Low Weber Number Research Articles

Spreading dynamics of the viscous droplet impacting on a spherical particle

The dynamic behavior of droplets in the impingement process with particles has attracted extensive interest due to its widespread industrial applications. In this study, collision experimentation was carried out to investigate the spatial and temporal variation of droplets on a target particle surface by utilizing high-speed photography. Energy conversion and force analysis were conducted through theoretical analysis. Moreover, we captured the microscopic and evolutionary features of droplets in detail by image processing. The dimensionless liquid film thickness in the maximum spreading state is primarily determined by its spreading area. The Weber number can be used to calculate the maximum spreading area of droplets for liquids with a specific viscosity and droplet-to-particle size ratio. The time-evolved liquid film morphology and spreading area of a highly viscous liquid show a different trend compared to that of the low-viscous liquid. When the liquid film is not broken, the maximum dimensionless spreading area is linearly related to the Weber number. At low Weber numbers and high Reynolds numbers, droplets exhibit more pronounced oscillation characteristics. The oscillation period of the collision is related to the droplet-to-particle size. The liquid film thickness decreases as the Weber and Reynolds numbers rise. As for the low-viscous liquid, a low Weber number leads to a periodic change in the dynamic contact angle. A decrease in the Reynolds number for the highly viscous droplets generates a greater dynamic contact angle. The recoiling of the liquid film results in a more significant reduction in the dynamic contact angle.

Energetics of spreading droplets and role of capillary waves at low Weber numbers below 10

In this study, we investigate the energy conversion and dissipation mechanisms of spreading droplets on a solid surface at a low Weber number regime, which neither conventional energy-balance-based theories nor empirical scaling laws can completely explain. The energetic analysis presented in this study shows that on a hydrophilic surface, the actual primary energy source driving the spreading process is the initial surface energy not the initial kinetic energy. The conventional energy-balance-based approaches are found to be valid only for the spreading process on a hydrophobic surface. Particular attention is also paid to the roles of the capillary waves. The capillary waves are found to play significant roles in all of the important flow physics, that is, the interfacial structure, the oscillatory motions and the rapid collapse of the liquid film, the onset of the viscous regime, and the energy loss mechanism. It is also shown that the energy dissipation caused by the capillary-wave-induced phenomena can be estimated to be 25%–35% and 55%–65% of the total energy loss for a hydrophilic and a hydrophobic surface, respectively, at the low Weber number regime.

A consistent volume-of-fluid approach for direct numerical simulation of the aerodynamic breakup of a vaporizing drop

A novel simulation framework has been developed in this study for the direct numerical simulation of the aerodynamic breakup of a vaporizing drop. The interfacial multiphase flow with phase change is resolved using a consistent geometric volume-of-fluid method. The bulk fluids are viscous and incompressible with surface tension at the interface. The newly-developed numerical methods have been implemented in the Basilisk solver, in which the adaptive octree/quadtree mesh is used for spatial discretization, allowing flexibility in dynamically refining the mesh in a user-defined region. The simulation framework is extensively validated by a series of benchmark cases, including the 1D Stefan and sucking problems, the growth of a 3D spherical bubble in a superheated liquid, and a 2D film boiling problem. The simulation results agree very well with the exact solution and previous numerical studies. 2D axisymmetric simulations were performed to resolve the vaporization of a moving drop with a low Weber number in a high-temperature free stream. The computed rate of volume loss agrees well with the empirical model of drop evaporation. Finally, the validated solver is used to simulate the aerodynamic breakup of an acetone drop at a high Weber number. A fully 3D simulation is performed and the morphological evolution of the drop is accurately resolved. The rate of vaporization is found to be significantly enhanced due to the drop deformation and breakup. The drop volume decreases nonlinearly in time and at a much higher rate than the empirical correlation for a spherical drop.

The spreading and sliding characteristics of droplet impingement on an inclined hydrophobic surface at low Weber numbers

Droplet sliding after impingement on an inclined superhydrophobic surface has very broad applications, but the investigations of this variable process are surprisingly lacking in the literature. In this paper, the spreading and sliding characteristics of a water droplet impinging on an inclined hydrophobic surface at low Weber numbers (We≲ 50) are investigated via a numerical approach. The influences of surface inclined angle α, impact velocity, and droplet diameter on droplet spreading diameter and maximum spreading diameter (Ds and Ds_max), sliding lengths of droplet, droplet leading and trailing edges (Ld, Llea and Ltra) are thoroughly examined. Results show that droplet displays markedly distinct behaviors when α≥ 60° as the maximum spreading diameter is noticeably larger and droplet oscillation becomes stronger and more unstable, which are contributed from the accumulated liquid mass at droplet head associated with the drastic growth of the tangential inertial force. However, the instability at α≥ 60° disappears when We≥ 40, resulting from the liquid mass at droplet head dominating the sliding process, which demonstrates that droplet motion at a high inclined angle is primarily determined by droplet leading edge. The nondimensionalized maximum spreading diameter (Ds_max∗) is found to scale with the normal Weber number (WeN) according to a power law (Ds_max∗∝WeNα1), and a universal scaling law of Ld_total∗∝WeT0.6 between the nondimensionalized total sliding length (Ld_total∗) and the tangential Weber number (WeT) is discovered. The relationship between the four distinct post-impingement regimes identified in this work and the Weber and Bond numbers are presented and explained. These results offer fresh insight into the currently insufficient understanding of droplet sliding process on inclined hydrophobic surfaces.

Experiments on the breakup and evaporation of small droplets at high Weber number

Shock-driven multiphase mixing is present in numerous physical systems such as detonation-driven propulsion engines, liquid–vapor cloud explosions, and hypersonic flight droplet impacts. At the microscale, droplets experience deformation, breakup, and evaporation under extreme conditions (high Weber and Reynolds regimes). For small droplets, these phenomena are simultaneous and highly transient, making their interactions and interdependencies warrant further investigation. In this study, experiments are conducted in a shock tube facility to investigate these simultaneous droplet-scale phenomena. An interface consisting of small acetone droplets (⌀ 10–40 [μm]) is impulsively accelerated by a strong planar shock wave (Mach ∼2.09). The droplet size distribution is well-characterized in-situ utilizing a Phase Doppler Particle Analyzer (PDPA) and shadowgraphy. The development of child droplet clouds is captured through an ensemble of Mie scattering images. A simplified model is developed to interpret the experimental results, combining deformation, breakup, and evaporation models. The results indicate that the breakup of small droplets at high Weber numbers is likely dominated by the Rayleigh–Taylor (RT) mechanism, aligning with previous empirical models for low Weber numbers.

Dynamics of Droplets Impacting on Aerogel, Liquid Infused, and Liquid-Like Solid Surfaces.

Droplets impacting superhydrophobic surfaces have been extensively studied due to their compelling scientific insights and important industrial applications. In these cases, the commonly reported impact regime was that of complete rebound. This impact regime strongly depends on the nature of the superhydrophobic surface. Here, we report the dynamics of droplets impacting three hydrophobic slippery surfaces, which have fundamental differences in normal liquid adhesion and lateral static and kinetic liquid friction. For an air cushion-like (super)hydrophobic solid surface (Aerogel) with low adhesion and low static and low kinetic friction, complete rebound can start at a very low Weber (We) number (∼1). For slippery liquid-infused porous (SLIP) surfaces with high adhesion and low static and low kinetic friction, complete rebound only occurs at a much higher We number (>5). For a slippery omniphobic covalently attached liquid-like (SOCAL) solid surface, with high adhesion and low static friction similar to SLIPS but higher kinetic friction, complete rebound was not observed, even for a We as high as 200. Furthermore, the droplet ejection volume after impacting the Aerogel surface is 100% across the whole range of We numbers tested compared to other surfaces. In contrast, droplet ejection for SLIPs was only observed consistently when the We was above 5-10. For SOCAL, 100% (or near 100%) ejection volume was not observed even at the highest We number tested here (∼200). This suggests that droplets impacting our (super)hydrophobic Aerogel and SLIPS lose less kinetic energy. These insights into the differences between normal adhesion and lateral friction properties can be used to inform the selection of surface properties to achieve the most desirable droplet impact characteristics to fulfill a wide range of applications, such as deicing, inkjet printing, and microelectronics.

Analysis of the Behavior of a Droplet that Impacts on Moving Surfaces at Low Weber Numbers

Analysis of the Behavior of a Droplet that Impacts on Moving Surfaces at Low Weber Numbers

Bulge formation of liquid film at the trailing edge: Scaling laws and particle removal assessment

We present numerical simulations and laboratory experiments to study bulge formation at the trailing edge of an inclined surface and its inhibitory effect on particle removal during surface cleaning. We investigate the spatial variations in liquid films near the trailing edge and find that the Weber number can be used as a dominant parameter to determine bulge occurrence over the trailing edge. We divide the film region near the trailing edge at which the bulge occurs into two: the region where the surface linearly grows and the surface tension is negligible, and the region where the surface tension force becomes dominant and the film surface is curved. In the investigated cases in which the Reynolds number is O(10) or greater, the viscous forces are negligible, which allows for the derivation of the scaling laws for the length of the two regions according to the condition that the bulge height scales with the capillary length. In the resulting scaling law, the length scales depend on the Froude number and the inclination angle. The proposed scaling law allows for the prediction of the bulge shape and the prediction agrees with the simulation results, particularly at low Weber numbers (i.e., We < 0.5). Moreover, we construct a particle removability map to assess the removal of particles of different sizes at specific locations on the substrate. The map reveals a reduction of the removability for small-size particles or particles located in the bulge.

Electrohydrodynamic instability and disintegration of low viscous liquid jet

The simple-jet mode can be developed from dripping, dripping faucet (transition), and jetting when subjected to a sufficient strong electric field caused by a needle-plated electrode. In the present work, the instability and disintegration of an electrified jet of anhydrous ethanol in the electrohydrodynamic simple-jet mode are visualized by a high-speed camera. A systematic analysis and description on the evolution of the breakup morphology of the charged jet in the simple-jet mode was carried out. In the dripping mode at low Weber number, the unstable simple-jet occurs, while the stable simple-jet mode happens as Weber number higher than critical value. In a dripping faucet and jetting regime, the simple-jet mode can easily take place, and jet undergoes typical breakup modes including varicose, whipping, whipping assisted bifurcation, and ramified instabilities as electric potential increases. The growth rate of the maximum perturbation of the zero-, first-, and second-order surface waves determines the type of the instabilities. The operating window of the simple-jet mode is presented to indicate that the simple-jet mode only operates in the suitable range of Weber number and electric Bond number. The spray characteristics, including envelope angle, droplet size, and the stable length of the electrified jet, have been explored to demonstrate that the uniform drops could be generated in the simple-jet regime. The evolution from the cone-jet to the simple-jet mode is also observed when a special hemispherical nozzle is used. For an almost stable electric potential, the cone-jet can gradually transform into the simple-jet with an increase in the liquid flow rate.

Flow and mixing characteristics of gas-liquid slug flow in a continuous Taylor-Couette flow reactor with narrow gap width

This work experimentally investigated flow characteristics and mixing properties of the two-phase flow in a continuous Taylor-Couette flow reactor (CTCR) with narrow gap width. In the device with a gap width of 0.5 mm, to the authors’ knowledge, the narrowest gap width device ever studied in CTCR systems, the present study revealed that slug flows with alternating gas and liquid phases, as observed in microchannels, appeared in a certain flow regime. This slug flow appeared mainly in the region of low rotational Reynolds and axial Weber numbers. The non-dimensional effective diffusion coefficient Deff/uL [-] of slug flow was also measured by a tracer response method. Deff/uL in the slug flow was on the order of 10−6 and it has been successfully found that ideal plug flow mixing characteristics could be achieved under this flow condition. Since the intensified and controllable mixing in the liquid phase due to Taylor vortex motion induced by the inner cylinder rotation is expected to be much better than the mixing in the liquid phase of slug flows in microchannels, this flow is expected to be one of the process intensification techniques that will outperform the performance of the conventional microreactors with microchannels.

Understanding the liquid jet break-up in various regimes at elevated pressure using a compressible VOF-LPT coupled framework

The present work implements a compressible Volume of Fluid (VOF) – Lagrangian Particle Tracking (LPT) coupled solver in OpenFOAM and utilizes it to simulate a liquid jet in crossflow (LJICF) numerically. This methodology helps accurately predict a complex primary breakup in the Eulerian framework and the secondary atomization of spherical droplets using a computationally efficient LPT method. The coupled solver with Adaptive Mesh Refinement (AMR) is rigorously validated for a liquid jet in crossflow at varying operating conditions – pressure, crossflow velocity, and inlet liquid jet velocity. We have further carried out a thorough investigation to study the effect of momentum flux ratio and weber number on the various flow features and liquid jet break-up phenomenon in a crossflow while identifying the stream-wise location of the liquid jet breakup region. At low momentum flux ratios in the bag breakup regime, the predictions reveal that the liquid jet breakup occurs due to the growth of similar instability as usually observed in the high-speed liquid sheet atomization. The short wavelength assumption of the inviscid dispersion relation resembles the Kelvin-Helmholtz type instability observed in this case, as opposed to Rayleigh-Taylor instability at high momentum flux ratio in the surface breakup regime. It is also proposed that the shear breakup along the transverse edges of the liquid column occurs due to the shear layer instability of the air passing around the liquid column. The simulation wavelength closely matches the Williamson correlation for shear layer instability around cylinders – a shape similar to the cross-section of the bottom of the liquid column. The results show a distinct streamer or bifurcation phenomenon at low momentum flux ratios and moderate weber numbers. Further investigation suggests that the internal liquid boundary layer and the three-dimensional flow field behind the liquid jet are responsible for streamer formation.

Numerical Study on Spreading and Vaporization Process of Liquid Nitrogen Droplet Impinging on Heated Wall

Micro-structured surfaces can affect heat transfer mechanisms because of enlarged specific surface areas. However, employing the Leidenfrost effect during liquid nitrogen (LN2) droplet cooling of a heated micro-structured surface possessing a fin with a spacing much smaller than the diameter of the droplet has not yet been explored. In the present work, a direct numerical simulation (DNS) is carried out to investigate heat transfer mechanisms of the LN2 droplet, whose diameter is sufficiently larger than the structured spacing of fin, impinging on a micro-structured surface with variable velocities. For a comparative study, a smooth surface is also employed in numerical simulations. The spreading mechanisms and vaporization behavior of the droplet along with liquid film morphology at various conditions are investigated. Results show that a smaller fin size inhibits LN2 in entering into the grooves between the fins and left the surface untouched by the droplet completely, and eventually, a thinner liquid film is spread out in contrast to the smooth surface. Notably, at a low Weber number, the droplet can be shrunk or even rebounded away from the wall after impinging on the wall. The fastest vaporization behavior for both surfaces, namely smooth and micro-structured, is obtained at a Weber number of 180. Additionally, an effective heat transfer upon the micro-structured surface is observed at a low impinging velocity of the droplet.

Maximum spreading of droplet-particle collision covering a low Weber number regime and data-driven prediction model

In the present study, the maximum spreading diameter of a droplet impacting with a spherical particle is numerically studied for a wide range of impact conditions: Weber number (We) 0–110, Ohnesorge number (Oh) 0.001 3–0.786 9, equilibrium contact angle (θeqi) 20°–160°, and droplet-to-particle size ratio (Ω) 1/10–1/2. A total of 2600 collision cases are simulated to enable a systematic analysis and prepare a large dataset for the training of a data-driven prediction model. The effects of four impact parameters (We, Oh, θeqi, and Ω) on the maximum spreading diameter (β*max) are comprehensively analyzed, and particular attention is paid to the difference of β*max between the low and high Weber number regimes. A universal model for the prediction of β*max, as a function of We, Oh, θeqi, and Ω, is also proposed based on a deep neural network. It is shown that our data-driven model can predict the maximum spreading diameter well, showing an excellent agreement with the existing experimental results as well as our simulation dataset within a deviation range of ±10%.

Intense deformation and fragmentation of two droplet collision at high Weber numbers

Droplet collision is common in nature and industrial processes, while understanding of the phenomena at high Weber numbers is not as comprehensive as the one at low Weber numbers. We experimentally investigate the collision dynamics of two droplets at high We ranging from 200 to 2000 by adopting pure water and pure ethanol droplets. Four distinct droplet collision regimes in the order of increasing deformation and fragmentation are reflection separation, finger separation, stretching separation and splashing. Due to the larger inertia force and lower viscous force, water droplet collision is easier to form finger separation and splashing regimes than ethanol droplets under the same We. The reinforcement of viscous force and surface tension weakens droplet deformation and fragmentation, and further changes droplet collision regime. Due to the viscous dissipation ratio of droplet at the maximum spread moment as a function of Ohnesorge number (Oh), the model of the maximum spread factor of coalesced droplet is different under various Oh. For a low Oh, a new formula of the maximum spread factor is proposed by optimizing the maximum spread time in the viscous dissipation model based on energy conservation, which agrees well with the experiment. For a high Oh, the formula of the maximum spread factor is developed by using the viscous dissipation ratio, and it is consistent with the experiment as the viscous dissipation ratio is 0.8.

The characteristic length scales in direct mode of drop formation from the edges of spinning discs with different surface wetting characteristics

The characteristic length-scales responsible for primary and secondary drop size in direct droplet mode are probed using discs of different surface wettability through exposure controlled high resolution static imaging. The primary drop formation from the edge of a hydrophobic disc is found to be mechanistically different from its hydrophilic counterparts. The disc wetting factor (w) influences primary drop size only at low Weber number (We). A single mean primary/secondary drop size-We relationship represents discs of different surface characteristics. It points to insignificance of surface wettability. For hydrophilic discs the mean primary drop size and the mean incipient bulge size vary as We-0.47 and We-0.5 respectively, which hints at incipient bulge size dependence of primary drop size. The two capillary wave-based length scales tested as characteristic length scales for secondary drops show high size variability, which is a distinctive feature of secondary drops..

On the interfacial dynamics and capillary waves during impingement of a drop on liquid pool: A background-oriented schlieren study at low Weber numbers

Understanding the dynamics of a droplet impinging on a liquid pool and the associated phenomena have been of interest due to its prevalence in nature as well as in technical applications. This paper aims toward studying the characteristics of the capillary waves generated due to the low Weber number droplet interactions with the liquid pool. In this direction, experiments have been carried out for six different pool heights varying from h = 1.4 to 12 mm, encompassing thin liquid film, shallow pool, and deep pool regimes. Due to its wide usage, water has been chosen as the fluid of interest for droplet as well as for pool liquid. The study is focused on droplets impinging on the liquid pool at low Weber number ranging from 1 to 100. In order to characterize the post-impact perturbations in the liquid, background oriented schlieren (BOS) technique has been employed which offers real-time, non-intrusive whole-field measurements of the perturbations in the liquid pool. Measurements from BOS have been validated against the side-view projection of the impact. The transient variations of the air–water interface for different pool regimes and Weber numbers have been delineated. Results evince the formation of secondary wave at impact followed by the formation of primary wave after the crater retraction. The wave formation was faster and had higher amplitude in thin liquid regime for droplets with the same Weber number compared to the other regimes, but the perturbations were reduced through higher dissipation. The formation of the Worthington jet was seen in shallow and deep pool regimes for droplets with higher Weber number (We = 100), and its effect on the capillary wave is also discussed.

From impact to solidification in drop-on-demand metal additive manufacturing using MetalJet

Drop-on-demand metal jetting is a promising additive manufacturing (AM) technology that is gaining interest due to its capability to directly print complex single and multi-material components at high resolutions. It also has key advantages over other metal AM techniques, such as avoiding powder handling and extensive post-processing. In this method, parts are built via spatially controlled deposition of individual molten droplets onto a substrate. Therefore, the success of the process entirely depends on the behaviour of these single droplets from deposition to solidification including their interactions with the substrate, which is scarcely investigated to date. To fill this research gap, the in-house MetalJet platform was used to investigate the spreading and solidification of metallic micro-droplets at low Weber numbers. This was undertaken onto various substrates using a range of jetting and substrate temperatures through an integrated experimental, analytical, and computational approach. This study reports that increasing the substrate temperature enhanced the diffusion between the droplet and substrate, hence improving the bonding. Moreover, ripples forming on a droplet’s periphery during solidification disappeared at elevated substrate temperatures, resulting in improved inter-droplet bonding. Furthermore, the significant role of the substrate wettability and thermal properties, which control the droplet’s dynamics and solidification behaviour, respectively, is elucidated. This highlights the importance of substrate material selection using this technology. The results presented in this article underpin the optimal process conditions under which the 3D structures produced with this technology can exhibit reliable integrity and consistency. This represents a step forward in the direct metal printing of high resolution functional multi-material components.

Prediction of Newtonian Droplet Breaking Time from a Capillary at Low Weber Numbers.

Droplet formation and growth processes have numerous scientific and industrial applications. Experimental and numerical studies on the formation, growth, and breaking of droplet are carried out in present work. The numerical results are in good agreement with the experiment. This work focused on the effect of different Weber numbers (We) on the droplet breaking time. The results show when We < 0.05, the length and volume of the droplet increase, and the breaking time decreases rapidly. The resultant force acting on the main droplet suddenly drops around the critical breaking time. The difference rate between the time tn (when the resultant force is zero) and the breaking time tb is less than 8.49%. For the dimensional analysis of the numerical results, a prediction formula of breaking time on the Weber number is modeled as aWeb + c for We < 0.5.

Penetration characteristics of a liquid droplet impacting on a narrow gap: Experimental and numerical analysis

Experimentalists are limited in the amount of information they can derive from drop impact experiments on porous surfaces because of the short timescales involved and the normally opaque nature of porous materials. Numerical simulations can supplement experiments and provide researchers with previously unattainable information such as velocity and pressure profiles, and quantification of fluid volume flow rates into the pores. Ethanol drops, 2.0 mm in diameter, are impacted on a narrow gap at Weber numbers that match the impact of water drops, also 2.0 mm in diameter, on the same gap size in a previous study. The experiments show the ethanol drops cleaving at all Weber numbers tested, while the water drops completely enter the gap at low Weber numbers and only cleave at higher Weber numbers. A volume of fluid numerical model of the experiments is constructed in OpenFOAM and used to probe the interior of the drops during impact. For the water drop, a high-pressure region fills the drop during impact which continuously drives liquid into the gap. For the ethanol drops, the high-pressure region is smaller and quickly attenuates, which results in a near-zero vertical velocity at the entrance of the gap. Compared to water, the lower surface tension of ethanol causes these drops to spread further upon impact, recoil less, and overall have less liquid over the gap, which promotes cleaving. Against a superficial thought, when the penetration of liquids into porous materials is to be maximized, a higher surface tension liquid is therefore desirable.

Effect of phase change on jet atomization: a direct numerical simulation study

Atomization is often accompanied by phase change, which could significantly affect performance parameters such as the cooling efficiency and combustion efficiency of atomization. Nevertheless, the effect of phase change on jet atomization is rarely numerically studied due to the complexity of the coupling of the aerodynamics and the thermodynamics as well as the modelling difficulty caused by the cross-scale flow. In this study, comprehensive direct numerical simulations were carried out to evaluate the effects of phase change on the primary breakup and secondary atomization. Two methods dealing with phase-interface movement and mass transfer across the interface are built to meet the requirements of different modelling scales and Weber numbers. Simulation results indicate that phase change affects the flow behaviours and volume distribution of broken droplets in the primary breakup. In the secondary atomization, phase change leads to significantly different deforming morphologies of droplets with low Weber number and a more thorough breakup of droplets with high Weber number.