## Abstract

In the present study, using a quasi 3D analytical simulation, air concentration distribution in ski jump generated jet is calculated. A numerical simulation is also performed to verify the results of the analytical model in parallel with the available experimental and other analytical data. By solving continuity and momentum equations in the case of air-water flow for three different cases, it was confirmed that the air concentrations along the ski jet are uniquely linked to the relative black water core length. Results showed that the black water core length is also influenced by the approach flow depth, Froude number, geometrical parameters of ski jump and the chute bottom angle. Finally, an analytical equation is proposed to predict the air concentration distribution along the ski jump jet regarding different hydraulic and geometric parameters. By calculating the velocity profiles along the jet, it showed that increasing the air concentration reduces the jet velocity profile.

## HIGHLIGHTS

A quasi model of a ski jump-generated jet is developed for calculating air concentration in the flow.

Continuity and momentum equations were solved analytically in the whole flow domain.

A numerical simulation is also performed to verify the results of the analytical model.

It was found that the air concentrations along the jet are uniquely linked to the relative black water core length.

### Graphical Abstract

## NOTATIONS

The following symbols are used in this paper:

*A*surface deformation area (m

^{2})*b*jet width (m)

*C*air concentration (-)

*C*_{air}calibration parameter

*Fr*_{o}approach flow Froude number (-)

*g*acceleration due to gravity (m/s

^{2})*h*_{o}approach flow depth (m)

*L*_{b}black water core length (-)

*k*turbulent kinetic energy

*P*pressure (Pa)

mechanical stress production (N/m)

*R*bucket radius (m)

*Re*_{o}approach flow Reynolds number (-)

*t*bucket height (m)

*u*,_{s}*v*,_{s}*w*_{s}air bubbles rising velocity in x, y, z (m/s)

*V*_{o}approach flow velocity (m/s)

*We*_{o}approach flow Weber number (-)

*x*stream wise coordinate (m)

*X*normalized horizontal stream wise coordinate (–)

*Y*_{o}initial jet width (m)

*z*coordinate perpendicular to

*x*(m)*z*_{l}lower water (m)

*α*geometrical take-off angle relative to horizontal (°)

*α*_{j}virtual take-off angle relative to horizontal,

*α*or_{U}*α*(°)_{L}*β*total bucket deflection angle (°)

*ρ*_{w}density of water (kg/m)

*σ*surface tension of water (N/m)

*μ*dynamic viscosity of water [kg/(ms)]

*ε*,_{tx}*ε*_{ty}_{,}*ε*_{tz}turbulence viscosity coefficient (m/s

^{2})*A*_{s}height of the disturbances

*A*,_{x}*A*,_{y}*A*_{z}area fraction (m

^{2})*D*dissipation function (-)

*C*_{nu}constant

*F*fluid volume (m

^{3})*f*,_{x}*f*,_{y}*f*_{z}viscous acceleration (m/s

^{2})buoyancy generation (N/m)

*L*_{t}characteristic size of turbulence eddies (m)

*L*_{T}characteristic turbulent length scale

*P*_{t}turbulent kinetic energy per unit volume (m

^{2}/s^{2})*P*_{d}surface tension energy (m

^{2}/s^{2})Richardson flux number (-)

radius of free surface deformation curvature (m)

*Q*_{l}specific turbulent kinetic energy (m

^{2}/s^{2})- u
_{max} characteristic velocity scale (m/s)

*u, v, w*velocity in the direction of the

*x*,*y*,*z*(m/s)volume fraction of the flow (m

^{3})*x’*horizontal stream wise coordinate (m)

*δ*Vvolume of air entrained per unit time (m

^{3})*z’*vertical coordinate, perpendicular to

*x*(m)*Z*normalized jet thickness (-)

*z*_{u}upper water (m)

*φ*chute bottom angle (°)

*δ*equivalent deflector angle (°)

*τ*Viscous stress tensor (Pa)

*ρ*_{mix}average density (kg/m

^{3})*σ*_{t}Schimidet Number(-)

*υ*kinematic viscosity of water (m

^{2}/s)*μ*_{t}eddy viscosity jet [kg/(ms)]

*ε*,_{dx}*ε*_{dy}_{,}*ɛ*_{dz}turbulence diffusion coefficient (m/s

^{2})

## INTRODUCTION

Air-water flow in hydraulic structures is an undesirable phenomenon that occurs during aeration from the water surface, naturally. Some researchers tried to find mechanism of entering air into the water and characterize it in hydraulic engineering by studying rising velocity of air bubbles (Zarrati & Hardwick 1991; Zarrati 1994), developing computational methods to investigate air concentration distribution and pressure fluctuations in rapid flows (Javanbarg *et al.* 2007; Samadi *et al.* 2021), considering an analytical model for calculating air concentration in air-water flows (Ahmadpour *et al.* 2015). Some researchers also studied the air-water flow, practically (Canepa & Hager 2003; Pagliara *et al.* 2006; Bombardelli & Chanson 2009; Ma *et al.* 2010; Chanson 2013; Nazari *et al.* 2015). Juon & Hager (2000) worked on ski jumps with a literature review on past studies, and physical model investigation on the plane and spatial features of ski jump jets. Heller *et al.* (2005) considered the 2D ski jump trajectories based on experimental modelling. Toombes & Chanson (2007) described the air transport characteristics of jets downstream of bottom outlets. Another study was given by Wahl *et al.* (2008), stating that prototype jet trajectories frequently differ from the often applied trajectory parabola. Schmocker *et al.* (2008) presented a model study of plane ski jump jets resulting particularly in a description of the jet air features, including preliminary general air profiles. Steiner *et al.* (2008) presented a physical model study considering deflectors as the jet-generating element. Heller & Pfister (2009) pointed to the effect of the effective take-off angle, which differs from the geometrical angle, and to the turbulence effect. Pfister & Hager (2009) achieved air concentration characteristics of generated jets by drop and deflector. Guha *et al.* (2010) performed numerical simulation of high-speed turbulent water jets in air using the Eulerian multiphase equations and the *k*–*ε* turbulence models, plus a novel numerical model for mass and momentum transfer. Their results reasonably predict the flow physics of high-speed water jets in air. Pfister & Hager (2012) studied the effect of pre-aerated approach flow on generated jets by deflector. They demonstrated that pre-aeration of the approach flow upstream of a jet-generating deflector influences the jet air features, as the jet black-water core length is reduced. Pfister *et al.* (2014) studied trajectories and air flow features of ski jump-generated jets. Ahmadpour *et al.* (2017) calculated air concentration profiles of jet generated by drop and deflector with analytical and numerical models. They presented equations for air concentration profiles of the generated jet.

Present research analyses air concentration characteristics of the jet downstream of a ski jump regarding different hydraulic and geometric conditions. To do so, three different cases were selected from the previous researches and new equations were developed to predict the air concentration distributions along the jet respected to the hydraulic and geometric parameters. To verify the analytical results, numerical simulations were also performed for three different cases. Finally, the obtained analytical and numerical results were compared with experimental and another analytical data in different conditions. Then, using the analytical model, the jet velocity profiles are plotted.

## MATERIALS AND METHODS

### Experimental data collection

In this study, different ski jump generated jets in different hydraulic and geometric conditions were considered to solve the analytical equations. To do so, three different conditions from the experimental data of Schmocker *et al.* (2008) and Balestra (2012) were used (Table 1).

Case . | φ (^{o})
. | R (m)
. | β (^{o})
. | α (^{o})
. | h_{o} (m)
. | Fr_{o} (-)
. | δ (^{o})
. | T (m)
. | L (m)
. _{b} | We_{o} (-)
. | Re_{o} × 10^{5} (-)
. |
---|---|---|---|---|---|---|---|---|---|---|---|

1 (J) | 30 | 0.4 | 12 | −18 | 0.046 | 9.56 | 6.0 | 0.009 | 0.318 | 161 | 2.65 |

2 (F) | 12 | 0.4 | 42 | 30 | 0.044 | 6.06 | 21 | 0.103 | 0.151 | 97 | 1.56 |

3 (Q) | 0 | 0.4 | 30 | 30 | 0.045 | 7.92 | 15 | 0.054 | 0.275 | 131 | 2.13 |

Case . | φ (^{o})
. | R (m)
. | β (^{o})
. | α (^{o})
. | h_{o} (m)
. | Fr_{o} (-)
. | δ (^{o})
. | T (m)
. | L (m)
. _{b} | We_{o} (-)
. | Re_{o} × 10^{5} (-)
. |
---|---|---|---|---|---|---|---|---|---|---|---|

1 (J) | 30 | 0.4 | 12 | −18 | 0.046 | 9.56 | 6.0 | 0.009 | 0.318 | 161 | 2.65 |

2 (F) | 12 | 0.4 | 42 | 30 | 0.044 | 6.06 | 21 | 0.103 | 0.151 | 97 | 1.56 |

3 (Q) | 0 | 0.4 | 30 | 30 | 0.045 | 7.92 | 15 | 0.054 | 0.275 | 131 | 2.13 |

where *h _{o}* = approach flow depth and

*Fr*and

_{o}, Re_{o}*We*

_{o}*=*approach flow Froude, Reynolds and Weber numbers, respectively. The geometry of the ski jump is given by

*R*= radius and

*β*= total deflection angle. The stream wise lip angle relative to the horizontal, similar to the geometrical take off angle, is thus

*α*

*=*

*β−*

*φ*, which is typically limited to 30° (USBR 1987). A geometrical parameter related to the ski jump is the equivalent deflector angle tan

*δ*= (1

*−*cos

*β*)/sin

*β*. The height of the ski jump is

*t*

*=*

*R*(1

*−*cos

*β)*, defined perpendicular to the chute bottom and

*L*= black water core length.

_{b}### Analytical model

*ρ*=

_{mix}*ρ*(1 −

_{w}*C*), where

*ρ*is the average density of the mixture,

_{mix}*ρ*is the water density and

_{w}*C*is the air concentration in the water (Javanbarg

*et al.*2007). To determine the air concentration profile during the jet, diffusion equation was used and so 3D continuity equation for air volume flow can be written:where

*ε*,

_{dx}*ε*and

_{dy}*ε*are turbulence diffusivity coefficients,

_{dz}*u*,

*v, w*are jet flow velocities,

*u*,

_{s}*v*,

_{s}*w*are air bubbles rising velocities in

_{s}*x*,

*y*,

*z*directions, respectively. Ahmadpour

*et al.*(2017) solved Equation (1) using separation of variables method as follows:

*c*is constant and

*Z*is the centre of the jet in the

_{C}*z*direction. Zarrati & Hardwick (1991) achieved that the bubble rising velocity was not very sensitive to water flow velocity and is in a similar range to that of a single air bubble rising in still water. Studies indicate that rising velocity of bubbles increases as air concentration increases since bigger bubbles are formed at higher air contents. Therefore, a value of 12 cm/s was considered for for air concentration less than 5%, a value of 19 cm/s for air concentration between 5 and 10% and a value of 25 cm/s for concentrations over 10% (

*w*> >

_{s}*u*and

_{s}*v*). Ahmadpour

_{s}*et al.*(2017) achieved jet velocity profiles of continuity and momentum equations in 3D turbulent flow.where

*θ*is the angle change,

*X*is movement along the jet in the

*x*direction, , is average jet flow velocity,

*u*is the maximum flow velocity and is characteristic velocity scale. Due to the force of gravity, momentum flux is not fixed in the jet, as momentum flux in the throwing jet can be obtained by using the velocity changes and angle change (Ahmadpour

_{max}*et al.*2017). Regarding the Reynolds shear stresses in momentum equation, the eddy viscosity parameter can be defined as:

*ρ*: water density, and : flow velocity fluctuations in

*x*and

*y*directions. Eddy viscosity, , has the same dimension as the

*μ*and the relationship between turbulence viscosity coefficient, , and turbulence diffusion coefficient, , can be represented as:where represents Schmidt number in the range of 0.5 to 2 (Gibson & Launder 1978). To account the effect of buoyancy on , a function of the Richardson flux number,

*R*, was made. This is a measurement of stability in stratified flows and is expressed as:where and are mechanical stress production and buoyancy generation, respectively. To do so, following correlation equation for suggested by Turner (1973) for stratified flows, was used:

_{f}A minimum value of 0.5 and a maximum value of 2 were imposed on this equation and therefore, *σ _{t}* was assumed 0.5.

### Numerical model

*,*

*,*are the area fraction in the direction of the

*x, y, z*; are the acceleration of gravity in the respective directions, , are the viscous acceleration in the direction

*x, y, z; t*is time,

*P*is pressure and

*ρ*is the fluid density. Structure and configuration of the fluid were determined by VOF. This function is as follows (Hirt & Nichols 1981; Hirt 2003):

The position of fluid in the fluid volume function, *F(x, y, z, t)*, is defined. This function describes VOF per unit volume of fluid.

There are six different boundary conditions for the simulated numerical models: upstream (x_{min}) = specified velocity with height of flow; downstream (x_{max}) = outflow condition; floor and side walls = wall condition; upper side = symmetry condition. A uniform grid was generated with mesh dimensions of 2 cm.

## RESULTS AND DISCUSSIONS

By applying different initial and boundary conditions in the numerical and analytical models regarding three selected experimental cases, hydraulic parameters were extracted by solving the governing equations in the domain.

### Air concentration along the Jet

*z*was normalized to the jet thickness () at each x section:resulting in cross-sectional air concentration profiles between the upper,

*Z*= 1 and the lower,

*Z*= 0 trajectories (Pfister & Hager 2009), thus

*Z*is 0.5. Ervine

_{c}*et al.*(1995) reported that deflectors generate additional turbulence within jets, dependent on In addition, slightly influences

*L*, because the gravity vector is almost perpendicular to the flow of flat chutes and jet break-up is enhanced.

The results of the numerical and analytical models were compared with experimental data for air concentration profiles of generated jets by ski jump. Figures 1–4 show *C*(*z*) variations with *X* in Case (J) and Table 2 presents deviations between the analytical and experimental and numerical results.

X . | Experimental data . | Numerical data . |
---|---|---|

0.90 to 1.08 | 4.2% | 3.9% |

1.80 to 2.20 | 12% | 10.2% |

3.60 to 4.40 | 6% | 5.3% |

7.36 to 8.76 | 4.6% | 4% |

X . | Experimental data . | Numerical data . |
---|---|---|

0.90 to 1.08 | 4.2% | 3.9% |

1.80 to 2.20 | 12% | 10.2% |

3.60 to 4.40 | 6% | 5.3% |

7.36 to 8.76 | 4.6% | 4% |

According to Table 2, *X* = 1.80 to 2.20 has the greatest difference. Figure 2, for *X* = 1.80 to 2.20, shows that the analytical method predicts less air concentration into the water jet. Moreover, regarding Figures 1–4, air concentration in the jet centre core *X* = 0.9 to 1.08 is close to zero and while *X* = 7.36 to 8.76 is close to 0.6. It shows that the jet black-water core length is less, and all three methods (experimental, analytical, numerical) predict this result. Figures 5–8 show *C*(*z*) variations with *X* in Case (F) and Table 3 presents deviations between the analytical and experimental and numerical data.

X . | Experimental data . | Numerical data . |
---|---|---|

0.9 to 1.08 | 4.1% | 3.8% |

1.8 to 2.20 | 7.3% | 7.4% |

3.60 to 4.40 | 9.3% | 8.1% |

7.36 to 8.76 | 7.7% | 7.8% |

X . | Experimental data . | Numerical data . |
---|---|---|

0.9 to 1.08 | 4.1% | 3.8% |

1.8 to 2.20 | 7.3% | 7.4% |

3.60 to 4.40 | 9.3% | 8.1% |

7.36 to 8.76 | 7.7% | 7.8% |

Table 3 and Figure 7 show that in *X* = 3.60 to 4.40 the greatest difference has been achieved where the analytical method predicts less air concentration into the water jet. Figures 5–8 illustrate that air concentration in the jet centre core *X* = 0.9 to 1.08 is close to zero and while *X* = 7.36 to 8.76 is close to 0.5. This result shows that jet black-water core length is low and all three methods predict this result. Figures 9–12 show *C*(*z*) variations with *X* in Case (Q) and Table 4 presents deviations between the analytical and experimental and numerical methods.

X . | Experimental data . | Numerical data . |
---|---|---|

0.9 to 1.08 | 7.11% | 7.8% |

1.8 to 2.20 | 10.62% | 10.2% |

3.60 to 4.40 | 10.8% | 9.8% |

7.36 to 8.76 | 2.57% | 3% |

X . | Experimental data . | Numerical data . |
---|---|---|

0.9 to 1.08 | 7.11% | 7.8% |

1.8 to 2.20 | 10.62% | 10.2% |

3.60 to 4.40 | 10.8% | 9.8% |

7.36 to 8.76 | 2.57% | 3% |

In Table 4, *X* = 3.60 to 4.40 and *X* = 1.8 to 2.20 have the greatest difference. According to Figures 10 and 11 for *X* = 1.8 to 2.20 and *X* = 3.60 to 4.40, the analytical method predicts less air concentration into the water jet. Figures 9–12 illustrate that air concentration in the jet centre core *X* = 0.9 to 1.08 is close to zero and while *X* = 7.36 to 8.76 is close to 0.5. It shows that jet black-water core length is low, and all three methods predict this result.

In Cases (J) and (Q), experimental and numerical diagrams are similar in all sections, but the analytical diagram is slightly different. In Case (F), experimental and numerical and analytical diagrams are similar in all sections. Comparing the three Cases (J), (F), (Q), in Cases (F) and (J) the air concentration at the edges of the jet becomes reduced and the core of the jet is more with the front of the jet, but in case (Q), the air concentration at the edges of the jet does not change much to the jet forward. As well, in Case (J) the amount of aeration is greater than in the other cases. However, in Case (Q), the amount of aeration is less than the rest. The results show that in all models, the air concentrations along the jet are uniquely linked to the relative black-water core length. The air concentration profiles in the three cases show that air entrains into the water jet and the entire process contributes to spread the jet and the core of the jet disappears.

### Velocity profile along the Jet

Velocity profiles for a generated jet of ski jump have been achieved using Equation (3) for Cases (Q), (J), and (F). Figure 13 shows the analytical results of calculating jet velocity profile for Case (J).

As can be seen from Figure 13(a), the velocity profile has a peak point at *Z* = 0.6, and this point shows the core of the jet. The velocity profile in the edges of the jet (Z = 0.9 to 1 and Z = 0 to 0.2) is zero. There is a great velocity difference between the core of the jet and the edges of the jet. Figure 13(b) illustrates that the peak point in the velocity profile is wider by increasing *X*. Figure 13(c) shows that the velocity profile is curved and velocity has increased in the edges of the jet by increasing *X*. Regarding Figure 13(d), the velocity profile in the core and edges of the jet are virtually identical and linear. There is an insignificant velocity difference between the core of the jet and the edges of the jet. By comparing the graphs of Case (J), the velocity of the core of the jet is 9 in section *X* = 0.9 to 1.08 and the velocity of the core of the jet is 6.5 in section *X* = 1.8 to 2.20 and the velocity is 4.4 in section *X* = 3.60 to 4.4 and velocity is 2 in section *X* = 7.36 to 8.76. Figure 14 shows the analytical results of calculating jet velocity profiles for Case (F).

According to Figure 14(a), the velocity profile has a peak point at *Z* = 0.6, and this point shows the core of the jet. The velocity profile in the edges of the jet (Z = 0.9 to 1 and Z = 0 to 0.3) is zero. There is a great velocity difference between the core of the jet and the edges of the jet. As shown in Figure 14(b), the peak point in the velocity profile is wider by increasing *X*. Figure 14(c) illustrates that the velocity profile is curved, that it is maximum in Z = 0.5 to 0.7 and velocity has increased in the edges of the jet by increasing *X*. Regarding Figure 14(d), the velocity profile in the core and edges of the jet are virtually identical and linear and differences are ignorable. By comparing the graphs of Case (F), the velocity of the core of the jet is 6.8 in section *X* = 0.9 to 1.08, the velocity of the core of the jet is 5.5 in section *X* = 1.8 to 2.20, velocity is 3 in section *X* = 3.60 to 4.4, and velocity is 1.4 in section *X* = 7.36 to 8.76. Figure 15 shows the analytical results of calculating jet velocity profiles for Case (Q).

From Figure 15(a) it can be concluded that the velocity profile has a peak point at *Z* = 0.5, that this point shows the core of the jet. The velocity profile in the edges of the jet (Z = 0.85 to 1 and Z = 0 to 0.15) is zero. There is a great velocity difference between the core of the jet and the edges of the jet. As shown in Figure 15(b), the peak point in the velocity profile is wider by increasing *X*. Figure 15(c) illustrates that the velocity profile is curved and velocity has increased in the edges of the jet by increasing *X*. From Figure 15(d), in the core and edges of the jet, the velocity profiles are virtually identical and linear. By comparing the graphs of Case (Q), the velocity of the core of the jet is 4.8 in section *X* = 0.9 to 1.08 and the velocity of the core of the jet is 3.5 in section *X* = 1.8 to 2.20 and velocity is 2.5 in section *X* = 3.60 to 4.4, and velocity is 1.1 in section *X* = 7.36 to 8.76.

Using the results obtained about aeration in the generated jet, the aeration in Case (J) was more than in the other cases, and velocity profile in Case (J) was compared to the others, where the core of the jet velocity change is much greater. Cases (J) and (Q) have a high Froude number, but aeration in Case (Q) was less than in others, therefore the core of the jet velocity change is less than in the other cases.

## CONCLUSIONS

Air concentration profiles in the generated jets from a ski jump in three dimensions were calculated by analytical and numerical methods. In the analytical model, by solving multiphase equations, air concentration profiles have been obtained along the generated jet from the ski jump in three different cases. In this article, in each section of the jet thrown from the ski jump, air concentration and jet velocity can be predicted. To perform the numerical simulations, for modelling free surface and turbulence, VOF and RNG models were used. Finally, results of analytical and numerical models were compared with experimental data and good agreement was achieved. Generally, it was verified that hydraulic and geometric conditions have a significant effect on jet black-water core length. Moreover, using analytical approach, the jet velocity profiles along the jet have been achieved and results showed that the velocity profile has decreased along increasing *X* and this prevents scouring at the end of the jet. Air entrains into the water jet and the entire process contributes to jet spreading and subsequent pressure decay.

## DATA AVAILABILITY STATEMENT

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

MSc Thesis