Parameters used in calculation.
One of most import noise sources in a jet powered aircraft is turbulent boundary layer (TBL) induced structural vibration. In this chapter, the general model for the prediction of TBL-induced plate vibration and noise is described in detail. Then numerical examples for a typical plate are illustrated. Comparisons of plate vibration and radiated noise between numerical results and wind tunnel test are presented. The effects of structural parameters on modal-averaged radiation efficiency and therefore the radiated noise are discussed. The result indicates that an increment of flow velocity will increase the acoustic radiation efficiency 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 higher radiation efficiencies.
- turbulent boundary layer
- plate vibration
- radiated noise
- modal radiation efficiency
The interior noise level in a jet aircraft is mainly depend on noise which generated by turbulent boundary layers (TBL), if the rest of noise sources such as ventilation systems, fans, hydraulic systems, etc. have been appropriately acoustically treated. When the aircraft passes through the atmosphere, the turbulent boundary layer creates pressure fluctuations on the fuselage. These pressure fluctuations cause the aircraft fuselage to vibrate. The noise generated by the vibration is then transmitted to the cabin.
The noise emitted by the aircraft fuselage depends on the speed of the vibrating plate, which in turn depends on the speed of the aircraft, the geometry and size of the plates, and the loss or damping of the plates. It is obvious that the acoustic performances of the internal system, trim panels etc., will also affect the noise inside the aircraft. Graham  came up with a model in aircraft plates to predict TBL induced noise, in which the modal excitation terms were calculated by an analytical expression. In Graham’s another research , the advantages of various models describing the cross power spectral density induced by a flow or TBL across a structure was discussed. Han et al.  tried to use energy flow analysis to predict the noise induced by TBL. The method can better predict the response caused by the TBL excitation. However, the noise radiation caused by the flat panel cannot be predicted well. To avoid this deficiency, Liu et al. [4, 5, 6] described a model to predict TBL induced noise for aircraft plates. In their work, the modal excitation terms and acoustical radiation efficiency can be predicted properly and the predicted results are also compared with that of the wind tunnel and in-flight test. Rocha and Palumbo  further investigated the sensitivity of sound power radiated by aircraft panels to TBL parameters, and discussed the findings by Liu  that ring stiffeners may increase TBL induced noise radiation significantly.
The radiation efficiency of a plate plays an important role in vibro-acoustic problems. In recent related research, the sound medium around the fuselage of the aircraft is often considered to be stationary. Under this assumption, Cremer and Heckl  used a more concise formula to predict the acoustic radiation efficiency of an infinite plate. Wallace  derived an integral formula based on far-field acoustic radiation power to calculate the modal acoustic radiation efficiency of a finite plate. Kou et al.  proposed modifications to the classical formulas given by Cremer and Leppington, regarding the influence of structural damping on the radiation efficiency.
A comparison of the acoustic radiation of the plate with stationary fluid and convective fluid-loaded can be found in [11, 12, 13]. Graham  and Frampton  studied the influence of the mean flow on the modal radiation efficiency of a rectangular plate. They found that at high speeds, as the modal wave moves upstream, the increase of flow velocity would reduce the modal critical frequency. As a consequence, the acoustics radiation efficiency under the critical frequency of the plate would be higher. Kou et al.  also conducted a research for the effect of convection velocity in the TBL on the radiation efficiency. Kou et al. found that the modal averaged radiation efficiency will increase with the increase of the convection velocity below the hydrodynamic coincidence frequency. The study also showed that the increase of the structural loss factor could increase the modal average radiation efficiency at the subcritical frequencies, and the damping effect increases with the increase of the flow velocity.
For a plate subjected to a TBL fluctuation, although a large amount of research work used experimental and computational methods for the vibro-acoustical properties of plates, it is worth a chapter to introduce the prediction model and summarize recent findings for TBL induced plate vibrations and noise radiations. The following sections begins with a description of models for the wavenumber-frequency spectrum of TBL, and then a specific presentation of the calculation of vibro-acoustic responses of the wall plate excited by TBL is followed. In the end, the effect of flow velocity (
2. Models for the wavenumber-frequency spectrum of turbulent boundary layer fluctuating pressure
As for the research about wavenumber-frequency spectrum of turbulent boundary layer, Corcos , Efimtsov , Smolyakov-Tkachenko , Williams , Chase [18, 19] and other researchers put up with a series of widely used of wavenumber-frequency spectrum model. The models are established according to a large number of experimental data and statistical theory of turbulence. The following parts introduce some typical wavenumber-frequency spectrum models.
2.1 The Corcos model
The model proposed by Corcos during the last few decades has been widely used for many different types of problems. The model is applicable in the immediate neighborhood of the so-called convective ridge , as long as
The Fourier Transform of
So, the normalized wavenumber-frequency spectrum in wavenumber domain is
2.2 The generalized Corcos model
Caiazzo and Desmet  proposed a generalized model which based on the Corcos model. The model uses butterworth filter to replace exponential decay of
Analogously, the normalized wavenumber-frequency spectrum in wavenumber domain is
2.3 The Efimtsov model
The Efimtsov model assumes, as in the Corcos model, that the lateral and the longitudinal effects of the TBL can be separated. However, in the Efimtsov model the dependence of spatial correlation on boundary layer thickness,
The normalized wavenumber-frequency spectrum is
2.4 The Smolyakov-Tkachenko model
Like Efimtsov model, Smolyakov-Tkachenko model also takes the boundary layer thickness and scale space separation of boundary layer effect of fluctuating pressure into account. Based on the experimental results, the difference is that the Smolyakov-Tkachenko model amend the space scale function index to , in order to make the computing result is consistent with the experimental results.
The normalized wavenumber-frequency spectrum is
2.5 The Ffowcs-Williams model
Ffowcs-Williams using Lighthill acoustic analogy theory to deduce a frequency-wave spectrum model, in which the speed of the pneumatic equation is set as the source term by Corcos form. A number of parameters in the model and function need further experiments to determine, which is not widely used at present. Hwang and Geib  ignore compression factor of the influence of this model to put forward a simplified model. The normalized wavenumber-frequency spectrum is
2.6 The Chase model
Chase’s model is another model commonly used and believed to describe the low-wavenumber domain better than Corcos’s model, which has the same starting point with the Ffowcs-Williams model. The normalized wavenumber-frequency spectrum can be described as
2.7 Comparison of models
Figure 1 shows the comparison of the above models. In the figure, the parameters used by the Corcos model are
3. Calculation of vibro-acoustic responses of the wall plate excited by TBL
Consider a simply supported thin rectangular plate excited by TBL, as shown in Figure 2. In the figure,
Assume that point s on the plate is excited by a normal force
where = (
The impulse response
The impulse response can be expanded as
The modal amplitude of impulse response by using the Galerkin method can be described as
3.1 Vibro-acoustic responses of plate solved by spatial domain integration
Cross spectral density of displacement response for any two points on the plate can be defined as
In the above equation,
When using the Corcos model, the coordinate transformation of the quadruple integral in the modal excitation term can be obtained
As for vibration (
So, vibration energy and acoustic radiation energy can be expressed as
According to the definition, the modal average acoustic radiation efficiency excited by TBL of the thin plate is
3.2 Vibro-acoustic responses of plate solved by wavenumber domain integration
Another approach to obtain the cross spectral density of vibration response is to solve it directly by using the separable integral property of some turbulent boundary layer pulsating pressure models in the wavenumber domain .
The wavenumber-frequency spectrum of TBL satisfies the following relationship
where , .
The formula can be obtained by substituting the cross spectral density of the vibration response
Similarly, the spectral density of the vibration velocity can be obtained as
As for the Corcos model, we can obtain that
According to the residue theorem,
Vibration energy and sound radiation energy are
Compare the above two equations, it can be seen that
Finally, the modal average acoustic radiation efficiency can be obtained as
By observing the above equation, it can be found that only the modal excitation term in the modal averaged radiation efficiency is related to turbulence.
Figure 3 shows the comparison of two methods for calculating the modal averaged radiation efficiency excited by TBL. The size of the plate is 1.25 × 1.1 m, and the thickness is 4 mm, structural loss factor of aluminum plate is 1%, mach number is 0.5. Obviously, the accuracy of the two methods is equal. Computation speed of analytical method is much faster than integral method, but its range of application has limitations. Only the Corcos model and Efimtsov model can be used to separate integrals in the wave number domain.
The comparison of measured and predicted velocity spectral density and the radiated sound intensity of a plate (
3.3 Characteristic frequency in hydrodynamic coincidence
When the velocity of bending wave in the wall plate is close to the sound velocity in the air, the sound radiation efficiency reaches the maximum value. The corresponding frequency is the so-called critical frequency, and its expression is
In the case of flow, when the velocity of flexural wave propagation in the wall plate is close to the turbulent convection velocity, the wall plate is most excited by the fluctuating pressure of TBL. The corresponding frequency is defined as the hydrodynamic coincidence frequency
Similarly, for order (
In conclusion, the relationship between critical frequency and hydrodynamic coincidence frequency can be summarized as follows
In the above two equations,
4. Effect of flow velocity and structural damping on the acoustic radiation efficiency
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.
|Plate length||1.25 m|
|Plate width||1.1 m|
|Plate thickness||0.002 m|
|Plate surface density||5.4 kg/m2|
|Plate bending stiffness||52 Nm|
|Air density||1.21 kg/m3|
|Sound speed||340 m/s|
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 (
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 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 (
The reason for above phenomenon may be further explored through the modal excitation terms. As illustrated in Figure 6, the lateral modal excitation term (10log10
As an example, the hydrodynamic coincidence lines for different flow velocity (
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 (
The effect of structural damping on the modal averaged radiation efficiency can be qualitatively explained by Eq. (61)
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  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.
This chapter studies the vibro-acoustic characteristics of the wall plate structure 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 higher radiation efficiencies.
The modal averaged radiation efficiency increases with the increase of structural 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%.
Thanks to the financial support by the Taishan Scholar Program of Shandong (no. ts201712054).
Conflict of interest
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