## Abstract

Geometrical changes and high flow velocity cause flow separation and cavitation in the transition regions of hydraulic structures. A few studies have been conducted on the flow pressure and cavitation index in these regions, and the results need to be still improved. The present study examined the flow pressure and cavitation index variations for expansion angles between 0° and 10° and Froude numbers up to 20.1. Several relevant equations were also suggested to predict permissible angles in the transition regions. The results showed that negative pressure occurred at all lateral expansion angles except 0° when the Froude number was equal to or greater than ≥6.5. The cavitation phenomenon did not occur on the side walls for Froude number up to 4.49. However, the values of the cavitation index were reduced to less than the critical value for the Froude number of 14 when expansion angle was greater than 6°. The results also revealed that the side walls should not be expanded when Froude number was equal to or greater than 17.5. The occurrence of the cavitation on these walls substantially increased for Froude number of 20.1 even as expansion angle equals 0°.

## HIGHLIGHTS

The relevance among the Fr number, expansion angle, and cavitation index in transition regions.

The allowable expansion angle for designing the transition regions in open channels.

Cavitation index and pressure distribution in transition regions.

### Graphical Abstract

## NOTATIONS

- Fr
Froude number

lateral expansion angle

- Re
Reynolds number

density of water

the dynamic viscosity of water

*u*the average flow velocity in the conduit

*g*the acceleration of gravity

*d*flow depth

*P*flow pressure

cavitation index

minimum cavitation index

the allowable size of the lateral expansion angle

*A, B, C, D, H*constant factors

- S
_{1} section one

- S
_{2} section two

flow velocity at the entrance of the transition region

flow depth at the entrance of the transition region

## INTRODUCTION

*et al.*2012; Wang

*et al.*2017). To decrease discharge per-unit width and energy losses, the side walls of transition regions in bottom outlets should have the capacity for gradual expansion (Figure 2). When the flow velocity is accelerated, it cannot adapt itself to wall divergence. This process sets up an adverse pressure gradient, leading to the separation of the flow from the expanded sidewalls (Sreedhar

*et al.*2017; Sengupta

*et al.*2019; Ntintakis

*et al.*2020). Thus, it is more than likely that the local vortex and cavitation phenomenon occur in these regions (Zhang

*et al.*2010; Najafi Nejad Nasser 2011).

*et al.*2020). The potential occurrence of cavitation was measured by the cavitation dimensionless index (Equation (1)) introduced by Falvey (1990). In his study, Falvey (1990) observed that the bottom outlets with indices equal to or less than 0.2 held the highest risk of cavitation. However, there was no risk of cavitation for the bottom outlets with indices higher than 0.2.where

*p*and

*u*are the pressure and average flow velocity of the fluid; is the vapor pressure of water; is the density of water.

The occurrence of cavitation erosion in transition regions is strongly determined by hydraulic and geometric characteristics. Hence, it is required to thoroughly examine flow properties and conduit geometry. The hydraulic characteristics of different flow rates and expansion angles are complicated in transition regions (Jablonská *et al.* 2016). Asnaashari *et al.* (2015) investigated the effect of Froude numbers on the flow pattern in channel-expansive transitions. The results of this study indicated that bed-shear stress increased at the end of the transition region due to flow separation. Furthermore, turbulent strength was raised owing to a boost in the Froude number. Yamini *et al.* (2021) numerically studied Sardab Dam’s bottom outlet and Howell Bunger valve's hydraulic performance under different opening conditions. The results demonstrated various velocity values, ranging from 2 to 5 m/s at the branch entrance of the bottom outlet. Neither flow separation nor adverse conditions were observed in this area. In their study, Li *et al.* (2017) observed that the conduit wall encountered the cavitation phenomenon at all the lateral expansion angles for the Froude numbers of 5.77 and 6.59 (at any percentage of gate openings). Li *et al.* (2016a) also found that the cavitation index (*σ*) on the sidewalls of the conduit was less than 0.2 for Fr ≥ 3.89 at all the lateral expansion angles, even at *α* = 0°, when the gate was partially open. To prevent cavitation occurrence in transition regions, Thompson & Kilgore (2006) introduced the relevance of as a maximum expansion angle. Using this equation, the lateral expansion angle reached 4.9°, 3.2°, and 3° for the Froude numbers of 3.89, 5.77, and 6.59, respectively. Therefore, the results of Li *et al.* (2016a) are not in good agreement with those of Thompson & Kilgore (2006).

The *k*–*ε* (RNG) turbulence model and the Volume-Of-Fluid (VOF) method have been used in numerous studies to quantitatively measure high-speed flows in hydraulic structures. Yamini *et al.* (2022) studied the flow patterns, stream flow, vortex severities, pre-swirl principles, and performance acceptability criteria for pump intakes in different conditions. Jamali *et al.* (2022) explored the effect of flow conditions and chute geometry on shock waves on chute spillways. The impact of different slope ratios on the hydrodynamic characteristics of lateral withdrawal of water supply was examined by Zhaoa *et al.* (2022). Ebrahimnezhadian *et al.* (2020) investigated the impact of the flip bucket edge angle on flow hydraulic characteristics. The effects of pressure ratio conditions and plate stages on the characteristics of cavitation, energy transfer, and loss of perforated plates were numerically studied by Xiaogang *et al.* (2020). Li *et al.* (2016b) simulated aerated flows in curve-connective tunnels. Li *et al.* (2011) identified the hydraulic characteristics of flow downstream in radial gates whose conduit had a sudden lateral enlargement and a vertical bottom drop. Dargahi (2010) assessed the flow characteristics of bottom outlets with moving gates. Stamou *et al.* (2008) studied supercritical flows in gradual expansions. Krüger & Rutschmann (2006) modeled supercritical flows via an extended shallow water approach. High-speed flow in a spillway tunnel was also analyzed by Sha *et al.* (2006).

Based on what was mentioned, the results of previous studies on the allowable expansion angle need to be still improved. To contribute to this strand of research, the present study numerically explored the flow pressure distribution and cavitation index for various Froude numbers (2.56 ≤ Fr ≤ 20.1) and lateral wall expansion angles (0 ≤ *α* ≤ 10) at 10% gate opening using the RNG(*k*–*ε*) turbulence model and the VOF method in Flow three-dimensional (3D) software. Practical relevance for predicting the cavitation index and allowable lateral expansion angle in these areas is also introduced.

## BASIC EQUATIONS OF FLOW FIELDS

The pressure and velocity of flows are two fundamental parameters to describe the flows of transition regions. Since water is an incompressible Newtonian fluid, these parameters are determined by the classical Navier–Stokes equations. These equations are developed based on the physical laws of conservation of mass and momentum. The 3D Reynolds averaged Navier–Stokes (RANS) equations in the Cartesian coordinate system for incompressible and turbulent fluid flows are presented below (Chen *et al.* 2021):

*k*–

*ε*and Standard

*k*–

*ε*models. These models have similar basic structures. Nevertheless, the RNG

*k*–

*ε*model is a more appropriate model for simulating currents in this study because it has been modified several times and is computationally efficient and stable. It also yields more accurate results for swirling and rapidly strained flows compared to the Standard

*k*–

*ε*model. Since the renormalization group methods are utilized in the RNG

*k*–

*ε*model, the constants achieved from this model differ from those obtained from the Standard

*k*–

*ε*model. It also includes more factors, such as the interphase turbulent momentum transfer (Miidla 2012). The equations of

*k*(Equation (4)) and

*ε*(Equation (5)) are provided below:where and are the turbulent Prandtl numbers for

*k*and;

*G*is turbulence kinetic energy generated by the mean velocity gradients;

_{k}*G*is turbulence kinetic energy produced by buoyancy; is the user-defined source term; and

_{b}*C*

_{2}and are constants.where is the turbulent Prandtl number for energy and is the gravitational vector component in the

*i*direction.where

*S*is the modulus of the mean rate-of-strain tensorwhere is the component of flow velocity parallel to the gravitational vector and

*u*is the component of flow velocity perpendicular to the gravitational vector.

where is the mean rate-of-rotation tensor in a rotating reference frame with the angular velocity . In general, is determined by the mean strain and rotation rates, the angular velocity of the system, and the turbulence fields.

*F*(

*x, y, z, t*) that represents the volume of fluid #1 per-unit volume. This function satisfies the following equation:where is the volume fraction of flow (

*u, v, w*). The symbols of , , are the velocity components of the fractional area open to the flow along the coordinate directions of

*x, y*, and

*z*, respectively. When Cartesian coordinates are used,

*R*and are set to 1 and 0, respectively. In Equation (6), is the time rate of change in the fluid volume fraction #1 related to the mass source for fluid #1. In Equation (6), corresponds to the density source .

The value of *F* is interpreted based on the type of problem. For a single fluid, *F* represents the fluid volume fraction. The *F* values of 1 and 0 show the presence or absence of the fluid, respectively. The locations in which *F* equals 0 are called voids; in other words, voids are regions without fluid mass. Therefore, a uniform pressure is assigned to these areas. In practice, these regions are filled with vapor or gas because the density of vapor or gas is insignificant compared to fluid density (Hirt & Nichols 1981; Zhang *et al.* 2011).

## GENERAL CHARACTERISTICS OF SEYMAREH BOTTOM OUTLET

Seymareh Dam is a concrete double-arch dam located in Ilam Province, Iran, and is equipped with two bottom outlets. The entrances of the first and second bottom outlets are placed 620 and 640 m, respectively, above the sea level (20 and 40 m above the river bed). The plan and longitudinal parts of the first bottom outlet are depicted in Figure 2(a) and 2(b). This outlet is designed for a maximum discharge of 654/m^{3}/s at 111.5 m upstream water head. With respect to flow conditions, it has two parts: the under-pressure part and the free-surface flow part with lengths of 36.5 and 8.9 m, respectively. As seen in Figure 2, a transition region with a length of 3.2 m is located at the end of the conduit.

## NUMERICAL MODEL

*P*). The pressure outflow (

*O*) and walls (

*W*) were introduced to the outflow and walls of the model, respectively (see Figure 3). To include the Renormalization Group, rather than constant factors, the turbulent model of

*k*–

*ε*(RNG) was employed. The VOF method was utilized to calculate the free-surface profile of the flow downstream in the service gate. To find an optimal size for the reservoir, Jamali & Manafpour (2019) analyzed the velocity profiles at different sections of a bottom outlet. The results demonstrated no significant changes in the profiles when the length and width of the reservoir exceeded 30 and 29.53 m

^{2}, respectively. Hence, these dimensions were chosen for the reservoir in the present study (Figure 3).

Item . | Definition . |
---|---|

Model input | Specified pressure (P) |

Model output | Pressure outflow (O) |

Walls | Wall (W) |

Border between blocks | Symmetry (S) |

Total number of computational mesh | 1,503,019 |

Item . | Definition . |
---|---|

Model input | Specified pressure (P) |

Model output | Pressure outflow (O) |

Walls | Wall (W) |

Border between blocks | Symmetry (S) |

Total number of computational mesh | 1,503,019 |

The number of mesh increased as a result of applying the grid overlay boundary condition to the entrance of the bottom outlet, rather than to the upstream reservoir, in the investigated scenarios. Half of the flow was simulated due to the longitudinal symmetry of the conduit, which boosted the accuracy of flow simulation in the region. The grid size, ranging from 2 to 25 cm, decreased from the upstream toward the downstream of the conduit. This method generates a higher resolution for the transition regions. Table 2 indicates the numerical model sensitivity under various conditions.

Row . | Inlet boundary condition . | Simulated area . | The size of cubic meshes (m) (from upstream block to downstream of conduit) . | Total cells . | The computational time (s) . | R^{2}. | |||
---|---|---|---|---|---|---|---|---|---|

10% . | 30% . | 70% . | 100% . | ||||||

1 | Flow Pressure (Entrance of reserviour) | Whole bottom outlet conduit | 0.5–0.4–0.3 | 1,506,304 | 25 | 0.921 | 0.902 | 0.895 | 0.912 |

2 | Volume Flow Rate (Entrance of the bottom outlet) | Whole bottom outlet conduit | 0.4–0.3–0.25 | 1,556,654 | 25 | 0.881 | 0.876 | 0.861 | 0.862 |

3 | Flow Velosity (Entrance of the bottom outlet) | Whole bottom outlet conduit | 0.4–0.3–0.25 | 1,586,718 | 25 | 0.862 | 0.854 | 0.839 | 0.842 |

4 | Flow Pressure (Entrance of the reserviour) | Half of the bottom outlet conduit | 0.5–0.35–0.2–0.1 | 1,706,546 | 25 | 0.941 | 0.902 | 0.915 | 0.927 |

5 | Flow Pressure (Entrance of the reserviour) | Half of the bottom outlet conduit | 0.5–0.30–0.2–0.1–0.05 | 1,680,255 | 25 | 0.971 | 0.956 | 0.946 | 0.941 |

6 | Applying Grid Overally bondary condition instead of the reserviour | Half of the bottom outlet conduit | 0.25–0.10–0.05–0.02 | 1,806,546 | 25 | 0.991 | 0.968 | 0.966 | 0.962 |

Row . | Inlet boundary condition . | Simulated area . | The size of cubic meshes (m) (from upstream block to downstream of conduit) . | Total cells . | The computational time (s) . | R^{2}. | |||
---|---|---|---|---|---|---|---|---|---|

10% . | 30% . | 70% . | 100% . | ||||||

1 | Flow Pressure (Entrance of reserviour) | Whole bottom outlet conduit | 0.5–0.4–0.3 | 1,506,304 | 25 | 0.921 | 0.902 | 0.895 | 0.912 |

2 | Volume Flow Rate (Entrance of the bottom outlet) | Whole bottom outlet conduit | 0.4–0.3–0.25 | 1,556,654 | 25 | 0.881 | 0.876 | 0.861 | 0.862 |

3 | Flow Velosity (Entrance of the bottom outlet) | Whole bottom outlet conduit | 0.4–0.3–0.25 | 1,586,718 | 25 | 0.862 | 0.854 | 0.839 | 0.842 |

4 | Flow Pressure (Entrance of the reserviour) | Half of the bottom outlet conduit | 0.5–0.35–0.2–0.1 | 1,706,546 | 25 | 0.941 | 0.902 | 0.915 | 0.927 |

5 | Flow Pressure (Entrance of the reserviour) | Half of the bottom outlet conduit | 0.5–0.30–0.2–0.1–0.05 | 1,680,255 | 25 | 0.971 | 0.956 | 0.946 | 0.941 |

6 | Applying Grid Overally bondary condition instead of the reserviour | Half of the bottom outlet conduit | 0.25–0.10–0.05–0.02 | 1,806,546 | 25 | 0.991 | 0.968 | 0.966 | 0.962 |

The cavitation index and flow pressure were computed for 77 scenarios with 7 Froude numbers ranging from 2.56 to 20.1 and 11 expansion angles (0, 1, 2, …, 10) (see Table 3). Different Froude numbers were used to determine the possibility of cavitation and flow pressure at the entrance of the transition region. It is worth noting that the cross-section shape of this region was rectangular in all simulations.

Specified pressure (reservoir head) . | Froude number . | α . |
---|---|---|

5 7 15 35 50 80 100 (normal water level in Seymareh bottom Dam) | 2.56 4.49 6.30 10.90 14.00 17.50 20.10 | 0 1 2 3 4 5 6 7 8 9 10 |

Specified pressure (reservoir head) . | Froude number . | α . |
---|---|---|

5 7 15 35 50 80 100 (normal water level in Seymareh bottom Dam) | 2.56 4.49 6.30 10.90 14.00 17.50 20.10 | 0 1 2 3 4 5 6 7 8 9 10 |

## VALIDATION OF NUMERICAL MODEL

*R*

^{2}] in Figure 5(a)–5(d)) were in good agreement with the experimental data, which verifies the appropriate behavior of the numerical model.

## EVALUATION OF FLOW PRESSURE ON SIDE WALLS OF TRANSITION REGION

In their study, Zhang *et al.* (2010) observed large negative pressures on the sidewalls of the transition regions for partial gate openings. In a similar vein, Davis & Sorensen (1984) found that air concentration reached its maximum level in small gate openings (the powder flow state). Hence, in the current study, the hydraulic characteristics of flows were investigated in 10% service gate opening as the critical percentage of gate openings. Moreover, the free-surface condition was used for the flow downstream of the gate in all the simulations.

^{3}), the dynamic viscosity of water (N.s/m

^{2}), the average flow velocity

*u*(m/s), the acceleration of gravity

*g*(m/s

^{2}), the cross-section area (depth (

*d*) × width (

*b*)) (m

^{2}), the cross-section shape, and the lateral expansion angle

*α*. All these parameters are included in the following formula:

Viscous forces are insignificant at large-scale flows for large Reynolds numbers (Ting 2016). Therefore, in this study, as the Reynolds number (16.52 × 10^{6} < Re < 30.94 × 10^{6}) exceeds (Li *et al.* 2016a) was neglected. Additionally, the shape of the cross-sections was rectangular in all the simulations.

Pressure coefficients () were computed for various entrance Froude numbers () and lateral expansion angles (*α*) in S_{1} (the entrance of the transition region where the geometry started to change) along the *Z*-direction (Figure 2). The results demonstrated that this area was prone to flow separation and local vortices.

*α*, the pressure generally decreased for all the . Further analysis revealed negative pressures for the of 2.56 and 4.49 at and , respectively. Similarly, the pressure coefficient values for these reached −0.2 and −0.25, respectively, at

*α*= 10°.

Furthermore, there was a remarkable decline in the pressure for Fr_{0} = 6.30. To explain it more clearly, negative pressures were obtained for all the lateral expansion angles, except for *α* = 0°. The minimum pressure coefficient value was −0.3 at *α* = 10°. With the increment of (6.3 < Fr_{0} ≤ 20.10), the pressure reached stability.

## EVALUATION OF CAVITATION INDEX ON SIDE WALLS OF TRANSITION REGION

Considering the cavitation index relevance, as another pressure coefficient, the entrance of transition regions is prone to cavitation. Jamali & Manafpour (2019) found that the cavitation phenomenon happened in the initial parts of the transition region. Consequently, in the present study, the cavitation index was measured for two sections, namely, and (2 m after ), along the Z-direction on the side walls.

*σ*values for . As illustrated in this figure, the

*σ*values were far higher than the critical value (0.2) for ≤ 4.49 at all degrees of

*α*. The minimum values achieved for the cavitation index were 6 and 1.8 for = 2.56 and 4.49, respectively. These results are in good agreement with those of Thompson & Kilgore's (2006) study. Conversely, Li

*et al.*(2016a) reported cavitation occurrence for Fr

_{0}≥ 3.89 even at

*α*= 0°.

By increasing the to 14, the value of the cavitation index sharply decreased to less than 0.2 at , which suggests the high possibility of cavitation erosion. The minimum *σ* value was −0.15 for = 14 at *α* = 10°. By raising the from 4.49 to 14, the value reached 0.2 at *α* = 7°.

The values were substantially reduced to less than 0.2 for Fr_{0} = 17.5 at all the lateral expansion angles, except for *α* = 0°. In general, when the was boosted to 20.10, the side walls faced the risk of cavitation even at *α* = 0°.

_{0}≤ 14. The main reason for the increase in the values was the reduction of the flow velocity near the side walls as a result of conduit width expansion and energy losses (Ashrafizadeh & Ghassemi 2015; Zhang 2017; Tang

*et al.*2019). Nonetheless, the values declined to less than 0.2 for 14 ≤ Fr

_{0}< 20.10 at all the lateral expansion angles, except for

*α*= 0°. The side walls also faced the risk of cavitation occurrence for Fr

_{0}≥ 20.10 even at

*α*= 0°. These findings suggest that an increase in accelerates flow velocity, which leads to flow separation near side walls.

Further analysis indicated that there was a sharp decrease in the *σ* values at *α* > 7° for all the , which can be associated with several physical processes, such as flow separation and local vortex. Increasing *α* from 7° to 8° boosted the local vortex intensity in the flow field and strengthened the flow turbulence, which resulted in flow separation. However, further investigations into physical processes are required to address this issue more profoundly.

## PROPOSED RELEVANCE FOR PREDICTING σ FOR VARIOUS FR AND α VALUES

_{1}and S

_{2}showed that the values measured for these sections were approximately similar when Fr

_{0}≥ 14. However, for Fr

_{0}≤ 14, more critical cavitation indices were obtained for . Figure 9(a) and 9(b) illustrates the minimum values of achieved for different Fr and values in .

*et al.*(2016a), except for coefficients. The coefficient values obtained from Equations (12) to (15) are presented in Table 4. The theoretical curves of Equations (14) and (15) are also illustrated in Figure 10.

Fr . | α . | A . | B . | C . | D . | E . | R^{2}
. |
---|---|---|---|---|---|---|---|

18.879 | −0.264 | −0.280 | 0.780 | −0.031 | 0.990 | ||

152.631 | −87.87 | 12.112 | 0.0750 | 0.998 | |||

0.0745 | 0.703 | −11.808 | 15,126.758 | 0.988 | |||

17.921 | −17.26 | 300.66 | 656.25 | 0.973 |

Fr . | α . | A . | B . | C . | D . | E . | R^{2}
. |
---|---|---|---|---|---|---|---|

18.879 | −0.264 | −0.280 | 0.780 | −0.031 | 0.990 | ||

152.631 | −87.87 | 12.112 | 0.0750 | 0.998 | |||

0.0745 | 0.703 | −11.808 | 15,126.758 | 0.988 | |||

17.921 | −17.26 | 300.66 | 656.25 | 0.973 |

*σ*values achieved from Equations (12) to (15) and the numerical data, as displayed in Figures 11 and 12, reveals that the results of the proposed relevance are in good agreement with the numerical data. To be more precise, the correlation coefficient values obtained from Equations (12) to (15) were 0.990, 0.998, 0.988, and 0.973, respectively (Table 4). Likewise, in Li

*et al.*(2016a)’s study, the

*σ*value of 0.987 was obtained through these equations.

## ALLOWABLE SIZE OF LATERAL EXPANSION ANGLE FOR VARIOUS FROUDE NUMBERS

_{a}) in order to prevent cavitation occurrence. As shown in Figure 8(a)–8(g), the values of the cavitation index dropped to the critical amount (0.2) for Fr

_{0}≥ 6.30. The values of for Fr

_{0}≥ 6.30 are provided in Figure 13. As can be seen, this value was reduced slightly from 7° to 6° for , followed by a sharp decrease to 0° for Fr

_{0}= 17.5. The comparison of the results of this study with those of previous studies indicates that the obtained values exceed those reported in the literature for . On the contrary, as for , the computed values are lower than those mentioned in previous studies.

## CONCLUSION

In this study, the flow pressure distribution and cavitation index on the side walls of the transition region for various Froude numbers and lateral expansion angles were numerically examined, and the possible relationship of with and was explicated. Ultimately, the allowable lateral expansion angle was measured. Based on the results, the following conclusions can be drawn:

The comparison of the cavitation indices of S

_{1}(the entrance of the transition region) and (2 m after ) revealed that S_{1}was an area prone to local vortices, flow separation, and consequently, cavitation.By increasing the expansion angle (

*α*), the pressure decreased for all the . Negative pressure was observed for Fr_{0}= 2.56 at . For Fr_{0}≥ 4.49, negative pressure was reported at all the lateral expansion angles, except for*α*= 0°.Pressure coefficient values reached −0.2 and −0.25 for Fr

_{0}= 2.26 and 4.49, respectively, at*α*= 10°. By raising the from 6.3 to 20.1, the pressure values stabilized and the minimum pressure coefficient value was obtained (−0.3).The probability of cavitation formation on the side walls of the transition region was very low at all the lateral expansion angles for Fr

_{0}≤ 4.49. The obtained results are well aligned with those of Thompson & Kilgore (2006). However, they differ from the results of Li*et al.*(2016a) who observed the occurrence of the cavitation phenomenon for Fr_{0}≥ 3.89 even at*α*= 0° in partial gate openings.With an increase in the to 14, the values of the cavitation index dropped to less than 0.2 at and reached 0.2 at

*α*= 7°. To put it in other terms, the probability of cavitation erosion was very high under these conditions.Given that the values of the cavitation index were less than 0.2 at all the lateral expansion angles, the expansion of side walls in transition regions should be prevented for Fr

_{0}≥ 17.5. Furthermore, the side walls held the risk of cavitation for Fr_{0}= 20.1 even at*α*= 0.The increased local vortex intensity and flow separation led to a dramatic reduction in the

*σ*values at*α*> 7° for all the Froude numbers.

These findings have great practical implications for designing and constructing transition regions. In this study, the capability of the CFD model to predict some flow characteristics, such as pressure distribution and flow separation, in transition regions was confirmed. However, gaining a deeper understanding of the issue requires carrying out experimental investigations under the aforementioned conditions at various gate openings (in the current research, the percentage of gate openings remained constant [10%]). Moreover, this study was limited to the case of rectangular cross-sections. Future research should scrutinize other shapes for cross-sections (e.g., the trapezoidal shape).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*An Experimental Investigation of Flow Energy Losses in Open- Channel Expansions*