Abstract

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.

Keywords: turbulent boundary layer, plate vibration, radiated noise, modal radiation efficiency

### 1. Introduction

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 [1] 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 [2], 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. [3] 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. [4–6] 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 [7] further investigated the sensitivity of sound power radiated by aircraft panels to TBL parameters, and discussed the findings by Liu [4] that ring stiffeners may increase TBL induced noise radiation significantly.

expression δ is the thickness of the boundary layer and U<sup>∞</sup> the velocity of the flow well away from the structure. The flat-plate boundary layer is taken to lie in the x-y plane of a Cartesian coordinate system, with mean flow in the direction of the x-axis. Corcos assumes that the cross power spectral density, between the pressures

where Φpp(ω) is the auto-power spectral density of turbulent boundary layer fluctuating pressure, kc = ω/Uc is the convection wave number. γ<sup>1</sup> and γ<sup>3</sup> can be obtained by fitting experimental data, γ<sup>1</sup> and γ<sup>3</sup> are 0.11–0.12 and 0.7–0.12

The Fourier Transform of ξ<sup>x</sup> and ξ<sup>y</sup> can obtain wavenumber-frequency spectrum

So, the normalized wavenumber-frequency spectrum in wavenumber domain is

<sup>Φ</sup>ppð Þ <sup>ω</sup> Spp kx; ky;<sup>ω</sup> � �

ð Þ kx=kc � <sup>1</sup> <sup>2</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup>

Caiazzo and Desmet [21] proposed a generalized model which based on the Corcos model. The model uses butterworth filter to replace exponential decay of x and y direction in the Corcos model. It can make the wavenumber-frequency spectrum attenuation rapidly near the convection wave number by adjusting the

� � exp j kxξ<sup>x</sup> <sup>þ</sup> kyξ<sup>y</sup>

ð Þ kx � kc <sup>2</sup> <sup>þ</sup> <sup>γ</sup>ð Þ <sup>1</sup>kc <sup>2</sup> � <sup>2</sup>γ3kc

1

Q�1 q¼0

PQ <sup>γ</sup>ð Þ <sup>1</sup>kc <sup>2</sup>P�<sup>1</sup>

Q�1 q¼0 e jθq

ð Þ kx � kc <sup>2</sup><sup>P</sup> <sup>þ</sup> <sup>γ</sup>ð Þ <sup>1</sup>kc <sup>2</sup><sup>P</sup> h i <sup>∑</sup>

� � � � � � � � exp �jkcξ<sup>x</sup>

h i � � <sup>d</sup>ξxdξ<sup>y</sup>

<sup>y</sup> <sup>þ</sup> <sup>γ</sup>ð Þ <sup>3</sup>kc <sup>2</sup>

3

k2

� <sup>2</sup>γ<sup>3</sup> ky=kc � �<sup>2</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup>

� �

exp <sup>j</sup> <sup>θ</sup><sup>q</sup> <sup>þ</sup> <sup>γ</sup>1kc <sup>ξ</sup><sup>x</sup> j j � � � �

P�1 p¼0 e jθ<sup>p</sup>

� � (1)

(2)

(3)

(4)

(5)

at two different positions separated by the vector n can be expressed as

ð ð Spp <sup>ξ</sup>x; <sup>ξ</sup>y;<sup>ω</sup>

<sup>¼</sup> <sup>Φ</sup>ppð Þ <sup>ω</sup> <sup>2</sup>γ1kc

c

<sup>¼</sup> <sup>2</sup>γ<sup>1</sup>

� � ¼ �Φppð Þ <sup>ω</sup> sin ð Þ <sup>π</sup>=2<sup>P</sup> sin ð Þ <sup>π</sup>=2<sup>Q</sup> exp �jkcξ<sup>x</sup>

exp <sup>j</sup> <sup>θ</sup><sup>p</sup> <sup>þ</sup> <sup>γ</sup>1kc <sup>ξ</sup><sup>x</sup> j j � � � � � <sup>∑</sup>

c π2

ky

where θ<sup>p</sup> = (π/2P)�(1 + 2p), θ<sup>q</sup> = (π/2Q)�(1 + 2q). When P = Q = 1, Eq. (4) is equal

Analogously, the normalized wavenumber-frequency spectrum in wavenumber

� <sup>Q</sup> <sup>γ</sup>ð Þ <sup>3</sup>kc <sup>2</sup>Q�<sup>1</sup>

� �<sup>2</sup><sup>Q</sup> <sup>þ</sup> <sup>γ</sup>ð Þ <sup>3</sup>kc <sup>2</sup><sup>Q</sup> h i <sup>∑</sup>

� � <sup>¼</sup> <sup>Φ</sup>ppð Þ <sup>ω</sup> exp �γ1kc <sup>ξ</sup><sup>x</sup> ð Þ j j exp �γ3kc <sup>ξ</sup><sup>y</sup>

Spp ξx; ξy;ω

respectively for smooth rigid siding.

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

Spp kx; ky;<sup>ω</sup> � � <sup>¼</sup>

^

2.2 The generalized Corcos model

Spp ξx; ξy;ω

to the Corcos model.

domain is

29

Spp kx; ky;<sup>ω</sup> � � <sup>¼</sup> <sup>k</sup><sup>2</sup>

parameters. Expression of this model is as follows

Spp kx; ky;<sup>ω</sup> � � ¼ � <sup>k</sup><sup>2</sup>

� ∑ P�1 p¼0

^

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 [8] used a more concise formula to predict the acoustic radiation efficiency of an infinite plate. Wallace [9] 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. [10] 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 [11–13]. Graham [11] and Frampton [12] 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. [13] 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 wavenumberfrequency 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 (Mc) and structural damping on the modal averaged radiation efficiency is discussed.
