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

^{*}

## Abstract

**:**

## 1. Introduction

^{2})—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.

## 2. Experimental

^{2}, which is exclusively used in this study as an operating parameter).

^{®}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

#### 2.2. Time-Series Data Analysis

^{®}2019b) to obtain the corresponding (to the latter) spectrum:

## 3. Results and Discussion

#### 3.1. Structure of Liquid Fountain with Mist

^{2}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.

#### 3.2. Axial Growth and Breakup Dynamics of Liquid Fountain

^{2}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).

#### 3.3. Time-Dependent Characteristics of Fountain Swinging

^{2}, 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.

^{2}, respectively. The same FFT analysis at ${\theta}_{T}=$ 5° for 5.0 W/cm

^{2}provides a dominant frequency of 18 Hz (not shown in the figure).

#### 3.4. Model-Evaluated Periodicity of Fountain Swinging

- ⊳
- 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$${L}_{F}-{L}_{F}^{\mathrm{CM}}=\frac{{m}_{1}\left[\left(1-a\right){L}_{1}+{L}_{2}\right]g+{m}_{2}\left(1-b\right){L}_{2}g}{{m}_{1}+{m}_{2}}$$

## 4. Concluding Remarks

^{2}, 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.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Compound Pendulum Modeling

- (1)
- The fountain, while altering its local shape dynamically, consists of two parts during the growth-and-breakup period, on average, simplified in geometry by a “cone” and cigar-shape or more simply a “cylinder.” The former involves the FR and BSR, the latter the FSSR which accompanies mist spreading (Figure 2);
- (2)
- (3)
- When the tilting is increased beyond ${\theta}_{T}=$ 4 or 5°, this single entity will experience its bending at the “connecting” region between the cone and the cylinder (Figure 3);
- (4)
- Under the most probable complex situation, the fountain will behave as a two-body entity which exhibits two “separate” swinging motions with two pivotal regions—one near the base of the FR and the other around the connecting region—resembling the so-called “double compound” pendulum [40].

- (5)
- The waveform of fountain-swinging, exhibited in Figure 5b, particularly demonstrates the periodicity of the fountain’s lateral dynamics with a peculiar nature of asymmetricity on the positive side of the angle ${\theta}_{F}$. This experimental trend of skewness signifies its frequency would be twice as high as an assumed symmetric (i.e., regular sinusoidal—given by the red, dashed smooth) waveform (cf. [41,42]);
- (6)
- The axial growth−breakup time-series data, on the other hand, with frequencies as high as 5–8 times that of the lateral will lead to the assumed representation of time-invariant fountain height throughout the oscillations.

- (7)
- The “tension-like” force(s) required to sustain the pendulum straight in the direction of its axis on the equilibrium position—specified by a prescribed angle—should be the acoustic strength, or acoustic radiation pressure/force(s) (denoted by ${F}_{AR}$), which could be balanced by the gravitational contribution (${m}_{F}g$) where ${m}_{F}$ is the apparent mass of the pendulum (i.e., fountain) as a whole;
- (8)
- The fountain, modeled by a dual-compound pendulum, is assigned to have the lower body of cone-shape of mass ${m}_{1}$ (Body 1, spanning the FR and BSR) and the upper body of cylindrical-shape of mass ${m}_{2}$ (Body 2, spanning the FSSR);
- (9)
- The two bodies thus simplified geometrically will rotate about their specific pivotal locations—Bodies 1 and 2 around Pivots 1 and 2, respectively—where Pivot 2 serves as the joint connecting the two bodies.

**Visualized structure and dynamics “converted” into quantitative expressions (Figure 2, Figure 3 and Figure 7)**: The swinging fountain as a dual-part entity consists of one part rotating around the base center of the fountain FR (Pivot 1, ${\mathrm{P}}_{1}$) with ${\theta}_{F}={\theta}_{F1}$ (covering FR + BSR) and the other around the connecting region or point (Pivot 2, ${\mathrm{P}}_{2}$) with ${\theta}_{F}={\theta}_{F2}$ (FSSR), occasionally bent at Pivot 2 satisfying the relation $\left|{\theta}_{F1}\left|\le \right|{\theta}_{F2}\right|$. The fountain will thus exhibit—as a double-compound pendulum—two separate (but mutually dependent/interactive) swinging motions characterized by the angular accelerations, $\frac{{d}^{2}{\theta}_{F1}}{d{t}^{2}}$ and $\frac{{d}^{2}{\theta}_{F2}}{d{t}^{2}}$, around Pivots 1 and 2, respectively [inferred from ASM (1), (3) and (4)].

**Time-series characteristics “converted” into quantitative relationships (Figure 5, Figure 6 and Figure 7)**: The axial fluctuations of the fountain, being regarded as time-invariant, would lead to its total length (tilted into the apparent height) which is a sum of individual parts, i.e., ${L}_{F}={L}_{1}+{L}_{2}\ne f\left(t\right)$, while the lateral fluctuations are to be given by slower oscillations that are asymmetric in nature, specified as

**Figure A1.**Forces diagram of conceptual double-compound pendulum inferred from modeled dual-part/body fountain depicted in Figure 7.

**Further simplifications “idealized” into model equations (Figure A1)**: The UsA fountain exerted by the “acoustic radiation” force(s), ${F}_{AR}$, tries by itself to maintain its “axial” direction “equilibrated” with the gravity component, i.e.,

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**Figure 1.**Schematic diagram of the experimental system for visual observation of ultrasonic atomization process for inclined liquid fountain.

**Figure 2.**Time course of liquid-fountain dynamics signifying its axial variations, based on the defined fountain’s breakup and attained heights; its lateral fluctuations are specified by ${\theta}_{F}$ (+ ${\theta}_{T}$).

**Figure 3.**Images of liquid fountain obtained for different transducer setting angles at input power density of 5.0 W/cm

^{2}.

**Figure 4.**Effects of transducer setting angle on (

**a**) breakup and attained heights, (

**b**) breakup frequency, and (

**c**) growth rate of liquid fountain for input power densities of 3.5, 5.0, and 6.5 W/cm

^{2}.

**Figure 5.**Time variations (

**a**) in fountain inclined angle, associated with mist generation, for transducer setting angles ${\theta}_{T}$ of 2° and 5° at 5.0 W/cm

^{2}, and (

**b**) in fountain inclined angle and height for ${\theta}_{T}=$ 2° over selected time interval with (

**c**) some images of fountain for representative phases of oscillations.

**Figure 6.**(

**a**) Time variations in fountain inclined angle, associated with mist generation, for transducer setting angle of 2° at input power densities of 3.5, 5.0, and 6.5 W/cm

^{2}, and (

**b**) corresponding outcomes of FFT analysis for fluctuating inclined angle.

**Figure 7.**(

**a**) General configuration of acoustic liquid fountain, proposed by Tsuchiya et al. [19], based on dynamics of four-region structure realized on typical fountain image, and (

**b**) schematic representation of comprehensive definitions as well as specifications towards swinging fountain model.

**Figure 8.**Resonance frequencies of various pendulum-like (or rocking), shape-deforming, and disturbed surrounding-flow oscillations represented by dimensionless form being correlatable via a simple expression proposed by Fan and Tsuchiya [37]; data adopted are from Wang et al. [18] for beads fountain and from Fan and Tsuchiya [37] for single dispersed entity.

Input Power (W/cm ^{2}) | ${\mathit{L}}_{1}$ (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 m ^{2})
| ${10}^{11}{\mathit{I}}_{2}$ (kg m ^{2})
| 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 |

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

Liquid fountain(W/cm^{2}) | ${d}_{\mathbf{base}}$(mm) | ${f}_{\mathbf{fount}}$(Hz) | ${f}_{F}$(Hz) | ${U}_{\mathbf{capil}}$(m/s) | $R{e}_{\mathbf{fount}}$ | $S{r}_{F}$ | $S{r}_{\mathbf{fount}}$ | $T{a}_{\mathbf{fount}}$ | $TaS{r}_{F}^{1/2}$ | $TaS{r}_{\mathbf{fount}}^{1/2}$ |

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 |

Beadsfountain(MHz) | ${d}_{\mathbf{bead}}$(mm) | ${f}_{2}$(kHz) | ${f}_{4}$(kHz) | ${U}_{\mathbf{capil}}$(m/s) | $R{e}_{\mathbf{bead}}$ | $S{r}_{2}$ | $S{r}_{4}$ | $T{a}_{\mathbf{bead}}$ | $TaS{r}_{2}^{1/2}$ | $TaS{r}_{4}^{1/2}$ |

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 |

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**MDPI and ACS Style**

Wang, X.; Tsuchiya, K.
Frequency Specificity of Liquid-Fountain Swinging with Mist Generation: Effects of Ultrasonic Irradiation Angle. *Fluids* **2022**, *7*, 306.
https://doi.org/10.3390/fluids7090306

**AMA Style**

Wang X, Tsuchiya K.
Frequency Specificity of Liquid-Fountain Swinging with Mist Generation: Effects of Ultrasonic Irradiation Angle. *Fluids*. 2022; 7(9):306.
https://doi.org/10.3390/fluids7090306

**Chicago/Turabian Style**

Wang, Xiaolu, and Katsumi Tsuchiya.
2022. "Frequency Specificity of Liquid-Fountain Swinging with Mist Generation: Effects of Ultrasonic Irradiation Angle" *Fluids* 7, no. 9: 306.
https://doi.org/10.3390/fluids7090306