# Riverbed Morphologies Induced by Local Scour Processes at Single Spur Dike and Spur Dikes in Cascade

^{1}

^{2}

^{*}

## Abstract

**:**

^{3}. However, the scour rates increased to a high degree, with the scour depths tending to match those at the first spur dike.

## 1. Introduction

_{L}, in a straight channel and in regime conditions is given (in dimensional form) by 0.47·(Q/f)

^{1/3}, where Q is the discharge and f is Lacey’s silt factor, thus approximating to 1.76·(d

_{50})

^{0.5}, with d

_{50}being the median grain size of the bed sediment. Inglis [3] highlighted how spur dikes projected from banks yield varying scour values, ranging from 1.7 to 3.8D

_{L}, depending on the severity of the river curvature and the length, angle, and position of the spur dike relative to the flow attack.

_{50}of 1.60 mm, and gravel with a d

_{50}of 5.78 mm. The tests were conducted under both clear-water and live-bed regimes.

_{50}, were used: coarse sand with a d

_{50}of 1.5 mm, and fine sand with a d

_{50}of 0.9 mm. The runs lasted 6 h and were performed under both clear-water and live-bed regimes. Single spur dikes with an inclination of 90° were tested by varying their width on the three values of 0.33, 0.67, and 1.00 ft. The main results from the study by Gill [7] were as follows: (i) the scour depth at the equilibrium stage was affected by the size of the bed material; (ii) the scour rate for fine sand was higher than that for coarse sand; (iii) the scour depths were dependent on the approach flow depth; and (iv) the maximum scour occurred for τ/τ

_{c}= 1 (i.e., at the sediment transport inception along the approaching bed). Therefore, the distinction between clear-water scour and scour caused by bed-load transporting flows can be ignored.

_{50}of 0.8 mm and sediment gradation of 1.37. All experiments were conducted under a clear-water regime and lasted for approximately 30 h. In 17 of the 21 cases, the spur dike was submerged. Three spur dike angles were considered: 45° (repelling spur dike), 90° (deflecting spur dike), and 135° (attracting spur dike).

_{50}of 4.3 mm and sediment gradation of 2.35. In the case of a singular spur dike, 36 runs were performed for up to approximately 11 days, considering three angles: 45°, 90°, and 135°. In the case of spur dikes in cascade, 24 runs were performed for up to approximately 52 days, considering three angles: 60°, 90°, and 120°.

_{50}of 0.27 mm and sediment gradation of 1.23.

_{sm}= σ

_{g}

^{−1/3}F

_{d}, the dimensionless ratio b/d

_{50}, and the dimensionless time T, as defined by Oliveto and Hager [13]. The merit of this study mainly consists in considering sand–gravel mixtures as a mobile bed. Thirty-two experimental runs under clear-water regime were carried out in a rectangular channel 24 m long, 1.0 m wide, and 0.50 m deep. The working test section had dimensions of 4.0 m × 1.0 m × 0.4 m, starting 12 m from the flume inlet. Each experiment lasted 20 h.

_{50}of 0.473 mm with some perplexities about the viscosity effects at the flow–sediment interface. However, the results regarding the spur dike spacings would appear interesting: it was found that the maximum scour depth formed in the vicinity of the first spur dike head, and with the increase in the spur dike spacing, the shielding effect of the first spur dike was weakened. In the case of a group of four spur dikes with a spacing of 2b, an integral scour zone was formed in the mainstream area. With a spacing of 3b, the scouring range of spur dike II was largely covered by the scour range of spur dike I. Finally, with a spacing of 4b, the shielding effect of spur dike I only partially affected a section of spur dike II. The local scour morphology of each spur dike remained relatively complete, forming four largely independent scour holes.

## 2. Empirical Equations from Literature

_{eq}is the (maximum) scour depth at the equilibrium stage, both of which are measured on the basis of the free surface; and m and n are coefficients that depend on the diameter of the bed sediment. The author does not explain whether live-bed conditions were utilised in the approach section; however, the data presented and the aspect of the curves of the temporal trend of the scour depth indicate that most of the tests, which served as the basis for Equation (1), were carried out under a clear-water regime [6].

_{s}:

_{0})

^{0.5}, where V is the average flow velocity, g is the gravitational acceleration, and h

_{0}is the approach flow depth; and n is the exponent of the Froude number. The Froude number, opening ratio, angle of inclination of the spur dike, and average drag coefficient of the sediment particle C

_{D}adequately represent the influence of the flow, spur dike, and sediment characteristics on the maximum scour depth. C

_{D}was defined as (4/3)·[(s − 1)gd

_{50}/(ρw

_{s}

^{2})], where s is the specific density ρ

_{s}/ρ, and w

_{s}is the settling velocity of the sediment. The effect of the spur dike inclination was only preliminarily investigated. However, under the given hydraulic and sedimentological conditions, the maximum scour depth attained the greatest magnitude for a spur dike inclination of 90°. The scour depth was smaller for all other inclinations, both upstream and downstream.

_{c}is the bed shear stress at the approaching bed when sediment transport inception occurs.

_{s}(t):

_{s}(t)/L

_{R}, (where L

_{R}is the reference length (sb

^{2})

^{1/3}with s as the spur dike height); N = 1.25 (i.e., value of the shape coefficient N for a vertical abutment or spur dike); σ is the sediment gradation; and F

_{d}is the densimetric Froude number. The densimetric Froude number F

_{d}is defined as V/(g’d

_{50})

^{0.5}, where g’ = g[(ρ

_{s}− ρ)/ρ], ρ

_{s}is the sediment density, ρ is the water density, and g is the gravitational acceleration. T is dimensionless time, defined as [σ

^{1/3}(g’d

_{50})

^{0.5}]·t/L

_{R}.

_{e}is the excess spur dike Froude number, (V − ξV

_{c})/(gb)

^{0.5}, with V average approach flow velocity; V

_{c}is the approaching flow velocity at the sediment transport inception; ξ is a shape coefficient equal to 0.5 for vertical wall abutments and spur dikes; and g is the gravitational acceleration.

## 3. Experiments

#### 3.1. Experimental Setup

_{50}of 1.7 mm, sediment gradation σ = (d

_{84}/d

_{16})

^{0.5}of 1.5, and a relative density Δ = (ρ

_{s}− ρ)/ρ of 1.65 was used for the mobile bed, where ρ

_{s}is the sediment density and ρ is the water density. The experiments were performed by testing either a single spur dike or spur dikes in cascade (three, five, and seven, respectively) under clear-water conditions. A rectangular plexiglass plate with a thickness of 10 mm and width b of 0.25 m was used as impermeable spur dike. The inclination angle of the dike was maintained at 90° in all experiments. According to Gisonni and Hager [20], a typical spacing of 3b between the elements was selected for the spur dikes in cascade. Figure 1 shows a scheme for the arrangement of the spur dikes in this study.

^{3}/s, representative of a moderate approach flow intensity and 0.075 m

^{3}/s, representative of a higher approach flow intensity. The corresponding approach flow depth, h

_{0}, was approximately 0.16 and 0.23 m, respectively. Water discharge was measured with an accuracy of ±5% by means of an orifice plate installed in the water-feeding circuit. However, the discharge was also accurately measured by a volumetric method using the tank at the end of the channel. Hereafter, the discharge values from the volumetric method will be considered. The water surface was measured using a conventional point gauge with an accuracy to the nearest millimetre. The bed topography was accurately surveyed with an accuracy of the order of the grain size (i.e., ±2 mm) using a shoe gauge with a horizontal plate (4 mm × 2 mm) at its base. Accurate measurements of the bed morphological patterns (i.e., both eroded and mounded areas) were performed during and at the end of each run. More than 500 bed-level data points were collected for each survey, moving the shoe gauge according to an adaptive grid (from 0.02 m × 0.02 m to 0.1 m × 0.1 m), which followed the bed changes. Therefore, the time t associated with a given survey during the run was considered as the average time representative of the temporal interval in which the survey was completed. At the earlier scour stage (before 8 h from the run starting), the survey was completed within around 1 h, while in the later stages (after 8 h from the run starting), the survey was completed within around 2 h.

_{di}is the densimetric Froude number at the inception of sediment transport along the approach section, as defined by Oliveto and Hager [10]. When the ratio of F

_{d}to F

_{di}is ≤ 1, the approaching flow conditions develop under a clear-water regime, whereas for F

_{d}/F

_{di}> 1, a live-bed regime occurs.

#### 3.2. Scale Effects

_{50}of 1.7 mm and was 12 m long; this median grain size was used to minimise the viscosity effects at the flow–sediment interface. The two criteria highlighted by Oliveto and Hager [13] for avoiding viscosity effects are: d

_{50}/h

_{0}> 0.002 (in the present case, d

_{50}/h

_{0}is greater than 0.007) and d

_{50}> 0.8 mm. However, the length of the mobile bed allowed the complete development of the bed morphological patterns downstream of the last spur dike.

^{5}) to be considered turbulent. Moreover, the distance between the first spur dike and channel inlet was 4 m, which is sufficient to allow a fully developed turbulent flow because the approaching flow conditions developed primarily over a granular bed surface. Some sporadic measurements of vertical velocity profiles supported this issue. Figure 2 shows the experimental data points collected at the approaching section 1 m upstream from the first spur dike for runs C-I 1 and C-II 1. A micro-propeller current meter, from Nixon Instruments Ltd., was used to measure the longitudinal velocity component. The micro-propeller had a cage diameter of 15 mm and an accuracy of ±1.5% of the true velocity. The experimental data points were compared with the logarithmic law of the wall v

_{x}/u

_{∗}= (1/k)·ln(z/z

_{0}), where v

_{x}is the longitudinal velocity component at the distance z from the bed surface, u

_{∗}is the shear velocity, k is the von Kármán constant (k ≈ 0.4), and z

_{0}is the distance from the boundary at which the idealised velocity given by the law of the wall goes to zero. z

_{0}is equal to 0.115ν/u

_{∗}for hydraulically smooth flow, where ν is the kinematic viscosity. Though the bed surface was granular in this study, the grain sizes were of the order of 1−2 mm; therefore, the bed surface was assumed to be a smooth surface. Moreover, the shear velocity was estimated considering that the bed surface was nearly horizontal (i.e., slope S << 1). As can be seen, the approaching flow was almost two-dimensional (i.e., the vertical velocity profiles at different transverse section are nearly the same) and the experimental data points fit with the law of the wall satisfactorily.

#### 3.3. Experimental Observations

^{3}/s, approach flow depth of 0.158 m, and sand bed with a median grain size d

_{50}of 1.7 mm and sediment gradation σ of 1.5. Notably, the local scour processes at various elements developed in an evident manner at different times. At 1.5 h, local scour developed primarily at the tip of the first spur dike, and the amount of removed sediment tended to propagate downstream, stopping only upstream of the second and third elements. Conversely, the bed levels around the fourth and fifth spur dikes, as well as further downstream, were undisturbed. After 23 h, the scour hole around the first spur dike was magnified, creating erosion on the part of the bank just upstream of the first spur dike itself. Interestingly, the removed sediments appeared almost uniformly distributed between the various spur dikes and, more generally, along the part of the channel in which the spur dikes were present. After 49 h, the bed morphology patterns became more complex; the local scour hole at the first spur dike tended to elongate toward the tip of the second spur dike. Sediment deposition tended to be concentrated immediately upstream of the third, fourth, and fifth elements, primarily around their tips. During this stage, some deposition occurred downstream of the spur dike. After 72 h, local scouring patterns were observed around the second and third spur dikes. However, local scour at the second spur dike appeared primarily because of the expansion of the scour hole at the first spur dike. Interestingly, the sediment deposition areas observed in the previous stage became more pronounced and leaned toward the bank. Finally, after 145 h, local scour and aggradation areas appeared well distinguished: local scour holes were evident around all five spur dikes. Similarly, aggradation zones were discernible along the right bank, and more between the second and third elements, between the third and fourth elements, between the fourth and fifth elements, and just downstream of the fifth element.

^{3}/s and h

_{0}= approximately 0.157 m.

## 4. Data Analysis and Results

#### 4.1. Uncertainty and Statistical Analysis

_{d}and log(T) from the experiments in the study were 1.58 and 5.67, respectively.

_{d}and log(T). The error can also be expressed as the relative error (RE = ΔY/Y) and as the relative sensitivity (RS = (X ΔY)/(Y ΔX)) [12]. The error ΔY is the deviation in input Y by the deviation ΔX being the error in input X. The sensitivity is investigated by ±10% in each input parameter and the results are recorded in Table 3. The results listed in Table 3 specify that F

_{d}is the most sensitive input parameters among them. The relative sensitivity of F

_{d}is 1.47 for log(T) in 10% increment in X and 1.54 in 10% decrement in X, respectively. Therefore, the computing accuracy of the proposed approach by Oliveto and Hager [10] greatly depends on F

_{d}. Thus, the dimensionless scour depth Z at the first spur dike is significantly dependent on the densimetric Froude number, F

_{d}.

#### 4.2. Comparison of the Experimental Data to Literature Formulas for Scour at the Equilibrium Stage

_{c}less than 0.5, as in the present study.

#### 4.3. Temporal Evolution of Scour Depth

^{0.5}/F

_{d}

^{1.5}, T according to the dimensionless variables introduced by Oliveto and Hager [10]. The reason for the use of this plane is behind the structure of Equation (5). In fact, when a semi-logarithmic plot is used with the time T on the x-axis and the y-axis on a linear scale, the experimental datapoints should collapse on a single line independently of F

_{d}and σ. Figure 7a shows the temporal scour evolution around the first spur dike, and Figure 7b extends the data analysis to the local scour around the other spur dikes. The predictions according to Oliveto and Hager [10] are represented by the straight line with ±25% prediction-bands error, as suggested by the authors.

## 5. Discussion

^{4}, instead presenting an underestimation. Conversely, the formula by Gill [7] is more conservative, but underperforms in the case of floods of limited duration.

^{3}. However, the scour rates increased to a high degree, with the scour depths tending to match the observed values at the first spur dike. Finally, the scour depths at the fifth, sixth, and seventh dikes were found to be approximately zero, with fluctuating trends owing to aggradation effects from upstream. Only these elements can be considered as effectively protected from upstream elements.

## 6. Conclusions

- -
- Some limitations of the formulas for the equilibrium scour depth at the first spur dike reported in the literature were emphasised. Underestimations (i.e., the ratio between the actual and predicted values) up to 160% and overestimation (i.e., the ratio between the predicted and actual values) up to 200% at the earlier scour stages were found.
- -
- The temporal evolution of scour depth at the first spur dike was satisfactorily predicted with a coefficient of correlation (CC), mean absolute error (MAE), and mean square error (MSE) of 0.91, 0.085, and 0.0097, respectively.
- -
- Similar scour hole geometries around the first spur dike were observed in all runs, with the maximum scour depth (at the end of each run) remaining almost unchanged, but strictly increasing as the number of spur dikes increased.
- -
- The scour depths on the second spur dike, although slightly increasing over time, were significantly lower than those at the first spur dike. Therefore, this experimental study confirmed some literature findings that also found that the scouring range at the second spur dike is largely covered by the deposition of sediment flowing from the first spur dike for a spacing of 3b.
- -
- For the third, fourth, and fifth spur dikes, the scour processes were delayed and started at a dimensionless time T greater than approximately 10
^{3}. However, the scour rates increased to a high degree, with scour depths tending to match the observed values at the first spur dike. In the case of the first spur dike, as in the case of the other spur dikes around which the local scour process was quite advanced, the position of the maximum scour depth migrated from the spur dike head to the channel wall. - -
- The scour depths at the fifth, sixth, and seventh spur dikes were found to be approximately zero; therefore, only these elements could be considered effectively protected from the upstream ones.
- -
- Three distinct mounds of decreasing size, moving downstream, and interlaced by eroded areas were observed for a single spur dike. The primary and secondary mounds reached the bank, requiring protection, and extended transversally to the channel axis. As the number of spur dikes increased, the aggradation areas, although of a more modest size, tended to be confined between adjacent dikes, leaving the central region of the channel less disturbed.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Channel cross section at spur dike location (scheme on the left) and spacing for spur dikes in cascade (scheme on the right).

**Figure 2.**Vertical velocity profiles at the approaching section 1 m upstream from the first spur dike for runs C-I 1 (diagram on the left) and C-II 1 (diagram on the right). Transverse sections (●) at the centre line, (+) at a distance of 0.25 m from the channel wall where the spur dike is fixed, and (✕) at a distance of 0.75 m from the same channel wall.

**Figure 3.**Flow patterns during the run CI-5 (

**a**) and vortices visualisation using small spheres of polystyrene as tracers during CI-7 (

**b**). The spheres of polystyrene had a diameter of about 4 mm and they were buoyant. Therefore, they described surface flow structures and attempted establishing a connection to the bed surface effects.

**Figure 4.**Contour maps, from top to bottom, at 1.5, 23, 49, 72, and 145 h of the observed morphological bed patterns for run CI-5. Dimensions are in cm. The red dot indicates the position of the maximum scour depth at each spur dike.

**Figure 5.**Contour maps at 145 h for the bed morphology of runs, from top to bottom, CI-1, CI-3, CI-5, and CI-7. Dimensions are in cm. The red dot indicates the position of the maximum scour depth at each spur dike.

**Figure 7.**Experimental data points (

**a**) for the first spur dike (symbols and colours are the same as those used in Figure 6) and (

**b**) for all spur dikes in the case of run CI-7.

**Table 1.**Summary of the main test conditions. Regarding the label associated with each run, “CI” refers to the runs with Q = 0.042 m

^{3}/s (and F

_{d}= 1.62 on average), “CII” to the runs with Q = 0.075 m

^{3}/s (and F

_{d}= 1.96 on average), and “No.” refers to the number of dikes.

Run | No. of Dikes | Q (m ^{3}/s) | h_{0}(m) | F_{d}(-) | F_{di}(-) | t (hrs) | R_{e}·10^{5}(-) |
---|---|---|---|---|---|---|---|

CII-1 | 1 | 0.075 | 0.226 | 2.000 | 3.565 | 25 | 1.812 |

CII-3 | 3 | 0.075 | 0.225 | 2.007 | 3.564 | 26 | 1.814 |

CII-5 | 5 | 0.075 | 0.236 | 1.916 | 3.582 | 26 | 1.788 |

CII-7 | 7 | 0.075 | 0.234 | 1.932 | 3.569 | 25 | 1.793 |

CI-1 | 1 | 0.042 | 0.150 | 1.688 | 3.391 | 144 | 1.134 |

CI-3 | 3 | 0.042 | 0.162 | 1.563 | 3.425 | 150 | 1.113 |

CI-5 | 5 | 0.042 | 0.158 | 1.602 | 3.414 | 145 | 1.120 |

CI-7 | 7 | 0.042 | 0.157 | 1.613 | 3.411 | 318 | 1.122 |

Instruments | No. of Dikes |
---|---|

Point gauge | ±1 mm |

Shoe gauge | ±2 mm |

Low-speed probe | ±1.5% true velocity |

Orifice plate | ±5% |

**Table 3.**Sensitivity analysis for the approach by Oliveto and Hager [10].

Percentage of Change | X | ΔX | ΔY | AS | RS | RE |
---|---|---|---|---|---|---|

+10% increase | F_{d} | 0.158 | 0.120 | 0.759 | 1.332 | 0.133 |

log(T) | 0.567 | 0.078 | 0.138 | 0.909 | 0.091 | |

−10% increase | F_{d} | 0.158 | −0.114 | −0.723 | −1.712 | −0.171 |

log(T) | 0.567 | −0.078 | −0.138 | −1.111 | −0.111 |

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## Share and Cite

**MDPI and ACS Style**

Aung, H.; Onorati, B.; Oliveto, G.; Yu, G.
Riverbed Morphologies Induced by Local Scour Processes at Single Spur Dike and Spur Dikes in Cascade. *Water* **2023**, *15*, 1746.
https://doi.org/10.3390/w15091746

**AMA Style**

Aung H, Onorati B, Oliveto G, Yu G.
Riverbed Morphologies Induced by Local Scour Processes at Single Spur Dike and Spur Dikes in Cascade. *Water*. 2023; 15(9):1746.
https://doi.org/10.3390/w15091746

**Chicago/Turabian Style**

Aung, HtayHtay, Beniamino Onorati, Giuseppe Oliveto, and Guoliang Yu.
2023. "Riverbed Morphologies Induced by Local Scour Processes at Single Spur Dike and Spur Dikes in Cascade" *Water* 15, no. 9: 1746.
https://doi.org/10.3390/w15091746