5. Conclusion

To clarify the contributions of each term to the acoustic field, four hybrid simulations were performed for each Mach number on the basis of total Lighthill's acoustic sources computed by the direct simulation, only the first term, only the second term, and only the third term.

Figure 18 shows the polar plots of the sound pressure level at the frequency of the vortex shedding predicted at r/D = 30.0 by the hybrid simulations for M = 0.4. It was clarified that the sound pressure levels predicted by the hybrid simulation based on all terms agree well with those based on only the first term. The sound pressure level based on the second term and that on the third term is negligibly weaker than that based on the first term. Meanwhile, the intensity of the second term is in itself comparable to that of the first term as mentioned above. This indicates that the radiation efficiency of the second term is weaker than that of

Figure 19 shows the predicted sound pressure level at r/D = 30.0 in the direction of the abovementioned acoustic propagation angle. The results clarified that the first term is the dominant acoustic source for all the Mach numbers. The difference between the sound pressure level based on the first term and that based on the second or third term was more than 30 dB for all the Mach numbers. This result shows that the momentum (the first term) of Lighthill's acoustic source is the dominant acoustic source for all the Mach numbers for cylinder flows, while it has

Figure 18. Polar plots of sound pressure levels predicted by decoupled simulations based on all, first, second, and third

terms of Lighthill's acoustic sources at the frequency of vortex shedding at r/D = 30.0 [24].

the first term.

344 Numerical Simulations in Engineering and Science

Aeroacoustic simulations composed of hybrid and direct simulations were introduced. The effects of the freestream Mach number on the flow and acoustic fields around a square cylinder were investigated. The Mach number was varied from 0.2 to 0.6. The Reynolds number based on the side length was 150. These results indicate the effectiveness and limit of the hybrid simulations.

It was found that the Strouhal number of vortex shedding, which is based on the side length, becomes lower as the freestream Mach number becomes higher. The Strouhal number for M = 0.2 is 0.151 and that for M = 0.6 is 0.144. As the Mach number increases, the velocity fluctuations of the vortices shed from the cylinder intensifies and the wake widens. The possible reason the velocity fluctuations of the vortices intensify is that the acoustic feedback exists like that in the oscillations in cavity flows. These effects can be found by the direct simulations.

The sound pressure level at the frequency of the vortex shedding in the direction of the acoustic propagation angle is proportional to M<sup>7</sup> for M > 0.3, while that is proportional to M<sup>5</sup> for M ≤ 0.3. The decomposition of scattered and direct sound showed that the direct sound is too intense to neglect for M ≥ 0.4. This indicates that the direct sound needs to be taken into consideration when predicting the flow-induced sound around the cylinder for M ≥ 0.4. Also, the directivity of the acoustic field cannot be the above-mentioned modified Curle's equation for such a high Mach number.

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Direct and Hybrid Aeroacoustic Simulations Around a Rectangular Cylinder

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Moreover, to clarify the contributions of each term of Lighthill's acoustic source to the acoustic field, acoustic simulations were performed using Lighthill's acoustic sources computed by the direct simulations. As a result, the momentum (the first term) of Lighthill's acoustic source was found to be dominant for all the Mach numbers while it has been clarified in the past research that the entropy (the second term) also needs to be taken into consideration for high-speed jets such as M = 0.9. Also, it was confirmed that only the first term needs to be taken into consideration independently of the freestream Mach number when the sound radiating from a cylinder flow is predicted on the basis of the Lighthill's acoustic analogy.

The present study has provided useful guidelines for predicting the aerodynamic sound on the basis of the Lighthill's acoustic analogy.
