Keywords

1 Introduction

Pneumatic sluicing is an efficient dredging technology which can be used in approach channel, ports, or lake (Pan and Wu 2019). The air-injection of pneumatic sluicing can be employed to stir the bottom nutrients or sediments and take them to the downstream or other place (Ding et al. 2019). It is significant to study the rising process of bubble plume and air-water interaction in open channel in order to improve the transportation efficiency of sands or nutrients.

There are lots of scholars focusing on the movement of bubble plume in water, the gas-liquid characteristics such as gas holdup, bubble velocities, bubble size, and entrained plume flux (Qiang et al. 2018; Yao et al. 2019; Wan et al. 2017; Cheng et al. 2020). For example, Liang and Peng (2005) proposed a formula to calculate the velocity of rising bubble plume in quiescent water with gas discharge, the density of liquid-gas, and height. Song et al. (2011) studied the distribution characteristics of bubble void fraction by physical model experiment and image processing technology. The results show that when the aspect ratio of model’s height and width is 1.0, the bubble plume structure is less affected by the pressure, and the plume structure is stable. With the increase of aspect ratio and pressure, the plume structure is unstable.

With the development of multi-phase numerical simulation, there are lots of researchers adopt air-water two phase flow to study the air plumes in water. And Euler-Euler model is used wildly in multiphase flow modeling with high void fraction (Duguay et al. 2021). For example, Liu and Li (2018) study the bubbly flows in water, and the sensitivity of different turbulence models and the scale-adaption of Euler-Euler model had been conducted. It found that the mesh size should be taken account in the model. Fleck and Rzehak (2019) studied the dynamic flow phenomena of bubble plume with Euler-Euler two-fluid model, and the periodically oscillating bubble plume was been simulated, which shown a good precision about the plume oscillation period. Godino et al. (2020) studied the air-water dispersed and segregated multiphase flows with experiments and Euler-Euler numerical model. Five cases with different flow regimes were simulated using the same set of interfacial force models, and the local rheology of the flow in different interfacial models was considered by a linear blending method. And so on.

Existing studies have gained many achievements in bubble plume and its application. Nevertheless, most of the studies only focus on the rising characteristics of bubble plume and its influence on surrounding waters in quiescent waters. And the study of bubble plume in open channels is insufficient, which the bubble plume is obviously affected by cross flow (Qiang et al. 2018). Based on numerical simulation, this paper uses Eulerian-Eulerian approach for air-water flows, and the population balance model (PBM) is applied to describe the distribution of bubbles, then the gas holdup, size distribution of bubbles, and the flow field of aerated area are studied.

2 Simulation Model

In CFD-PBM model, the turbulent dissipation rate, gas holdup and flow field are calculated by the two-fluid CFD model. And the results are used to calculate the bubble coalescence and breaking rate. Then the PBM equations are solved for the bubble size distribution, flow pattern, interphase force and the turbulence source term caused by bubble turbulence in the improved two-fluid model.

2.1 Euler-Euler Model

Euler-Euler model is used to simulate gas-liquid two phase flow. The model regards the bubble as a continuous phase and runs through water phase. It can simulates the distribution of gas holdup well, and greatly promotes the calculation speed. The continuity equations and momentum equations are represented as (Ranganathan and Sivaraman 2011):

$$ \frac{\partial }{\partial t}\left( {\alpha_i \rho_i } \right) + \nabla \cdot \left( {\alpha_i \rho_i {{\bf{u}}}_i } \right) = 0, \, i = g,l $$
(1)
$$ \frac{\partial }{\partial t}\left( {\varepsilon_i \rho_i u_i } \right) + \nabla \cdot \left( {\alpha_i \rho_i {{\bf{u}}}_i {{\bf{u}}}_i } \right) = - \nabla P^{\prime} + \left( {\alpha_i \mu_{eff} \left( {\nabla {{\bf{u}}} + \nabla {{\bf{u}}}^T } \right)} \right) + F_{i,j} + \alpha_i \rho_i g $$
(2)

In these equations, ρ, αi, u, t stand for density, the ith group volume, fraction velocity vector, time, and the subscript i represents gas (g) or liquid (l). \( P^{\prime}\), F, μeff and g, are modified pressure, interphase force, effective viscosity, and acceleration of gravity, respectively. \( P^{\prime}\) can be calculated by Eq. (3):

$$ P^{\prime} = P + \frac{2}{3}\mu_{eff,l} \nabla \cdot {{\bf{u}}}_l + \frac{2}{3}\rho_l k_l $$
(3)

The effective viscosity of liquid phase μeff,l can be calculated as

$$ \mu_{eff,l} = \mu_l + \mu_{t,l} + \mu_{b,l} $$
(4)

in which μl is the liquid viscosity, μtl and μtg represent the turbulence viscosity of liquid phase and the turbulence viscosity induced by gas phase, as given by Eq. (5) and (6):

$$ \mu_{t,l} = \rho_l C_\mu \frac{k_l^2 }{{\varepsilon_l }} $$
(5)
$$ \mu_{b,l} = C_{\mu b} \rho_l \alpha_g d_b \left| {u_g - u_l } \right| $$
(6)

where kl is turbulent kinetic energy, Cμb is empirical parameters with a value of 0.6 (Ranganathan and Sivaraman 2011).

The k–ε model is used for turbulence model. Its advantage is that this model has the wide applicability and is verified useful in CFD-PBM model.

2.2 The Drag Force

During the pneumatic sluicing process, momentum exchange occurs at the interface between gas and liquid phases, and the interphase force plays an important role in the accuracy of simulation results. The gas-liquid interphase forces include the drag force, lift force, virtue mass force, turbulent dispersion force, and wall lubrication force. Compared the drag force, the other interphase forces can be neglected because of their less significance in the interaction. The drag force represents the interfacial momentum transfer caused by the gas-liquid phase velocity slip. It can be described by Schiller-Naumann formulas shown in Eq. (7) (Olmos et al. 2001):

$$\begin{array}{*{20}l} {F_D = \frac{3}{4}\frac{{C_d }}{{d_b }}\alpha _g \rho _l \left| {u_g - u_l } \right|\left( {u_g - u_l } \right)} \hfill \\ {C_D = \left\{ {\begin{array}{*{20}l} {24\left( {1 + 0.15\text{Re} ^{0.687} } \right)/\text{Re} } \hfill & {{\text{Re}} \le {\text{1000}}} \hfill \\ {0.44} \hfill & {{\text{Re}} > {\text{1000}}} \hfill \\ \end{array} } \right.} \hfill \\ \end{array}$$
(7)

where the CD is the drag coefficient of a bubble size db.

2.3 PBM Model

PBM model is a general method to describe the distribution of dispersed phase size in multiphase flow system (Hulburt and Katz 1964). In gas-liquid multiphase flow system, PBM model can be used to consider the influence of bubble coalescence and breakup on the distribution of bubble size, so as to study the mechanism of two-phase interaction in gas-liquid system. For the gas-liquid system, the group equilibrium equation can be expressed as:

$$ \begin{gathered} \frac{{\partial n\left( {v,t} \right)}}{\partial t} + \nabla \cdot \left[ {n\left( {v,t} \right)} \right] = \underbrace {\frac{1}{2}\int_0^v {a\left( {v - v{\rm{^{\prime}}},v{\rm{^{\prime}}}} \right)} n\left( {v - v{\rm{^{\prime}}},t} \right)n\left( {v{\rm{^{\prime}}},t} \right){\rm{d}}v{\rm{^{\prime}}}}_{{\text{birth}}\;{\text{term}}\;{\text{due}}\;{\text{to}}\;{\text{coalescence}}} - \hfill \\ \underbrace {\int_0^\infty {a\left( {v,v{\rm{^{\prime}} }} \right)} n\left( {v,t} \right)n\left( {v{\rm{^{\prime}}},t} \right)\rm{d}v{\rm{^{\prime}} }}_{{\text{death}}\;{\text{term}}\;{\text{due}}\;{\text{to}}\;{\text{coalescence}}} + \underbrace {\int_\Omega {vb\left( {v{\rm{^{\prime}}}} \right)\beta \left( {v|v{\rm{^{\prime} }}} \right)n\left( {v{\rm{^{\prime}} },t} \right){\rm{d}}v{\rm{^{\prime}} - }b\left( v \right)n\left( {v,t} \right)} }_{{\text{birth}}\;{\text{term}}\;{\text{due}}\;{\text{to}}\;{\text{breakup}}} - \underbrace {b\left( v \right)n\left( {v,t} \right)}_{{\text{death}}\;{\text{term}}\;{\text{due}}\;{\text{to}}\;{\text{breakup}}} \hfill \\ \end{gathered} $$
(8)

In which, v and v′ is the bubble volume, n(v, t) is the distribution function of bubble size, a(v, v′) is the bubble coalescence rate function, b(v) is the bubble breaking rate function, β(v|v′) denotes the probability density function of bubble breaking up into daughter bubbles with the volume from v to v′. In this paper, the discrete method is employed, and transport equation of the bubbles in the kth group can be expressed by Eq. (9):

$$ \frac{\partial }{\partial t}\left( {\rho_g \alpha_k } \right) + \nabla \cdot \left( {\rho_g \alpha_k v_k } \right) = \rho_g \left( {B_{k,c} - D_{k,c} + B_{k,b} - D_{k,b} } \right) $$
(9)

In which, ρ is the density of gas phase, αk is the volume fraction of the bubbles in the kth group, as shown in Eq. (10). The source terms of bubble generation and extinction caused by coalescence and breakup are represented by Eq. (11)–(14).

$$ \alpha_k = N_k V_k \quad \quad { (}k = 0,1,...,N - 1{)} $$
(10)
$$ \overline{B}_{ag,k} = \frac{1}{2}\sum_{i = 1}^N {w_i } \sum_{j = 1}^N {w_j } \left( {L_i^3 + L_j^3 } \right)^{k/3} a\left( {L_i ,L_j } \right) $$
(11)
$$ \overline{D}_{ag,k} = \sum_{i = 1}^N {L_i^k w_i } \sum_{j = 1}^N {w_j } a\left( {L_i ,L_j } \right) $$
(12)
$$ \overline{B}_{br,k} = \sum_{i = 1}^N {w_i } \int_0^\infty {L_k g\left( {L_i } \right)} \beta \left( {L/L_i } \right) $$
(13)
$$ \overline{D}_{br,k} = \sum_{i = 1}^N {w_i L_i^k } g\left( {L_i } \right) $$
(14)

In this study, Luo’s bubble breaking and coalescence model (Luo and Svendsen 1996) is used to investigate the distribution of bubble size in liquid.

The model mentioned above is employed in bubble columns and aeration tanks. Gas holdup, bubble breakup rate and the distribution of daughter bubble size are in good agreement with the tests data (Cheng et al. 2020).

2.4 Simulation Model

The sketch of simulation model is shown in Fig. 1. The model is 10 m long and 0.7 m high. The 0.1 m aeration zone is set in the middle bottom of the channel located at x is from 0.45 m to 0.55 m. The water depth is kept in 0.5 m. The left side of the model is velocity inlet and the right side is set as free outflow. The quadrilateral mesh is adopted with the size of 1 mm, and the total number is 70,000 (as shown in Fig. 1(a)). The gas enters from the inlet with the uniform velocity of 0.2 m/s and initial gas diameter of 0.03 mm, as shown in Fig. 1(b). The cross water flow velocity ranges from 0 to 1 m/s, as shown in Table 1. All the simulations are carried out on the ANSYS Fluent platform. SIMPLE algorithm is used for pressure-velocity equations. QUICK scheme is employed for momentum equations. The volume fraction equations are discretized by the first-order upwind format, and the relaxation factors are set with the default values. The time step in the calculation is 0.002s, and the process ends when air bubble plume is steady (Table 2).

Fig. 1.
figure 1

Computational domain mesh and initial condition: (a) computational domain mesh, (b) initial condition

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Table 1. Simulation conditions (unit: m/s)
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Table 2. Size of discrete bubbles
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3 Results

3.1 Gas Holdup

Figures 2 and 3 show the distribution contours of gas holdup at different flow velocities and gas holdup curves at different heights. It can be seen that gas holdup decreases gradually along the radial direction, and gas holdup in the middle of bubble plume is higher than that at the sides. With cross flow, the bubble plume is incline to the direction of velocity. And with the increase of cross velocity, the bubble plume is wider and gas hold up is lower at the same height than that with a small velocity.

Fig. 2.
figure 2

Volume fraction of gas phase in different conditions

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Fig. 3.
figure 3

Volume fraction of gas phase at h = 0.4 m (vl = 1.0 m/s)

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3.2 Bubble Size Distribution

Figure 4 shows the distribution of bubble size with different flow velocities. It can be seen that with the uplifting of the bubbles, the diameter of bubbles gradually increases, and the diameter reaches the maximum at the water surface. The coalescence of bubbles is effected by cross flow velocity, the smaller the flow velocity is, and the larger the diameter of bubble flow is.

Fig. 4.
figure 4

Air diameters of different conditions

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3.3 The Influence of Bubbles on Water Flow

The water velocity of the y direction in bubble plume is the main influencing factor for the pneumatic sluicing.

Figure 5 shows water vertical velocity of the y direction near bubbles at different heights with different flow velocities. Results show that in different conditions, the greater the cross flow velocity, the smaller the vertical velocity of water phase at the same height. Therefore, a smaller velocity of cross flow can bring the sediment or nutrients higher, but a larger flow velocity can take the masteries to the farther downstream.

Fig. 5.
figure 5

Water velocity of the y direction in different conditions

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4 Conclusions

The uplifting characteristics of bubble plume is a main influence factor of the pneumatic sluicing. There are little studies focus on the bubble plume with cross flow. Based on CFD-PBM model, this paper studied the bubble plume with different cross flow velocities, and the results show that:

  1. (1)

    The width of bubble plume is increasing with the increase of cross velocity, and the gas hold up is reducing with the cross velocity.

  2. (2)

    The bubble diameter is enlarged during the uplifting of bubbles and the bubble growth to maximum at the water surface. The larger the cross flow velocity, the smaller the bubble diameter at the same water level.

  3. (3)

    The uplifting force is decreased with the cross flow velocity, and the greater the cross flow velocity, the smaller the vertical velocity of water phase.

The results indicate that the cross flow velocity has great influence on the bubble plume as well as the lifting effectiveness of pneumatic sluicing, which provides references for the further studies.