An Yu, Yi-fu Wang, Qing-hong Tang, Da-qing Zhou

1. College of Energy and Electrical Engineering, Hohai University, Nanjing 210098, China

2. College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China

Abstract: In this paper, the flow field around NACA0015 hydrofoil is calculated under the condition of two-phase cavitating flow to provide a theoretical guidance for reducing the cavitation noise. A modified turbulence model coupled with the Zwart cavitation model is used to calculate the flow field. According to the computed sound source data, the dipole sound pressure distribution diagrams for the extremely short time and the complete time around the hydrofoil under different working conditions are obtained.The sound pressure distribution and the sound pressure amplitude are analyzed in detail. In addition, the field points around the hydrofoil are selected to generate the corresponding frequency response function curves, which are analyzed from the aspects of the trend, the variation and the extreme value. The results show that the dipole characteristics are gradually diminished and finally disappear with the increase of the frequency. At the frequency of the vapor volume fraction fluctuation, the noise level at the field point will have an extreme value.

Key words: Cavitating flow, dipole noise, sound field distribution, frequency response

Introduction

The cavitation is a phase transition process between a liquid and its vapor in the liquid or at the interface of the liquid and the solid. It often happens in the ship engineering, the underwater submarine, the water conservancy engineering, the water conservancy machinery and other fields. The cavitation can cause some damages, such as pitting on the surface of the solid, and vibration and noise, leading to the reduction of the hydraulic unit efficiency, the damage of the propeller and the turbine blades, and the exposure of the underwater submarine. Therefore, it is of great significance to study the cavitation and its mechanism.In recent decades, the cavitation was widely studied for the turbines, the propellers, the water pumps with many research results.

Yu et al.[1]studied the ventilation cavitation around an NACA0015 hydrofoil by using a new simulation method for the numerical simulation. The results showed that when the ventilation volume is large, it has a strong inhibitory effect on the natural cavitation, and when the ventilation volume is small,the vapor volume level does not change much.Wimshurst et al.[2]studied the effect of the cavitation on the performance of tidal turbines, with the cavitation analysis performed by the blade resolved computations, and with consideration of the blockage and the spanwise flow. Zhang et al.[3]investigated the influences of bubble size distribution on propagation of acoustic waves in dilute polydisperse bubbly liquids. Ji et al.[4]used the non-uniform wake to analyze the unsteady cavitation turbulent flows around a conventional propeller and compared them with experimental results. In addition, the cavity volume was also calculated and analyzed, and it was concluded that the acceleration caused by the cavity volume changes is the main source of the pressure fluctuations caused by the propeller cavitation. Ji also did the experimental and numerical investigations of transient cavitating vortical flow in the jet pump[5]and propeller with vortex generator[6]. Wake and Riedelbauch[7]studied and analyzed the occurrence process of the cavitation vortices between blades under the dead load of a Francis turbine. Konno et al.[8]carried out model experiments and numerical simulations under different conditions to study the cavitation fracture caused by the propeller tip vortex.The unsteady cavitation flow in a pump turbine was simulated by Liu et al.[9], and it was found that the opening of the guide vane had a great influence on the cavitation. Huang et al.[10]reviewed the recent year’s investigations of transient flow structure and unsteady mechanism of cavitating flow.

Aktas et al.[11]studied the cavitation noise of the propellers, with cavitation tests in a medium-sized tunnel, and found that the underwater radiated noise(URN) spectra, based on the tunnel tests, are more consistent with the full-scale URN in the low and medium frequency ranges than in the high frequency range. Wei et al.[12]studied the non-cavitation noise produced by propellers running in the tail of a submarine. He indicated the importance of the submarine scattering effect on the non-cavitation noise of propellers based on the prediction results of the acoustic field of submarine propellers. Lafeber et al.[13]discussed the computational methods and experimental methods for predicting the underwater radiation noise caused by the cavitation of the ship propeller, and analyzed the prediction results of different models. Zhang et al.[14]studied the effect of the gas content in the water on the propeller cavitation noise level, by measuring the radiated noise level of a ship’s propeller model. Bao et al.[15]proposed an improved empirical mode decomposition (EMD)technique, which realized the adaptive and effective extraction of the cavitation noise with the modulation from the ship-radiated noise and alleviated the adverse effect of the noise in the process of extracting the modulation information. Lee and Seo[16]discussed and verified the applicability of the spectral kurtosis to the identification and the detection of the tip vortex cavitation in the propeller. Zhang et al.[17]presented a theoretical model to describe the cavitation noise of the ship propeller with the random acoustic pulse sequence, derived a general spectral formula for the propeller cavitation noise and introduced a temporal random sequence to explain the phenomena in the noise spectrogram of the propeller cavitation. Yasui et al.[18]performed a numerical simulation of the cavitation noise under a certain condition. The results showed that the temporal fluctuation of the number of bubbles will lead to the generation of the broad-band noise. Lee et al.[19]proposed a multi-parameter inversion scheme to determine the positions and the intensity of the cavitation noise sources, and based on this, established a hull pressure model. The inversion results are in good agreement with the results of the propeller cavitation dynamics analysis, and the model data are basically consistent with the experimental data. Zhang et al.[20]studied the cavitation noise of the water jet, collected and analyzed the cavitation noise signals from different nozzles and nozzle exits,calculated the maximum Lyapunov exponent and reached related conclusions. Bertetta et al.[21]proposed a coupling method of the algorithm and the panel code to design the CPP propellers with different pitch distances, so as to reduce the cavitation during their operations and thus reduce the cavitation noise.This method was experimentally validated. Wang et al.[22]conducted experimental studies of the cavitation characteristics and the induced noise of different types of fluidic pump cavitation reactors, analyzed the influence of the throat length, the throat type and the diffusion angle, with related conclusions, which provided a guidance for the design and the application of the jet pump cavitation reactor (JPCR). Wittekind and Schuster[23]proposed an acoustic ship model and the physical process of studying the cavitation noise by observing the broadband part of the spectrum of the low-frequency noise for full-size ships, and estimated the possible influence of the cavitation noise on the background noise in the sea. According to the application of the cavitation and acoustic phenomena in the large gas cleaning, Camerotto et al.[24]studied the effects of the low volume surface tension and different oxygen concentrations on the bubble activity ,with related conclusions. Bao et al.[25]presents an adaptive method to extract the modulated cavitation noise by combining the empirical mode decomposition and the singular value decomposition,and the practicability of the method was verified by simulation results.

This paper mainly focuses on the cavitation noise problem and studies the influence of the cavitation number on the cavitation induced noise. In this study,the filter-based turbulence model (FBM) coupled with the Zwart cavitation model is used to simulate the cavitation characteristics around the hydrofoil to obtain the acoustic field data. The acoustic field data are imported into the virtual lab and the direct boundary element method is used to calculate the acoustic field distribution to study the relationship between the different cavitation numbers and the acoustic field distribution

1. Mathematical model

1.1 The fundamental equation

In the vapor-liquid two-phase flow where the cavitation occurs, the continuity equation and the momentum equation are as follows:

where u is the flow velocity of the fluid,mρ is the density of the mixed fluid,Tμ is the turbulent viscosity coefficient andmμ is the dynamic viscosity coefficient of the mixed phase.

1.2 FBM turbulence model

The standard k-ε model correlates the turbulent kinetic energy, the eddy viscosity coefficient and the dissipation rate of the turbulent kinetic energy.The governing equation of the standard k-ε model is as follows:

wheretP is the turbulent kinetic energy generation term,ijτ is the Reynolds stress tensor, k is the turbulent kinetic energy, ε is the turbulence dissipation rate,Tμ is the turbulence viscosity coefficient andijδ is the Kronecker number. The coefficients in the above model formula: Cε1=1.44,Cε2=1.92, σε=1.3 and σk=1.0. The turbulence viscosity coefficient is:

The standard k-ε model is based on the steady average flow, cannot be used to accurately predict the turbulence viscosity coefficient of the flow field, and the results are often too large, which will lead to some deviations when dealing with some complex turbulence flow field structures, resulting in inaccurate predictions of the cavitation. In the numerical calculation under the NACA0015 hydrofoil cloud cavitation conditions, the LES can capture the details of the periodic collapse of the cloud cavitation[26].However, compared with other turbulence models, the large eddy simulation (LES) requires a very refined mesh for the calculation, which will consume enormous computer resources. In order to solve the above problems, the FBM model is used in this study.In the FBM model, the turbulence viscosity coefficient of the k-ε turbulence model is modified as follows:

where fFBMis the filtering function, λ is the filtering scale.

1.3 Zwart-Gerber-Belamri cavitation model

The vapor mass transport equation can be expressed as follows

wherevα is the vapor volume fraction,vρ is the vapor density.

The Zwart cavitation model is a built-in model of ANSYS CFX 18.0. It is based on the Rayleigh-Plesset equation describing the cavity dynamics. It is also based on the Kubota cavitation model. Zwart indicated that the Kubota model was incorrect for the vaporization, because one of its assumptions was that the cavitation bubbles did not interact with each other.However, in the middle and late stages of the cavitation, the density of the nucleation site must decrease with the increase of the gas volume fraction.In this case, he substituted rnuc(1 - α) for α[27]. Its evaporation source term and the condensation source term are as follows:

where Feis the empirical error coefficient of the evaporation source term, which is taken as 50 here,rnucis the nucleation site volume fraction, which is 5.0×10-4in the above solution software,vP stands for the cavitation pressure, Fcis the empirical error coefficient of the condensation source term, which is set as 0.01 and R is the cavity radius and is set as 1.0×10-6m.

1.4 FW-H equation

The FW-H Equation is the basis of the flow-induced noise prediction method and is derived from the continuity equation in conjunction with the N-S equation. The FW-H Equation is as follows

where δ(f) is the Dirac-δ function, p′ is the acoustic pressure and Tijis the Lighthill stress tensor.The three terms on the right side of the FW-H Equation are the quadrupole acoustic source term, the dipole acoustic source term, and the monopole acoustic source term. The high frequency noise radiated by the vortex shedding and the hydrofoil boundary layer is equivalent to the quadrupole acoustic source. The low frequency noise produced by the unsteady pulsating force is caused by the non-uniform flow field around the hydrofoil and the pulsating turbulent flow field, and is equivalent to the dipole source. The noise produced by the bubble development and collapse is equivalent to the monopole source. This paper mainly studies the dipole acoustic source term.

1.5 Acoustic wave Helmholtz equation

The Helmholtz wave equation is derived from the equation of continuity, the equation of motion, and the equation of state of acoustic waves. The acoustic Helmholtz wave equation is as follows

where2∇ is the Lagrangian operator, p is the acoustic pressure, k=w/ c is the wave number, w is the angular frequency, c is the velocity of the acoustic waves in a fluid medium and q is the volume velocity per unit volume.

2. Computational model and boundary conditions

In this study, the NACA0015 hydrofoil is selected. Its chord length C = 0.07 m and the span wise is 0.015 m. Figure 1 shows the calculation model and the boundary conditions. At the front and the back of the computational domain, the symmetric surface boundary conditions are used to simulate the free flow of the borderless flow field. The distance from the inlet to the center of the hydrofoil is 3.5C, the distance from the outlet to the center of the hydrofoil is 6.5C, and the distances from the upper and lower walls to the center of the hydrofoil are both 1.5C.The boundary conditions of the velocity inlet and the pressure outlet are used in the computation, and the cavitation number is adjusted by varying the outlet pressure. The inlet velocity is fixed at 7.2 m/s, the Reynolds number R e = 5.62× 1 05. The inlet angle of attack is 6°. The hydrofoil surface and the water tunnel wall are both under non-slip boundary conditions.

Fig. 1 The calculation model and boundary conditions

The structural mesh is used in the computation,and the mesh is refined near the hydrofoil to make the calculation more accurate. In order to ensure the computing speed, the mesh density at the edge of the model is reduced appropriately, and the total mesh nodes are set as 541 110. Figure 2 shows the mesh around the hydrofoil and the refined mesh at the hydrofoil leading edge and trailing edge. Figure 3 shows the Yplus value on the hydrofoil surface. It can be seen from the figure that the value is basically between 5 and 15, meeting the requirements of the calculation.

Fig. 2 (Color online) Mesh around NACA0015 at close view of the leading edge region (a), an angle of attack α=6°(b), close view of the trailing edge region (c)

Fig. 3 The Yplus value on the hydrofoil surface

3. Results and discussions

In this study, three different cavitation numbers of different cavitation types are selected to study the cavitation induced dipole noise. The different cavitation types include the non-cavitation, sheet cavitation and the cloud cavitation.

3.1 Non-cavitation

ANSYS CFX is used to simulate the flow around the hydrofoil. The calculated time step is 0.0005 s.

In the study of the non-cavitation, the cavitation number is 3.30. Based on the source data obtained, the uncoupled indirect boundary element method is used to calculate the dipole noise levels around the NACA0015 hydrofoil, and eight representative frequencies are selected for the study. The distributions of the hydrofoil dipole noise levels at these frequencies are shown in Fig. 4.

Fig. 4 (Color online) Dipole noise distribution (dB (rms)) at eight frequencies under non-cavitation condition

As can be seen from Fig. 4, at lower frequencies,such as 4.92 Hz and 7.38 Hz, the acoustic field distribution around the hydrofoil has obvious dipole characteristics. At 4.92 Hz, the maximum acoustic pressure amplitude reaches 44.6 dB (root mean square(rms)) (rms can be simply understood as the effective value here. The same will be done afterwards). The farther away from the hydrofoil, the sound pressure amplitude decreases to 38.8 dB, 33.1dB, and down to-13.1 dB. At 7.38 Hz, the maximum acoustic pressure amplitude reaches 41.0 dB and gradually decreases to-14.1 dB. With the increase of the study frequency, at 44.28 Hz, the acoustic pressure amplitude distribution around the hydrofoil no longer has the dipole feature,and only retains the following feature: that the closer to the hydrofoil is, the higher the noise level is, and at the place closest to the hydrofoil, the acoustic pressure amplitude is 61.2 dB.

At the frequencies of 61.50 Hz, 71.34 Hz and even 140.22 Hz, 201.72 Hz and 332.10 Hz, the noise level distribution around the hydrofoil is more chaotic.At 332.10 Hz, the maximum acoustic pressure amplitude reaches 44.8 dB and gradually decreases to-53.4 dB. It is worth mentioning that, with the increase of the frequency, although the acoustic pressure distribution no longer has obvious dipole characteristics, it does have a basically symmetric distribution on the hydrofoil. The farther away the hydrofoil is, the slower the acoustic pressure amplitude decreases. The chaos might be caused by the insufficient sound source data to calculate the dipole noise level. In addition, the complete boundary conditions including the pressure, the acoustic impedance and the velocity are required to solve the Helmhotz equation, but the dipole noise level is obtained only by the pressure pulsation, which may also be a reason for the chaotic feature.

In order to study the dipole noise caused by the hydrofoil more specifically, and compare it with that in the conditions of the sheet cavitation and the cloud cavitation, three points near the hydrofoil are selected for the study. Under the conditions of the sheet cavitation and the cloud cavitation, a cavitation cavity will appear in the positions of these three points,which is conducive to the comparison. The selected points near the hydrofoil are shown in Fig. 5. When the sheet cavitation occurs, the cavity will appear at points 1 and 2. When the cloud cavitation occurs, the cavity will appear at points 1-3.

Under the non-cavitation condition, the frequency response function curves at points 1-3 are shown in Fig. 6. As can be seen from the figure, the frequency response functions at these three points have basically the same trend, and with the change of the frequency,the noise at other frequencies basically fluctuates within the range of 20.0 dB-60.0 dB, and there are no special cases such as with the high extremum. Thus, in the case of non-cavitation, the dipole noise generated by the flow of the fluid around the NACA0015 hydrofoil is relatively similar at different positions around the hydrofoil, without much variation.

Fig. 5 (Color online) The selected flow field point locations

Fig. 6 (Color online) The frequency response function curves of the three field points in the case of non-cavitation

3.2 Sheet cavitation

3.2.1 Sheet cavitation short time dipole noise

In the study of the sheet cavitation, the cavitation number is 1.08. Figure 7 shows the selected timing of the study, and Fig. 8 shows the comparison between the numerical simulation and the experimental results in the case of the sheet cavitation. It can be seen that the sheet cavitation is attached to the leading edge stably, and the sheet cavitation hardly changes with time.

Fig. 7 The selected timing for the sheet cavitation study

Fig. 8 (Color online) Comparison between numerical results and experiment data

Figures 9-11, respectively, show the pressure amplitudes generated by the surface pressure on the hydrofoil with three time steps (0.0015 s) nearT0,T1andT2. It can be seen from a single figure that the pressure amplitude above and below the hydrofoil is higher than that at the further position except near the outlet. This indicates that the cavitation produces additional noise. Above and below the hydrofoil, the pressure amplitude is increased in the sheets. Near the outlet, the pressure amplitude near the upper and lower walls is more concentrated, and near the middle it is more dispersed. By comparing the acoustic pressure amplitude in these three extremely short time periods, it can be found that the distribution and the value of the acoustic pressure are almost the same,which indicates that the dipole noise generated by the sheet cavitation is relatively stable. This is consistent with the fact that the water vapor volume fraction changes little.

Fig. 9 (Color online) Acoustic pressure amplitude dB (rms) at T0

Fig. 10 (Color online) Acoustic pressure amplitude dB (rms) at T1

Fig. 11 (Color online) Acoustic pressure amplitude dB (rms) at T2

3.2.2 Sheet cavitation global dipole noise

The dipole noise levels in the flow field at different times are calculated, along with the dipole acoustic pressure distributions at eight typical frequencies. The results are shown in Fig. 12.

Fig. 12 (Color online) Dipole noise level distribution (dB (rms))at eight frequencies in the case of sheet cavitation

It can be seen from Fig. 12, that at lower frequencies as 2.64 Hz and 6.60 Hz, the noise level distribution around the hydrofoil has obvious dipole characteristics. The closer to the hydrofoil, the greater the dipole noise level is. At 2.64 Hz, the maximum acoustic pressure amplitude reaches 28.6 dB and gradually decreases to -42.2 dB. At 11.88 Hz,although the dipole noise characteristics can still be seen, but near the hydrofoil, the dipole noise level distribution sees a slight deformation. At the trailing edge of the hydrofoil, there are two regions with a low acoustic pressure amplitude, which affects the integrity of the dipole. At the point nearest the hydrofoil, the acoustic pressure amplitude is 60.3 dB.

When the frequency reaches 48.85 Hz, the dipole characteristics of the noise level distribution can no longer be seen from the figure, but the acoustic pressure distribution still retains its symmetry. The acoustic pressure amplitude decreases gradually from 49.9 dB to -45.9 dB. At the frequencies from 71.29 Hz to 332.67 Hz, the dipole noise level distribution becomes more chaotic, with, however, a basically symmetric distribution on the hydrofoil.

The three sites in the non-cavitation study are selected again, to obtain their frequency response function curves in the cases of the sheet cavitation, as is shown in Fig. 13. According to Fig. 13, the trend of the curves at the three positions is basically the same.The noise level basically fluctuates between 0 dB and 40.0 dB. Unlike the cases of the non-cavitation, there is an extreme value of the noise level in the region from 13.00 Hz to 16.00 Hz, which is about 65.0 dB,more than 50% higher than the noise levels at other frequencies. As can be seen from Fig. 7, during the sheet cavitation, the cycle of the water vapour volume fraction completing a fluctuation is about 73.5 ms, and the corresponding frequency is calculated as 13.61 Hz,within the frequency range where the extreme value of the above noise exists. In other words, at the frequency of the fluctuation of the water vapour volume fraction of the sheet cavitation, the noise level at the three field points around the hydrofoil has an extreme value of 62.0 dB. At other frequencies, the noise level is relatively low, basically within 0 dB-40.0 dB.

Fig. 13 (Color online) The frequency response function curves at the three field points in the cases of sheet cavitation

3.3 Cloud cavitation

3.3.1 Cloud cavitation short time dipole noise

In the study of the cloud cavitation, the cavitaton number is 0.65. Figure 14 shows the cavity morphology at the typical moments of the cloud cavitation. It can be seen that the calculated results are closely consistent with the experimental results.

Fig. 14 (Color online) Comparison of numerical simulation and experimental results at typical cloud cavitation moments

Figure 15 shows the research points for the noise level distribution in a very short time of the cloud cavitation. Figures 16-20 show the pressure amplitudes generated by the surface pressure on the hydrofoil with three time steps near the study timing.After calculation, T1=T0+24.0%T, T2=T0+47.6%T,T3=T0+71.4%T, T4=T0+83.3%T. As can be seen from Figs. 16-20, the surface dipole sound field distribution with the three time steps near T1and T4is very similar to the sheet cavitation, with only a slight difference in the sound pressure amplitude.According to Fig. 14, at time T1, the cavity at the leading edge of the hydrofoil is developing towards the trailing edge, which has not been completely developed. At the time T4, the cavity at the trailing edge of the hydrofoil is in the state of falling off.

Fig. 15 The selected time points for the study

Therefore, when the cavity is on the leading edge of the hydrofoil, the dipole noise level distribution around it in a very short time is not different, and the influence of the cavity off the hydrofoil is not significant. Near the timeT0, as shown in Fig. 16, the sound pressure amplitude is almost higher than those at other moments, reaching the maximum about 202.0 dB in the middle of the model, and it is distributed in a dense and wide area. At the timeT2,the cavity in the latter half of the hydrofoil is moving towards the trailing edge. As shown in Fig. 18, the overall sound pressure amplitude is relatively low, and there is basically no high sound pressure area in the calculation domain. However, at the leading edge of the hydrofoil, there is a wave-like region with a high sound pressure amplitude of about 160.0 dB to 170.0 dB. At the timeT3, the cavity at the trailing edge of the hydrofoil is falling off. As shown in Figure 19, the sound pressure distribution at the timeT3is similar to that at the timeT2, and there is also a wave-like region similar to that in Fig. 18 at the leading edge of the hydrofoil.

Fig. 16 (Color online) Acoustic pressure amplitude dB (rms) at T0

Fig. 17 (Color online) Acoustic pressure amplitude dB (rms) at T1

Fig. 18 (Color online) Acoustic pressure amplitude dB (rms) at T2

Fig. 19 (Color online) Acoustic pressure amplitude dB (rms) at T3

Fig. 20 (Color online) Acoustic pressure amplitude dB (rms) at T4

The comparison shows that the sound pressure distributions near T0, T2and T3are different from those at other moments. It is found through observation that these moments are the extreme points in Fig. 15, namely, the derivative of the volume fraction of the water vapor with respect to the number of steps (time) is 0. It is also worth mentioning that the distribution of the sound pressure at the tail end of the model in a very short time is basically the same and similar to that in the case of the sheet cavitation.

3.3.2 Cloud cavitation global dipole noise

The dipole noise level in the flow field at all times is calculated, and the dipole acoustic pressure distributions at eight typical frequencies are selected.The results are shown in Fig. 21.

Fig. 21 (Color online) Dipole noise level distribution (dB (rms))at eight frequencies in the case of cloud cavitation

At 3.18 Hz and 7.94 Hz, the sound field distribution around the hydrofoil has obvious dipole characteristics. At 3.18 Hz and 7.94 Hz, the sound pressure amplitude reaches 118.0 dB and 124.0 dB at the places closest to the hydrofoil, much higher than those in the cases of the non-cavitation and the sheet cavitation. At 49.21 Hz, the sound field distribution shows no obvious dipole characteristics. The sound pressure amplitude decreases gradually from 162.0 dB to 82.3 dB with the increase of the distance from the hydrofoil. At higher frequencies from 79.37 Hz to 333.33 Hz, the sound field distribution becomes more complex and chaotic, with just a weak symmetric distribution on the hydrofoil.

The same three field points as in the cases of the sheet cavitation and the non-cavitation are selected to generate their frequency response function curves. As shown in Fig. 22, similar to the cases of the noncavitation and the sheet cavitation, their trends are basically the same. At different frequencies, the noise level fluctuates within the range of 100.0 dB -150.0 dB, and with the increase of the frequency, the noise level shows a downward trend. According to Fig. 15,the period of the growth, the development, the shedding and the collapse of the cloud cavitation bubbles is 63 ms and the frequency is 15.87 Hz. In Fig.22, a noise level extreme value close to 160.0 dB is corresponding to a frequency of 15 Hz, which is very close to 15.87 Hz. And there is a large extreme value of the noise level around 45.00 Hz, 100.00 Hz and 120.00 Hz.

Fig. 22 (Color online) The frequency response function curves at the three field points in the case of cloud cavitation

4. Conclusions

In this paper, the flow field around the NACA0015 hydrofoil is calculated under the conditions of the non-cavitation (σ = 3.30), the sheet cavitation (σ = 1.08) and the cloud cavitation(σ = 0.65), and the dipole sound pressure distribution diagrams in a very short time and a complete time under different working conditions are generated according to the sound source data obtained in the calculation. The sound pressure distribution and the sound pressure amplitude are analyzed in detail. The field points around the hydrofoil are selected to generate the corresponding frequency response function curves, which are analyzed from the aspects of the trend, the variation and the extreme value. The details can be summarized as follows:

(1) Under the condition of the non-cavitation,with the increase of the frequency, the dipole characteristics of the noise level around NACA0015 hydrofoil become less obvious, with only the symmetry kept on the hydrofoil. For the frequency response function curves of the selected field points,the trend is basically the same, the noise level basically fluctuates in the range of 20.0 dB-60.0 dB(rms), without high extreme values and other special cases.

(2) Under the condition of the sheet cavitation,three extreme points in the time-varying curve of the water vapor volume fraction in one cycle are selected,and the cavity form is basically unchanged. At the same time, the noise level distribution at 0.0015 s near the selected time is basically the same, which shows that the noise level distribution in a very short time does not change with time in the case of the sheet cavitation. The distribution of the global dipole noise level at different frequencies is similar to that in the case of the non-cavitation. In addition, the trend of the frequency response function curves of the field points in the case of the sheet cavitation is basically the same,and the noise level amplitude is basically within 0 dB-40.0 dB at different frequencies. However, near the frequency of the fluctuation of the volume fraction of the water vapor, there is an extreme value of the noise level around the hydrofoil, which is about 65.0 dB.

(3) Under the condition of the cloud cavitation,five typical moments are selected in one cycle of the water vapor volume fraction fluctuation. It is found that the dipole noise level in very short time near the vapor volume fraction reaches its extreme value and is different from that in the general case, when there is a cavity at the leading edge of the hydrofoil. In addition,the distributions of the dipole noise level at different frequencies are almost the same as those in the cases of the non-cavitation and the sheet cavitation. In the frequency response function curves at the same field points, near the frequency of the cloud cavitation water vapor volume fraction fluctuation, i.e., around 15.87 Hz, there is an extreme value of the noise level around the hydrofoil, which is about 160.0 dB, and with the increase of the frequency, the noise level shows a downward trend. Unlike the cases of the non-cavitation and the sheet cavitation, the frequency response function curves in the case of the cloud cavitation also have higher noise level value at other frequencies, which indicates that the cloud cavitation is a kind of cavitation with obvious unsteady characteristics.