#### 5.1. Vibration Analysis

Figure 12 shows the vibration contrast between the simulation and experiment under 1.0

Q_{d}. The APF is 24.67 Hz, the first blade passing frequency (BPF

_{1}) is 148 Hz, and the secondary blade passing frequency (BPF

_{2}) is 172 Hz. Obvious peak values of vibration appear at the APF, BPF

_{1}, and BPF

_{2}, and the maximum appears at BPF

_{2}. The simulated value at the APF is slightly higher than the experimental value, but it is still in good agreement. The tendency of the vibration simulation is almost consistent with the experimental results, and the deviation is small which implies that the numerical simulation method is feasible. In addition, peak values emerge at BPF

_{1}, BPF

_{2}, and other frequencies, which could be attributed to the combined interference of the rotor and background noise. On the other hand, the BPF

_{2} of the standard

k-ε model is higher than that of the RNG

k-ε and SST

k-ω models, and the high-frequency band is supposed to be distinct, the reason for this may be the intricate environmental factors. In subsequent research, the reality factor should also be considered.

#### 5.2. Internal Sound Field Analysis

Figure 13 shows the sound pressure level

L_{p} at suction and discharge of the pump which is defined as

where

p_{ref} is the referenced sound pressure, which is generally taken as 1 × 10

^{−6} Pa, d

f is the frequency interval (1.162 Hz), and

f_{max} is the analysis frequency (2000 Hz).

Table 2 shows the deviation of the internal sound field between the turbulence models and the experiments. It can be seen that the experimental values are significantly smaller than the simulated values. The choice of the turbulence model is therefore important. The deviations between the standard

k-ε model and the experimental values are the largest with a range of 2.76%~7.35%, and the deviations between the RNG

k-ε model and the experiments are the smallest with 0.83%~3.13%. The deviations of

L_{p} at BPF and the harmonic frequency between the RNG

k-ε model and the experimental values are the smallest.

In

Figure 13,

L_{p} in the suction region stays at 108 dB and at 114 dB in the discharge region. At 1.2

Q_{d}, the deviations within the simulations cannot be neglected. This could be explained by cavitation occurring at suction, where the bubbles begin to collapse and perish under the extrusion force when flowing toward a high-pressure area. Then the center of the bubbles produces high frequency shock waves which induce cavitation noise. The reason why

L_{p} of the SST

k-ω model is higher than that of the RNG

k-ε model may be that the turbulent kinetic energy of the SST

k-ω model is sensitive to free boundary conditions at the exterior margin of the boundary layer and the tiny turbulent frequency

ω of the boundary would result in an overestimation of turbulent kinetic energy

k.

Figure 14 shows the comparison of the noise spectrum of the five-stage pump at the suction part for the simulations and experiments. It was found that at 0.8

Q_{d}, the peak value of sound pressure level (SPL) appears at BPF

_{1}, BPF

_{2,} and their harmonic frequency in the spectrum, which attributes to the RSI function. The sound field result based on all three models could be obtained by the variation tendency of noise, and the maximum error relative to the experimental data is less than 5%. At 1.0

Q_{d}, the curve with the RNG

k-ε model is most similar to the experiment, because when the pump was working at the design operating point, the flow field of the pump was quite stable and smooth and the flow in the flow passage components is rather uniform. The errors of the standard

k-ε, RNG

k-ε, and SST

k-ω models are all relatively small, however, the curves based on the RNG

k-ε and SST

k-ω models are closer to the experimental results. At 1.2

Q_{d}, the internal noise enlarges and approaches the maximum which is attributed to the combined action of cavitation and intense pressure pulsation. The most appropriate model is once more the RNG

k-ε model.

The large deviation of the results of the standard

k-ε model could be explained as follows: in this turbulence model, the dynamic eddy viscosity coefficient

μ_{t} is assumed to be isotropic scalar, however, it should be an anisotropic tensor in most cases. This leads to certain distortions when applied to a strong swirl. However, a turbulent vortex with a three-dimensional, unsteady and complex property is considered in the RNG

k-ε turbulence model [

30,

31], and the RNG

k-ε model involves extra terms to calculate

k and

ε on the basis of the standard

k-ε model, affecting the eddy factor and low-Reynolds action. In contrast, the SST

k-ω model takes advantage of the boundary layer under various pressure gradients, and the SST

k-ω model employs

k-ω pattern in the internal region near the wall and results in higher resolution with more physical significance. In general, the fluid field calculated by the RNG

k-ε model is meticulous, and swirling flows are taken into account by modifying the eddy viscosity.

SPL at the discharge part of the pump is shown in

Figure 15. The peak values appear at BPF

_{1}, BPF

_{2}, and their harmonic frequency due to the RSI function. Compared with the noise in the suction region, the noise is larger. The dipole source acts mainly downstream and the pressure pulsation signal caused by the RSI moves downstream as well. Because of the rotation effect of the impeller, it is difficult for the sound waves in the pump to separate in the upstream direction. The interference between the jet-wake structure and the static components is equivalent to a secondary source and downstream propagation.

At 0.8 Q_{d}, in the range of 0–1000 Hz, the disparity between the results obtained by the RNG k-ε and SST k-ω models becomes apparent. Under this low flow rate, the speed of outflow from the last-stage impeller would rapidly decay, because the constant of the RNG k-ε model is based on the standard k-ε model and further obtained by theoretical derivation, not an experimental method, which modifies turbulent viscosity and its dissipation rate equation considering the influence of the averaged strain rate on the dissipation term.

At 1.0 Q_{d}, the maximum SPL appears at BPF_{2} because the subsequent four-stage impellers have seven blades, and when the fluid flows through the four-stage impellers, the characteristic frequency of BPF_{2} is enhanced due to the interaction between the impeller and diffuser, which leads to the promotion of the noise energy at BPF_{2}.

At 1.2 Q_{d}, the peak value of each characteristic frequency and broadband noise level were increased significantly, because the radial force in the impeller raises relatively. This leads to exaggeration of the pressure impact loaded on the flow components, the pressure fluctuation of fixed and rotating wall varies violently, and eventually the noise intensity increases.

To sum up, the error of the SPL found in the experiments with the three turbulence models agrees within 10%. The RNG k-ε turbulence model is the most suitable one.

#### 5.3. External Sound Field

Figure 16 shows the simulated and experimental radiated noise of different monitoring points.

Figure 16 indicates that

L_{p} is maintained at 79–87 dB. The noise intensity becomes minimal under the design flow rate and the SPL at 1.2

Q_{d} is the largest. The simulated

L_{p} is higher than the experimental one. It was found that at 1.0

Q_{d} and 1.2

Q_{d}, the simulated values with the three turbulence models are in good agreement with the experimental results, and the deviation is less than 5%. At 0.8

Q_{d}, the deviation reaches 7.7%, because the fluid force obtained by the flow simulation lacks high accuracy. Compared with the standard

k-ε and SST

k-ω models, the deviation from the RNG

k-ε model is minimal which illustrates that the RNG

k-ε model is the most appropriate for the external noise of the pump as well.

To visually understand the suitability of the three turbulence models,

Table 3 shows the deviations of the results of turbulence models with experimental data in the external sound field at M3, the most representative point out of the five points.

Table 3 shows that the deviation between the standard

k-ε model and the experimental values is smallest with a range of 2.34%~5.35%, and the corresponding values for the RNG

k-ε model are 1.85%~3.14%.

Figure 17 shows the distribution of the radiated sound field under various turbulence models for 1.0

Q_{d}. The SPL at the last impeller was the smallest, while the largest one was observed at the first impeller, due to a combination of cavitation and inlet whirl. There is an obvious reduction at the middle section of the pump. In addition, the simulated values are consistently higher than the experimental values. The distributions of the simulation values are all similar, whilst the specific values are distinct. The largest noise energy was obtained by the standard

k-ε model, and the discrepancy between the values of the RNG

k-ε and SST

k-ω models are very small.

The measured external noise could not reflect the details of the radiated noise, due to interference frequencies. Thus, a numerical simulation has been conducted to investigate this point further. The spectrum weighting method adopted the A weighting filter, which is the most relevant one for the human ear. To obtain the radiated noise power, a standard ISO-surface mesh was established outside the pump.

Figure 18 shows the spectrum of the radiated sound power.

The characteristic frequencies include BPF

_{1}, BPF

_{2}, and their harmonic frequencies,

f_{1},

f_{2} are also revealed. Additionally, some inspective frequencies exist, such as 15 APF, 25 APF, 26 APF, 29 APF, 32 APF, 33 APF, 34 APF, 37 APF, 38 APF, and 39 APF, which are attributed to the complicated interfere-action. At 1.0

Q_{d},

Figure 18 shows that the simulated curves of noise are similar. The peak level of noise at the characteristic frequency decreases significantly in the low-frequency range, and the descent range is about 5–8 dB. The noise level in the higher frequencies has changed. At 1.2

Q_{d}, the peak value in the low-frequency range changes slightly compared with that at 1.0

Q_{d}. However, the SPL at high frequencies increases clearly, which may be attributed to the combination of the large flow rate and the occurrence of cavitation. As the flow rate increases, the noise induced by the fluid source rises and the noise energy subsequently enhances when it spreads outward through the pump body. The fluid force exerted on the pump body increases and initiates vibration in the pump structure, finally the noise energy raises, which radiates towards the external space through air.

To obtain the sensitivity of the frequency band and turbulence model, the octave spectrum of the radiated noise under various flow rates and turbulence models was analyzed. The maximal noise value in the noise band is taken as the criterion, and the specific noise at each frequency is normalized with this maximum, as shown in

Figure 19. The minimum is represented by 0 and 1 represents the maximum value.

First of all, when the flow rate changes, the red pattern of the contour is negligible, which means the influence of the change of the flow rate is rather small. Meanwhile, the dark parts were obvious when the frequency varies, which indicates that the ratio of noise energy changes under different frequency bands. On the other hand, the differences between the three models are not significant. It can be seen that there exists two red light-bands at 128–256 Hz, which correspond to BPF

_{1} (148 Hz) and BPF

_{2} (172 Hz), and indicates that the energy at BPF occupies a dominant position in the external sound field. However, the gap between BFP

_{1} and BPF

_{2} is more obvious in

Figure 19b compared with the others. This confirms the conclusion mentioned before, namely that the RNG

k-ε model is the most appropriate. In the low noise-band, the color within the standard

k-ε model is deeper, which means that the calculated noise level by the standard

k-ε model is larger and agrees with previous conclusions as illustrated in

Section 5.2 and

Section 5.3.