Research on the artificial control method of the gas nuclei spectrum in the small-scale experimental pool under atmospheric pressure

1 Introduction

From a microscopic perspective, the occurrence of cavitation behavior in water can be explained as that, in a short period, the flow field is disturbed, the pressure of the flow field in the water is reduced, and the micrometer-sized gas nuclei in the water rapidly grow into millimeter or centimeter-sized cavitation bubbles. The natural environment waters have the characteristics of many microorganisms, complex geological structures, and polytropic water quality conditions. These characteristics together create a mesoscale and concentration-rich gas nuclei group in the waters, which provides a good environmental basis for cavitation [1,2]. In the prototype experiment, due to the large size of the prototype model and the large volume of disturbed fluid, the cavitation occurrence area increases accordingly, and the cavitation phenomenon is obvious. Under laboratory conditions, the purity of experimental water is generally higher than that of natural water. The experimental water body is generally not replaced, and the experimental water body is reused for a long time. Compared with the water body in the natural environment, gas content in the water body after repeated use will be reduced. The aforementioned factors lead to the fact that the cavitation experiment in the laboratory is weaker than the prototype experiment in natural waters, and the scale effect occurs [3–5]. Therefore, in order to ensure that the cavitation experiment meets the requirements of the scale effect, artificial gas nucleus seeding must be carried out in the laboratory waters before the formal experiment to ensure that the water environment between the experimental waters and the natural waters is similar.

Before and after the artificial seeding of gas nuclei, it is necessary to master the content and difference of gas nuclei in laboratory waters and natural waters. In order to characterize the difference in gas nucleus content, the concept of gas nucleus spectrum is introduced. The gas nucleus spectrum shows the size of the gas nucleus in water and the number of gas nuclei of the corresponding size in water. At present, the measurement methods of the gas nucleus spectrum in water mainly include hydrodynamic measurement methods [6,7], optical measurement methods [8–10], and acoustic measurement methods [11–15]. The hydrodynamic measurement method uses a variable-diameter cavitator to rapidly change the velocity of the flow field, so that cavitation is more likely to occur. By measuring the pressure changes upstream and downstream of the cavitation area, the gas nucleus spectrum in water is solved. This measurement method can effectively avoid the influence of solid impurities in water on cavitation and improve the measurement accuracy of the gas nucleus spectrum in water. However, this method has higher requirements on the layout of the measuring pipeline and the measurement accuracy of the pressure. The optical measurement method takes a picture of the area to be measured, and the gas nucleus is highlighted in the water. The number of gas nuclei in water is counted by image grayscale processing and automatic counting methods. There are two main disadvantages of this method. First, the image of solid impurities in water is similar to that of the gas nucleus, which requires the water quality in the measurement area to be relatively pure. Second, the picture has a depth of field, which determines the depth of the measurement area and directly affects the number of gas nucleus. Therefore, the use of optical measurement methods requires some experience. The acoustic measurement method uses the resonance characteristics of the gas nucleus under the excitation of the acoustic signal to establish the relationship between the acoustic signal and the size and quantity of the gas nucleus. The attenuation of the acoustic signal is measured by the transducer and the hydrophone, so as to solve the gas nucleus spectrum in the measurement area. This method can quickly measure the gas nucleus spectrum in water and avoid the influence of solid impurities in water [16].

With the continuous development of gas nucleus measurement technology, some scholars have carried out research on the natural gas nucleus spectrum in water. Johnson et al. measured the gas nuclei spectrum in water in natural sea areas and found that the number of gas nuclei in water increases with the increase of wind speed and the decrease of water depth [17–25]. Norris et al. found that when the free surface was covered with ice, the number of gas nuclei in water decreased significantly, which further indicated that the number of gas nuclei in water would increase under the action of wind, wave, and current. Similarly, some scholars began to pay attention to the distribution of gas nuclei spectrum in laboratory waters [26,27]. Billet, Gates, and Venning et al. explored the relationship between the number of gas nuclei and the gas content in the circulating water tunnel in the unsaturated state and the saturated state [28]. The study found that when the gas content in the water tunnel is not saturated, the number of gas nuclei increases with the increase of the gas content in the water tunnel, but the increase in the number of gas nuclei is smaller than that of the gas content. When the gas content in the water tunnel is saturated, the number of gas nuclei increases sharply [29,30].

Due to the different lengths of experiments, the “survival time” of gas nuclei in the experimental waters varies, so the stability of the gas nuclei in the experimental waters directly affects the experimental results. Lida et al. used the laser diffraction method in the sonochemical reaction vessel to study the gas nuclei spectrum under the action of sound [31]. Studies have shown that the size and number of gas nuclei in water gradually increase with the increase of ultrasonication time. Khoo et al. used a cavitation sensitivity measuring instrument to study the gas nuclei spectrum in different periods and regions of water in circulating water tunnels in Australia and Japan [6,32]. The study found that the distribution of the gas nuclear spectrum is different in different regions. When the gas content in the water tunnel is not saturated, the gas nuclei spectrum is less affected by the gas content; when the gas content in the water tunnel is saturated, the gas nuclei concentration increases significantly. Since the distribution of the gas nuclei spectrum is significantly affected by the area and gas content, in order to ensure that the gas nuclei spectrum in the water area meets the experimental requirements, it is necessary to manually control the gas nuclei spectrum in the experimental pool. (Follow-up, it is emphasized that the gas nuclei spectrum needs to increase gas nuclei, because of the scaling effect and the difference between the gas nuclei spectrum of the experimental waters and the natural waters.)

In terms of artificial aeration research, Venning et al. used the jet shear method to explore the difference in the gas nuclei spectrum in water by changing the jet pressure and tried to use the physical properties of gas nuclei as particles in the PIV measurement process [33]. Russell et al. carried out artificial aeration experiments in a circulating water tunnel, using Mie scattering imaging technology to measure the gas nuclei spectrum in water [9]. The study found that the flow velocity in the water tunnel increased, resulting in a shorter dissolution time of the gas nuclei in the water and a larger concentration of the gas nuclei. At the same time, it was found in the experiment that different areas in the water tunnel have different pressure values, resulting in different gas nuclei spectrum distributions [34–36].

It was clearly pointed out in the 23rd ITTC conference that different water tunnels with the same gas content have different distributions of the gas nuclei spectrum in the water tunnels. Gas nuclei survive longer in larger water tunnels. Moreover, due to the different basic settings and functions of each water tunnel, the research conclusion of a single water tunnel cannot be directly applied to other water tunnels or pools, which also determines that the gas nuclei spectrum distribution in different structural water tunnels or pools needs to be studied separately. In the cross-medium vehicle experiment, the vehicle model sailed from the bottom of the pool to the air and experienced several stages of water navigation, penetrating the free liquid surface, and air navigation [13,37–39]. Such experiments often require a large span in the depth direction of the pool and a certain time span in the water navigation process of the vehicle, so it is very necessary to ascertain time-space distribution characteristics of the gas nuclei spectrum in the water area of the cross-medium vehicle experiment.

In the previous studies on cavitation, many cavitation experiments have the following problems. The speed of the scale model is limited, and the decompression method is used in the experiment to meet the equal cavitation number of the prototype and the scale. However, various sensors are loaded in the model shell, and the decompression operation takes a long time, which increases the risks of water leakage, sensor failure, and circuit failure. With the mature application of high-efficiency gas compression, high-precision PLC control, high-sensitivity solenoid valve, large-scale pool, and other technologies and devices, the foundation has been laid for the realization of ultra-high-speed technology for scale models. At the same time, with the realization of ultra-high speed, it also indicates that the tedious decompression operation can be avoided and the operation time can be reduced, thus improving the experimental efficiency. In view of the above characteristics of cavitation experiments, in order to better carry out cavitation experimental research, explore the experimental effects and load characteristics under different gas nuclei environments based on square water tank, micron gas nuclei generator, and acoustic nuclei measurement system, this article changes the aeration time, measures the gas nuclei spectrum in water, and explores the distribution law of gas nuclei spectrum in water.

The existing research has explored acoustic measurement methods for gas nuclei spectrum in water. This article uses acoustic measurement methods to explore the effect of aeration duration on the gas nuclei spectrum in water. By using the basic theory of statistics, construct a gas nucleus spectrum distribution function to form a simple measurement method and improve measurement efficiency.

2 Experiment setup

In this article, an artificial aeration and gas nuclei spectrum measurement system is designed, as shown in Figure 1. This experiment uses a symmetrical square pool as the experimental carrier. The side length of the square pool is 1 m, and the water depth is 0.9 m. Transducers and hydrophones are arranged at different depths in the water. The transducers emit acoustic signals of different frequencies. Under the interference of the gas nuclei, the acoustic signals will attenuate, and the attenuated acoustic signals will be measured by the hydrophones. The transmission and reception of the signal are collected by the data collector to the computer. The process of adding gas nuclei in water is as follows. The gas nuclei generator sucks the liquid in the pool through the water intake, and the sucked liquid enters the booster pump inside the aerator. Under the entrainment action of the impeller of the booster pump, the fluid is broken and micron-sized gas nuclei are generated, thereby forming a gas–liquid two-phase flow mixed with the gas nuclei and the fluid, and the gas nuclei flow returns to the square pool from the water outlet.

Figure 1

Experimental system. (1) square pool, (2) Transducer, (3) hydrophone, (4) data collector, (5) computer, (6) the gas nuclei generator, (7) the water intake, and (8) the water outlet.

The measurement of the gas nuclei spectrum adopts the acoustic measurement method, and the acoustic bubble spectrometer produced by Dynaflow is used to measure and analyze the acoustic signal under the influence of the gas nuclei. Experiments in small-scale square pools are often small-scale experimental models and relatively small experimental speeds. In such experiments, gas nuclei need time and a low-pressure environment to grow into large-sized cavitation bubbles, and the gas nuclei with a size of 5–15 μm have the most obvious effect on cavitation. Therefore, the focus of this article is on the gas nuclei within this scale. Here, the size of the gas nuclei will be expanded to be between 0 and 100 μm, and the acoustic detection method of gas nuclei spectroscopy will be verified. Corresponding to the gas nuclei within this size range, it is necessary to select transducers and hydrophones with appropriate parameters. When the gas nuclei are excited by an acoustic signal of a specific frequency, it will resonate. Research shows that the relationship between the radius of the gas nuclei and the excitation frequency is as follows [40]:

where c is the sound speed in pure water, b is the damping, ω is the frequency of sound, r is the radius of nuclei, ω 0 is the natural frequency of nuclei, and m ₁ is the kernel of the equation.

According to formula (1), some frequencies are selected to solve and plot, and Figure 2 can be obtained:

Figure 2

The kernel function m is plotted against the bubble radius for different sound frequencies.

It can be seen from Figure 2 that the peak values of different frequency curves correspond to the radius value of the resonant gas nuclei. The higher the frequency of the acoustic excitation signal, the smaller the size of the measured gas nuclei; and during the change process, the frequency change speed increases with the increase of the radius. According to this calculation idea, the required frequency range of the acoustic signal can be converted according to the size of the micrometer-sized gas nuclei to be measured, and finally, the full-scale measurement of the micron-sized gas nuclei can be realized.

3 Experimental results and discussion

3.1 Gas nuclei spectrum solution

According to the layout requirements of the experimental device, transducers and hydrophones are arranged in the pool, and micron gas nuclei, whose medium is air, are mixed into the water through the gas nuclei generator. By controlling the length of aeration, the purpose of controlling the concentration of gas nuclei in water is achieved. The acoustic signal originates from the transducer, is transmitted, and is captured by the hydrophone. Taking the aeration time of 5 and 20 s as an example, the signal emitted by the transducer and the signal curve received by the hydrophone are shown in Figure 3. It can be found from the figure that during this transmission process, the existence of gas nuclei generates a signal, and there is a time difference between received signals. And with the increase in the concentration of gas nuclei, the time for the acoustic signal to reach the hydrophone is later. It shows that the propagation velocity of the underwater acoustic signal decreases with the increase of the concentration of the gas nuclei, and the gas nuclei “block” the propagation of the acoustic signal underwater.

Figure 3

Measurement results of acoustic signals in water under different gas nuclei concentrations.

At the same time, under the action of the gas nuclei in the water, the sound intensity of the received signal is obviously weaker than that of the transmitted signal. During the experiment, in order to ensure that the acoustic signal penetrates the high-concentration gas-nuclei water smoothly, it is captured by the hydrophone. As the concentration of gas nuclei increases, the emission sound intensity must be increased appropriately. The amplitude difference between the transmitted signal and the received signal after attenuation increases with the increase of aeration time. In addition to visualizing the difference between peaks, this attenuation effect needs to be expressed numerically through the sound attenuation coefficient. To sum up, the “impeding effect” of gas nuclei on the propagation of underwater acoustic signals is manifested in two aspects. On the one hand, the increase in the concentration of gas nuclei will reduce the speed of sound in water; on the other hand, the increase in the concentration of gas nuclei will dissipate the acoustic energy.

As shown in Figure 4, the underwater acoustic signal is processed to obtain the mean square value of the sound pressure of the transmitted signal and the received signal, which paves the way for further solving the gas nuclei spectrum. It is not difficult to find from the figure that with the increase of the duration/concentration of gas nuclei in water, the mean square difference of the sound pressure between the transmitted signal and the received signal also increases. This rule is consistent with the rule obtained in Figure 3, which once again verifies the hindering effect of the gas nuclei on the propagation of the acoustic signal [41]

(2) u ( ω ) = c c ̇ c = c Δ l / Δ t ,

where u ( ω ) is the acoustic factor, c ̇ c is the sound speed in the mixture water, Δ l is the distance between transducer and hydrophone, and Δ t is the sound propagation duration.

(3) v ( f ) = − c ln ( p ¯ 2 ( f ) / p ¯ ref 2 ( f ) ) 4 π f Δ l ,

where v ( f ) is the sound attenuation coefficient, p ¯ 2 ( f ) is the MSA of a pressure signal at a distance Δ l in the bubbly medium, p ¯ ref 2 ( f ) is the MSA of a pressure signal at a distance Δ l in the pure fluid, and f is the frequency of the acoustic signal.

Figure 4

The sound pressure mean square value of the transmitted signal and received signal in water with different concentrations of gas nuclei. (a) Aeration time 5 s and (b) aeration time 20 s.

Using formulas (2) and (3), the sound velocity coefficient and sound attenuation coefficient of the acoustic signal in water under the aerated state are solved, and Figure 5 is obtained. As shown in the figure, the speed of sound coefficient increases with frequency, first decreases suddenly, then increases, and then decreases in stages. The sound attenuation coefficient increases and decreases with the frequency. It decreases first, then increases, and gradually decreases in the later period.

Figure 5

Variation curve of sound velocity coefficient and sound attenuation coefficient with frequency (aeration time is 5 s).

Referring to the processing method in the literature [42], Eq. (1) can be transformed into the following formula:

(4) 4 π c 2 ∫ r 1 r 2 b ω r ( ω 0 2 − ω 2 ) 2 + 4 b 2 ω 2 N ( r ) d r = u v ,

where r ₁ is the minimum nuclei radius, r ₂ is the maximum nuclei radius, and N ( r ) is the number of gas nuclei corresponding to the gas nuclei with a radius of r in a unit volume of the water sample.

To solve Eq. (4), the discretization is made by the complex trapezoidal formula and then solved by the Tikhonov regularization method. The gas nuclei spectrum equations were constructed and solved, and the gas nuclei number density distribution function formula (5) was stipulated:

(5) f ( r j ) = N ( r j ) N ,

where f ( r j ) is the number density of gas nuclei, N is the total number of gas nuclei in a unit volume of water sample, and N ( r j ) is the number of gas nuclei corresponding to the gas nuclei with a radius of r j in a unit volume of water sample.

Finally, when the aeration time is 5 s, the gas nuclei spectrum distribution map in water is obtained. It is not difficult to find from Figure 6 that the gas nuclei spectrum has a single peak at the position of the number of small-sized gas nuclei, and the overall distribution is exponential. This is because as the size of the gas nuclei increases, the buoyancy force is greater than the tension force, and the gas nuclei float up to the free liquid surface and rupture or directly undergo positive mass diffusion in the liquid, resulting in rupture.

Figure 6

Gas nuclei spectrum in water when aerated for 5 s.

3.2 Analysis of aeration results

The purpose of the aerated nuclei in water is to create a cavitation experiment environment to ensure that the experiment meets a similar scale, so as to realize the mutual conversion between the model and the prototype experiment load. However, in order to complete the experiment within the limited “survival” time of high concentration gas nuclei environment, it is necessary to simplify the experimental process and coordinate the experimental process. At the same time, in order to avoid the leakage risk of the transducer and hydrophone in the decompression state, it is necessary to master the relationship between the gas nuclei spectrum and the aeration time in the water before the formal experiment. Therefore, in the formal experiment process, as long as the aeration time is controlled, the gas nuclei spectrum in water can be controlled, so as to reduce the use of transducers and hydrophones, simplify the experimental process, and reduce the experimental time.

By controlling the aeration time, the gas nuclei spectrum under different aeration times is obtained, as shown in Figure 7. From the diagram, it is not difficult to find that regardless of the length of aeration time, the gas nuclei spectrum in water always presents an exponential distribution. With the increase of aeration time, the increase of small-size gas nuclei concentration is greater than that of large-size gas nuclei. This also means that small-sized nuclei are more affected by tension than buoyancy and gravity, so small-sized nuclei can survive longer and accumulate in water than large-sized nuclei. Overall, it is feasible to control the gas nuclei spectrum by controlling the time length.

Figure 7

Gas nuclei spectrum in water with different aeration times.

3.3 Simple measurement and prediction method of the gas nuclei spectrum in water

Through the above research, it is found that the acoustic measurement method is effective and reliable for the measurement of the gas nuclei spectrum in water, and the distribution of the obtained gas nuclei spectrum satisfies a certain rule. At the same time, it can be seen from practice that the characteristics of expensive transducers and hydrophones limit the frequency emission and measurement range of acoustic signals. In view of the above two points, this article attempts to predict the gas nuclei spectrum for the limited experimental range of transducers and hydrophones, a limited number of gas nuclei spectrum data and numerical methods.

Affected by the physical characteristics of the gas nuclei, the gas nuclei spectrum has the following outstanding characteristics. Due to the buoyancy effect, the large gas nuclei are more likely to float up, resulting in collapse during the floating process or after reaching the free surface. In other words, the survival probability of small gas nuclei is greater, the peak of the gas nuclei spectrum must appear in the position of small-size gas nuclei, and the curve of the gas nuclei spectrum will be biased toward small-size gas nuclei. However, the gas nuclei spectrum characterizes the number of gas nuclei of a certain size in water, so the gas nuclei spectrum will not be less than zero. In the past, the research object of the gas nuclei distribution function was the gas nuclei in the natural survival state. In this article, combined with the characteristics of artificial aeration, two previous research results were selected and these equations were constrained. The specific equations and constraints are as follows:

(6) n ( r j ) = N ( r j ) N = k 1 ⋅ r j r , model 1 n ( r j ) = N ( r j ) N = k 2 e − β / r j ⋅ r j α , model 2 ,

(7) n ( r ) ≥ 0 , r > 0 n ( r ) → 0 , r → ∞ ,

where k ₁, k ₂, α, β are proportional coefficients. It is worth noting that k ₂ is a normalized coefficient in previous studies, and its value is related to α. This article modifies it to an independent proportional coefficient. According to formulas (6) and (7), the experimental results are compared with the calculation results of the mathematical model, and then Figure 8 is drawn.

Figure 8

The prediction results of different models for the gas nuclei spectrum in water under aeration. (a) The fitting results of the gas nuclei spectrum when the aeration time is 5 s. (b) The fitting results of the gas nuclei spectrum when the aeration time is 20 s.

From Figure 8, it can be found that the two mathematical models are in good agreement with the experimental values. This shows that the artificial aeration scheme can effectively change the spectrum of gas nuclei in water, and this change is consistent with the distribution of gas nuclei of different sizes in nature. In the two models, the modified model 2 is in good agreement with the experimental results, especially at less than 10 μm, and the gas nuclei in this size range have obvious influence on cavitation. At the same time, combined with the better experimental and model comparison results, and the analysis of formula (5), it can be known that based on the known three or more experimental data, it can be fitted by the formula to obtain the number of gas nuclei of other unmeasured sizes and improve the experimental efficiency.

Through research, it is found that the artificial aeration method can effectively change the gas nuclei spectrum in water and increase the concentration of gas nuclei. By using the underwater acoustic measurement method, the change in the gas nuclei spectrum can be captured sensitively. The gas nuclei spectrum in water after aeration is in good agreement with the mathematical model of the gas nuclei spectrum under non-artificial intervention, which shows that the distribution of gas nuclei in water under artificial aeration is similar to that under non-artificial intervention, which is conducive to the occurrence of cavitation. At the same time, it shows that the combination of experiment and numerical method can reduce the measuring state, reduce the measurement cost, and improve the measurement efficiency.

4 Conclusions

In this article, the square pool is taken as the research object, and the aerator is used to change the gas nuclei content in the water. Using underwater acoustic equipment such as transducers and hydrophones and using acoustic measurement methods to explore the feasibility of changing the gas nuclei spectrum in water by artificial aeration under normal pressure. The gas nuclei spectrum distribution function based on the natural state is used for reference and modified by predecessors, so as to explore the difference between the gas nuclei spectrum in the state of artificial aeration and the distribution of gas nuclei in natural waters. Finally, the following conclusions can be obtained:

1. In the process of the gas nuclei spectrum in water measurement and solving, the following problems will cause errors and uncertainties, including (a) the sound propagation distance should be the chip position of the transducer and the hydrophone, not the distance between the transducer and the hydrophone shell; (b) formula (4) is a pathological equation, and the constraint solution must be carried out using regular methods during the solution process to reduce uncertainties.

2. In pure experimental waters, the gas nuclei in the water can hardly be detected by the acoustic measurement method. The method of artificial aeration can effectively increase the gas nuclei in water, change the distribution of gas nuclei in water, and improve the environment of cavitation experiments.

3. Gas nuclei in water will hinder the propagation of acoustic signals in water, and the obstruction will increase with the increase of the concentration of gas nuclei.

4. The gas nuclei spectrum function in traditional naturally existing waters can predict the distribution of gas nuclei in the state of artificial aeration. However, the modified distribution function is more consistent with the artificially improved gas nuclei distribution, and the distribution law of the gas nuclei spectrum under the artificial aeration state is similar to the distribution law of the gas nuclei spectrum under the natural state.

5. The revised gas nuclei spectrum distribution function provides the possibility to improve the gas nuclei spectrum measurement technology in water and the measurement efficiency.