Numerical simulation of two-dimensional binary droplets collision outcomes using the level set method

Towards a deterministic composite collision outcome model for surface-tension dominated droplets

We have established a parallel, adaptive interface-tracking framework in order to conduct, based on the framework, direct simulation of binary head-on droplet collision in the high-Weber-number regime (from 200 to 1500) that exhibits complex topological changes and substantial length scale variations. The overall algorithms include a combined Eulerian and Lagrangian solver to track moving interfaces, conservative Lagrangian mesh modification and reconstruction, cell-based unstructured adaptive mesh refinement (AMR) in the Eulerian solver, and associated Eulerian and Lagrangian domain partitions to minimize communication overhead. Based on the combined computational and experimental efforts, we have resolved for the first time the free-surface instabilities of the colliding droplets at such high Weber number. We detail the characteristics of coalescence, stretch, end pinching, fingering, free-surface movement and drop breakup. The Taylor–Culick rim is present soon after the collision. Furthermore, we observe two types of longitudinal instabilities on the rim, namely, the Rayleigh–Taylor (RT)-type instability in the initial deceleration phase of the circular sheet right after droplet coalescence, and later the Rayleigh–Plateau (RP) instabilities. As the Taylor–Culick rim disintegrates in the retraction phase, fingering effect is profound and resulting in wider droplet size distribution.

The recently introduced entropic lattice Boltzmann method (ELBM) for multiphase flows is extended here to simulation of droplet collisions. Thermodynamically consistent, non-linearly stable ELBM together with a novel polynomial equation of state is proposed for simulation large Weber and Reynolds number collisions of two droplets. Extensive numerical investigations show that ELBM is capable of accurately capturing the dynamics and complexity of droplet collision. Different types of the collision outcomes such as coalescence, reflexive separation, and stretching separation are identified. Partition of the parameter plane is compared to the experiments and excellent agreement is observed. Moreover, the evolution of the shape of a stable lamella film is quantitatively compared with experimental results. The end pinching and the capillary-wave instability are shown to be the main mechanisms behind formation of satellite droplets for near head-on and off-center collisions with high impact parameter, respectively. It is shown that the number of satellite drops increases with increasing Weber number, as predicted by experiments. Also, it is demonstrated that the rotational motion due to angular momentum and elongation of the merged droplet play essential roles in formation of satellite droplets in off-center collisions with an intermediate impact parameter.

Three-dimensional binary droplet collisions are studied using a multiphase cascaded lattice Boltzmann method (LBM). With this model it is possible to simulate collisions with a Weber number of up to 100 and a Reynolds number of up to 1000, at a liquid to gas density ratio of over 100. This is made possible by improvements to the collision operator of the LBM. The cascaded LBM in three dimensions is introduced, in which additional relaxation rates for higher order moments, defined in a co-moving reference frame, are incorporated into the collision operator. It is shown that these relaxation rates can be tuned to reduce spurious velocities around curved phase boundaries, without compromising the accuracy of the simulation results. The range of attainable Reynolds numbers is therefore increased. Different outcomes from both head-on and off-centre collisions are simulated, for both equal and unequal size droplets, including coalescence, head-on separation, and off-centre separation. For head-on collisions the critical Weber number between coalescence and separation is shown to decrease with decreasing ambient gas pressure. The variation of critical Weber number with droplet size ratio is also studied. Comparisons are made with the theoretical predictions of Tang et al. [“Bouncing, coalescence, and separation in head-on collision of unequal-size droplets,” Phys. Fluids 24, 022101 (2012)], and the effect of ambient gas pressure is again considered. For off-centre collisions, boundaries between different collision outcomes are accurately defined and quantitative comparisons are made with the theoretical predictions of Rabe et al. [“Experimental investigation of water droplet binary collisions and description of outcomes with a symmetric Weber number,” Phys. Fluids 22, 047101 (2010)]. While general agreement between the simulated and theoretical boundaries is presented, deviations due to varying liquid viscosity are observed. Finally, the prediction of the independence of regime boundaries with varying droplet size ratio, when using the symmetric Weber number as defined by Rabe et al., is discussed. Simulation results showing qualitative agreement are presented, although some discrepancies are reported.

PurposeShape optimization using the level-set method is one of the most effective automatic design tools for electromagnetic machines. While level-set method has the advantage of being able to suppress unfeasible shape, it has a weakness of being unable to handle complex topology changes such as perforate at material region. With this method, it is only possible to define simple connected topology, and it is difficult to determine the optimal shape which has holes. Therefore, it is important to efficiently expand the searching area in the optimization process with level-set method.Design/methodology/approachIn this paper, the authors introduce the newly defined hole sensitivity which is based on concept of topological derivatives, and combine it with level-set method to effectively create holes in the search process. Furthermore, they consider a variable bandwidth of gray scale, which indicates the transition width between air and magnetic body and combine it with the hole creation method described above. With these methods, the authors aim to expand the searching area in comparison with the conventional level-set method.FindingsAs a result of applying the proposed methods to a magnetic shielding problem, the multi-layered shielding which effectively reduces the magnetic flux in the target area, is successfully produced.Originality/valueThe proposed methods enable us to effectively create a hole and to expand the searching area in the topology optimization process unlike in the case of conventional level-set method.

Numerical and experimental investigation of induced flow and droplet–droplet interactions in a liquid spray

The investigations of the collision outcome and dynamics of mass transfer process of unequal-sized binary droplets collision are preformed computationally and theoretically. All the possible collision outcomes, bouncing, coalescence, reflexive and stretching separation are obtained numerically. The present study focuses on water droplet interactions in atmospheric air over a wide range of Weber numbers and impact parameters with two droplet diameter ratio, 0.5 and 0.25. The numerical formulation is based on the complete set of conservation equations for both the liquid and the surrounding gas phases. An improved volume-of-fluid (VOF) technique, augmented by an adaptive mesh refinement (AMR) algorithm, is used to track the liquid/gas interfaces. A novel thickness-based refinement methodology is also developed to resolve the extremely thin gas film during early approaching of droplets. Numerous high-fidelity bouncing cases are simulated using proposed efficient methodology. The smallest cell size is up to O(10 -4 ) of the small droplet diameter with minimum grid size of 0.02 μm. In addition, an advanced visualization technique using the Ray-tracing methodology is implemented to gain direct insight into the detailed physics of droplet interaction and mass transfer process. A simple theory based on the geometric relation of the unequal-sized droplet collision system is established to predict mass transfer ratio of stretching separation outcome. Good agreement is achieved with simulation results.

Size effects on dynamics of nanodroplets in binary head-on collisions

The level set method (LSM) has been widely used in image segmentation due to its intrinsic nature which allows handling complex shapes and topological changes easily. We propose a new level set algorithm, which is based on probabilistic c mean objective function which incorporates intensity inhomogeneity in image and robust to noise. The computational complexity of the proposed LSM is greatly reduced by using highly parallelizable lattice Boltzmann method (LBM). So the proposed algorithm is effective and highly parallelizable. The proposed LSM is implemented using Experimental results demonstrate the performance of the proposed method.

A complete knowledge of the effect of droplet viscosity on droplet-droplet collision outcomes is essential for industrial processes such as spray drying. When droplets with dispersed solids are dried, the apparent viscosity of the dispersed phase increases by many orders of magnitude, which drastically changes the outcome of a droplet-droplet collision. However, the effect of viscosity on the droplet collision regime boundaries demarcating coalescence and reflexive and stretching separation is still not entirely understood and a general model for collision outcome boundaries is not available. In this work, the effect of viscosity on the droplet-droplet collision outcome is studied using direct numerical simulations employing the volume of fluid method. The role of viscous energy dissipation is analysed in collisions of droplets with different sizes and different physical properties. From the simulations results, a general phenomenological model depending on the capillary number (Ca, accounting for viscosity), the impact parameter (B), the Weber number (We), and the size ratio (Δ) is proposed.

We present a simple yet effective method for modeling of object decomposition under combustion. A separate simulation models the flame production and generates heat from a combustion process, which is used to trigger pyrolysis of the solid object. The decomposition is modeled using level set methods, and can handle complex topological changes. Even with a very simple flame model on a coarse grid, we can achieve a plausible decomposition of the burning object.

Direct numerical simulation of droplet collision with stationary spherical particle: A comprehensive map of outcomes

A spray model and a droplet collision model are implemented into a radio frequency inductively coupled plasma model. The discrete parcel technique combined with the stochastic Monte Carlo method is used to solve the spray equation and determine the outcomes of droplet collisions in dense sprays. Plasma--spray interactions are considered by adding source terms to the conservation equations of mass, momentum and energy of the plasma phase. Two types of the outcomes of water droplets collisions, coalescence and grazing, are predicted and compared to the experimental and analytical results. The agreement is quite good. The effects of droplet collisions on droplet size distribution of the spray and the spray evaporation are investigated. It is found that the droplet collisions can increase the average droplets size of the spray. For the mono-disperse spray, the collisions can lead to a delay on the spray evaporation. However, for the poly-disperse spray, the effect of droplet collisions on the spray evaporation could not be predicted before the calculation due to the randomness of droplet collisions.

Understanding the mechanics and outcome of droplet and bubble collisions is central to a range of processes from emulsion stability to mineral flotation. The atomic force microscope has been shown to be sensitive and accurate in measuring the forces during such interactions; in combination with a suitable model framework, a powerful tool is obtained for understanding surface forces and droplet stability in dynamic systems. Here we demonstrate for the first time that this process is not limited to linear motion, and that accelerating, decelerating and cyclical droplet velocities can be used to explore the collisions between droplets and bubbles in ways that much more closely mimic real systems. In particular, the motion of droplets experiencing oscillating pumping pressures is explored, providing insight into fluid handling for microfluidics. By modelling a range of processes in which drops collide and deform, and sometimes coalesce, the validity of the theoretical model – which accounts for deformation, surface forces and dynamic lubrication – is demonstrated. Further, it is shown how this model can be used as a predictive tool to determine whether a given droplet collision will be stable or coalescent.

Studying energy conversion provides critical insights into the abnormal behavior transition from separation to coalescence following fingering (CFF) in binary droplet collisions at high Weber numbers (We). This study investigates the energy conversion dynamics of binary droplet collisions with an Ohnesorge number (Oh) of approximately 0.001 and We=30–650 through three-dimensional phase-field lattice Boltzmann simulation. Key findings reveal that the classical spreading theory [Jiang et al., “An experimental investigation on the collision behaviour of hydrocarbon droplets,” J. Fluid Mech. 234, 171–190 (1992)] leads to a significant overestimation of viscous dissipation by approximately 25% at high We, primarily due to the neglect of kinetic energy of internal flow at the spreading end point. A crucial finding reveals that the ratio of viscous dissipation rate to surface energy release rate during most of the merging stage is maintained at one-third, contrasting with the established value of one-half observed in soap film rupture studies. Based on this finding, a quantitative analysis for CFF further found that the additional dissipation caused by fingering needs to reach about onefold of the initial surface energy to cause the reflexive separation behavior to transform into coalescence behavior. Furthermore, this study attributes the low onset We of CFF in the gallium–indium–tin (GaInSn) droplets–gas system to its properties of significantly high density ratio and oscillation-induced surface tension reduction, challenging prior explanations based solely on low Oh.