# Discharge Flow Rate for the Initiation of Jet Flow in Sky-Jump Spillways

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction and Background

_{f}(with decreasing flow). A hydraulic jump is then re-established on the flip bucket, which works again as a stilling basin. Therefore, a hysteresis phenomenon occurs, since the flow rates for the initiation and finishing of the jet flow, Q

_{i}and Q

_{f}, are different, being Q

_{i}slightly higher than Q

_{f}.

_{i}and Q

_{f}. They applied the momentum Equation to the control volume between two sections, the first located upstream of the obstacle, with a flow depth y

_{0}, and the second on the obstacle, where a section with the critical flow depth y

_{c}was assumed. The formula obtained for Q

_{f}, neglecting the horizontal component of the friction resistance, considering uniform velocity, parallel flow, and assuming a linear distribution of the hydrostatic pressure in the initial and final sections of the control volume, was:

_{0}is the flow depth and F

_{0}the Froude number both in the initial section, upstream of the obstacle, and z is the vertical height of the obstacle, equivalent to the depth of the flip bucket.

_{c}, they obtained the formula for the initiation of the jet flow:

_{0}, and a hydrostatic distribution over the obstacle corresponding to the critical flow depth y

_{c}. The formula obtained for the end of the jet flow was:

## 2. General Approach

_{0}and the number of Froude F

_{0}at the lowest point of the flip bucket that must be met for jet flow to initiate, and it is the relationship that allows to determine the flow rate for the initiation of the jet flow.

_{0}and the flow rate Q. The flow rate for the initiation of the jet flow Qi, and the corresponding flow depth y

_{0i}are given by the point of intersection of both characteristic curves. The approach is analogous to that of determining a pump operating point, when the system consists of the pump, or group of pumps, and the pipe. The characteristic curves link the pump head or head loss and the pumped flow.

_{0}and Q (which can be determined from the relationship between z/y

_{0}and F

_{0}obtained by the momentum Equation), and the chute characteristic curve, which expresses the ratio between y

_{0}and the flow rate Q compatible with the energy loss that occurs along the channel, for being physically feasible.

_{0}= f(Q) may be constructed point to point by applying the Bernoulli theorem to successive sections of the chute, or by using a Computational Fluid Dynamic numerical model. We followed the second option, using the commercial software Flow3D. Three different tools were used along the workflow: Experimental work at the hydraulic laboratory, numerical Computational Fluid Dynamic modeling, and analytical deduction based on the momentum theorem (Figure 1). An empirical flow rate for the initiation of the jet flow was obtained from the physical models at the laboratory. The results of these laboratory tests were used to calibrate and validate the numerical models performed with Flow3D, regarding the flow characteristics along the chute. Once validated, Flow3D models served to elaborate the chute characteristic curves and to determine the flow rate for the initiation of the jet flow in the absence of air within the flow.

## 3. New Formula for The Initiation of the Jet Flow

_{0}at the lowest point of the flip bucket (Figure 2). As the supercritical flow from the chute increases, the hydraulic jump moves towards the lip of the flip bucket until the energy of the flow is enough to completely sweep the hydraulic jump out of the flip bucket and the jet flow occurs. When the stream of water jumps, the flow regime is supercritical all along the chute–bucket system. Therefore, the flow depth y

_{0}at the lowest point of the flip bucket is determined by the upstream conditions, with control section at the ogee crest of the spillway.

_{0}is the flow depth at the bottom of the flip bucket, where the corresponding speed is v

_{0}. In section (B) the flow is critical (depth y

_{c}) and the corresponding speed v

_{c}. The radius of curvature R and depth of the flip bucket z are also relevant parameters.

_{0}, and the parameter z/y

_{0}. For that, we divide Equation (9) by y

_{0}and substitute for:

## 4. The Method of the Characteristic Curves

_{0}and the flow rate Q. The operating point defines the flow rate for the initiation of the jet flow and the flow depth at the lowest point of the flip bucket at that instant.

_{0}, where z is the depth of the flip bucket, and the Froude number F

_{0}at the lowest point of the bucket. This Equation can also be expressed as a function of the unit flow rate q and the flow depth at the lowest point of the flip bucket y

_{0}, taking into consideration Equations (10)–(12):

_{0}= s(q) can be built point by point using Equation (14), under the hypothesis of cylindrical flip bucket and neglecting the effect of the flow aeration. This is the curve we call flip bucket characteristic curve.

_{0 =}f(q) can be built using a CFD numerical model with greater accuracy than using Bernoulli theorem. We determined the value of y

_{0}for different unit flow rates q using the commercial CFD code Flow3D.

## 5. Experimental Work with Physical Models

## 6. Numerical Models: Chute Characteristic Curve and Flow Rate for The Initiation of the Jet Flow

_{i}, x

_{j}, x

_{z}) and for incompressible fluid (fluid density constant), these Equations are:

_{i}, u

_{j}, u

_{k}) are velocity component in Cartesian coordinates (x

_{i}, x

_{j}, x

_{k}), A

_{i}is fractional area in the i-direction, A

_{j}and A

_{k}are similar area fractions in the j and k direction, respectively, (Gx

_{i}, Gx

_{j}, Gx

_{k}) are body acceleration and (fx

_{i}, fx

_{j}, fx

_{k}) are viscous acceleration. In Equation (19) A is the average flow area, U is the average velocity and F is the volume flow function. When the cell is filled with fluid, the value of F is 1, and when it is empty, F is 0.

_{t}is turbulent dynamic viscosity and P

_{k}is production of turbulence kinetic energy. The remaining terms C

_{1}

_{ε}, C

_{2}

_{ε}

_{,}σ

_{k}and σ

_{ε}are model parameters whose values can be found in Yakhot et al. [31]. Finally, the turbulence viscosity can be computed using the parameter Cµ = 0.085 in the Equation (22):

_{0}

_{exp}) was used, since the proposed analytical formula does not consider the effect of aeration, and the purpose is to compare both results.

## 7. Results and Discussion

_{p}and flow depth y

_{0ip}were determined, and compared to the results of the numerical simulation (Qi

_{sim}, y

_{0i sim}) and to the values experimentally observed (Qi

_{exp}, y

_{0i exp}). The main results of the different phases of the research are summarized in (Table 8).

_{0}and the Froude number F

_{0}, both at the lowest point of the flip bucket (Equations (1)–(6)). It can also be expressed as a relationship between y

_{0}and the unit flow rate for the initiation of the jet flow q (Table 12). This is useful for quantifying the flow rate for the initiation of the jet flow using the proposed method of the characteristic curves.

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

A | average flow area; |

AR | absolute error; |

α | flip angle; |

C_{1}_{ε}, C_{2}_{ε} | k-ε turbulence model parameters; |

σ_{k}, σ_{ε} | k-ε turbulence model parameters; |

F | is the volume flow function; |

f_{xi}, f_{xj}, f_{xk} | viscous acceleration; |

F_{0} | Froude number; |

g | gravity acceleration; |

G_{xi}, G_{xj}, G_{xk} | body acceleration; |

H | spillway total height; |

La | horizontal distance from the upstream spillway vertical wall to the point where the straight part of the chute ends; |

Lc | horizontal distance from the upstream spillway vertical wall and measurement point “p”; |

Lt | horizontal distance between the measurement point “p” and the flip bucket’s lip; |

MAE | mean absolute error; |

MRE | mean relative error; |

R | radius of curvature of flip bucket; |

RE | relative error; |

P | distance from the bottom of the flip bucket to the ground; |

p | pressure; |

P_{k} | production of turbulence kinetic energy; |

Q | flow rate; |

q | unit flow rate; |

Qi | flow rate for the initiation of the jet flow; |

Qi_{exp} | experimental flow rate for the initiation of the jet flow; |

Qi_{p} | theoretical flow rate for the initiation of the jet flow; |

Qi’_{p} | theoretical flow for the initiation of the jet flow not considering R; |

Qi_{sim} | numerical flow rate for the initiation of the jet flow; |

Qf | flow rate for the finishing of the jet flow; |

y_{c} | critical flow depth; |

y_{0} | flow depth; |

y_{0i} | flow depth for the initiation of the jet flow; |

y’_{0iexp} | experimental flow depth for the initiation of the jet flow with aerated upper area; |

y_{0iexp} | experimental flow depth for the initiation of the jet flow without aerated upper area; |

y_{0ip} | theoretical flow depth for the initiation of the jet flow; |

y_{0i sim} | numerical flow depth for the initiation of the jet flow; |

U | average velocity; |

u_{i}, u_{j}, u_{k} | velocity component; |

v_{0} | velocity; |

v_{c} | critical velocity; |

z | vertical height of the obstacle or depth of the flip bucket |

x_{i}, x_{j}, x_{k} | Cartesian coordinates; |

k | turbulence kinetic energy; |

ε | rate of turbulence energy dissipation; |

µ | dynamic viscosity; |

µ_{t} | turbulent dynamic viscosity; |

t | time; |

ρ | fluid density. |

## References

- Godon, R. Le barrage et l’usine hydro-électrique de Marèges sur la Dordogne. Techniques des Travaux
**1936**, 12, 101–110. [Google Scholar] - Rhone, T.J.; Peterka, A.J. Improved tunnel spillway flip buckets. J. Hydraul. Div.
**1959**, 85, 53–76. [Google Scholar] - Vischer, D.L.; Hager, W.H. Dam Hydraulics; Wiley: Chichester, UK; New York, NY, USA, 1998. [Google Scholar]
- Rajan, B.H.; Shivashankara Rao, K.N. Design of trajectory buckets. Water Energy Int.
**1980**, 37, 63–76. [Google Scholar] - Bollaert, E.F.R.; Duarte, R.; Pfister, M.; Schleiss, A.J.; Mazvidza, D. Physical and numerical model study investigating plunge pool scour at Kariba Dam. In Proceedings of the 24th ICOLD Congress, Kyoto, Japan, 2–8 June 2012; pp. 241–248. [Google Scholar]
- Peterka, A.J. Hydraulic Design of Stilling Basins and Energy Dissipators; U.S. Dept. of the Interior, Bureau of Reclamation, Technical Service Center: Denver, CO, USA, 1964; pp. 199–205.
- Schleiss, A.J. Scour evaluation in space and time—The challenge of dam designers. In Rock Scour due to Falling High-Velocity Jets; A A Balkema Publishers: Rotterdam, The Netherlands, 2002; pp. 3–22. [Google Scholar]
- Pfister, M.; Schleiss, A.J. Ski jumps, jet and plunge pools. In Energy Dissipation in Hydraulic Structures; Hubert Chanson; Taylor & Francis Group: London, UK, 2015; pp. 105–140. [Google Scholar]
- Rouve, G. Some observation on flow over spillway flip buckets. In Proceedings of the 6th Symposium of Civil and Hydraulic Engineering Department, High Velocity Flows, Indian Institute of Science, Bangalore, India, 18–20 January 1967. [Google Scholar]
- Baines, P.G. A unified description of two-layer flow over topography. J. Fluid Mech.
**1984**, 146, 127–167. [Google Scholar] [CrossRef] - Lawrence, G.A. Steady flow over an obstacle. J. Hydraul. Eng. ACE
**1987**, 8, 981–991. [Google Scholar] [CrossRef] - Pratt, L.J. A note on nonlinear flow over obstacles. Geophys. Astrophys. Fluid Dyn.
**1983**, 24, 63–68. [Google Scholar] [CrossRef] - Baines, P.G.; Whitehead, J.A. On multiple states in single-layer flows. Phys. Fluids
**2003**, 15, 298–307. [Google Scholar] [CrossRef][Green Version] - Mehrotra, S.C. Hysteresis effect in one-and two fluid system. In Proceedings of the V Australian Conference on Hydraulics and Fluid Mechanics, University of Canterbury, Christchurch, New Zeland, 9–13 December 1974; Volume 2, pp. 452–461. [Google Scholar]
- Austria, P.M. Catastrophe model for the forced hydraulic jump. J. Hydraul. Res.
**1987**, 25, 269–280. [Google Scholar] [CrossRef] - Abecasis, F.M.; Quintela, A.C. Hysteresis in the Transition from Supercritical to Subcritical Flow; memoria n.523; Laboratorio National de Engeharia Civil: Lisbon, Portugal, 1979. [Google Scholar]
- Abecasis, F.M.; Quintela, A.C. Hysteresis in steady free-surface flow. Water Power
**1964**, 4, 147–151. [Google Scholar] - Muskatirovic, D.; Batinic, D. The influence of abrupt change of channel geometry on hydraulic regime characteristics. In Proceedings of the 17th IAHR Congress, Baden Baden, Germany, 15–19 August 1977; pp. 397–404. [Google Scholar]
- Heller, V.; Hager, W.H.; Minor, H.E. Ski jump hydraulics. J. Hydraul. Eng.
**2005**, 131, 347–355. [Google Scholar] [CrossRef] - Arangoncillo, V. Análisis del Funcionamiento con Pequeños Caudales, del Deflector de los Aliviaderos Trampolín; Trabajo de Suficiencia Investigadora; Universidad Politécnica de Madrid: Madrid, Spain, 2011. [Google Scholar]
- Falvey, H.T. Engineering Monograph No.42: Cavitation in Chutes and Spillways; US Department of the Interior, Bureau of Reclamation: Denver, CO, USA, 1990.
- U.S. Bureau of Reclamation. Design of Small Dams; U. S. Government Printing Office: Washington, DC, USA, 1977.
- Flow Science, Inc. FLOW-3D User Manual Release 11.0.3; Flow Science, Inc.: Santa Fe, NM, USA, 2014. [Google Scholar]
- Hirt, C.H.; Sicilian, J.M. A porosity technique for the definition of obstacles in rectangular cell meshes. In Proceedings 4th International Conference on Numerical Ship Hydrodynamics; National Academy of Science: Washington, DC, USA, 1985; pp. 1–19. [Google Scholar]
- Nichols, B.D.; Hirt, C.W. Methods for calculating multidimensional, transient free surface flows past bodies. In Proceedings of 1st Int.Conf. Ship Hydrodynamics; Schot, J.W., Salvesen, N., Eds.; Naval Ship Research and Development Center: Bethesda, MD, USA, 1975; pp. 253–277. [Google Scholar]
- Nichols, B.D.; Hirt, C.W.; Hotchkiss, R.S. Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries; Rep. LA-8355; Los Alamos Scientific Lab.: Los Alamos, NM, USA, 1980. [Google Scholar]
- Hirt, C.W.; Nichols, B.D. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys.
**1981**, 39, 201–225. [Google Scholar] [CrossRef] - Jakeman, A.J.; Letcher, R.A.; Norton, J.P. Ten iterative steps in development and evaluation of environmental models. Environ. Modell. Softw.
**2006**, 21, 602–614. [Google Scholar] [CrossRef][Green Version] - Blocken, B.; Gualtieri, C. Ten iterative steps for model development and evaluation applied to Computational Fluid Dynamics for Environmental Fluid Mechanics. Environ. Model. Softw.
**2012**, 33, 1–22. [Google Scholar] [CrossRef] - Chanel, P.G. An Evaluation of Computational Fluid Dynamics for Spillway Modeling. Master’ Thesis, University of Manitoba, Winnipeg, MB, Canada, 2008. [Google Scholar]
- Yakhot, V.; Orszag, S.; Thangam, S.; Gatski, T.; Speziale, C. Development of turbulence models for shear flows by a double expansion technique. Fluid Dyn.
**1992**, 4, 1510–1520. [Google Scholar] [CrossRef][Green Version] - Speziale, C.G.; Thangam, S. Analysis of an RNG based turbulence model for separated flows. Int. J. Eng. Sci.
**1992**, 30, 1379–1388. [Google Scholar] [CrossRef] - Pope, S.B. Turbulent Flows; Cambridge University Press: Cornell University, New York, NY, USA, 2000. [Google Scholar]

**Figure 3.**Head of pressure laws on the initial and final sections at the moment of the jet flow initiation.

**Figure 4.**Section of a spillway with a flip bucket with a radius of curvature R and with a depth z, and flow depth y

_{0}at the lowest point of the flip bucket.

**Figure 6.**Sky-jump spillway in the testing channel. Case: radius of curvature 0.4 m; flip angle 45°.

**Figure 7.**Geometrical configuration of the physical models: radius of curvature R, depth of the flip-bucket z, flip angle α, and parameter z/R are variable parameters.

**Figure 8.**Flow at the lower part of the flip-bucket for case C2, with a discharge of 43.56 L/s. The upper part of the flow is intensely aerated. The lowest point of the flip bucket is “p” and the flow depth without intensely aerated upper area is y

_{0i}

_{exp}.

**Figure 10.**Outline of the geometric configuration used to obtain the chute characteristic curves using numerical modeling.

**Figure 11.**Three chute characteristic curves, with radius of 0.2 m (

**A**), 0.3 m (

**B**) and 0.4 m (

**C**), are visually overlapped because they only differ slightly on the initial curved part.

**Figure 13.**Characteristic curves and operating point of the chute-flip bucket system: (

**a**) A1 (R = 0.2 m; α = 30°); (

**b**) A2 (R = 0.2 m; α = 45°); (

**c**) B1 (R = 0.3 m; α = 30°); (

**d**) B2 (R = 0.3 m; α = 45°); (

**e**) C1 (R = 0. 4 m; α = 30°); (

**f**) C2 (R = 0.4 m; α = 45°).

**Figure 14.**Flow rate for the initiation of the jet flow based on z and R. Qi

_{p}theoretical flow rate considering R; Qi’

_{p}theoretical flow rate not considering R; Qi

_{exp}experimental flow rate; Qi

_{sim}numerical flow rate.

**Figure 15.**Flow rate for the initiation of the jet flow, obtained by the proposed methodology and formula for different values of R, Qi

_{p}, for a constant value of z = 0.054 m.

**Figure 16.**Comparison of the results obtained using the formulas of various authors. The flip bucket characteristic curve is represented according to the formula of different authors. For the previous formulas, it is indicated with “i” the initiation condition of the jet flow, and with “f” the end condition of the jet flow: (

**a**) A1; (

**b**) A2; (

**c**) B1; (

**d**) B2; (

**e**) C1; (

**f**) C2.

**Figure 17.**Flow rate for the initiation of the jet flow Qi obtained using the formula of various authors, the proposed formula, numerical modeling and physical models in the laboratory.

Instrumentation | Measuring Range | Accuracy |
---|---|---|

FLUXUS-ADM7407 | 0.01–25 m/s | ± 1.6 % of reading ±0.1 m/s |

Ultrasonic distance measuring system UAS | 0.3 m–2 m | >=1 mm |

**Table 2.**Experimental results: Flow rate for the initiation of the jet flow Qi

_{exp}(L/s) for each physical model and total flow depth y’

_{0i exp}(mm) and flow depth without intensely aerated upper area y

_{0i exp}(mm).

Physical Model | ^{1}A1 | ^{2}B1 | ^{3}C1 | ^{4}A2 | ^{5}B2 | ^{6}C2 |
---|---|---|---|---|---|---|

Qi_{exp} (L/s) | 1.65 | 2.75 | 3.70 | 4.80 | 7.65 | 11.95 |

y’_{0i}_{exp} (mm) | - | - | - | - | - | *14.50 |

y_{0i}_{exp} (mm) | - | - | - | - | - | **13.00 |

^{1}(R = 0.2 m, α = 30°);

^{2}(R = 0.3 m and α = 30°);

^{3}(R = 0.4 m α = 30);

^{4}(R = 0.2 m α = 45°);

^{5}(R = 0.3 m α = 45°);

^{6}(R = 0.4 m α = 45° (C2) The total flow depth y’

_{0i exp}(mm) with aerated upper area is * 14.50 mm and the flow depth without intensely aerated upper area y

_{0i exp}(mm) is ** 13.00 mm, corresponding to the measured flow rates for the flip bucket (C2), at the lowest point of the flip bucket.

**Table 3.**Calibration results (roughness variation) and mesh-sensitivity analysis (cell side variation).

Roughness (mm) | Q_{sim} (L/s) | y_{0}_{sim cell 1 mm} (mm) | y_{0}_{sim cell 1.25 mm} (mm) | y_{0}_{sim cell 2.5 mm} (mm) |
---|---|---|---|---|

0.03 | 11.34 | 12.98 | 13.01 | 14.78 |

0.10 | 12.21 | 14.00 | 14.10 | 15.35 |

0.25 | 12.64 | 14.25 | 14.36 | 15.48 |

0.50 | doesn’t jump | doesn’t jump | doesn’t jump | doesn’t jump |

Q_{exp} (L/s) | Q_{sim} (L/s) |
*AE_{Q} (L/s) | **RE_{Q} (%) | y’_{0 exp} (mm) | y_{0 exp} (mm) | y_{0 sim cell 1.25} (mm) | ***AEy’_{0 exp- sim} (mm) | ****REy’_{0 exp-sim} (%) |
---|---|---|---|---|---|---|---|---|

11.95 | 11.34 | 0.61 | 5.07 | 14.50 | 13.00 | 13.01 | 1.49 | 10 |

_{Q}(Absolute Error) and ** RE

_{Q}(Relative Error) between experimental flow rate and numerical flow rate. *** AEy

_{0}(Absolute Error) and **** REy’

_{0}(Relative Error) between experimental flow depth considering the intensely aerated area and numerical flow depth.

Q_{exp} (L/s) | Q_{sim} (L/s) | *AE_{Q} (L/s) | **RE_{Q} (%) | y’_{0 exp} (mm) | y_{0 exp} (mm) | y_{0 sim} (mm) | ***AEy’_{0 exp-sim} (mm) | ****REy’_{0 exp-sim} (%) |
---|---|---|---|---|---|---|---|---|

43.56 | 42.86 | 0.70 | 1.61 | 42.00 | 38.00 | 38.22 | 3.78 | 9 |

_{Q}(Absolute Error) and **RE

_{Q}(Relative Error) between experimental flow rate and numerical flow rate. ***AEy

_{0}(Absolute Error) and ****REy’

_{0}(Relative Error) between experimental flow depth considering the intensely aerated area and numerical flow depth.

**Table 6.**Modeled points (Q

_{sim}, y

_{0 sim}), to draw the chute characteristic curves with radius 0.2 m (A), 0.3 m (B), 0.4 m (C).

Q_{sim}_A (L/s) | y_{0 sim} _A (mm) | Q_{sim}_B (L/s) | y_{0 sim}_B (mm) | Q_{sim}_C (L/s) | y_{0 sim}_C (mm) |
---|---|---|---|---|---|

0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

0.701 | 2.041 | 0.701 | 2.043 | 0.701 | 2.044 |

1.188 | 2.791 | 1.188 | 2.798 | 1.188 | 2.873 |

1.581 | 3.409 | 1.581 | 3.492 | 1.581 | 3.474 |

1.994 | 3.899 | 1.994 | 3.799 | 1.994 | 3.892 |

2.952 | 4.806 | 2.952 | 4.858 | 2.952 | 4.861 |

3.276 | 5.126 | 3.276 | 5.155 | 3.276 | 5.211 |

3.839 | 5.666 | 3.839 | 5.666 | 3.839 | 5.712 |

4.844 | 6.358 | 4.844 | 6.358 | 4.844 | 6.557 |

5.075 | 6.465 | 5.075 | 6.525 | 5.075 | 6.707 |

7.239 | 8.054 | 7.239 | 8.094 | 7.239 | 8.094 |

8.505 | 8.890 | 8.505 | 8.890 | 8.502 | 8.977 |

9.877 | 9.877 | 9.877 | 9.877 | 9.877 | 9.917 |

11.247 | 10.973 | 11.247 | 10.991 | 11.247 | 11.027 |

14.335 | 13.500 | 14.335 | 13.500 | 14.335 | 13.433 |

**Table 7.**Results of flow rate for the initiation of the jet flow Qi

_{sim}and flow depth y

_{0i sim}at the lowest point, obtained with numerical simulation for each geometrical configuration.

Tested Geometry | R (m) | α (°) | z (m) | Qi_{sim} (L/s) | y_{0i sim} (mm) |
---|---|---|---|---|---|

A1 | 0.2 | 30 | 0.027 | 1.12 | 2.61 |

B1 | 0.3 | 30 | 0.040 | 2.03 | 4.01 |

C1 | 0.4 | 30 | 0.054 | 2.86 | 4.61 |

A2 | 0.2 | 45 | 0.059 | 3.79 | 5.38 |

B2 | 0.3 | 45 | 0.088 | 5.76 | 7.06 |

C2 | 0.4 | 45 | 0.117 | 8.49 | 9.08 |

**Table 8.**The flow rate for the initiation of the jet flow and the flow depth at the lowest point of the flip bucket are showed.

Tested Geometry | R (m) | α (°) | z (m) | Qi_{p} (L/s) | Qi_{sim} (L/s) | Qi_{exp} (L/s) | y_{0ip} (mm) | y_{0i}_{sim} (mm) | y’_{0i}_{exp} (mm) | y_{0i exp} (mm) |
---|---|---|---|---|---|---|---|---|---|---|

A1 | 0.2 | 30 | 0.027 | 1.20 | 1.12 | 1.65 | 2.81 | 2.61 | - | - |

B1 | 0.3 | 30 | 0.040 | 2.18 | 2.03 | 2.75 | 4.20 | 4.01 | - | - |

C1 | 0.4 | 30 | 0.054 | 2.93 | 2.86 | 3.70 | 4.84 | 4.61 | - | - |

A2 | 0.2 | 45 | 0.059 | 3.86 | 3.79 | 4.80 | 5.67 | 5.38 | - | - |

B2 | 0.3 | 45 | 0.088 | 5.97 | 5.76 | 7.65 | 7.17 | 7.06 | - | - |

C2 | 0.4 | 45 | 0.117 | 8.84 | 8.49 | 11.95 | 9.21 | 9.08 | 14.50 | 13.00 |

_{p}, y

_{0ip}) is obtained using the proposed methodology and formula, (Qi

_{sim}, y

_{0i sim}) is obtained by numerical simulation and (Qi

_{exp}, y

_{0i exp}) experimentally in laboratory. Experimental flow depth was measured with enough accuracy only for the C2 flip bucket, being y

_{0i exp}the flow depth excluding the intensely aerated upper area.

Tested geometry | Mean Value | AE_{Qi (p-sim)} (L/s) | RE_{Qi (p-sim)} (%) | AEy_{0i} _{(p-sim)} (mm) | REy_{0i} _{(p-sim)} (%) | AE_{Qi (exp-sim)} (L/s) | RE_{Qi} _{(exp-sim)} (%) |
---|---|---|---|---|---|---|---|

A1 | 0.08 | 6.66 | 0.20 | 7.11 | 0.53 | 32.12 | |

B1 | 0.15 | 6.88 | 0.19 | 4.52 | 0.72 | 26.18 | |

C1 | 0.07 | 2.39 | 0.24 | 4.95 | 0.84 | 22.70 | |

A2 | 0.07 | 1.81 | 0.29 | 5.11 | 1.01 | 21.04 | |

B2 | 0.21 | 3.52 | 0.12 | 1.67 | 1.89 | 24.70 | |

C2 | 0.35 | 3.96 | 0.13 | 1.41 | 3.46 | 28.95 | |

*MAE-MRE | 0.15 | 4.20 | 0.19 | 4.12 | 1.41 | 22.84 |

_{p}and the Qi

_{sim}flow rate determined with Flow3D; between the flow rate obtained experimentally Qi

_{exp}and the numerical flow rate Qi

_{sim}; and between the flow depth obtained by the proposed formulation y

_{0ip}and the flow depth y

_{0i sim}determined with Flow3D.

**Table 10.**Comparison using the Absolute Error and the Relative Error of the numerical flow rate and theoretical flow rate for the initiation of the jet flow considering or not R.

Tested Geometry | Qi_{p} (L/s) | Qi’_{p} (L/s) | Qi_{sim} (L/s) | AE_{Qi (sim-p)} (L/s) | RE_{Qi (sim-p)} (%) | AE_{Qi’(sim-p)} (L/s) | RE_{Qi’ (sim-p)} (%) | |
---|---|---|---|---|---|---|---|---|

A1 | 1.20 | 0.95 | 1.12 | 0.08 | 7.14 | 0.17 | 15.18 | |

B1 | 2.18 | 1.80 | 2.03 | 0.15 | 7.39 | 0.23 | 11.33 | |

C1 | 2.93 | 2.50 | 2.86 | 0.07 | 2.45 | 0.36 | 12.59 | |

A2 | 3.86 | 3.00 | 3.79 | 0.07 | 1.85 | 0.79 | 20.84 | |

B2 | 5.97 | 4.47 | 5.76 | 0.21 | 3.65 | 1.29 | 22.40 | |

C2 | 8.84 | 6.45 | 8.49 | 0.35 | 4.12 | 2.04 | 24.30 | |

*MAE-MRE | 0.15 | 4.43 | 2.44 | 17.80 |

_{p}and Qi’

_{p}, respectively, considering or not the term that depends on the radius, and the flow rate for the initiation of the jet flow obtained by numerical simulation Qi

_{sim}.

**Table 11.**Comparison using the Absolute Error and the Relative Error of the flow rate for the initiation of the jet flow, obtained by the proposed methodology and formula and considering whether or not the radius-dependent term, respectively Qi

_{p}and Qi’

_{p}, for z = 0.054 m.

R (m) | z (m) | α (°) | Qi_{p} (L/s) | Qi’_{p} (L/s) | AE_{Q-Q’} (L/s) | RE_{Q-Q’} (%) |
---|---|---|---|---|---|---|

0.4 | 0.054 | 30 | 2.93 | 2.55 | 0.38 | 12.97 |

0.3 | 0.054 | 34.78 | 3.09 | 2.55 | 0.54 | 17.50 |

0.2 | 0.054 | 42.95 | 3.44 | 2.55 | 0.89 | 25.87 |

**Table 12.**Relationship between z/y

_{0}and the Froude number F

_{0}at the lowest point of the flip bucket, expressed as a function of the flow depth y

_{0}and the unit flow rate q.

Authors | Initiation Condition | End Condition |
---|---|---|

Abecasis-Quintela (Equations (1) and (2)) | ${\mathit{y}}_{\mathbf{0}}^{\mathbf{3}}-{\mathit{y}}_{\mathbf{0}}\left(\mathbf{3}\frac{{\mathit{q}}^{\mathbf{4}/\mathbf{3}}}{{\mathit{g}}^{\mathbf{2}/\mathbf{3}}}+\mathbf{2}\mathit{z}\frac{{\mathit{q}}^{\mathbf{2}/\mathbf{3}}}{{\mathit{g}}^{\mathbf{1}/\mathbf{3}}}\right)+\mathbf{2}\frac{{\mathit{q}}^{\mathbf{2}}}{\mathit{g}}=\mathbf{0}$ | ${\mathit{y}}_{\mathbf{0}}^{\mathbf{3}}-\mathit{z}\text{}{\mathit{y}}_{\mathbf{0}}^{\mathbf{2}}-{\mathit{y}}_{\mathbf{0}}\left(\mathbf{3}\frac{{\mathit{q}}^{\mathbf{4}/\mathbf{3}}}{{\mathit{g}}^{\mathbf{2}/\mathbf{3}}}+\mathit{z}\frac{{\mathit{q}}^{\mathbf{2}/\mathbf{3}}}{{\mathit{g}}^{\mathbf{1}/\mathbf{3}}}\right)+\mathbf{2}\frac{{\mathit{q}}^{\mathbf{2}}}{\mathit{g}}=\mathbf{0}$ |

Muskatirovic-Batinic (Equations (3) and (4)) | $\frac{{\mathit{y}}_{\mathbf{0}}}{\mathbf{2}}\left(\sqrt{\mathbf{1}+\frac{\mathbf{8}{\mathit{q}}^{\mathbf{2}}}{{\mathit{y}}_{\mathbf{0}}^{\mathbf{3}}\mathit{g}}}-\mathbf{1}\right)+\frac{{\mathit{q}}^{\mathbf{2}}}{\mathit{g}{\mathit{y}}_{\mathbf{0}}^{\mathbf{2}}\left(\mathbf{1}+\frac{\mathbf{4}{\mathit{q}}^{\mathbf{2}}}{\mathit{g}{\mathit{y}}_{\mathbf{0}}^{\mathbf{3}}}-\sqrt{\mathbf{1}+\frac{\mathbf{8}{\mathit{q}}^{\mathbf{2}}}{\mathit{g}{\mathit{y}}_{\mathbf{0}}^{\mathbf{3}}}}\right)}\text{}-\text{}\frac{\mathbf{3}{\mathit{q}}^{\raisebox{1ex}{$\mathbf{2}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{3}$}\right.}}{{\mathit{g}}^{\raisebox{1ex}{$\mathbf{1}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{3}$}\right.}}+\mathit{z}=\mathbf{0}$ | ${\mathit{y}}_{\mathbf{0}}^{\mathbf{3}}-{\mathit{y}}_{\mathbf{0}}\left(\mathbf{3}\frac{{\mathit{q}}^{\mathbf{4}/\mathbf{3}}}{{\mathit{g}}^{\mathbf{2}/\mathbf{3}}}+\mathbf{2}\mathit{z}\frac{{\mathit{q}}^{\mathbf{2}/\mathbf{3}}}{{\mathit{g}}^{\mathbf{1}/\mathbf{3}}}+{\mathit{z}}^{\mathbf{2}}\right)+\mathbf{2}\frac{{\mathit{q}}^{\mathbf{2}}}{\mathit{g}}=\mathbf{0}$ |

Heller-Hager-Minor (Equations (5) and (6)) | ${\mathit{y}}_{\mathbf{0}}^{\mathbf{3}}\mathit{g}-\mathbf{2}\mathit{g}{\mathit{y}}_{\mathbf{0}}\left[\left(\frac{\mathbf{3}}{\mathbf{2}}\frac{{\mathit{q}}^{\mathbf{4}/\mathbf{3}}}{{\mathit{g}}^{\mathbf{2}/\mathbf{3}}}\right)+\left(\mathit{z}\text{}\frac{{\mathit{q}}^{\mathbf{2}/\mathbf{3}}}{{\mathit{g}}^{\mathbf{1}/\mathbf{3}}}\right)+\left(\frac{{\mathit{z}}^{\mathbf{2}}}{\mathbf{2}}\right)\right]+\mathbf{2}{\mathit{q}}^{\mathbf{2}}=\mathbf{0}$ | ${\mathit{y}}_{\mathbf{0}}^{\mathbf{3}}\mathit{g}-\mathbf{2}\mathit{g}{\mathit{y}}_{\mathbf{0}}\left[\left(\frac{{\mathit{q}}^{\mathbf{2}}}{\mathit{g}\left(\frac{{\mathit{q}}^{\mathbf{2}/\mathbf{3}}}{{\mathit{g}}^{\mathbf{1}/\mathbf{3}}}+\mathit{z}\right)}\right)+\left(\frac{{\mathit{q}}^{\mathbf{4}/\mathbf{3}}}{\mathbf{2}{\mathit{g}}^{\mathbf{2}/\mathbf{3}}}\right)+\left(\frac{{\mathit{z}}^{\mathbf{2}}}{\mathbf{2}}\right)+\left(\frac{\mathit{z}{\mathit{q}}^{\mathbf{2}/\mathbf{3}}}{{\mathit{g}}^{\mathbf{1}/\mathbf{3}}}\right)\right]+\mathbf{2}{\mathit{q}}^{\mathbf{2}}=\mathbf{0}$ |

**Table 13.**Flow rate for the initiation of the jet flow Qi and flow depth at the lowest point of the flip bucket y

_{0i}obtained using the formula of different authors, by means of the proposed formula and by numerical simulation Absolute Error and Relative Error, taking as the reference the flow rate obtained with the numerical simulation Qi

_{sim}.

Tested Geometry | Author | Initial Condition | Qi (L/s) | y_{0i} (mm) | AE_{Qi (p-sim)} (L/s) | RE_{Qi (p-sim)} (%) |
---|---|---|---|---|---|---|

A1 | Abecasis-Quintela | final | 0.95 | 2.35 | 0.17 | 15.18 |

A1 | Abecasis-Quintela | initial | 1.89 | 3.86 | 0.77 | 40.74 |

A1 | Muskaritovic-Batinic | final | 3.69 | 5.56 | 2.57 | 69.65 |

A1 | Muskaritovic-Batinic | initial | 3.24 | 5.92 | 2.12 | 65.43 |

A1 | Heller-Hager-Minor | final | 2.93 | 4.88 | 1.81 | 61.77 |

A1 | Heller-Hager-Minor | initial | 3.69 | 5.56 | 2.57 | 69.65 |

A1 | Proposed Model | initial | 1.20 | 2.81 | 0.08 | 6.66 |

A1 | Numerical Model | initial | 1.12 | 2.61 | - | - |

A1 | Experimental | initial | 1.65 | - | 0.53 | 32.12 |

B1 | Abecasis-Quintela | final | 1.80 | 3.66 | 0.23 | 12.78 |

B1 | Abecasis-Quintela | initial | 3.18 | 5.17 | 1.15 | 36.16 |

B1 | Muskaritovic-Batinic | final | 6.02 | 7.21 | 3.99 | 66.28 |

B1 | Muskaritovic-Batinic | initial | 6.90 | 8.67 | 4.87 | 70.58 |

B1 | Heller-Hager-Minor | final | 4.74 | 6.29 | 2.71 | 57.17 |

B1 | Heller-Hager-Minor | initial | 6.02 | 7.21 | 3.99 | 66.28 |

B1 | Proposed Model | initial | 2.18 | 4.20 | 0.15 | 6.88 |

B1 | Numeric Model | initial | 2.03 | 4.01 | - | - |

B1 | Experimental | initial | 2.75 | - | 0.72 | 26.18 |

C1 | Abecasis-Quintela | final | 2.5 | 4.45 | 0.36 | 14.40 |

C1 | Abecasis-Quintela | initial | 4.73 | 6.46 | 1.87 | 39.53 |

C1 | Muskaritovic-Batinic | final | 9.11 | 9.38 | 6.25 | 68.61 |

C1 | Muskaritovic-Batinic | initial | 19.76 | 17.50 | 16.90 | 85.53 |

C1 | Heller-Hager-Minor | final | 7.36 | 8.18 | 4.50 | 61.14 |

C1 | Heller-Hager-Minor | initial | 9.11 | 9.38 | 6.25 | 68.61 |

C1 | Proposed Model | initial | 2.93 | 4.84 | 0.07 | 2.39 |

C1 | Numeric Model | initial | 2.86 | 4.61 | - | - |

C1 | Experimental | initial | 3.70 | - | 0.84 | 22.70 |

A2 | Abecasis-Quintela | final | 3.0 | 5.01 | 0.79 | 26.33 |

A2 | Abecasis-Quintela | initial | 4.91 | 6.50 | 1.12 | 22.81 |

A2 | Muskaritovic-Batinic | final | 10.20 | 10.14 | 6.41 | 62.84 |

A2 | Muskaritovic-Batinic | initial | 19.51 | 17.33 | 15.72 | 80.57 |

A2 | Heller-Hager-Minor | final | 8.09 | 8.61 | 4.30 | 53.15 |

A2 | Heller-Hager-Minor | initial | 10.20 | 10.14 | 6.41 | 62.84 |

A2 | Proposed Model | initial | 3.86 | 5.67 | 0.07 | 1.81 |

A2 | Numeric Model | initial | 3.79 | 5.38 | - | - |

A2 | Experimental | initial | 4.80 | - | 1.01 | 21.04 |

B2 | Abecasis-Quintela | final | 4.47 | 6.07 | 1.29 | 28.86 |

B2 | Abecasis-Quintela | initial | 8.18 | 6.68 | 2.42 | 29.58 |

B2 | Muskaritovic-Batinic | final | 23.68 | 20.42 | 17.92 | 75.58 |

B2 | Muskaritovic-Batinic | initial | 19.68 | 17.40 | 13.92 | 70.73 |

B2 | Heller-Hager-Minor | final | 16.35 | 14.99 | 10.59 | 64.77 |

B2 | Heller-Hager-Minor | initial | 23.68 | 20.42 | 17.92 | 75.68 |

B2 | Proposed Model | initial | 5.97 | 7.17 | 0.21 | 3.52 |

B2 | Numeric Model | initial | 5.76 | 7.06 | - | - |

B2 | Experimental | initial | 7.65 | - | 1.89 | 24.70 |

C2 | Abecasis-Quintela | final | 6.45 | 7.59 | 2.04 | 31.63 |

C2 | Abecasis-Quintela | initial | 13.95 | 13.08 | 5.46 | 39.14 |

C2 | Muskaritovic-Batinic | final | 50.10 | 39.73 | 41.61 | 83.05 |

C2 | Muskaritovic-Batinic | initial | 19.95 | 17.59 | 11.46 | 57.44 |

C2 | Heller-Hager-Minor | final | 29.25 | 24.57 | 20.76 | 70.97 |

C2 | Heller-Hager-Minor | initial | 50.10 | 39.73 | 41.61 | 83.05 |

C2 | Proposed Model | initial | 8.84 | 9.21 | 0.35 | 3.96 |

C2 | Numeric Model | initial | 8.49 | 9.08 | - | - |

C2 | Experimental | initial | 11.95 | 13.00 | 3.46 | 28.95 |

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**MDPI and ACS Style**

Pellegrino, R.; Toledo, M.Á.; Aragoncillo, V.
Discharge Flow Rate for the Initiation of Jet Flow in Sky-Jump Spillways. *Water* **2020**, *12*, 1814.
https://doi.org/10.3390/w12061814

**AMA Style**

Pellegrino R, Toledo MÁ, Aragoncillo V.
Discharge Flow Rate for the Initiation of Jet Flow in Sky-Jump Spillways. *Water*. 2020; 12(6):1814.
https://doi.org/10.3390/w12061814

**Chicago/Turabian Style**

Pellegrino, Raffaella, Miguel Ángel Toledo, and Víctor Aragoncillo.
2020. "Discharge Flow Rate for the Initiation of Jet Flow in Sky-Jump Spillways" *Water* 12, no. 6: 1814.
https://doi.org/10.3390/w12061814