# Investigating the Influence of the Relative Roughness of the Riverbanks to the Riverbed on Equilibrium Channel Geometry in Alluvial Rivers: A Variational Approach

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Huang’s Variational Model

_{s}, representing the bedload transport discharge within a channel, and ζ, which denotes the channel’s cross-sectional shape factor, specifically its width-to-depth ratio.

_{m}, which corresponds to the maximum bedload transport discharge point, Q

_{smax}. This maximum discharge point is achieved under well-defined flow energy conditions, encompassing a flow discharge and an energy gradient, as well as the channel’s boundary composition, particularly related to sediment size. The implications of this relationship are significant, shedding light on the complex dynamics governing bedload transport in alluvial open channels.

_{s}) and the width-to-depth ratio (ζ) offers a comprehensive portrayal of the underlying dynamics. When the bedload transport discharge reaches its maximum value, denoted as Q

_{s}= Q

_{smax}, under specific conditions comprising flow discharge, energy slope and sediment size, a unique value for the width-to-depth ratio, ζ

_{m}, emerges. This specific state is known as maximum flow efficiency (MFE), a term introduced by Huang and Nanson [51]. In cases where the bedload transport discharge is less than the maximum, i.e., Q

_{s}< Q

_{smax}, a single value for bedload transport discharge can be achieved through two distinct width-to-depth ratios, one smaller and the other larger than ζ

_{m}. In the broader framework of physics, MFE signifies the state of stationary equilibrium in river channel flow. It embodies the concept of the most efficient utilization of available energy by the flow for transporting a given bedload, resulting in the maximum bedload discharge while expending the specified energy quantity. In contrast, the other states represent dynamic equilibrium states, wherein the flow has the flexibility to choose between two channel cross-sections. These cross-sections offer varying resistance levels, allowing for the expenditure of more than the minimal energy required. These concepts were further elucidated by Huang et al. (2004) and Nanson and Huang (2008) [4,48].

#### 2.2. Definition of the Relative Roughness of Riverbank to the Riverbed

^{3}), $\rho $ means the density of water (1000 kg/m

^{3}), and $g$ means the gravity acceleration (9.8 m/s

^{2}).

## 3. Mathematical Analysis of the Influence of Relative Roughness of Riverbanks to a Riverbed on River Channel Equilibrium Form

## 4. Effects of Relative Roughness of Riverbanks to a Riverbed on Equilibrium Hydraulic Geometry

#### 4.1. Equilibrium Hydraulic Geometry in the State of ${\tau}_{0}={\tau}_{c}$

^{3}, and $\rho $ at 1000 kg/m

^{3}. Subsequently, we varied the coefficient $\lambda $ across values of 0.1, 0.5, 1, and 2 to represent the changing relative roughness of riverbanks to a riverbed. Analysis of the results presented in Table 2 demonstrated that as $\lambda $ transitioned from lower to higher values, coefficients a, b, and c exhibited distinct variations within a narrow range. In more precise terms, with $\lambda $ values of 0.1, 0.5, 1, and 2, a assumed the respective values of 4.4642, 3.9633, 3.7919, and 3.6405, while b took on values of 1.4266, 1.6015, 1.8960, and 2.2524. The coefficient c registered values of 2.6736 × 10⁻⁵, 2.6267 × 10⁻⁵, 1.8698 × 10⁻⁵, and 1.3268 × 10⁻⁵. Notably, as $\lambda $ increased, a gradually decreased, while b steadily increased. This implies that in scenarios where a river operates under a lower threshold state, a smaller $\lambda $ value leads to adopting a narrower and deeper cross-sectional configuration for sediment transport.

#### 4.2. Averaged Equilibrium Hydraulic Geometry at the State of ${\tau}_{0}>{\tau}_{c}$

^{5}, 8.1486 × 10

^{5}, 7.4526 × 10

^{5}, and 5.8931 × 10

^{5}representing a decrease of up to 36.64%. Meanwhile, ${K}_{D}$ gradually increased, with values of 0.0121 × 10

^{7}, 0.0124 × 10

^{7}, 0.0139 × 10

^{5}, and 0.0155 × 10

^{7}, respectively. However, ${K}_{S}$ did not demonstrate a specific trend, with values ranging from 0.2172 to 0.2905, 0.2269, and 0.3152, respectively. This suggests that the coefficient ${K}_{S}$ remains relatively stable or experiences minimal change and can thus be considered a constant. In summary, as the relative roughness of riverbanks to a riverbed increases, the river tends to adjust itself by adopting a wider and shallower cross-section for sediment transport and vice versa. However, the relative roughness of riverbanks to a riverbed has little influence on ${K}_{S}$, indicating a weak correlation between the two.

## 5. Comparative Analysis between This Study and Previous Studies

#### 5.1. Hydraulic Geometric Relationships in the State of ${\tau}_{0}={\tau}_{c}$

#### 5.2. Hydraulic Geometric Relationships in the State of ${\tau}_{0}>{\tau}_{c}$

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Relationships between coefficients ${K}_{W}$, ${K}_{D}$, and ${K}_{S}$ in Equation (38) and $\lambda $.

**Figure 4.**Relationships between coefficients ${K}_{W}^{\prime}$, ${K}_{D}^{\prime}$, and ${K}_{V}^{\prime}$ in Equation (40) and $\lambda $.

$\mathit{\lambda}$ | ${\mathit{\zeta}}_{\mathit{m}}$ | $\frac{{\mathit{\zeta}}_{\mathit{m}}\mathbf{-}{\mathit{\zeta}}_{\mathit{m}\mathbf{0}}}{{\mathit{\zeta}}_{\mathit{m}\mathbf{0}}}$ (%) | ${\mathit{Q}}_{\mathit{s}\mathbf{max}}$ (m^{3}/s) | $\frac{{\mathit{Q}}_{\mathit{s}\mathbf{max}}\mathbf{-}{\mathit{Q}}_{\mathit{s}\mathbf{max}\mathbf{0}}}{{\mathit{Q}}_{\mathit{s}\mathbf{max}\mathbf{0}}}$ (%) |
---|---|---|---|---|

1 | 61.48 | 0 | 0.0741 | 0 |

0.1 | 89.2 | 45.09 | 0.0729 | −1.62 |

0.5 | 73.36 | 19.32 | 0.0734 | −0.94 |

2 | 29.8 | −51.53 | 0.0763 | 2.97 |

$\mathit{\lambda}$ | ${\mathit{\zeta}}_{\mathit{m}}$ | Values of a, b, and c |
---|---|---|

0.1 | 3.12 | $\begin{array}{l}a=4.4622\\ b=1.4266\\ c=2.6736\times {10}^{-5}\end{array}$ |

0.5 | 2.47 | $\begin{array}{l}a=3.9633\\ b=1.6015\\ c=2.6267\times {10}^{-5}\end{array}$ |

1 | 2 | $\begin{array}{l}a=3.7919\\ b=1.8960\\ c=1.8698\times {10}^{-5}\end{array}$ |

2 | 1.62 | $\begin{array}{l}a=3.6405\\ b=2.2524\\ c=1.3268\times {10}^{-5}\end{array}$ |

$\mathit{\lambda}$ | ${\mathit{\zeta}}_{\mathit{m}\mathbf{max}}$ | $\mathbf{Varying}\mathbf{Range}\mathbf{of}{\mathit{\zeta}}_{\mathit{m}}$ |
---|---|---|

0.1 | 1000 | 4~1000 |

0.5 | 833 | 3~833 |

1 | 625 | 3~625 |

2 | 215 | 2~215 |

$\mathit{\lambda}$ | ${\mathit{\zeta}}_{\mathit{m}}$ | Averaged Equilibrium Channel Relationships |
---|---|---|

0.1 | $4\le {\zeta}_{m}\le 1000$ | $\begin{array}{l}W=9.2753\ast {10}^{5}{d}^{1.0769}{(nQ)}^{0.3916}{Q}_{s}^{0.1515}=9.2753\ast {10}^{-5}{d}^{1.0769}{(n)}^{0.3916}{\left(\frac{{Q}_{s}}{Q}\right)}^{0.1515}{Q}^{0.5431}\\ D=0.0121\ast {10}^{7}{d}^{1.6946}{(nQ)}^{0.6162}{Q}_{s}^{-0.3351}=0.0121\ast {10}^{7}{d}^{1.6946}{(n)}^{0.6162}{\left(\frac{{Q}_{s}}{Q}\right)}^{-0.3351}{Q}^{0.2811}\\ S=0.2172{d}^{0.2148}{(nQ)}^{-0.8310}{Q}_{s}^{0.8005}=0.2172{d}^{0.2148}{n}^{-0.8310}{\left(\frac{{Q}_{s}}{Q}\right)}^{0.8005}{Q}^{-0.0305}\end{array}$ |

0.5 | $3\le {\zeta}_{m}\le 833$ | $\begin{array}{l}W=8.1486\ast {10}^{5}{d}^{1.0772}{(nQ)}^{0.3916}{Q}_{s}^{0.1515}=8.1486\ast {10}^{5}{d}^{1.0772}{n}^{0.3916}{\left(\frac{{Q}_{s}}{Q}\right)}^{0.1515}{Q}^{0.5431}\\ D=0.0124\ast {10}^{7}{d}^{1.6941}{(nQ)}^{0.6160}{Q}_{s}^{-0.3347}=0.0124\ast {10}^{7}{d}^{1.6941}{(n)}^{0.6160}{\left(\frac{{Q}_{s}}{Q}\right)}^{-0.3347}{Q}^{0.2813}\\ S=0.2905{d}^{0.2209}{(nQ)}^{-0.8288}{Q}_{s}^{0.7957}=0.2905{d}^{0.2209}{n}^{-0.8288}{\left(\frac{{Q}_{s}}{Q}\right)}^{0.7957}{Q}^{-0.0331}\end{array}$ |

1 | $3\le {\zeta}_{m}\le 265$ | $\begin{array}{l}W=7.4526\ast {10}^{5}{d}^{1.0773}{(nQ)}^{0.3918}{Q}_{s}^{0.1512}=7.4526\ast {10}^{5}{d}^{1.0773}{(n)}^{0.3918}{\left(\frac{{Q}_{s}}{Q}\right)}^{0.1512}{Q}^{0.5430}\\ D=0.0139\ast {10}^{7}{d}^{1.6967}{(nQ)}^{0.6170}{Q}_{s}^{-0.3368}=0.0139\ast {10}^{7}{d}^{1.6967}{(n)}^{0.6170}{\left(\frac{{Q}_{s}}{Q}\right)}^{-0.3368}{Q}^{0.2802}\\ S=0.2269{d}^{0.2159}{(nQ)}^{-0.8306}{Q}_{s}^{0.7996}=0.2269{d}^{0.2159}{(n)}^{-0.8306}{\left(\frac{{Q}_{s}}{Q}\right)}^{0.7996}{Q}^{-0.0310}\end{array}$ |

2 | $2\le {\zeta}_{m}\le 215$ | $\begin{array}{l}W=5.8931\ast {10}^{5}{d}^{1.1158}{(nQ)}^{0.4058}{Q}_{s}^{0.1207}=5.8931\ast {10}^{5}{d}^{1.1158}{n}^{0.4058}{\left(\frac{{Q}_{s}}{Q}\right)}^{0.1207}{Q}^{0.5265}\\ D=0.0155\ast {10}^{7}{d}^{1.6684}{(nQ)}^{0.6067}{Q}_{s}^{-0.3145}=0.0155\ast {10}^{7}{d}^{1.6684}{n}^{0.6067}{\left(\frac{{Q}_{s}}{Q}\right)}^{-0.3145}{Q}^{0.2922}\\ S=0.3152{d}^{0.2577}{(nQ)}^{-0.8154}{Q}_{s}^{0.7667}=0.3152{d}^{0.2577}{(n)}^{-0.8154}{\left(\frac{{Q}_{s}}{Q}\right)}^{0.7667}{Q}^{-0.0487}\end{array}$ |

$\mathit{\lambda}$ | ${\mathit{\zeta}}_{\mathit{m}}$ | Averaged Hydraulic Geometry Relationships |
---|---|---|

0.1 | $4\le {\zeta}_{m}\le 1000$ | $\begin{array}{l}W=199.0{d}^{-0.0407}{(nQ)}^{0.5489}{S}^{0.1893}\\ D=0.068{d}^{0.0899}{(nQ)}^{0.2683}{S}^{-0.4186}\\ V=0.0739{d}^{-0.0492}{n}^{-0.8172}{Q}^{0.1828}{S}^{0.2293}\end{array}$ |

0.5 | $3\le {\zeta}_{m}\le 833$ | $\begin{array}{l}W=165.2629{d}^{-0.0421}{(nQ)}^{0.5494}{S}^{0.1904}\\ D=0.0754{d}^{0.0929}{(nQ)}^{0.2674}{S}^{-0.4206}\\ V=0.0803{d}^{-0.0508}{n}^{-0.8168}{Q}^{0.1832}{S}^{0.2302}\end{array}$ |

1 | $3\le {\zeta}_{m}\le 625$ | $\begin{array}{l}W=158.4510{d}^{-0.0408}{(nQ)}^{0.5489}{S}^{0.1891}\\ D=0.0784{d}^{0.0909}{(nQ)}^{0.2671}{S}^{-0.4212}\\ V=0.0805{d}^{-0.0501}{n}^{-0.8160}{Q}^{0.1840}{S}^{0.2321}\end{array}$ |

2 | $2\le {\zeta}_{m}\le 215$ | $\begin{array}{l}W=78.4891{d}^{-0.0406}{(nQ)}^{0.5188}{S}^{0.1574}\\ D=0.1279{d}^{0.1057}{(nQ)}^{0.2722}{S}^{-0.4102}\\ V=0.0996{d}^{-0.0651}{n}^{-0.7910}{Q}^{0.2090}{S}^{0.2528}\end{array}$ |

Channel Geometry Factor | This Study | Fan and Huang (2020) [51] | Threshold Theory (Lane, 1952) [58] |
---|---|---|---|

Width ($W$) | $W\propto {Q}^{0.46}$ | $W\propto {Q}^{0.46}$ | $W\propto {Q}^{0.46}$ |

Depth ($D$) | $D\propto {Q}^{0.46}$ | $D\propto {Q}^{0.46}$ | $D\propto {Q}^{0.46}$ |

Slope ($S$) | $S\propto {Q}^{-0.46}$ | $S\propto {Q}^{-0.46}$ | $S\propto {Q}^{-0.46}$ |

Width/depth ratio ($W/D$) | 1.62–3.12 | 2–4 | 7.05–8.61 |

Hydraulic Geometry Factor | This Study | Fan and Huang (2020) [51] | Hydraulic Geometry Model (Rhodes, 1987) [59] | Huang and Warner (1995) [60] |
---|---|---|---|---|

Width ($W$) | $W\propto {Q}^{0.5188~0.5494}$ | $W\propto {Q}^{0.5431~0.5489}$ | $W\propto {Q}^{0.3~0.6}$ | $W\propto {Q}^{0.5}$ |

Depth ($D$) | $D\propto {Q}^{0.2671~0.2722}$ | $D\propto {Q}^{0.2655~0.2692}$ | $D\propto {Q}^{0.2~0.5}$ | $D\propto {Q}^{0.3}$ |

Velocity ($V$) | $V\propto {Q}^{0.1828~0.2090}$ | $V\propto {Q}^{0.1840~0.1876}$ | $V\propto {Q}^{0.0~0.3}$ | $V\propto {Q}^{0.2}$ |

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**MDPI and ACS Style**

Fan, J.; Luo, Q.; Bai, Y.; Liu, X.; Li, R.
Investigating the Influence of the Relative Roughness of the Riverbanks to the Riverbed on Equilibrium Channel Geometry in Alluvial Rivers: A Variational Approach. *Water* **2023**, *15*, 4029.
https://doi.org/10.3390/w15224029

**AMA Style**

Fan J, Luo Q, Bai Y, Liu X, Li R.
Investigating the Influence of the Relative Roughness of the Riverbanks to the Riverbed on Equilibrium Channel Geometry in Alluvial Rivers: A Variational Approach. *Water*. 2023; 15(22):4029.
https://doi.org/10.3390/w15224029

**Chicago/Turabian Style**

Fan, Jinsheng, Qiushi Luo, Yuchuan Bai, Xiaofang Liu, and Renzhi Li.
2023. "Investigating the Influence of the Relative Roughness of the Riverbanks to the Riverbed on Equilibrium Channel Geometry in Alluvial Rivers: A Variational Approach" *Water* 15, no. 22: 4029.
https://doi.org/10.3390/w15224029