Study of spreading,vibration,and fracture behavior of double droplets after positive collision

Abstract

The droplet collision phenomenon is a more complex heat and mass transfer phase transition phenomenon, which is subject to the joint action of kinetics and thermodynamics. During the collision process, the mutual fusion interference of double droplets makes the kinetic mechanism after droplet collision more complicated, and its in-depth study can provide important theoretical support for the fields of engineering applications, industrial production and wetted wall design. In order to investigate the kinetic behavior of double droplets positive collision, this paper mainly combines experimental and numerical simulation methods to investigate the spreading, vibration and fracture characteristics of double droplets of the same volume after collision. Firstly, the rebound vibration of the fused droplet and single droplet is equivalent to a single-degree-of-freedom damped vibration system, and the spreading and vibration characteristics of the single droplet and the double droplets after collision under the same collision velocity are analyzed comparatively by experimental methods. The results show that when the droplet does not fracture, the spreading factor and damping coefficient of single droplet and double droplets gradually increase with the increase of collision velocity, and the vibration time gradually decreases. The damping coefficient and vibration time of the double droplets are higher than that of the single droplet, while the spreading factor is lower than that of the single droplet. Then, the double droplets positive collision phenomenon is studied in depth, and it is found that the spreading factor of the fused droplet increases with the increase of the droplet diameter, the collision velocity, and the wall contact angle. Affected by the low wall temperature, the fused droplet undergoes a phase transition, which affects the bottom flow of the droplet, leading to an increase in the damping coefficient and a decrease in the vibration time. With the decrease of the collision velocity and wall contact angle, the damping coefficient gradually increases and the vibration time decreases. Finally, the numerical simulation method reveals that rebound fracture and spreading fracture phenomena occur after double droplets positive collision, and the critical values of the collision velocity required for the occurrence of rebound fracture and spreading fracture are found. This provides a reliable theoretical basis for the study of the heat and mass transfer process after the collision of multiple droplets on the wall.

Introduction

The droplet collision phenomenon is widely used in aerospace field, industry and agricultural production.^1–5 In different applications, there are different requirements for the motion patterns of droplets after collision. For example, in inkjet printing, it is required that no deposition occurs after the droplet collides with the paper. In aerospace, icing occurs when water molecules in clouds collide with the cold surface of the wing. This requires the droplets to detach quickly from the wing surface without icing. In the field of spraying, a thin liquid film is required to be formed after droplet collision. Therefore, in order to better control the morphology change after droplet collision, it is very meaningful to study the law of motion of double droplets after positive collision.

In recent years, many scholars have conducted in-depth studies on the kinetic properties of single droplet.^6–9 Yin et al. ¹⁰ used numerical methods to study the spreading and slipping characteristics of a single droplet colliding an inclined hydrophobic surface. It was found that the maximum spreading diameter of the droplet becomes larger as the inclination angle of the wall increases, and the oscillation of the droplet becomes stronger and more unstable. Lipson and Chandra ¹¹ studied the phenomenon of a single droplet colliding a rough stainless steel surface. It was found that the droplets diffuse into the porous surface during the collision process, first forming a film and then being attracted to the pores by capillary forces. The larger the surface roughness, the smaller the degree of droplet diffusion. Zhao and Ding ¹² studied the single liquid nitrogen droplet collision by numerical simulation method, and found that under the condition of low velocity, the single liquid nitrogen droplet with smaller volume was easy to rebound after the collision. And increasing the droplet volume and collision velocity, the degree of spreading increased and even generated splashing phenomenon. Yang et al. ¹³ experimentally investigated that the spreading area of ethanol droplets after collision showed a tendency of decreasing and then increasing with increasing concentration. Shi et al. ¹⁴ experimentally found that single droplet colliding a hydrophilic wall would have a longer motion wave resulting in more energy dissipation during the oscillation process. Manglik et al. ¹⁵ and Ravi et al. ¹⁶ equated the vibration of a single droplet to a spring-damped system, used a triangular fitting function to describe the droplet height change, gave a damped oscillation equation, and verified Fedorchenkod’s droplet oscillation cycle theory. Yang et al. ¹⁷ derived the period of single droplet height oscillation on a hydrophobic surface through a single-degree-of-freedom vibration model.

In addition, the phenomenon of double droplets collision has been studied by many scholars. Ahmad et al. ¹⁸ numerically simulated the process of double droplets successively colliding on the tilted wall by using the Lattice Boltzmann method, and it was shown that the tilting of the surface resulted asymmetric spreading of the droplets and that an increase in the surface tilt angle resulted in faster downward expansion and reduced lateral expansion. Fujimoto et al.^19,20 carried out an experimental study of double droplets successively colliding on a solid wall and studied the effects of the size of the double droplets, the spacing, the impingement velocity, the wall temperature, and the angle of the wall tilt on the spreading process. Cheng et al. ²¹ used the fluid volume method to numerically simulate the collision of double droplets of different sizes against a superhydrophobic wall, and found that when a large droplet collided with a small droplet, the fused droplet was more likely to rebound away from the wall. The kinetic energy of rebound came from the surface energy released after the coalescence of large and small droplets. The fused droplets did not rebound off the wall due to the obvious gravity, and their capillary waves played an important role in the kinetic properties of the fused droplets. Graham et al. ²² carried out a numerical simulation of the spreading process for a range of different wettability and offset of the double droplets collision problem were studied both experimentally and numerically. It was found that the maximum spreading diameter decreased with increasing hydrophobicity and offset, but increased with increasing droplet collision velocity. Summarizing the literature, it is found that although scholars at home and abroad have deeply studied the influence of the collision parameters on the morphology of droplet motion, and also carried out experimental and numerical simulation studies for single droplet and double droplets. However, there are few studies on the phenomenon of double droplets positive collision, and there is no detailed mechanistic description of the rebound vibration process and fracture process after its collision on the wall. Therefore, in order to better provide a data base for droplet collision engineering applications, it is necessary to conduct a comparative study of the kinetic properties of single and double droplets positive collision.

In this paper, we mainly investigate the dynamics of spreading, vibration, and fracture of double droplets after positive collision by combining experimental and numerical simulation methods. Section “Experiment method” describes the experimental system of double droplets positive collision. In Section “Comparison of single and double droplets spreading and vibration characteristics,” the spreading and vibration characteristics of single droplet collide wall are compared and analyzed with those of double droplets positive collision by experimental methods. In Section “Comparison of single and double droplets spreading and vibration characteristics,” the spreading and vibrational properties of single droplet colliding with the wall are analyzed in comparison with those of double droplets after positive collision by experimental methods. In Section “Kinetic process analysis of double droplets collision,” the spreading and vibration characteristics of fused droplets under different conditions are studied by experimental method. In Section “Numerical simulation,” after verifying the accuracy of the numerical model, the mechanism of the fusion droplets undergoing rebound fracture and spreading fracture phenomena is analyzed.

Experiment method

The experimental system, shown in Figure 1, is mainly composed of a droplet generation system, a high-precision image and data acquisition system, a low temperature cooling circulatory system, and a droplet collision platform. Among them, the droplet generation system is mainly composed of a microfluidic syringe pump, a piezoelectric actuator composed of multilayer piezoelectric sheets and a nozzle. After the piezoelectric actuator applies voltage, the piezoelectric sheet generates displacement to push a small amount of water out of the nozzle to form droplets. By adjusting the voltage and the displacement of the piezoelectric sheet, the force of the droplet can be changed to obtain droplets with different velocities, and the size of the nozzle can be changed to obtain droplets with different diameters. The image acquisition system consists of two high-precision high-velocity cameras and a calculator. The data acquisition system consists of a data collector for reading data, a surface measuring instrument for observing the contact angle of the wall surface, and a temperature thermocouple for measuring the temperature of the wall surface. The low temperature cooling circulatory system consists of a vacuum pipeline and a low temperature cooling circulation tank, whose main function is to be responsible for the heat exchange on the wall surface. The droplet collision platform consists of a copper wall surface, a constant temperature environmental chamber and a semiconductor cooling system.

Figure 1.

Schematic diagram of the experimental system.

In order to avoid the influence of water vapor in the air on the results of the experiment, first, nitrogen, which is in a stable state at room temperature, was passed before the beginning of the experiment. The air in the constant temperature environment chamber was expelled to reduce the experimental error. Then, a static droplet was formed by dropping onto the purple copper wall using a dosing syringe. The falling droplet with the same volume as the stationary droplet was obtained through the droplet generation system to complete the experiment of double droplets positive collision. Finally, the kinetic process of droplet collision was recorded by a high-velocity camera in the image acquisition system, with a shooting velocity of 1200 fps and an image resolution of 1024 × 768 pixels. The captured images were processed by software to obtain the droplet height and spreading diameter at different moments. In order to avoid the chance of the experimental results, the controlled variable method was used for several experiments to obtain the average value to ensure the reliability of the data.

Comparison of single and double droplets spreading and vibration characteristics

Collision process comparison

Figures 2(a) and (b) show the kinetic process of single droplet collides on the wall and double droplets positive collision with the same volume, in which the wall temperature T_w = 5°C, wall contact angle θ = 55°, droplet 2 diameter d₂ = 2 mm, and the collision velocity v = 0.2 m/s. It can be seen in Figure 2(a) that the process of single droplet collision on the wall can be divided into two processes: the spreading and vibration processes. The droplets spread rapidly after contacting the wall and reach the maximum spread in a short time (Figures 2(a) i and ii), defined as the initial moment. Then the droplets start to rebound and vibrate with the help of surface tension protection, showing a reciprocal change from the edge to the middle of the aggregation and then to the edge of the diffusion, so that the droplets spreading diameter and height are constantly changing. Eventually, the droplets come to a standstill at the wall surface (Figures 2(a) iii–vi). Compared with the single droplet collision process on the wall, the double droplets positive collision process has an additional process of collision and fusion with the stationary droplet 1, as shown in Figure 2(b). Droplet 2 hits droplet 1 and fuses with droplet 1 to form a new droplet before hitting the wall. The new fused droplet quickly reaches the maximum spread on the wall (Figures 2(b) i and ii), which is set as the initial moment of the double droplets positive collision system and the end of the collision and fusion process. Then, it enters the same rebound vibration process as the single droplet phenomenon. After a short period of oscillation, the kinetic energy of the droplet is gradually dissipated through viscous dissipation and finally comes to rest on the wall (Figures 2(b) iii–vi). Compared to the single droplet collides on the wall, the double droplets positive collision has a higher height at maximum spreading and a longer oscillation time, but the spreading area at maximum spreading is similar.

Figure 2.

Kinetic processes of droplets collide: (a) single droplet collides on the wall and (b) positive collision of double droplets.

Comparison of spreading characteristics

Figure 3 shows the schematic diagram of double droplets positive collision. In order to study the degree of spreading of single and double droplets on the wall, a spreading factor K is introduced here. For single droplet: $K=d_{max}/d_2$ , for double droplets: $K=D_{max}/d_1$ .

Figure 3.

Schematic diagram of double droplets positive collision.

Figure 4 shows the comparison curves of spreading factors under different collision velocities. From the figure, it can be seen that the spreading factors of both single and double droplets increase gradually with the increase of the collision velocity of droplet 2, and the spreading factors of single droplet are higher than double droplets. The primary function is obtained by linear fitting of the data, and the growth rate (slope value of the primary function) of the single droplet is higher than the double droplets, indicating that the increase of the collision velocity promotes the spreading of the single droplet better than the double droplets. This is due to the fact that after droplet 2 collides with droplet 1, part of the energy is used for its own spreading and the other part of the energy is used to overcome the work done by the viscous force of droplet 1. This makes the spreading of the fused droplet less than the single droplet collision.

Figure 4.

Comparison curves of spreading factors of single droplet and double droplets at different collision velocities.

Comparison of vibration characteristics

Figure 5 shows the vibration height curves of single and double droplets, where d₂ = 2 mm, T_w = 5°C, θ = 55°, and v = 0.2 m/s. As shown in the figure, the maximum amplitude and amplitude variation of single droplet are obviously larger than that of double droplets during vibration, but the time consumed in the vibration process is shorter than double droplets.

Figure 5.

Comparison curves of vibration height of single droplet and double droplets.

Figure 6(a) shows the variation curves of vibration damping coefficients of single and double droplets under the same collision velocity, as shown in the figure, the damping coefficients of double droplets have been higher than those of single droplet in the range of 0.1–0.3 m/s. With the increase of collision velocity, the damping coefficients of single and double droplets are gradually reduced. And with the increase of the collision velocity, the damping coefficients of both single and double droplets decrease gradually, and the damping coefficients of double droplets decrease more than that of single droplet. Figure 6(b) shows the variation curves of the vibration time of single and double droplets under the same collision velocity. As shown in the figure, the vibration time of single and double droplets gradually increases with the increase of the collision velocity, and the vibration time of double droplets is longer than that of single droplet.

Figure 6.

Variation curves of vibration damping coefficients and vibration time of single droplet and double droplets: (a) damping coefficient c and (b) vibration time t.

By comparing the spreading and vibration characteristics of single and double droplets, it can be seen that the spreading factors and damping coefficients of single and double droplets have the same trend with the change of droplet 2 collision velocity, but the difference in amplitude and numerical value is very large. It can be seen that the complexity of the spreading and vibration mechanism of double droplets collision and the need for in-depth systematic research.

Kinetic process analysis of double droplets collision

Figure 7 shows the kinetic visualization processes of double droplets collision under different conditions. As shown in Figure 7(a), droplet 2 collides droplet 1, which is stationary on the wall, and then rapidly fuses with droplet 1 to form a new droplet. Due to the falling inertial force, the fused droplet quickly reaches the maximum spreading. Then the fused droplet rebound vibration phenomenon occurs by the reaction of surface tension, and starts to protrude upward, in the shape of “bullet head,” driving the two sides of the reciprocating rebound vibration. Until the kinetic energy introduced by droplet 2 is dissipated through viscosity, and comes to a standstill on the wall surface. Figure 7(a) compared to (b), the collision velocity is smaller, the maximum spreading area of the fused droplet, and the time consumed for resting are smaller. Figure 7(a) compared to (c), smaller wall contact angle, less time consumed and smaller spreading area of fused droplet at rest.

Figure 7.

Visualization of the kinetic processes of double droplets collision: (a) θ=70°, v=0.15 m/s, d₂=2 mm, T_w=-5°C, (b) θ=70°, v=0.3 m/s, d₂=2 mm, T_w=-5°C and (c) θ=80°, v=0.15 m/s, d₂=2 mm, T_w=-5°C.

Spreading characteristics of double droplets after collision

Influence of collision velocity on spreading factor

Figure 8 shows the comparison curves of the spreading factors under different collision velocities, where d₂ = 2 mm and T_w = −5°C. As can be seen from the figure, the spreading factors of the fused droplets gradually increase with the increase of the collision velocity of droplet 2. This is due to the fact that under the condition that the droplet 2 is of the same diameter, the greater the collision velocity, the greater the inertial force with which they are endowed. The greater the lateral force that is converted into squeezing the droplets 1 during collision is also greater, promoting the spreading of the fused droplets. In addition, by increasing the wall contact angle, the spreading factors are similarly increased. This is due to the fact that the better the wall wettability is at the wall with a smaller contact angle, the greater the spreading area presented by droplet 1. This causes the droplet 2 to be subjected to more viscous resistance of the droplet 1 during lateral spreading, and the degree of spreading is reduced.

Figure 8.

Comparison curves of spreading factors under different collision velocities.

Influence of droplet 2 diameter on spreading factor

Figure 9 shows the comparison curves of the spreading factors under different droplet 2 diameters, where v = 0.1 m/s and T_w = −5°C. From the figure, it can be seen that the spreading factors of the fused droplets gradually increase with the increase of droplet 2 diameter and wall contact angle. This is due to the fact that under the condition of consistent collision velocity, the larger the droplet 2 diameter is, the larger the inertia force it has when falling, which will promote the spreading of the fused droplet.

Figure 9.

Comparison curves of spreading factors under different droplet 2 diameters.

Vibration characteristics of double droplets after positive collision

Figure 10 shows the force diagram of the fused droplet during the vibration process. The fused droplet is affected by the inertia force F _t , viscous force F _μ and surface tension F _σ . The fused droplet undergoes rebound vibration on the wall under this internal force.

Figure 10.

Force diagram of a fused droplet during vibration.

The displacement of the spreading radius R(t) relative to the radius R at rest is r. The inertial force F _t direction pointing to the center of the circle, as the droplet vibration velocity changes continuously with the radius, the correction factor N₁ is introduced to simplify the calculation, and its expression is

$$F_{\text{t}}=N_1\times m\times a=N_{1}m\frac{d^{2}r}{d{t}^2}$$

(1)

where m is the droplet mass; a is the vibration acceleration; and t is the vibration duration. The combined force F_σ of the surface tension, with the direction pointing to the center of the circle and introducing the correction factor N₂, is expressed as

$$F_{\sigma }=N_2\times \sigma \times 2\pi r$$

(2)

where σ is the surface tension coefficient. The viscous force F _μ , which is in the opposite direction to the droplet vibration and whose magnitude is proportional to the vibration velocity, is introduced as a correction factor N₃, whose expression is

$$F_{\mu }=N_3\times \mu \frac{d_r}{d_t}$$

(3)

where μ is the coefficient of viscosity. Under the action of each of the above forces, the droplet vibration equation is

$$F_t+F_{\sigma }+F_{\mu }=0$$

(4)

Substituting equations (1) to (3) into equation (4) yields

$$N_{1}m\frac{d^{2}r}{d{t}^2}+N_{2}2\pi \sigma r+N_3\mu \frac{d_r}{d_t}=0$$

(5)

To further describe the vibration process, a theoretical model of a single-degree-of-freedom damped vibration system is used to analyze the droplet vibration behavior, and the droplet center height variation can be expressed by the damped vibration equation

$$m{\zeta ″}_0+c\zeta {\prime }_0+k{\zeta }_0=0$$

(6)

where ${\zeta ″}_0$ and $\zeta {\prime }_0$ are the first and second order derivatives of the true height ${\zeta }_0$ of the droplet, respectively. The damping coefficient c = 2 mε. The general solution is obtained by calculating the second order linear chi-square differential equation

$${\zeta }_0={\zeta }_{\mathrm{eq}}+[\mathrm{Acos}(\mathrm{wt})+\mathrm{Bsin}(\mathrm{wt})]e^{-{\epsilon }_{0}t}$$

(7)

$${\zeta }_0={\zeta }_{\mathrm{eq}}+C_0{e}^{-{\epsilon }_{0}t}$$

(8)

where ${\zeta }_{\mathrm{eq}}$ is the droplet rest height; C₀ and ε₀ are the fitting coefficient; w is the angular frequency; A and B are the two coefficients determined by the initial conditions; and t is the vibration time.

In order to more accurately express the numerical variation of the droplet’s vibration height, a dimensionless parameter is defined here – the ratio of the droplet’s vibration height $\zeta$ , the droplet’s true height ${\zeta }_0$ to its resting height ${\zeta }_{\mathrm{eq}}$ . Substituted into equation (8) as

$$\zeta =1+C{e}^{-\epsilon t}$$

(9)

where C and ε are the fitting coefficient; and t is the vibration time, ms (all the vibrations of a single cycle of the droplet are in milliseconds, so milliseconds are used for the calculation).

During the rebound vibration of a fused droplet, the top region is subjected to heat transfer from the bottom while heat transfer with the adjacent waters occurs, which can reduce the efficiency of heat transfer. If the vibration damping coefficient of the droplet is small, the amplitude and vibration duration during the vibration will be larger. The time and amount of heat transfer between the top of the droplet and the adjacent waters will increase, which reduces the heat transfer efficiency and prolongs the time for the droplet to be in complete phase transition.

Influence of collision velocity on vibration characteristics

The vibration height curves and fitting the curves of the fused droplets under different collision velocities are given in Figure 11, where d₂ = 2 mm, T_w = −5°C, θ = 80°, and v = 0.1, 0.15, 0.2, and 0.3 m/s. It can be seen from the figure that the maximum amplitude and the vibration time duration of the fused droplets increase gradually with the increase of the collision velocity. This is due to the fact that in the case of the same droplet volume and kinetic viscosity, the larger the droplet 2 collision velocity is, the more kinetic energy is introduced into the fused droplet, which promotes the droplet spreading. And with more energy into the rebound vibration, so that the amplitude becomes larger and the vibration time are prolonged.

Figure 11.

Vibration height curves and fitting the curves of fused droplets under different collision velocities.

Figure 12(a) shows the variation curves of the vibration damping coefficients of the fused droplets with the collision velocity under different contact angles. From the figure, it can be seen that the vibration damping coefficients of the fused droplets decrease with the increase of the collision velocity and the wall contact angle. Figure 12(b) shows the variation curves of fused droplets vibration time with collision velocity at different contact angles. As can be seen from the figure, the vibration time of the fused droplets gradually increase with the increase of droplet 2 collision velocity and wall contact angle. This phenomenon is mainly caused by the gradual increase of kinetic energy introduced into the fused droplet by droplet 2.

Figure 12.

Variation curves of vibration damping coefficients and vibration time with collision velocity for fused droplets at different contact angles: (a) c versus v and (b) t versus v.

Influence of wall temperature on vibration characteristics

Figure 13 shows the vibration height curves and fitting the curves of the fused droplets under different wall temperature conditions, where v = 0.2 m/s, d₂ = 2 mm, θ = 80°, and T_w = −10, −5, 0 and 5°C. As can be seen from the figure, the vibration time of the fused droplets decrease with the decrease of the wall temperature, while the maximum amplitude does not change significantly. This is due to the fact that a low wall temperature leads to an increase in the viscosity coefficients of the fused droplets and a gradual decrease in the vibration time. If the wall temperature reaches the temperature that drives the phase transition at the bottom of the droplet, the droplet undergoes a phase transition that affects the bottom flow of the droplet.

Figure 13.

Vibration height curves and fitted the curves of fused droplets at different wall temperatures.

Figure 14(a) shows the variation curves of the vibration damping coefficients of the fused droplets with wall temperature at different contact angles. From the figure, it can be seen that the vibration damping coefficients decrease gradually with the increase of wall temperature and contact angle. Figure 14(b) shows the variation curves of the vibration time of double droplets with wall temperature at different contact angles. From the figure, it can be seen that increasing the wall contact angle and wall temperature, the vibration time of the droplets gradually increase.

Figure 14.

Variation curves of vibration damping coefficients and vibration time with wall temperature for fused droplets at different contact angles: (a) c versus T_w and (b) t versus T_w.

Numerical simulation

In order to investigate the fracture behavior of double droplets of the same volume after positive collision, it is investigated by developing a numerical model. The numerical model is calculated using a 2D model, which can capture the movement changes of the gas-liquid free interface more clearly and reduce the amount of calculation. A 2D area of 20 × 20 mm is created to set up a circular area with droplets 2 and droplets 1 presenting different sizes on walls with different wall contact angles. During the numerical calculations, the free interface tracking model and the control equations are used to capture the morphological changes of the droplets. The phase transition behavior of the droplets is controlled by the solidification and melting model.

Numerical model and method

Free interface tracking model

In order to more accurately analyze the morphological changes generated by the double droplets positive collision process, the VOF coupled Level-set method is used to track the gas-liquid free interface for calculation. ²³ The flow field solution is dominated by the idea of VOF, and the Level-set method is used as an auxiliary when considering the interface effect. Setting air as the main phase and liquid water as the secondary phase, the continuity equation for the volume fraction of the secondary phase is:

$$\frac{1}{{\rho }_{\mathrm{s}}}[\frac{\partial ({\alpha }_s{\rho }_s)}{{\partial }_t}+\mathrm{\Delta }({\alpha }_s{\rho }_{s}v)]=0$$

(10)

where $\rho$ is the density, kg·m³; ${\alpha }_s$ is the volume fraction; v is the velocity, m/s; and t is the time, s. The main phase volume fraction is:

$${\alpha }_p=1-{\alpha }_s$$

(11)

where ${\alpha }_p$ is the principal volume fraction; and ${\alpha }_s$ is the secondary phase volume fraction.

Control equations

The main control equations applied in this numerical simulation of the double droplets positive collision process include:

(1) Mass conservation equation

For the motion of an in compressible fluid, the total mass of the process remains constant and the continuity equation can be expressed as

$$\frac{\partial \rho }{\partial t}+\nabla \left(\rho v\right)=S_m$$

(12)

where $\rho$ is the droplet density (the droplet is affected by heat transfer from the wall during the collide, so the density varies); $v$ is the velocity; $t$ is the time; and $S_m$ is the mass source term.

(2) Conservation of momentum equation

$$\rho \frac{\partial (v)}{\partial t}+\partial \rho \nabla v^2=-\nabla p+\nabla [\mu (\nabla v+\nabla v^T)]+\rho g+F_{\mathrm{bf}}$$

(13)

where $\mu$ is the kinetic viscosity; $p$ is the pressure; $g$ is the gravitational acceleration; and $F_{\mathrm{bf}}$ is the tension source term.

(3) Energy conservation equation

$$\frac{\partial (\rho T)}{\partial t}+\rho \nabla (\mathrm{vT})=\nabla (\frac{\lambda }{c_p}\nabla T)S_h$$

(14)

where $T$ is the temperature; $\lambda$ is the heat transfer coefficients; $c_p$ is the specific heat capacity; and $S_h$ is the energy source term.

Solidification-melting model

An enthalpy-porosity phase change model was used to simulate the solidification-melting phase change process inside the droplet, ²⁴ and the energy equation was calculated and solved as

$$\frac{\partial }{\partial t}(\rho H)+\nabla (\rho \nu H)=\nabla (k\nabla T)$$

(15)

where $\rho$ is the density, H is the enthalpy; v is the collision velocity; and k is the thermal diffusion coefficients. After considering the phase change, the enthalpy in the energy equation is the sum of sensible heat and latent heat, $H=c_{p}T+\mathrm{\Delta }H$ . Where the latent heat part is related to the latent heat of solidification/melt phase change of water as

$$\mathrm{\Delta }H=L_f\gamma$$

(16)

where $c_p$ is the specific heat capacity; and L _f is the latent heat. The enthalpy-porosity phase change model treats the ice-water mixing region as a porous structure, and the porosity is the volume fraction $\gamma$ of the liquid phase, which takes the value interval [0,1] and is related to the temperature as

$$\gamma =\{\begin{array}{cc}1\hfill & T\ge T_{\mathrm{liquid}}\hfill \\ \multicolumn{1}{c}{\frac{T-T_{\mathrm{solid}}}{T_{\mathrm{liquid}}-T_{\mathrm{solid}}}}& T_{\mathrm{solid}}\le T<T_{\mathrm{liquid}}\hfill \\ \multicolumn{1}{c}{0}& T<T_{\mathrm{solid}}\hfill \end{array}$$

(17)

where $T$ is the temperature; $T_{\mathrm{solid}}$ is the freezing temperature; and $T_{\mathrm{liquid}}$ is the melting temperature. Assuming that the motion within the ice-water mixing zone obeys Darcy’s law for flow within a porous medium and conforms to the Carman-Koseny assumption, the source term of the momentum equation can be introduced as

$$S_{\mathrm{F}}=\frac{{(1-\gamma )}^2}{{\gamma }^3+\eta }{C}_{\mathrm{mush}}v$$

(18)

where $\eta$ is a very small value ( $\eta$  = 0.001) to avoid a zero denominator; and $C_{\mathrm{mush}}$ is the viscosity coefficients, which is related to the morphology of the porous medium. A reasonable value of $C_{\mathrm{mush}}$ constant is taken to maintain the flowability when the liquid phase ratio is high, but also to suppress the flow rate when the liquid phase ratio is reduced.

From the expression of the momentum source term, it can be seen that in the liquid phase region, that is, $\gamma$  = 1, the momentum source term is zero, and the momentum equation is consistent with the form of the fluid N-S equation. After solidification begins to become an ice-water mixture, the volume fraction $\gamma$ of the liquid phase decreases, and the weight of the source term in the momentum equation gradually exceeds that of the transient, convection and diffusion terms. Until the velocity is basically zero after complete transformation into the solid phase.

Calculation method

Calculate the material setup air and liquid water, setup droplet physical properties (see Table 1). After an irrelevance check of the mesh, the pressure-velocity coupling was performed using the PISO method suitable for non-constant flow problems, and the gas-liquid free interface reconstruction was performed using the Geo-Reconstruct method. The partial differential equations are discretized by the finite volume method and the Body Force Weighted is used in the pressure discretization. Finally convergence calculations with residuals less than 10⁻⁶ are performed.

Table 1.

Physical properties of liquid droplets.

Viscosity ( $\mu$ )	0.001 kg/(m·s)
Density ( $\rho$ )	1,000 kg/m3
Surface tension coefficient ( $\sigma$ )	0.0728 N/m
Specific heat ( $c_p$ )	4182 J/kg·K
Thermal conductivity ( $\lambda$ )	0.6 W/m·K
Melting latent heat (L f )	333,000 J/kg
Droplet temperature (Twater)	10°C
Phase change temperature (Tice)	0°C

Comparison and validation of numerical simulation and experimental methods

Figure 15 shows the comparison of the kinetic processes of double droplets positive collision under the experimental observation and numerical simulation method, where d₂ = 4 mm, θ = 70°, v = 0.15 m/s, and T_w = −5°C. The maximum spreading moment of the fused droplet is the initial moment. From the figure, it can be seen that the droplet motion patterns of experimental observation and numerical simulation are consistent at the same time.

Figure 15.

Comparison of experimentally observed and numerically simulated kinetic processes of the double droplets positive collision.

Figure 16 shows the comparison curves of the vibration height of the fused droplets from experimental observation and numerical simulation. As shown in the figure, the droplets vibration height changes consistently with time, and the error of the vibration time is about 6.35%, and the maximum error of the peak value is about 5.21%, which verifies the reliability of the numerical model.

Figure 16.

Comparison curves between experimentally observed and numerically simulated vibration height of droplets.

Study of fracture phenomena

Rebound fracture characteristics

Figure 17 shows the kinetic pressure cloud of the rebound fracture after the positive collision of the double droplets, where d₂ = 4 mm, θ = 90°, T_w = −5°C, and v = 0.6 m/s. As shown in the figure, the convex droplet 2 bottom and droplet 1 top start to stretch to both sides after the positive collision (Figures i and ii). The pressure at the bottom of droplet 2 and the top of droplet 1 is maximum on both sides, and this state continues until the fused droplet reaches the moment of maximum spreading (Figure iii). Then the fused droplet is counteracted by surface tension and starts to enter the rebound vibration period (Figure iv). During the rebound period, the waters on both sides and the maximum pressure gradually converge towards the center (Figures v and vi). The waters on both sides continuously stretch the center waters, making the center waters gradually lower and the waters on both sides gradually higher. If the collision velocity of droplet 2 increases, the inertia force when falling will increase, which makes the fused droplet spread more. The height when spreading is lower and the pulling force of rebound on both sides by the action of surface tension will be larger. During the rebound period, if the height of the center waters of the fused droplet is lower as well as the pull of rebound of the waters on both sides is greater, it is more likely to lead to rebound fracture of the center waters (Figure vii).

Figure 17.

Kinetic pressure cloud of rebound fracture.

Figure 18 shows the critical collision velocity comparison curves for the occurrence of rebound fracture phenomenon under different wall contact angles. It can be seen from the figure that the critical collision velocity required for the occurrence of rebound fracture phenomenon of the fused droplet decreases with the increase of the diameter of droplet 2. This is due to the fact that the larger the diameter (inertia force) of droplet 2, the larger the pulling force on both sides of the rebound, which makes the center of the water domain gradually lower and more likely to rebound fracture. As the wall contact angle decreases, the critical collision velocity required for a fused droplet to undergo rebound fracture also decreases. This is due to the fact that the larger the wall contact angle, the less the wall is wetted. The fused droplet spreads on the wall to the maximum moment of the droplet height is higher, the fused droplet in the rebound of the center of the water domain height is higher, less likely to break. Therefore greater collision velocities are required for rebound fracture to occur.

Figure 18.

Comparison curves of critical collision velocities for occurrence of rebound fracture phenomenon.

Spreading fracture characteristics

Figure 19 shows the kinetic pressure cloud of spreading fracture after double droplets positive collision, where d₂ = 4 mm, θ = 90°, T_w = −5°C, and v = 0.8 m/s. As shown in the figure, on the basis of the occurrence of rebound fracture, continuing to increase the collision velocity accelerates the process of the collide of droplet 2 (Figures i–v), which fuses to produce the spreading fracture phenomenon. This is due to the fact that the increase in the collision velocity (inertial force) of droplet 2 increases the lateral force that squeezes droplet 1 and its own tensile force that spreads to the sides. During spreading, the fused droplet is continuously subjected to excessive tensile forces on both sides, resulting in spreading breakage of the fused droplet (Figure vi).

Figure 19.

Kinetic pressure cloud of spreading fracture.

Figure 20 shows the critical collision velocity comparison curves for the occurrence of spreading fracture phenomenon under different wall contact angles. It can be seen from the figure that the critical collision velocity required for the fused droplet spreading fracture phenomenon gradually decreases with the increase of droplet 2 diameter. This is due to the droplet 2 diameter (inertia force) is larger, in the process of collide extrusion droplet 1 lateral force and its own to the sides of the spreading of the tension is also the larger, more likely to tear the center of the waters, the spreading fracture phenomenon. As the wall contact angle decreases, the critical collision velocity required for the fused droplet to undergo spreading fracture also decreases gradually. This is due to the fact that the larger the wall contact angle, the poorer the wall wetting, the fused droplet spreading on the wall is smaller, the spreading height is higher, and it is not easy to tear the center waters. Therefore, a larger collision velocity (lateral force) is needed to stretch the center waters to both sides for spreading fracture to occur.

Figure 20.

Comparison curves of critical collision velocities for the occurrence of spreading fracture phenomenon.

Conclusion

In this paper, the kinetic characteristics of double droplets are investigated after the positive collision with the wall. Firstly, the spreading and vibration characteristics of single droplet and double droplets after positively colliding with the wall are compared and analyzed by experimental methods, and the vibration characteristics of the fused droplets are investigated by using the theory of single-degree-of-freedom damped vibration. Then, after verifying the accuracy of the model through experimental data, the rebound fracture and spreading fracture phenomena of the double droplets positive collision are analyzed to find the critical value and influencing factors needed for the occurrence of fracture. The main conclusions are as follows:

• (1) Comparing the kinetic properties of single droplet and double droplets, it is found that the double droplets positive collision process has more collision and fusion with stationary droplet 1. And with the increase of the collision velocity, the spreading factor and vibration time of both single droplet and double droplets gradually increase, and the damping coefficient gradually decreases. However, the damping coefficient and vibration time of the double droplets are higher than the single droplet, while the spreading factor is lower.

• (2) The spreading factor of the fused droplet increases gradually with the increase of droplet 2 diameter, collision velocity and wall contact angle during the spreading process.

• (3) In the vibration process, the theoretical model of single-degree-of-freedom damped vibration system is used to analyze the vibration behavior of the fused droplet. With the decrease of the collision velocity and wall contact angle, the vibration damping coefficient of the fused droplet gradually increases and the vibration time decreases. The influence of low wall temperature causes the viscosity coefficient of the droplet to increase. If the cold wall surface drives the droplet to phase change, it will affect the flow at the bottom of the droplet, resulting in an increase in the damping coefficient and a decrease in the vibration time.

• (4) Increasing the collision velocity of the droplets, rebound fracture and spreading fracture phenomena will occur sequentially after the fusion of double droplets positive collision. And along with the increase of the droplet diameter and the decrease of the wall contact angle, the critical collision velocity required for the fusion droplet to fracture gradually decreases.