## Abstract

Hydraulic experiments on installing a permeable spur dike at three positions (1/4, 1/2, and 3/4) on the concave bank of the bend of a spillway chute with three angles (45°, 60° and 75°) were carried out for studying the backwater condition in front of the permeable spur dike. Results show that the maximum backwater height occurs at the cross-section where the permeable spur dike meets the concave bank of the bend. A formula for the maximum backwater height was derived by the employment of the principle of momentum conservation, and the formula indicates that the height is influenced by the geometric parameters of the permeable spur dike and the bend, the layout of the spur dike in the bend, and the inflow discharge. Based on experimental data, a regression analysis was implemented on the water depth coefficient in the formula. Furthermore, the maximum backwater height can be obtained through the water depth at the concave bank of the end of bend when the parameters and installing pattern of the permeable spur dike are determined.

## INTRODUCTION

In the bend of a river, a spur dike can be defined as an elongated obstruction having one end on the bank and the other end projecting into the current, and it is considered to be a river training structure with great importance in bank protection and river regulation which has now been extensively used in all countries to enhance navigation, to improve the control of floods (Huthoff *et al.* 2012), and to protect banks from erosion (Ouyang & Lu 2016). The permeable spur dike is a novel type of river training structure which has developed rapidly since the 1970s (Liu *et al.* 2008) relative to the conventional impermeable spur dike (Safarzadeh *et al.* 2016; Vaghefi *et al.* 2016; Karami *et al.* 2017). The permeable spur dike possesses a certain permeable capacity due to the existence of water-pervious holes in it, and a part of the inflow can be blocked in front of the dike while other parts of the flow can go through the water-pervious holes, which decrease the elevation of the backwater in front of the dike and the pressure difference between the front and rear surfaces of the dike. Additionally, the permeable spur dike is simple in structure and reliable in operation and it also has advantages such as relatively easy maintenance, better stability and less investment than the traditional impermeable spur dike. So far, relatively limited studies have been conducted on the permeable spur dike in a natural channel and regrettably, similarly to the solid spur dike, the focus of these studies has also been mainly on bank erosion (Kang *et al.* 2011), sediment transport (Gu *et al.* 2011) and improving the effect of water–sediment regulation (Zey *et al.* 2018), while no research on the application of a permeable spur dike in the spillway drainage channel has been performed.

The spillway drainage channel is different from the natural channel: the flow in it falls within supercritical flow and it is often built with a bend section due to the influence of topography, geology and other conditions. Additionally, because of the effect of transverse circulation inside the bend, the convex bank and concave bank water-level decreases and increases, respectively, which makes the water surface distribution in the bend uneven and flow conditions more complicated and more drastic; it even affects the energy dissipation in the stilling basin immediately downstream of the drainage channel. So far, various methods for improving the water surface effect of a bend have been explored by researchers, such as the method of bed transverse fan-shaped elevation (Northwest Hydraulic Research Institute of China 1961), a sloping ridge (Zhou *et al.* 2014) and a guide wall (Zhang *et al.* 2016). Yang *et al.* (2018) studied the improving effect of the water surface in the bent spillway with a permeable spur dike using an indicator named water surface uniformity, and the results demonstrate that the water surface effect in the bend can be significantly improved when a permeable spur dike is installed at an appropriate angle and location in the bend. Nevertheless, what is worth considering in engineering design is the height of the backwater in front of the permeable spur dike, which affects the height of the side walls of the bend, when it is installed in the bend. Therefore, a study is necessary to investigate the backwater height in front of a permeable spur dike, which motivates this work.

On the research results of Yang *et al.* (2018), this paper carried out an analysis of the backwater height in front of the permeable spur dike, then, based on the analysis, a theoretical derivation for the calculation of the maximum backwater height in front of the permeable spur dike was performed in terms of the principle of momentum conservation. The results obtained can be a reference for designing the height of side walls of the bend in a chute.

## MATERIAL AND METHODS

### Experimental arrangement

The experiment was carried out in the water conservancy laboratory of Shandong Agricultural University. The experimental system consists of a pump, a high water pond, a water valve, a model test area, water supply pipelines, a flow-steadying grid, a tail water pond, backwater channels, an underground reservoir and an electromagnetic flow meter, as shown in Figure 1. In this system, the water valve and electromagnetic flow meter are used to control the measurement of experimental flow rate and the model test area is for placing the experimental model and measuring water depth and flow velocity. The model test area and tail water pond are in a big model pond with a length and a width of 20 m and 6 m, respectively. The water volume of the underground reservoir is 400 m^{3}.

### Experimental model

The experimental model is made of PVC plastic sheets with a thickness of 8 mm, and comprises an intake section 3 m long, an upstream straight section 0.2 m long, a bend section with center line and axial radius of 1.5 m and 1.2 m respectively, and a downstream straight section 2 m long. The bottom slope of the whole chute is 0.02. The chute has a rectangle profile with a net width of 0.5 m. The permeable spur dike is also made of PVC plastic sheets with a thickness of 10 mm. The water-pervious holes are circular and arranged symmetrically on the spur dike. The layout of the bend of the chute with spur dikes and the profile of the permeable spur-dike are shown illustratively in Figure 2.

### Experimental schemes

This paper mainly analyzed the maximum backwater height in front of a permeable spur dike, then, based on the analysis, a calculation formula for the maximum backwater height was theoretically derived. Therefore, when analyzing the maximum backwater height in front of a permeable spur dike, the installing location and installing angle of the dike in the bend are mainly taken into account irrespective of the influence of the size of permeable spur dike on the maximum backwater height. Different sizes and water permeabilities of the permeable spur dike are adopted when deriving the calculation formula for the maximum height of the backwater in front of it. The geometry of the permeable spur dike for different purposes is shown in Table 1. The installing locations of the dike are respectively at 1/4, 1/2 and 3/4 of the concave bank of the bend. The installing angle of the dike represents the angle between the spur dike and the tangent of the concave bank of the bend. According to other former experiments related to the installing angle of the spur dike, the permeable spur dike at each installing location is installed with three angles of 45°, 60°, 75°, respectively. The experimental scheme is shown in Table 2, where scheme 10 is a control scheme. The experimental discharges of 50, 80, 100, 120, and 150 m^{3}/h are utilized in each scheme.

Purpose | The geometry of the permeable spur dike (m) | ||||
---|---|---|---|---|---|

Length | Height | Thickness | Diameter of the water-pervious hole | Permeable rate | |

Analysis of the maximum backwater height | 0.2 | 0.06 | 0.01 | 0.014 | 38.5% |

Calculation of the maximum backwater height | 0.2 | 0.06 | 0.01 | 0.014 | 38.5% |

0.2 | 0.08 | 0.01 | 0.014 | 38.5% | |

0.25 | 0.08 | 0.01 | 0.014 | 36.9% | |

0.25 | 0.06 | 0.01 | 0.014 | 36.9% | |

0.2 | 0.06 | 0.01 | 0.01 | 19.6% | |

0.2 | 0.08 | 0.01 | 0.01 | 19.6% | |

0.25 | 0.08 | 0.01 | 0.01 | 18.8% | |

0.25 | 0.06 | 0.01 | 0.01 | 18.8% |

Purpose | The geometry of the permeable spur dike (m) | ||||
---|---|---|---|---|---|

Length | Height | Thickness | Diameter of the water-pervious hole | Permeable rate | |

Analysis of the maximum backwater height | 0.2 | 0.06 | 0.01 | 0.014 | 38.5% |

Calculation of the maximum backwater height | 0.2 | 0.06 | 0.01 | 0.014 | 38.5% |

0.2 | 0.08 | 0.01 | 0.014 | 38.5% | |

0.25 | 0.08 | 0.01 | 0.014 | 36.9% | |

0.25 | 0.06 | 0.01 | 0.014 | 36.9% | |

0.2 | 0.06 | 0.01 | 0.01 | 19.6% | |

0.2 | 0.08 | 0.01 | 0.01 | 19.6% | |

0.25 | 0.08 | 0.01 | 0.01 | 18.8% | |

0.25 | 0.06 | 0.01 | 0.01 | 18.8% |

Case no. | The location of spur dike | Arrangement angle of spur dike (°) |
---|---|---|

1 | 1/4 of the concave bank | 45 |

2 | 60 | |

3 | 75 | |

4 | 1/2 of the concave bank | 45 |

5 | 60 | |

6 | 75 | |

7 | 3/4 of the concave bank | 45 |

8 | 60 | |

9 | 75 | |

10 | without spur dikes |

Case no. | The location of spur dike | Arrangement angle of spur dike (°) |
---|---|---|

1 | 1/4 of the concave bank | 45 |

2 | 60 | |

3 | 75 | |

4 | 1/2 of the concave bank | 45 |

5 | 60 | |

6 | 75 | |

7 | 3/4 of the concave bank | 45 |

8 | 60 | |

9 | 75 | |

10 | without spur dikes |

### Experimental measurement

Five longitudinal measuring lines were arranged symmetrically along the direction of water flow in the bend, including a center line, two lines with 1/4 width of the bend from both sides of the center line, and two lines close to the left and right side walls. Thirteen lateral measuring lines were uniformly arranged along the bend. The water-depth measuring points, amounting to 65, are the intersection points of the longitudinal and lateral measuring lines in the bend.

The experimental gauge elements primarily focus on water depth and discharge. The discharge was controlled by a water valve and was measured by the E-magC-type electromagnetic flowmeter produced by China Kaifeng Instrument Co., Ltd. The digital water-level point gauge was adopted for the measurement of water depth, which was displayed on a screen with a precision of 0.01 mm. Compared with the traditional water-level point gauge, the digital one can display the value of water level on a small screen, which has a higher accuracy and efficiency than the traditional one.

## RESULTS AND DISCUSSION

### The improving effect of a single permeable spur dike on the water surface in a bend

In order to quantitatively reflect the improving effect of a permeable spur dike on the water surface in a bend, water surface uniformity with different layouts of the permeable spur dike is calculated using the water depth measured in this experiment according to the formula for water surface uniformity of a bend proposed by Zhang *et al.* (2016). Figure 3 shows the surface uniformity for different installing angles and locations of the permeable spur dike under different unit width discharges.

As can be seen from Figure 3, the water surface uniformity increases a lot with a permeable spur dike in the bend. For instance, compared with the case of no permeable spur dike in the bend, when the permeable spur dike is installed with the angle of 75° at 1/2 of the bend, the water surface uniformity increases by 11.2% for the unit width discharge equaling 200 m^{2}/h.

Figure 3(a)–3(c) also indicate that for large discharges, no matter what the installing location of the permeable spur dike in the bend is, the water surface uniformity in the bend is always the largest with the installing angle being 75°, while 45° is the smallest, from which it can be concluded that for the same installing angle of permeable spur dike, the installing location of the dike is not responsible for the improving effect of the water surface in the bend. When the installing angle is 75°, the water surface improving effect is the best, and 60° is the second best. Nevertheless, the improving effect of the water surface varies when the permeable spur dike is installed with the same angle at different installing locations. For example, when the permeable spur dike is installed with the angle of 45°, installing at 1/2 of the bend has the best improving effect on the water surface and when installed with the angle of 75°, installing at 3/4 of the bend has the best effect.

### Analysis on the height of upstream backwater of a permeable spur dike

When a permeable spur dike is installed in the bend of the spillway chute, the partial flow in front of the dike will be obstructed by it, which gives rise to a decrease and an increase in the flow velocity and the water level in the front of the spur dike; meanwhile, the backwater will occur within a certain distance upstream of the spur dike. In terms of the upstream backwater height measured in this experiment, contour maps of backwater height in front of the permeable spur dike for different installing angles and locations in the bend under the unit width discharge of 200 m^{2}/h are vividly illustrated in Figure 4, where ‘1/4–45°’ denotes the contour map produced by the spur dike at 1/4 of the bend with 45° and the rest are denoted in the same manner.

As can be seen from these figures, the maximum backwater height in front of the permeable spur dike occurs at the cross-section where the spur dike meets the concave bank of the bend. The range of backwater caused by the permeable spur dike varies with the change of the installing angles and locations of the dike. Under the conditions of the same installing location, the greater the installing angles of the permeable spur dike, the greater the range of backwater occurs. For example, in Figure 4(a)–4(c), the permeable spur dike is installed at 1/4 of the bend, and the installing angles corresponding to the largest and smallest range of backwater are 75°and 45° respectively.

### Calculation of the maximum backwater height in front of the permeable spur dike

From the above analysis, it can be concluded that the maximum height of the backwater in front of the permeable spur dike occurs at the cross-section where the spur dike meets the concave bank of the bend. Based on the experimental results, a theoretical derivation was conducted using the principle of momentum conservation for the calculation of the maximum backwater height in front of the permeable spur dike.

### Fundamental principle

#### Basic assumption

Figure 5(a) shows the momentum analysis in the bend; , represent the length and width of the permeable spur dike, respectively, represents the permeable rate of the permeable spur dike, and therefore the effective area of the permeable spur dike . The installing angle of the spur dike in the bend is , the deflection angle of installing position of the spur dike in the bend is , the deflection angle of the bend is , the width of the bend is *b*, and the radius of the concave bank of the bend is . Section I is the anterior section of the permeable spur dike, represents the flow velocity calculating the flow momentum of section I, and represents the maximum backwater height. Section II is the end-section of the bend, represents the flow velocity calculating the flow momentum of section II, and represents the water depth of the concave bank of section II. For the convenience of deriving the calculation model, the assumptions are as follows.

(1) In order to facilitate the analysis and solution of the change of the water flow momentum at the concave bank in the curved channel, the curved side wall is simplified into a straight side wall whose length is the distance *AB* in Figure 5(a), represented by *l*, from the concave bank at the end of the bend to the intersection of the extension line at the concave bank of the bend end and the tangent line of the point where the permeable spur dike meets the concave bank of bend, and thereby .

(2) The permeable spur dike only affects the flow momentum within the scope of its height, and the flow momentum higher than the height of the permeable spur dike is not affected by it. In addition, the flow momentum affected by the permeable spur dike continues to be affected by the downstream concave bank of the bend.

(3) After the complex action of the permeable spur dike and the side wall on the flow in the bend, the direction of the flow momentum at the end-section of the bend has changed, but the magnitude remains the same, which equals the magnitude of the flow momentum at section I, namely .

(4) When calculating the flow momentum of section I, is taken as the water depth for computing the flow momentum and is the corresponding flow velocity. Likewise, when calculating the flow momentum of section II, is taken as the water depth for computing the flow momentum and is the corresponding flow velocity. When calculating the flow momentum affected by the permeable spur dike, is taken as the water depth for computing the flow momentum. However, due to the flow affected by the permeable spur dike at section I, take . , , as water depth adjustment coefficient for momentum calculation of the corresponding computed cross-sections, respectively. Figure 5(b) is formed by moving the above-mentioned vectors together by the shifting method.

(5) The change of the flow momentum in the bend is only caused by the permeable spur dike. So, the side wall of the concave bank in the bend and the effect of other factors on the flow momentum are neglected.

#### Calculation of the maximum backwater height in front of the permeable spur dike

### Simplification of equations

The above Equation (17) is the formula for calculating the maximum height of the backwater in front of a permeable spur dike.

### Calculation of the water depth adjustment coefficient

In Equation (17), the water depth adjustment coefficient affecting the solution of the maximum backwater height is unknown. So the calculation of is critical.

*p*-value (

*p*-value = 1 × 10

^{−122}) is less than 0.05, which indicates that the original hypothesis should be rejected under a significance level of 0.05, and the regression equation should be considered as acceptable. The

*p*-values of each constant term and linear term in the regression equation are all less than 0.05, so all the constant terms and linear terms in the regression equation are regarded as satisfactory. Therefore, the regression equation of the water depth coefficient

*α*is:

#### Error test of the formula of water depth adjustment coefficient

*RE*is the relative error,

*X*is the maximum backwater height obtained by substituting Equation (19) into Equation (17), and

*Y*is the experimental measured maximum backwater height.

Through substituting the maximum backwater height obtained by Equation (19) and by the experimental measurement into Equation (20), the relative errors of the maximum backwater height for every group were obtained. According to the calculated results, 187 out of 216 groups of data have a relative error of less than , accounting for 86.1%. Only nine groups of data have a relative error of greater than , accounting for 2.3%. Furthermore, the maximum relative error is 16.4% and the overall average error is 2.88%. Therefore, this indicates that the depth adjustment coefficient can be calculated by Equation (19). The values of observation vs prediction of the model are shown in Figure 6.

## CONCLUSIONS

This work performed an analysis and a calculation of the maximum backwater height in front of the permeable spur dike in terms of the experimentally measured data. According to the preceding analysis and discussions, the following conclusions are drawn:

The range of backwater caused by the permeable spur dike varies with the change of installing patterns of the spur dike in the bend and the maximum backwater range occurs when the spur dike is installed at 1/2 of the bend with the installing angle of 75°. Additionally, the maximum backwater height occurs at the cross-section where the spur dike meets the concave bank of the bend.

The formula for calculating the maximum backwater height in front of the permeable spur dike was obtained based on the principle of momentum conservation, and the formula demonstrates that the maximum backwater height is influenced by those factors of the geometric parameters of the permeable spur dike and the bend, the layout of the permeable spur dike in the bend, and the inflow discharge.

Finally, a regression analysis was performed for the water depth adjustment coefficient in the formula of the maximum backwater height according to the measured experimental data, and after the regression, the maximum backwater height in front of the permeable spur dike can be obtained when knowing the water depth at the concave bank of the end of bend, which can provide a reference for practical engineering.

## REFERENCES

## Author notes

Jinmeng Yang, Jing Zhang, and Qinghua Zhang are co-first authors for this work.