# Applications of Two-Dimensional Spatial Routing Procedure for Estimating Dispersion Coefficients in Open Channel Flows

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Estimation of Dispersion Coefficients

#### 2.1.1. Velocity-Based Method

#### 2.1.2. Concentration-Based Method

#### Routing Procedure Using Concentration-Time Curves

#### Routing Procedure Using Spatial Distributions of Concentration

#### 2.2. Model Descriptions

^{th}particle position in i-direction; ${\tilde{x}}_{i}^{k}$ is the particle position after shear advection; ${t}_{sa}$ is the time after shear advection; $z$ is the vertical position of each particle; $\Delta t$ is the time step; $R$ is the random number following the Gaussian distribution; ${\epsilon}_{h}$ is the horizontal turbulent diffusion coefficient in the assumption of isotropic turbulence. The second term of the right-hand side in Equation (11) indicates the deterministic transport by vertically varied velocity profiles, and the third term is the transport of random fluctuations by the turbulent diffusion. The solute column is stretched on the horizontal planes by the vertical profiles of longitudinal (${u}_{s}$) and transverse velocities (${u}_{n}$), as shown in Figure 2. The vertical velocity profiles were generated using theoretical and empirical formulas, which can be reproduced using the depth-averaged flow fields. In the longitudinal direction, the log-profile suggested by Rozovskii [33] was employed to generate the vertical profile as

#### 2.3. Generation of Concentration Fields for Estimating Routing Procedures

#### 2.3.1. Idealized Concentration Fields to Evaluate the Frozen Cloud Assumption

#### 2.3.2. Generation of Concentration Fields in M2 Channel Using the PDM-2D Simulation Results

## 3. Results

#### 3.1. Evaluations of the Frozen Cloud Assumption

#### 3.2. Estimations of the Dispersion Coefficients in Meandering Channel

## 4. Discussions

#### 4.1. Validity of the Frozen Cloud Assumption

#### 4.2. Applicability to Non-Fickian Mixing

## 5. Conclusions

- For a solute mixing problem under the unsteady flow condition, the 2D S-RP provides quite accurate estimation results of dispersion coefficients when the mixing shows the Fickian dispersion.
- The temporal concentration curves present the non-Fickian mixing due to the unsteady flow condition even though the solute cloud shows the Fickian dispersion. Thus, the dispersion coefficients by the 2D ST-RP using the temporal data contain errors due to violation of the frozen cloud assumption.
- Both the dispersion coefficients calculated by the 2D ST-RP and the 2D S-RP showed errors against to the results using the velocity-based method for the solute mixing in the initial period where the non-Fickian mixing occurs. These discrepancies showed that the two routing methods were derived based on the Fickian dispersion model.
- The results obtained using the 2D S-RP in the meandering channel more accurately exhibited the spatial variability along the meander cycle of dispersion coefficients compared to the 2D ST-RP.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Conceptual diagram of routing procedures using spatial and temporal concentration measurements.

**Figure 5.**Outlines of M2 channel and the solute transport simulation results by the PDM-2D: (

**a**) velocity contour; (

**b**) particle distribution; (

**c**) concentration distribution.

**Figure 6.**Comparisons of concentration-time curves at y/W = 0.5 for checking the time step dependence: (

**a**) S6; (

**b**) S9.

**Figure 8.**Comparisons of the analytic solution and the predicted concentration fields by the 2D S-RP: (

**a**) t = 4 s; (

**b**) t = 5 s; (

**c**) t = 7 s; (

**d**) t = 11 s.

**Figure 9.**Comparisons of the analytic solution and the concentration-time contours by the 2D ST-RP: (

**a**) S2; (

**b**) S3; (

**c**) S4; (

**d**) S5.

**Figure 10.**Comparisons of predicted concentration fields using the 2D S-RP: (

**a**) S3; (

**b**) S5; (

**c**) S7; (

**d**) S9.

**Figure 11.**Comparisons of predicted concentration-time distributions using the 2D ST-RP: (

**a**) S3; (

**b**) S5; (

**c**) S7; (

**d**) S9.

**Figure 12.**Comparisons of the spatial variabilities of the dispersion tensor: (

**a**) ${D}_{xx}/h{u}^{*}$; (

**b**) ${D}_{xy}/h{u}^{*}$; (

**c**) ${D}_{yy}/h{u}^{*}$.

Q (m ^{3}/s) | h (m) | U (m/s) | ${\mathit{u}}^{*}$ (m/s) | ${\mathit{C}}_{0}$ (ppm) | Δt (s) | L | No. of Particles |
---|---|---|---|---|---|---|---|

0.06 | 0.4 | 0.15 | 0.0078 | 100,000 | 0.5 | 300 | 30,000 |

Section No. | y/W | ${\mathit{C}}_{\mathit{p}}/{\mathit{C}}_{0}$ | ${\mathit{t}}_{\mathit{p}}\left(\mathbf{s}\right)$ | ${\mathit{\mu}}_{\mathit{t}}\left(\mathbf{s}\right)$ | ${\mathit{\xi}}_{\mathit{t}}$ | ||||
---|---|---|---|---|---|---|---|---|---|

Exp. | Sim. | Exp. | Sim. | Exp. | Sim. | Exp. | Sim. | ||

S5 | 0.30 | 0.013 | 0.011 | 35.3 | 36.0 | 34.9 | 36.1 | 0.72 | 0.63 |

0.45 | 0.022 | 0.021 | 37.1 | 38.0 | 36.6 | 38.3 | 0.43 | 0.59 | |

0.70 | 0.030 | 0.006 | 39.8 | 41.0 | 40.5 | 43.8 | 0.92 | 0.36 | |

S6 | 0.30 | 0.014 | 0.009 | 44.8 | 42.0 | 44.4 | 42.5 | 0.29 | 0.62 |

0.45 | 0.020 | 0.015 | 40.5 | 45.0 | 43.9 | 45.3 | 0.60 | 0.68 | |

0.70 | 0.008 | 0.011 | 50.1 | 51.0 | 50.7 | 52.1 | 0.16 | 0.54 | |

S9 | 0.30 | 0.008 | 0.008 | 67.5 | 0.44 | 68.1 | 71.6 | 0.44 | 0.41 |

0.45 | 0.015 | 0.013 | 65.3 | 0.56 | 68.2 | 69.3 | 0.56 | 0.62 | |

0.70 | 0.008 | 0.013 | 69.3 | 0.62 | 72.0 | 68.5 | 0.62 | 0.58 |

**Table 3.**Calculation results of the dispersion coefficients using the 2D S-RP and the 2D ST-RP in the test conditions.

Routing Span | ${\mathit{D}}_{\mathit{L}}({\mathbf{m}}^{2}/\mathbf{s})$ | ${\mathit{D}}_{\mathit{T}}({\mathbf{m}}^{2}/\mathbf{s})$ | ||||
---|---|---|---|---|---|---|

2D ST-RP | 2D S-RP | Analytic Solution | 2D ST-RP | 2D S-RP | Analytic Solution | |

S1-S2 | 0.135 | 0.163 | 0.163 | 0.0048 | 0.0041 | 0.0041 |

S1-S3 | 0.166 | 0.163 | 0.0047 | 0.0041 | ||

S1-S4 | 0.178 | 0.163 | 0.0051 | 0.0041 | ||

S1-S5 | 0.136 | 0.163 | 0.0033 | 0.0041 | ||

S1-S6 | 0.165 | 0.163 | 0.0027 | 0.0041 | ||

Avg. | 0.156 | 0.163 | 0.0041 | 0.0041 | ||

S2-S3 | 0.153 | 0.163 | 0.0042 | 0.0041 | ||

S2-S4 | 0.196 | 0.163 | 0.0057 | 0.0041 | ||

S2-S5 | 0.141 | 0.163 | 0.0050 | 0.0041 | ||

S2-S6 | 0.173 | 0.163 | 0.0051 | 0.0041 | ||

Avg. | 0.166 | 0.163 | 0.0050 | 0.0041 | ||

S3-S4 | 0.195 | 0.163 | 0.0058 | 0.0041 | ||

S3-S5 | 0.134 | 0.163 | 0.0050 | 0.0041 | ||

S3-S6 | 0.170 | 0.163 | 0.0050 | 0.0041 | ||

Avg. | 0.166 | 0.163 | 0.0053 | 0.0041 | ||

S4-S5 | 0.144 | 0.163 | 0.0038 | 0.163 | ||

S4-S6 | 0.092 | 0.163 | 0.0042 | 0.0041 | ||

Avg. | 0.118 | 0.163 | 0.0040 | 0.0041 | ||

S5-S6 | 0.208 | 0.163 | 0.0044 | 0.0041 |

Section No. | ${\mathit{D}}_{\mathit{L}}/\mathit{h}{\mathit{u}}^{*}$ | ${\mathit{D}}_{\mathit{T}}/\mathit{h}{\mathit{u}}^{*}$ | ||
---|---|---|---|---|

2D ST-RP | 2D S-RP | 2D ST-RP | 2D S-RP | |

2 | 0.89 | 3.33 | 2.19 | 0.22 |

3 | 0.08 | 1.49 | 0.11 | 0.31 |

4 | 0.09 | 1.37 | 1.09 | 0.52 |

5 | 1.81 | 1.14 | 0.27 | 0.56 |

6 | 5.95 | 2.52 | 0.09 | 0.43 |

7 | 4.73 | 2.64 | 0.27 | 0.47 |

8 | 1.38 | 2.44 | 0.34 | 0.46 |

9 | 3.80 | 2.73 | 0.02 | 0.39 |

10 | 4.33 | 2.85 | 0.11 | 0.38 |

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**MDPI and ACS Style**

Shin, J.; Rhee, D.; Park, I. Applications of Two-Dimensional Spatial Routing Procedure for Estimating Dispersion Coefficients in Open Channel Flows. *Water* **2021**, *13*, 1394.
https://doi.org/10.3390/w13101394

**AMA Style**

Shin J, Rhee D, Park I. Applications of Two-Dimensional Spatial Routing Procedure for Estimating Dispersion Coefficients in Open Channel Flows. *Water*. 2021; 13(10):1394.
https://doi.org/10.3390/w13101394

**Chicago/Turabian Style**

Shin, Jaehyun, Dongsop Rhee, and Inhwan Park. 2021. "Applications of Two-Dimensional Spatial Routing Procedure for Estimating Dispersion Coefficients in Open Channel Flows" *Water* 13, no. 10: 1394.
https://doi.org/10.3390/w13101394