Frequency Specificity of Liquid-Fountain Swinging with Mist Generation: Effects of Ultrasonic Irradiation Angle

Abstract

Atomization of liquid into the air attained through submerged ultrasound irradiation will involve the formation of liquid fountain, which exhibits a sequence of oscillating and/or intermittent characteristics/events: its vertical/axial growth and breakup; its lateral “compound swinging”; and its associated dynamics of mist formation and spreading. This study attempts to provide a mechanistic view of ultrasonic atomization (UsA) process in terms of the swinging periodicity of water fountain and to specifically examine the influence of ultrasonic irradiation (i.e., transducer installation) angle on the liquid-fountain oscillations with mist generated intermittently. Through high-speed visualization, it was qualitatively found that as the extent of tilt (from the vertical direction) in the irradiation angle was increased, the degree of occurrence of mist generation and the amount of identifiable mist being generated tended to decrease. This trend was associated with reductions in both the growth rate and breakup frequency of the fountain on the tilt. It was further found, through the analysis of time variation in the resulting angle of liquid-fountain inclination, that the swinging fountain fluctuated periodically in an asymmetric manner and its periodicity could be fairly predicted based on a proposed simple “pendulum” model. An optimum value of the transducer installation angle was observed and judged to be 2° from the viewpoint of effective mist generation as well as fluid dynamic stability of the UsA liquid fountain.

1. Introduction

Ultrasonic atomization (UsA) techniques have been employed with an ultimate goal being a flexible green alternative for energy-efficient processes, including gas–liquid mass transfer enhancement/intensification [1,2], wastewater treatment [3,4], air purification [5,6,7,8], drug delivery [9,10], solvent–solute (or particles) separation [11,12,13], and/or its concentration [14,15].

Among those applications, a typical technique of installing the ultrasonic transducer is to set it, in the form of an oscillating disk, within a liquid and to irradiate ultrasonic wave almost vertically upward, but with a slight tilt [16,17]. In such a configuration, as the bulk liquid is irradiated at a high driving frequency (≈800 kHz or higher, [18]), a fountain of liquid will emerge from its free surface and micro- to nano-sized droplets could be generated—as mist—from the fountain.

If the installation angle/direction is exactly vertical, namely, the transducer’s flat disk parallel to (or its angle being 0° vs.) the horizontal free surface, the acoustic fountain will be disturbed rather profoundly, either directly by relatively large drops of liquid separated from the fountain tip and then falling or indirectly by the associated undulation of the surrounding free surface, e.g., [19], which ideally should be avoided. The mist of droplets, on the other hand, have been investigated as mass-transfer or aerosol carriers by many researchers, in particular, regarding the effects of a variety of operating parameters on the droplet size distribution (DSD), such as the driving frequency, oscillating-disk diameter, input power, liquid depth, liquid surface tension and viscosity, and the liquid-phase temperature, e.g., [11,20,21,22,23].

Whether tilted or not, the acoustic liquid fountain was proposed—based on a visualization study—to have a general configuration characterized by dynamics of the four-region structure consisting of (A) Foundation, (B) Bumpy-Surface, (C) Finely-Structured Surface and (D) Lumped-Crest Regions [19]. While such qualitative information needs to be clarified based on mechanistic support, it is quite a difficult task to reproduce or at least verify the fountain’s dynamic structure over a diversity of the above-said operational parameters to be taken into account.

In regard to the mechanistic description of the fountain structure, the information available in the literature, e.g., [24,25,26,27] is mostly limited to a mere precursory structure or mound-like fountain. Xu et al. [24] conducted numerical simulation to realize the shape of the acoustic fountain with different ultrasonic parameters, through rigorous formulation of the theoretical bases. A notable finding is that the steady-attained height of acoustic fountain “step-increased with an increase in the acoustic pressure,” with a step increment of half the ultrasound wavelength and the fountain surface tending to be stable over an “anti-resonant pressure” field. Kim et al. [25] experimentally investigated the mechanism controlling the stability of water protrusion; for better spatiotemporal resolution and control, particle image velocimetry (PIV) was applied to the induced flow field around the “ultrasonic focal spot,” resulting from focused ultrasound at two distinct frequencies of ultrasound, 0.55 and 1.1 MHz. By varying the pressure level of the transducers, they observed three different regimes, namely, weak, intermediate (stable) and highly forced (explosive) fountains, with the detection of “the fountain height undergoing a stepwise change between the regimes” as reported by Xu et al. [24].

More specific experimental findings were reported by Aikawa and Kudo [26], including the relationship between thresholds for free radical generation and atomization under ultrasound irradiation, which should provide useful insights into the structural variation of dynamic fountains to be examined in the present study. The former is closely related to (or an outcome of) the cavitation collapsing process; the latter to the shape transition of free surface from a protuberance to a fountain of aspect ratio greater than unity. They detected the essentially identical threshold level in the increased transducer driving voltage—or equivalently, input power density (of 2.5 W/cm²)—for all these four phenomena. Utilizing focused shadowgraphy, they further detected the spotty-shaped high-intensity nodes—created by the ultrasonic waves incident on and reflected from the “pressure-release” interface [28] of shape resembling a parabolic mirror; its convergence effect would promote the generation of cavitation bubbles inside the protuberance; the induction of cavitation promoted by such high-intensity nodes in turn could trigger the rapidly growing or “abruptly shape-transitioning” fountain that leads to the atomization.

When it comes to the structure and dynamics of acoustic fountains to be examined in this study, a typical fountain is no longer a mere protrusion, nor stable, and associated with mist spreading (out of the Finely-Structured Surface Region) as well as large drops and/or liquid ligaments formed as a result of tip breakup (the Lumped-Crest Region). Kawase et al. [29] and Yasuda et al. [30] used a 2.4-MHz ultrasonic transducer with installation angle of 5° at input powers of 20 and 18 W, respectively, to investigate the effects of carrier gas specifically and various operating conditions including the liquid properties on attaining efficient collection of UsA-concentrated alcohols. They provided a preferred flow mode of the carrier as well as some optimum conditions but no specific information regarding the fountain characteristics other than those of the mist as well as drops.

In this study, we apply high-speed imaging/image processing to observe/analyze detailed dynamics of the acoustic fountain by changing the angle of ultrasonic irradiation. The study involves mainly two objectives: to find an optimum irradiation angle, if it indeed exists, from the viewpoint of operability/stability of and effective mist generation out of the UsA fountain; and to quantify its periodicity both in the axial “growth-breakup” cycle and lateral “swinging.” The expected outcome of novelty lies in attempting to reveal factor(s) triggering mist emergence via experimental frequency analysis (fast Fourier transform) and theoretical prediction (pendulum model).

The paper is organized, after giving useful introductory literature information, as the experimental specifics on the system used, along with visual-image and time-series data analyses (Section 2); the resulting structural dynamics of the fountain both in the axial and lateral directions, followed by their frequency as well as time-series characterization, and modeling efforts—elaborated in the Appendix A—with its outcome in predicting the fountain-swinging periodicity (Section 3); and ends with itemized concluding remarks.

2. Experimental

The experimental setup used for visualizing the UsA process is shown in Figure 1. An ultrasonic transducer (Honda Electronics HM-2412, Toyohashi, Japan: 2.4 MHz) was placed on the bottom of a square vessel with its dimensions, 200 × 200 × 185 (height) mm. The transducer element (oscillating disk) itself was 20.0 mm in diameter, having an effective oscillating diameter of 16 mm. The installation or setting angle, ${\theta }_T$, of the ultrasonic transducer was tested over the range 0–10° with a 1° increment. The input power applied to the transducer was changed over 7–13 W (input power density examined then ranged 3.5–6.5—or three specific values of 3.5, 5.0, and 6.5—W/cm², which is exclusively used in this study as an operating parameter).

The distance from the center of the oscillator to the free surface of the liquid (effective liquid depth) was kept 20 mm, or 27.5 mm above the vessel bottom. The liquid used was ion-exchanged water (Merck Millipore Elix^® Essential UV3, Darmstadt, Germany); its temperature was adjusted, before each run, to 25 °C in a constant temperature bath. The liquid properties are presumed to be kept invariant during the UsA over a maximum 10-s period of operation.

2.1. High-Speed Visualization and Image Analysis

The visualization was carried out using a high-speed camera (RedLake MotionPro X4, Tokyo, Japan: frame rate(s) up to 5000 fps, with a resolution of at least 512 × 512 pixels, and an exposure time as short as 197 µs) under backlighting using a metal halide lamp (Lighterrace MID-25FC, Takatsuki, Japan). The imaging was started 5 s after the transducer was turned on, and a 1-s period of data was recorded. A sheet of light diffuser was installed between the liquid fountain and the lamp, thus reducing the non-uniformity of backlighting.

Figure 2 depicts—through a typical sequence of images—how two important “instantaneous” aspects of the liquid-fountain dynamics are determined visually: (1) the axial variations of its tip, confined by breakup and attained heights of the liquid fountain, and (2) its lateral fluctuations characterized by the fountain swinging. Note that the very initial period of 0.6 ms or so covers the non-periodic startup growth phase, which should be excluded from the cyclic growth-and-breakup phase (see the 2nd to 5th frames covering one cycle). An image analysis software (DITECT Dipp Motion) was used to track the breakup and attained liquid heights; the vertical distance spanned over the period between the moment of fountain breakup and the succeeding one signifies the growth rate of the fountain, while the number of breakup events per unit time defines the breakup frequency.

With the aid of a video analysis software (KEYENCE Movie Editor), the instantaneous state of swinging liquid fountain was specified in terms of its angle of inclination, ${\theta }_F$, as depicted in Figure 2 (see the far right): first, the line perpendicular to the transducer disk (tilted by ${\theta }_T$) is drawn, which is assigned to represent the “virtual axis” of the fountain emanating—acoustically radiating—from the center of the transducer; second, two tangential lines are drawn along the profile of the fountain surface to approximate the fountain outline as an equilateral triangle; the angle between the center line of the triangle and the virtual axis specifies the “net” inclined angle of fountain.

2.2. Time-Series Data Analysis

Once specified in its instantaneous state, or phase, the lateral dynamics of the acoustic fountain were analyzed in terms of time variation in its inclined angle ${\theta }_F=f\left(t\right)$. If the fountain exhibits a swinging motion, the resulting time-series data/signals $f\left(t\right)$ should contain periodic nature. Such data can be quantitatively processed based on the principle of frequency analysis. Since the current study aims to obtain the frequency characteristics—hopefully extracting the dominant frequency(ies) if any—the Fourier-transform analysis will be adopted; in its fast, discrete form of algorithm, fast Fourier transform (FFT), a given time-domain signal $f\left(t\right)$ is to be decomposed into a frequency-domain spectrum $F\left(\omega \right)$. In the FFT, the corresponding (to the former) sum of discrete elements, $f_{j+1}\left(t\right)$ $\left( j=0, 1, \dots , N-1\right)$, sampled at equal intervals were processed through a fast, efficient algorithm (MathWorks MATLAB^® 2019b) to obtain the corresponding (to the latter) spectrum:

$$F_{k+1}\left(\omega \right)={{\displaystyle \sum }}_{j=0}^{N-1}{\Psi }^{jk}{f}_{j+1}\left(t\right)$$

(1)

where the term specifying the inner product on the right-hand side is given by the $jk$-th power of $\Psi =e^{-2\pi i/N}$ [31], and the thus-evaluated left-hand side provides the power spectrum, ${\left|F\left(\omega \right)\right|}^2$, which signifies the spectrum intensity at each frequency [18].

3. Results and Discussion

The effects of the angle (${\theta }_T$) of the ultrasonic irradiation, or transducer installation/setting, are presented below on dynamic (oscillating and/or intermittent) characteristics of the liquid fountain for the given input power densities; either axial or lateral direction will be prescribed to each characteristic under discussion.

It is noted here that the input power density may reflect the so-called acoustic strength, which could be a measure of acoustic radiation pressure/force—depicted schematically in Figure 2 (designated as $F_{AR}$, a sort of “tension” force trying to maintain the fountain straight along a given direction, presumably the transducer axis)—in a traveling-wave field or acoustic streaming in a standing-wave field [32,33,34]. These two competing factors would contribute to the liquid surface being pushed upwards, leading to the eventual formation of liquid fountain, e.g., [27]. It is also speculated that the more extensive these contributions are, the faster the fountain growth may be, and the greater the amount of resulting atomization would be.

3.1. Structure of Liquid Fountain with Mist

The images obtained of liquid fountain and/or some associated mist visible for different transducer setting angles ${\theta }_T=$ 0°, 2°, 4°, 5°, 6°, and 7° at the applied power density of 5.0 W/cm² are shown in Figure 3. When the ultrasonic transducer is placed horizontally (0°), the fountain will be formed vertically along the direction of the transducer axis; individual small (visible) droplets along with the mist are observed to be scattered irregularly/intermittently (see Supplementary Material, Video S1) from the surface around the so-called Finely-Structured Surface Region (FSSR, [19,35]) mostly below the liquid fountain tip, or the Lumped-Crest Region (LCR). As the fountain ruptures vertically, the (large) separated tip drop—the remnant of the lumped crest—will not only disrupt the oscillation state of the fountain, often ending up merging with the remaining fountain, but also cause such disturbed fluctuations of the fountain surface to affect the direction of a higher part—including the FSSR as well as LCR—of the fountain in turn.

With an increase in ${\theta }_T$, the liquid fountain will be inclined correspondingly and may be bent in between—prior to reaching its tip—under the influence of gravity, while the axial interference from the separated drop(s) will be gradually reduced. In the range of ${\theta }_T$ increasing from 0° to 5°, the visually identifiable mist decreases gradually, but noticeably above 6°; and not identifiable/detected beyond 7°. The upper limit of ${\theta }_T$ in properly operating the UsA transducer is thus confined from the viewpoint of generating the mist.

3.2. Axial Growth and Breakup Dynamics of Liquid Fountain

The liquid fountain will repeat its dynamic phases between growth and breakup processes on a ms-time scale. Based on the image-analysis procedure described in Section 2.1, the breakup/attained heights, breakup frequency, and growth rate of the fountain were determined for different input power densities. Three sets of data are provided under each condition to show the degree of reproducibility for each parameter.

Figure 4a shows the variations in the liquid-fountain breakup and attained heights for 3.5, 5.0, and 6.5 W/cm² with increasing transducer setting angle ${\theta }_T$. Both the fountain heights increase with input power density. Over the range of ${\theta }_T$ examined, both the heights are regarded to be essentially invariant with the angle ${\theta }_T$. As shown in Figure 4b,c, respectively, the breakup frequency and the growth rate of the fountain take maximum values at 2°; the former being clearly exhibited (by sharp peaks) but the latter being indicated by the moderate hills. Figure 4c, in particular, signifies the presence of a transition range (from 4° to 6°) that demarcates the high growth-rate range (0–4°) and the low-to-negligible range (6° or higher).

Notably, the appreciable reduction in the breakup frequency near ${\theta }_T=$ 0° (≤1°) could stem from the disruption of the oscillation “rhythm” of the fountain due to vertically interacting drops mentioned above; a slightly tilted transducer setting angle (0–3°) appears to have little impact on the fountain growth rate. Based on the image analysis, atomization is observed to occur both before and after the fountain breakup rather cyclically. It is sufficient to say here that the transducer installation angle should be set 2° in terms of effective mist generation and fluid dynamic stability of the liquid fountain.

3.3. Time-Dependent Characteristics of Fountain Swinging

The amplitude and frequency of fountain oscillations for two representative transducer setting angles, ${\theta }_T$ of 2° and 5°, were analyzed together with the amount of mist “determined” from the visual appearance on the relevant images such as the ones shown in Figure 3 (also see Supplementary Material, Videos S2 and S3).

Figure 5a shows, for the above two values of ${\theta }_T$ at 5.0 W/cm², time variations of the (net) fountain inclined angle ${\theta }_F$ along with the associated mist spreading detected. The position of red triangles in the figure specifies the moment (and the angle) of mist generation; the size of each triangle signifies the amount of atomization—judged to be either small or large, limited by the visual information only (see the inset images). While the fountain oscillations in ${\theta }_F$ are occasionally disrupted by tip-separated drops for ${\theta }_T=$ 2°, the quasi-periodicity in the ${\theta }_F$ variation over time is still evident. Regarding the atomization, it is mostly triggered in cycle as the fountain inclined angle is shifted from positive to negative in the 0° ± 5° range; particularly noticeable is multiple, sequential occurrence of misting (regarded as a single event) when the amplitude of oscillations in ${\theta }_F$ is larger. For the larger ${\theta }_T=$ 5°, the fountain is no longer directly disturbed by the separated falling drops. In comparison to the smaller ${\theta }_T$, the amplitude of ${\theta }_F$ oscillations, the angular range of triggering atomization, and more importantly/drastically, the degree of occurrence as well as the amount of atomization will all decrease. In addition, during one swing cycle, essentially only one-time mist generation can be detected.

The periodic (or at least quasi-cyclic) natures in the lateral and axial dynamics described above can here be contrasted more specifically in terms of their apparent frequencies. As shown in Figure 5b, selected portions of the fountain-swinging and growth–breakup time-series data would exhibit their own frequencies. The latter fluctuates more quickly than the former, with the frequency ratio of roughly 5 or greater—signifying the multiple (5–8 times) occurrence of axial growth and breakup in one lateral oscillation period. This frequency “gap” indicates that the fountain height could be regarded as effectively constant, averaged over one cycle of the fountain swinging. It should be noted here that even though the amplitude of oscillations in ${\theta }_F$ is appreciable during the swinging, the Foundation Region (FR) always formed in line with the direction of ultrasonic irradiation (see the inset $t_3$). Of particular importance, the heights of the FR and Bumpy-Surface Region (BSR) are essentially time-invariant (below the dashed line), and the occurrence of mist generation is mostly confined to the time-varying (but cyclic) FSSR. In this sense, the fountain swinging could be regarded as a motion of a dual-body (double-compound) pendulum with the FR as the pivoting region ([36,37]; to be discussed more extensively in Section 3.4).

Figure 6a,b shows the effects of the input power on the swinging characteristics at ${\theta }_T=$ 2° and the corresponding FFT-analysis results, respectively. It is important to keep in mind that, on each time-series data given in Figure 6a, virtual/auxiliary data points (marked by small dots) are added, obtained via Lagrange interpolation; these points are necessary to perform FFT which requires equal intervals in the time coordinate. Some quasi-periodicity in the variations of the fountain inclined angle over time is apparent for all the input power densities. With increasing input power density, the amplitude of oscillations in ${\theta }_F$ tends to decrease, while the degree of occurrence and amount of atomization do increase. The FFT analysis has revealed that—while quasi-periodic—the swinging fountain has a dominant frequency of 21, 21, and 24 Hz for 3.5, 5.0, and 6.5 W/cm², respectively. The same FFT analysis at ${\theta }_T=$ 5° for 5.0 W/cm² provides a dominant frequency of 18 Hz (not shown in the figure).

3.4. Model-Evaluated Periodicity of Fountain Swinging

As described in Section 3.2 and Section 3.3, the acoustic liquid fountain will exhibit respectively quasi-periodic oscillations with recurring growth-and-breakup in the axial direction and swinging in the lateral direction—the latter, in particular, if the ultrasound is irradiated on the tilt—under the influence of gravity. The latter dynamics could then be regarded as a motion of a planar compound pendulum. In such phenomenological, intuitive modeling, a sequence of assumptions are needed for its mathematical formulation and predictive realization of the experimental findings described so far. The relevant details are provided in the Appendix A. A schematic configuration towards such modeling efforts is given in Figure 7.

As elaborated on—and ending with the final simplification—in the Appendix A, the resulting model system of swinging fountain could be represented by a simple pendulum and would then be expressed mathematically as

$$I_F\frac{d^2{\theta }_F}{d{t}^2}+\left(F_{AR}{L}_F-m_F{L}_F^{\mathrm{CM}}g\right){\theta }_F=m_F{L}_F^{\mathrm{CM}}g{\theta }_T$$

(2)

This is essentially a 2nd-order differential equation for a simple harmonic oscillation, which can be solved for the system angular frequency ${\omega }_F$—with the gravity-equilibrated $F_{AR}$ given by Equation (A2) and the small-angle approximation for ${\theta }_T$, i.e., $\cos {\theta }_T\cong 1$—to yield the natural frequency of (harmonic) oscillation:

$${\omega }_F^2\equiv {(2\pi f_F)}^2=\frac{m_F\left(L_F-L_F^{\mathrm{CM}}\right)g}{I_F}$$

(3)

which signifies the ratio of the torque imposed by gravity on the mass of the pendulum to the resistance to acceleration defined by the moment of inertia. The swinging frequency of the acoustic fountain in this study is then specified via Equation (A1) as

$$f_{\mathrm{fount}}=2f_{\mathrm{pend}} \left(\equiv 2f_F\right)=\frac{1}{\pi }\sqrt{\frac{m_F\left(L_F-L_F^{\mathrm{CM}}\right)g}{I_F}}$$

(4)

In this rather general expression, each parameter on the right-hand side could be provided as follows:

⊳

The total mass moment of the fountain consisting of Bodies 1 and 2, the geometries of which are respectively a cone of the base radius $R_1$ and the height $L_1$, and a cylinder of the radius $R_2$ and the height $L_2$:

$$\begin{array}{ll}{I}_F& =I_1+m_1{\left(a{L}_1\right)}^2+I_2+m_2{\left(L_1+b{L}_2\right)}^2\\ & =m_1\left(\frac{3}{10}{R}_1^2+\frac{1}{9}{L}_1^2\right)+m_2\left(\frac{1}{2}{R}_2^2+L_1^2+L_1{L}_2+\frac{1}{4}{L}_2^2\right)\end{array}$$

⊳

The total mass itself of the fountain:

$$m_F=m_1+m_2={\rho }_L\pi \left(\frac{1}{3}{R}_1^2{L}_1+R_2^2{L}_2\right)$$

⊳

The apparent center of mass for the fountain as a whole ($L_F^{\mathrm{CM}}$) as well as the fountain length itself ($L_F$) combined as

$$L_F-L_F^{\mathrm{CM}}=\frac{m_1\left[\left(1-a\right)L_1+L_2\right]g+m_2\left(1-b\right)L_{2}g}{m_1+m_2}$$

or

$$m_F\left(L_F-L_F^{\mathrm{CM}}\right)=\left(\frac{2}{3}{L}_1+L_2\right)m_{1}g+\frac{1}{2}{L}_2{m}_{2}g$$

Comparison(s) of the model-evaluated frequency $2f_F$ range, along with specific/representative values of the model parameters, determined based on the images obtained in this study, with the experimentally found (FFT-evaluated, dominant) frequencies, are summarized in

Table 1. Comparison of model-predicted and FFT-evaluated values for fountain-swinging frequency.

Input Power (W/cm2)	${\mathit{L}}_1$ (mm)	${\mathit{L}}_2$ (mm)	${\mathit{R}}_1$ (mm)	${\mathit{R}}_2$ (mm)	${10}^4{\mathit{m}}_1$ (kg)	${10}^5{\mathit{m}}_2$ (kg)	${10}^{10}{\mathit{I}}_1$ (kg m2)	${10}^{11}{\mathit{I}}_2$ (kg m2)	Pred [Equation (4)] $2{\mathit{f}}_{\mathit{F}}$ (Hz)	Exp (FFT) Main Freq. (Hz)
3.5 @2°	10.8	4.8	2.9	0.8	0.95	0.96	2.4	0.31	19.1–21.5	21
5.0 @2°	16.6	5.2	3.5	1.0	2.10	1.60	7.7	0.81	16.2–17.9	21
5.0 @5°	14.7	6.2	3.1	0.7	1.50	0.95	4.3	0.23	18.4–19.0	18
6.5 @2°	18.5	8.2	3.0	1.0	1.70	2.60	4.6	1.30	13.4–17.3	24

. Reasonable agreement between these values appear to be obtained.

As inferred from the above modeling outcome, the (asymmetric) fountain swinging observed in this study is of a resonant nature, i.e., occurring at or (if not completely undamped, or ${\beta }_i\ne 0$) near the natural frequency of the fountain as a whole. In dimensionless form, it can be expressed in terms of the resonance Strouhal number ($Sr$) based on the resonance frequency $f_{\mathrm{fount}}$ (or $f_F$ in view of purely modeling perspective) together with some characteristic length and velocity. The latter two characteristics were selected in this study to be the basal diameter of the fountain, $d_{\mathrm{base}}$ ($=2R_1$), and the propagation velocity of a capillary waveform [37] or the so-called capillary−inertial velocity [38] along the acoustic fountain surface. Such a specification will lead to the Strouhal number defined by

$$S{r}_F=\frac{f_{F }{d}_{\mathrm{base}}}{\sqrt{2\sigma /{\rho }_l{d}_{\mathrm{base}}}}=\frac{f_{F }{l}_c}{\sqrt{2\sigma /{\rho }_l{l}_c}}{\left(\frac{d_{\mathrm{base}}}{l_c}\right)}^{\frac{3}{2}}$$

(5)

Note in this equation that the capillary length $l_c$ defined by

$$l_c=\sqrt{\sigma /{\rho }_{l}g}$$

(6)

is explicitly introduced into the expression, along with the reduced basal diameter.

Further evaluation of the current model performance in providing the dominant frequency of the laterally oscillating liquid acoustic fountain in the air—besides comparing it with the FFT-evaluated values—was attempted in terms of thus-defined Strouhal number. To do so, a well-accepted convention, in the multiphase fluid mechanics area, of plotting this dimensionless number against the Reynolds number ($Re$) was adopted in the present study; specifically utilized is a simple relationship/correlation proposed by Fan and Tsuchiya [37]—and well-supported by, e.g., Sankaranarayanan et al. [39]—given by the following form:

$$\frac{Sr}{S{r}_0}={\left(1-\frac{T{a}_0}{Ta}\right)}^2$$

(7)

where $Ta$ is the Tadaki number defined by $Ta\equiv Re M{o}^{0.23}$ with $Re\equiv d_{\mathrm{base}}\sqrt{2\sigma /{\rho }_l{d}_{\mathrm{base}}}/\left({\mu }_l/{\rho }_l\right)$—in this particular phenomenon of UsA fountain oscillations—and the Morton number which depends only on the liquid properties, $Mo\equiv {\mu }_l^{4}g/\left({\sigma }^3{\rho }_l\right)$. The two correlative parameters, $S{r}_0$ and $T{a}_0$, are system-specific.

Figure 8 depicts, first, the present results for estimated values of the resonance $S{r}_F$, obtained based on the above idealized model as well as the FFT-supported values of $S{r}_{\mathrm{fount}}=2S{r}_{\mathrm{pend}} \left(\equiv 2S{r}_F\right)$, each in four data points. To be compared with this liquid fountain “pendulum” is a single (gas) bubble associated with its wake—more specifically, the primary wake closely following the (zigzag) rising bubble in a stationary liquid—or their combined entity, “bubble−wake pendulum” exhibiting a similar oscillating/rocking motion [36,37]. The figure includes some selected data (seven) points along with the specifically “tuned” correlation—linear in the given coordinates—of such a gas (dispersed)−liquid system [37]. It should be noted that both the pendulum systems compare reasonably well with each other in terms of not only the linearly increasing trend but also the range of ordinate ($Ta S{r}^{1/2}$) values—an encouraging outcome.

Additionally shown in Figure 8 are the acoustic liquid fountain consisting of a chain of beads/drops in contact, or the corrugated jet of liquid, studied specifically by Wang et al. [18], which exhibits characteristic oscillations in individual bead shapes; and fixed solid spheres exhibiting the surrounding fluid (gas or liquid) oscillations, known as vortex shedding behind the sphere (or in its wake), some data of which are selected from the compilation presented by Fan and Tsuchiya [37]. The former involves two modes of shape oscillations—assigned by $n$, the order corresponding to the representative shape of an oscillating drop (here, its vertical cross section being an ellipse: $n=2$, and a diamond: $n=4$; [18]). While not the same type of oscillations—pendulum swinging vs. shape fluctuations—their frequencies in terms of $Ta S{r}^{1/2}$ demonstrate a similar range as well as trend. The latter essentially falls in the same range as well, despite—again—the different type of (surrounding flow) oscillations. The fact that all these comparisons (see

Table 2. Comparison of model-predicted Strouhal numbers in different aspects of oscillating phenomena of UsA fountain.

Liquid fountain (W/cm2)	$d_{\mathbf{base}}$ (mm)	$f_{\mathbf{fount}}$ (Hz)	$f_F$ (Hz)	$U_{\mathbf{capil}}$ (m/s)	$R{e}_{\mathbf{fount}}$	$S{r}_F$	$S{r}_{\mathbf{fount}}$	$T{a}_{\mathbf{fount}}$	$Ta S{r}_F^{1/2}$	$Ta S{r}_{\mathbf{fount}}^{1/2}$
3.5 @2°	2.9	21.5	10.7	0.16	1030	0.39	0.79	2.07	1.30	1.84
5.0 @2°	3.5	17.9	9.0	0.14	1130	0.44	0.87	2.27	1.50	2.12
5.0 @5°	3.1	19.0	9.5	0.15	1060	0.39	0.77	2.14	1.33	1.88
6.5 @2°	3.0	17.3	8.7	0.16	1040	0.34	0.67	2.10	1.22	1.72
Beads fountain (MHz)	$d_{\mathbf{bead}}$ (mm)	$f_2$ (kHz)	$f_4$ (kHz)	$U_{\mathbf{capil}}$ (m/s)	$R{e}_{\mathbf{bead}}$	$S{r}_2$	$S{r}_4$	$T{a}_{\mathbf{bead}}$	$Ta S{r}_2^{1/2}$	$Ta S{r}_4^{1/2}$
1.0	1.47	0.13	0.38	0.21	114	0.92	2.68	1.28	1.22	2.09
2.0	0.74	0.36	1.08	0.29	81	0.91	2.72	0.91	0.86	1.49
3.0	0.49	0.66	1.99	0.36	66	0.90	2.70	0.74	0.70	1.21

for a detailed list of the relevant data sets selected) apparently lead to the similar trend and range in the dimensionless frequency signifies that the relationship between $Re$ and $Sr$, or more specifically, the linear relationship between $Ta$ and $Ta S{r}^{1/2}$, should provide quite a robust predictive scheme for a variety of resonance frequencies.

4. Concluding Remarks

For ultrasound that is irradiated out of the transducer used with its installation angles ranging 0–6° or higher (up to 10° examined) and input power densities of 3.5–6.5 W/cm², the UsA fountain will exhibit a sequence of oscillating/intermittent features. Our main findings are summarized below:

(1)

A slight tilt in irradiation of ultrasound should be advantageous in both operational and mist-generating performances. The acoustic liquid fountain will become improved in its operability/stability by avoiding the disruption of oscillation rhythm of the fountain from vertically interacting tip-separated drops;

(2)

The atomization, or mist generation, is mostly—almost exclusively—triggered and enhanced as the laterally swinging fountain comes across the direction of irradiation, more specifically, with its inclined angle being shifted from positive to negative in the 0° ± 5° range. The degree of occurrence of mist generation and the amount of identifiable mist generated would decrease, associated with reductions in both the growth rate and breakup frequency of the fountain on the tilt;

(3)

In line with such axial extents of growth and breakup of the fountain, both taking maximum values at the transducer installation angle of 2°, its optimum value should be recommended to be slightly tilted 2° from the viewpoint of stability of the UsA fountain and not to exceed 5° from that of effective mist generation;

(4)

A mechanistic view of the UsA process is provided in terms of the swinging periodicity of liquid fountain with mist generated intermittently, in particular, if the ultrasound is irradiated on the tilt—under the influence of gravity. The periodicity of both the axial and lateral oscillations has been quantified—the latter in particular being model-predicted based on a simple planar pendulum concept proposed, partly confirmed via FFT-evaluated dominant frequency.