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 [14], Efimtsov [15], Smolyakov-Tkachenko [16], Williams [17], 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 [20], as long as ωδ/U<sup>∞</sup> > 1. In this

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

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

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

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

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

2. Models for the wavenumber-frequency spectrum of turbulent

ing parts introduce some typical wavenumber-frequency spectrum models.

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 [20], as long as ωδ/U<sup>∞</sup> > 1. In this

As for the research about wavenumber-frequency spectrum of turbulent boundary layer, Corcos [14], Efimtsov [15], Smolyakov-Tkachenko [16], Williams [17], 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 follow-

may increase TBL induced noise radiation significantly.

Boundary Layer Flows - Theory, Applications and Numerical Methods

increases with the increase of the flow velocity.

boundary layer fluctuating pressure

radiation efficiency is discussed.

2.1 The Corcos model

28

efficiency.

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

$$\mathcal{S}\_{pp}\left(\xi\_{\mathbf{x}},\xi\_{\mathbf{y}},\alpha\right) = \Phi\_{pp}(\alpha)\exp\left(-\gamma\_1 k\_c |\xi\_{\mathbf{x}}|\right)\exp\left(-\gamma\_3 k\_c \left|\xi\_{\mathbf{y}}\right|\right)\exp\left(-jk\_c \xi\_{\mathbf{x}}\right) \tag{1}$$

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 respectively for smooth rigid siding.

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

$$\begin{split} \mathbb{S}\_{pp}(\boldsymbol{k}\_{\rm x},\boldsymbol{k}\_{\rm y},\boldsymbol{\alpha}) &= \int \left[ \mathbb{S}\_{pp} \left( \boldsymbol{\xi}\_{\rm x}, \boldsymbol{\xi}\_{\rm y}, \boldsymbol{\alpha} \right) \exp \left[ j \left( \boldsymbol{k}\_{\rm x} \boldsymbol{\xi}\_{\rm x} + \boldsymbol{k}\_{\rm y} \boldsymbol{\xi}\_{\rm y} \right) \right] d \boldsymbol{\xi}\_{\rm x} d \boldsymbol{\xi}\_{\rm y} \\ &= \Phi\_{pp}(\boldsymbol{\alpha}) \frac{2 \gamma\_{1} \boldsymbol{k}\_{\rm c}}{\left( \boldsymbol{k}\_{\rm x} - \boldsymbol{k}\_{\rm c} \right)^{2} + \left( \gamma\_{1} \boldsymbol{k}\_{\rm c} \right)^{2}} \cdot \frac{2 \gamma\_{3} \boldsymbol{k}\_{\rm c}}{\boldsymbol{k}\_{\rm y}^{2} + \left( \gamma\_{3} \boldsymbol{k}\_{\rm c} \right)^{2}} \end{split} \tag{2}$$

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

$$\begin{split} \hat{S}\_{pp}(\mathbf{k}\_{\mathbf{x}}, \mathbf{k}\_{\mathbf{y}}, \boldsymbol{\alpha}) &= \frac{\mathbf{k}\_{\mathbf{c}}^{2}}{\Phi\_{pp}(\boldsymbol{\alpha})} \mathbf{S}\_{pp}(\mathbf{k}\_{\mathbf{x}}, \mathbf{k}\_{\mathbf{y}}, \boldsymbol{\alpha}) \\ &= \frac{2\boldsymbol{\gamma}\_{1}}{\left(\mathbf{k}\_{\mathbf{x}}/\mathbf{k}\_{\mathbf{c}} - \mathbf{1}\right)^{2} + \boldsymbol{\gamma}\_{1}^{2}} \cdot \frac{2\boldsymbol{\gamma}\_{3}}{\left(\mathbf{k}\_{\mathbf{y}}/\mathbf{k}\_{\mathbf{c}}\right)^{2} + \boldsymbol{\gamma}\_{3}^{2}} \end{split} \tag{3}$$

#### 2.2 The generalized Corcos model

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 parameters. Expression of this model is as follows

$$\begin{split} S\_{pp}\left(\xi\_{\mathbf{x}},\xi\_{\mathbf{y}},\omega\right) &= -\Phi\_{pp}(\omega)\sin\left(\pi/2\mathcal{P}\right)\sin\left(\pi/2\mathcal{Q}\right)\exp\left(-jk\_{c}\xi\_{\mathbf{x}}\right) \\ &\times \sum\_{p=0}^{P-1} \exp\left[j\left(\theta\_{p} + \gamma\_{1}k\_{c}|\xi\_{\mathbf{x}}|\right)\right] \times \sum\_{q=0}^{Q-1} \exp\left[j\left(\theta\_{q} + \gamma\_{1}k\_{c}|\xi\_{\mathbf{x}}|\right)\right] \end{split} \tag{4}$$

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 to the Corcos model.

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

$$\begin{split} \hat{S}\_{pp}(k\_{\rm x},k\_{\rm y},\alpha) &= -\frac{k\_{\rm c}^{2}}{\pi^{2}} \frac{P\mathcal{Q}(\boldsymbol{\chi}\_{1}\boldsymbol{k}\_{\rm c})^{2\mathcal{P}-1}}{\left[\left(\boldsymbol{k}\_{\rm x}-\boldsymbol{k}\_{\rm c}\right)^{2\mathcal{P}} + \left(\boldsymbol{\chi}\_{1}\boldsymbol{k}\_{\rm c}\right)^{2\mathcal{P}}\right]\_{p=0}^{p-1}} \\ &\times \frac{\mathcal{Q}(\boldsymbol{\chi}\_{3}\boldsymbol{k}\_{\rm c})^{2\mathcal{Q}-1}}{\left[\left(\boldsymbol{k}\_{\rm y}\right)^{2\mathcal{Q}} + \left(\boldsymbol{\chi}\_{3}\boldsymbol{k}\_{\rm c}\right)^{2\mathcal{Q}}\right]\_{q=0}^{Q-1}} \end{split} \tag{5}$$

#### 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, δ, as well as spatial separation is taken into account. Correlation length 1/γ1kc and 1/γ3kc in Corcos model are replaced with Λ<sup>x</sup> and Λy. The Efimtsov model gives the cross power spectral density of the pressure at two different positions separated by the vector ξ as

$$\mathcal{S}\_{pp}\left(\xi\_{\mathbf{x}},\xi\_{\mathbf{y}},\alpha\right) = \Phi\_{pp}(\alpha)\exp\left(-|\xi\_{\mathbf{x}}|/\Lambda\_{\mathbf{x}}\right)\exp\left(-\left|\xi\_{\mathbf{y}}\right|/\Lambda\_{\mathbf{y}}\right)\exp\left(-jk\_{c}\xi\_{\mathbf{x}}\right) \tag{6}$$

where

$$\Lambda\_{\mathbf{x}} = \delta \left[ \left( \frac{a\_1 \text{Sh}}{U\_c / U\_\tau} \right)^2 + \frac{a\_2^2}{\text{Sh}^2 + \left( a\_2 / a\_3 \right)^2} \right]^{-1/2} \tag{7}$$

<sup>h</sup>ð Þ¼ <sup>ω</sup> <sup>1</sup> � <sup>m</sup>1<sup>A</sup>

F kx; ky;<sup>ω</sup> � � <sup>¼</sup> <sup>A</sup><sup>2</sup> <sup>þ</sup> ð Þ <sup>1</sup> � kx=kc <sup>2</sup> <sup>þ</sup>

1:005 m<sup>1</sup>

where δ\* is the thickness of boundary layer, which is also set as δ\* = δ/8.

Ffowcs-Williams using Lighthill acoustic analogy theory to deduce a frequencywave 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 [22] ignore compression factor of the influence of this model to put forward a simplified model. The normalized wavenumber-frequency spectrum is

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

1

xK�<sup>5</sup>

<sup>M</sup> <sup>þ</sup> CTj j <sup>k</sup> <sup>2</sup>

<sup>3</sup><sup>ω</sup> <sup>1</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> ð Þ ð Þ CMFM <sup>þ</sup> CTFT (21)

<sup>T</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> <sup>α</sup><sup>2</sup>

<sup>M</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> <sup>α</sup><sup>2</sup>

K�<sup>5</sup> T � � (18)

<sup>þ</sup> j j <sup>k</sup> <sup>2</sup> <sup>þ</sup> ð Þ bM<sup>δ</sup> �<sup>2</sup> (19)

<sup>þ</sup> j j <sup>k</sup> <sup>2</sup> <sup>þ</sup> ð Þ bT<sup>δ</sup> �<sup>2</sup> (20)

<sup>M</sup> � <sup>1</sup> � � � � <sup>3</sup>=<sup>2</sup> (22)

<sup>T</sup> � <sup>1</sup> � � � � <sup>3</sup>=<sup>2</sup> (23)

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

3

<sup>Δ</sup>F kx; ky;<sup>ω</sup> � � <sup>¼</sup> <sup>0</sup>:995 1 <sup>þ</sup> <sup>A</sup><sup>2</sup> <sup>þ</sup>

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

2.5 The Ffowcs-Williams model

^

spectrum can be described as

^

2.6 The Chase model

where

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

Spp kx; ky;<sup>ω</sup> � � <sup>¼</sup> ð Þ <sup>2</sup><sup>π</sup> <sup>3</sup>

K2

K2

FM <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup>

<sup>T</sup> <sup>þ</sup> <sup>μ</sup><sup>2</sup> <sup>3</sup>α<sup>2</sup>

FT <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>α</sup><sup>2</sup>

31

kc

� �<sup>2</sup> 2γ<sup>1</sup>

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

> ρk<sup>2</sup> cU<sup>3</sup> τ <sup>Φ</sup>ð Þ <sup>ω</sup> CMk<sup>2</sup>

<sup>M</sup> <sup>¼</sup> ð Þ <sup>ω</sup> � Uckx <sup>2</sup> h2 U2 τ

<sup>T</sup> <sup>¼</sup> ð Þ <sup>ω</sup> � Uckx <sup>2</sup> h2 U2 τ

<sup>M</sup> <sup>þ</sup> <sup>μ</sup><sup>4</sup> <sup>α</sup><sup>2</sup> <sup>M</sup> � <sup>1</sup> � � � � <sup>=</sup> <sup>α</sup><sup>2</sup>

<sup>T</sup> � <sup>1</sup> � � <sup>þ</sup> <sup>2</sup>μ<sup>4</sup> <sup>α</sup><sup>2</sup> <sup>T</sup> � <sup>1</sup> � � � � <sup>=</sup> <sup>α</sup><sup>2</sup>

ρ<sup>2</sup>hU<sup>4</sup> τ

<sup>Φ</sup>ð Þ¼ <sup>ω</sup> ð Þ <sup>2</sup><sup>π</sup> <sup>2</sup>

α2

<sup>m</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>A</sup><sup>2</sup>

6:515 ffiffiffiffi G p � ��<sup>1</sup>

<sup>6</sup>:<sup>45</sup> � � � � �3=<sup>2</sup>

<sup>1</sup>:<sup>025</sup> <sup>þ</sup> <sup>A</sup><sup>2</sup> (13)

� �<sup>2</sup> � <sup>m</sup><sup>2</sup>

1

<sup>G</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>A</sup><sup>2</sup> � <sup>1</sup>:005m<sup>1</sup> (14)

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

� � n o �3=<sup>2</sup>

ky=kc

(12)

(15)

(16)

(17)

$$\Lambda\_{\mathcal{V}} = \begin{cases} \delta \left[ \left( \frac{a\_4 Sh}{U\_\epsilon/U\_\tau} \right)^2 + \frac{a\_5^2}{Sh^2 + (a\_5/a\_6)^2} \right]^{-1/2}, & M\_\infty < 0.75\\ \delta \left[ \left( \frac{a\_4 Sh}{U\_\epsilon/U\_\tau} \right)^2 + a\_7^2 \right]^{-1/2}, & M\_\infty > 0.9 \end{cases} \tag{8}$$

where Sh is the Strouhal number and equal to Sh = ωδ/U<sup>τ</sup> and U<sup>τ</sup> the friction velocity which varies with the Reynolds number but is typically of the order 0.03 U∞–0.04 U∞. At high frequencies these expressions correspond to a Corcos model with γ<sup>1</sup> = 0.1 and γ<sup>3</sup> = 0.77. Coefficient a1–a<sup>7</sup> are 0.1, 72.8, 1.54, 0.77, 548, 13.5 and 5.66 respectively. When 0.75 < M<sup>∞</sup> < 0.9, the Λ<sup>y</sup> can be determined by numerical interpolation. At high frequency, the Efimtsov model and the Corcos model are equal while γ<sup>1</sup> = 0.10 and γ<sup>3</sup> = 0.77.

The normalized wavenumber-frequency spectrum is

$$\hat{S}\_{pp}(\mathbf{k}\_{\mathbf{x}}, \mathbf{k}\_{\mathbf{y}}, \boldsymbol{\alpha}) = \frac{2\Lambda\_{\mathbf{x}}^{-1}}{\left(\mathbf{k}\_{\mathbf{x}}/\mathbf{k}\_{\mathbf{c}} - 1\right)^{2} + \left(\Lambda\_{\mathbf{x}}\mathbf{k}\_{\mathbf{c}}\right)^{-2}} \cdot \frac{2\Lambda\_{\mathbf{y}}^{-1}}{\mathbf{k}\_{\mathbf{y}}^{2} + \Lambda\_{\mathbf{y}}^{-2}} \tag{9}$$

#### 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

$$\exp\left[-\left(\left|\xi\_{\mathbf{x}}\right|/\Lambda\_{\mathbf{x}} + \left|\xi\_{\mathbf{y}}\right|/\Lambda\_{\mathbf{y}}\right)\right] \text{ to } \exp\left[-\sqrt{\left(\xi\_{\mathbf{x}}^2/\Lambda\_{\mathbf{x}}^2 + \xi\_{\mathbf{y}}^2/\Lambda\_{\mathbf{y}}^2\right)}\right], \text{ in order to make}$$

the computing result is consistent with the experimental results.

The normalized wavenumber-frequency spectrum is

$$\hat{S}\_{pp}(k\_{\mathbf{x}},k\_{\mathcal{I}},\boldsymbol{\alpha}) = \mathbf{0}.\mathcal{G}74A(\boldsymbol{\alpha})h(\boldsymbol{\alpha})\left[F(k\_{\mathbf{x}},k\_{\mathcal{I}},\boldsymbol{\alpha}) - \Delta F(k\_{\mathbf{x}},k\_{\mathcal{I}},\boldsymbol{\alpha})\right] \tag{10}$$

where

$$A(\omega) = 0.124 \left[ 1 - \frac{1}{4k\_c \delta^\*} + \left( \frac{1}{4k\_c \delta^\*} \right)^2 \right]^{1/2} \tag{11}$$

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

2.3 The Efimtsov model

Spp ξx; ξy;ω � �

Λ<sup>y</sup> ¼

8 >>><

>>>:

where

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, δ, as well as spatial separation is taken into account. Correlation length 1/γ1kc and 1/γ3kc in Corcos model are replaced with Λ<sup>x</sup> and Λy. The Efimtsov model gives the cross power spectral density of the pressure at two different positions separated by the vector ξ as

þ

þð Þ <sup>a</sup>5=a<sup>6</sup> <sup>2</sup>

where Sh is the Strouhal number and equal to Sh = ωδ/U<sup>τ</sup> and U<sup>τ</sup> the friction velocity which varies with the Reynolds number but is typically of the order 0.03 U∞–0.04 U∞. At high frequencies these expressions correspond to a Corcos model with γ<sup>1</sup> = 0.1 and γ<sup>3</sup> = 0.77. Coefficient a1–a<sup>7</sup> are 0.1, 72.8, 1.54, 0.77, 548, 13.5 and 5.66 respectively. When 0.75 < M<sup>∞</sup> < 0.9, the Λ<sup>y</sup> can be determined by numerical interpolation. At high frequency, the Efimtsov model and the Corcos

> x ð Þ kx=kc � <sup>1</sup> <sup>2</sup> <sup>þ</sup> ð Þ <sup>Λ</sup>xkc �<sup>2</sup> � <sup>2</sup>Λ�<sup>1</sup>

> > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 4kcδ <sup>∗</sup>

� �<sup>2</sup> " #<sup>1</sup>=<sup>2</sup>

r� � � �

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

> ξ2 x=Λ<sup>2</sup> <sup>x</sup> <sup>þ</sup> <sup>ξ</sup><sup>2</sup> y=Λ<sup>2</sup> y

Spp kx; ky;<sup>ω</sup> � � <sup>¼</sup> <sup>0</sup>:974Að Þ <sup>ω</sup> <sup>h</sup>ð Þ <sup>ω</sup> F kx; ky;<sup>ω</sup> � � � <sup>Δ</sup>F kx; ky;<sup>ω</sup> � � � � (10)

<sup>4</sup>kc<sup>δ</sup> <sup>∗</sup> <sup>þ</sup>

" #�1=<sup>2</sup>

� � � � � �=Λ<sup>y</sup> � �

a2 2 Sh<sup>2</sup> <sup>þ</sup> ð Þ <sup>a</sup>2=a<sup>3</sup>

2

, M∞>0:9

, M<sup>∞</sup> < 0:75

y

, in order to make

k2 <sup>y</sup> <sup>þ</sup> <sup>Λ</sup>�<sup>2</sup> y

exp �jkcξ<sup>x</sup>

� � (6)

(7)

(8)

(9)

(11)

¼ Φppð Þ ω exp � ξ<sup>x</sup> ð Þ j j=Λ<sup>x</sup> exp � ξ<sup>y</sup>

Uc=U<sup>τ</sup> � �<sup>2</sup>

> <sup>þ</sup> <sup>a</sup><sup>2</sup> 5 Sh<sup>2</sup>

� ��1=<sup>2</sup>

<sup>þ</sup> <sup>a</sup><sup>2</sup> 7

� ��1=<sup>2</sup>

The normalized wavenumber-frequency spectrum is

Spp kx; ky;<sup>ω</sup> � � <sup>¼</sup> <sup>2</sup>Λ�<sup>1</sup>

to exp �

the computing result is consistent with the experimental results. The normalized wavenumber-frequency spectrum is

<sup>A</sup>ð Þ¼ <sup>ω</sup> <sup>0</sup>:124 1 � <sup>1</sup>

<sup>Λ</sup><sup>x</sup> <sup>¼</sup> <sup>δ</sup> <sup>a</sup><sup>1</sup> Sh

Boundary Layer Flows - Theory, Applications and Numerical Methods

δ <sup>a</sup><sup>4</sup> Sh Uc=U<sup>τ</sup> � �<sup>2</sup>

δ <sup>a</sup><sup>4</sup> Sh Uc=U<sup>τ</sup> � �<sup>2</sup>

model are equal while γ<sup>1</sup> = 0.10 and γ<sup>3</sup> = 0.77.

^

exp � ξ<sup>x</sup> j j=Λ<sup>x</sup> þ ξ<sup>y</sup>

^

where

30

2.4 The Smolyakov-Tkachenko model

� � � � � �=Λ<sup>y</sup>

h i � �

$$h(\alpha) = \left[1 - \frac{m\_1 A}{6.515\sqrt{G}}\right]^{-1} \tag{12}$$

$$m\_1 = \frac{1 + A^2}{1.025 + A^2} \tag{13}$$

$$G = \mathbf{1} + A^2 - \mathbf{1}.005m\_1\tag{14}$$

$$F(k\_{\mathbf{x}}, k\_{\mathcal{I}}, o) = \left[ A^2 + \left( 1 - k\_{\mathbf{x}}/k\_{\epsilon} \right)^2 + \left( \frac{k\_{\mathcal{Y}}/k\_{\epsilon}}{6.45} \right) \right]^{-3/2} \tag{15}$$

$$\Delta F(k\_{\rm x}, k\_{\rm y}, \alpha) = 0.995 \left[ 1 + A^2 + \frac{1.005}{m\_1} \left\{ (m\_1 - k\_{\rm x}/k\_c)^2 + \left(k\_{\rm y}/k\_c\right)^2 - m\_1^2 \right\} \right]^{-3/2} \tag{16}$$

where δ\* is the thickness of boundary layer, which is also set as δ\* = δ/8.

### 2.5 The Ffowcs-Williams model

Ffowcs-Williams using Lighthill acoustic analogy theory to deduce a frequencywave 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 [22] ignore compression factor of the influence of this model to put forward a simplified model. The normalized wavenumber-frequency spectrum is

$$\hat{S}\_{pp}\left(\mathbf{k}\_{\mathbf{x}},\mathbf{k}\_{\mathbf{y}},\boldsymbol{\omega}\right) = \left(\frac{|\mathbf{k}|}{k\_{\mathbf{c}}}\right)^{2} \frac{2\boldsymbol{\gamma}\_{1}}{\left(\mathbf{k}\_{\mathbf{x}}/k\_{\mathbf{c}}-\mathbf{1}\right)^{2} + \boldsymbol{\gamma}\_{1}^{2}} \cdot \frac{2\boldsymbol{\gamma}\_{3}}{\left(\mathbf{k}\_{\mathbf{y}}/k\_{\mathbf{c}}\right)^{2} + \boldsymbol{\gamma}\_{3}^{2}} \tag{17}$$

#### 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

$$\hat{\mathbf{S}}\_{pp}(\mathbf{k}\_{\mathbf{x}},\mathbf{k}\_{\mathbf{y}},\boldsymbol{\alpha}) = \frac{(2\pi)^3 \rho k\_c^2 U\_\mathbf{r}^3}{\Phi(\boldsymbol{\alpha})} \left( \mathbf{C}\_M k\_\mathbf{x}^2 K\_M^{-\mathbf{s}} + \mathbf{C}\_T |\mathbf{k}|^2 K\_T^{-\mathbf{s}} \right) \tag{18}$$

where

$$K\_M^2 = \frac{\left(\rho - U\_c k\_x\right)^2}{h^2 U\_\tau^2} + \left|k\right|^2 + \left(b\_M \delta\right)^{-2} \tag{19}$$

$$K\_T^2 = \frac{\left(\rho - U\_c k\_x\right)^2}{h^2 U\_\tau^2} + |\mathbf{k}|^2 + \left(b\_T \delta\right)^{-2} \tag{20}$$

$$\Phi(\alpha) = \frac{\left(2\pi\right)^2 \rho^2 h U\_\tau^4}{3\alpha \left(1 + \mu^2\right)} \left(\mathcal{C}\_M F\_M + \mathcal{C}\_T F\_T\right) \tag{21}$$

$$F\_M = \left[\mathbf{1} + \mu^2 a\_M^2 + \mu^4 \left(a\_M^2 - \mathbf{1}\right)\right] / \left[a\_M^2 + \mu^2 \left(a\_M^2 - \mathbf{1}\right)\right]^{3/2} \tag{22}$$

$$F\_T = \left[1 + a\_T^2 + \mu^2 \left(3a\_T^2 - 1\right) + 2\mu^4 \left(a\_T^2 - 1\right)\right] / \left[a\_T^2 + \mu^2 \left(a\_T^2 - 1\right)\right]^{3/2} \tag{23}$$

#### Figure 1.

A comparison of models for different wavenumber-frequency spectrum of turbulent boundary layer fluctuating pressure, reproduced from Ref. [23].

$$a\_M^2 = \mathbf{1} + \left(b\_M k\_\epsilon \delta\right)^{-2}, \quad a\_T^2 = \mathbf{1} + \left(b\_T k\_\epsilon \delta\right)^{-2} \tag{24}$$

$$
\mu = h \mathbf{U}\_{\mathbf{r}} / \mathbf{U}\_{\mathbf{c}} \tag{25}
$$

Assume that point s on the plate is excited by a normal force F at points, and the

The modal amplitude of impulse response by using the Galerkin method can be

Wð Þ¼ r;ω Hð Þ� r;s;ω Fð Þ s;ω (27)

Hmnð Þ ω Ψ mnð Þr Ψ mnð Þs (29)

DKmnð Þ� <sup>1</sup> <sup>þ</sup> <sup>j</sup><sup>η</sup> msω<sup>2</sup> (30)

<sup>Ψ</sup> mnð Þ r1 <sup>Ψ</sup> mnð Þ r2 <sup>J</sup>mnð Þ <sup>ω</sup> (31)

Sppð Þ s1 � s2 Ψ mnð Þ s1 Ψ mnð Þ s2 ds1ds2 (32)

mn � � (33)

<sup>D</sup>ð Þ <sup>1</sup> <sup>þ</sup> <sup>j</sup><sup>η</sup> <sup>∇</sup><sup>4</sup> � msω<sup>2</sup> � �Hð Þ¼ <sup>r</sup>;s;<sup>ω</sup> <sup>δ</sup>ð Þ <sup>r</sup> � <sup>s</sup> (28)

vibration displacement response at point rcan be calculated by

Schematic diagram of simply supported thin rectangular plate excited by TBL.

The impulse response H satisfies the following governing equation

M m¼1 ∑ N n¼1

Hmnð Þ¼ <sup>ω</sup> <sup>1</sup>

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

j j Hmnð Þ <sup>ω</sup> <sup>2</sup>

When using the Corcos model, the coordinate transformation of the quadruple

1 km J 3 mn þ 1 kn J 4

<sup>S</sup>Sppð Þ s1 � s2;<sup>ω</sup> <sup>H</sup><sup>∗</sup> ð Þ r1;s1;<sup>ω</sup> <sup>H</sup>ð Þ r2;s2;<sup>ω</sup> <sup>d</sup>s1ds2

where s = (xo, y0), r = (x, y).

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

described as

Figure 2.

can be defined as

where

33

SWWð Þ¼ r1; r2;<sup>ω</sup> <sup>Ð</sup>

S Ð

Jmnð Þ¼ ω

¼ Φppð Þ ω ∑

ð S ð S

integral in the modal excitation term can be obtained

S

<sup>J</sup>mnð Þ¼ <sup>ω</sup> <sup>4</sup>

In the above equation, Jmn(ω) is called modal excitation term.

1 kmkn J 1 mn þ J 2 mn þ

M m¼1 ∑ N n¼1

The impulse response can be expanded as

Hð Þ¼ r;s;ω ∑

$$C\_M = 0.0745, C\_T = 0.0475, b\_M = 0.756, \quad b\_T = 0.378, \quad h = 3.0 \tag{26}$$

#### 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 γ<sup>1</sup> = 0.116, γ<sup>3</sup> = 0.77, the order of Generalized Corcos model is (P = 1, Q = 4). From the comparison among those models, it can be seen that the Generalized Corcos model attenuates quickly in the vicinity of the convective wave number, and its order is adjustable, which can effectively control the computational accuracy. The model can obtain more accurate prediction results by adjusting parameters. In addition, the Chase model is considered to be able to better describe the pressure characteristics of TBL pulsation at low wave number segment, while other models have some defects at low wave number segment. However, Corcos model is the most commonly used in practical application. Because the model is simple in form and has clear physical significance, a simple calculation formula can usually be obtained when solving the structural vibration and sound response induced by turbulent boundary layer. It should be noted that the structure radiated sound predicted by Corcos model tends to be larger at low wave number.
