# Modifying Elder’s Longitudinal Dispersion Coefficient for Two-Dimensional Solute Mixing Analysis in Open-Channel Bends

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Velocity Profile in Bends

#### 2.2. Derivation of Longitudinal Dispersion Coefficient

#### 2.3. Experiments in Two Open Channels

^{3}/s [31]. Tracer tests were performed under the same flow conditions to elucidate the dispersion characteristics. A salt solution (NaCl) was used as the tracer.

^{3}/s, and the average water depth was 0.487 m in the channel. The average cross-sectional velocity was measured to be 0.49–0.64 m/s in the A315 reach and 0.39–0.47 m/s in the A317 reach. A fluorescent substance (rhodamine WT) was used as the tracer in the mixing experiment. The tracer was released at the injection point (shown in Figure 4) in each reach.

## 3. Results and Discussion

#### 3.1. Flow Characteristics in Meandering Channels

#### 3.2. Longitudinal Dispersion Coefficient in Meandering Channels

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

$1.\text{}{y}^{\prime}=\frac{y}{d}\phantom{\rule{0ex}{0ex}}2.\text{}{u}^{\prime}=\frac{{u}_{*}}{\kappa}\left(1+\mathrm{ln}{y}^{\prime}\right)+asi{n}^{2}\pi {y}^{\prime}\phantom{\rule{0ex}{0ex}}3.\text{}\mathsf{\epsilon}:\mathrm{constant}$ |

${{\displaystyle \int}}_{0}^{y}{u}^{\prime}dy={{\displaystyle \int}}_{0}^{y}\left\{\frac{{u}_{*}}{\kappa}\left(1+\mathrm{ln}{y}^{\prime}\right)+asi{n}^{2}\pi {y}^{\prime}\right\}dy\dots \left(1\right)$ |

$(1)=\mathrm{d}{{\displaystyle \int}}_{0}^{{y}^{\prime}}\left\{\frac{{u}_{*}}{\kappa}\left(1+\mathrm{ln}{y}^{\prime}\right)+asi{n}^{2}\pi {y}^{\prime}\right\}d{y}^{\prime}=\mathrm{d}{{\displaystyle \int}}_{0}^{{y}^{\prime}}\left\{\frac{{u}_{*}}{\kappa}\left(1+\mathrm{ln}{y}^{\prime}\right)+\frac{a}{2}\left(1-cos2\pi {y}^{\prime}\right)\right\}d{y}^{\prime}\phantom{\rule{0ex}{0ex}}=\mathrm{d}\left\{\frac{{u}_{*}}{\kappa}{{y}^{\prime}\mathrm{ln}{y}^{\prime}|}_{0}^{{y}^{\prime}}+\frac{a}{2}{\left({y}^{\prime}-\frac{1}{2\pi}sin2\pi \right)|}_{0}^{{y}^{\prime}}\right\}=\mathrm{d}\left\{\frac{{u}_{*}}{\kappa}{y}^{\prime}\mathrm{ln}{y}^{\prime}+\frac{a}{2}\left({y}^{\prime}-\frac{1}{2\pi}sin2\pi \right)\right\}\phantom{\rule{0ex}{0ex}}{{\displaystyle \int}}_{0}^{y}\frac{1}{\epsilon}{{\displaystyle \int}}_{0}^{y}{u}^{\prime}dydy=\frac{d}{\epsilon}{{\displaystyle \int}}_{0}^{y}\left\{\frac{{u}_{*}}{\kappa}{y}^{\prime}\mathrm{ln}{y}^{\prime}+\frac{a}{2}\left({y}^{\prime}-\frac{1}{2\pi}sin2\pi \right)\right\}dy=\frac{{d}^{2}}{\epsilon}{{\displaystyle \int}}_{0}^{{y}^{\prime}}\left\{\frac{{u}_{*}}{\kappa}{y}^{\prime}\mathrm{ln}{y}^{\prime}+\frac{a}{2}\left({y}^{\prime}-\frac{1}{2\pi}sin2\pi \right)\right\}d{y}^{\prime}\dots \left(2\right)$ |

$\left(2\right)=\frac{{d}^{2}}{\epsilon}\left\{\frac{{u}_{*}}{\kappa}{\left(\frac{1}{2}{{y}^{\prime}}^{2}\mathrm{ln}{y}^{\prime}-\frac{1}{4}{{y}^{\prime}}^{2}\right)|}_{0}^{{y}^{\prime}}+\frac{a}{2}{\left(\frac{1}{2}{{y}^{\prime}}^{2}+\frac{1}{4{\pi}^{2}}cos2\pi {y}^{\prime}\right)|}_{0}^{{y}^{\prime}}\right\}\phantom{\rule{0ex}{0ex}}=\frac{{d}^{2}}{\epsilon}\left\{\frac{{u}_{*}}{\kappa}\left(\frac{1}{2}{{y}^{\prime}}^{2}\mathrm{ln}{y}^{\prime}-\frac{1}{4}{{y}^{\prime}}^{2}\right)+a\left(\frac{1}{4}{{y}^{\prime}}^{2}+\frac{1}{8{\pi}^{2}}\left(cos2\pi {y}^{\prime}-1\right)\right)\right\}\phantom{\rule{0ex}{0ex}}-\frac{1}{d}{{\displaystyle \int}}_{0}^{d}{u}^{\prime}{{\displaystyle \int}}_{0}^{y}\frac{1}{\epsilon}{{\displaystyle \int}}_{0}^{y}{u}^{\prime}dydydy\phantom{\rule{0ex}{0ex}}=-\frac{1}{d}\times \frac{{d}^{2}}{\epsilon}{{\displaystyle \int}}_{0}^{d}\left\{\frac{{u}_{*}}{\kappa}\left(1+\mathrm{ln}{y}^{\prime}\right)+\frac{a}{2}\left(1-cos2\pi {y}^{\prime}\right)\right\}\left\{\begin{array}{c}\frac{{u}_{*}}{\kappa}\left(\frac{1}{2}{{y}^{\prime}}^{2}\mathrm{ln}{y}^{\prime}-\frac{1}{4}{{y}^{\prime}}^{2}\right)+\\ a\left(\frac{1}{4}{{y}^{\prime}}^{2}+\frac{1}{8{\pi}^{2}}\left(cos2\pi {y}^{\prime}-1\right)\right)\end{array}\right\}dy\dots \left(3\right)$ |

$\left(3\right)=-\frac{{d}^{2}}{\epsilon}{{\displaystyle \int}}_{0}^{1}\left\{\frac{{u}_{*}}{\kappa}\left(1+\mathrm{ln}{y}^{\prime}\right)+\frac{a}{2}\left(1-cos2\pi {y}^{\prime}\right)\right\}\left\{\begin{array}{c}\frac{{u}_{*}}{\kappa}\left(\frac{1}{2}{{y}^{\prime}}^{2}\mathrm{ln}{y}^{\prime}-\frac{1}{4}{{y}^{\prime}}^{2}\right)+\\ a\left(\frac{1}{4}{{y}^{\prime}}^{2}+\frac{1}{8{\pi}^{2}}\left(cos2\pi {y}^{\prime}-1\right)\right)\end{array}\right\}d{y}^{\prime}\phantom{\rule{0ex}{0ex}}=-\frac{{d}^{2}}{\epsilon}\{{{\displaystyle \int}}_{0}^{1}{\left(\frac{{u}_{*}}{\kappa}\right)}^{2}\left(1+\mathrm{ln}{y}^{\prime}\right)\left(\frac{1}{2}{{y}^{\prime}}^{2}\mathrm{ln}{y}^{\prime}-\frac{1}{4}{{y}^{\prime}}^{2}\right)d{y}^{\prime}\phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int}}_{0}^{1}\frac{a{u}_{*}}{\kappa}\left(1+\mathrm{ln}{y}^{\prime}\right)\left(\frac{1}{4}{{y}^{\prime}}^{2}+\frac{1}{8{\pi}^{2}}\left(cos2\pi {y}^{\prime}-1\right)\right)d{y}^{\prime}\phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int}}_{0}^{1}\frac{a{u}_{*}}{\kappa}\left(\frac{1-cos2\pi {y}^{\prime}}{2}\right)\left(\frac{1}{2}{{y}^{\prime}}^{2}\mathrm{ln}{y}^{\prime}-\frac{1}{4}{{y}^{\prime}}^{2}\right)d{y}^{\prime}\phantom{\rule{0ex}{0ex}}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{{\displaystyle \int}}_{0}^{1}{a}^{2}\left(\frac{1-cos2\pi {y}^{\prime}}{2}\right)\left(\frac{1}{4}{{y}^{\prime}}^{2}+\frac{1}{8{\pi}^{2}}\left(cos2\pi {y}^{\prime}-1\right)\right)d{y}^{\prime}\}\dots \left(4\right)$ |

$\mathrm{A}={{\displaystyle \int}}_{0}^{1}{\left(\frac{{u}_{*}}{\kappa}\right)}^{2}\left(1+\mathrm{ln}{y}^{\prime}\right)\left(\frac{1}{2}{{y}^{\prime}}^{2}\mathrm{ln}{y}^{\prime}-\frac{1}{4}{{y}^{\prime}}^{2}\right)d{y}^{\prime}\phantom{\rule{0ex}{0ex}}\mathrm{B}={{\displaystyle \int}}_{0}^{1}\frac{a{u}_{*}}{\kappa}\left(1+\mathrm{ln}{y}^{\prime}\right)\left(\frac{1}{4}{{y}^{\prime}}^{2}+\frac{1}{8{\pi}^{2}}\left(cos2\pi {y}^{\prime}-1\right)\right)d{y}^{\prime}\phantom{\rule{0ex}{0ex}}\mathrm{C}={{\displaystyle \int}}_{0}^{1}\frac{a{u}_{*}}{\kappa}\left(\frac{1-cos2\pi {y}^{\prime}}{2}\right)\left(\frac{1}{2}{{y}^{\prime}}^{2}\mathrm{ln}{y}^{\prime}-\frac{1}{4}{{y}^{\prime}}^{2}\right)d{y}^{\prime}\phantom{\rule{0ex}{0ex}}\mathrm{D}={{\displaystyle \int}}_{0}^{1}{a}^{2}\left(\frac{1-cos2\pi {y}^{\prime}}{2}\right)\left(\frac{1}{4}{{y}^{\prime}}^{2}+\frac{1}{8{\pi}^{2}}\left(cos2\pi {y}^{\prime}-1\right)\right)d{y}^{\prime}$ |

$\left(4\right)=-\frac{{d}^{2}}{\epsilon}\left\{A+B+C+D\right\}$ |

$\mathrm{A}={{\displaystyle \int}}_{0}^{1}{\left(\frac{{u}_{*}}{\kappa}\right)}^{2}\left(1+\mathrm{ln}{y}^{\prime}\right)\left(\frac{1}{2}{{y}^{\prime}}^{2}\mathrm{ln}{y}^{\prime}-\frac{1}{4}{{y}^{\prime}}^{2}\right)d{y}^{\prime}\phantom{\rule{0ex}{0ex}}={\left(\frac{{u}_{*}}{\kappa}\right)}^{2}\left\{{\left({y}^{\prime}\mathrm{ln}{y}^{\prime}\right)\times \left(\frac{1}{2}{{y}^{\prime}}^{2}\mathrm{ln}{y}^{\prime}-\frac{1}{4}{{y}^{\prime}}^{2}\right)|}_{0}^{1}-{{\displaystyle \int}}_{0}^{1}{\left({y}^{\prime}\mathrm{ln}{y}^{\prime}\right)}^{2}d{y}^{\prime}\right\}\phantom{\rule{0ex}{0ex}}=-{\left(\frac{{u}_{*}}{\kappa}\right)}^{2}{{\displaystyle \int}}_{0}^{1}{\left({y}^{\prime}\mathrm{ln}{y}^{\prime}\right)}^{2}d{y}^{\prime}-{\left(\frac{{u}_{*}}{\kappa}\right)}^{2}\left\{{\left(\frac{1}{3}{{y}^{\prime}}^{3}{\left(\mathrm{ln}{y}^{\prime}\right)}^{2}\right)|}_{0}^{1}-{{\displaystyle \int}}_{0}^{1}\frac{1}{3}{{y}^{\prime}}^{3}\times 2\mathrm{ln}{y}^{\prime}\times \frac{1}{{y}^{\prime}}d{y}^{\prime}\right\}\phantom{\rule{0ex}{0ex}}={\left(\frac{{u}_{*}}{\kappa}\right)}^{2}\times \frac{2}{3}\times {{\displaystyle \int}}_{0}^{1}{{y}^{\prime}}^{2}\mathrm{ln}{y}^{\prime}d{y}^{\prime}\phantom{\rule{0ex}{0ex}}={\left(\frac{{u}_{*}}{\kappa}\right)}^{2}\times \frac{2}{3}\times \left(\frac{-1}{{3}^{2}}\text{}\right)=-\frac{2}{27}{\left(\frac{{u}_{*}}{\kappa}\right)}^{2}=-0.0741{\left(\frac{{u}_{*}}{\kappa}\right)}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{B}={{\displaystyle \int}}_{0}^{1}\frac{a{u}_{*}}{\kappa}\left(1+\mathrm{ln}{y}^{\prime}\right)\left(\frac{1}{4}{{y}^{\prime}}^{2}+\frac{1}{8{\pi}^{2}}\left(cos2\pi {y}^{\prime}-1\right)\right)d{y}^{\prime}\phantom{\rule{0ex}{0ex}}=\frac{a{u}_{*}}{\kappa}\left\{{\left(1+\mathrm{ln}{y}^{\prime}\right)\left(\frac{1}{12}{{y}^{\prime}}^{3}+\frac{1}{8{\pi}^{2}}\left(\frac{1}{2\pi}sin2\pi {y}^{\prime}-{y}^{\prime}\right)\right)|}_{0}^{1}-{{\displaystyle \int}}_{0}^{1}\frac{1}{{y}^{\prime}}\left(\frac{1}{12}{{y}^{\prime}}^{3}+\frac{1}{8{\pi}^{2}}\left(\frac{1}{2\pi}sin2\pi {y}^{\prime}-{y}^{\prime}\right)\right)d{y}^{\prime}\right\}\phantom{\rule{0ex}{0ex}}=\frac{a{u}_{*}}{\kappa}\left\{\left(\frac{1}{12}-\frac{1}{8{\pi}^{2}}\right)-{{\displaystyle \int}}_{0}^{1}\left(\frac{1}{12}{{y}^{\prime}}^{2}+\frac{1}{8{\pi}^{2}}\left(\frac{1}{2\pi {y}^{\prime}}sin2\pi {y}^{\prime}-1\right)\right)d{y}^{\prime}\right\}\phantom{\rule{0ex}{0ex}}=\frac{a{u}_{*}}{\kappa}\left\{\frac{1}{18}-{{\displaystyle \int}}_{0}^{1}\frac{sin2\pi {y}^{\prime}}{16{\pi}^{3}{y}^{\prime}}d{y}^{\prime}\right\}=\frac{a{u}_{*}}{\kappa}\left\{\frac{1}{18}-\frac{1}{16{\pi}^{3}}{{\displaystyle \int}}_{0}^{1}\frac{sin2\pi {y}^{\prime}}{{y}^{\prime}}d{y}^{\prime}\right\}\phantom{\rule{0ex}{0ex}}=\frac{a{u}_{*}}{\kappa}\left\{\frac{1}{18}-\frac{1}{16{\pi}^{3}}{\displaystyle \sum}_{n=1}^{\infty}\frac{{\left(2\pi \right)}^{2n-1}{\left(-1\right)}^{n-1}}{\left(2n-1\right)!\left(2n-1\right)}\right\}=\frac{0.0527a{u}_{*}}{\kappa}\phantom{\rule{0ex}{0ex}}\mathrm{C}={{\displaystyle \int}}_{0}^{1}\frac{a{u}_{*}}{\kappa}\left(\frac{1-cos2\pi {y}^{\prime}}{2}\right)\left(\frac{1}{2}{{y}^{\prime}}^{2}\mathrm{ln}{y}^{\prime}-\frac{1}{4}{{y}^{\prime}}^{2}\right)d{y}^{\prime}\phantom{\rule{0ex}{0ex}}=\frac{a{u}_{*}}{\kappa}\left[{\left(\frac{1}{2}{y}^{\prime}-\frac{sin2\pi {y}^{\prime}}{4\pi}\right)\left(\frac{1}{2}{{y}^{\prime}}^{2}\mathrm{ln}{y}^{\prime}-\frac{1}{4}{{y}^{\prime}}^{2}\right)|}_{0}^{1}-{{\displaystyle \int}}_{0}^{1}\left(\frac{1}{2}{y}^{\prime}-\frac{sin2\pi {y}^{\prime}}{4\pi}\right)\left({y}^{\prime}\mathrm{ln}{y}^{\prime}\right)d{y}^{\prime}\right]\phantom{\rule{0ex}{0ex}}=\frac{a{u}_{*}}{\kappa}\left[-\frac{1}{8}-{{\displaystyle \int}}_{0}^{1}\frac{1}{2}{{y}^{\prime}}^{2}\mathrm{ln}{y}^{\prime}d{y}^{\prime}+\frac{1}{4\pi}{{\displaystyle \int}}_{0}^{1}sin2\pi {y}^{\prime}\left({y}^{\prime}\mathrm{ln}{y}^{\prime}\right)d{y}^{\prime}\right]\phantom{\rule{0ex}{0ex}}=\frac{a{u}_{*}}{\kappa}\left[-\frac{1}{8}-\frac{1}{2}\times \left(\frac{-1}{{3}^{2}}\right)+\frac{1}{4\pi}{{\displaystyle \int}}_{0}^{1}sin2\pi {y}^{\prime}\left({y}^{\prime}\mathrm{ln}{y}^{\prime}\right)d{y}^{\prime}\right]\phantom{\rule{0ex}{0ex}}=\frac{a{u}_{*}}{\kappa}\left[-\frac{5}{72}+\frac{1}{4\pi}\times {\displaystyle \sum}_{n=1}^{\infty}\frac{{\left(2\pi \right)}^{2n-1}{\left(-1\right)}^{n}}{\left(2n-1\right)!{\left(2n+1\right)}^{2}}\right]=-0.0723\frac{a{u}_{*}}{\kappa}\phantom{\rule{0ex}{0ex}}\mathrm{D}={{\displaystyle \int}}_{0}^{1}{a}^{2}\left(\frac{1-cos2\pi {y}^{\prime}}{2}\right)\left(\frac{1}{4}{{y}^{\prime}}^{2}+\frac{1}{8{\pi}^{2}}\left(cos2\pi {y}^{\prime}-1\right)\right)d{y}^{\prime}\phantom{\rule{0ex}{0ex}}=\frac{{a}^{2}}{2}{{\displaystyle \int}}_{0}^{1}\left(1-cos2\pi {y}^{\prime}\right)\left(\frac{1}{4}{{y}^{\prime}}^{2}-\frac{1}{8{\pi}^{2}}\left(1-cos2\pi {y}^{\prime}\right)\right)d{y}^{\prime}\phantom{\rule{0ex}{0ex}}=\frac{{a}^{2}}{2}\left[{{\displaystyle \int}}_{0}^{1}\frac{1}{4}{{y}^{\prime}}^{2}\left(1-cos2\pi {y}^{\prime}\right)d{y}^{\prime}-\frac{1}{8{\pi}^{2}}{{\displaystyle \int}}_{0}^{1}{\left(1-cos2\pi {y}^{\prime}\right)}^{2}d{y}^{\prime}\right]\phantom{\rule{0ex}{0ex}}=\frac{{a}^{2}}{2}\left[{{\displaystyle \int}}_{0}^{1}\frac{1}{4}{{y}^{\prime}}^{2}d{y}^{\prime}-\frac{1}{4}{{\displaystyle \int}}_{0}^{1}{{y}^{\prime}}^{2}cos2\pi {y}^{\prime}d{y}^{\prime}-\frac{1}{8{\pi}^{2}}{{\displaystyle \int}}_{0}^{1}\left(\frac{3}{2}-2cos2\pi {y}^{\prime}+\frac{1}{2}cos4\pi {y}^{\prime}\right)d{y}^{\prime}\right]\phantom{\rule{0ex}{0ex}}=\frac{{a}^{2}}{2}\left[\frac{1}{12}-\frac{1}{4}{{\displaystyle \int}}_{0}^{1}{{y}^{\prime}}^{2}cos2\pi {y}^{\prime}d{y}^{\prime}-\frac{3}{16{\pi}^{2}}\right]\phantom{\rule{0ex}{0ex}}=\frac{{a}^{2}}{2}\left[\frac{1}{12}-\frac{1}{8{\pi}^{2}}-\frac{3}{16{\pi}^{2}}\right]=\frac{{a}^{2}}{2}\left[\frac{1}{12}-\frac{5}{16{\pi}^{2}}\right]=0.0258{a}^{2}$ |

${D}_{L}=-\frac{{d}^{2}}{\epsilon}\left\{A+B+C+D\right\}\phantom{\rule{0ex}{0ex}}=-\frac{{d}^{2}}{\epsilon}\left\{-0.0741{\left(\frac{{u}_{*}}{\kappa}\right)}^{2}+0.0527\frac{a{u}_{*}}{\kappa}-0.0723\frac{a{u}_{*}}{\kappa}+0.0258{a}^{2}\right\}\phantom{\rule{0ex}{0ex}}=\frac{{d}^{2}}{\epsilon}\left\{-0.0258{\left(a-0.38\frac{{u}_{*}}{\kappa}\right)}^{2}+0.0778{\left(\frac{{u}_{*}}{\kappa}\right)}^{2}\right\}$ |

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**Figure 1.**Comparison of vertical distribution for longitudinal flow velocity in straight and curved channels (adapted from Mozaffari et al. [28]).

**Figure 3.**Plan view of small-scale meandering channel in laboratory (adapted from Baek et al. [3]).

**Figure 4.**Plan view of mid-scale nature-like channel in Andong River Experiment Center in Korea (adapted from Shin et al. [37]).

**Figure 5.**Comparison between measured velocity (blue line) and that estimated by Equation (7) (red line) in laboratory meandering channel at Section U1 (y denotes lateral coordinate from left to right bank).

**Figure 6.**(

**A**) Secondary flow patterns at representative cross-sections for case 402 in laboratory meandering channel (y denotes lateral coordinate from left to right bank, and z denotes vertical coordinate from bottom to water surface). (

**B**) Secondary flow patterns at each cross-section of AMC 315 in nature-like channel.

**Figure 7.**Comparison between measured velocity (dot) and that estimated by Equation (7) (line) at Section S4 of AMC 315 in nature-like channel (y denotes lateral coordinate from left to right bank).

**Figure 8.**Comparison between measured velocity (dot) and that estimated by Equation (7) (line) at Section

**S5**of AMC 315 in nature-like channel.

**Table 1.**Comparison between observed dispersion coefficients and estimated ones for laboratory and nature-like channels.

Channel | Case Number | a (Maximum Value) | ${\mathit{D}}_{\mathit{L}}/\mathit{d}{\mathit{u}}_{*}$ | ||
---|---|---|---|---|---|

This Study (Maximum Value) | Adapted from Elder [5] (a = 0) (Minimum Value) | Observed Value | |||

Lab. | 301 | 0.0059 | 6.22 | 5.93 | 9.2 |

303 | 0.0178 | 6.22 | 4.5 | ||

402 | 0.0092 | 6.22 | 4.3 | ||

Nature-like | A315 | 0.092 | 6.89 | 4.72~8.58 | |

A317 | 0.103 | 6.90 | 5.38~9.82 |

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Baek, K.O.; Seo, I.W.
Modifying Elder’s Longitudinal Dispersion Coefficient for Two-Dimensional Solute Mixing Analysis in Open-Channel Bends. *Water* **2022**, *14*, 2962.
https://doi.org/10.3390/w14192962

**AMA Style**

Baek KO, Seo IW.
Modifying Elder’s Longitudinal Dispersion Coefficient for Two-Dimensional Solute Mixing Analysis in Open-Channel Bends. *Water*. 2022; 14(19):2962.
https://doi.org/10.3390/w14192962

**Chicago/Turabian Style**

Baek, Kyong Oh, and II Won Seo.
2022. "Modifying Elder’s Longitudinal Dispersion Coefficient for Two-Dimensional Solute Mixing Analysis in Open-Channel Bends" *Water* 14, no. 19: 2962.
https://doi.org/10.3390/w14192962