## Abstract

In order to study the water level at the convex and concave banks after installing a guide wall in a spillway chute bend, with the original condition that the Fr at the entrance of the channel bend is larger than 1.0 (supercritical flow) when there is no the guide wall, systematic experiments with the guide wall were conducted for three radii (2.4*B*, 3.2*B* and 4*B*; *B* is the width of the channel), bottom slopes (0.01, 0.005 and 0.02), and discharges (50, 100 and150 m^{3} h^{−1}). Results show that, firstly, after installing a guide wall, the Fr becomes smaller and even lower than 1.0, which means the flow status changes from supercritical to subcritical in some conditions with the help of the guide wall. Secondly, the water depth at the convex bank decreases with the increase of the relative axial radius while this presents to be adverse at the concave bank. Thirdly, for water surface differences in cross-sections, the maximum value decreases with the increase of the relative axial radius, and increases with the increase of the discharge per unit width or the bottom slope. Additionally, a novel formula for calculating the maximum water surface difference was obtained in this article.

## NOTATION

*B*the net width of the channel (m)

*C*the Chezy coefficient (-)

*g*the gravitational acceleration (m s

^{−2})*h*the average water depth at entrance of the bend (m)

*i*the bottom slope (-)

*J*_{r}the water surface transverse gradient in the bend (-)

*k*the Karman constant (-)

*R*the axial radius of the bend (m)

*v*the average velocity at entrance of the bend (m s

^{−1})*V*_{cp}the longitudinal vertical velocity (m s

^{−1})*Q*the discharge (m

^{3}s^{−1})*q*the discharge per unit width (m

^{2}s^{−1})*α*_{0}the velocity distribution coefficient or correction factor (-)

- Δ
*z*the maximum water surface difference (m)

## INTRODUCTION

The flow in the bend of a spillway chute is generally a supercritical flow and has characteristics including large fluctuations along the flow direction, high and low water levels on cross-sections, a cross-shock wave with diamond-shape on the free surface, and obvious turbulent character in three dimensions. Many researchers have made achievements in the hydraulics in the spillway or the hydraulics of supercritical flow. Reinauer & Hager (1997) studied the flow pattern, surface profiles and typical velocity fields of the supercritical bend flow with both theoretical and experimental methods. Hessaroeyeh & Tahershamsi (2009) highlighted the prediction of free flow surface profile and classical cross-sections in the supercritical bend flow. Parsaie *et al.* (2015) analyzed the hydraulic characteristics of flow over a dam spillway and the impact of guide wall shape on the flow pattern in a numerical method based on data from Kamal-Saleh dam. Dehdar-behbahani & Parsaie (2016) investigated an engineering case (Balaroud dam) with both experimental and numerical methods and found a proper model for simulating the flow pattern around the guide wall. Jahani *et al.* (2018) studied the impact of guide wall and pier geometry on the hydraulic characteristics of a dam spillway in a numerical way and they gave the conclusion that the vertical inclination of the guide walls and piers was the main affecting factor.

In order to eliminate the cross-shock wave in the bend and reduce water surface differences on cross-sections, it is necessary to take engineering measures to adjust the bending flow. Knapp (1951) firstly suggested several methods to improve hydraulic characteristics in the river bend, and many technologies were then studied, like the fan-shaped bottom to balance the centrifugal force (Northwest China Institute of Hydraulic Research 1961), water flaps (Beltrami *et al.* 2007) and a convex corner at the inner bend wall (Jaefarzadeh *et al.* 2012), the sill (Pagliara *et al*. 2016) and the spur dike (Lee & Jang 2016; Vaghefi *et al.* 2018).

A guide wall installed in the bend in the spillway chute is a hydraulic structure designed to protect banks by controlling the flow direction and velocity, which has been broadly applied in engineering practices. In China, Dongzhou reservoir, Huangqian reservoir and Xinzhuang reservoir in Shandong province have adopted this technology and have shown beneficial effects. The guide wall along the axis or two sides of the axis in a bending channel divides the bend into two or more chutes, thereby reducing the water surface differences between the concave and convex sides of the cross-section, making the flow more even. At present, research on this technology has mainly achieved the following results: Odgaard & Kennedy (1983) made a preliminary discussion on the improvement of the flow-control mechanism of a submerged vane in a bending channel. Rao & Prabhu (2004) investigated the effect of a guide vane and a group of guide vanes on the pressure-drop distribution in a 180° bend. In an engineering case, Yan *et al.* (2010) studied the effect of a guide wall in ski jump dissipation. Han *et al*. (2011) used Fluent to analyze the reduction of secondary flow in a bend with one vane and three vanes. Zhai (2014) conducted experimental research on the flow characteristics of different guide wall setting schemes of a bending channel in a spillway. Based on a model test, Zhang *et al.* (2015) analyzed the dynamic pressure on the two sides of the guide wall in a bending channel. Wang *et al.* (2016) analyzed the effect of the guide wall on the water surface difference of cross-sections in a bending channel. Zhang *et al.* (2016) used experimental data from the bending flow in a spillway chute with one guide wall to calculated the evenness of the water surface and reduction rate of unevenness of the water surface at the concave and convex banks, and then analyzed the effect of axial radius, bottom slopes and discharges on the improvement of flow characteristics. Dey *et al.* (2017) focused on the submerged vane and studied the optimum angle in a 180° channel bend. Akhtari & Seyedashraf (2018) studied the effect of middle vanes with sharp 60° bends with both experimental and numerical methods.

The above methods can all be used to reduce the shock wave and improve the flow characteristics in a bend. However, some of them are only efficient at one designed flow discharge or one water depth, and some are difficult to construct or consume large engineering quantities.

In this paper, the hydraulic model test of the spillway chute with a guide wall installed through the axis was carried out, and the main purpose was to find out how the water level changes at the concave bank and convex bank. Also, the correction coefficient of the maximum water surface difference of the cross-section was calculated by statistical methods. The formula for calculating the maximum water surface difference was developed, which is suggested for application in the engineering design process.

## METHODS

### Experimental arrangement and model

Experiments were carried out at the Hydraulic Laboratory of Shandong Agricultural University. The model was performed with an underground reservoir, a pump, a high headwater pond, a flowmeter, a gate valve, a conveyance channel, a tailwater pond, a model test area and a backwater channel. Figure 1 shows the experimental setup.

The spillway chute model has a rectangular profile and includes a straight section and a bending section, whose net width is 0.5 m, and the axial length of the chute is 1.5 m. The guide wall is located on the axis line of the bend, which has a height of 0.07 m. A design diagram of the experimental model is shown in Figure 2. According to different relative radii of the bend, three models were applied, which were *R/B* = 2.4, 3.2*,* and 4 (*R* is the axial radius of the bending channel and *B* is the net width of the channel) and the turning angles were 71.62°, 53.71° and 42.97° respectively. Each model was divided into three test conditions in which the bottom slopes (*i*) of the chute were 0.005, 0.01, and 0.02, respectively. The experimental model entity was made of PVC sheets, with a thickness of 0.008 m.

### Experimental schemes

Based on different relative radii, bottom slopes of the chute, and with or without a guide wall, the experimental schemes were divided into nine schemes (Table 1).

Scheme no. | Experimental schemes | ||
---|---|---|---|

R/B | i | With or without a guide wall | |

1 | 2.4 | 0.005 | with |

2 | 0.01 | with | |

3 | 0.02 | with | |

4 | 3.2 | 0.005 | with |

5 | 0.01 | with | |

6 | 0.02 | with | |

7 | 4 | 0.005 | with |

8 | 0.01 | with | |

9 | 0.02 | with |

Scheme no. | Experimental schemes | ||
---|---|---|---|

R/B | i | With or without a guide wall | |

1 | 2.4 | 0.005 | with |

2 | 0.01 | with | |

3 | 0.02 | with | |

4 | 3.2 | 0.005 | with |

5 | 0.01 | with | |

6 | 0.02 | with | |

7 | 4 | 0.005 | with |

8 | 0.01 | with | |

9 | 0.02 | with |

### Experimental measurements

- (1)
The measurement of the discharge

In this work, a gate valve was used to control discharge, and the value of the discharge could be read on the screen of the electromagnetic flowmeter (type E-magC). The electromagnetic flowmeter was produced by Kaifeng Instrument Co., Ltd. The guide wall in the chute worked under three hydraulic conditions, which were: water level was lower than, equal to or higher than the guide wall, so three discharges (represented by *Q*: 50 m^{3} h^{−1}, 100 m^{3} h^{−1} and 150 m^{3} h^{−1}) were applied corresponding to these three conditions.

- (2)
The measurement of the water depth

The digital water level indicator (type SX40-1) produced by Chongqing Huazheng Hydrometric instrument Co., Ltd was applied for measuring water level in the test area, with a precision of 10^{−5} m and a measurement range of 1–400 mm, while the maximum error is ±0.04 mm when the measurement value range is 0–200 mm, and the maximum error is ±0.06 mm when the measurement value range is 200–400 mm. In detail, upstream and downstream of the weir were measured, and each measuring section had three measuring lines (at center, left bank, right bank), and the average value represented the average water depth of each section.

- (3)
The measurement of the velocity

The Acoustic Doppler Velocimeter (ADV, which has two types of probe: the down-looking probe and the side-looking probe) was applied for measurement of 3D velocity. The device was produced by Nortek AS Instrument Co., Ltd. The measuring range is ±0 ∼ 4 m/s, the sampling point is 50 mm from the probe and the diameter is 6 mm. The measurement error is ±0.5% of the measured value, and the highest it can reach is ±1 mm/s. The sampling frequency was set at 50 Hz and the sample time was set at 6 s in this work. Since the data collected by ADV is dynamic data for a certain period of time, the collected data show a fluctuating character, as shown in Figure 3(a) (the uppermost line is the *x*-axis velocity, the middle line is the *y*-axis velocity, and the bottom line is the *z*-axis velocity). In order to eliminate the impact of data fluctuations, the collected data is processed by Explore Pro (post-processed software). The peak filtering threshold is 0.8, and the processed data is shown in Figure 3(b). The post-processed velocity for a certain period represents the measured velocity.

- (4)
The measuring sections

The measuring area was classified into 12 equal parts and a total of 13 cross-sections were designed. Based on the demand of the experimental research, six cross-sections were designed with six sounding verticals including left and right banks, left and right sides of the guide wall, and at a distance of 0.25*B* from the left and right banks at each section. The measuring sections are shown in Figure 4, in which 3–15 are cross-sections, and A, B, C1, C2, D, and E are profile sections.

## RESULTS AND DISCUSSION

### Flow pattern

Flow pattern is an important and direct way to see how the guide wall impacts on the characteristics of the bend flow. A camera was applied to record experimental phenomena, and the flow pattern before and after the installation of a guide wall is shown in Figure 5.

Figure 5 shows an experimental photograph of the curve flow of the spillway with and without the guide wall; the relative radius (*R/B*) of the spillway curve is 2.4, the bottom slope is *i* = 0.02, and the discharge per unit width is 300 m^{2} h^{−1}. It can be seen from the figure that in the absence of the guide wall, obviously changes happen in the flow pattern. The water surface at the concave bank is much higher than at the convex bank, thus forming a large water surface difference, and in addition, clear diamond waves can be seen on the water surface. After installation of the guide wall, the wall divides the bend flow into two branches. Under the action of the guide wall, the water depth that is smaller than the height of the guide wall cannot flow from the convex bank to the concave bank under the action of centrifugal force. Thus, the water surface difference between the left and right banks is reduced, then the bend flow is adjusted, and the diamond waves are almost avoided.

The Froude number is also a classical parameter to represent the influence of the guide wall. In comparing the Froude number at the entrance of the upstream channel in each test, Table 2 can be compiled.

With or without a guide wall | i = 0.005 | i = 0.01 | i = 0.02 | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Discharge per unit width (m^{2} h^{−1}): | 100 | 200 | 300 | 100 | 200 | 300 | 100 | 200 | 300 | |

without | Fr | 1.09 | 1.08 | 1.03 | 1.17 | 1.12 | 1.02 | 1.24 | 1.15 | 1.12 |

with | Fr | 0.77 | 0.83 | 0.83 | 0.79 | 0.83 | 0.83 | 1.05 | 1.02 | 1.01 |

With or without a guide wall | i = 0.005 | i = 0.01 | i = 0.02 | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Discharge per unit width (m^{2} h^{−1}): | 100 | 200 | 300 | 100 | 200 | 300 | 100 | 200 | 300 | |

without | Fr | 1.09 | 1.08 | 1.03 | 1.17 | 1.12 | 1.02 | 1.24 | 1.15 | 1.12 |

with | Fr | 0.77 | 0.83 | 0.83 | 0.79 | 0.83 | 0.83 | 1.05 | 1.02 | 1.01 |

R/B | i | α_{0} | ||
---|---|---|---|---|

100 m^{2}h^{−1} | 200 m^{2}h^{−1} | 300 m^{2}h^{−1} | ||

2.4 | 0.005 | 1.4543 | 1.7155 | 1.9428 |

0.01 | 1.3557 | 1.4060 | 1.8135 | |

0.02 | 1.1769 | 1.1045 | 1.5152 | |

3.2 | 0.005 | 1.4064 | 1.5462 | 1.9463 |

0.01 | 1.2579 | 1.4109 | 1.6735 | |

0.02 | 1.2754 | 1.2112 | 1.6441 | |

4 | 0.005 | 1.4867 | 1.4100 | 1.7394 |

0.01 | 1.1562 | 1.3079 | 1.6647 | |

0.02 | 0.9806 | 1.1400 | 1.3828 |

R/B | i | α_{0} | ||
---|---|---|---|---|

100 m^{2}h^{−1} | 200 m^{2}h^{−1} | 300 m^{2}h^{−1} | ||

2.4 | 0.005 | 1.4543 | 1.7155 | 1.9428 |

0.01 | 1.3557 | 1.4060 | 1.8135 | |

0.02 | 1.1769 | 1.1045 | 1.5152 | |

3.2 | 0.005 | 1.4064 | 1.5462 | 1.9463 |

0.01 | 1.2579 | 1.4109 | 1.6735 | |

0.02 | 1.2754 | 1.2112 | 1.6441 | |

4 | 0.005 | 1.4867 | 1.4100 | 1.7394 |

0.01 | 1.1562 | 1.3079 | 1.6647 | |

0.02 | 0.9806 | 1.1400 | 1.3828 |

It can be seen from Table 2 that in the absence of a guide wall, the flow is a supercritical flow (Fr > 1.0), and this is the original experimental condition. With the installation of a guide wall, the Froude number at the entrance becomes smaller, and the flow changes into subcritical flow (Fr < 1.0) except for the bottom slope *i* = 0.02 (Fr > 1.0). In other words, when *i* = 0.005 and 0.01, with the installation of the guide wall, the flow changed from supercritical to subcritical, so the guide wall can help to adjust the flow hydraulics and was proved to change the flow status in some conditions.

### Analysis of the water level at the concave and convex banks

The changes in water level due to the guide-wall presence occurred at both banks, and different conditions in the bend resulted in different water characteristics at each bank. Water levels at the two banks versus bottom slopes, relative radii and discharges per unit width, respectively, are recorded in Figure 6.

#### The water levels at the concave bank

Figures 6(a)–6(c) show the water level changes under different conditions:

- (1)
Figure 6(a) presents the effect of the bottom slope on the water level at the concave bank. The water level with

*i**=*0.005 is higher than the water level with*i**=*0.01, while the water level with*i**=*0.02 is the lowest of these three slopes. This indicates that with the same discharge per unit width and relative radius, the steeper the bottom slope, the lower the water level at the concave bank. - (2)
Figure 6(b) presents the effect of the relative radius on the water level at the concave bank. The water level with

*R/B*= 2.4 is higher than the water level with*R/B**=*3.2, while the water level with*R/B**=*4 is the lowest of these three relative radii. This indicates that with the same discharge per unit width and bottom slope, the bigger the relative radius, the lower the water level at the concave bank. - (3)
Figure 6(c) presents the effect of the discharge per unit width on the water level at the concave bank. The water level with

*q*= 300 m^{2}h^{−1}is higher than the water level with*q**=*200 m^{2}h^{−1}, while the water level with*q**=*100 m^{2}h^{−1}is the lowest of these three discharges. This indicates that with the same relative radius and bottom slope, the bigger the discharge per unit width, the higher the water level at the concave bank.

#### The water levels at the convex bank

Figures 6(d)–6(f) show the water level changes under different conditions:

- (1)
Figure 6(d) presents the effect of the bottom slope on the water level at the convex bank. The water level with

*i**=*0.005 is higher than the water level with*i**=*0.01, while the water level with*i**=*0.02 is the lowest of these three slopes. This indicates that with the same discharge per unit width and relative radius, the steeper the bottom slope, the lower the water level at the concave bank. This changing law is consistent with the concave bank. - (2)
Figure 6(e) presents the effect of the relative radius on the water level at the concave bank. The water level with

*R/B*= 4 is higher than the water level with*R/B*= 3.2, while the water level with*R/B*= 2.4 is the lowest of these three relative radii. This indicates that with the same discharge per unit width and bottom slope, the bigger the relative radius, the higher the water level at the convex bank. This changing law is contrary to the convex bank. - (3)
Figure 6(f) presents the effect of the discharge per unit width on the water level at the concave bank. The water level with

*q**=*300 m^{2}h^{−1}is higher than the water level with*q**=*200 m^{2}h^{−1}, while the water level with*q*= 100 m^{2}h^{−1}is the lowest. This indicates that with the same relative radius and bottom slope, the bigger the discharge per unit width, the higher the water level at the convex bank. This changing law is consistent with the concave bank.

It should be noted that with the installation of the guide wall, the flow status may change from supercritical to subcritical, and in this article, this change happened when *i* = 0.005 and 0.01, while the flow was still supercritical when *i* = 0.02 (as discussed in the section above on flow pattern). Considering the original condition is supercritical flow at the entrance of the channel bend in this article, the following discussion is not classified according to the changed water flow characteristics after installing the guide wall.

### The maximum water surface difference and the correction coefficient

#### Analysis of influencing factors of maximum water surface differences

When water flows into the bend, under the effect of centrifugal force, water levels at the concave and convex banks in the same cross-section present differently. Generally, the water level at the concave bank is greater than that at the convex bank. In this paper, the water surface difference between the two is called the cross-section water surface difference. The water surface differences of each cross-section are different and the largest one in the bend is called the maximum water surface difference.

The images (Figure 7) show the results of the maximum water surface difference against different bottom slopes, relative radii, and discharges per unit width.

In Figure 7(a), with the same bottom slope and discharge per unit width, a larger relative radius results in a decrease in the maximum water surface difference. Similarly, in Figure 7(b), with the same bottom slope and relative radius, a larger discharge per unit width results in an increase in the maximum water surface difference. However, in Figure 7(c), with the same relative radius and discharge per unit width, a steeper bottom slope results in an increase in the maximum water surface difference. Comparisons of maximum water surface differences in these three figures also indicate that the bottom slope has less effect on the maximum water surface difference than the other two factors for the flow conditions experimented. The above analyses are consistent with the conclusion in other literature (Wang *et al.* 2016; Zhang *et al.* 2016).

#### The correction coefficient of the maximum water surface difference in the bend with a guide wall

- (1)
The calculation of the maximum water surface difference

*J*

_{r}is the water surface transverse gradient in the bend,

*k*is the Karman constant,

*C*is the Chezy coefficient,

*V*

_{cp}is the longitudinal vertical velocity (m s

^{−1}),

*g*is the gravitational acceleration (m s

^{−2}), Δ

*z*is the maximum water surface difference (m), and

*α*

_{0}is the velocity distribution coefficient or correction factor.

The biggest difference with Equations (1)–(3) is the value *α*_{0}. None of Equations (1)–(3) can be used directly to describe the maximum water surface difference with a guide wall, so *α*_{0} should be further studied in this work.

- (2)
Experimental results on the correction coefficient

According to the experimentally measured Δ*z* and *V*_{cp} (replaced by the average flow velocity at the beginning of the bend for calculation purposes) and the known axial radius *R* and chute width *B*, the calculated correction coefficients with a guide wall for each experimental scheme are shown in Table 3.

- (3)
Analysis of influencing factors of the maximum water surface difference correction coefficient

By analyzing the collected data on water level, the correction coefficients in different conditions can be shown in Figure 8.

(1) The effect of the axial radius on the correction coefficient.

Figures 8(a)–8(c) demonstrate the maximum water surface difference correction coefficient with different axial radii.

The histograms show that with the same discharges per unit width and the same bottom slopes, different relative radii result in different correction coefficients. This physically means that the relative radius plays an important role in influencing the correction coefficient. Generally, the smaller relative radius results in an increase in the correction coefficient.

(2) The effect of the bottom slope on the correction coefficient.

Figures 8(d)–8(f) demonstrate the maximum water surface difference correction coefficient with different bottom slopes.

The histograms show that with the same discharges per unit width and same relative radii, different bottom slopes result in different correction coefficients. This physically means that the bottom slope plays an important role in influencing the correction coefficient. Generally, a lower bottom slope results in an increase in the correction coefficient.

(3) The effect of the discharge on the correction coefficient.

Figures 8(g)–8(i) demonstrate the maximum water surface difference correction coefficient with different discharges per unit width.

The histograms show that with the same discharges per unit width and same relative radii, different discharges per unit width result in different correction coefficients. This physically means that the discharge per unit width plays an important role in influencing the correction coefficient. Generally, the larger discharge per unit width results in an increase in the correction coefficient.

Additionally, it can be obtained from Figures 8(a)–8(c) that the maximum influencing value of different relative radii on the correction coefficient is 0.3055 (comparing all the correction coefficient differences for relative radii (*R*/*B*) of 4 and 2.4, and finding the scheme with the greatest difference; that is Figure 8(b), with discharge per unit width 200 m^{2} h^{−1}). Similarly, it can be obtained from Figures 8(d)–8(f) that the maximum influencing value of different discharges per unit width on the correction coefficient is 0.611. From Figures 8(g)–8(i), the maximum influencing value of different bottom slopes on the correction coefficient is 0.5399. This means that the correction coefficient is governed mostly by the bottom slopes and also influenced by discharge per unit width and relative radii, while the relative radius has the smallest effect among these three factors.

(4) The calculation of the maximum water surface difference correction coefficient.

The above analysis shows that, with installation of a guide wall, the maximum water surface difference correction coefficient is related to the discharge per unit width, the relative radius, and the bottom slope. However, the discharge per unit width and the bottom slope have a clear relationship between water depth and average velocity of the entrance in the bend. Therefore, in order to further determine variables in the correction efficient regression equation, correlation analysis and regression analysis are performed on the discharge per unit width, the bottom slope, the water depth and the average velocity of the entrance in the bend. Additionally, considering the factor of the bend relative radius (*R/B*) exists in Equation (4), this factor is not taken into account in the equation of the correction coefficient again. The correction impact factor correlation analysis results are shown in Table 4; *h* is the average water depth at the entrance of the bend, and *v* is the average velocity at the entrance of the bend.

q | i | h | v | |
---|---|---|---|---|

Correction coefficient | 0.685** | −0.574** | 0.799** | 0.296 |

q | i | h | v | |
---|---|---|---|---|

Correction coefficient | 0.685** | −0.574** | 0.799** | 0.296 |

*Note*: **represents that the correlation is significant while the confidence measurement is 0.01.

It is observed in Table 4 that the discharge per unit width and the average water depth at the entrance of the bend are positively correlated with the correction coefficient, but the bottom slope is negatively correlated with the correction coefficient, and the average velocity at the entrance of the bend is not related to the correction coefficient.

*h*is the average water depth at the entrance of the bend, while

*h/R*is the relative water depth.

Equation (5) has an *r* (coefficient of association) of 0.868 > = 0.273 (0.01 is the correlation level and *N* is the number of the observation data). The *F* (checking value) = 36.8 > = 3.4 (0.05 is the correlation level and *m* is the number of the independent variable). The sig (significance testing) = 0, which means there is a correlation with the curve fitting Equation (5). For the regression variable, the sig value in the *t*-test of the independent variables *i* and *h* in the regression Equation (5) is 0, indicating that the independent variables *h* and *i* are significantly correlated with the dependent variable. So *h* and *i* can effectively predict the variation of the dependent variable. Equation (5) can be used to calculate the maximum water surface difference correction coefficient (*α*_{0}) in the bend of a spillway chute with a guide wall in the center of the channel.

## CONCLUSIONS

With the original condition that the Fr in the entrance of the channel bend is larger than 1.0 without the guide wall (supercritical flow), the major findings of this work are as follows:

- (1)
With the installation of the guide wall, the Fr becomes smaller and even lower than 1.0, which means the flow status changes from supercritical to subcritical in some conditions, and so the hydraulics changed with the help of the guide wall.

- (2)
According to the observation and data analysis, with the installation of a guide wall, the effect of the bottom slope and the discharge per unit width on the water depth at the concave bank and convex bank are similar while the influence of the relative radius on water depth at the concave bank and convex bank are opposite, which are that the smaller relative radius results in larger water depth at the concave bank while the larger one results in larger water depth at the convex bank.

- (3)
By means of a guide wall, the smaller relative radius, the larger discharge per unit width, and the larger bottom slope result in a larger maximum water surface difference.

- (4)
The bottom slope, the discharge per unit width and the relative radius have effects on the maximum water surface difference correction coefficient, and experiments showed that the correction coefficient was governed mostly by the bottom slopes and also influenced by discharge per unit width and relative radii, while relative radii had the smallest effect among these three factors.

- (5)
Equation (5) can be used to calculate the maximum water surface difference correction coefficient, and Equation (6) can be used to calculate the maximum water surface differences. Regression analysis of the correction coefficient proves that these formulas can be successfully applied to investigate flow characteristics in a bend with a guide wall.

## FUNDING

This work is supported by the CRSRI Open Research Program (Program SN: CKWV2018460/KY).