4.1 Effect of convection velocity on the modal averaged radiation efficiency

The specific parameters and dimensions used in the calculation are listed in Table 1.

The increment of vibration power and acoustic radiation energy are different with the increase of the velocity, which indicates that the changing of velocity can affect the modal averaged radiation efficiency. The modal averaged radiation efficiency of the aluminum plate at three flow velocities (Mc = 0.5; 0.7; 0.9) is shown in Figure 5. It can be seen that when the Mc increases from 0.5 to 0.9, the modal averaged radiation efficiency will increase by 3–7 dB below the hydrodynamic coincidence frequency. And the corresponding hydrodynamic coincidence frequencies (fh) are 1482, 2905, and 4802 Hz, respectively. The results show that the modal averaged radiation efficiency increases in the frequency range below the hydrodynamic coincidence frequency. The increase of the modal averaged radiation efficiency indicates that with the increase of flow velocity, the increment of the radiated sound power is larger than that of the mean square velocity.

The phenomenon that the modal averaged radiation efficiency increases with the flow velocity can be explained by the hydrodynamic coincidence effect. For the lateral incoming flow problem, the hydrodynamic coincidence is mainly


#### Table 1.

<sup>f</sup> <sup>h</sup> <sup>¼</sup> <sup>U</sup><sup>2</sup> c 2π

Measured and predicted velocity auto spectrum and the radiated sound intensity of the plate with the size of

<sup>f</sup> c,mn <sup>¼</sup> <sup>c</sup><sup>0</sup>

<sup>f</sup> h,mn <sup>¼</sup> Uc

<sup>f</sup> <sup>h</sup> <sup>¼</sup> <sup>M</sup><sup>2</sup>

coincidence frequency can be summarized as follows

a � b = 0.62 � 0.3 m. Narrow band analysis in per Hz. Flow speed 86 m/s.

Boundary Layer Flows - Theory, Applications and Numerical Methods

fh,mn ≈ Uckm/2π, where km = mπ/a is lateral modal wave number.

In conclusion, the relationship between critical frequency and hydrodynamic

In the above two equations, Mc = Uc/c<sup>0</sup> is mach number. Subsonic turbulence is generally considered, so the hydrodynamic coincidence frequency is always less than the critical frequency of the plate. It is important to note that the characteristics of frequency is a reference value which is based on the infinite plate hypothesis. Actually, the characteristics frequency of the limited plate slightly higher than a reference value. In addition, for the transverse flow problem, modal power line frequency can be thought of only related to the transverse mode. That is to say,

cidence frequency are

Figure 4.

38

Similarly, for order (m, n) mode, its critical frequency and hydrodynamic coin-

ffiffiffiffiffi ms D r

<sup>2</sup><sup>π</sup> kmn (57)

<sup>2</sup><sup>π</sup> kmn (58)

<sup>c</sup> � f <sup>c</sup> (59)

f h,mn ¼ Mc � f c,mn (60)

(56)

Parameters used in calculation.

#### Figure 5.

Effect of the convective Mach number on the modal averaged radiation efficiency of the finite aluminum plate. Reproduced from Ref. [23].

determined by the lateral modal trace speed and the convection velocity. When the bending wave velocity of the lateral mode is the same as the turbulent flow velocity (Uc = 2πf/km), the corresponding hydrodynamic coincidence frequency is f = mUc/2a. Thus a higher convection velocity at the same frequency will lead the TBL excitation to coincide with a lower order lateral mode.

critical frequency, a lower order lateral mode always has higher modal averaged radiation efficiency than that of a higher order lateral mode with the same n, since the modal critical frequency moves to lower frequency. So plate with higher flow

As an example, the hydrodynamic coincidence lines for different flow velocity (Mc) and the modal radiation efficiencies of mode (m, 1) are illustrated in Figure 7.

velocity is supposed to have higher modal averaged radiation efficiency.

The black solid lines in the figure are the hydrodynamic coincidence line corresponding to the mode order and frequency. It can be seen that at a certain frequency, the modal averaged radiation efficiency of the hydrodynamic coincidence modes at higher velocity is always greater than that of the low velocity. In a word, an increase of the flow velocity will increase the modal radiation efficiency of the coincided mode, and then results in the increase of the modal averaged radiation efficiency. Besides, owing to the low pass property of the modal excitation term, the increase of the modal radiation efficiency is restrained above the hydrodynamic coincidence frequency. As a consequence, the modal averaged radiation efficiency is great affected by the flow velocity which only occurs below the hydrodynamic

4.2 Effect of structural damping on modal averaged radiation efficiency

The modal averaged radiation efficiency changes with structural loss factors for different flow velocity (Mc), as shown in Figure 8. The reference value is calculated according to Leppington's formula [25]. Though Leppington's formula is widely used in statistical energy analysis, it does not take the flow and structural damping into account. Figure 8 indicates that an increase of the structural loss factor will increase the modal averaged radiation efficiency under the critical frequency, but the increments are different for different flow velocity. It is found that the modal averaged radiation efficiency is not sensitive to the change of structure loss factor at low Mach number. For example, for a typical high-speed train (Mc = 0.25), the increased modal averaged radiation efficiency is less than 2 dB in the frequency band below the critical frequency when the structural loss factor increases from 1 to 4%. In the case of high flow velocity, the effect of structure loss factor on the modal averaged radiation efficiency is much obvious. When Mc = 0.7, the modal averaged radiation efficiency will increase by about 5 dB if the structural loss factor has the same increment. The results show that the influence of structural damping on the

Hydrodynamic coincidence lines and variation of the modal radiation efficiency with the lateral mode number

and the frequency of a finite aluminum plate. m varies, n = 1. Reproduced from Ref. [13].

coincidence frequency.

TBL-Induced Structural Vibration and Noise DOI: http://dx.doi.org/10.5772/intechopen.85142

Figure 7.

41

The reason for above phenomenon may be further explored through the modal excitation terms. As illustrated in Figure 6, the lateral modal excitation term (10log10Λm(ω)) is plotted with the lateral mode number (m) and frequency for different flow velocity (Mc). In the figure, the peak of the lateral mode excitation term corresponds to the maximum excitation and its position depends on the hydrodynamic coincidence frequency. The black bold lines in the two sub graphs are the positions where the hydrodynamic coincidence occurs. It can be seen that the slope of the hydrodynamic coincidence line is inversely proportional to the flow velocity, and the higher the velocity is, the lower the order of a certain frequency is. In addition, the lateral modes near the hydrodynamic coincidence line are all strongly excited. As the frequency increases, the number of these modes increases, but the amplitude of their corresponding mode excitation term decreases. Below the

Figure 6. Variation of the lateral modal excitation term with the lateral mode number and the frequency of a finite aluminum plate. (a) Convective Mc = 0.5 and (b) convective Mc = 0.9. Reproduced from Ref. [13].

#### TBL-Induced Structural Vibration and Noise DOI: http://dx.doi.org/10.5772/intechopen.85142

determined by the lateral modal trace speed and the convection velocity. When the bending wave velocity of the lateral mode is the same as the turbulent flow velocity (Uc = 2πf/km), the corresponding hydrodynamic coincidence frequency is f = mUc/2a. Thus a higher convection velocity at the same frequency will lead the TBL excita-

The reason for above phenomenon may be further explored through the modal

excitation terms. As illustrated in Figure 6, the lateral modal excitation term (10log10Λm(ω)) is plotted with the lateral mode number (m) and frequency for different flow velocity (Mc). In the figure, the peak of the lateral mode excitation term corresponds to the maximum excitation and its position depends on the hydrodynamic coincidence frequency. The black bold lines in the two sub graphs are the positions where the hydrodynamic coincidence occurs. It can be seen that the slope of the hydrodynamic coincidence line is inversely proportional to the flow velocity, and the higher the velocity is, the lower the order of a certain frequency is. In addition, the lateral modes near the hydrodynamic coincidence line are all strongly excited. As the frequency increases, the number of these modes increases, but the amplitude of their corresponding mode excitation term decreases. Below the

Variation of the lateral modal excitation term with the lateral mode number and the frequency of a finite aluminum plate. (a) Convective Mc = 0.5 and (b) convective Mc = 0.9. Reproduced from Ref. [13].

tion to coincide with a lower order lateral mode.

Boundary Layer Flows - Theory, Applications and Numerical Methods

Figure 6.

40

critical frequency, a lower order lateral mode always has higher modal averaged radiation efficiency than that of a higher order lateral mode with the same n, since the modal critical frequency moves to lower frequency. So plate with higher flow velocity is supposed to have higher modal averaged radiation efficiency.

As an example, the hydrodynamic coincidence lines for different flow velocity (Mc) and the modal radiation efficiencies of mode (m, 1) are illustrated in Figure 7. The black solid lines in the figure are the hydrodynamic coincidence line corresponding to the mode order and frequency. It can be seen that at a certain frequency, the modal averaged radiation efficiency of the hydrodynamic coincidence modes at higher velocity is always greater than that of the low velocity. In a word, an increase of the flow velocity will increase the modal radiation efficiency of the coincided mode, and then results in the increase of the modal averaged radiation efficiency. Besides, owing to the low pass property of the modal excitation term, the increase of the modal radiation efficiency is restrained above the hydrodynamic coincidence frequency. As a consequence, the modal averaged radiation efficiency is great affected by the flow velocity which only occurs below the hydrodynamic coincidence frequency.

#### 4.2 Effect of structural damping on modal averaged radiation efficiency

The modal averaged radiation efficiency changes with structural loss factors for different flow velocity (Mc), as shown in Figure 8. The reference value is calculated according to Leppington's formula [25]. Though Leppington's formula is widely used in statistical energy analysis, it does not take the flow and structural damping into account. Figure 8 indicates that an increase of the structural loss factor will increase the modal averaged radiation efficiency under the critical frequency, but the increments are different for different flow velocity. It is found that the modal averaged radiation efficiency is not sensitive to the change of structure loss factor at low Mach number. For example, for a typical high-speed train (Mc = 0.25), the increased modal averaged radiation efficiency is less than 2 dB in the frequency band below the critical frequency when the structural loss factor increases from 1 to 4%. In the case of high flow velocity, the effect of structure loss factor on the modal averaged radiation efficiency is much obvious. When Mc = 0.7, the modal averaged radiation efficiency will increase by about 5 dB if the structural loss factor has the same increment. The results show that the influence of structural damping on the

#### Figure 7.

Hydrodynamic coincidence lines and variation of the modal radiation efficiency with the lateral mode number and the frequency of a finite aluminum plate. m varies, n = 1. Reproduced from Ref. [13].

Eq. (61) shows that the modal averaged radiation efficiency is equivalent to the weighted average function of the modal velocity response, and the weighted coefficient is the modal averaged radiation efficiency. In the frequency band below the critical frequency, the radiation efficiency of each mode varies in the range from 0 to 1. Due to this weighted effect of Eq. (61), the vibration energy (denominator in the equation) decreases more effectively than the acoustic radiation power (molecule in the equation). Thus the radiation efficiency increases in the frequency band below the critical frequency. However, the phenomenon that the radiation efficiency of a damped plate is enlarged with increment of flow velocity has not yet

Moreover, it is observed that the effect of structural damping on modal averaged radiation efficiency has a good agreement with the research of Kou [23] at low flow velocity. In their work, it is shown that the modal averaged radiation efficiency of heavily damped structures is sensitive to the change of structural loss factor without turbulent flow. It also implies that Leppington's equation is not applicable to the prediction of modal averaged radiation efficiency of damped structures at high flow

This chapter studies the vibro-acoustic characteristics of the wall plate structure

The modal averaged radiation efficiency increases with the increase of structural

Thanks to the financial support by the Taishan Scholar Program of Shandong

Figures 6–8 in this chapter are reproduced from an AIP Publishing journal paper written by the second and third authors, and all the figures are cited in this text. According to AIP webpage for Copyright and Permission to Reuse AIP materials, AIP Publishing allows authors to retain their copyrights (https://publishing.aip.org/

damping below the critical frequency band. The larger the flow rate, the more significant the effect of structural damping on acoustic radiation efficiency. In the case of low flow velocity, the modal averaged radiation efficiency is not sensitive to the change of structural damping. The structural damping increases from 1 to 4%, and the increase of modal averaged radiation efficiency less than 2 dB. In the case of high flow rate, the modal averaged radiation efficiency will increase by 5 dB when

the increment of the structural damping is from 1 to 4%.

excited by turbulent boundary layer (TBL). Based on the modal expansion and Corcos model, the formulas for calculating the modal averaged radiation efficiency are derived. The results indicate that an increment of flow rate will increase the vibration energy and the radiated sound energy of the structure. However, the amplitude of two cases varies with the velocity are not the same, and when the velocity increases, the acoustic radiation efficiency will increase below the hydrodynamic coincidence frequency range. The main reason for this phenomenon is that a higher convection velocity will coincide with lower order modes which have

been clearly interpreted.

TBL-Induced Structural Vibration and Noise DOI: http://dx.doi.org/10.5772/intechopen.85142

velocity.

5. Conclusion

higher radiation efficiencies.

Acknowledgements

(no. ts201712054).

43

Conflict of interest

Figure 8.

Effect of the structural loss factor on the modal averaged radiation efficiency of a finite aluminum plate. (a) Convective Mc = 0.25 and (b) convective Mc = 0.7. Reproduced from Ref. [13].

modal averaged radiation efficiency is related to the flow velocity, and the influence of structural damping can be enhanced by increasing the flow velocity.

The effect of structural damping on the modal averaged radiation efficiency can be qualitatively explained by Eq. (61)

$$\sigma\_{av} = \frac{\prod\_{t}^{t}}{\rho\_0 c\_0 S \langle V^2 \rangle} = \frac{\sum\_{m=1}^{\infty} \sum\_{n=1}^{\infty} \sigma\_{mn}(o) I\_{mn}(o) \left| V\_{mn}(o) \right|^2}{\sum\_{m=1}^{\infty} \sum\_{n=1}^{\infty} I\_{mn}(o) \left| V\_{mn}(o) \right|^2} \tag{61}$$

#### TBL-Induced Structural Vibration and Noise DOI: http://dx.doi.org/10.5772/intechopen.85142

Eq. (61) shows that the modal averaged radiation efficiency is equivalent to the weighted average function of the modal velocity response, and the weighted coefficient is the modal averaged radiation efficiency. In the frequency band below the critical frequency, the radiation efficiency of each mode varies in the range from 0 to 1. Due to this weighted effect of Eq. (61), the vibration energy (denominator in the equation) decreases more effectively than the acoustic radiation power (molecule in the equation). Thus the radiation efficiency increases in the frequency band below the critical frequency. However, the phenomenon that the radiation efficiency of a damped plate is enlarged with increment of flow velocity has not yet been clearly interpreted.

Moreover, it is observed that the effect of structural damping on modal averaged radiation efficiency has a good agreement with the research of Kou [23] at low flow velocity. In their work, it is shown that the modal averaged radiation efficiency of heavily damped structures is sensitive to the change of structural loss factor without turbulent flow. It also implies that Leppington's equation is not applicable to the prediction of modal averaged radiation efficiency of damped structures at high flow velocity.
