# River Channel Forms in Relation to Bank Steepness: A Theoretical Investigation Using a Variational Analytical Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Flow Relations in an Open Channel with Bedload Transport

^{3}and 1000 kg/m

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

^{2}), and so ${\gamma}_{s}={\rho}_{s}g$ and $\gamma =\rho g$.

## 3. Variational Analysis of the Effect of Channel-Form Adjustment on Bedload Transport

^{3}/s, channel slope or energy gradient $S$ is 2/10,000, sediment size $d$ is 0.3 mm, and $n$ is 0.03, the variations of ${Q}_{s}$ with a change in width–depth ratio $\zeta $ from 10 to 1000 are computed according to Equations (10) and (11) for bank slope $\theta $ to take each of the specific values of 0°, 30°, 45° and 60°. Figure 2 and Table 1 present the computed results. It can be seen from Figure 2 that for each of the specific values of $\theta $, an increase in channel width–depth ratio $\zeta $ from 10 to 1000 makes ${Q}_{s}$ increase gradually at the early stage, then reach a maximum and afterwards decline gradually. Importantly, it is seen clearly that in the situation of ${Q}_{s}<{Q}_{smaxm}$, a given ${Q}_{s}$${Q}_{s}$ can be satisfied with two values of channel width–depth ratio $\zeta $ and only when ${Q}_{s}$ equals the maximum, or ${Q}_{s}={Q}_{smaxm}$, does channel width–depth ratio $\zeta $ take a unique value of ${\zeta}_{m}$.

^{3}/s, respectively, while the corresponding optimal values of the width–depth ratio, or ${\zeta}_{m}$, become larger and larger, with the corresponding values of 101, 113, 128.7 and 168.5, respectively. When $\theta $= 30° is taken as a reference level, it can be found from Table 1 that with $\theta $ taking respective values of 0°, 45° and 60°, ${\zeta}_{m}$ varies in the wide range of from −10.62% to 49.12%, while ${Q}_{smax}$ varies in the narrow range of from –1.23% to 0.29%. This demonstrates clearly that bank slope $\theta $ exerts a much more significant influence on optimal width–depth ratio of river channels, or ${\zeta}_{m}$, than on the maximum sediment (bedload) transport discharge ${Q}_{smax}$.

## 4. Effects of Riverbank Steepness on Equilibrium Channel Geometry

#### 4.1. Equilibrium Channel Relations at the Lower Threshold

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

^{3}, the values of coefficients $a$, $b$ and $c$ in Equation (21) are then computed according to Equation (22) when bank angle $\theta $ takes respective values of 0°, 30°, 45° and 60°. Table 2 presented the computed results and it can be seen that with $\theta $ taking different values, the coefficients in the relations change in complex forms. Specifically, coefficient $a$ increases significantly with an increase of $\theta $, while coefficients $b$ and $c$ vary in complex forms within very small ranges and maintain the relationship of $b\cdot c\approx 4.6\cdot {10}^{-5}$. This demonstrates clearly that when flow in a river channel reaches the lower threshold, a decrease in riverbank steepness can result a wider and shallower channel cross-section, while channel depth and slope remain almost unchanged.

#### 4.2. Averaged Equilibrium Channel Relations

^{5}, 8.1238 × 10

^{5}, 9.6346 × 10

^{5}and 11.387 × 10

^{5}, respectively, with an increase of up to 52.6%. Nevertheless, ${K}_{D}$ varies in a gradually decreasing form, taking values of 1.3887 × 10

^{5}, 1.2739 × 10

^{5}, 1.1677 × 10

^{5}and 1.0892 × 10

^{5}, respectively, with a 21.6% decrease in down. However, ${K}_{S}$ varies not in a consistent form, increasing at the early stages and finally decreasing by taking respective values of 0.2268 to 0.2744, 0.3103 and 0.2471. The difference between the maximum and minimum values of ${K}_{S}$ is very small, with a value of 0.0835, and as such, ${K}_{S}$ can be regarded as a constant of about 0.27. These results demonstrate clearly that an increase in riverbank angle, that is a decrease in riverbank steepness, can result in a wider and shallower channel cross-section and vice versa. Nevertheless, such a change in bank angle $\theta $ exerts only insignificant influences on channel slope measured by the variation of coefficient ${K}_{S}$.

## 5. Comparison of Theoretical Results with Previous Studies

#### 5.1. Equilibrium Channel Relations at the Lower Threshold

#### 5.2. Averaged Equilibrium Channel Relations

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Relationships of bedload transport rate ${Q}_{s}$ against channel width/depth ratio $\zeta $ under different values of bank angle $\theta $.

**Figure 3.**Relationships between coefficients ${K}_{W}$, ${K}_{D}$ and ${K}_{S}$ in equilibrium channel relations in Equation (32) and riverbank angle $\theta $: (

**a**) Relationship between coefficients ${K}_{W}$ and $\theta $; (

**b**) Relationship between coefficients ${K}_{D}$ and $\theta $; and (

**c**) Relationship between coefficients ${K}_{S}$ and $\theta $.

**Figure 4.**Relationships between coefficients ${{K}^{\prime}}_{W}$, ${{K}^{\prime}}_{D}$ and ${{K}^{\prime}}_{V}$ in the averaged hydraulic geometry relations in Equation (34) and riverbank angle $\theta $: (

**a**) Relationship between coefficients ${{K}^{\prime}}_{W}$ and $\theta $; (

**b**) Relationship between coefficients ${{K}^{\prime}}_{D}$ and $\theta $; and (

**c**) Relationship between coefficients ${{K}^{\prime}}_{V}$ and $\theta $.

**Table 1.**Values of maximum sediment transport rate ${Q}_{smax}$ and optimal channel width/depth ratio ${\zeta}_{m}$ under different values of bank angle $\theta $.

$\mathit{\theta}$ | ${\mathit{\zeta}}_{\mathit{m}}$ | $\frac{{\mathit{\zeta}}_{\mathit{m}}-{\mathit{\zeta}}_{\mathit{m}0}}{{\mathit{\zeta}}_{\mathit{m}0}}\left(\mathbf{\%}\right)$ | ${\mathit{Q}}_{\mathit{s}\mathit{max}}$ (m^{3}/s) | $\frac{{\mathit{Q}}_{\mathit{s}\mathit{m}\mathit{a}\mathit{x}}-{\mathit{Q}}_{\mathit{s}\mathit{m}\mathit{a}\mathit{x}\mathbf{0}}}{{\mathit{Q}}_{\mathit{s}\mathit{m}\mathit{a}\mathit{x}\mathbf{0}}}\left(\mathbf{\%}\right)$ |
---|---|---|---|---|

0° | 101 | −10.62 | 0.08733 | 0.29 |

30° | 113 | 0 | 0.08708 | 0 |

45° | 128.7 | 13.89 | 0.08670 | −0.44 |

60° | 168.5 | 49.12 | 0.08601 | −1.23 |

**Table 2.**Equilibrium channel relations at the lower threshold under different values of bank angle $\theta $.

$\mathit{\theta}$ | ${\mathit{\zeta}}_{\mathit{m}}$ | Values of the Coefficients in Equation (22) |
---|---|---|

0° | 2 | $\begin{array}{l}a=3.5613\\ b=1.7807\\ c=2.6131\times {10}^{-5}\end{array}$ |

30° | 2.31 | $\begin{array}{l}a=4.3948\\ b=1.9025\\ c=2.4453\times {10}^{-5}\end{array}$ |

45° | 2.83 | $\begin{array}{l}a=5.2499\\ b=1.8551\\ c=2.5071\times {10}^{-5}\end{array}$ |

60° | 4 | $\begin{array}{l}a=6.7210\\ b=1.6803\\ c=2.7693\times {10}^{-5}\end{array}$ |

**Table 3.**Values and potential varying ranges of optimal channel width/depth ratio ${\zeta}_{m}$ under different values of bank angle $\theta $.

$\mathit{\theta}$ | ${\mathit{\zeta}}_{\mathit{m}}$ | Potential Varying Range of ${\mathit{\zeta}}_{\mathit{m}}$ |
---|---|---|

0° | 625 | 3–625 |

30° | 749.8 | 3–749 |

45° | 878.2 | 3–878 |

60° | 1000 | 5–1000 |

$\mathit{\theta}$ | ${\mathit{\zeta}}_{\mathit{m}}$ | Averaged Equilibrium Channel Relations |
---|---|---|

0° | $3\le {\zeta}_{m}\le 625$ | $\begin{array}{l}{W}_{m}=7.4642\ast {10}^{5}{d}^{1.0773}{(nQ)}^{0.3918}{Q}_{s}^{0.1512}=7.4642\ast {10}^{5}{d}^{1.0773}{n}^{0.3918}{\left(\frac{{Q}_{s}}{Q}\right)}^{0.1512}{Q}^{0.5430}\\ {D}_{m}=1.3887\ast {10}^{5}{d}^{1.6967}{(nQ)}^{0.6170}{Q}_{s}^{-0.3386}=1.3887\ast {10}^{5}{d}^{1.6967}{n}^{0.6170}{\left(\frac{{Q}_{s}}{Q}\right)}^{-0.3368}{Q}^{0.2802}\\ {S}_{m}=0.2268{d}^{0.2159}{(nQ)}^{-0.8306}{Q}_{s}^{0.7996}=0.2268{d}^{0.2159}{n}^{-0.8306}{\left(\frac{{Q}_{s}}{Q}\right)}^{0.7996}{Q}^{-0.0310}\end{array}$ |

30° | $3\le {\zeta}_{m}\le 749$ | $\begin{array}{l}{W}_{m}=8.1239\ast {10}^{5}{d}^{1.0804}{(nQ)}^{0.3928}{Q}_{s}^{0.1489}=8.1239\ast {10}^{5}{d}^{1.0804}{n}^{0.3928}{\left(\frac{{Q}_{s}}{Q}\right)}^{0.1489}{Q}^{0.5417}\\ {D}_{m}=1.2739\ast {10}^{5}{d}^{1.6961}{(nQ)}^{0.6168}{Q}_{s}^{-0.3363}=1.2739\ast {10}^{5}{d}^{1.6961}{n}^{0.6168}{\left(\frac{{Q}_{s}}{Q}\right)}^{-0.3363}{Q}^{0.2805}\\ {S}_{m}=0.2744{d}^{0.2210}{(nQ)}^{-0.8288}{Q}_{s}^{0.7957}=0.2744{d}^{0.2210}{n}^{-0.8288}{\left(\frac{{Q}_{s}}{Q}\right)}^{0.7957}{Q}^{-0.0331}\end{array}$ |

45° | $3\le {\zeta}_{m}\le 878$ | $\begin{array}{l}{W}_{m}=9.6346\ast {10}^{5}{d}^{1.0905}{(nQ)}^{0.3966}{Q}_{s}^{0.1407}=9.6346\ast {10}^{5}{d}^{1.0905}{n}^{0.3966}{\left(\frac{{Q}_{s}}{Q}\right)}^{0.1407}{Q}^{0.5373}\\ {D}_{m}=1.1677\ast {10}^{5}{d}^{1.6838}{(nQ)}^{0.6124}{Q}_{s}^{-0.3269}=1.1677\ast {10}^{5}{d}^{1.6838}{n}^{0.6124}{\left(\frac{{Q}_{s}}{Q}\right)}^{-0.3269}{Q}^{0.2855}\\ {S}_{m}=0.3103{d}^{0.2469}{(nQ)}^{-0.8194}{Q}_{s}^{0.7754}=0.3103{d}^{0.2469}{n}^{-0.8194}{\left(\frac{{Q}_{s}}{Q}\right)}^{0.7754}{Q}^{-0.044}\end{array}$ |

60° | $5\le {\zeta}_{m}\le 1000$ | $\begin{array}{l}{W}_{m}=11.387\ast {10}^{5}{d}^{1.0918}{(nQ)}^{0.3960}{Q}_{s}^{0.1420}=11.387\ast {10}^{5}{d}^{1.0918}{n}^{0.3960}{\left(\frac{{Q}_{s}}{Q}\right)}^{0.1420}{Q}^{0.5380}\\ {D}_{m}=1.0892\ast {10}^{5}{d}^{1.6848}{(nQ)}^{0.6140}{Q}_{s}^{-0.3286}=1.0892\ast {10}^{5}{d}^{1.6848}{n}^{0.6140}{\left(\frac{{Q}_{s}}{Q}\right)}^{-0.3286}{Q}^{0.2854}\\ {S}_{m}=0.2471{d}^{0.2446}{(nQ)}^{-0.8254}{Q}_{s}^{0.7884}=0.2471{d}^{0.2446}{n}^{-0.8254}{\left(\frac{{Q}_{s}}{Q}\right)}^{0.7884}{Q}^{-0.037}\end{array}$ |

$\mathit{\theta}$ | ${\mathit{\zeta}}_{\mathit{m}}$ | Averaged Hydraulic Geometry Relations |
---|---|---|

0° | $3\le {\zeta}_{m}\le 625$ | $\begin{array}{l}{W}_{m}=152.4097{d}^{-0.0408}{(nQ)}^{0.5489}{S}^{0.1891}\\ {D}_{m}=0.0805{d}^{0.0909}{(nQ)}^{0.2671}{S}^{-0.4212}\\ {V}_{m}=0.0815{d}^{-0.0501}{n}^{-0.8160}{Q}^{0.1840}{S}^{0.2321}\end{array}$ |

30° | $3\le {\zeta}_{m}\le 749$ | $\begin{array}{l}{W}_{m}=162.449{d}^{-0.0414}{(nQ)}^{0.5479}{S}^{0.1871}\\ {D}_{m}=0.0776{d}^{0.0934}{(nQ)}^{0.2655}{S}^{-0.4226}\\ {V}_{m}=0.0798{d}^{-0.0520}{n}^{-0.8134}{Q}^{0.1866}{S}^{0.2395}\end{array}$ |

45° | $3\le {\zeta}_{m}\le 878$ | $\begin{array}{l}{W}_{m}=171.5476{d}^{-0.0448}{(nQ)}^{0.5453}{S}^{0.1815}\\ {D}_{m}=0.0736{d}^{0.1041}{(nQ)}^{0.2669}{S}^{-0.4216}\\ {V}_{m}=0.0792{d}^{-0.0593}{n}^{-0.8122}{Q}^{0.1878}{S}^{0.2401}\end{array}$ |

60° | $5\le {\zeta}_{m}\le 1000$ | $\begin{array}{l}{W}_{m}=205.509{d}^{-0.0441}{(nQ)}^{0.5431}{S}^{0.1768}\\ {D}_{m}=0.0698{d}^{0.1019}{(nQ)}^{0.2692}{S}^{-0.4168}\\ {V}_{m}=0.0697{d}^{-0.0578}{n}^{-0.8123}{Q}^{0.1877}{S}^{0.2400}\end{array}$ |

Channel Geometry Factors | This Study | Threshold Theory (Lane, 1952) [66] |
---|---|---|

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

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

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

Width/depth ratio ($W$) | 2–4 | 7.05–8.61* |

Hydraulic Geometry Factors | This Study | Hydraulic Geometry Model (Rhodes, 1987) [68] | Huang and Warner (1995) [69] |
---|---|---|---|

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

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

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

**Table 8.**Comparison of riverbank steepness effects on river channel forms between the theoretical results of this study and the semi-empirical results by Huang and Nanson (1998) [13].

Theoretical Results of this Study | Results of Huang and Nanson (1998) [13] | ||||
---|---|---|---|---|---|

Bank angle $\theta $ | Channel width change (%) | Channel depth change (%) | Bank type | Channel width change (%) | Channel depth change (%) |

0° | 0 | 0 | Cohesive sand or gravels | 0 | 0 |

30° | 8.8 | −8.3 | |||

45° | 29.1 | −15.9 | |||

60° | 52.6 | −21.6 | Noncohesive sand | 62.2 | −26.7 |

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**MDPI and ACS Style**

Fan, J.; Huang, H.; Yu, G.; Su, T.
River Channel Forms in Relation to Bank Steepness: A Theoretical Investigation Using a Variational Analytical Method. *Water* **2020**, *12*, 1250.
https://doi.org/10.3390/w12051250

**AMA Style**

Fan J, Huang H, Yu G, Su T.
River Channel Forms in Relation to Bank Steepness: A Theoretical Investigation Using a Variational Analytical Method. *Water*. 2020; 12(5):1250.
https://doi.org/10.3390/w12051250

**Chicago/Turabian Style**

Fan, Jinsheng, Heqing Huang, Guoan Yu, and Teng Su.
2020. "River Channel Forms in Relation to Bank Steepness: A Theoretical Investigation Using a Variational Analytical Method" *Water* 12, no. 5: 1250.
https://doi.org/10.3390/w12051250