Numerical investigation of freely moving particle–droplet interaction with initial contact
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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.
KeywordsNumerical simulation Particle–droplet interaction Merging/Separation Size ratio
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 . In case of industrial fields, collision phenomena between droplets and particles can be observed in a wet scrubber, which purifies flue gas contaminated with SO2 and NOx by a droplet jetting device . 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 . 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.  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.  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.  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 ϕf 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 . 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 . 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
2.3 Solution process including fluid–structure interaction
Fluid–structure interaction was modeled using fictitious domain method (FDM) . 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 .
3 Validation with benchmarking
3.1 Impacting solid sphere on free surface
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/m3. 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 . 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. .
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.
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 Dr 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 Dr 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.
Evolution of the two forces was computed, as shown in Fig. 6. For the merging case, where Oh is 0.0262 and Dr 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 Dr 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 Dr where the particle and droplet separate from increased relative velocity and, thus, the inertial force acting on particle.
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, Dr, 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 .
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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