Dynamic behaviors of droplets impacting on ultrasonically vibrating surfaces

Journal Pre-proofs Dynamic behaviors of droplets impacting on ultrasonically vibrating surfaces Haixiang Zhang, Xiwen Zhang, Xian Yi, Feng He, Fenglei Niu, Pengfei Hao PII: DOI: Reference:

S0894-1777(19)30888-X https://doi.org/10.1016/j.expthermflusci.2019.110019 ETF 110019

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Experimental Thermal and Fluid Science

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3 June 2019 8 September 2019 5 December 2019

Please cite this article as: H. Zhang, X. Zhang, X. Yi, F. He, F. Niu, P. Hao, Dynamic behaviors of droplets impacting on ultrasonically vibrating surfaces, Experimental Thermal and Fluid Science (2019), doi: https:// doi.org/10.1016/j.expthermflusci.2019.110019

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Dynamic behaviors of droplets impacting on ultrasonically vibrating surfaces Haixiang Zhanga, Xiwen Zhanga, Xian Yib, Feng Hea, Fenglei Niuc and Pengfei Haoa * aDepartment

of Engineering Mechanics, Tsinghua University, Beijing, 100084, China

bKey

Laboratory of Icing and Anti/De-icing of Aerocraft, China Aerodynamics Research and Development Center, Mianyang Sichuan, 621000, China cSchool of Nuclear Science and Engineering, North China Electric Power University, Beijing, 102206, China Abstract Ultrasonic vibration has a wide application prospect in the fields of droplet atomization and aircraft anti/de-icing. The present work experimentally investigates the dynamic behaviors and antiicing characteristics of droplets, with a wide range of velocities, impacting on ultrasonically vibrating surfaces. Two kinds of splash mode are observed, including edge splash and surface splash, and a novel rebound mode, named sub-droplet rebound, is found in our experiments. The surface splash is mainly induced by the Faraday instability of surface capillary waves, and the appearance of a large number of bubbles proves that cavitation is also one of the influence factors. By analyzing the interaction of aerodynamic force and surface tension, the critical curve of the edge splash is obtained. The convergence of capillary waves at the apex contributes to the sub-droplet rebound. In addition, the influences of impact velocities and ultrasonic vibration amplitudes on the droplet spreading and the size distribution of secondary droplets are elucidated and discussed. A higher excitation amplitude results in a wider secondary droplet size distribution and a larger average size. We also investigate the dynamic process of supercooled water droplets impacting on ultrasonically vibrating surfaces with and without preformed frozen ice, respectively. The ultrasonic vibration could effectively promote the droplet splash and prevent the ice accumulation even in the state that the surface has been frozen. The anti-icing efficiency of the ultrasonic vibration increases with the increase of the ultrasonic vibration amplitude. The results of this work could provide insights for controlling droplet spreading, atomization, and anti-icing by using ultrasonic vibration. Key words: ultrasonic vibration, droplet, impacting, splash, anti-icing 1. Introduction The ultrasonic vibration could disintegrate liquid into fine droplets, which is known as ultrasonic atomization. [1, 2] [3] [4] [5] [6] This phenomenon has widespread application in diverse areas, such as spray cooling, nuclear reactors, ink-jet printing, pesticide dispersion, atmospheric studies, and de-icing on wind turbine blades or aircrafts. [7] [8] [9, 10] Since 19th century, many scholars have studied the mechanisms of the ultrasonic atomization[11-15] [16] [5]. There are two major hypotheses that explain the mechanism of liquid splash during ultrasonic atomization: capillary wave hypothesis and cavitation hypothesis. In 1831, Faraday firstly reported that the frequency of the surface waves is half of the excitation frequency[17]. Many studies have been carried out and believed that ultrasonic atomization work on the principle of faraday excitation wherein capillary waves are generated on the free surface of a liquid sheet[18, 19] [20] [21] [22-24] [25]. Lang experimentally evidenced that the mechanism of ultrasonic atomization involves the rupture of capillary surface waves and the subsequent ejection of the wave peaks from the surfaces as particles, and found the number-median diameter of the

particles is a constant fraction, 0.34, of the capillary wavelength[6]. Taking the effect of the liquidfilm thickness into account, Peskin and Raco obtained a correlation between drop size, frequency, amplitude, and liquid-film thickness, which is a modified form of Lang’s[26]. Afterwards, many other correlations were proposed by introducing the dimensionless numbers, incorporating the physico-chemical properties of the atomizing liquid and the operating parameters of the ultrasonic vibration[12, 15, 27]. Recently, Li and Umemura determined the surface-wave mode physically and studied the threshold condition for spray formation, and demonstrated that for a deep liquid layer, the threshold condition for the formation of a spray is determined only by the forcing strength. [20, 21] For a droplet rather than a liquid sheet, capillary wave hypothesis were usually employed in previous research [28] [29]. Numerous studies have been presented to demonstrate the dynamic behaviors of a static droplet, which is placed on an ultrasonically vibrating surface[11, 30, 31] [32, 33]. James et al. introduced different kinds of instabilities, droplet ejection, and drop bursting for a water drop placed on a vibrating surface, and found a well-defined critical value of the acceleration of the vibrating surface, above which small secondary droplets begin to be ejected from the freesurface wave crests[22, 23]. Deepu et al. experimentally investigated the dynamics of ultrasonic atomization of droplets, including the spreading, disintegration and atomization rate of droplets, and found that the spreading rate and the atomization rate are all very sensitive to changes in viscosity[11, 12, 33]. Based on the spherical Faraday instability, Liu et al. experimentally recorded the deformation and fragmentation process of a spherical droplet on a vertically vibrated plate with high speed camera, and validated that Lang’s equation is applicable to the spherical Faraday instability in a low frequency. [34, 35] The existing literature about the dynamics of ultrasonic atomization mainly focused on a liquid film or a static droplet, the dynamic behaviors of an impinging droplet on an ultrasonically vibrating surface are still unclear and remain to be studied sufficiently. Furthermore, ultrasonic vibration, one of the mechanical de-icing technologies, could remove adherent frozen water droplets from cold surface, which provides the possibilities for effective de-icing[36] [37] [38] [39] [40] [41] [42] [10]. Li and Chen experimentally studied the instantaneously shedding frozen water droplets from cold vertical surface by ultrasonic vibration, and proved the ultrasonic vibration is a highly potential deicing method for practical application. [43] [44] Palacios et al. conducted wind-tunnel testing of a NACA 0012 isotropic airfoil under the effects of ultrasonic excitation, and demonstrated the ultrasonic de-icing is a low-power approach, compared with the thermal de-icing systems. The enhanced de-icing ability was observed when employing multifrequency bursts in their subsequent study. [39] [45] In summary, ultrasonic vibration has been proved to be an effective de-icing approach[36] [46] [47] [7]. However, the anti-icing characteristic of supercooled droplets impacting an ultrasonically vibrating surface still needs further exploration. In this paper, we focus on the dynamic behaviors and anti-icing characteristics of impinging droplets with a wide range of velocities on ultrasonically vibrating surfaces with various amplitudes. We performed two sets of impact experiments at room temperature and subzero temperature, respectively. Two kinds of splash mode were observed, including edge-splash and surface-splash, and a novel rebound mode, named sub-droplet rebound, was found in our experiment. We explained the mechanisms behind each outcome in detail, and the critical curve of the edge splash was obtained by analyzing the interaction of aerodynamic force and surface tension. The appearance of large amount of bubbles during the collision process directly verified the cavitation hypothesis. We also 2

provided a simple analysis to elucidate and discuss the influence of impact velocities and amplitudes on the droplet spreading and the size distribution of secondary droplets. Furthermore, we also investigated the dynamic process of supercooled water droplets impacting on ultrasonically vibrating surfaces with and without preformed frozen ice, respectively, and demonstrated the antiicing efficiencies of the ultrasonic vibration. 2. Experimental Setup Two sets of impact experiments were performed in ambient environment, at room temperature (25±1℃) and subzero temperature (-7±2℃), respectively. The experimental setups are shown in Fig. 1a and 1b[48]. At room temperature, ultrapure water droplets (D0 = 2.6 mm ± 0.05 mm) with various velocities (U0 = 0.14 m/s ~ 4.47 m/s) were released from a fine capillary tube equipped with a syringe pump onto an ultrasonic vibration surface. The ultrapure water with electrical conductivity less than 0.1 μS/cm was generated by filtering and deionizing water using an ultrapure water system. The ultrasonic vibration device used in this experiment mainly consists of ultrasonic generator, piezoelectric transducer, amplitude transformer, and metal probe, as shown in Fig. 1c. The piezoelectric transducer could transform the ultrasonic electrical signal, produced by the ultrasonic generator, into ultrasonic mechanical vibration. Then, the amplitude of the ultrasonic mechanical vibration was amplified by the amplitude transformer, which also applied as the mechanical impedance converter. The frequency of the ultrasonic vibration device is a constant value (f = 27.99 KHz), and the amplitude (A0 = 5μm ~ 17μm) could be controlled by means of adjusting the input power of the signal transducer (see Fig. 1d). The metal probe of the ultrasonic vibration device was made of aluminum, and employed as the experimental substrate. In order to specify the wettability of the surface, the static contact angle was measured by using the standard contact angle goniometer (JC2000CD1, POWEREACH). The static contact angle was measured from the figure of a static drop, as Fig. 1e shows. The static contact angle of the ultrasonic probe surface is θ=82±1°, which was averaged from several measurements. A high-speed camera (FASTCAM Mini UX100, Photron) and two lenses were employed to measure the amplitude of various input power from side-view (frame rate: 32000fps, shutter speed: 1/50 000 s, high resolution tens: 1pix=1μm)and record the dynamical process of droplet impact from side-view (frame rate: 8000fps, shutter speed: 1/50 000 s, tele-macro lens: 1pix=20μm) and topview (frame rate: 16000fps, shutter speed: 1/50 000 s, tele-macro lens: 1pix=20μm). At subzero temperature, the impact experiments were done in a temperature controlled refrigerator, as shown in Fig. 1b. The deionized water was stored in a sterile syringe, which connected by a disposable sterile tube with a capillary tube. The most part of the disposable sterile was placed in the refrigerator, and the supercooled water droplets could generated by propelling the sterile syringe slowly. The experiments were performed by releasing supercooled water droplets (D0 = 2.6 mm ± 0.05 mm, T0 = -3℃ ± 1℃) on the ultrasonic vibration surface (Ts = -7℃ ± 2℃). An infrared thermometer and a temperature sensor were utilized to monitor the temperature of the supercooled water droplets and the ultrasonic vibration surface, respectively. The infrared thermometer was placed near the opening of the capillary tube, at which the supercooled water droplets were released, and the temperature sensor was attached on the side of the metal probe. Similarly, the dynamic and anti-icing process were recorded by the high-speed camera (FASTCAM Mini UX100, Photron, frame rate: 8000fps, shutter speed: 1/50 000 s, tele-macro lens: 1pix=20μm). Repeat experiments were performed at least three times at each condition. 3

Fig. 1. (a) Schematic diagram of the experimental setup at room temperature; (b) Schematic diagram of the experimental setup at subzero temperature; (c) The internal structure of the ultrasonic vibration device; (d) The relationship between the amplitude of the ultrasonic vibration and the input power of the signal transducer; (e) Droplet profile on the aluminum vibration substrate.

3. Results and Discussion 3.1 Impact dynamics of droplets on ultrasonically vibrating surfaces A droplet placed on an ultrasonically vibrating surface has been proved to burst into a fine spray of smaller secondary droplets or form a stable capillary wave on the surface[22, 23]. Nevertheless, we observed much more complicated phenomena, including stable capillary wave, surface splash, edge splash, sub-droplet rebound, etc., during the collision of droplets with ultrasonically vibrating surfaces. Fig. 2 illustrates the physical dynamics of droplets with different velocities impacting on ultrasonically vibrating surfaces with various amplitudes. At a low amplitude (A0=5μm), stable capillary waves appeared on the surface of the spreading liquid film, with few small secondary droplets slightly splashed from the surface. In the final phase, the droplet would undergo a retraction process and stand on the vibrating surface without any splash on the surface (shown in Fig. 2a). With the increase of the amplitude, the splash become more and more drastic, and the entire volume of the drop is almost ejected from the vibrating surface (see Fig. 2b, 2c and 2d). In addition, the edge splash also appears when droplets with high velocities first contact with the ultrasonically vibrating surface, as shown in the second snapshots of Fig. 2c and 2d. Notably, an intriguing phenomenon, named sub-droplet rebound, occurs under particular conditions, as shown in Fig. 2b and 2d. 4

Fig. 2. Impact dynamics of droplets on ultrasonically vibrating surfaces: (a) U0=1.4m/s, A0=5μm; (b) U0=1.4m/s, A0=14μm; (c) U0=4m/s, A0=8.43μm; (d) U0=4m/s, A0=17μm. The first and second lines of each image represent the selected high-speed snapshots from top-view and side-view, respectively. The snapshots at t=0.5ms in (c) and at t=0.25ms in (d) show the edge splash. The top-view snapshots at t=1ms in (b) and at t=0.5ms in (d) show amount of air bubbles generated at the inner of the droplet.

3.1.1 Capillary wave and cavitation There are two major hypotheses that explain the mechanism of liquid splash during ultrasonic atomization: cavitation hypothesis and capillary wave hypothesis[15]. For a static drop, most studies demonstrated that the splash is ultimately effected by surface wave breakup[11]. Cavitation hypothesis is generally applied to high frequency and high energy intensity systems as a random phase[49] [50] [51]. As shown in Fig. 2, both capillary waves and cavitation events are observed in our experiments, which directly suggests that both the mechanisms are simultaneously responsible for the splash in the collision process. Once the droplet contacts the vibrating surface, capillary waves will first appear at the edge of the liquid film, and gradually spread to the center. As the amplitude or the impact velocity increases, amount of air bubbles are generated at the inner of the droplet (see the top-view snapshots at t=1ms in Fig. 2b and at t=0.5ms in Fig. 2d). During the implosive collapse of these bubbles, high intensity hydraulic shocks are generated, which expedites the droplet splash. We present in Fig. 3 the possible patterns of droplet collisions on ultrasonically vibrating surfaces as a function of the impact velocity U0 and the amplitude of the ultrasonic vibration A0, including stable capillary wave, bursting (surface splash), edge splash, sub-droplet rebound, etc. 5

The details of these phenomena will be discussed in Section 3.1.2 and 3.1.3.

Fig. 3. The outcomes of droplet collisions on ultrasonically vibrating surfaces. The two sub-regions colored in yellow and cyan shadow indicate bursting (surface splash) and stable capillary wave, respectively. The black labels represent no-edge splash, and the blue labels represent edge splash. The square labels represent no-sub-drop rebound, and the pozidriv labels represent sub-droplet rebound. The red line represents the critical curve of the edge splash.

3.1.2 Two splash modes: surface splash and edge splash The drastic splash would appear under a higher amplitude, which was defined as bursting in the previous study [22]. For distinguishing with the edge splash, we called the phenomenon of bursting as surface splash. Many previous studies demonstrated that in order to break a droplet into small secondary droplets using ultrasonic vibration with a certain frequency, the amplitude of motion must exceed a certain limiting value Acrit [20] [52]:   0.33 (1) Acrit ~ ( )( )   f where, ρ is the liquid density, μ is the liquid viscosity, and σ is the surface tension. For a certain liquid (i.e. water in this work), the threshold amplitude Acrit only depends on the frequency of the ultrasonic vibration, which is also a constant. In our study, we also found that the critical state between stable wave and surface splash only relies on the amplitude of the ultrasonic vibration rather than the impact velocity (see the yellow and cyan shadow regions in Fig. 3). Nonetheless, once the splash has been taken place, the splash intensity is influenced by the combined effect of the amplitude and the impact velocity. The vertical vibration could induce the capillary waves with half of the excitation frequency [6]. The increase of the amplitude of the ultrasonic vibration leads to the rupture of the liquid column, and secondary droplets are jetted from the wave crest, which is known as the ‘Faraday instability’(see Fig. 4a). As mentioned in Section 3.1.1, cavitation events are observed in our experiments. Here, we believed that the transient cavitating bubbles could contribute to the intensity of the capillary waves, which results in the more intense surface splash. In particular, the higher vibration amplitude and the cavitation event could also result in significantly nonuniform distribution of the capillary waves and ligament corrugation, as Fig. 4c shows.

6

Fig. 4. (a) Schematic diagram the surface splash, the secondary droplets are generated by the rupture of the capillary waves; (b) Schematic diagram the edge splash, the rapid gas escape is the critical factor to induce the breakup of the lifting liquid film; (c) Schematic diagram of the nonuniform distribution of the capillary waves and the ligament corrugation.

Except for the surface splash induced by the Faraday instability and cavitation, the edge splash is also observed in our experiments. Commonly, a droplet will splash (i.e., break up at the rim and eject smaller pieces) when impacting a static solid surface at a high impact velocity[53, 54]. Similarly, we also observed the edge splash in the experiments (see the blue labels in Fig. 3). Compared with a static surface, the threshold value of impact velocity, above which droplets will splash at the edge, could be obviously reduced due to the vibrational dynamics of the ultrasonic probe. Various theories have been put forward to explore the mechanisms of the edge splash and try to obtain the critical criterion of droplet splash [55] [56] [57] [58] [59]. Except for the traditional instability theories, including Rayleigh-Taylor instability, Rayleigh-Plateau instability, and KelvinHelmhotlz instability, Riboux and Gordillo provided a new theory on the spreading and splashing of the impacting droplets, based on which other modified theories are also proposed [58] [60] [61] [59] [62]. In this work, the ultrasonic vibration could induce the gas vibration near the vibration surface, the splashing is largely affected by the gaseous atmosphere. Whether the K-H instability or the new theory proposed by Riboux and Gordillo, the droplet splash is dominated by the contribution coming from the overpressures of gas. The fast edge splash caused by the fast escape of gases between the impinging droplet and the vibrating surface (see Fig. 4b). Here we argue that the edge splash observed in our experiment is also induced by the balance of aerodynamic force ΣG and surface tension ΣL. The criterion for the droplet edge splash to occur was obtained as[55, 56]: G  L ~ P  M G

D0U 0 4 k BT

L 

(2)

where, P is the pressure of the gas around the droplet, γ is the adiabatic constant of the gas, MG is the gas molecular weight, kB is Boltzmann’s constant, T is the temperature, νL is the kinematic viscosity of liquid, and σ is the surface tension. In our experiments, the gas pressure is affected by the ultrasonic wave, and can be expressed as: P

2 G A0U C  Pa 2

(3)

here, Pa is the atmosphere pressure, UC is the acoustic velocity, ρG is the density of the gas. Notably, the acoustic pressure could also produce strong nonlinear effects. In this work, the temperature T, the diameter of the droplet D0, and the angular frequency ω are constants, i.e. the splashing threshold is mainly dependent on the impact velocity U0 and the amplitude of the ultrasonic probe A0. The criterion ΣG/ΣL can be represented as a function of U0 and A0:  Pa  G  L ~   

 M G D 0 L  4 k BT

 2  G U C   U 0      2

7

 M G D 0 L  4 k BT

  A0 

(4)

Therefore, the function of the critical curve can be derived as: A0  

U0  

(5)

where, α and β are fitting parameters. In this experiment, the values of the fitting parameters are α=-9×10-6 and β=1.8×10-5, respectively. The derived function of the critical curve (the red line in Fig. 3) can divide the experimental phenomena into two regions: one is the edge splash (ES) region (the blue labels in Fig. 3), another is the no-edge splash (n-ES) region (the black labels in Fig. 3). 3.1.3 Sub-droplet rebound Another intriguing phenomenon of sub-droplet rebound (DR) was found in our experiments, as shown in Fig. 2b. Similar to the Worthington jet, the sub-droplet rebound also occurs through the convergence of capillary waves at the apex [63]. As shown in Fig. 5, the capillary waves induced by the ultrasonic vibration converge from the edge to the center of the droplet, which is the main force to promote the droplet rebound.

Fig. 5. Schematic diagram of sub-droplet rebound. The black arrows in the first line and the yellow arrows in the second line represent the propagation of the capillary waves.

For the sub-droplet rebound to take place, two requirements need to be fulfilled simultaneously. The first one is that the capillary wave is strong enough to lift the sub-droplet, another is that the capillary wave could propagate to the center before the droplet spreads completely on the surface. The intensity of the wave is usually described by its energy flux density Ic:

1 (6) Ic  Lc2 A02Uc 2 where, ρL is the density of the water, ωc is the angular frequency of the capillary wave. Uc denotes the spreading speed of the capillary wave, and can be expressed as: U c  f c  c

(7)

The frequency of the capillary fc has been proved to be one half of the excitation frequency f, the wave length λc is obtained by the Kelvin’s equation:  c  (2   L f c2 )1 3

(8)

In terms of the energy conversion, the wave intensity is positively correlated with the input power of the ultrasonic vibration device and the kinetic energy of the falling droplet. The wave intensity mainly relies on the ultrasonic vibration amplitude and the impact velocity. Therefore, for a droplet with low impact velocity, with the increase of the impact velocity, the sub-droplet rebound could appear on the surface with a relatively lower vibration amplitude of the ultrasonic transducer tip. For a droplet with higher impact velocity, the higher spreading speed could restrain the propagation of the capillary wave, which may result in the consequence that the capillary wave 8

could not propagate to the center before the droplet spreads completely on the surface. Consequently, for a high-speed droplet, the threshold value of the critical amplitude increases with the increase of the impact velocity. With the increase of the impact velocity, the threshold value of the critical amplitude, above which the sub-droplet occurs, decreases first and then increases (see the black dotted line in Fig. 3). 3.2 Spreading factor of the droplet Many previous works have shown that the application of vertical vibrations on a solid surface could spread a static droplet, and the droplet increases its spreading on the surface as increases as the amplitude of vibration[32, 33]. A static droplet spreading to a thin film on an ultrasonically vibrating surface is dominated by the radial acoustic force of the ultrasonic standing wave. However, the collision process between a droplet and an ultrasonically vibrating surface is more complicated and dominated by the simultaneous effect of the inertia force, the viscous force, the surface tension and the acoustic force. In order to explore the influence of the ultrasonic vibration on the spreading of the impacting droplet, we define the maximum spreading factor βs= Dmax/D0, and Dmax denotes the maximum spreading diameter during the collision process. A universal scaling used to describe the maximum spreading of a liquid droplet impact on a static solid surface has been deduced by Lee et al., shown as [64]: (  s2   02 )1 2 Re 1 5  We1 2 (C  We1 2 )

(9)

where, β0 is the spreading factor at zero velocity (β0≈1.35 for the contact angle θ=82±1° of the vibration surface in this work), Re=U0D0/νL is the Reynolds number, We=ρLD0U02/σ is the Weber number, and C is the fitting parameter. Here, we also performed conventional impact experiments on the static surface (i.e. A0=0), and plotted the rescaling spreading factor with various vibration amplitude as a function of Weber number in Fig. 6a. The curve as predicted by Eq.9 describes the data of a static surface (A0=0) very well, and the fitting parameter C=8.3 is close to 7.6 obtained in [64]. At a low-amplitude (i.e. A0=5μm), the rescaled data are fitted by the rescaled correlation with a much larger standard error. Further, the rescaled correlation is not suitable for the data of high amplitudes, which means that except for the inertial force, surface tension and viscous force, the acoustic force induced by the ultrasonic vibration is also a main influence factor of droplet spreading.

Fig. 6. (a) The rescaled maximum spreading factor defined in [64] as a function of the Weber number in a log-log plot. The black line and the red line are fitting curves of the experiment data of A0=0 and A0=5μm, respectively. (b) Maximum spreading factor of impinging droplet with varied velocities as a function of the ultrasonic vibration amplitude. The green line indicates the most optimized amplitude Aop=8.4μm in the experiment. 9

Hence, we plotted the variation of the maximum spreading factor over the amplitude of the ultrasonic vibration in Fig. 6(b). The maximum spreading factor increases firstly and then decreases with the increasing amplitude at each impact velocity, which is distinctly different from a static droplet on a vibrating surface. At a certain impact velocity, a higher amplitude contributes to a stronger wave intensity, which results in a larger acoustic force. The acoustic force includes the radial acoustic radiation force, which pushes the droplet radially outwards and axial acoustic radiation force, which exerts an upward pull. The radial acoustic radiation force and the inertia force are all responsible on the spreading of the droplet. The rapid splash, including the edge splash and the surface splash, could significantly reduce the volume of the droplet during the spreading process, which results in a lower spreading factor. The most optimized amplitude, that promotes the spreading of the droplet, is Aop=8.4μm in our experiments. 3.3 Size distribution of the secondary droplets In order to investigate the size distribution of the secondary droplets, the image processing and analysis software---Image J was employed to measure the secondary droplet size. Fig. 7 shows the distribution frequency of the diameter of the secondary droplets, generated through vibrating a static droplet on the ultrasonic vibration surface. The secondary droplets could be recognized by means of the Image J software, as the two insets shows in Fig. 7. Fig. 7 indicates that a higher amplitude leads to a wider range of the size distribution of the secondary droplets.

Fig. 7. Distribution frequency of the diameter of the secondary droplets, generated by vibrating a static droplet on the ultrasonic probe. The two insets show the recognition of secondary droplets through the Image J software.

Utilizing the identical processing and analytical method, we could obtain the mean droplet size and the droplet-size distribution of the secondary droplets at various experiment conditions, as Fig. 8 and Fig. 9 illustrate. Similar to a static droplet, the average diameter of the secondary droplets of a falling droplet increases with the increasing amplitude of the ultrasonic vibration at each impact velocity (see Fig. 8a). While, the relationship between the size distribution and the impact velocity is more complex, the average diameter increases to the maximum and then decreases with the impact velocity (see Fig. 8b). Previous studies have shown that the mean diameter of the secondary droplets is related to the wavelength of the capillary wave [6] [34, 65], dm=ξ·λc, and the correlation coefficient ξ depends on the wave intensity. As analyzed in Section 3.1.3, the wave intensity Iw ∝ A02, i.e. the wave intensity depends only on the amplitude of the ultrasonic vibration at a certain vibration frequency and impact velocity. A higher amplitude contributes to a stronger wave intensity, which results in a larger droplet size. 10

Through modified the classic Lang’s equation [6], which ignored the effect of liquid thickness, the empirical formula was derived by Peskin and Raco [26]: 1

  A  h  3 dm  2    2 tanh  0   2 3   A0  L0 A0   dm  A0 

(10)

where, h is the liquid thickness on the vibration surface. If the liquid layer is deep enough, the empirical formula can transform into the classic Lang’s equation. This implicit function, which demonstrates the average diameter is inversely proportional to the thickness of the liquid film could help to explain the downward trend of the mean size with the impact velocity as shown in Fig. 8b. For an impinging droplet, a higher impact velocity results in a thinner spreading liquid film. The thickness of the liquid film have to be taken into account in this scenario. Therefore, the mean size of secondary droplets decreases with the increase of the impact velocity at the high-velocity region.

Fig. 8. (a) Average diameter of the secondary droplets as a function of the ultrasonic vibration amplitude under varied impact velocities. (b) Average diameter of the secondary droplets as a function of the impact velocity under varied ultrasonic vibration amplitude.

Furthermore, except for the mean droplet size, the droplet-size distribution is also importance for many applications. In the ideal case, the ligaments induced by the ultrasonic vibration are perfectly smooth, and homogeneously distribute at a certain wavelength (see Fig. 4a), leading to the uniform droplet size d~λc. Actually, the waves can be more or less spread in size, ligaments can be very corrugated, especially for an impinging droplet (see Fig. 4c). Therefore, the size distribution of the secondary droplets is determined by the ligament size distribution and the ligament corrugation. A number of previous studies have emphasized that gamma distributions are significantly better than other distributions (e.g. Poisson distribution and log-normal distribution) for fitting drop-size distribution data either in the spays or in the ultrasonic atomization[66, 67] [68]. The global droplet size distribution can be described by the two-parameter compound gamma distribution: ( mn )

( mn )

 d  2(mn) 2 x 2 Pm ,n  x    ( m) ( n ) dm  

1

 m  n (2 mnx )

(11)

where, dm is the average diameter of secondary droplets, the parameter m sets the order of the ligament size distribution and n the ligament corrugation, κm-n is a Bessel function of order m-n. A higher value of m indicates that ligaments are of more uniform size, while the low value of m indicates a broad dispersion of ligament size. Similarly, extremely corrugated ligaments correspond to lower values of n, inversely the most smooth ligaments would lead to n essentially infinite[67, 68]. Fig. 9 shows the rescaled secondary droplet-size distributions for various experimental conditions, 11

and the rescaled sizes are fitted with Eq. 11 to obtain the values of parameter m and n. As shown in Fig. 9a, the rescaled size for a low vibration amplitude (A0=7μm) with m=3.7, is much narrower distributed than the high vibration amplitude (A0=17μm) with m=1.46. On the one hand, the vibration amplitude affects the distribution of surface capillary waves, on the other hand, the cavitation effect also contributes to the nonuniform distribution of the capillary waves. At a certain vibration amplitude, the rescaled distributions with various impact velocities are almost identical with approximate parameter values, except for the much higher impact velocity (U0=4m/s), which leads to a thinner water film on the surface (see Fig. 9b). Compared to the capillary wave distributions, the effects of both vibration amplitude and impact velocity to the ligament corrugation are not sufficiently significant due to the secondary droplets are mainly induced by the rupture of the ligament rather than the ligament corrugation.

Fig. 9. The rescaled secondary droplet-size distributions for various experimental conditions. Each group of data is fitted by the two-parameter Gamma distribution Pm,n, and the solid line with same color represents the fitted curve. (a) Various ultrasonic vibration amplitude at a certain impact velocity, U0=1.41m/s; (b) Various impacte velocities at a certain ultrasonic vibration amplitude, A0=11.3μm.

3.4 Anti-icing characteristic of the ultrasonic vibration So far, a number of techniques used to anti-ice and/or de-ice the airfoils have been developed and tested, including antifreeze chemicals, surface coating, electrical resistance heating, pulse electrothermal de-icing, hot air circulation and manual chip-off, etc. [36] The heating method can remove ice effectively but consume a lot of energy. The superhydrophobic surfaces, low cost and no special lightning protection needed, are usually employed to prevent ice formation, because of their ability to promote the rebound and splash of impinging droplets[48]. However, once the ice nucleation occurs, the superhydrophobic surface would lose its anti-icing characteristic. The rapid splash of the impinging droplets, which doesn’t rely on the hydrophobicity of the surface, provides the potential possibility of the anti-icing characteristics of the ultrasonic vibration. Therefore, we performed the impact experiments of supercooled droplets (details of the experiments have been described in Section 2) and discussed the anti-icing characteristic of the ultrasonic vibration. Fig. 10 illustrates the impact behaviors between a supercooled water droplet (T0=-3℃, D0=2.6mm, U0=1.41m/s) and cooled ultrasonically vibrating surfaces (Ts= -8℃). On a surface with a low vibration amplitude, the impinging droplet adheres to the surface with a slight splash, and is gradually froze on the surface (see Fig. 10a). At a higher ultrasonic amplitude, the main part of the droplet could splash from the surface, only a thin liquid film is left on the surface (as shown in Fig. 10b). The ultrasonic vibration could effectively remove the impinging supercooled water droplets. In order to further explore the continuable anti-icing ability of the ultrasonic vibration, we also performed the impact experiment on the surface with a piece of preformed ice (shown in Fig. 10c). 12

Even though the surface has been iced, the ultrasonic vibration is still able to remove the impinging droplets, which is more applicable than the surface coating technology.

Fig. 10. Impact behaviors between a supercooled water droplet (T0=-3℃, D0=2.6mm, U0=1.41m/s) and cooled ultrasonically vibrating surfaces (Ts=-8℃). (a) A0=7μm; (b) A0=14μm; (c) A0=14μm, and the surface is with a piece of preformed ice.

Furthermore, the anti-icing efficiency of the ultrasonic vibration could also be quantificationally evaluated. Here, the anti-icing efficiency ε is defined as:

  (1 

Vi )  100% 1.11* V w

(12)

where, Vw is the volume of the falling droplet, Vi is the volume of the residual ice on the surface. Fig. 11 shows the anti-icing efficiency of the ultrasonic vibration with various amplitudes under different impact velocities. The amplitude of the ultrasonic vibration plays a significant role on the anti-icing efficiency of the ultrasonic vibration. The anti-icing efficiency increases with the increase of the amplitude of the ultrasonic vibration, and a higher impact velocity also leads to a slightly higher anti-icing efficiency. U0=1.41m/s

Anti-icing efficiency

80%

85%

U0=2m/s

76%

60%

51.2% 45.5% 40%

20%

16%

13%

7

11.29

17

Amplitude (μm)

Fig. 11. Anti-icing efficiency of the ultrasonic vibration with various amplitudes under different impact velocities.

4. Conclusions 13

To conclude, this work unveils the dynamic behaviors of droplets impacting on ultrasonically vibrating aluminum surfaces and provides an exploration on the anti-icing characteristic of the ultrasonic vibration for the first time. We presented two kinds of splash mode, including edge splash and surface splash, and found a novel sub-droplet rebound, caused by the convergence of capillary waves at the apex. According to analyzing the Faraday instability of surface capillary waves and the balance of the aerodynamic force and surface tension, the mechanisms behind the surface splash and edge splash are revealed. The transient cavitating bubbles observed between the droplet and the vibrating surface could also contribute to the intensity of the capillary waves, and the critical curve of the edge splash is obtained. The acoustic force induced by the ultrasonic vibration is also a main influence factor of droplet spreading, and the maximum spreading factor increases firstly and then decreases with the increasing amplitude at each impact velocity. A higher excitation amplitude results in a wider secondary droplet size distribution and a larger average size. As the impact velocity increases, the average size of the secondary droplets first increases and then decreases. The nonuniform distribution of the capillary waves is the key factor to determine the size distribution of secondary droplets, and the size distribution satisfies the Gamma distribution. More significantly, the ultrasonic vibration is able to effectively and continuously remove the impinging supercooled water droplets, even though the surface has been iced. The anti-icing efficiency of the ultrasonic vibration could reach more than 80% at a higher excitation amplitude, which provides the potential possibility of the application of the ultrasonic vibration in the anti-icing field. The results of this work could provide insights for controlling droplet spreading, atomization, and anti-icing by using the ultrasonic vibration. Acknowledgments This work was supported by the National Key R&D Program of China (Grant Nos. 2016YFC1100300 and 2017YFC0111100), the National Natural Science Foundation of China (Grant Nos.11635005 and 11632009), and the Open Subject of Key Laboratory of Icing and Anti/De-icing of Aerocraft of China (Grant No.AIADL20180101). References 1. Gholampour, N., D. Brian, and M. Eslamian, Tailoring Characteristics of PEDOT:PSS Coated on Glass and Plastics by Ultrasonic Substrate Vibration Post Treatment. Coatings, 2018. 8(10). 2. Yeo, L.Y. and J.R. Friend, Ultrafast microfluidics using surface acoustic waves. Biomicrofluidics, 2009. 3(1): p. 12002. 3. Barreras, F., H. Amaveda, and A. Lozano, Transient high-frequency ultrasonic water atomization. Experiments in Fluids, 2002. 33(3): p. 405-413. 4. Fogler, H.S. and K.D. Timmerhaus, Ultrasonic Atomization Studies. The Journal of the Acoustical Society of America, 1966. 39(3): p. 515-518. 5. Gaete-Garreton, L., et al., Ultrasonic atomization of distilled water. J Acoust Soc Am, 2018. 144(1): p. 222. 6. Lang, R.J., Ultrasonic Atomization of Liquids. The Journal of the Acoustical Society of America, 1962. 34(1): p. 6-8. 7. Gao, P., et al., Study on droplet freezing characteristic by ultrasonic. Heat and Mass Transfer, 2016. 53(5): p. 1725-1734. 8. O'Sullivan, J.J., et al., Atomisation technologies used in spray drying in the dairy industry: A review. Journal of Food Engineering, 2019. 243: p. 57-69. 14

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17

1. 2. 3. 4.

Highlights The dynamic behaviors of droplets impacting on the ultrasonically vibrating surfaces are revealed for the first time. A novel phenomenon of sub-droplet rebound is found in the experiment. Based on the balance of the aerodynamic force and surface tension, the critical curve of the edge splash is put forward. The potential anti-icing characteristics of the ultrasonic vibration are originally evaluated.

18