0.69 |

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**MDPI and ACS Style**

Kazem, M.; Afzalimehr, H.; Nazari-Sharabian, M.; Karakouzian, M.
Determination of Skin Friction Factor in Gravel Bed Rivers: Considering the Effect of Large-Scale Topographic Forms in Non-Uniform Flows. *Hydrology* **2022**, *9*, 58.
https://doi.org/10.3390/hydrology9040058

**AMA Style**

Kazem M, Afzalimehr H, Nazari-Sharabian M, Karakouzian M.
Determination of Skin Friction Factor in Gravel Bed Rivers: Considering the Effect of Large-Scale Topographic Forms in Non-Uniform Flows. *Hydrology*. 2022; 9(4):58.
https://doi.org/10.3390/hydrology9040058

**Chicago/Turabian Style**

Kazem, Masoud, Hossein Afzalimehr, Mohammad Nazari-Sharabian, and Moses Karakouzian.
2022. "Determination of Skin Friction Factor in Gravel Bed Rivers: Considering the Effect of Large-Scale Topographic Forms in Non-Uniform Flows" *Hydrology* 9, no. 4: 58.
https://doi.org/10.3390/hydrology9040058