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Aeration is an efficient way to prevent cavitation erosion of a discharge structure and alleviate damage to trajectory nappe on the downstream energy dissipator. Due to the complexity of the aeration jet, it is still a big challenge to understand the aeration mechanism at the water-air interface. This study applied the two-fluid model to simulate three-dimensional entrainment characteristics of jet flow, and also analyzed effects of bubble size, wall roughness, drag force model, turbulence model, and numerical model dimension on clear-water length. Results reveal that the two-fluid model can accurately and effectively simulate clear-water length, thickness variation, and average cross-section air concentration of jet flow with maximum error of 4.1%, 6.5% and 4.2% (x = L1, defining clear-water length with 1% air concentration as L1), respectively. Wall roughness height has a crucial influence on clear-water length. Taking L1 as an example, clear-water length decreases by 13.9% when roughness height increases from 0 to 3 mm. The drag models and bubble particle sizes affect clear-water length only when the profile aeration concentration is greater than 15%. Turbulence model and numerical model dimension have a minor effect on clear-water length.

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  • Two-fluid model can accurately simulate aerated deflector-generated jets.

  • The wall roughness height has a crucial influence on the clear-water length.

  • The drag models and bubble particle sizes affect clear-water length only when the profile aeration concentration is greater than 15%.

  • Turbulence model has a minor effect on the clear-water length.

Graphical Abstract

Graphical Abstract
Graphical Abstract
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Graphical Abstract
Graphical Abstract
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air concentration, clear-water length, multiphase flow, numerical simulation, the two-fluid model
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With rapid development of high dam construction in China, the maximum discharge of per unit width is more than 200 m3/(s·m), and the maximum discharge speed can reach as high as 50 m/s, which makes safety the primary issue. Skipping energy dissipator is a remarkable strategy to discharge jet flow to far downstream channels to guarantee safety of dams (Nazari et al. 2015). The movement of trajectory nappe is accompanied by intense aeration. When the trajectory nappe jumps into the air, high-speed water flow carries huge kinetic energy and vortices generated due to turbulence, resulting in huge momentum exchange between water and air, and consuming kinetic energy of the high-speed flow. Jet flow contains two flow patterns: water-carrying bubble flow (water phase as continuous phase) and discrete droplet flow (air phase as continuous phase). Jet flow can be divided into three areas (Figure 1): clear-water area, water-carrying bubble area, and discrete droplet area. The intense aeration and breaking of jet flow can affect the accurate prediction of trajectory distance (Miyashita et al. 2014). Due to the complexity of the aeration jet, it is still a big challenge to understand the aeration mechanism at the water-air interface (Zhang et al. 2010). Exploring underlying mechanisms and simulating deflector-generated jets in spillways accurately can prevent cavitation damage of flood discharge structures, and alleviate impact damage of the water jet on downstream energy dissipator, thus cutting cost and guaranteeing safety in hydraulic engineering (Pfister & Hager 2010; Bai et al. 2018; Wang et al. 2020).
Figure 1
Area division of aerated flow.
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Area division of aerated flow.

Figure 1
Area division of aerated flow.
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Area division of aerated flow.

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With advancement of numerical calculation, many computational fluid dynamics (CFD) methods have emerged, including grid method (e.g., finite volume method) and meshless method (e.g., smoothed particle hydrodynamics, Lattice Boltzmann method). Meshless method is challenging to simulate water-air multiphase flow accurately and efficiently with high Reynolds number in hydraulic engineering (Liu & Liu 2010). At present, for grid method, the VOF (volume of fluid) model, the two-fluid model, and the mixture model, are widely used in multiphase simulation. Among these, the mixture method is suitable for a small amount of discrete phase (air volume fraction is less than 15%) in continuous water phase (Bombardelli 2012; Chanson 2013). The VOF model is often applied to track air-water interface, yet the interaction between bubbles and water is not considered (Xue et al. 2013; Witt et al. 2015). The two-fluid model is suitable for multiphase simulation and studying complex hydrodynamic characteristics of water-air flow in hydraulic engineering; for example, Guha et al. (2010) simulated high-speed free jet and reasonably predicted velocity, pressure, and aeration concentration distribution of jet flow. Compared with the VOF model, the two-fluid model solves the governing equation of water and air simultaneously and considers the force of each phase in detail so that the flow field characteristics of each phase can be obtained more realistically and comprehensively. In this model, each phase is coupled through pressure and exchange coefficient and only the pressure is shared by water and air. Multiphase interaction can be reflected in momentum equation through the drag force coefficient. In addition, interaction force, resistance, virtual mass force and turbulent dispersion force of interphase are also considered in the model. Yang et al. (2019) improved the drag force model in the two-fluid model and the calculated cavity length and cross-section concentration distribution of jet flow were in good agreement with experimental results. Based on the two-fluid model, Teng et al. (2016) simulated a series of physical model tests conducted by Pfister & Schwindt (2014). However, effects of bubble size, wall roughness, drag force model, and turbulence model on clear-water length of aerated jet flow have been less discussed.

Due to high speed and severe mixing of water and air in hydraulic engineering, the volume fraction of discrete bubble in water flow is often more than 15%, so the VOF method or the two-fluid model is more suitable (Ahmadpour et al. 2022). In this paper, we aim to: (1) simulate three-dimensional (3D) aeration characteristics of the free jet using the two-fluid model and VOF model; (2) verify the effectiveness and accuracy of the two-phase model using the experimental results of Pfister & Schwindt (2014); and (3) explore the effects of bubble size, wall roughness, drag force model, and the turbulence model on the clear-water length of the jet flow based on one-at-a-time experiments. The schematic figure is shown in Figure 2.
Figure 2
Schematic figure.
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Schematic figure.

Figure 2
Schematic figure.
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Schematic figure.

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VOF model

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VOF method is a free surface tracking technology applied to the grid. In the calculation area, volume fraction of each phase in each calculation grid will be tracked. In each calculation unit, volume proportion of water is denoted as ⁠, and volume proportion of air (or bubbles) is denoted as ⁠. When ⁠, it means that the control unit is only air; and when ⁠, it means that the control unit is filled with water; when ⁠, it means that the control unit has both water and air. The density and viscosity of the mixed fluid in each control unit can be expressed as:
(1)
(2)
where, ρw, ρa, μw, μa denote the density and viscosity of water and air respectively. Transport control equation of water volume fraction can be expressed as:
(3)
where, t represents time, ui and xi represent velocity component and coordinate component respectively.

The two-fluid model

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Governing equation

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In the two-fluid model, the volume fraction of each phase is consistent with VOF method. The sum of volume fraction of water and gas is 1, that is:
(4)
continuous equation of each phase is:
(5)
momentum equation of each phase is:
(6)
where is momentum exchange. In the momentum equation of water, momentum exchange term ⁠, while in the momentum equation of air, ⁠. can be expressed as:
(7)
where, includes interphase drag force ⁠, external volume force ⁠, lift ⁠, wall lubrication ⁠, virtual mass force and turbulent diffusion ⁠. The latter four items can be expressed as:
(8)
(9)
(10)
(11)
where is lift coefficient; is wall lubrication coefficient; is the phase relative velocity component tangential to the wall surface; is unit normal pointing away from wall boundary; ⁠, and ⁠, ⁠, da is bubble diameter, yw is the distance to the wall boundary; ⁠; ⁠.

Interphase drag force

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In the two-fluid model, the interphase drag force is used to reflect the interaction force between water and air, which includes the comprehensive effects of friction, pressure and condensation, expressed as follows:
(12)
where, is the interphase exchange coefficient between water and air, and represent the velocity of water and air respectively. can be expressed as:
(13)
The momentum exchange coefficient is closely related to drag coefficient (CD) and interfacial area density (Ai). Interfacial area density is closely related to bubble diameter. For drag coefficient ⁠, many scholars have given different empirical models. Among these, the Schiller and Naumann model is expressed as:
(14)
where ⁠. In the Schiller and Naumann model, the calculation of Reynolds number only considers the viscosity coefficient of water and ignores the viscosity of air.
Kolev (2005) believed that in high-speed flow with severe aeration, bubbles had the characteristics of breaking and merging, while the expression of the Schiller and Naumann model was only based on a single bubble. Affected by the breaking and merging of a large number of bubbles, the actual CD value was greater than that of a single bubble. Therefore, to calculate drag coefficient CD in severe aeration flow, Yang et al. (2019) improved the effective viscosity coefficient of flow to consider the influence of aeration concentration. Then calculation of Reynolds number of aerated flow becomes: ⁠, and water-air momentum exchange is more truly reflected in the two-fluid model.
(15)
Depending on Reynolds number, drag coefficient CD can be expressed as:
(16)

For the symmetric model, the calculation of CD is the same with the Schiller and Naumann model, but the density and viscosity of the aerated flow are taken as their mixing density and mixing viscosity, which can be expressed as:

⁠, ⁠, ⁠. All ⁠, ⁠, and in Equations (12)–(14) are substituted by the mixing density ⁠, mixing viscosity and average diameter ⁠.

Turbulence model

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For the simulation of aerated flow, due to its high Reynolds number and wide simulation area, it is necessary to use the turbulence model to close the equation, such as turbulence model (Yang et al. 2019; Wei et al. 2020). When the two-fluid model is used to simulate aerated flow, the mixture turbulence model and per phase turbulence model can be solved directly.

The mixture turbulence model adopts mixing property and mixing velocity of water and air, which is effectively to capture important characteristics of turbulence. The equation and equation can be expressed as:
(17)
(18)
where, the turbulent viscosity of the mixture phase can be expressed as ⁠. is the turbulence generation term. and ⁠, where is 0.75, is 0.45. is the characteristic time induced by turbulence.

For per phase turbulence model, equations and ε equations are solved for each phase. The form of the equation is similar to Equations (17) and (18), but the physical properties and velocity of each phase are used to replace mixture properties and mixing velocity. In this paper, for the two-fluid model, the mixture turbulence model and per phase turbulence model are used to close the control equation.

The simulations with the models introduced above are carried out by ANSYS FLUENT 16.0.

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Verification of aeration characteristics of deflector-jets

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The Laboratory of Hydraulics, Hydrology and Glaciology (VAW) in Zurich carried out a large number of physical model tests on aerated deflector-generated jets (Pfister & Hager 2009; Pfister & Schwindt 2014), and analyzed trajectory and air concentration distribution of deflector-generated jets. The experiment was carried out in a flume with a width of 0.3 m and a length of 6.0 m. The structure parameters of the deflector are shown in Figure 3: the slope angle of the chute bottom is 12°, the deflector angle is 11.3°, and the deflector height t is 0.013 m. Numerical simulation is an important supplement and extension of the model test, which can obtain the field characteristics and aeration concentration distribution of air-water flow in detail. This paper takes the initial Froude number of upstream inflow Fr0 = 8.9, upstream water depth h0 = 0.066 m, and inflow velocity V0 = 7.16 m/s (Pfister & Schwindt 2014). The two-fluid model is used to carry out a 3D numerical simulation of average cross-section air concentration distribution, thickness variation and clear-water length (L) of the aerated jet. At the same time, bubble size, turbulence model, drag force model, and wall roughness that affect aeration characteristics are analyzed.
Figure 3
Definition sketch of deflector and aerated jet (Blue: clear-water area).
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Definition sketch of deflector and aerated jet (Blue: clear-water area).

Figure 3
Definition sketch of deflector and aerated jet (Blue: clear-water area).
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Definition sketch of deflector and aerated jet (Blue: clear-water area).

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Computational domain and meshing

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A 3D mathematical model is established based on the physical model test of Pfister and Hager (Figure 4). The required mesh size is the millimeter level because of small bubble size, which increases the number of grids exponentially. The width of the computational domain takes 0.1 m to reduce grid numbers, the chute length is 0.6 m, and the width of the downstream computational domain is 1.5 m. The Gambit is used to divide the grid of computational domain, all grids are regular hexahedron, the free jet area is encrypted by adaptive encryption method, and grid size of the computational domain is reduced by halves when the aeration concentration is less than 90%, and convergence of numerical grids is checked.
Figure 4
Computational domain and meshing.
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Computational domain and meshing.

Figure 4
Computational domain and meshing.
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Computational domain and meshing.

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Boundary conditions and model settings

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In the two-fluid model, water is set as main phase and air is set as second phase, in which bubble size is set as 1 mm. The upstream inflow boundary conditions are set as velocity inlet. The downstream outlet is set as pressure outlet, the default pressure is standard atmospheric pressure, and reflux volume fraction of air is set as 1, that is, the downstream outlet is simulated as free outflow, the air inlet boundary is set as air pressure inlet, the default pressure is standard atmospheric pressure, and its volume fraction of air is set as 1. The roughness height is set as 0 due to smooth surface of organic glass material. The near wall surface is treated by the standard wall function method. Adaptive method of wall grid is used to meet the requirements of the turbulence model. Transient method is adopted and calculation time step is set as 0.0001 s. When the water flow at the inlet and outlet tends to be stable, it is considered that the calculation has reached a stable state.

For interphase interaction in the equation, virtual mass force cannot be ignored due to the large water-air density ratio, and virtual mass coefficient takes 0.5. The wall lubrication force adopted the Antal model and the coefficient ⁠, ⁠. Drag force model adopts the symmetric model, the interface area density model adopts the symmetric model, the turbulence model uses the mixture turbulence model.

Effects of grid size and boundary conditions on clear-water length

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Due to the small bubble size, the required mesh size is millimeter level to better simulate the aeration characteristics of aerated flow, which will require huge computational resources. To obtain the most efficient mesh size, we divided mesh size from 1.8 to 6 mm into six sets of meshes, corresponding to number of meshes from 454,000 to 3,237,000. To analyze the influence of boundary conditions on both sides of downstream on aeration characteristics of the aerated jet flow, boundary conditions on the left and right sides of the downstream are taken as both pressure boundaries or both wall boundaries. Pfister & Schwindt (2014) defined clear-water length with profile air concentrations of 1, 5 and 10% as L1, L5 and L10, respectively. The variation of clear-water length with the grid number under different boundary conditions is shown in Figure 5. It can be seen that clear-water length with different aeration concentrations under different boundary conditions increases with the increase of grid numbers. Considering that a large number of working conditions need to be calculated in subsequent analysis, the calculation results of 1,988,000 grids are used, which can not only take into account discrete error but also appropriately improve calculation efficiency.
Figure 5
Effects of grid size and boundary conditions on clear-water length.
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Effects of grid size and boundary conditions on clear-water length.

Figure 5
Effects of grid size and boundary conditions on clear-water length.
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Effects of grid size and boundary conditions on clear-water length.

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Because the depth of incoming flow is much smaller than the width, the upper and lower sides air penetrates the trajectory nappe before the left and right sides air, determining clear-water length. When the left and right sides of boundary conditions change from wall boundary to pressure boundary, jet flow through the deflector entering the downstream can aerate not only from the upper and lower sides of downstream but also from the left and right sides, clear-water length decreases slightly compared with wall boundary conditions on both sides (Figure 5). Aeration of the left and right sides aggravates aeration ability of the upper and lower sides of the jet, which will shorten clear-water length slightly.

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Verification of calculation results

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Clear-water length

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Pfister & Schwindt (2014) conducted a large number of physical model tests with inflow depth h0 of 0.047 ∼ 0.086 m and initial Froude number Fr0 of 5.9 ∼ 10.4, and obtained empirical formulas for clear-water length L1, L5 and L10:
(19)
where i takes 1, 5 and 10, then Li and Bi represent the characteristic length and the corresponding characteristic coefficient of clear-water profile with air concentration of 1, 5 and 10%, respectively. Fitting by a large number of experimental data, B1, B2 and B3 are 73, 105 and 130, respectively. Taking inflow water depth h0 = 0.066 m and initial Froude number Fr0 = 8.9 as an example, aeration characteristics of trajectory nappe are numerically simulated.
Figure 6 compares aerated flow patterns simulated by the two-fluid model and the VOF method. Using the two-fluid model, aeration of jet flow is relatively intense, and there is an obvious clear-water area in the initial stage. After a certain distance, when trajectory nappe is completely penetrated by surrounding air, the clear-water disappears. This phenomenon cannot be observed using VOF model. By VOF model, with clear water-air interface, surrounding air is always unable to penetrate trajectory nappe, and clear-water continuously exists. Taking into account interaction between bubbles and water, the two-fluid model can accurately simulate floating and convective diffusion of bubbles in jet flow, thus the simulated clear-water area and water-air mixing area of aerated jet flow are in good agreement with those of physical model test. The VOF method shares the velocity field and pressure field between air and water, but ignores the interphase interaction, leading to inaccurate calculation of aeration characteristics of jet flow and inconsistent results with physical reality. Therefore, the subsequent section adopted the two-fluid model for aerated flow simulation. The comparison between clear-water length of the two-fluid model and the measured value is shown in Table 1, suggesting that the simulation results are in good agreement with experimental results, and the maximum error is only 4.1%. The simulation of clear-water length of the aerated jet by the two-fluid model is reliable.
Table 1

Comparison of clear-water length between the two-fluid model and experiments

Clear-water length (m)Experiments (Pfister & Schwindt 2014)Two-fluid modelError
L1 0.60 0.62 3.9% 
L5 0.86 0.83 3.5% 
L10 1.06 1.02 4.1% 
Clear-water length (m)Experiments (Pfister & Schwindt 2014)Two-fluid modelError
L1 0.60 0.62 3.9% 
L5 0.86 0.83 3.5% 
L10 1.06 1.02 4.1% 
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Figure 6
The simulated flow pattern of aerated jet (red represents water phase and blue represents air phase) under condition of slope angle of chute bottom = 12°, deflector angle = 11.3°, deflector height t = 0.013 m, inflow water depth h0 = 0.066 m and initial Froude number Fr0 = 8.9. Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/ws.2022.428.
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The simulated flow pattern of aerated jet (red represents water phase and blue represents air phase) under condition of slope angle of chute bottom = 12°, deflector angle = 11.3°, deflector height t = 0.013 m, inflow water depth h0 = 0.066 m and initial Froude number Fr0 = 8.9. Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/ws.2022.428.

Figure 6
The simulated flow pattern of aerated jet (red represents water phase and blue represents air phase) under condition of slope angle of chute bottom = 12°, deflector angle = 11.3°, deflector height t = 0.013 m, inflow water depth h0 = 0.066 m and initial Froude number Fr0 = 8.9. Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/ws.2022.428.
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The simulated flow pattern of aerated jet (red represents water phase and blue represents air phase) under condition of slope angle of chute bottom = 12°, deflector angle = 11.3°, deflector height t = 0.013 m, inflow water depth h0 = 0.066 m and initial Froude number Fr0 = 8.9. Please refer to the online version of this paper to see this figure in colour: http://dx.doi.org/10.2166/ws.2022.428.

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Thickness variation of aerated jet

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Taking aeration concentration of 10% as interface between water and air, thickness variation of the aerated jet is obtained. Based on a large number of experimental data, Pfister & Schwindt (2014) found that thickness variation along the aerated jet direction can be expressed as Equation (20), where h0 is initial inflow depth, and L1 is clear-water length (air concentration is 1%).
(20)
The numerical simulation results are consistent with experimental results, and the maximum error is only 6.5% (Figure 7). Before the clear-water area disappeared, flow thickness obtained by numerical simulation was slightly smaller than that of experimental results; when the clear-water disappeared, flow thickness obtained by numerical simulation was slightly larger than that of experimental data.
Figure 7
Thickness variation of aerated jet.
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Thickness variation of aerated jet.

Figure 7
Thickness variation of aerated jet.
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Thickness variation of aerated jet.

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Average cross-section air concentration along jet flow

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Pfister & Hager (2009) and Pfister & Schwindt (2014) conducted a series of tests on aeration concentration of the aerated jet based on the physical model, and obtained Equations (21) and (22) respectively:
(21)
(22)
The numerical simulation results are in good agreement with experimental data (Figure 8). At x = L1, where the clear-water area disappeared, average cross-section aeration concentration of aerated jet was 38.0% for both empirical equations, while the numerical simulation results were 36.4%, and the error was 4.2%. Generally, the two-fluid model can accurately and effectively reflect aeration characteristics of deflector-generated jets.
Figure 8
Distribution of average cross-section aeration concentration along the jet flow.
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Distribution of average cross-section aeration concentration along the jet flow.

Figure 8
Distribution of average cross-section aeration concentration along the jet flow.
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Distribution of average cross-section aeration concentration along the jet flow.

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Analysis of effects on clear-water length

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Effect of different turbulence models on clear-water length

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Different turbulence models (mixture of standard turbulence model, mixture of Realizable turbulence model, mixture of RNG turbulence model and per phase of RNG turbulence model) are conducted to simulate clear-water length of trajectory nappe. Results suggest that different turbulence models have little effect on clear-water length with all air concentrations profile cases (Figure 9).
Figure 9
Effect of turbulence model on clear-water length.
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Effect of turbulence model on clear-water length.

Figure 9
Effect of turbulence model on clear-water length.
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Effect of turbulence model on clear-water length.

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Effect of drag force models on clear-water length

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This paper adopted five interphase drag force models to explore their effect on clear-water length, including the Schiller and Naumann model, symmetric model, Yang's model and two self-proposed models. The first self-proposed model (Model 1) modified calculation formula of effective viscosity based on the Schiller and Naumann model using User-Defined Function (UDF). Effective viscosity is the same as Yang's model, which considers the factors of aeration concentration, thus could reduce Reynolds number and increase drag force coefficient (Equations (15) and (16)). Subsequently, the second self-proposed model (Model 2) modified drag coefficient based on the symmetric model using UDF. Drag coefficient is calculated using Yang's model (Equation (16)), considering the aeration concentration influence. When Re is greater than 500, drag coefficient is no longer a constant value, but depends on aeration concentration. When Re is less than 16, drag coefficient CD is proportional to 1/Re.

These five drag force models are conducted to simulate aeration characteristics of the trajectory nappe. Their simulation results are shown in Figure 10. It suggests that different drag force models have little effect on clear-water length L1, L5 and L10. Although it is found that different drag force models are closely related to water-air interaction and the escape of bubbles, the influences on clear-water length L1, L5 and L10 are not obvious. Only when profile aeration concentration is greater than 15%, does it show weak effect on clear-water length.
Figure 10
Effects of drag force model on clear-water length.
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Effects of drag force model on clear-water length.

Figure 10
Effects of drag force model on clear-water length.
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Effects of drag force model on clear-water length.

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Effect of bubble diameters on clear-water length

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In the two-fluid model, bubble diameter is commonly used to characterize the bubble distribution with different diameters. The momentum exchange between water and air is greatly affected by bubble size. The formula of momentum exchange coefficient K (Equation (13)) reflects that the larger the bubble diameter, the more severe the momentum exchange. The bubble diameter directly affects aeration characteristics such as bubble floating velocity and aeration concentration distribution in water. Therefore, it is crucial to study the effect of bubble diameter on clear-water length of jet flow.

According to test results, Ye et al. (2014) found that bubble size of aerated flow was mainly distributed in the 0.1–1.0 mm range. Yang et al. (2019) found through experiments that 85% of bubble sizes in water were between 0.5 and 4 mm. In this paper, six bubble diameters of 0.4 mm, 0.6 mm, 0.8 mm, 1, 2 and 4 mm are selected to study clear-water length of aerated deflector jets. The simulation results are shown in Figure 11. It suggests that bubble diameter has little effect on length L1, L5 and L10 of jet flow, and conclusions are consistent with the simulation results of Teng et al. (2016) and Yang et al. (2019). With the increase of aeration concentration, when profile aeration concentration is greater than 15%, bubble size will have a slight impact on clear-water length, and with the increase of bubble size, clear-water length tends to decrease slightly.
Figure 11
Effect of bubble diameters on clear-water length.
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Effect of bubble diameters on clear-water length.

Figure 11
Effect of bubble diameters on clear-water length.
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Effect of bubble diameters on clear-water length.

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Effect of wall roughness on clear-water length

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When jet flow velocity is 3–4 m/s, the surface wave will be broken. With the increase of the flow velocity, flow turbulence is enhanced, and air entrainment phenomenon is intensified. The increase of wall roughness can increase aeration concentration. When flow velocity reaches above 6–7 m/s, the number of water droplets jumping into air increases, resulting in increasing air concentration accordingly (Qi 2017). The air-carrying capacity of water flow is closely related to turbulence intensity. Due to severe turbulence and shear effect of the interface of water and air, a large number of vortices are formed at the interface. Numerous pits are formed with the strong pulsation of vortex in the surface. And mingling of air into water during closure would form an aerated jet.

In this paper, eight groups of wall boundary conditions with different roughness heights are selected to simulate are aerated jet. The results are shown in Figure 12. With the increase of roughness height, clear-water length decreases for all air concentration profile cases. Taking clear-water length L1 as an example, when roughness height increases from 0 to 3 mm, clear-water length decreases by 13.9%. Therefore, roughness height has a significant influence on clear-water length.
Figure 12
Effect of wall roughness on clear-water length.
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Effect of wall roughness on clear-water length.

Figure 12
Effect of wall roughness on clear-water length.
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Effect of wall roughness on clear-water length.

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Effect of 2D and 3D model on clear-water length

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Based on the two-fluid model, many scholars have carried out 2D numerical simulations of aerated flow (Pfister & Schwindt 2014; Teng et al. 2016; Yang et al. 2019) and achieved results consistent with experimental results. When the two-fluid model is used to calculate clear-water length of the aerated jet, mesh must be fine enough. 3D numerical simulation of clear-water length of the trajectory nappe in large-scale water conservancy projects is still difficult to achieve. Therefore, it is necessary to evaluate the reliability of the 2D model for calculation of clear-water length, which would be valuable in numerical simulation for aerated jet flow in hydraulic engineering.

To study the reliability of the 2D model for clear-water length calculation, this paper compared 2D and 3D model simulation results on clear-water length based on the two-fluid model. The simulated clear-water area with different air concentration profile is shown in Figure 13. And clear-water length with different aeration concentration profile is shown in Figure 14. It suggests that shapes of clear-water area with different air concentration profile calculated by 2D model and 3D model are almost similar, as is clear-water length. The maximum difference of clear-water length between the 2D and 3D models is 3.8%.
Figure 13
Effect of 2D and 3D model on the shape of clear-water area.
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Effect of 2D and 3D model on the shape of clear-water area.

Figure 13
Effect of 2D and 3D model on the shape of clear-water area.
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Effect of 2D and 3D model on the shape of clear-water area.

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Figure 14
The comparison of 2D and 3D models on clear-water length.
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The comparison of 2D and 3D models on clear-water length.

Figure 14
The comparison of 2D and 3D models on clear-water length.
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The comparison of 2D and 3D models on clear-water length.

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Discussion on influence factors of clear-water length

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This study explored influence factors of clear-water length based on six bubble sizes, eight wall roughnesses, five drag force models, and four turbulence models using one-at-a-time experiments. Numerical simulation showed that the drag models and bubble sizes would affect clear-water length only when the aeration concentration was greater than 15%. Previous studies showed that bubble particle sizes had no effect on distribution of aeration concentration when profile aeration concentration was smaller than 5% (Teng et al. 2016; Yang et al. 2019). Yang et al. (2019) found the drag force model and bubble size had an impact on the distribution of aeration concentration along the jet flow when profile aeration concentration was greater than 20%, which was consistent with this study. Our results also showed that turbulence model and numerical model dimension had a minor effect on clear-water length. The wall roughness height had a crucial influence on the clear-water length.

The aeration mechanism of high-velocity flow is extremely complex. The physical model test is an important method to study the aeration of jet flow. Hydraulic model test that meets the similarity of gravity mainly considers gravity effect and reflects the macroscopic phenomenon of water flow, but viscous force and surface tension of water flow cannot fully meet the similarity. As aeration is a kind of microscopic flow phenomenon with a scale of millimeters or even microns, it is difficult to reflect both macroscopic and microscopic flow phenomena in the model test. Therefore, the aeration characteristics obtained from the model tests have significant scale effects. There is no consistent understanding for the quantification of scale effects. The numerical simulation in this paper is based on the experimental results of Pfister & Schwindt (2014). It is still a challenge to simulate the prototype aerated nappe with fine grid size. In the future, CFD parallel computing technology and massively parallel computing platforms need to be improved to achieve systematic research on the hydraulic characteristics of prototype aerated flow.

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The numerical simulations are used to calculate aeration characteristics of deflector-generated jets in spillways. The effectiveness and accuracy of the two-fluid model is systematically verified, and factors of clear-water length are discussed and analyzed.

  • (1)

    The two-fluid model can accurately simulate floating and convective diffusion of bubbles in jet flow. The maximum error of clear-water length is 4.1%, the maximum error of aerated jet thickness is 6.5%, and the error of average cross-section aeration concentration of the aerated jet where clear-water area (L1) has disappeared is 4.2%.

  • (2)

    Compared with wall boundary condition on left and right sides of downstream, clear-water length is shorter with pressure boundary on left and right sides. Aeration of the left and right sides will intensify aeration ability of the upper and lower sides of the trajectory nappe, resulting in a slight reduction in clear-water length.

  • (3)

    The turbulence model type and the numerical model dimension show a minor effect on clear-water length, and drag models and bubble diameters affect clear-water length only when profile aeration concentration is greater than 15%. Wall roughness height has a great influence on clear-water length; taking L1 as an example, when roughness height increases from 0 to 3 mm, clear-water length decreases by 13.9%.

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Ahmadpour
R.
,
Sarkardeh
H.
 &
Azamathulla
H.
2022
Mathematical simulation of air-water flow along Ski Jump Jet
.
Water Science & Technology: Water Supply
22
(
2
),
2093
–
2105
.
Google Scholar
 
Bai
R.
,
Liu
S.
,
Tian
Z.
,
Wang
W.
 &
Zhang
F.
2018
Experimental investigation of air-water flow properties of offset aerators
.
Journal of Hydraulic Engineering
144
(
2
),
04017059.1
–
04017059.10
.
Google Scholar
Crossref
Search ADS
 
Bombardelli
F.
2012
Computational multi-phase fluid dynamics to address flows past hydraulic structures
. In:
Proceedings of the 4th IAHR International Symposium on Hydraulic Structures
,
Porto, Portugal
.
Google Scholar
 
Chanson
H.
2013
Hydraulics of aerated flows: Qui pro quo?
Journal of Hydraulic Research
51
,
223
–
243
.
Google Scholar
Crossref
Search ADS
 
Guha
A.
,
Barron
R. M.
 &
Balachandar
R.
2010
Numerical simulation of high-speed turbulent water jets in air
.
Journal of Hydraulic Research
48
(
1
),
119
–
124
.
Google Scholar
Crossref
Search ADS
 
Kolev
N. I.
2005
Multiphase Flow Dynamics 2
, 2nd edn.
Springer-Verlag
,
Berlin
.
Google Scholar
 
Liu
G.
 &
Liu
M.
2010
Smoothed particle hydrodynamics (SPH): an overview and recent developments
.
Archives of Computational Methods in Engineering
17
(
1
),
25
–
76
.
Google Scholar
Crossref
Search ADS
 
Miyashita
T.
,
Sugawa
O.
,
Imamura
T.
,
Kamiya
K.
 &
Kawaguchi
Y.
2014
Modeling and analysis of water discharge trajectory with large capacity monitor
.
Fire Safety Journal
63
(
2
),
1
–
8
.
Google Scholar
 
Nazari
O.
,
Jabbari
E.
 &
Sarkardeh
H.
2015
Dynamic pressure analysis at chute flip buckets of five dam model studies
.
International Journal of Civil Engineering
13
(
1A
),
45
–
54
.
Google Scholar
 
Pfister
M.
 &
Hager
W.
2009
Deflector-generated jets
.
Journal of Hydraulic Research
47
(
4
),
466
–
475
.
Google Scholar
Crossref
Search ADS
 
Pfister
M.
 &
Hager
W.
2010
Chute aerators. II: hydraulic design
.
Journal of Hydraulic Engineering
136
(
6
),
360
–
367
.
Google Scholar
Crossref
Search ADS
 
Pfister
M.
 &
Schwindt
S.
2014
Air concentration distribution in deflector-jets
. In:
5th IAHR International Symposium on Hydraulic Structures
.
Google Scholar
Crossref
Search ADS
 
Qi
C.
2017
Intensive Study of Aeration and Ventilation Characteristics of Aeration Devices and Air Supply System for Hydraulic Structures
.
Ph.D. Thesis
,
Tianjin University
,
Tianjin
(In Chinese)
.
Google Scholar
 
Teng
P. H.
,
Yang
J.
 &
Pfister
M.
2016
Studies of two-Phase flow at a chute aerator with experiments and CFD modelling
.
Modelling and Simulation in Engineering
2016
,
4729128
.
Google Scholar
Crossref
Search ADS
 
Wang
H.
,
Liu
X.
,
Bai
R.
 &
Liu
S.
2020
Modified bottom Air entrainment estimate by chute offset aerator
.
Journal of Hydraulic Engineering
146
,
06020016
.
Google Scholar
Crossref
Search ADS
 
Wei
W.
,
Deng
J.
 &
Xu
W.
2020
Numerical investigation of air demand by the free surface tunnel flows
.
Journal of Hydraulic Research
59
,
158
–
165
.
Google Scholar
Crossref
Search ADS
 
Witt
A.
,
Gulliver
J.
 &
Shen
L.
2015
Simulating air entrainment and vortex dynamics in a hydraulic jump
.
International Journal of Multiphase Flow
72
,
165
–
180
.
Google Scholar
Crossref
Search ADS
 
Xue
H.
,
Diao
M.
,
Yue
S.
 &
Xu
L.
2013
3-D numerical simulation of the jet nappe in the beveled flip bucket of the spillway
.
Journal of Hydraulic Engineering
44
(
6
),
703
–
709
(In Chinese)
.
Google Scholar
 
Yang
J.
,
Teng
P.
 &
Zhang
H.
2019
Experiments and CFD modeling of high-velocity two-phase flows in a large chute aerator facility
.
Engineering Applications of Computational Fluid Mechanics
13
(
1
),
48
–
66
.
Google Scholar
Crossref
Search ADS
 
Ye
M.
,
Wu
P.
,
Wang
B.
 &
Zhou
Q.
2014
Numerical simulation of aerated flow in hydraulic tunnel
.
Journal of Hydroelectric Engineering
33
(
4
),
105
–
111
(in Chinese)
.
Google Scholar
 
Zhang
F.
,
Xu
W.
 &
Zhu
Y.
2010
Experimental study on formation of air bubbles in self-aerated open channel flows
.
Journal of Hydraulic Engineering
41
(
3
),
343
–
347
(In Chinese)
.
Google Scholar
 
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