# A Simplified Simulation Method for Flood-Induced Bend Scour—A Case Study Near the Shuideliaw Embankment on the Cho-Shui River

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{3}/s, respectively. Moreover, the adjacent irrigation channels were damaged during these flood events.

## 2. Field Data Collection and Formula Derivation

#### 2.1. Site Description

^{2}. The aerial photograph in Figure 2a shows two concave embankments located near the Mingchu Bridge: one on the right-hand side upstream (but facing downstream) of the Mingchu Bridge, and the other on the left-hand side downstream of the bridge, where the Shuideliaw Embankment is located.

_{16}= 1.7 mm, median size d

_{50}= 108 mm, and d

_{84}= 329 mm.

#### 2.2. Field Data

**Q**ranged from 164 to 5309 m

_{p}^{3}/s and the flow intensity (unit peak flow discharge

**q**) ranged from 1.31 to 25.37 m

_{p}^{2}/s. The channel slope

**s**was also an important factor, with a range from 0.0063 to 0.0153, and the median particle size

_{0}**d**of the bed materials was from 3.43 to 168 mm. The characteristic parameter for bend scour is the outer radius of the bend R

_{50}_{0}, which was recorded to range from 169 to 800 m. Finally, the measured maximum scour depth in a river bend was from 0.22 to 5.23 m. With these measured data, a reliable bend scour computation equation was developed, as described in Section 2.3. Furthermore, the measured data for two flood events, Typhoon Trami and Typhoon Usagi, were adopted in the present study to verify the accuracy of the proposed method, as described in Section 4.2. Figure 4 shows the flow hydrographs released from the Chi-Chi Weir for Typhoon Trami (August 2013) and Typhoon Usagi (September 2013).

#### 2.3. Proposed Bend Scour Computation Equation

_{bs}is the bend scour depth; q is the unit discharge; ρ is the density of fluid; g is the gravitational acceleration; ${s}_{0}$ is the channel slope; d

_{50}is the median particle diameter; σ

_{g}is the geometric standard deviation of the particle size distribution; ρ

_{s}is the density of sediment particles; and R

_{0}(= R

_{c}+ 0.5 W) is the outer radius of the bend. Based on Buckingham’s π theorem, bend scour depth can be written in the following dimensionless form:

## 3. Method for Embankment Toe Scour Simulation

#### 3.1. 2D Finite-Volume Hydraulic Model

**Q**is the vector of conserved variables;

**F**

_{I}and

**G**

_{I}are the inviscid flux vectors in the x- and y-directions, respectively;

**F**

_{V}and

**G**

_{V}are the viscous flux vectors in the x- and y-directions, respectively;

**S**is the source term; h is the water depth; u and v are the depth-averaged velocity components in the x- and y-directions, respectively; ρ is the density of flow; T

_{xx}, T

_{xy}, and T

_{yy}are the depth-averaged turbulent stresses; g is the gravitational acceleration; s

_{0x}and s

_{0y}are the bed slopes in the x- and y-directions, respectively; and s

_{fx}and s

_{fy}are the friction slopes in the x- and y-directions, respectively.

**T**(θ)

^{−1}is the inverse of the rotation matrix corresponding to the m side; θ is the angle between the outward unit vector

**n**and the x-axis;

**n**is the outward unit vector normal to the boundary of the control volume; L

^{m}is the length of the m side for the cell; $\mathbf{F}(\overline{\mathbf{Q}})={\mathbf{F}}_{I}(\overline{\mathbf{Q}})-{\mathbf{F}}_{V}(\overline{\mathbf{Q}})$ is the numerical flux; ${\mathbf{F}}_{I}(\overline{\mathbf{Q}})$ represents the inviscid numerical flux; ${\mathbf{F}}_{V}(\overline{\mathbf{Q}})$ denotes the viscous numerical flux; and $\tilde{\mathbf{S}}$ is the integral form of the source terms. To resolve discontinuous shock waves or hydraulic jumps, many different types of approximate Riemann solvers have been developed for estimating the inviscid numerical flux ${\mathbf{F}}_{I}(\overline{\mathbf{Q}})$. This paper employs an upstream flux-splitting finite-volume scheme [24] to obtain the inviscid numerical flux. In addition, the Jacobian transformation method [25] is used to estimate the viscous numerical flux.

- The cell interface is the dry edge, when ${h}_{L}\le {h}_{tol}$ and ${h}_{R}\le {h}_{tol}$, in which ${h}_{L}$ and ${h}_{R}$ are the left and right water depths at the center of the cell interface, respectively. There is no flux estimation.
- The cell interface is the wet edge, when ${h}_{L}>{h}_{tol}$ and ${h}_{R}>{h}_{tol}$. The upstream flux-splitting finite-volume scheme coupled with the hydrostatic reconstruction method [24] is adopted to estimate the well-balanced numerical fluxes.
- The cell interface is the partially wet edge with flux, when ${h}_{L}>{h}_{tol}$, ${h}_{R}\le {h}_{tol}$, and ${h}_{L}+{({z}_{b})}_{L}>{h}_{R}+{({z}_{b})}_{R}$, where z
_{b}is the bed elevation. The momentum flux is set at 0 and the mass flux (hu)_{LR}at cell interface LR is estimated as:$$\delta \eta ={h}_{L}+{({z}_{b})}_{L}-{h}_{R}-{({z}_{b})}_{R}$$$${(hu)}_{LR}=1.42\delta \eta \sqrt{\left|\delta \eta \right|}.$$ - The cell interface is the partially wet edge without flux, when ${h}_{L}>{h}_{tol}$, ${h}_{R}\le {h}_{tol}$, and ${h}_{L}+{({z}_{b})}_{L}\le {h}_{R}+{({z}_{b})}_{R}$. According to the bed elevation condition, there is no flux across the cell interface. Thus, no flux estimation is required.

#### 3.2. General Scour Computation Equation

_{gs}denotes the short-term general scour depth. The accuracy of Equation (15) has been verified to give satisfactory results when estimating the short-term general scour depth in the Cho-Shui River [22].

#### 3.3. Method for Simulating Bend Scour Depth Evolution

- Inputting flow hydrographs (simulation time step is 1 h), setting model parameters (i.e., numerical time step and Manning roughness coefficients), and then simulating the 2D flow field in the bend reach through the proposed 2D finite-volume hydraulic model (Section 3.1). At the end of each simulation time step, the water depth and velocity in each computational cell are outputted.
- After 2D flow simulation, one obtains the approach flow conditions (i.e., entire hydrographs) at the specific computational cell, including water depth, velocities, water surface width, and the centerline radius of the bend.
- For each simulation time step (1 h), one substitutes the approach flow conditions into the general scour computation equation (i.e., Equation (15)) to achieve the estimated general scour depth. After the entire general scour simulation, the evolution of the estimated general scour depth is thus obtained.
- For each simulation time step (1 h), one re-evaluates the water depth by adding the water depth and the estimated general scour depth to obtain the revised water depth: $\tilde{h}=h+{d}_{gs}$.
- For each simulation time step (1 h), one re-evaluates velocity through unit flow discharge divided by revised water depth and obtains the revised velocity as $\tilde{v}=q/\tilde{h}$.
- For each simulation time step (1 h), one substitutes the revised flow conditions (i.e., ${h}_{u}=\tilde{h}$) into Equations (1)–(3) and (6) to obtain the bend scour depth. After entire bend scour simulation, the evolution of the estimated bend scour depth is thus obtained.

## 4. Flow Field and Embankment Toe Scour Simulations

#### 4.1. Verifications of 2D Finite-Volume Model

_{p}is the peak water level error; ET

_{p}represents the error of time to peak water level; R

^{2}denotes the coefficient of determination; ${\eta}_{p}^{sim}$ and ${\eta}_{p}^{mea}$ denote the simulated and measured peak water levels, respectively; ${T}_{p}^{sim}$ and ${T}_{p}^{mea}$ are the simulated and measured time to peak water level, respectively; ${\eta}_{i}^{sim}$ and ${\eta}_{i}^{mea}$ are the simulated and measured water levels at each time; and ${\overline{\eta}}^{sim}$ and ${\overline{\eta}}^{mea}$ denote the average simulated and measured water levels. The results for the two examined flood events, for which the Eη

_{p}results were 0.13% and 0.46%, respectively, are provided in Table 2. Because the overall performance of Eη

_{p}is less than 0.5%, the results demonstrate that the model can provide high numerical accuracy for the peak water level in modeling 2D flow with variable bed topography. In addition, the average value of R

^{2}coefficient for two flood events is 0.88, close to the best value of 1.0, indicating that the proposed 2D model can achieve reasonable performance for water-level hydrograph.

#### 4.2. Verifications of Proposed Method for Bend Scour Depth

#### 4.3. Practical Embankment Safety Curve

^{3}/s. By using the proposed equation ${d}_{bs}=0.0005Q$, when discharge is equal to 6122 m

^{3}/s, the total bend scour depth is estimated at 3.06 m, which is larger than the depth of the embankment foundation (3.0 m). Therefore, the results demonstrate that the proposed simplified equation could be used to correctly assess embankment toe safety. To further provide advantageous assessment information, the discharge warning value can be achieved from this equation fairly simply. Moreover, the depth of the embankment foundation can be substituted into this equation, resulting in 6000 m

^{3}/s waring discharge. The results herein suggest that the study embankment will be unstable and may further fail if discharge from the Chi-Chi Weir is greater than 6000 m

^{3}/s. By contrast, the numerical experiments demonstrated that the maximum bend scour depth near the Shuideliaw Embankment is greater than the designed foundation depth, because the flow discharge is greater than 6000 m

^{3}/s (which is between the flood discharges for the two- and five-year return periods, similar to the bankfull discharge). This similarity between the flow and bankfull discharge denotes a significant change in the river bed, and indicates why embankment failure usually occurs during bankfull discharge.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Aerial view of the study site before (

**a**) and after (

**b**) Typhoon Saoloa. The location of the scour measurement is also illustrated (

**c**).

**Figure 3.**Size distribution of sediment particles at the study site where the survey was conducted in 2013.

**Figure 5.**Definition sketch showing the bend scour: (

**a**) top view; (

**b**) cross-sections looking downstream.

**Figure 9.**Comparisons of simulated and measured water level hydrographs for (

**a**) Typhoon Trami and (

**b**) Typhoon Usagi.

**Figure 10.**2D contours showing simulated velocities under peak flood condition for (

**a**) Typhoon Trami (Q

_{p}= 4186 m

^{3}/s) and (

**b**) Typhoon Usagi (Q

_{p}= 5309 m

^{3}/s).

**Figure 11.**The simulated time variations of bend scour depths using four approaches for (

**a**) Typhoon Trami and (

**b**) Typhoon Usagi.

**Figure 12.**The relationship between the bend scoured depth, scoured bed level, and flow discharge at Chi-Chi Weir (

**a**). The location of Shuideliaw Embankment and adjacent bed change at cross section 105 are also illustrated (

**b**).

River | Site | Flood Event | Q_{p} (m^{3}/s) | q_{p} (m^{2}/s) | s_{0} | d_{50} (mm) | R_{0} (m) | d_{bs} (m) | Remark |
---|---|---|---|---|---|---|---|---|---|

Cho-Shui River | Cho-Shui Embankment | 1. Typhoon Nanmadol (August 2011) | 164 | 1.48 | 0.0063 | 3.43 | 605 | 0.67 | Lin and Lin [30] |

Fanziliao Embankment | 2. Typhoon Talim (June 2012) | 346 | 1.86 | 0.0071 | 23.5 | 418 | 0.22 | ||

3. Typhoon Saola (August 2012) | 475 | 2.97 | 0.0057 | 7.5 | 418 | 0.8 | |||

Linan Embankment | 4. Typhoon Nanmadol (August 2011) | 186 | 1.31 | 0.0066 | 85 | 558 | 0.36 | ||

5. Typhoon Saola (August 2012) | 2279 | 13.21 | 0.0066 | 34.5 | 558 | 2.36 | |||

Shanhan Embankment | 6. Typhoon Talim (June 2012) | 562 | 4.44 | 0.0153 | 19.5 | 471 | 1.9 | ||

Aiguoqiao Embankment | 7. Typhoon Talim (June 2012) | 1182 | 11.87 | 0.0112 | 48 | 169 | 0.47 | ||

8. Typhoon Saola (August 2012) | 1837 | 14.94 | 0.0148 | 25.5 | 169 | 2.45 | |||

Shuideliaw Embankment | 9. Typhoon Trami (August 2013) | 4186 | 25.37 | 0.00527 | 108 | 800 | 1.71 | Lu [31] | |

10. Typhoon Usagi (September 2013) | 5309 | 18.31 | 0.00527 | 108 | 800 | 3.30 | |||

Da-Chia River | Fengzhou Embankment | 11. Typhoon Soulik (July 2013) | 6692 | 22.01 | 0.00755 | 168 | 500 | 5.23 | |

12. Typhoon Trami (August 2013) | 2393 | 10.64 | 0.00755 | 168 | 500 | 0.55 |

Events | Three Criteria | ||
---|---|---|---|

Eη_{p} (%) | ET_{p} (%) | R^{2} | |

Typhoon Trami (August 2013) | 0.1335 | 8.69 | 0.874 |

Typhoon Usagi (September 2013) | 0.4600 | 8.33 | 0.889 |

**Table 3.**The simulated maximum bend scour depths and the relative errors by four approaches for Typhoon Trami and Typhoon Usagi.

Methods | Typhoon Trami (Measured = 1.71 m) | Typhoon Usagi (Measured = 3.3 m) | ||
---|---|---|---|---|

Simulated | $\mathit{E}{\mathit{d}}_{\mathit{b}\mathit{s}}(\%)$ | Simulated | $\mathit{E}{\mathit{d}}_{\mathit{b}\mathit{s}}(\%)$ | |

Galay et al. [27] | 1.52 | −11.11 | 2.22 | −32.73 |

Thorne [28] | 3.04 | 77.78 | 4.85 | 46.97 |

U.S. Army Corps of Engineering [29] | 3.25 | 90.06 | 4.37 | 32.42 |

Proposed method | 1.90 | 11.11 | 2.78 | −15.76 |

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**MDPI and ACS Style**

Guo, W.-D.; Hong, J.-H.; Chen, C.-H.; Su, C.-C.; Lai, J.-S. A Simplified Simulation Method for Flood-Induced Bend Scour—A Case Study Near the Shuideliaw Embankment on the Cho-Shui River. *Water* **2017**, *9*, 324.
https://doi.org/10.3390/w9050324

**AMA Style**

Guo W-D, Hong J-H, Chen C-H, Su C-C, Lai J-S. A Simplified Simulation Method for Flood-Induced Bend Scour—A Case Study Near the Shuideliaw Embankment on the Cho-Shui River. *Water*. 2017; 9(5):324.
https://doi.org/10.3390/w9050324

**Chicago/Turabian Style**

Guo, Wen-Dar, Jian-Hao Hong, Cheng-Hsin Chen, Chih-Chiang Su, and Jihn-Sung Lai. 2017. "A Simplified Simulation Method for Flood-Induced Bend Scour—A Case Study Near the Shuideliaw Embankment on the Cho-Shui River" *Water* 9, no. 5: 324.
https://doi.org/10.3390/w9050324