Evolution of cavity size and energy conversion due to droplet impact on a water surface

Wan Xu ,Dekui Yuan,Hongguang Sun ,Tong Guo ,Fengze Zhao ,Huimin Ma ,Changgen Liu

1 Department of Mechanics, Tianjin University, Tianjin 300350, China

2 State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China

3 State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China

ABSTRACT The deformation characteristics of the cavity due to droplet impact on a water surface are experimentally investigated.Dimensional analysis shows that the characteristic values of time,depth,and horizontal diameter can be taken as 10-3 times the ratio of surface tension to the product of viscosity coefficient and gravitational acceleration,the maximum depth,and the maximum horizontal diameter,respectively.The evolutions of the dimensionless cavity sizes for different values of Weber number (We) coincide for 220 < We <686.A partial-sphere model of cavity is established based on experimental observations.Energy models are then derived,and the energy conversions are calculated to identify the relationship between these conversions and cavity deformation.It is found that the kinetic energy model established under the hypothesis proposed by Leng is no longer applicable when the dimensionless time t* <3.5,owing to deviations from the geometric model.

1.Introduction

Droplet impact on a liquid surface is a common phenomenon in science and engineering.In nature,the splashing of raindrops is important for soil erosion with dispersal of seeds and microorganisms[1].In the metallurgical industry,droplet impact is of interest in a variety of processes,such as metal casting and spray coating[2,3].In agricultural irrigation,the atomization of impacted droplets is responsible for uniformity [4].In addition,droplet impact is witnessed in many other areas,such as inkjet printing[5],spray cooling[6],fire extinguishing,and meteorite impact.Further study of the characteristics and mechanism of this phenomenon is therefore of great importance.

A series of phenomena,including cavity,crown,central jet,secondary droplet,and secondary central jet formation can be seen as a result of droplet impact.Worthington [7] studied the phenomenon of droplet impact on a solid surface for the first time.He performed experiments with mercury and milk droplets falling from different heights and hitting a smoked glass plate,and he plotted the characteristics under different conditions.Another experiment was carried out by Worthington and Cole[8] to study the splashing that occurred after droplet impact on a liquid surface.They obtained images of the cavity and central jet by using the then novel technique of short-exposure photography.Rodriguez and Mesler[9]observed the phenomena of droplet floating,bouncing,coalescence,splashing,and vortex ring formation in a series of experiments and pointed out the existence of a boundary between splashing and vortex ring formation.From a series of experiments on 2.3-mm-diameter droplets,Rein [10] concluded that the transition phenomena between coalescence and splashing included a thin high-speed jet,bubble entrapment,and a thick central jet.Edgerton and Killian [11] found that the region above bubble entrapment included a splashing crown and a thick jet,with one or two large drops becoming detached.The formation of a vortex ring was first studied by Chapman and Critchlow [12],who concluded that the travel depth of the vortex ring depended on the impact velocity.Zheng [13] suggested that the formation of vortex ring was related to droplet vibration and presented an equation for the droplet falling height under this circumstance.Cresswell and Morton [14] argued that the viscous resistance at the free surface was the main cause of vortex ring formation.Moreover,Riobooet al.[15] focused on crown formation and performed experiments at different Weber numbersWeand Ohnesorge numbersOh,from which they were able to determine the crown–splash and deposition–crown limits.Okawaet al.[16]studied the number of secondary droplets detached from therim of the crown and pointed that the ratio of the mass of the secondary droplets to the initial droplet was in the range 0–1.Fanet al.[17] conducted experiments on water droplet impacting on high-temperature oil surface and concluded three typical regimes,including secondary jet,bubble and vapor explosion.Dhuperet al.[18]focused on jet dynamics and observed four different jet regimes such as short jet,singular jet,slender jet,and compound thin thick jet,which were closely related to the collapse of cavities.

The formation and development of a cavity have also been widely investigated,since they play an important role in the impingement process.By performing a regression analysis on experimental data,Oguz and Prosperetti [19] found that the time to reach the maximum cavity depth was proportional to drop diameter times drop velocity to the third power.Van Hinsberget al.[20] used an optical sensor to monitor the thickness of the liquid film between the cavity and a solid wall to further analyze the effects of various parameters on the time evolution of cavity depth.Furthermore,dimensionless numbers were also taken into consideration.Castillo-Orozcoet al.[21] presented the evolution of cavity depth and central jet height for different values ofWeandOh,while Dhuperet al.[18] discussed the evolution of cavity sizes at different Froude numbersFr.Maet al.[22] and Guoet al.[23] concluded that the maximum cavity depth and horizontal diameter were both correlated linearly withWe.

To obtain a better understanding of the mechanism of droplet impingement,a number of studies have been conducted on energy conversion during this process.The gravitational potential energy and the surface energy of the crown and the cavity at the maximum cavity expansion were derived by Engel [24].He assumed that the sum of gravitational potential energy and surface energy of the targeted liquid was equal to half of the initial droplet kinetic energy and obtained an equation for the maximum cavity depth that gave values in good agreement with experimental results.Pumphrey and Elmore[25]derived an equation for the maximum cavity diameter by assuming that the initial droplet kinetic energy was fully converted into gravitational potential energy of the cavity.However,Leng[2]argued that only 28%of the kinetic energy of the impacting drop was converted into gravitational potential energy at the maximum cavity.Fedorchenko and Wang [26] considered the surface energy on the basis of Pumphrey and Elmore’s research [25] and also derived an equation for the maximum cavity depth,although errors were still present because the energy of the crown and the viscous dissipation energy were not considered.In addition,Xuet al.[27]studied the energy conversion at different pool temperatures and found that the conversion rate of cavity energy to central jet energy increased with the temperature.Maet al.[28] pointed that the total energy loss of the processes involved increased as the initial droplet kinetic energy increased.Efforts have been made to study the energy conversion involved in the impact process,but further work is still needed since no consensus has yet been reached.

The complexity of this impact process means that there is still insufficient understanding of the characteristics and mechanisms of the various phenomena involved.Although the evolution of cavity sizes and the characteristics of the cavity at its maximum expansion have been thoroughly studied,studies on the quantitative relationship between cavity sizes and time are still lacking.Therefore,the aim of the experimental study reported here is to investigate the quantitative relationship between cavity dimensions and time,and furthermore to establish energy models for the cavity and discuss the energy conversion during impact.By exploring the similarity of cavity evolutions under different parameters,the current study could lay a foundation for future researches and applications.

2.Experimental Setup and Method

The experimental setup for a single droplet impacting on a water surface is shown in Fig.1.The setup included a glass water tank(of dimensions 400 mm×400 mm×400 mm)filled with distilled water,an LED background light source,a metal shelf,a peristaltic pump controlling the liquid flow accurately,a needle providing initial droplets,a computer,a synchronizer,and two high-speed cameras (SpeedSense 9072,Dantec Dynamics,Germany),each with a frame rate of 1000 Hz and a resolution of 1280 pixels × 800 pixels.The two cameras were placed orthogonally so that the average measured values could be used.Two sets of experiments were performed,with both cameras being used in the first set (Experiment 1 [22]),and only one camera (#1) in the second (Experiment 2 [23]).

Studies have shown that the entire impact process is affected by the Weber number [22,23,28],which represents the ratio of inertial force to surface tension and is defined as

where ρ and σ are the density and surface tension of the experimental liquid,anddand v are the diameter and impact velocity of the initial droplet.To investigate the influence ofWeon the impingement process,a total of eight different falling heights were set up in Experiment 1,in which 10 groups of experiments were carried out at each height.In Experiment 2,three different falling heights were chosen,and three groups of experiments were conducted at each height.The experimental parameters can be found in Table 1.In Experiment 1,the liquid depth was 163 mm and the environment temperature was 28.0 °C,while in Experiment 2,the liquid depth was 40 mm and the environment temperature was 17.0 °C.The atmospheric pressure was standard atmospheric pressure.

To vary the initial droplet impact velocity,the experiments were conducted by releasing droplets from different heights into the water tank.The impact velocity could be obtained by measuring the relative displacement of the droplet in two images prior to the moment of impact and dividing by the time interval of 3 ms between them.The droplet diameter could be adjusted by the needle diameter and the peristaltic pump,but could not be directly obtained.A calibration target with a dot pitch of 2 mm was used to provide the calibration coefficient.The geometrical sizes were obtained by multiplying the calibration coefficient by the corresponding sizes in pixels,which were obtained from an image processing algorithm.By using the average measured values from both high-speed cameras as the final data,the experimental error could be reduced.Each group of experiments was carried out quickly,and the distilled water was replaced after each group,to minimize the influence of surface impurities on the experimental results.

3.Evolution of Cavity Sizes

In this study,the three typical phenomena of cavity,central jet,and secondary droplet formation were observed after droplet impact on the water surface.IfWeis large enough,further phenomena such as a secondary central jet and second-secondary droplets can be observed.The present study focuses on the characteristics of the cavity phenomenon.

3.1.Characteristics of cavity shape and related experimental observations

The experimental observations indicated that within the range of Weber numbers studied (200

Given the limited ranges of the parameters,the experimental observations in this study are not comprehensive.However,phenomena at lowWe(We<200)were observed by Leng[2]in similar experiments on droplet impact.That study showed cavity behavior at lowWediffered in some respects from that at higher values.First,the droplet did not form a splashing crown after impact.Second,the cavity was conical in shape during its contraction,which resulted in bubble entrapment.Third,a thin high-speed jet wasformed,with several small droplets pinching off.In addition,the duration of the whole process was shorter at lowWe.

3.2.Approximate geometric model of cavity

Based on the experimental observations,it is assumed that the appropriate geometric model of the cavity is a partial sphere [29].The geometry and the characteristic sizes are shown in Fig.3(a),whereLis the cavity’s horizontal size (diameter),His its vertical size (depth),Ris the radius of the sphere,RCis the distance from the center of the sphere to any point on the cavity wall,and θ is the angle between the line from the edge of the cavity to the center of the sphere and the horizontal plane.L,H,andRCare all obtained from experimental data,andRis calculated fromLandH.

The cavity depthHand horizontal diameterL,and the distancesRCfrom the center of the sphere to any five points on the cavity wall were measured with a time interval of 0.005 s,for four different values ofWe(220,340,487,and 621) in Experiment 1 and three different values (287,469,and 668) in Experiment 2.Values ofRC/Rat different moments were calculated,and a statistical analysis was performed,as illustrated in Fig.3(b).A total of 175 data with a mean value of 0.9986 and a standard deviation of 0.0226 were obtained from Experiment 1,and a total of 110 data with a mean value of 0.9917 and a standard deviation of 0.0248 were obtained from Experiment 2.Therefore,the partial-sphere model of the cavity can be considered reasonable.

3.3.Evolution of cavity dimensional sizes for different We

To analyze the quantitative relationship between cavity sizes and time,eight values ofWe(220,280,340,412,487,567,621,and 686) were selected from Experiment 1 and three values(287,469,and 668) from Experiment 2.Cavity depthHand horizontal diameterLwere measured during the impact process with an interval of 0.001 s.Measurements were started at the moment of impact and stopped when the central jet appeared.All measuredvalues of cavity sizes are plotted against time for different values ofWein Fig.4.

Fig.3. Geometric model of cavity and its verification:(a)partial-sphere model and(b)statistical analysis on RC/R.Groups from Experiment 1 are unmarked and groups from Experiment 2 are marked with an asterisk.

Fig.4(a) shows the variation of cavity depth with time during the process of cavity deformation.It can be seen that the depth increases with time until it reaches a maximum value,and then it decreases gradually,but it is always greater than 0 mm since the cavity still exists when the timing stops.In addition,the slopes of the curves indicate that the contraction of the cavity is slower than its expansion,which can be attributed to the energy dissipation caused by the generation and propagation of capillary waves.Furthermore,the rate of cavity expansion and the maximum cavity depth,as well as the duration of the cavity,increase asWeincreases.

Fig.4(b) shows the evolution of cavity horizontal diameter for different values ofWe.The horizontal diameter also increases with time,and the rate gradually slows down,but it does not decrease consistently in the contraction process.Aftert=0.025 s,it is obvious that the horizontal diameter is fluctuating,which can be attributed to the propagation of capillary waves.Additionally,the findings show that the rate of diameter change,the maximum horizontal diameter,and the duration of the cavity increase with increasingWe.

From Fig.4,it can also be seen that data from Experiment 2 shows the same changes in pattern.However,the duration of the cavity is shorter,and the horizontal diameter gradually increases in the later stage of contraction,which is mainly because the temperature and the dynamic viscosity of the liquid are different in the two experiments.In fact,further discussion in Section 3.4 will illustrate that after selection of reasonable characteristic values,the variations in dimensionless sizes from the two experiments are quantitatively the same.

The results show that the maximum vertical size and the maximum horizontal size of the cavity both increase with increasingWewithin the experimental range (220

3.4.Evolution of cavity dimensionless sizes for different We

The variations of the cavity dimensional sizes with time exhibit clear regularities,as shown in Fig.4.If the appropriate dimensionless parameters are selected,a unified relationship between cavity sizes and time may be obtained.Thus,the vertical and horizontal sizes of the cavity,as well as the time,are nondimensionalized,to allow an analysis of the relationship between dimensionless sizes and dimensionless time.

According to the present experiment results and those in the literature [1,2,8],the process of a droplet impacting on a liquid surface is related to the Weber numberWe=ρdv2/σ,the Reynolds numberRe=ρdv/μ,and the Froude numberFr=v2/gd.Based on the π theorem for dimensional analysis,with σ,μ,andgchosen as the fundamental quantities,the characteristic time can be derived ast′=σ/μg.Considering that the magnitude relationship between cavity sizes and time should be consistent,the characteristic time is taken as

Furthermore,it has been noted that the maximum cavity dimensions vary withWe,and thus the characteristic vertical and horizontal sizes can be respectively taken as

As a result,the dimensionless time,dimensionless depth,and dimensionless horizontal diameter are respectively

where σ and μ are the surface tension and the dynamic viscosity of the experimental liquid,andgis the gravitational acceleration.HmaxandLmaxare the maximum cavity depth and maximum cavity horizontal diameter,respectively.From Eq.(5),it can be seen that the dimensionless time can also be expressed as

wheredand v are the diameter and impact velocity of the initial droplet.Eq.(8) indicates that the impact process is jointly controlled byWe,Re,andFr,which is in agreement with other studies.

Fig.5. Evolutions of cavity dimensionless sizes for different values of We: (a) cavity depth;(b) cavity horizontal diameter.Groups from Experiment 1 are unmarked and groups from Experiment 2 are marked with an asterisk.Data from Experiment 2 are included for verification.

Similarly,the dimensionless cavity depths and the dimensionless cavity horizontal diameters from Experiment 1 are plotted against the dimensionless time in Fig.5,respectively,forWe=220,280,340,412,487,567,621,and 686.It can be seen that,ignoring the propagation stage of capillary waves,the eight curves for different values ofWealmost fall on the same dimensionless curve.In other words,it can be concluded that the evolutions of the dimensionless cavity sizes for different values ofWecoincide when 220

Fig.5 also suggests that the curves of dimensionless horizontal diameter diverge earlier than those of dimensionless depth,which is attributable to the presence of capillary waves.These waves first appear at the rim of the cavity when the annular film of the crown spreads outward,and they then propagate to the base of the cavity.This propagation of capillary waves affects the deformation of the cavity,resulting in the unstable changes in cavity sizes and the divergence of the curves in Fig.5.Consequently,the divergence time of the dimensionless horizontal diameter does not coincide with that of the dimensionless depth.

In order to facilitate subsequent analysis,the least squares method is used to fit all the data of Experiment 1,and the fitted curve is plotted in Fig.5(a).The regression equation is

The fitted dimensionless curve of horizontal diameter against time is plotted in Fig.5(b),and the corresponding regression equation is

The fitted curves are in poorer agreement with experimental results during the propagation of capillary waves.It can therefore be concluded that within the experimental range(220

For data from Experiment 2,plots of the dimensionless cavity sizesvsdimensionless time are also shown in Fig.5.In Fig.5(a),it can be seen that the dimensionless depths from Experiment 2 more or less coincide with the other eight groups of data whent*<4.0,and from Fig.5(b)that the dimensionless horizontal diameters also coincide with the data from Experiment 1 whent*<3.5.Moreover,most of the data fall on the fitted curves,which confirms that the characteristic dimensions and characteristic time previously selected are appropriate.

It should be noticed that the current work is just to fit the curves in form,so as to reflect the similarity of cavity evolutions with different parameters.A unified model for the cavity evolution is presented in Fig.5 and can be simply described by Eqs.(9) and(10).At this stage,the empirical equations did describe the similarity of the different curves.However,the theoretical model and its specific physical mechanisms still require further researches to explore.

4.Energy Conversion of Cavity

The impact of a single droplet on a liquid surface is accompanied by conversion of various forms of energy,such as surface energy,gravitational potential energy,kinetic energy,and thermal energy.It has been shown that during the formation and development of a cavity,the main components of the total energy are the surface energy,gravitational potential energy,and kinetic energy[26,27],and hence the conversion of these three types of energy will be discussed here.

4.1.Surface energy, gravitational potential energy, and kinetic energy of cavity

On the basis of the partial-sphere geometric model of the cavity shown in Fig.3(a),the radius of the sphere can be expressed as

The surface energy is the product of the net generated surface area and the surface tension [20].Denoting the surface area of the initial liquid surface byA0and the surface area of the liquid surface during cavity deformation byA,the surface energy of the cavity is given by

The gravitational potential energy of the cavity is equivalent to the work required to lift the liquid content of the cavity to the original undisturbed liquid surface [21],which is expressed by

The kinetic energy of the cavity is the total kinetic energy of the fluid around the cavity.If the potential φ of the fluid is known,the kinetic energy of the cavity can be obtained as follows [2]:

where dSis an area element of the cavity surface.To derive the kinetic energy of the hemispherical cavity,Leng [2] assumed that the velocity potential of the surrounding fluid was φ=C/r,and the resultant velocity at the cavity wall was then given by dR/dt=C/R2.By solving forC,the velocity potential was deduced as

Substituting φ into Eq.(14)gives the kinetic energy of the hemispherical cavity.Based on Leng’s deduction,the kinetic energy of the partial-spherical cavity can be expressed as

It should be noted that when the kinetic energy is calculated according to Eq.(16),it can be derived from Eq.(11) that

where dH/dtand dL/dtare obtained from the regression Eqs.(9)and(10),respectively.Eq.(17)is the differential form of Eq.(11),meaning the rate of change ofRwith respect to time.

4.2.Variation of cavity dimensionless energy with dimensionless time

To discuss the relationship between the dimensionless energy and the dimensionless time,the surface energyS,gravitational potential energyP,and kinetic energyKof the cavity are calculated according to Eqs.(12),(13),and (16),respectively,forWe=220,340,487,and 621.The dimensionless time is obtained from Eq.(5) and the dimensionless energy by dividing the energy by the kinetic energy of the initial drop,which is given bywhere ρ is the density of the liquid,d0is the diameter of the initial drop,and v0is the impact velocity of the initial drop.Eq.(18)could be obtained from the kinetic energy theorem.The variations of the cavity dimensionless energiesS/K0,P/K0,andK/K0with dimensionless time are shown in Fig.6.

Fig.6. Evolutions of cavity dimensionless energies S/K0, P/K0,and K/K0 for: (a) We=220,(b) We=340,(c) We=487,and (d) We=621.

It can be seen from Fig.6 that there is continuous conversion of the cavity surface energy,gravitational potential energy,and kinetic energy after droplet impact on the water surface,which corresponds to the deformation of the cavity.In the initial stage of the expansion,the surface energy and gravitational potential energy increase slowly,while the kinetic energy increases rapidly,as the initial droplet enters the liquid.When the kinetic energy of the droplet is fully converted into the energy of the cavity,the surface energy and gravitational potential energy continue to increase,while the kinetic energy of the cavity begins to decrease.Then,at the maximum depth cavity,the surface energy and gravitational potential energy reach their peak values,and the kinetic energy reaches its minimum.

During contraction,the surface energy and gravitational potential energy decrease,while the kinetic energy increases.However,a sharp increase can be observed in the kinetic energy whent* >4,and the sum of the three forms of energy exceeds the kinetic energy of the initial droplet,which is obviously unreasonable.For the case ofWe=482,the deformation of the cavity whent* >4(t>35 ms) is shown in Fig.7.In the later stage of contraction,the vertical depth of the cavity rapidly decreases,while the horizontal diameter remains almost unchanged,resulting in a deviation from the assumed geometric model.Therefore,the calculated values ofRand dR/dtare too large,and the kinetic energy increases sharply to an excessively large value.Meanwhile,the variations of surface energy and gravitational potential energy are comparatively reasonable,since their calculations do not involve the value ofR.In other words,the models of the surface energy and gravitational potential energy are applicable to the whole process,whereas the current kinetic energy model is only applicable whent* <3.5,given that the curves begin to diverge whent* >3.5,as shown in Fig.5(b).

Fig.6 also illustrates that the total energy of the cavity at its maximum depth is around 50%–80% of the initial kinetic energy,and meanwhile the kinetic energy of the cavity is still greater than zero and is about 5%–10%of the initial kinetic energy.This is basically consistent with the numerical results obtained by Maet al.[28],and further verifies the regression equations and energy models described above.In addition,there is a decrease in the ratio of the total energy absorbed by the cavity to the initial kinetic energy asWeincreases,indicating that the energy dissipated during the impact process increases with the kinetic energy of the initial droplet.

The above analysis shows that the energy models deduced in this study have some deficiencies.With the assumption of a partial-spherical model,errors will occur in the calculations of the cavity energies when the cavity shape differs greatly from that of the model.Additionally,other forms of energy,such as the viscous dissipation energy and the energy spent on splash,are not considered in this study,and this can also introduce errors.

5.Conclusions

In this study,experimental data on a single droplet impacting on a water surface have been processed and analyzed in a dimensionless manner.Besides,energy models of the cavity have been theoretically deduced and energy conversions have been calculated.The main findings are as follows:

(1)When 200

(2) When the characteristic values of time,depth,and horizontal diameter are respectively taken asthe maximum depth,and the maximum horizontal diameter,the variations in the dimensionless cavity sizes with dimensionless time for different values ofWecoincide.A unified pattern of cavity evolution is presented and the regression equations are in good agreement with the experimental results whent* <3.5.

(3)During the impact process,the surface energy,gravitational potential energy,and kinetic energy of the cavity are continuously converted,which corresponds to deformation of the cavity.The velocity field model proposed by Leng [2] has been employed in the calculation of kinetic energy,but this is no longer applicable whent* <3.5 since the cavity shape clearly deviates from the partial-sphere model.In a follow-up study,it will be necessary to carry out further experiments and numerical simulations to obtain a more accurate model.

Data Availability

Data will be made available on request.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study has been supported by the National Natural Science Foundation of China(11872271)and the Major Scientific and Technological Projects of Tianjin (18ZXRHSF00270).