# Numerical investigation of freely moving particle–droplet interaction with initial contact

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## Abstract

We simulated freely moving particle–droplet interaction with initial contact. Fluid–structure interaction was modeled by fictitious domain method, and two-fluid interface was tracked using Level Contour Reconstruction Method. For tracking movement of the solid object, additional object distance function was calculated at Eulerian grid center. Since simple geometry, i.e., circle, was used in this study, object distance function can be easily computed from center location updated by averaged velocity to constrain solid movement. The interaction phenomenon was simplified as center-to-center contact without initial velocity. The gravitational acceleration was also ignored. We choose the size ratio and Ohnesorge (Oh) number as main parameters. Two characteristic behaviors were captured: the merging and separation case. Each velocity of the particle and droplet was shown to see the detailed evolution for merging and separation. In addition, two major forces acting on the particle, a capillary force and inertial force, were analyzed.

## Keywords

Numerical simulation Particle–droplet interaction Merging/Separation Size ratio## 1 Introduction

Collisions between liquid droplet and particle can be easily identified in nature and also applicable to various industrial fields. The most well-known natural phenomenon would be aerosol. Aerosol, acting as a condensation nucleus in the clouds, interacts with the droplet in a variety of ways, which affects the water cycle [1]. In case of industrial fields, collision phenomena between droplets and particles can be observed in a wet scrubber, which purifies flue gas contaminated with SO_{2} and NO_{x} by a droplet jetting device [2]. In recent years, a study has been carried out to analyze the effect on the concentration of fine dust from the installation of the automatic sprinkler system on the roadside [3]. However, it was less effective than natural phenomena from precipitation.

To improve its performance in applications related to the interaction between particle and droplet, various physical phenomena must be clearly identified. In many previous studies [4, 5, 6], droplet collision phenomena with particle, one of which is stationary, have been experimentally or numerically analyzed, but the case is still rare for studying the interaction between freely moving particle and droplet. Duburovsky et al. [7] conducted an experimental study on the collision of small particle with large droplets and classified results into four processes: particle capture, shooting through with satellite droplet formation, shooting through with gas bubble formation, and droplet destruction. Pawar et al. [8] investigated the collision between freely moving particle and droplet and classified the results into three cases: agglomeration, stretching separation, and separation with satellites droplet based on Weber number and impact parameter. Most studies were focused on relative velocity between particle and droplet. Recently, Yang et al. [9] simulated the interaction between freely moving particle and droplet, and analyzed the results according to size and eccentricity ratio. It was carried out in plane 2D geometry which can be quite different from real physics. It is hard to find relative studies on spherical particle–droplet interaction with different size ratio.

In this study, we will focus on interaction between spherical particle and droplet with initial contact aiming first tryout for full 3D simulation. Driving force would be the surface tension, since relative velocity between particle and droplet will be zero at the start. Without initial impact velocity, geometry becomes simple, so 2D axi-symmetric version of the code was utilized. In addition, size of the droplet as well as size ratio between particle and droplet were changed to see its effect on overall interaction.

## 2 Numerical formulation

### 2.1 Tracking two-fluid interface and contact model

We used the level contour reconstruction method (LCRM) [10, 11, 12] to track the moving two-fluid interface over time. The LCRM is a fusion form of front tracking and level set method. It has very efficient procedure tracking interface motion, because it forms a level set field with the distance function *ϕ*_{f} calculated from the interface and reconfigures the interface periodically and automatically. For further details, refer to the previous studies [10, 11, 12].

Since the LCRM tracks additional Lagrangian interface, the information of the contact line can be accurately calculated. Infinite shear stress from nonslip boundary condition on the wall has been relieved using the Navier slip boundary condition [13]. An extended surface was constructed from the contact line into the solid surface to account for contact angle hysteresis by forcing dynamic contact angle. Detailed explanation regarding contact model can be found in [12]. Since distance function for non-deforming solid can be easily computed, implementation of contact dynamics for the moving sphere becomes very straightforward. Contact angle is restricted between given advancing and receding angle. For more specific information about the contact model and two-fluid interface tracking method, see previous studies [10, 11, 12].

### 2.2 Tracking moving solid object

*ϕ*

_{s}. At the beginning of the simulation, the solid area is created by calculating the distance function \(\phi_{\text{s}}^{0}\) at the cell center of the Eulerian grid \(\left( {x_{\text{i}}^{0} ,\;y_{\text{j}}^{0} } \right)\) with the theoretical equation (circle in current study, See Fig. 1). During the simulation, the distance function \(\phi_{\text{s}}^{n}\) need to be updated at \(\left( {x_{\text{i}}^{n} ,\;y_{\text{j}}^{n} } \right)\). To compute \(\phi_{\text{s}}^{n}\) at the cell center with the movement of the solid object, the center position of the solid should be tracked with time.

*u*

_{s}as follows:

*M*

_{s},

*u*

_{s},

*X*

_{s}, and d∀ represent mass, translational velocity, displacement of the solid object, and the solid volume, respectively. This procedure could be efficient compared to two-fluid interface tracking, since only center position of the solid object can be traced. For more detailed information of tracking moving, solid object can be found in the previous study [14].

*I*, which equal to the value 0 in one phase and 1 in the other:

### 2.3 Solution process including fluid–structure interaction

Fluid–structure interaction was modeled using fictitious domain method (FDM) [15]. FDM is basically a very simple method, assuming a moving solid as one of fluids. In other words, Navier–stokes equation was applied to the entire flow field including the solid region. Rigid body constraints were applied to the solid region with the averaged velocities (Eq. 1). Considering a nonphysical slip condition at the solid–liquid interface, an additional high viscosity coefficient is applied to the solid region [16].

*u*represents the velocity,

*P*, the pressure, and

*g*, the gravitational acceleration, and

*F*is the local surface tension force at the interface, which can be described by the hybrid formulation as follows:

*σ*is the surface tension coefficient which is assumed to be constant in the present study, and interface curvature,

*κ*, unit normal vector,

*n*, a parameterization of the interface and the geometric information,

*x*

_{f}, and area of the interface element, dA, are computed directly from the Lagrangian interface.

*u*

_{f}represents the Lagrangian interface velocity. The governing equations are computed using the Chorin’s projection method [17] on a staggered grid. For more detailed solution, procedure can be found in the previous studies [10, 11, 12].

## 3 Validation with benchmarking

### 3.1 Impacting solid sphere on free surface

^{3}and the viscosity coefficient was 100 times that of water. To verify the grid convergence, simulations were performed while changing the number of grids as 50 × 70, 100 × 140, 200 × 280, and 400 × 560 (time step: 10 μs). The depth of sphere bottom was plotted in Fig. 2a with time. The solution converged at the grid resolution of 400 × 560, and the results agree well with reference data [18].

### 3.2 Droplet impact on a stationary spherical particle

Additional benchmarking simulation, droplet impact on a stationary spherical particle, was performed to test the accuracy of moving fluid interface on curved wall. The schematic of the simulation is depicted in Fig. 2b. A stationary brass particle with a diameter of 10 mm is on top of the spherical particle in the computational domain filled with air. The size of the domain is 10 mm in both radial and axial dimensions. Open condition is applied at the upper and right-side boundary, and a symmetry condition is used at the lower boundary. A 100 times viscosity of water was used for solid region and the density of brass particle is 8600 kg/m^{3}. A water droplet with diameter of 3.1 mm was impacting on a brass particle at a velocity of 0.436 m/s. Advancing and receding angles were set to 120° and 40°, respectively, from reference data [19]. Considering the convergence behavior of previous benchmarking test, grid resolution of 200 × 280: 50 CPD (cells per diameter) was used (time step: 1 μs). The maximum diameter of the contact point is compared with the other numerical simulation and experimental results (see Fig. 2b). Although there is small time lag, current simulation shows better performance matching experimental maximum diameter compared to simulation from Mitra et al. [19].

## 4 Results and discussion

The actual phenomena involving particle–droplet interaction such as wet scrubber or collision between raindrops and aerosol would be affected by various factors such as relative velocity, impact direction, and distance between center axes, etc. However, we simplified the simulation geometry considering the initial research status. First, the distance between the center axis of the particle and the droplet was set to zero (center-to-center interaction). In addition, the shapes of the particle and the droplet were assumed to be sphere at the start. Since we are focusing on surface tension-driven interface interacting with freely moving solid, droplet was initially placed on top of the solid with contact angle of 170°. Water advancing and receding angles were set to 65° and 35° [20, 21], respectively. The gravity was also ignored and the viscosity of solid particle is 100 times of water similar to benchmarking setting.

^{3}, surface tension of 0.0728 N/m, and dynamic viscosity of 1.00 × 10

^{−3}Pa s were used. With respect to the size of the particle and the droplet, we defined the size ratio (

*D*

_{r}) and additional Ohnesorge number (Oh) was chosen as the main simulation variables:

*D*

_{d}represents the diameter of water droplet and

*D*

_{p}is the diameter of solid particle.

*D*

_{r}of 0.5, 1.0, 1.5, and 2.0. In addition, for each

*D*

_{r}, Oh was changed to 0.00370, 0.00302, and 0.00262. As can be seen from Fig. 3, simulation results can be categorized as merging (Fig. 4a) and separation cases (Fig. 4b). Interestingly, two cases are heavily related to size ratio,

*D*

_{r}, not Oh. When

*D*

_{r}was 0.5 and 1, the particle and droplet merged and moved together (merging), and when

*D*

_{r}was 1.5 and 2, the particle and droplet moved apart (separation).

The detailed sequences of particle–droplet interaction are shown in Fig. 4a with pressure field for the merging case, where Oh is 0.00262 and *D*_{r} is 1.0. The initial high pressure inside the droplet (0.1 ms) would be defined by the Young–Laplace equation. At the beginning of the simulation, the droplet started to move due to the difference between the initial and the equilibrium contact angles of the droplet. At this time, the droplet attached to the particle, moved downward together. Even the surface tension tries to pull the solid particle upward, the increased pressure field inside the droplet makes asymmetric pressure distribution around the solid particle. This increased pressure at the top of the solid particle compared to low pressure at the bottom will force the particle motion downward. For the separation case, as shown in Fig. 4b, where Oh is 0.00262 and *D*_{r} is 1.5, increased initial pressure inside the droplet from surface tension can also be observed. Similarly, the droplet began to move together due to the difference between the initial and the equilibrium contact angles. However, the particle escaped from the droplet.

*D*

_{r}is shown in Fig. 5. For merging cases (

*D*

_{r}of 0.5 and 1.0), we noticed that the velocities of the particle and droplet were both increased with

*D*

_{r}(see Fig. 5a). And the velocity of the droplet is oscillating around the velocity of the particle, and it ultimately converges to the velocity of the particle as one body movement. For separation cases (

*D*

_{r}of 1.5 and 2.0), the velocity of the droplet is converged after a specific time (detachment time), but the velocity of the particle keeps increasing due to the increased pressure by small satellite droplet on top of the particle (see 3 ms in Fig. 4b). Thus, relative velocity between the particle and the droplet increases over time after detachment.

*F*

_{surf}) and a downward force (inertial force,

*F*

_{p}) caused by the asymmetrically increased pressure inside the droplet as mentioned above (see Fig. 4). Those forces can be formulated as follows:

*m*

_{p}is mass of the particle, and

*u*

_{r}is the relative velocity between the particle (

*u*

_{p}) and the droplet (

*u*

_{d}).

Evolution of the two forces was computed, as shown in Fig. 6. For the merging case, where Oh is 0.0262 and *D*_{r} is 1.0, the capillary force was always greater than the inertial force during the whole simulation (see Fig. 6a). For the separation case, where Oh is 0.0262 and *D*_{r} is 1.5, there was a moment (near time of 4 ms) when the inertial force is greater than the capillary force (see Fig. 6b). This means that the downward force pushing particle away from the droplet instantaneously overcomes the force pulling the particle back into the droplet. At this moment, the particle escapes from the droplet. Similar behaviors were also observed for different *D*_{r} where the particle and droplet separate from increased relative velocity and, thus, the inertial force acting on particle.

## 5 Conclusions

We simulated freely moving particle–droplet interaction with Fictitious Domain Method (FDM) integrated into Level Contour Reconstruction Method (LCRM). Prior to simulation of particle–droplet interaction, benchmarking simulations of water entry and droplet impact on a stationary particle case were carried out to validate the numerical code. To simplify the phenomenon considering initial research status, we focused on center-to-center interaction with initial contact without gravitational acceleration and initial velocity. As a result, the interaction regimes were divided into two cases, i.e., merging and separation heavily related to size ratio, *D*_{r}, not Oh number.

We compared the velocity of the particle and the droplet to see the behavior of merging and separation clearly. The relative velocity between the particle and the droplet for the separation case was measured greater than the merging case. For the merging case, relative velocity becomes smaller ultimately approaching to one velocity indicating single-body movement. We also analyzed the two forces acting on the particle, an upward force (capillary force) and a downward force (inertial force). In the merging case, the capillary force was always greater than the inertial force during entire simulation, whereas, in the separation case, there was a moment when the inertial force was higher than the capillary force, and the particle escaped from the droplet at that moment.

As already mentioned, this was first attempt to see the basic interaction behavior with initial contact of droplet and particle. Thus, gravitational effect has been ignored. In the future, we are trying to include the effect from initial impact speed from droplet or particle as well as gravity. Eccentricity, finite distance between impact direction of the droplet and particle would be also considered with numerical update to massively parallel three-dimensional solver [22].

## Notes

### Acknowledgements

This work is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03028518).

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