## Abstract

Siphon-shaft spillways can discharge large amounts of water down to the crest level in a narrow reservoir thanks to the siphonic pressures. However, cavitational pressures in the siphon-shaft limit the operating head of these spillways. An enlargement at the shaft mouth can reduce the vacuum pressures and velocities within the siphonic flows, thereby removing the risk of cavitation occurring. In this study, four different enlarging shaft profiles with various coning angles of 0°, 10°, 15°, and 20° were applied to a siphon-shaft spillway model to eliminate cavitation pressures in the shaft. These models are analyzed in three dimensions by computational fluid dynamics based on the RANS turbulence model coupled with the Volume of Fluid (VOF) method to simulate fluid motion. The numerical uncertainties of the simulation were calibrated with some experimental results and techniques in the literature. The results showed that the enlargement in the shaft mouth with a conical profile significantly decreased the vacuum pressures and velocities in the siphon-shaft. Thus, the use of conical profiles considerably reduced the cavitation numbers along the shaft surface and increased the discharge performance by about 11%.

## HIGHLIGHTS

A new methodological and hydraulic approach to siphon-shaft weirs is presented.

The models applied in the study significantly reduced the cavitation pressure.

With the applied models, the spillway performance of the siphon-shaft weirs has been significantly improved.

## INTRODUCTION

Spillways can be grouped into controlled (gated) and uncontrolled (free overflow) spillways in general. Thanks to the controlled spillways, the water level can be adjusted precisely or kept constant above the crest level. Thus, a better reservoir operation can be achieved by minimizing the level changes in the reservoir. However, the biggest problem of this type of spillway is the maintenance, repair, and operating costs of the moving hydraulic elements, although they cannot be operated very often. In the other type, free overflow spillway, water is allowed to pass freely over a fixed spillway profile when it exceeds the crest level. However, since it is not possible to control the water level, large level changes occur during floods and thus an effective reservoir operation cannot be achieved. Siphon-shaft spillways can offer the advantages of these two spillway types together. Siphon-shaft spillways do not need any movable control elements, and water level changes in reservoirs do not affect the spillway discharge much. They also have the capability to sluice below the crest level (like bottom outlets). Thus, more effective flood control can be achieved in the reservoir when necessary. Since siphon-shaft spillways are placed inside the reservoir, they can be used safely even in embankment and arch dams where spillways have a placement problem. However, it is still recommended to build a free overflow second spillway due to the operating modes and limited capacities of these spillways. One of the most important problems to be considered in the design of these spillways is cavitation. Siphon vacuum pressures need to be controlled and therefore their operating heads are limited. Another problem is limited sluice capacities.

The use of siphons in dam spillways has attracted the attention of researchers since the beginning of the 20th century. Davis & Stickney (1914), Stickney (1922), and Lawaczeck (1930) developed different siphon designs for dam spillways. Rousselier & Blanchet (1951), Head (1971), Charlton (1971), Ackers & Thomas (1975), Head (1975), Unser (1975), Ali & Pateman (1980), Ervine & Oliver (1980), and Bollrich (1994) conducted various theoretical and experimental studies on the design and performance of siphon spillways. Since the beginning of the 21st century, many researchers have tried to determine the performances by analyzing siphon spillways in different designs and hydraulic conditions with experimental and numerical methods (Babaeyan-Koopaei *et al.* 2002; Houichi *et al.* 2006; Yücel 2008; Jourabloo 2010; Ghafourian *et al.* 2011; Musavi-Jahromi 2011; Tadayon & Ramamurthy 2013; Aydın *et al.* 2015; Petaccia & Fenocchi 2015).

Although there are many studies on both siphon and shaft spillways, there are limited studies on siphon-shaft type spillways, as a hybrid spillway. The idea of equipping a shaft spillway with a siphon was first put forward by Iyer (V.G.) in 1933–1934 in order to rapidly increase and maximize the flow of a shaft spillway. Later, Binnie (1938) applied a siphon to a bell-mouthed shaft spillway. This type of spillway, which will be called siphon-shaft, was first applied at Marconahali (India) dam and then again at Hirebhasagar dam in India. Binnie (1938) conducted experiments on the siphon-shaft weir that he created by adding a hood to the shaft weir to be used as a spillway in the existing diversion tunnel. Ağıralioğlu (1977) suggested that the section above the cavitation critical height could be enlarged in order to reduce vacuum pressures in high-head siphon-shaft weirs. Khatsuria (2005) pointed out that some precautions must be taken to prevent the formation of cavitation by vacuum pressures in a siphon spillway. Aydin & Ulu (2021) studied the aeration performance of the aeration holes to prevent cavitational pressures based on the idea of Ağıralioğlu & Müftüoğlu (1989). This study focuses on the enlargement effect of the siphon-shaft profiles on the vacuum pressures in the shaft, which is able to trigger cavitation. Aydın & Ulu (2023a), inspired by Ağıralioğlu (1977), developed a pressure-controlled siphon-shaft spillway profile based on Bernoulli's principle. A disadvantage of this profile, which can adjust cavitation pressures very effectively, is that it creates a large footprint. In this study, the aim is to examine the expansion effect at the shaft mouth by using linear profiles instead of non-linear profiles (Wagner's profile and the pressure-controlled profile offered by Aydın & Ulu (2023a)). For this, some conical profiles with different enlarging angles were applied to a siphon-shaft spillway and their effects on cavitation pressure and discharge performance were investigated.

## SHAFT PROFILES

According to the nappe head on the crest, the spillway flow is controlled by three different flow conditions: free flow, submerged, and full orifice. In all three cases, hydraulic sections do not operate at full capacity due to effects such as swirling, vibration, turbulence, and air intake. The hydraulic performance of shaft spillways depends on the hydraulic profile at the entrance of the shaft. While using the Wagner profile that fits the free flow jet suggested by USBR on a solid shaft spillway, crest shapes such as daisy, labyrinth, and piano-key were also tried in addition to morning-glory in order to increase the discharge capacity (Aydin & Ulu 2023b). Siphon-shaft spillways, which are formed by covering the top of the shaft spillways with a hood and operating them with the siphon effect, can be operated with pressure and full hydraulic capacity even at low water heads or reservoir levels below the crest. However, the low siphonic pressures of these spillways must be kept under control. For this purpose, a shaft profile specifically for siphon-shaft spillways was first developed by Aydın & Ulu (2023a). Developed based on the Bernoulli principle, this pressure-controlled siphon-shaft profile offers a very effective performance in terms of high discharge and pressure control, while giving a wide footprint.

*α*= 10°, 15°, and 20°) were applied to the cone in the shaft mouth together with a linear shaft profile (

*α*= 0°, Aydın & Ulu 2023a). The cases with coning angles of 0°, 10°, 15°, and 20° will be called P1, P2, P3 and P4, respectively (Figure 3). The main purpose of these profiles is to increase the pressure by increasing the flow cross-section toward the crest where the siphon pressure is the lowest. In the applied model in Figures 2(a)–2(d) are the geometric parameters of the siphon-shaft,

*H*is the operating head (difference between the upstream head and downstream head),

*H*is the piezometric head at the outlet,

_{d}*H*is the crest height from the upstream water surface,

_{o}*R*

_{s}is the radius of the shaft mouth,

*R*is the shaft outlet radius,

*P*is the crest height from the reservoir bottom. While the upstream water level in the reservoir was variable, the tail-water level at the spillway outlet was kept constant under a certain piezometric head (

_{c}*H*) to ensure the full siphon action.

_{d}*R*= 2.00 m,

*D*= 4.00 m,

*H*= 6.00 m,

_{d}*a*= 7.58 m,

*b*= 1.62 m,

*c*= 1.50 m,

*d*= 1.52 m were taken as constant. The sharp edges adjacent to the flow were hydraulically rounded with a curve radius of 4 m to prevent separation zones in flow. The operating head is taken into account as a variable in the range of

*H*= 10–20 m and

*R*is taken as 3.0, 4.18, 4.80, and 5.44 m (from P1 to P4 case), respectively, depending on the coning angle.

_{s}## METHOD

*et al.*2015, 2019, 2020, 2022; Aydin & Isik 2015; Aydin 2016a, 2016b; Parsaie

*et al.*2016; Aydin & Ulu 2021, 2023a, 2023b). FLOW-3D which is a general-purpose CFD software uses the finite volume method as a discretization method to solve the following governing equations in three dimensionless Cartesian coordinates for incompressible single-phase fluid flows.where

*t*refers to the time,

*ρ*refers to the fluid density,

*V*refers to the volume fraction open to flow,

_{f}*U*indicates the mean velocity vectors,

_{i}*P*is the hydrostatic pressure,

*A*is the fractional area open to the flow in

_{i}*i*direction,

*G*is the mass acceleration,

_{i}*f*is the viscous acceleration or Reynolds stress. The Volume of Fluid (VOF) estimates the interface in free-surface flows described by Hirt & Nichols (1981) for Cartesian coordinates as follows.

_{i}*F*represents the volume fraction occupied by the fluid for a single fluid. Physically, the voids represent regions of very low density compared to liquid, such as air. A modified volume of fluid method (TruVOF) is used in the free-surface tracking based on the VOF formulation. The

*k–ε*turbulence model which is prepared in the solutions consists of two transport equations for the turbulent kinetic energy and its dissipation (Flow-Science 2019). As an application on the CFD on the scope, Aydin

*et al.*(2015) successfully analyzed the hydrodynamics of a siphon side weir using a CFD technique. All the CFD simulations were carried out according to the flow chart in Figure 4.

### Numerical accuracy

Ağıralioğlu (1977) conducted a series of physical model studies in a hydraulic laboratory to develop a head shape for a siphon-shaft spillway and to verify the mathematical model results he obtained. Some data of this experimental model are used as a benchmark (for *c* = 75, *a* = 160, *d* = 110 mm) to validate the numerical model. In the benchmark model, the shaft profile was described based on the USBR (1987) design principles for the submerged flow condition of the shaft spillway. A standard profile (Wagner profile) was applied to the shaft entrance in order to keep the head dimensions small. Since the shaft inlet is always full during a siphonic flow, it was stated that it would be appropriate to determine the shaft inlet profile according to the submerged state of sharp-edged shaft weirs. The shaft profile for the physical model was obtained from Wagner (1956) tables for *Q* = 60 l/s, *R _{s}* = 0.16 m,

*H*

_{o}= 0.12 m,

*P*

_{c}= 0.32 m, operating head

*H*= 0.124 m, and

*H*/

*R*= 0.775. The shaft diameter (

_{s}*D*) was obtained as

*D*= 0.18 m by trial and error, assuming 75% full at the flow rate at the project operating head, so that the difference between the crest level and the base outlet elevation was 1.0 m. According to the recommendation of USBR (1987), the radius of curvature of the elbow was chosen greater than 1.5 times the diameter of the gallery, and the radius of curvature of the inner wall was taken as 2

*D*= 0.36 m and thus the radius of curvature of the elbow center was taken as 0.45 m.

*Q*is the spillway discharge,

*A*is the area of outlet galley,

_{o}*g*is the gravitational acceleration,

*H*is the operating head. The relative errors between the CFD and the experimental model (Benchmark) data were in the range of 1.2–4.4% for the flow discharges (Figure 5(a)), and 4.5–7.5% for the discharge coefficients (Figure 5(b)). These results indicate that the numerical model results have sufficient reliability.

An uncertainty analysis, Grid Convergence Index (GCI), was also performed to reveal the grid sensitivity on the numerical solutions based on the ASME procedure (Roache *et al.* 1986; Freitas 1993; Celik *et al.* 2008). Three grid sizes (fine, medium, and coarse grids) were applied to the numerical domain with a grid refinement factor of 1.4. The maximum numerical uncertainty for the velocity profiles in the fine-grid solution was reported as 0.16%, which corresponds to ±0.0026 m/s for the centerline, and 3.1%, which corresponds to 0.052 m/s. The global average of the local order of accuracy was determined as 18.02 for shaft-centerline and 8.01 for shaft-edge, which indicates good calculations. The maximum relative errors are 0.8 and 2.8%, respectively. It can be noticed that all uncertainties are within reasonable limits with respect to the grid size.

### Hydraulic performance of shaft profiles

*H*= 10 m). The lowest pressures occur at the shaft crest and in the shaft transition region (at the edge points where the contraction stops). In the P2–P4 profiles, the siphonic pressures dismiss significantly, especially at the critical shaft crest. This also indicates that the risk of cavitation is significantly reduced. Figure 7 presents the velocity contours to evaluate the hydraulic efficiency of the profiles. As the conical angle increases, the velocities above the shaft decrease, which means that the pressures decrease. The stagnation zones in the inner middle and sides of the hood reduce the hydraulic performance of the flow. Therefore, the section with rounded edges can be improved in terms of hydraulics performance, but this is beyond the scope of this study.

*Z*= 0 which indicates transition edge. This increase in conical shafts is more pronounced at a maximum operating head (Figure 8(b)). The P1 profile exposes the upper critical areas of the shaft to low vacuum pressures that are risky for cavitation, but the conical profiles significantly increase the pressures in the critical region, removing them from cavitation pressures. At a low operating head (

*H*= 10 m), the minimum pressure for P1 is −53 KPa, and 30, 34, and 37 KPa for P2, P3, and P4, respectively. The average pressures are 28, 56, 58, and 64 KPa for P1, P2, P3, and P4, respectively. For

*Z*≥ 0, the average of 7–9 times increase in pressures in the shaft was determined with conical profiles (P2–P4) for the low operating head. These increases are up to 11–15 times at the high operating head.

In Figures 8(c) and 8(d), the velocity distributions near the shaft surface are given for the minimum and the maximum operating heads, respectively. As seen in these plots, the conical profiles reduce flow velocities toward the crest from 15 to 2 m/s for *H _{min}* = 10 m, and from 20 to 3 m/s for

*H*= 20 m. The average decreases in flow velocities are between 25 and 50% with P2–P4 conical profiles for

_{max}*Z*≥ 0. For

*Z*< 0, an average increase of 20–25% was observed at the velocities in the shaft due to the increase in the discharge capacity with the conical profiles. In general, the coning angle has an increasing effect on the pressure distribution and a decreasing effect on the velocity distribution along the conical shaft. These effects are more pronounced for high operating heads (Figure 8).

*P*refers to the reference absolute pressure,

*P*refers to the absolute vapor pressure of water,

_{v}*ρ*refers to the density of water,

*U*refers to the flow velocity in the cavitation zone. At the water temperature of 20 °C,

*ρ*= 998.2 kg/m

^{3}and

*P*= 2330 Pa. Although cavitation can theoretically start when the pressure drops below the vapor pressure at a normal water temperature, it can start at higher pressure due to dissolved gas and/or particles in suspension. On the other hand, it may occur even at unexpected average pressures due to pressure fluctuations in turbulent flows (Novak

_{v}*et al.*2007). In the abstract, it is considered that cavitation is not formed when the cavitation number is greater than 1.0. However, the critical value may be smaller than this value because of friction, irregularities in the flow pattern. According to Ağiralioğlu (1977), in practice, the critical cavitation index at which cavitation begins is considered to be 0.7. This value corresponds to 7 m of water column. The USBR (1987) recommends the sub-atmospheric (vacuum) pressure at the crest to be 6.70 m for sea level to avoid cavitation. Houichi

*et al.*(2006) also defined this limit value as 7.92 m for the siphon crest at sea level. Time-dependent pressure fluctuations can also generate cavitation due to high turbulence or eddies/vortices formation in the flow (Tullis 1989).

*Z*> 0, it is seen that the conical profiles have a significant increasing effect on the cavitation number, but for

*Z*< 0, the cavitation numbers are around the critical value. For

*Z*≥ 0 and

*H*= 10 m, the cavitation number increases on average from 0.01 of P1 to 2.59–10.85 with the use of P2–P4, respectively. This increase is from 0.05 to 2.81–12.58 with P2–P4, respectively, for

*H*= 20 m. Cavitation numbers at the crest, which is the most critical point with respect to cavitation, increase from

*σ*= −0.92 of P1 at minimum operating head to

*σ*= 1.63,

*σ*= 3.57 and

*σ*= 6.12 for P2, P3, and P4, respectively. The cavitation numbers increase up to 4.86–25.57 for

*H*and 5.76–31.59 for

_{min}*H*around the shaft mouth (Z = 5.5 m). These values prove that the conical profiles (P2–P4) expanding toward the crest eliminate the risk of cavitation in the shaft. On the other hand, there is no significant change in the down regions of the shaft (

_{max}*Z*< 0) that are exposed to lower cavitation pressures. In order to eliminate the risk of vacuum pressures in corners in the shaft and hood, attention should be paid to the flow separation of these regions.

*H/D*). With this capacity increase, the discharge coefficient of the siphon-shaft spillway increases from the average

*C*= 0.81 to 0.89, 0.90, and 0.91, respectively. The increase in the discharge performances on an average is 10, 11, and 12% for P2, P3, and P4, respectively. The linear-expanding P2–P4 profiles presented in this study offer a discharge performance close to that of the non-linear pressure-controlled profile presented by Aydın & Ulu (2023a) (about

_{o}*C*= 0.92).

_{o}## CONCLUSIONS

This study focuses on the effect of a linear expanding shaft profile (conical shaft) on the hydraulic performance of siphon-shaft spillways. For this purpose, conical shaft profiles with three different coning angles were used together with a linear shaft profile. The prepared models were analyzed with a three-dimensional CFD technique, the accuracy of which was tested by a reliable procedure in the literature. In the model calibration tests, the results of the CFD models showed a maximum deviation of 4.4% from the experimental results in terms of spillway discharges. The maximum grid uncertainty for the velocity profiles was calculated as 3.1%, which corresponds to an edge velocity of ±0.052 m/s. Numerical analysis details are shown that increasing the coning angle significantly decreases the vacuum pressures and velocities in the siphon-shaft. Therefore, it is noted that the use of conical shaft mouths in siphon-shaft spillways significantly reduces the risk of cavitation, as well as increasing discharge performance. It also provides an average 11% increase in discharge efficiency. In addition, it can provide economy thanks to a lower footprint, but attention should be paid to the streamlining capability due to sudden turns in the transition sections. In terms of better hydraulic efficiency, the pressure-controlled profile developed for siphon-shaft spillways by Aydın & Ulu (2023a) may be a more suitable option despite a larger footprint. It should also be stated that this study was carried out under certain limitations with three different cone angles (0 ≥ *α* ≥ 20°), an operating head range of 10 m ≥ *H* ≥ 20 m, and a shaft diameter of *D* = 4 m. It is recommended to carry out a detailed hydraulic modeling study where dimensionless parameters will be applied so that the study can be applied to different dimensions and hydraulic conditions.

## ACKNOWLEDGEMENTS

This study was supported by the Scientific and Technological Research Council of Turkey (TUBITAK) with the project No. 219M006.

## 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.

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