Numerical Analysis of Dual Droplet Simultaneous Oblique Impact on a Water Film

Abstract

The simultaneous oblique impact of multiple droplets on a liquid film is an intricate phenomenon prevalent in diverse natural and industrial processes. However, previous studies have primarily focused on single droplet impact, while an in-depth understanding of the more complex multi-droplet scenario remains lacking. The current study aims to numerically investigate the simultaneous oblique impact of two droplets on a water film using a three-dimensional Volume of Fluid (VOF) model. The effects of the Weber number and the impact angle on the crown behavior are carefully analyzed. The results demonstrate that increasing the Weber number enhances the central uprising sheet height but has minor influences on the upstream crown radius and central sheet radius. In contrast, the increase in the impact angle leads to a decreased upstream crown radius and an increased central sheet radius, while the central sheet height remains relatively unaffected. In addition, the splashing threshold for the dual droplet impact cases is significantly lower than that of the single droplet impact cases due to the interactions between the adjacent crowns. The present results provide novel insights into the underlying physics and useful supports in developing predictive models for the intricate multi-droplet impact phenomenon.

1. Introduction

The impact of multiple droplets on a liquid film is a complex and important process in nature and in industrial applications, such as during raining, surface cooling, fuel injection, and aircraft icing [1,2,3,4,5]. The impact process is highly complex due to the multiphase and interfacial flows involved. During the droplet impact process, various physical phenomena may take place, such as air entrapment and turbulence, which significantly affect the dynamics of the system. By continuing to explore the underlying physical mechanisms and developing predictive models, researchers can improve the efficiency and reliability of droplet-based technologies and pave the way for the development of new applications.

In the past, researchers have mainly focused on studying the impact process of a single droplet on a liquid film [6,7,8,9,10,11,12,13,14,15,16,17]. For example, Cossali et al. [6] conducted experiments to set an empirical correlation for the splashing limit. Thoroddsen [8] provided valuable insights into the dynamics of droplet impact and the influence of fluid properties on the behavior of the ejecta sheet. Coppola et al. [14] performed simulations by the VOF method to study the ejecta sheet. Josserand et al. [17] used scaling analysis and numerical simulations to investigate the dynamics during the splash formation.

The impact of multiple droplets is highly complex due to interactions among the droplets [18]; these interactions can lead to the formation of complex flow patterns. While single droplet impact research does provide a logical foundation for understanding the physics of droplet impact, it is important to recognize that the dynamics of multi-droplet impact are more complex due to the interactions that occur between the droplets [19]. Therefore, it is necessary to develop new models and correlations based on multi-droplet impact studies to understand the physical mechanisms involved in this phenomenon.

In general, the impact of multiple droplets on a liquid surface can be classified into three major categories, including simultaneous impact, non-simultaneous impact, and successive impact [20]. First, simultaneous droplet impact refers to the scenario where multiple droplets strike a liquid surface concurrently. In the process of simultaneous droplet impact, previous studies have revealed lower splashing thresholds due to crown collisions [21], distinct crown deformation patterns [22], and impact dynamics influenced by droplet spacing [23]. For example, Wang et al. [24] discovered that the temperature distribution and heat transfer rate were highly dependent on droplet size, velocity, and spacing during the impact of double droplets on a moving film. Liang et al. [20] carried out a numerical simulation to explore the heat transfer mechanism during the impact of multiple droplets on a liquid film. The research demonstrated that the heat transfer rate was strongly affected by the location of the impact. Fest-Santini et al. [18] employed direct numerical simulations to investigate the behavior and dynamics of the interactions between multiple droplets and a liquid film. Liu et al. [25] utilized the CLSVOF method to simulate and analyze the dynamic behavior of two hollow droplets on a wet surface. Their results revealed significant differences in the impact dynamics between hollow droplets and continuous dense droplets. Second, non-simultaneous impact, involving droplets hitting the liquid surface at different times, has been explored by emphasizing the significant effects of impact location on droplet behaviors [26]. Third, successive impact involves subsequent droplets hitting a liquid film after the initial impact. Liang et al. [27] discovered that the deformation of the liquid film caused by leading droplets could significantly influence the impact behavior of the following droplets. Bao et al. [28] investigated the interface evolution during successive oblique impacts of two droplets on a liquid film. They showed that the resulting film deformation was highly dependent on the Weber number.

While previous studies have examined the impact process of multiple droplets on a liquid film, they have predominantly focused on normal impact process [18,20,21,22,23,24,25], leaving the simultaneous oblique impact scenarios relatively unexplored. Among various oblique impact scenarios, the simultaneous oblique impact of dual droplets on a liquid film, as examined in the present study, represents a scenario that has been overlooked in previous studies. Given the common occurrence of simultaneous oblique multi-droplet impingement in nature and industrial applications, it is necessary to conduct in-depth studies to reveal its impact process and mechanism.

The present study aims to explore the simultaneous oblique impact process of two droplets on a water film through numerical simulations. By taking into account the inherent three-dimensional nature of the impact process, a three-dimensional model is used for the simulation. The interface evolution and field distributions are presented in detail. The effects of Weber number and impact angle on the crown behavior are carefully analyzed. This investigation is aimed at increasing our understanding of the fundamental mechanism of the multi-droplet impact process.

2. Physical Model and Numerical Method

2.1. Physical Model

A three-dimensional simulation was performed with Basilisk, which is an open-source program for fluid flow. The physical model of two droplets impacting a liquid film is illustrated in Figure 1. As can be seen, two identical oblique droplets are positioned above a horizontal liquid film. The physical properties of water and ambient air used in the simulations are shown in

Table 1. Physical properties of water and ambient air used in the simulation.

Fluid	Density	Dynamic Viscosity	Surface Tension
	ρ [kg/m3]	μ [Pa·s]	σ [N/m]
Air	1.204	1.813 × 10−5	-
Water	998	1.002 × 10−3	0.073

. D₀ is the diameter of the droplet. θ is the impact angle. The liquid film has a thickness of h and its dimensionless thickness is represented as h⁺ = h/D₀. The fluid is incompressible, with constant viscosity and surface tension. The initial time was set at the moment when the two droplets first contacted the liquid film. The distance between the centers of the two droplets at this moment is denoted as s. Heat transfer was not considered in the simulation. The computational domain used in the simulation has a dimension of 8D₀ in each direction. The bottom boundary is a no-slip wall and the other boundaries are outflow. Direct numerical simulation (DNS) was performed by solving the NavierStokes equations for incompressible flows using the Volume of Fluid (VOF) method. No turbulence model was employed. Convergence was achieved by minimizing the estimated errors associated with the volume of fluid tracers and velocity components, with tolerances of 10⁻⁶ and 10⁻³, respectively [29]. The viscous terms were discretized using the Crank–Nicholson scheme, while the convective terms are computed using the Bell–Colella–Glaz second-order unsplit upwind scheme [30].

2.2. Numerical Method

In the present study, the Volume of Fluid (VOF) method [31] was used to simulate the process of multiple droplets falling on a liquid film. The VOF method is a numerical technique used to model the interface between two or more immiscible fluids. The VOF method works by dividing the computational domain into small control volumes and tracking the volume fraction (f) of each fluid within each cell. In each cell, the sum of the volume fractions of all fluids must be unity. Cells with a volume fraction of zero represent one fluid (e.g., air), while cells with a volume fraction of one represent the other fluid (e.g., water). Cells with a volume fraction value between zero and one contain the interface between the two fluids.

$$\begin{array}{c}f=\left\{\begin{array}{cc}0,& \mathrm{the} \mathrm{gas} \mathrm{phase} \hfill \\ 0<f<1,& \mathrm{the} \mathrm{interface} \hfill \\ 1.& \mathrm{the} \mathrm{liquid} \mathrm{phase} \hfill \end{array}\right.\end{array}$$

(1)

The density of the effective fluid is represented by ρ, with ρ₁ and ρ₂ denoting the densities of the two immiscible fluids. μ is the effective viscosity, and μ₁ and μ₂ are the viscosities of the two fluids, as follows:

$$\rho ={\rho }_{1}f+(1-f){\rho }_2$$

(2)

$$\mu ={\mu }_{1}f+(1-f){\mu }_2$$

(3)

By knowing the volume fraction of each fluid, the location of the interface between the fluids can be determined and the resulting flow behavior can be simulated. In this study, NavierStokes equations [32,33,34,35] for incompressible fluids were used to describe the fluid dynamics [14], as follows:

$$\nabla ·V=0$$

(4)

$$\rho \frac{\partial V}{\partial \mathrm{t}}+\rho \nabla ·\left(VV\right)=-\nabla P+\nabla \cdot \left(2\mu D\right)+F_s$$

(5)

$$\frac{\partial f}{\partial \mathrm{t}}+V·\nabla f=0$$

(6)

In the present study, emphasis was placed on two key parameters, namely, the Weber number and the impact angle. The Weber number (We = ρV²D₀/σ, where ρ is the density of the fluid, V is the velocity of the droplet, D₀ is the diameter of the droplet, and σ is surface tension) is a dimensionless parameter representing the ratio of the inertial force to the surface tension. The impact angle, which represents the inclination of the droplet velocity relative to the liquid film, determines the direction of energy transfer between the droplet and the liquid film.

2.3. Grid Independency and Model Validation

Cossali et al. [10] utilized high-speed imaging to study single droplet impact on a liquid film, examining the influence of time on impact parameters. The crown diameter D_ue (Figure 2) is an important parameter for evaluating the result of the impact process, researchers have often used these data to validate numerical simulation results [27,28]. In this study, the crown diameter D_ue from Cossali et al. [10] is also selected as a quantitative benchmark to verify grid independence and physical model.

Figure 3 shows the results of the grid independence study for the crown diameter (D_ue) at three specific moments, namely 3.0 ms, 6.0 ms, and 9.0 ms. Several grid resolutions were tested, with the grid size ranging from coarse to fine. Specifically, the grid sizes are equivalent to 16, 32, and 64 grids per droplet diameter. According to the obtained results, when the grid size increases from 16 to 32 grids per droplet diameter, the crown diameter decreases significantly. However, when the grid size increases from 32 to 64 grids per droplet diameter, the change in the crown diameter is negligible, which indicates that the sufficient grid accuracy has been achieved. A similar study by Bao et al. [28] suggests that a grid size of around 25 grids per droplet diameter is sufficient to accurately capture the fluid dynamics involved in this type of problem. In this paper, a grid size of 64 grids per droplet diameter was chosen to simulate the process of droplet impact.

Figure 4 presents a quantitative comparison between the numerical results and experimental measurements by Cossali et al. [10]. The strong agreement between the simulation data and experimental measurements indicates that the model can accurately predict the droplet impact phenomenon.

3. Results and Discussion

3.1. The Impact Process

Figure 5 shows the impact process of two oblique droplets on a water film, in which D₀ = 4 mm, We = 600, θ = 30°, h⁺ = 0.6, and s* = s/D₀ = 2. In the present study, a dimensionless time τ is defined as τ = t/(D₀/V). At the early stage of the impact, a very thin liquid jet called the ejecta sheet [8] is formed in the neck region. The flow pushes the surrounding liquid outward, creating a crown-shaped structure around the droplet. The height and radius of this crown structure grow with time as the fluid continues to be displaced by the impact. The rim of the crown subsequently leads to the formation of the secondary droplets. Such a phenomenon is identified as splashing. When multiple droplets impact a surface, the crowns generated by each droplet can interact with each other, leading to a complex interaction region between the crowns. In this case, the crowns generated from each droplet interact with each other at τ = 0.5. The interaction region between the crowns generated from each droplet can grow into a central uprising sheet due to the motion of the droplets. As the central uprising sheet grows higher, it becomes unstable and eventually breaks up into secondary droplets.

The velocity vector distribution is presented in Figure 6a, while the velocity vector distribution of the interface of water and air is shown in Figure 6b (We = 600, θ = 30°, h⁺ = 0.6, s* = 2 τ = 2.0). The appearance of a central uprising sheet in the middle can be explained by the interaction between the adjacent crown liquid sheets. The interaction region between the two droplets primarily moves in the central direction as a result of the motion of the droplets.

This study focuses on the quantitative analysis of crown parameters, including upstream crown radius R_u, central uprising sheet radius R_c, and central uprising sheet height H_c. The crown parameters were normalized as follows: R_u^* = R_u/D₀, R_c^* = R_c/D₀, H_c^* = H_c/D₀. The definitions of these crown parameters are shown in Figure 7.

3.2. Effects of Weber Number

Figure 8 shows the impact outcomes with different Weber numbers. Specifically, droplet impacts at Weber numbers 100, 300, and 600 were simulated to investigate how the changes in Weber number affect the impact dynamics. In addition, in these simulations, D₀ = 4 mm, θ = 30°, h⁺ = 0.6, and the distance between the centers of the droplets is 2D₀. As the Weber number increases, the droplet impact process becomes more violent, resulting in greater amounts of liquid mass being separated from the uprising sheet. This is because the Weber number represents the ratio of inertial forces to surface tension forces. At higher Weber numbers, the inertial forces become increasingly dominant over surface tension forces, leading to the formation of larger and more energetic liquid structures during the impact process.

In the study of droplet impact on liquid films, the dimensionless parameter K is used to indicate the condition for splashing occurrence [36], the value of K indicates whether the flow inertia can overcome surface tension and viscous effects, which in turn determines the occurrence of splashing, as follows:

$$K=We{Oh}^{-0.4}$$

(7)

where We is Weber number and Oh is Ohnesorge number.

$$We=\frac{\rho V^2{D}_0}{\sigma },Oh=\frac{\mu }{\sqrt{\rho \sigma D_0}}$$

(8)

Based on the study of normal single droplet impact on a liquid film, Okawa et al. [12] pointed out that, for low viscosity liquids (e.g., water), the critical value K_s for splashing is 2100. Cossali et al. [6] found that the thickness of the liquid film is also an important parameter determining the critical value for splashing. The thinner the liquid film, the smaller the critical splashing value, and thus splashing occurs more easily. The splashing criterion they obtained is as follows:

$$K>K_{\mathrm{S}}=2100+5880{h^{+}}^{1.44}$$

(9)

h⁺ is the dimensionless thickness of the liquid film. Motzkus et al. [15] also found a similar relationship, and proposed the splashing condition as:

$$K>K_{\mathrm{S}}=2100+2000{h^{+}}^{1.44}$$

(10)

In the present study, splashing occurs when We is 100, corresponding to a K value of 1238, which is much lower than the critical splashing values given in the above studies. Liang et al. [20] believed that when multiple droplets impact on a liquid film, the interaction of crown will reduce the critical splashing value. Therefore, the splashing criteria for single droplet impact cannot be simply extended to situations involving multiple droplet impact.

3.3. Effects of Impact Angle

Figure 9 Shows the oblique impact results of two droplets on a film with different impact angles, droplet impacts with 15°, 30°, and 45° impact angles are simulated while D₀ = 4 mm, We = 600, h⁺ = 0.6, and s* = 2. the impact angle can significantly affect the droplet interactions. For smaller impact angles, the droplets will have a lower horizontal velocity component and a weaker interaction. On the other hand, for larger impact angles, the droplets will have a higher horizontal velocity component, which can lead to more complex and energetic interactions between the droplets.

3.4. The Upstream Crown Radius

Figure 10a displays influence of the Weber number on the upstream crown radius R_u^* (defined in Figure 7). While the effects of the Weber number on splashing behavior can be significant, the influence on the upstream crown radius is not significant. Figure 10b displays the influence of impact angle θ on the upstream crown radius R_u^*. The upstream crown radius was found to increase with the increment of Weber number.

To understand how the crown radius of a droplet evolves during impact, many researchers have conducted experimental and quantitative parametric investigations. Most models and correlations are developed for the vertical impact of a single droplet on a film. The relationship between the dimensionless crown splash diameter D* and dimensionless time τ proposed by Yarin and Weiss [37] is as follows:

$$D*=C{\left(\tau -{\tau }_0\right)}^n$$

(11)

where C and n are constants, n = 1/2. However, in many cases, attempts to fit data using this formula are unsuccessful, meaning that this simple square root relationship cannot describe the evolution of all types of crowns [19]. In addition, Cossali et al. [10] found that, although the coefficient C is independent of liquid film thickness, it is not a constant but is slightly correlated with impact velocity. The relationship they proposed between the dimensionless crown diameter D^* and dimensionless time τ is as follows:

$$D*=C{\left(\tau -{\tau }_0\right)}^{0.43\pm 0.03}$$

(12)

Liang et al. [38] studied the relationship between crown diameter and the Weber number We and Reynolds number Re of the water droplet. Their results showed that within a certain range, the diameter of the crown is independent of We and Re. The evolution of the non-dimensional crown radius r_B by Gao and Li [39], which has been validated for many cases, can be expressed as follows:

$$r_B=\beta \sqrt{\tau }+\frac{1}{\sqrt{6{\mathrm{h}}^{+}}}-{\left(\frac{1}{3{\mathrm{h}}^{+}}-\frac{1}{\sqrt{6{\mathrm{h}}^{+}}}\right)}^{1/2}$$

(13)

Figure 10b shows the evolution of the upstream crown radius R_u* of numerical simulation (We = 600, h⁺ = 0.6) and the non-dimensional crown radius (r_B) by the model of Gao and Li [39]. When two droplets impact the liquid film vertically, the crown radius agrees well with the model by Gao and Li. As the impact angle increases, the model tends to overestimate the crown radius. In oblique multi-droplet impact scenarios, the modeling approach should take the impact angle and the effects of droplet interaction into consideration.

Figure 10. Influence of Weber number and impact angle on the upstream crown radius R_u^* (h⁺ = 0.6, s* = 2): (a) Weber number (We = 100, 300, 600, θ = 30°); and (b) impact angle (We = 600, θ = 0°,15°, 30°, 45°) [39].

Figure 10. Influence of Weber number and impact angle on the upstream crown radius R_u^* (h⁺ = 0.6, s* = 2): (a) Weber number (We = 100, 300, 600, θ = 30°); and (b) impact angle (We = 600, θ = 0°,15°, 30°, 45°) [39].

As the droplet impacts the liquid film, its kinetic energy is transformed into crown energy [40]. Analyzing the variations in total kinetic energy (E_k) and vertical kinetic energy (E_kv) provides valuable insights into the dynamics of this phenomenon:

$$E_{\mathrm{k}}=\frac{1}{2}m{V}^2, E_{\mathrm{k}\mathrm{v}}=\frac{1}{2}m{V}_v^2=\frac{1}{2}m{V}^2{\mathrm{c}\mathrm{o}\mathrm{s}}^2\theta$$

(14)

As depicted in Figure 11, m represents the mass of the droplet, V denotes the velocity of the droplet, and V_v is the vertical velocity of the droplet, V_{v =} Vcos θ, θ is the impact angle.

Our modified model for evolution of the upstream crown radius R_u* is as follows:

$$R_u^{\mathrm{*}}=b{r}_B\frac{E_{\mathrm{k}\mathrm{v}}}{E_{\mathrm{k}}}=b{r}_B{cos}^2\theta$$

(15)

where r_B is the non-dimensional crown radius by Gao and Li [39], b is the modified coefficient (b = 0.9) to fit the curve. At small impact angles (e.g., θ = 15° and 30°), the modified model exhibits good agreement with the simulation results (Figure 10b). However, as the impact angle increases, the difference between the model and simulation results tends to increase. This might be due to the fact that, at larger impact angles, the interactions between droplets are likely to become stronger and more complex. As a result, additional modifications and adjustments may be required to improve the accuracy of the model.

3.5. The Central Uprising Sheet Radius

Figure 12a displays influence of the Weber number on the central uprising sheet radius R_c^* (defined in Figure 7), similar to the upstream crown radius, the influence of the Weber number on the central uprising sheet radius is not significant. As the Weber number increases at a fixed droplet size, the kinetic energy of the droplet increases. However, for the range of Weber numbers investigated here, the additional kinetic energy does not appear to substantially affect the outward expansion of the central uprising sheet. Figure 12b displays the influence of impact angle θ on the central uprising sheet radius R_c^*, the central uprising sheet radius increases with the increment of impact angle. At higher angles, the horizontal momentum of the droplets leads to stronger transverse interactions between the spreading crowns. This enhances the outward motion of the central liquid sheet formed by the merging crowns, resulting in larger R_c^* values.

3.6. The Central Uprising Sheet Height

Figure 13a displays influence of the Weber number on the central uprising sheet height H_c^* (defined in Figure 7), the central uprising sheet height increases with the increment of the Weber number. Cossali et al. [10] studied the process of single droplet impact on a liquid film and believed that the evolution of the dimensionless crown height has a strong correlation with the Weber number. After a droplet impacts a liquid film, the height of the crown gradually increases over time, reaching a maximum value and then decreasing gradually as the crown collapses. The maximum dimensionless height of the crown is H^*_max, occurring at dimensionless time τ_max, both of which are related to Weber number, as follows:

$$H_{\max }=A_{1}W{e}^p$$

(16)

$${\tau }_{max}=A_{2}W{e}^p$$

(17)

where A₁ and A₂ are constants, p = 0.65–0.75. Thus, higher Weber numbers lead to larger H^*_max and τ_max. When a droplet impacts a liquid surface, part of its kinetic energy transforms into the potential energy of the crown. The higher the Weber number, the greater the kinetic energy carried by the droplet into the impact process, and correspondingly, the greater the potential energy transferred to the crown, resulting in an increase in the crown height. Figure 13b displays influence of impact angle θ on the central uprising sheet height H_c^*. Unlike the Weber number, the influence of the impact angle on the central uprising sheet height is not significant. This may be because the central uprising sheet height is dictated primarily by the initial kinetic energy; the impact angle does not substantially alter the initial kinetic energy of the droplets.

4. Conclusions

In this study, the simultaneous oblique impact of dual droplets on a water film was investigated using a three-dimensional numerical simulation employing the Volume of Fluid (VOF) method. The results show that the splashing threshold for the dual droplet simultaneous oblique impact cases is significantly lower than that of the single droplet impact cases due to the interactions between the adjacent crowns. The effects of the Weber number and impact angle on the crown behavior were carefully analyzed. The increase in the Weber number enhanced the central uprising sheet height but had minor influences on the upstream crown radius and the central sheet radius. In contrast, a larger impact angle led to a decreased upstream crown radius and an increased central sheet radius, while the central sheet height remained relatively unaffected. The present results provide novel insights into the underlying physics of this complex multi-droplet impact phenomenon, which is crucial for developing predictive models and improving various droplet-based technologies.

Due to the relatively high computational costs, this study only investigated the early stage of the dual droplet simultaneous oblique impact process on the water film. Future studies can extend the simulation scope to cover the later stages of the impact process, which are also crucial for a comprehensive understanding of this phenomenon. In addition, the present simulations were performed for a limited range of impact angles and Weber numbers. This does not cover the entire spectrum of possible conditions in industrial and natural settings, potentially limiting the generalizability of the findings. Therefore, expanding the range of impact angles and Weber numbers in the simulations will provide a more comprehensive understanding of the impact processes. In addition, this study focuses on a fixed droplet size and a fixed film thickness. However, in practical applications, these parameters can vary widely. The effects of the droplet size and the film thickness on the impact process should also be investigated in the future.