# Geometric Characteristics of Spur Dike Scour under Clear-Water Scour Conditions

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup and Procedures

_{50}is 0.2 mm, 0.7 mm, and 1.0 mm respectively; the non-uniform coefficient ${\sigma}_{g}$ = 1.14~1.3. The schematic diagram of experimental plane and profile are shown in Figure 1.

_{c}= 0.85, the upstream flow velocity U was measured by an acoustic Doppler velocimeter (ADV); U

_{c}is the incipient velocity of sediment, calculated using the Shamov formula [16]. The design of flow, sediment characteristics, and working conditions are shown in Table 1.

_{50}is the median diameter of sediment; h is the depth of water.

## 3. Results and Discussion

#### 3.1. Scour Depth

_{s}/L = 2K

_{s}; for intermediate spur dikes, d

_{s}/(Lh)

^{0.5}= 2K

_{s}K

_{ϴ}. K

_{s}is the structural shape coefficient, and K

_{ϴ}is the alignment angle coefficient. However, when θ = 90°, L = L’; when θ ≠ 90°, L’ = L × sin θ. The L’ is the length of the spur dike projection. In view of this, for the local scour depth of the short spur dikes, the alignment angle coefficient also needs to be considered. According to the results of experimental observations of various working conditions, the adjustment laws of local depth of erosion are shown in Figure 2.

_{s}/L’ increases as L’/h increases gradually. Regression analysis shows that the slope of the linear relationship is 1.5, close to 2K

_{s}, and for the straight-wall spur dikes, K

_{s}= 0.75. Since the short spur dike only considers the structural shape coefficient, and does not consider the alignment angle coefficient, and when θ = 90°, K

_{θ}= 1.0; θ = 60°, K

_{θ}= 0.96; θ = 30°, K

_{θ}= 0.92. After calculation and correction, it can be seen that the adjustment law is similar to the θ = 90° arrangement. For the intermediate spur dikes, the upper limit is close to 2K

_{s}K

_{ϴ}as the L’/h is gradually increased, and is still 1.5.

#### 3.2. Prediction of Plane Area and Volume of Scour Hole

_{s}~ d

_{s}

^{2}and V

_{s}~ d

_{s}

^{3}are usually used to discuss or predict the plane area and scour hole-volume by the maximum scour depth [5,20,21,22]. In order to facilitate the discussion of this problem, here are the formulas of A

_{s}~ d

_{s}

^{2}; V

_{s}~ d

_{s}

^{3}is written as the power function relation, that is, A

_{s}= C

_{1}d

_{s}

^{2}; V

_{s}= C

_{2}d

_{s}

^{3}, where C

_{1}, C

_{2}is the undetermined coefficient, respectively, and its value is related to the influencing factors. For any form of alignment, taking into account the experimental observation results and the existing research results (Table 1 and Table 2), C

_{1}, C

_{2}values and relative coarseness adjustment characteristics can be obtained, as shown in Figure 3.

_{50}= 57 ~ 1094; h/L’ = 0.2~21.5. Affected by this, C

_{1}and C

_{2}have a slightly larger range of fluctuations, the averages are 20.5 and 8.0 respectively, as shown in Figure 3. With regard to the classification of spur dike type [1], it is considered that this result is also applicable to short and intermediate spur dikes.

_{2}= 12.11; Rodrigue [22] considered C

_{2}= 3.87, and the difference is slightly larger. According to the C

_{1}and C

_{2}adjustment characteristics, C

_{1}= 20.5 and C

_{2}= 8.0 are selected, and the plane area and volume of the scour holes are predicted according to the maximum scour depth.

_{s}= 20.5 d

_{s}

^{2}and V

_{s}= 8.0 d

_{s}

^{3}can be used to predict the plane area and volume of scour holes by the maximum scour depth.

_{50}= 0.2 mm). Where, ignored the impact of these small scour holes, which are basically within the range of ±15%, which is considered reasonable; see Figure 5.

#### 3.3. The Morphology of Scour Holes

_{s}~ A

_{s}d

_{s}, but the morphology of the scour hole is not regular in geometry. The ratio between V

_{s}and A

_{s}d

_{s}is still indistinct. Based on the results of experimental observation and previous research, the regulation laws of V

_{s}and A

_{s}d

_{s}, under the influence of various factors, are discussed and defined. See Table 1 and Figure 7.

_{s}and A

_{s}d

_{s}. Under the influence of those factors, the slope of the linear relationship is the constant of V

_{s}/A

_{s}d

_{s}, and regression analysis showed that its value was 0.32; in addition, we easily found that C

_{1}and C

_{2}mean ratio is also closer to this constant. Therefore, it can be concluded that the scour hole-volume has a proportional constant with the product of the plane area and the maximum scour depth, and this characteristic also reflects the geometric similarity of the scour hole morphology, which can also be seen from Figure 6.

#### 3.4. The Profile Morphology of the Scour Holes

_{I}is the azimuth, i = 1, 2, 3; α

_{1}is defined as scour hole upstream; α

_{2}is defined as along the spur dike axis direction; and α

_{3}is defined as the downstream of the scour hole. R

_{i}is the radius of the scour hole corresponding to each azimuthal angle, that is, the width of the scour hole plane.

#### 3.5. The Profiles Slope of the Scour Holes

_{ij}= arctan (Δds/ΔR

_{ij}), where, Δds is the vertical height difference between adjacent points of scour hole profiles; ΔR

_{ij}is the horizontal distance between two adjacent points; φ

_{ij}is the slope value of any point on the slope of the scour holes, i = 1, 2, 3, respectively, corresponding to α

_{1}, α

_{2}, α

_{3}; and j is the number of calculation, j = 1, 2, 3, …, n.

_{50}= 0.2 mm, φ = 33.1°; d

_{50}= 0.7 mm, φ = 34.8°; d

_{50}= 1.1 mm, φ = 35.4°; and where φ is the sediment angle of repose.

_{ij}/φ undoubtedly reflects the difference between the profile slopes and the angle of repose of sediment; the dimensionless parameter R

_{ij}/R

_{i}is normalized to the radius of each scour hole. Therefore, the dimensionless parameter R

_{ij}/R

_{i}and φ

_{ij}/φ relationship pare reflect the azimuth of the profile slopes distribution characteristics. The profile slope distribution of each azimuth is shown in Figure 10.

_{ij}/R

_{i}and φ

_{ij}/φ shows that although the slope distribution is slightly different, but presents from small to large, and then the trend is reduced, showing an inverted “U” distribution. This shows that profile slopes distribution of scour hole also has geometric similarity.

_{ij}/φ ratio indicates that the slope of a certain distance in the scour hole is approximately equal to the sediment repose angle, and the average value is smaller than the sediment repose angle. Zhang [13] pointed out that the slope ratio of upstream and downstream angles of the scour hole is constant, about 0.5. The experimental results further indicate that the ratio of the average slope upstream and along the axis direction of spur dike, and the average slope downstream of the scour hole, with a mean of 0.6, and the discussion results, are relatively close.

## 4. Conclusions

_{1}= 20.5 and C

_{2}= 8.0, it is reasonable to predict the plane area and volume of the scour hole by maximum scour depth. There is a fixed proportional relationship between the product of the plane area, and the maximum scour depth and the scour hole-volume, and the constant is 0.32, which has geometric similarity. With the decrease of alignment angle, the position of maximum scour depth gradually approached the head of spur dike. The arrangement of the spur dike significantly changed the position of the local maximum scour depth and the plane shape of the scour hole. With the decrease of alignment angle, there is a gradual transition from an approximate ellipse to an approximately triangular shape. The position of maximum scour depth gradually approached the head of spur dikes.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic diagram of flume alignment; (

**b**) Schematic diagram of profiles alignment; (

**c**) Geometric characteristics parameters.

**Figure 4.**(

**a**) The relationship between maximum scour depth and plane area of scour hole;

**(b**) The relationship between maximum scour depth and scour hole-volume.

**Figure 5.**(

**a**) The error distribution of scour holes’ plane area; (

**b**) The error distribution of scour hole-volume.

**Figure 6.**Three-dimensional structure of scour holes in typical conditions (axis unit: m). (

**a**) d

_{50}= 1.1 mm; L’/d

_{50}= 182; ϴ = 90°; (

**b**) d

_{50}= 0.2 mm; L’/d

_{50}= 520; ϴ = 60°; (

**c**) d

_{50}= 0.2 mm; L’/d

_{50}= 300; ϴ = 30°; (

**d**) d

_{50}= 0.7 mm; L’/d

_{50}= 171; ϴ = 90°; (

**e**) d

_{50}= 0.7 mm; L’/d

_{50}= 149; ϴ = 60°; (

**f**) d

_{50}= 0.7 mm; L’/d

_{50}= 86; ϴ = 30°.

**Figure 9.**The profile morphological characteristics of each azimuthal of the scour holes. (

**a**) d

_{50}= 1.1 mm; L’/d

_{50}= 182; θ = 90°; (

**b**) d

_{50}= 0.2 mm; L’/d

_{50}= 520; θ = 60°; (

**c**) d

_{50}= 0.2 mm; L’/d

_{50}= 300; θ = 30°; (

**d**) d

_{50}= 0.7 mm; L’/d

_{50}= 171; θ = 90°; (

**e**) d

_{50}= 0.7 mm; L’/d

_{50}= 149; θ = 60°; (

**f**) d

_{50}= 0.7 mm; L’/d

_{50}= 86; θ = 30°.

**Figure 10.**The profile slope distribution of each azimuth of the scour holes. (

**a**) d

_{50}= 1.1 mm; L’/d

_{50}= 182; θ = 90°; (

**b**) d

_{50}= 0.2 mm; L’/d

_{50}= 520; θ = 60°; (

**c**) d

_{50}= 0.2 mm; L’/d

_{50}= 300; θ = 30°; (

**d**) d

_{50}= 0.7 mm; L’/d

_{50}= 171; θ = 90°; (

**e**) d

_{50}= 0.7 mm; L’/d

_{50}= 149; θ = 60°; (

**f**) d

_{50}= 0.7 mm; L’/d

_{50}= 86; θ = 30°.

Case | Parameter | Non-Dimensional Parameter | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Length L (m) | Alignment Angle θ (°) | Sediment Size d_{50} (mm) | Flow Depth h (mm) | Scour Depth d_{s} (cm) | Area of Plane A_{S} (cm^{2}) | Volume V_{S} (cm^{3}) | L’/d_{50} | L’/h | A_{S}/d_{s}^{2} | V_{S}/d_{s}^{3} | |

A1 | 0.12 | 90 | 0.2 | 0.1 | 17.0 | 2589.85 | 12,608.42 | 600 | 1.2 | 8.961 | 2.566 |

A2 | 60 | 13.5 | 2393.81 | 11,565.47 | 520 | 1.04 | 13.135 | 4.701 | |||

A3 | 30 | 7.0 | 897.66 | 1963.36 | 300 | 0.6 | 18.320 | 5.724 | |||

A4 | 90 | 0.15 | 8.5 | 798.87 | 2186.84 | 600 | 0.8 | 11.057 | 3.561 | ||

A5 | 60 | 7.9 | 1108.11 | 3101.92 | 520 | 0.69 | 17.755 | 6.291 | |||

A6 | 30 | 2.8 | 271.91 | 315.05 | 300 | 0.4 | 34.682 | 14.352 | |||

A7 | 90 | 0.3 | 5.7 | 301.72 | 593.44 | 600 | 0.4 | 9.287 | 3.204 | ||

A8 | 60 | 4.8 | 436.11 | 992.11 | 520 | 0.35 | 18.928 | 8.971 | |||

A9 | 30 | 2.1 | 141.97 | 100.65 | 300 | 0.2 | 32.193 | 10.868 | |||

A10 | 0.2 | 90 | 12.3 | 1692.79 | 6278.40 | 1000 | 0.67 | 11.189 | 3.374 | ||

A11 | 60 | 8.4 | 870.28 | 3178.40 | 865 | 0.57 | 12.334 | 5.363 | |||

A12 | 30 | 3.4 | 399.91 | 494.00 | 500 | 0.3 | 34.594 | 12.569 | |||

B1 | 0.2 | 90 | 0.7 | 0.1 | 19.5 | 4337.44 | 22,029.48 | 286 | 2.0 | 11.407 | 2.971 |

B2 | 60 | 13.9 | 2796.7 | 13,687.77 | 247 | 1.73 | 14.475 | 5.097 | |||

B3 | 30 | 7.2 | 1610.18 | 4722.7 | 143 | 1.0 | 31.061 | 12.653 | |||

B4 | 0.12 | 90 | 0.15 | 15.3 | 2085.74 | 10,079.63 | 171 | 0.8 | 8.910 | 2.814 | |

B5 | 60 | 14.3 | 2216.03 | 10,156.05 | 149 | 0.69 | 10.837 | 3.473 | |||

B6 | 30 | 5.2 | 1011.51 | 1736.57 | 86 | 0.4 | 37.408 | 12.350 | |||

B7 | 0.12 | 90 | 0.2 | 12.6 | 1701.08 | 6287.63 | 171 | 0.6 | 137.408 | 3.143 | |

B8 | 60 | 12.1 | 2112.84 | 6397.93 | 149 | 0.52 | 14.431 | 3.601 | |||

B9 | 30 | 9.6 | 1793.41 | 5732.78 | 86 | 0.3 | 27.999 | 6.480 | |||

B10 | 0.05 | 90 | 2.6 | 82.89 | 103.11 | 57 | 0.25 | 12.262 | 5.867 | ||

S1 | 0.20 | 90 | 1.1 | 0.08 | 20.2 | 6169.60 | 30,927.71 | 182 | 2.50 | 15.120 | 3.752 |

S2 | 0.20 | 90 | 0.12 | 18.6 | 2922.98 | 18,230.19 | 182 | 1.67 | 8.449 | 2.833 | |

S3 | 0.20 | 90 | 0.15 | 17.8 | 3634.43 | 20,715.28 | 182 | 1.33 | 11.471 | 3.673 |

Case | Parameter | Non-Dimensional Parameter | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Length L (m) | Alignment Angle ϴ (°) | Sediment Size d_{50} (mm) | Flow Depth h (mm) | Scour Depth d_{s} (cm) | Area of Plane A_{S} (cm^{2}) | Volume V_{S} (cm^{3}) | L’/d_{50} | L’/h | A_{S}/d_{s}^{2} | V_{S}/d_{s}^{3} | |

F1 | 140 | 90 | 1.28 | 6.5 | 40.7 | 50,100 | 776,000 | 1094 | 21.5 | 30.245 | 11.510 |

F2 | 125 | 6.6 | 19.9 | 7770 | 34,000 | 977 | 18.9 | 19.621 | 4.314 | ||

F3 | 125 | 6.9 | 29.4 | 25,370 | 189,000 | 977 | 18.1 | 29.351 | 7.437 | ||

F4 | 125 | 6.6 | 37.2 | 41,160 | 595,000 | 977 | 18.9 | 29.743 | 11.558 | ||

F5 | 109 | 7.0 | 16 | 4730 | 19,000 | 852 | 15.6 | 18.477 | 4.639 | ||

F6 | 109 | 6.9 | 27.3 | 22,340 | 170,000 | 852 | 15.8 | 29.975 | 8.355 | ||

F7 | 109 | 6.6 | 35.9 | 36,690 | 480,000 | 852 | 16.5 | 28.468 | 10.374 | ||

F8 | 94 | 6.7 | 33.4 | 34,480 | 385,000 | 734 | 14.0 | 30.908 | 10.333 | ||

F9 | 94 | 7.0 | 24.3 | 15,920 | 103,000 | 734 | 13.4 | 26.961 | 7.178 | ||

F10 | 94 | 7.0 | 13.1 | 3050 | 9000 | 734 | 13.4 | 17.773 | 4.003 | ||

F11 | 79 | 6.9 | 31.2 | 26,250 | 284,000 | 617 | 11.4 | 26.966 | 9.351 | ||

F12 | 79 | 7.1 | 23.1 | 13,550 | 88,000 | 617 | 11.1 | 25.393 | 7.139 | ||

F13 | 79 | 7.1 | 10.4 | 3090 | 6000 | 617 | 11.1 | 28.569 | 5.334 | ||

F14 | 64 | 7.0 | 29.7 | 23,170 | 236,000 | 500 | 9.1 | 26.267 | 9.008 | ||

F15 | 64 | 7.0 | 8 | 850 | 2000 | 500 | 9.1 | 13.281 | 3.906 | ||

F16 | 64 | 7.2 | 20.8 | 12,450 | 69,000 | 500 | 8.9 | 28.777 | 7.668 | ||

K1 | 30.5 | 45 | 0.8 | 30.2 | 18.99 | / | 106,700 | 381 | 1.39 | / | 15.581 |

K2 | 30.5 | 45 | 18.6 | 22.41 | / | 113,800 | 381 | 0.86 | / | 10.112 | |

K3 | 15.2 | 45 | 30.66 | 16.68 | / | 67,630 | 190 | 2.84 | / | 14.573 | |

K4 | 15.2 | 45 | 30.7 | 26.69 | / | 185,280 | 190 | 2.84 | / | 9.745 | |

K5 | 15.2 | 45 | 18.45 | 27.98 | / | 166,720 | 190 | 1.71 | / | 7.611 | |

K6 | 15.2 | 45 | 18.49 | 17.16 | / | 55,690 | 190 | 1.71 | / | 11.021 | |

K7 | 30.5 | 90 | 18.56 | 22.15 | / | 135,730 | 381 | 0.61 | / | 12.490 | |

K8 | 30.5 | 90 | 30.0 | 25.68 | / | 197,600 | 381 | 0.98 | / | 11.668 | |

K9 | 15.2 | 90 | 18.42 | 15.45 | / | 99,160 | 190 | 1.21 | / | 26.888 | |

K10 | 15.2 | 90 | 18.63 | 9.29 | / | 26,500 | 190 | 1.23 | / | 33.052 | |

K11 | 15.2 | 90 | 30.23 | 13.04 | / | 52,470 | 190 | 1.99 | / | 23.663 | |

K12 | 15.2 | 90 | 30.72 | 16.2 | / | 109,130 | 190 | 2.02 | / | 25.668 | |

K13 | 30.5 | 135 | 18.28 | 25.13 | / | 143,020 | 381 | 0.84 | / | 9.012 | |

K14 | 15.2 | 135 | 18.38 | 30 | / | 202,510 | 190 | 1.70 | / | 7.500 | |

K15 | 15.2 | 135 | 18.4 | 17 | / | 54,350 | 190 | 1.70 | / | 11.062 | |

K16 | 15.2 | 135 | 30.43 | 20.9 | / | 95,530 | 190 | 2.82 | / | 10.464 | |

K17 | 15.2 | 135 | 30.46 | 28.47 | / | 260,370 | 190 | 2.82 | / | 11.283 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, L.; Wang, P.; Yang, W.; Zuo, W.; Gu, X.; Yang, X.
Geometric Characteristics of Spur Dike Scour under Clear-Water Scour Conditions. *Water* **2018**, *10*, 680.
https://doi.org/10.3390/w10060680

**AMA Style**

Zhang L, Wang P, Yang W, Zuo W, Gu X, Yang X.
Geometric Characteristics of Spur Dike Scour under Clear-Water Scour Conditions. *Water*. 2018; 10(6):680.
https://doi.org/10.3390/w10060680

**Chicago/Turabian Style**

Zhang, Li, Pengtao Wang, Wenhai Yang, Weiguang Zuo, Xinhong Gu, and Xiaoxiao Yang.
2018. "Geometric Characteristics of Spur Dike Scour under Clear-Water Scour Conditions" *Water* 10, no. 6: 680.
https://doi.org/10.3390/w10060680