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  • View in gallery

    Comparison the entrainment factors from experiments (Sujith et al. 1997) and calculations with different water droplet sizes and sound frequencies.

  • View in gallery

    The motion and force condition of droplet with d = 50 μm, f = 50 Hz, and SPL = 123.4 dB: (a) velocity, (b) displacement, (c) Stokes force, (d) Basset force, (e) pressure gradient force, and (f) additional mass force.

  • View in gallery

    Variation curve of droplet velocity with time under the action of sound waves for (a) different droplet sizes with the sound field with frequency f = 50 Hz and SPL = 123.4 dB, (b) different SPL with the droplet diameter d = 50 μm and the sound frequency f = 50 Hz, and (c) different frequencies with the droplet diameter d = 50 μm and SPL = 123.4 dB.

  • View in gallery

    Displacement of a cloud droplet with the size of 10 µm with sound waves of f = 50 Hz and SPL = 123.4 dB, and without sound waves.

  • View in gallery

    (top) Maximum displacement difference between the cases with and without sound waves for different SPLs and droplet sizes, in which the sound frequency f = 50 Hz: (a) the relationships between the maximum displacement difference and different droplet sizes under different SPLs and (b) the relationships between the maximum displacement difference and droplet sizes for different SPLs. (bottom) Maximum displacement difference between the cases with and without sound waves for different frequencies and droplet sizes, in which SPL = 123.4 dB: (c) the relationships between the maximum displacement difference and droplet sizes for different frequencies and (d) the relationships between the maximum displacement difference and frequencies for different droplet sizes.

  • View in gallery

    Curves of droplet velocity. (a) Change with sound frequency for a cloud droplet diameter of 10 μm with SPL = 123.4 dB. (b) Change with SPL for a cloud droplet diameter of 10 μm with sound frequency f = 50 Hz. (c) Change with droplet sizes with sound frequency f = 50 Hz and SPL = 123.4 dB.

  • View in gallery

    (a) Curves of acceleration amplitude generated by the four forces as a function of frequency for a cloud droplet diameter of 10 μm due to the sound wave with SPL = 123.4 dB. (b) Curves of acceleration at the equilibrium position generated by Stokes force as a function of frequency for cloud droplet of 10 μm due to the sound wave with SPL = 123.4 dB. (c) Curves of acceleration amplitude change with SPLs generated by different forces for cloud droplet of 10 μm due to the sound wave with frequency f = 50 Hz. (d) Curves of acceleration amplitude change with droplet sizes generated by different forces due to the sound wave with frequency f = 50 Hz and SPL = 123.4 dB.

  • View in gallery

    Curve of the ratio of the magnitude of each force to time in the stable oscillation stage due to the sound wave with frequency f = 50 Hz and SPL = 123.4 dB for a droplet size of (a) d = 10 μm and (b) d = 100 μm.

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Mechanism of Cloud Droplet Motion under Sound Wave Actions

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  • 1 College of Water Resources and Civil Engineering, China Agricultural University, Beijing, China
  • | 2 State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, and State Key Laboratory of Plateau Ecology and Agriculture, Qinghai University, Xining, China
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Abstract

Sound waves have proven to be effective in promoting the interaction and aggregation of droplets. It is necessary to theoretically study the motion of particles in a sound field to develop new acoustic technology for precipitation enhancement. In this paper, the motion of cloud droplets due to a traveling sound wave field emitted from the ground to the air is simulated using the motion equation of point particles. The force condition of the particles in the oscillating flow field is analyzed. Meanwhile, the effects of droplet size, sound frequency, and sound pressure level (SPL) on the velocity and displacement of the droplets are also investigated. The results show that Stokes force and gravity play a dominant role in the falling process of cloud droplets, and the effect of the sound wave is mainly reflected in the fluctuation of velocity and displacement, which also promotes the displacement of cloud droplets to a certain extent. The maximum displacement increments of cloud droplets of 10 µm can reach 9200 µm due to the action of sound waves of 50 Hz and 143.4 dB. The SPL required for a noticeable velocity fluctuation for droplets of 10 µm with frequency of 50 Hz is 88.2 dB. When SPL < 100 dB and frequency > 500 Hz, the effect is negligible. The cloud droplet size plays a significant role in the motion, and the sound action is weaker for larger particles. For a smaller sound frequency and higher SPL, the effect of the sound wave is more prominent.

Corresponding author: Jun Qiu, aeroengine@tsinghua.edu.cn

Abstract

Sound waves have proven to be effective in promoting the interaction and aggregation of droplets. It is necessary to theoretically study the motion of particles in a sound field to develop new acoustic technology for precipitation enhancement. In this paper, the motion of cloud droplets due to a traveling sound wave field emitted from the ground to the air is simulated using the motion equation of point particles. The force condition of the particles in the oscillating flow field is analyzed. Meanwhile, the effects of droplet size, sound frequency, and sound pressure level (SPL) on the velocity and displacement of the droplets are also investigated. The results show that Stokes force and gravity play a dominant role in the falling process of cloud droplets, and the effect of the sound wave is mainly reflected in the fluctuation of velocity and displacement, which also promotes the displacement of cloud droplets to a certain extent. The maximum displacement increments of cloud droplets of 10 µm can reach 9200 µm due to the action of sound waves of 50 Hz and 143.4 dB. The SPL required for a noticeable velocity fluctuation for droplets of 10 µm with frequency of 50 Hz is 88.2 dB. When SPL < 100 dB and frequency > 500 Hz, the effect is negligible. The cloud droplet size plays a significant role in the motion, and the sound action is weaker for larger particles. For a smaller sound frequency and higher SPL, the effect of the sound wave is more prominent.

Corresponding author: Jun Qiu, aeroengine@tsinghua.edu.cn

1. Introduction

Over the past century, the continuous exploration of sound waves has broadened its application scope steadily. Sound aggregation is an effective way to utilize a sound wave, which is widely used to remove inhalable dust such as PM2.5 (Zhou et al. 2016; Zu et al. 2017) and other harmful droplets (Sadighzadeh et al. 2018).

Various studies on the movement of particles using sound waves have been published to reveal the principle of sound aggregation and guide its application, such as the forces acting on particles in fluid by a sound wave, the interaction between the particles and the fluid, and the interaction between the individual particles. Cleckler et al. (2012) used the motion equation of point particles to simulate the motion of heavy spherical particles caused by sound waves. Zhou et al. (2015) studied the motion law of particles with various particle sizes affected by different sound fields by numerical simulation. Sujith et al. (1999) attempted to analyze the relative motion among particles in a sound field and between particles and fluids by numerical integration and spectral analysis to explain the phenomena of momentum and heat transfer in multiphase flow. González et al. (2000) observed the movement of glass beads with a diameter of 7.9 µm due to the sound wave and compared the experimental results with theoretical results. Setayeshgar et al. (2015) studied the movement of polydisperse particles in large-scale and multiwavelength sound chambers. Zhou et al. (2017) studied the motion characteristics of particles in the traveling and standing waves and compared the experimental results with the existing theory. Fan et al. (2013) explored the time of collision between two particles considering particle size, distance, sound pressure level (SPL), and frequency, accounting for the interactions between particles, and between the particles and the fluid. To facilitate coagulation of two particles in fluid with a certain distance, a critical relative or convergence speed between them is required, and thus Dong et al. (2006) proposed the concept of effective aggregation length (i.e., the maximum distance of effective coagulation between particles) and calculated the values of two particles with different sizes, frequencies, and SPLs.

Sound waves produce an oscillating fluid. Many studies on particle motion in an oscillating fluid also have been presented. Hassan and Kawaji (2008) studied the particle motion induced by vibration in a viscous fluid both theoretically and experimentally and analyzed the relationship between the particle motion amplitude and fluid viscosity and vibration amplitude and frequency. Baird et al. (1967) studied the effect of vertical vibration on the velocity of plastic balls in water and found that the liquid oscillation led to a decline in velocity. The experimental results of Blekhman et al. (2013) showed that under certain external oscillation conditions, bubbles sank in the fluid. Ikeda and Yamasaka (1989) calculated the falling velocity of a single spherical particle moving in a vertical oscillating fluid.

In general, most of the previous studies on the motion of a single particle with sound waves focus on the effects of the sound wave, fluid, and particle properties on the motion process. Both theoretical and experimental studies have found that sound waves with higher SPL and lower frequency have stronger carrying effects on smaller particles. However, due to the differences in experimental settings and research objects, it is difficult to generalize the effect of sound waves, especially the mechanism of sound waves on cloud droplets, which is crucial for developing new technologies for the exploitation of atmospheric water resources.

According to atmospheric physics, the mechanisms of cloud droplet growth primarily include condensation and coagulation. Sound aggregation uses the air, which vibrates with the sound propagation, to carry particles of different sizes to vibrate with various amplitudes and cause them to agglomerate through the relative motion. This rule is also applicable to the coagulation process of micron-scale cloud droplets. Therefore, in recent years, scientists have proposed the concept of acoustic promoted precipitation (i.e., accelerating cloud droplet condensation and enhancing precipitation by emitting sound waves into the atmosphere) (Tulaikova and Amirova 2015). Compared with the traditional artificial precipitation methods, sound waves have the advantages of less environmental pollution, strong directivity, and simple operation.

When thunderstorms occur accompanied by rapid rain, there are often bursts of thunder. It is possible that the air oscillation caused by the thunder promotes the condensation of cloud droplets and thus, enhances precipitation. The SPL of thunder is generally 90 dB and the maximum is approximately 120 dB (Bodhika et al. 2018; Bhartendu 1971; Vavrek et al. 2020). With the distance increase from the thunder source, the energy of thunder decreases gradually. The thunder signal generally concentrates in low frequencies with the range of 0–2000 Hz. Bhartendu (1969) recorded the change in the sound pressure of thunder, and the spectrum showed that 52 and 96 Hz were the most dominant values. Bodhika et al. (2018) indicated that the peak sound frequency is between 225 and 425 Hz, while the experimental results of Yuhua and Ping (2012) showed that the peak sound frequency is between 210 and 280 Hz. The properties of thunder are similar to those of human-made sound waves, and therefore, the study attempts to predict if the atmospheric fluctuations caused by thunder can promote condensation and the growth of micron-scale cloud droplets in the atmosphere.

In this study, the motion characteristics of cloud droplets in the vertical sound field are discussed microscopically. The purpose is to clarify the effect of sound waves on the motion of cloud droplets to lay a foundation for predicting the effect of acoustic precipitation enhancement macroscopically. The size of natural cloud droplets in atmospheric clouds varies from 1 to 100 µm. According to atmospheric cloud physics, the typical diameter of cloud droplets is 10 µm, with the typical number density of 109 m−3, and the boundary between the cloud droplet and the raindrop is 100 µm (McDonald 1958). Clouds can be divided into high, medium, and low clouds according to their height, and the clouds that form rainfall are primarily low (WMO 2019). The diameter of cloud droplets in cumulonimbus clouds, which are most likely to form precipitation, ranges from 2 to 100 µm with the typical size of 20 µm. The probability of precipitation from a stratus cloud with the size of 0.5–30 µm is much smaller than that of cumulonimbus.

Additionally, the lower the cloud layer is, the larger the size of the cloud droplets, and the higher the density of cloud droplets is, the more favorable the precipitation. It is necessary to gain a better understanding of the effect of air vibrations on cloud droplets to clarify the effect of sound waves on precipitation. Nevertheless, most of the previous studies have focused on the general law between particle motion and acoustic properties, as well as particle size. This paper presents a specific scene to discuss the movement of cloud droplets under the action of sound waves similar to the natural thunder. Its purpose is to study the mode and degree of the influence that sound waves exert on cloud droplets, so as to better guide the practical application. Therefore, in this paper, “particles” were reified as cloud droplets in the atmosphere, and their motion characteristics due to the action of vertical sound waves were explored, accounting for the influence of gravity. The movement process of droplets of different sizes under sound wave action was analyzed in this study, simulating the properties of cloud droplets and thunder under natural conditions. The results provide preliminary theoretical guidance for acoustic precipitation enhancement in the future.

2. Methodology

a. Sound waves

The sound wave is assumed as a one-way traveling wave propagating along the vertical direction. The formula of fluid velocity oscillation is shown in Eq. (1):

uf=usin(ωtkx),

where uf is the velocity of flow field; u is the velocity amplitude of fluid oscillation driven by the sound wave; ω = 2πf is the angular velocity of sound field oscillation; f is the frequency of vibration; k = 2πf/c is the number of sound waves in sound field; c is the propagation velocity of a sound wave in air, which equals to 340 m s−1; x is the position in sound field; and t is the time.

The range of the SPL of the artificial sound wave is between 80 and 150 dB, and the SPL corresponding to each u can be derived combining Eqs. (2) and (3):

u=2I/ρgc,
I=I0×10SPL/10,

where I is the sound field intensity (W m−2); I0 is the standard reference value, which is set to 10−12 W m−2; ρg = 1.293 kg m−3 is the density of air; and SPL is the sound pressure level (dB).

Each SPL corresponds to a certain u. According to the given range of the SPL, the velocity oscillation amplitude (u) is between 6.74 × 10−4 and 2.13 m s−1. Table 1 shows the SPL corresponding to different flow fields. Through literature research (Bodhika et al. 2018; Bhartendu 1971; Vavrek et al. 2020), many ground observational studies have found that the maximum sound pressure level of thunder is around 120 dB under natural conditions, so we think that the sound pressure level in the cloud is likely to be above 120 dB.

Table 1.

Velocity amplitudes of different sound fields and the corresponding SPL.

Table 1.

b. Motion equation

A series of studies on the forces acting on spherical particles in a flow field have been carried out, forming a mature theoretical framework. Mei et al. (1991) improved the force analysis of a single particle in the oscillating flow field with the finite Reynolds number. Maxey and Riley (1983) established the motion equation of a rigid sphere in an inhomogeneous compressible fluid. Basset (1889) and Oseen (1927) analyzed the unsteady motion of spherical particles in an incompressible viscous fluid. Based on reference (Kim et al. 2000), the forces acting on particles in fluids are primarily composed of the following parts:

f1=CDρg|ufυ|(ufυ)2πd24,
f2=32πd2μ0tdds(ufυ)πυ(ts)ds,
f3=12ρgVd(ufυ)dt,
f4=ρgVDufDt,
f5=πd36(ρpρg)g,
mpdυdt=f1+f2+f3+f4+f5,

where υ is the velocity of cloud droplets; d is the diameter of cloud droplets; CD is the drag coefficient; μ = 17.9 × 10−6 Pa s is dynamic viscosity; υ = μ/ρg is kinetic viscosity; V is the volume of the cloud droplets; Duf/Dt is the total differential of the velocity field uf to t; g = 9.81 m s−2 is gravitational acceleration; ρp = 1000 kg m−3 is the density of cloud droplets; f1 indicates the viscous resistance acting on the particles caused by the relative motion between the particles and the fluid; f2 is Basset force, which indicates the increased resistance on the particles from the variable motion in the flow field, reflecting the influence of the historical movement; f3 is the force promoting the accelerated motion of the surrounding fluid, called additional mass force; f4 is the pressure gradient force acting on the sphere; and f5 is the gravity minus buoyancy.

Equation (9) is an equation based on Newton’s second law. The value of the drag coefficient CD is related to Reynolds number Red = ρgd|ufυ|/μ and is calculated by Eq. (10) (Seinfeld and Pandis 1998):

CD={24/Red,Red224Red(1+0.15Red0.687),2<Red5000.44,Red>500,

When CD = 24/Red, the viscous resistance can be simplified to f1 = 3πdμ(ufυ). Since the Reynolds number is small with the maximum value less than 12, the viscous resistance is calculated by f1 = 3πdμ(ufυ) to simplify the calculation and improve the accuracy and stability of the algorithm.

c. Numerical solution

The Lagrangian method is used to track the motion pathway of particles. The downward motion of cloud droplets due to gravity is considered, and the sound field around the spherical particles is assumed not to be disturbed by the motion of particles. The fourth-order Runge–Kutta method is applied to solve the motion process of cloud droplets shown in the nonlinear Eq. (9). For each step, the historical integral term is discretized, and the acceleration in each time step in the past is considered to be a constant. The Barthel force at the current time step is obtained, as shown in Eqs. (11) and (12). According to the numerical solution, the variation in velocity and displacement with time can be obtained. Therefore, the forces can be calculated. The typical size of cloud droplets is 10 µm, and the boundary between cloud droplets and raindrops is 100 µm. Therefore, the simulated size range of cloud droplets is 5–120 µm. The frequency of the sound wave oscillation is between 20 and 2000 Hz, and the amplitude of sound field velocity is chosen between 0.0001 to 1.5 m s−1, corresponding to SPL ∈ (63.42, 146.94) dB. In the numerical simulation, it is assumed that the initial droplet velocity is 0, and the droplet begins to drop at t = 0 from x = 0. The experiment only simulates the 1D movement of cloud droplets in the vertical direction with downward as the positive direction:

f2=32πd2μ0tndds(ufυ)πυ(tns)ds=32πd2μ0t1a1πυ(tns)ds+t1t2a2πυ(tns)ds++tn1tnanπυ(tns)ds,
an=(ufυ)n(ufυ)n1tntn1,

where the subscripts n and n + 1 represent two adjacent moments.

3. Results and discussion

a. Reliability verification

The entrainment factor is defined as the ratio of the droplet velocity amplitude to the amplitude of the sound field velocity, which is a dimensionless number, indicating the ability of the sound field to drive the particle. The larger the entrainment coefficient is, the more significant the effect of the sound wave. Compared with the results of experiments which investigate of the behavior of water droplets in axial acoustic fields by Sujith et al. (1997), the calculated solutions are shown in Fig. 1, where the scatter points are the experimental data, and the curves are our calculated results through using Eq. (9). The calculated curves and the experimental data agree well, indicating that the theoretical equations and numerical method adopted in this paper is reasonable and reliable.

Fig. 1.
Fig. 1.

Comparison the entrainment factors from experiments (Sujith et al. 1997) and calculations with different water droplet sizes and sound frequencies.

Citation: Journal of Atmospheric and Oceanic Technology 37, 9; 10.1175/JTECH-D-19-0210.1

b. A typical example

Figure 2 shows typical simulation results of the cloud droplet velocity and the displacement and acceleration caused by various forces with a particle size of 50 µm, a frequency of 50 Hz, and as SPL of 123.4 dB (typical values selected according to the characteristics of real thunder and cloud droplets). In the traveling wave field, the velocity and acceleration of the cloud droplet will eventually reach a stable oscillation phase in which the velocity and the Stokes force will be balanced at a nonzero position, while the Basset force, the pressure gradient force, and the additional mass force fluctuate up and down around zero. The displacement of the cloud droplets shows a tendency to rise with the fluctuations. Because the cloud droplets are forced to vibrate due to the action of the sound waves, the cloud droplet motion frequency is the same as that of the sound wave, and the oscillation amplitude is related to the cloud droplet property, the sound wave intensity, the local air density, and the air pressure.

Fig. 2.
Fig. 2.

The motion and force condition of droplet with d = 50 μm, f = 50 Hz, and SPL = 123.4 dB: (a) velocity, (b) displacement, (c) Stokes force, (d) Basset force, (e) pressure gradient force, and (f) additional mass force.

Citation: Journal of Atmospheric and Oceanic Technology 37, 9; 10.1175/JTECH-D-19-0210.1

c. Initial and stable stages

Because of the action of the sound field, the motion of a single cloud droplet mainly goes through two stages. The first is an unstable start-up period, and the second is a stable oscillation period with a constant velocity. The start-up time increases with the increase in the droplet size, as shown in Fig. 3a. Figure 3b shows the SPL variation with the time, which shows that the length of the start-up time does not change with the SPL. As with the SPL, Fig. 3c shows that the duration of the start-up period does not change with the sound frequency. For the maximum droplet with the size of 100 µm in the simulation, after a transition period of 0.3 s, the velocity of the droplets is also substantially maintained in a stable oscillation state, indicating that the cloud droplets respond more sensitively to sound waves. Regardless of gravity, or due to the action of horizontal sound waves, studies (Zhou et al. 2017) have shown that the velocity of the droplets oscillates around zero. Because of gravity, the equilibrium position of the velocity oscillation of the cloud droplet after the transition period is generally greater than zero. Therefore, in all cases, the downward displacement of the cloud droplets increases with time. The start-up period only exists in the initial stage of the cloud droplet’s movement, so the following research in this paper is carried out for the stable oscillation period with strong regularity.

Fig. 3.
Fig. 3.

Variation curve of droplet velocity with time under the action of sound waves for (a) different droplet sizes with the sound field with frequency f = 50 Hz and SPL = 123.4 dB, (b) different SPL with the droplet diameter d = 50 μm and the sound frequency f = 50 Hz, and (c) different frequencies with the droplet diameter d = 50 μm and SPL = 123.4 dB.

Citation: Journal of Atmospheric and Oceanic Technology 37, 9; 10.1175/JTECH-D-19-0210.1

d. Displacement of cloud droplet

Figure 4 shows the displacement of a cloud droplet with the size of 10 µm with and without sound waves (i.e., uf = 0). In the case without sound waves, the displacement of the cloud droplet is approximately a straight line through the minimum values of the case with sound waves (i.e., the lower boundary of the displacement fluctuation and the average changing rates of the displacement in two cases are the same). It can be seen from Fig. 4 that an oscillating change process exists for the displacement difference between the two cases. When sound waves are absent, the droplet’s velocity increases rapidly and then flattens out in a straight line. From Fig. 3c we can infer that the droplet velocity fluctuates dramatically in the initial stage due to the presence of sound waves. During this period, the area enclosed by the velocity–time curve is larger than that of the smooth curve (no acoustic wave), resulting in an increase in displacement.

Fig. 4.
Fig. 4.

Displacement of a cloud droplet with the size of 10 µm with sound waves of f = 50 Hz and SPL = 123.4 dB, and without sound waves.

Citation: Journal of Atmospheric and Oceanic Technology 37, 9; 10.1175/JTECH-D-19-0210.1

The maximum value of the amplitude of the droplet displacement fluctuation during the complete period of the acoustic wave is defined as the maximum displacement difference. Figure 5a shows the results of the maximum difference between displacements calculated with and without acoustic wave fields for different droplet sizes, indicating that the maximum displacement difference tends to increase exponentially with the increase of SPL. The smaller the droplet size of the cloud droplet is, the higher the maximum displacement difference will be. For the droplet size of 10 µm and the SPL of 143.4 dB, the increase in the cloud droplet displacement due to the action of sound waves can reach 9200 µm compared with the case when there is no sound wave action; however, for the cloud drops of 100 μm, the calculated maximum displacement increase is 3700 µm. It can be seen from Fig. 5b that as the droplet size increases, the displacement difference decreases significantly and finally converges. As suggested in Fig. 5a, when the SPL is less than 100 dB, the effect of sound waves on the displacement of the cloud droplets is no longer significant. When the SPL is less than 100 dB, the maximum displacement difference of different particle sizes is close to 0, and the displacement difference has an obvious upward trend from 100 dB.

Fig. 5.
Fig. 5.

(top) Maximum displacement difference between the cases with and without sound waves for different SPLs and droplet sizes, in which the sound frequency f = 50 Hz: (a) the relationships between the maximum displacement difference and different droplet sizes under different SPLs and (b) the relationships between the maximum displacement difference and droplet sizes for different SPLs. (bottom) Maximum displacement difference between the cases with and without sound waves for different frequencies and droplet sizes, in which SPL = 123.4 dB: (c) the relationships between the maximum displacement difference and droplet sizes for different frequencies and (d) the relationships between the maximum displacement difference and frequencies for different droplet sizes.

Citation: Journal of Atmospheric and Oceanic Technology 37, 9; 10.1175/JTECH-D-19-0210.1

Figures 5c and 5d show the relationship between the maximum displacement difference of the droplets and the frequency of the sound waves. With a higher frequency, the displacement difference is smaller. It can be seen from Fig. 5c that when the frequency is greater than 500 Hz, the effect of sound waves on the droplet displacement is significantly reduced. For the sound wave of the frequency f = 20 Hz, the maximum displacement difference does not change with the various droplet sizes when the droplet diameter is less than 20 μm. In Fig. 5d, when the sound frequency is greater than 1000 Hz, the curves are substantially coincident and close to zero, indicating that the sound wave is unable to promote the displacement of the cloud droplet. The results verify that the existence of sound waves not only causes the fluctuation of the cloud droplets, but also increases their displacement, which is beneficial to accelerate the falling of raindrops, and the effect is more pronounced with a higher SPL and lower frequency.

Dong et al. (2006) proposed the concept of effective agglomeration length Le, which is defined as the maximum distance for effective collisions of droplets. They also calculated the effective aggregation length values of two droplets for various droplet sizes due to different sound frequencies and SPLs. Because of the action of sound waves, the motion radius of each cloud droplet increases, which is equivalent to the increase in the effective aggregation length of cloud droplets Le−AC = Le + LAC, where Le−AC represents the effective agglomeration length due to the action of sound waves, and LAC indicates the oscillation amplitude caused by the action of the sound waves. The increase in the effective aggregation length of the cloud droplets affected by the action of sound waves indicates that the probability of collision between the two droplets that are farther apart is increased, which is equivalent to reducing the distance between the droplets. This result is another explanation of the aggregation effect of the sound waves on cloud droplets.

e. Velocity of cloud droplet

Figure 6a shows the curve representing the change in the cloud drop velocity with frequency, where the black line represents the maximum value in the stable oscillation process of the cloud drop velocity (i.e., the upper boundary), while the red line represents the lower boundary, and the blue line represents the values at the equilibrium position called “equilibrium velocity” υb in this study. The calculation results show that when the droplet size is 10 µm, the equilibrium velocity is approximately 0.003 038 m s−1. The equilibrium velocity increases with the increase in the droplet size. For small droplets, the gravity is easily balanced by the resistance. As the droplet size increases, the advantage of the gravity as a volume force gradually manifests itself. As an area force, the Stokes resistance can only be balanced with gravity at a large equilibrium velocity, which can be seen in the corresponding relationships between the reaction droplet size and the equilibrium velocity in Fig. 6c, as well as the typical droplet size range of the raindrops.

Fig. 6.
Fig. 6.

Curves of droplet velocity. (a) Change with sound frequency for a cloud droplet diameter of 10 μm with SPL = 123.4 dB. (b) Change with SPL for a cloud droplet diameter of 10 μm with sound frequency f = 50 Hz. (c) Change with droplet sizes with sound frequency f = 50 Hz and SPL = 123.4 dB.

Citation: Journal of Atmospheric and Oceanic Technology 37, 9; 10.1175/JTECH-D-19-0210.1

It can be seen from Fig. 6a that as the frequency increases, the rapidly changing flow field has more difficulty driving the cloud droplet, and the oscillation range of the cloud droplet velocity is reduced. However, the equilibrium velocity remains constant with the frequency variation. Figure 6b shows the relationship between cloud drop velocity and SPL. When SPL is small, the velocity oscillation is quite weak; while when SPL becomes larger, the range of the velocity oscillation is rapidly expanded. Therefore, for a cloud droplet of 10 µm, only when the SPL of the sound wave is greater than a critical value, can significant fluctuations of velocity be generated, resulting in a significant oscillation of displacement. Similarly, with the frequency, the magnitude of the equilibrium velocity does not change with SPL as shown in Figs. 6a and 6b. When the frequency of sound wave expands from 20 to 200 Hz, the ratio of the cloud drop velocity amplitude to the equilibrium velocity is reduced from 33 to 30, and when SPL increases from 83.4 to 143.4 dB, this ratio is increased from 0.33 to 326. The ratio comparison indicates that the velocity of the cloud droplets is more sensitive to SPL than to frequency. In this study, the concept of “critical SPL,” SPLc, is proposed (i.e., the sound pressure level at which the velocity begins to fluctuate significantly). For a certain droplet, a critical velocity oscillation threshold is given, which is υamplitude = υb/2, where υamplitude is the velocity amplitude of the cloud drop. When the amplitude of its velocity is no less than this value, the droplet fluctuation is considered significant. The SPLc values are different for cloud droplets of various droplet sizes, and for the droplet of 10 µm, SPLc = 88.2 dB.

Droplets in still air settle because of gravity and quickly reach a constant velocity called the terminal velocity (Finlay 2001). In the simulation, it is assumed that the air is still (i.e., uf = 0) and the terminal velocity for droplets with different diameters is calculated, as indicated by the green dot in Fig. 6c, which also shows the curves of the droplet velocity changing with different droplet sizes due to the action of sound waves. The figure indicates that equilibrium velocity increases as the droplet size increases, but the fluctuation range of the velocity rapidly decreases, and the ratio of the amplitude to the equilibrium velocity decreases from 132 at 5 µm to 0.039 at 100 µm. As the droplet size increases, the effect of sound waves becomes weaker, and the terminal velocity and the equilibrium velocity coincide. This phenomenon indicates that the influence of the sound waves on the downward movement of the cloud droplets is primarily manifested by the formation of a certain oscillation based on the terminal velocity, the amplitude of which is related to the frequency and SPL of the sound wave, as well as the droplet size. Such conclusion is contrary to that in reference (Hwang 1990), which shows that in an oscillating flow, the falling velocity of the droplets is smaller than the final velocity of the same droplets in the stationary fluid.

f. Attribution of forces

The cloud droplets in the air are subjected to four forces from the air, including Stokes force, Basset force, pressure gradient force, and additional mass force. These forces also exhibit oscillatory properties when the fluid oscillates due to the action of the sound field. Over time, the forces transit into a relatively stable oscillation state, in which the Stokes force oscillates up and down at a nonzero equilibrium position, while other forces gradually move closer to zero. Figure 7a shows the trend of the acceleration amplitude generated by the four forces as a function of frequency. The Stokes force produces the largest amplitude of acceleration, while the additional mass force produces the smallest amplitude of acceleration. As the frequency increases, the amplitudes of the four forces gradually increase. When the frequency is increased from 20 to 200 Hz, the amplitude of the acceleration generated by the Stokes force is increased by 4.5 times, while the Basset force, the pressure gradient force, and the additional mass force are increased by 27.8, 9.0, and 89 times, respectively. The Stokes force grows the slowest with the increase in frequency, the added mass force increases the fastest, and the growth of each force tends toward convergence. However, as the frequency increases, the acceleration at the equilibrium position generated by the Stokes force remains near a constant value of −9.79 m s−2, as shown in Fig. 7b. The acceleration generated by the gravity after subtracting the buoyancy is (ρpρg) × g/ρp ≈ 9.80 m s−2. These two values are approximately opposite to each other, indicating that the cloud droplet is mainly resisted by the Stokes force. Such phenomenon can also explain the simulation results in which the equilibrium velocity is basically consistent with the terminal velocity.

Fig. 7.
Fig. 7.

(a) Curves of acceleration amplitude generated by the four forces as a function of frequency for a cloud droplet diameter of 10 μm due to the sound wave with SPL = 123.4 dB. (b) Curves of acceleration at the equilibrium position generated by Stokes force as a function of frequency for cloud droplet of 10 μm due to the sound wave with SPL = 123.4 dB. (c) Curves of acceleration amplitude change with SPLs generated by different forces for cloud droplet of 10 μm due to the sound wave with frequency f = 50 Hz. (d) Curves of acceleration amplitude change with droplet sizes generated by different forces due to the sound wave with frequency f = 50 Hz and SPL = 123.4 dB.

Citation: Journal of Atmospheric and Oceanic Technology 37, 9; 10.1175/JTECH-D-19-0210.1

Figure 7c shows the changing curves of the acceleration amplitude generated by Stokes force, Basset force, pressure gradient force, and additional mass force with SPLs, respectively. As can be seen from Fig. 7c, the four curves are straight parallel lines, indicating that as the SPL increases, the forces increase exponentially. When SPL = 143.4 dB, the amplitude of the acceleration oscillation generated by each force is nearly 103 times that when SPL = 83.4 dB and the acceleration amplitude is increased by about 100.5 times for each SPL increment of 10 dB. Meanwhile, the acceleration at the equilibrium position generated by Stokes does not change with SPL; the value of which is similar to the gravitational acceleration. It can be seen from the results that the influence of the sound wave frequency and the SPL on the forces on the cloud droplets is primarily reflected by the amplitude of each force during the oscillation, which increases with the growth of sound frequency and SPL but stays constant at the equilibrium position.

Figure 7d shows the curves of the amplitude of the acceleration generated by the four forces with the size of the cloud droplets. Each force responds differently to droplet sizes. With the increase in droplet sizes, the amplitude of the acceleration generated by the Stokes force gradually decreases, and the Basset force first increases and then decreases. When the droplet size is 40 µm, the Basset force reaches the maximum value, 1.5365 m s−2. Because the acceleration generated by the pressure gradient force is only related to the fluid velocity field uf, the pressure gradient force does not change with the droplet size. For the additional mass force, the amplitude of the generated acceleration first increases rapidly and then tends to converge with the increment of the droplet size.

To better show the change in force, Fig. 8 shows the curve of the ratio of the magnitude of each force to time in the stable oscillation stage, as well as the cloud droplet velocity, fluid velocity, and the relative velocity between the two. As can be seen from Fig. 8, the Stokes force and gravity dominate the cloud droplet, and the fluctuations of the Basset force and Stokes force are not synchronized. The Stokes force fluctuates synchronously with the velocity difference between the fluid and the droplets. Figure 8a shows the case with the cloud droplet size of 10 µm, while Fig. 8b shows the case with the droplet size of 100 µm, and there is a significant difference between the two cases. For a cloud droplet of 10 µm, its velocity is almost identical to that of the fluid, except that the phase is slightly behind the fluid, resulting in the difference in the fluctuation of velocity with time. Because the magnitude of the viscous resistance is related to the velocity difference (ufυ), the fluctuations of viscous resistance and velocity difference are synchronized. For a cloud droplet of 100 µm, the fluctuation characteristics of the droplet velocity are not prominent, but since the variation of the fluid velocity is relatively large, the velocity difference is increased, and the change in the velocity difference is almost synchronized with the fluid velocity. Such phenomenon, in which the droplet velocity significantly lags behind the fluid as the droplet size increases, is consistent with the conclusions of the reference (Sujith et al. 1999).

Fig. 8.
Fig. 8.

Curve of the ratio of the magnitude of each force to time in the stable oscillation stage due to the sound wave with frequency f = 50 Hz and SPL = 123.4 dB for a droplet size of (a) d = 10 μm and (b) d = 100 μm.

Citation: Journal of Atmospheric and Oceanic Technology 37, 9; 10.1175/JTECH-D-19-0210.1

4. Conclusions

In this study, a numerical simulation of the cloud droplet movement due to the simultaneous action of the sound field and gravity field is carried out. Various forces acting on the droplet resulting from the oscillating fluid are considered to analyze the motion process of droplets. The effects of the sound waves with different parameters, mainly referring to its frequency and SPL, on the motion process of droplets with different sizes are investigated, and the following conclusions can be derived:

The fluctuation of cloud droplet velocity caused by the sound waves promotes the relative motion between droplets. Meanwhile, the different displacement increments of cloud droplets of different sizes caused by the sound waves also reduce the distance between droplets and increase the effective aggregation length, which is beneficial to the coagulation of droplets. Therefore, it is believed that the sound waves can be used for precipitation enhancement based on these theoretical analyses. Some large-scale numerical simulations of the movement of cloud droplets swarm considering the interaction between fluids and droplets due to sound waves is still required to better guide field practices in the future.

Acknowledgments

This research was supported by the Integration Program of the Major Research Plan of the National Natural Science Foundation of China (91847302) and the National Natural Science Foundation of China (51879137, 51979276).

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