## Abstract

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 = L_{1}, defining clear-water length with 1% air concentration as L_{1}), respectively. Wall roughness height has a crucial influence on clear-water length. Taking L_{1} 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.

## HIGHLIGHTS

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

## INTRODUCTION

^{3}/(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).

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.

*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.

## METHODOLOGY

### VOF model

*ρ*,

_{w}*ρ*,

_{a}*μ*,

_{w}*μ*denote the density and viscosity of water and air respectively. Transport control equation of water volume fraction can be expressed as:where,

_{a}*t*represents time,

*u*and

_{i}*x*represent velocity component and coordinate component respectively.

_{i}### The two-fluid model

#### Governing equation

*d*

_{a}is bubble diameter,

*y*

_{w}is the distance to the wall boundary; ; .

#### Interphase drag force

*C*) and interfacial area density (

_{D}*A*). 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: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.

_{i}*C*value was greater than that of a single bubble. Therefore, to calculate drag coefficient

_{D}*C*in severe aeration flow, Yang

_{D}*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.

For the symmetric model, the calculation of *C*_{D} 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:

### Turbulence model

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.

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.

## NUMERICAL SETTINGS

### Verification of aeration characteristics of deflector-jets

*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

*Fr*

_{0}= 8.9, upstream water depth

*h*

_{0}= 0.066 m, and inflow velocity

*V*

_{0}= 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.

### Computational domain and meshing

### Boundary conditions and model settings

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

*L*

_{1},

*L*

_{5}and

*L*

_{10}, 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.

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.

## RESULTS AND DISCUSSION

### Verification of calculation results

#### Clear-water length

*h*

_{0}of 0.047 ∼ 0.086 m and initial Froude number

*Fr*

_{0}of 5.9 ∼ 10.4, and obtained empirical formulas for clear-water length

*L*

_{1},

*L*

_{5}and

*L*

_{10}:where

*i*takes 1, 5 and 10, then

*L*

_{i}and

*B*

_{i}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,

*B*

_{1},

*B*

_{2}and

*B*

_{3}are 73, 105 and 130, respectively. Taking inflow water depth

*h*

_{0}= 0.066 m and initial Froude number

*Fr*

_{0}= 8.9 as an example, aeration characteristics of trajectory nappe are numerically simulated.

Clear-water length (m) . | Experiments (Pfister & Schwindt 2014) . | Two-fluid model . | Error . |
---|---|---|---|

L_{1} | 0.60 | 0.62 | 3.9% |

L_{5} | 0.86 | 0.83 | 3.5% |

L_{10} | 1.06 | 1.02 | 4.1% |

Clear-water length (m) . | Experiments (Pfister & Schwindt 2014) . | Two-fluid model . | Error . |
---|---|---|---|

L_{1} | 0.60 | 0.62 | 3.9% |

L_{5} | 0.86 | 0.83 | 3.5% |

L_{10} | 1.06 | 1.02 | 4.1% |

#### Thickness variation of aerated jet

*h*

_{0}is initial inflow depth, and

*L*

_{1}is clear-water length (air concentration is 1%).

#### Average cross-section air concentration along jet flow

*x*=

*L*

_{1}, 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.

### Analysis of effects on clear-water length

#### Effect of different turbulence models on clear-water length

#### Effect of drag force models on clear-water length

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 *C*_{D} is proportional to 1/*Re*.

*L*

_{1},

*L*

_{5}and

*L*

_{10}. 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

*L*

_{1},

*L*

_{5}and

*L*

_{10}are not obvious. Only when profile aeration concentration is greater than 15%, does it show weak effect on clear-water length.

#### Effect of bubble diameters on clear-water length

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.

*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

*L*

_{1},

*L*

_{5}and

*L*

_{10}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.

#### Effect of wall roughness on clear-water length

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.

_{1}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.

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

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.

#### Discussion on influence factors of clear-water length

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.

## CONCLUSIONS

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 (

*L*_{1}) 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

*L*_{1}as an example, when roughness height increases from 0 to 3 mm, clear-water length decreases by 13.9%.

## DATA AVAILABILITY STATEMENT

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

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

*Intensive Study of Aeration and Ventilation Characteristics of Aeration Devices and Air Supply System for Hydraulic Structures*

*Ph.D. Thesis*