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, U<sup>c</sup> is turbulent flow velocity, and the direction of the incoming flow is parallel to the X-axis. In this chapter, vibro-acoustic responses are solved by modal superposition method [23].

Figure 2. Schematic diagram of simply supported thin rectangular plate excited by TBL.

Assume that point s on the plate is excited by a normal force F at points, and the vibration displacement response at point rcan be calculated by

$$W(\mathbf{r}, \alpha) = H(\mathbf{r}, \mathbf{s}, \alpha) \cdot F(\mathbf{s}, \alpha) \tag{27}$$

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

The impulse response H satisfies the following governing equation

$$\left[D(\mathbf{1} + j\eta)\nabla^4 - m\_s a^2\right]H(\mathbf{r}, \mathbf{s}, a) = \delta(\mathbf{r} - \mathbf{s})\tag{28}$$

The impulse response can be expanded as

$$H(\mathbf{r}, \mathbf{s}, \alpha) = \sum\_{m=1}^{M} \sum\_{n=1}^{N} H\_{mn}(\alpha) \Psi\_{mn}(\mathbf{r}) \Psi\_{mn}(\mathbf{s}) \tag{29}$$

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

$$H\_{mn}(o) = \frac{1}{DK\_{mn}(1+j\eta) - m\_{\rm r}o^2} \tag{30}$$

#### 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

$$\begin{split} S\_{WW}(\mathbf{r\_1}, \mathbf{r\_2}, \boldsymbol{\omega}) &= \int\_{\mathcal{S}} \int\_{\mathcal{S}} \mathbb{S}\_{pp}(\mathbf{s\_1} - \mathbf{s\_2}, \boldsymbol{\omega}) H^\* \left( \mathbf{r\_1}, \mathbf{s\_1}, \boldsymbol{\omega} \right) H(\mathbf{r\_2}, \mathbf{s\_2}, \boldsymbol{\omega}) d\mathbf{s\_1} d\mathbf{s\_2} \\ &= \Phi\_{pp}(\boldsymbol{\omega}) \sum\_{m=1}^{M} \sum\_{n=1}^{N} |H\_{mn}(\boldsymbol{\omega})|^2 \Psi\_{mn}(\mathbf{r\_1}) \Psi\_{mn}(\mathbf{r\_2}) J\_{mn}(\boldsymbol{\omega}) \end{split} \tag{31}$$

where

α2

solved by modal superposition method [23].

2.7 Comparison of models

pressure, reproduced from Ref. [23].

Figure 1.

wave number.

by TBL

32

<sup>M</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> ð Þ bMkc<sup>δ</sup> �<sup>2</sup>

Boundary Layer Flows - Theory, Applications and Numerical Methods

, α<sup>2</sup>

A comparison of models for different wavenumber-frequency spectrum of turbulent boundary layer fluctuating

CM ¼ 0:0745, CT ¼ 0:0475, bM ¼ 0:756, bT ¼ 0:378, h ¼ 3:0 (26)

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

3. Calculation of vibro-acoustic responses of the wall plate excited

Figure 2. In the figure, U<sup>c</sup> is turbulent flow velocity, and the direction of the incoming flow is parallel to the X-axis. In this chapter, vibro-acoustic responses are

Consider a simply supported thin rectangular plate excited by TBL, as shown in

<sup>T</sup> <sup>¼</sup> <sup>1</sup> <sup>þ</sup> ð Þ bTkc<sup>δ</sup> �<sup>2</sup> (24)

μ ¼ hUτ=Uc (25)

$$J\_{\rm mn}(\boldsymbol{\omega}) = \int\_{\mathcal{S}} \int\_{\mathcal{S}} \mathbb{S}\_{\rm pp}(\mathfrak{s}\_{\mathbf{1}} - \mathfrak{s}\_{\mathbf{2}}) \Psi\_{\rm mn}(\mathfrak{s}\_{\mathbf{1}}) \Psi\_{\rm mn}(\mathfrak{s}\_{\mathbf{2}}) \, d\mathfrak{s}\_{\mathbf{1}} d\mathfrak{s}\_{\mathbf{2}} \tag{32}$$

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

When using the Corcos model, the coordinate transformation of the quadruple integral in the modal excitation term can be obtained

$$J\_{\rm mn}(\alpha) = \frac{4}{\mathcal{S}} \left( \frac{1}{k\_m k\_n} J\_{mn}^1 + J\_{mn}^2 + \frac{1}{k\_m} J\_{mn}^3 + \frac{1}{k\_n} J\_{mn}^4 \right) \tag{33}$$

Where

$$\begin{Bmatrix} f\_{mn}^{1} \\ f\_{mn}^{2} \\ f\_{mn}^{3} \\ f\_{mn}^{4} \\ f\_{mn}^{6} \end{Bmatrix} = \int\_{0}^{b} \int\_{0}^{a} \begin{Bmatrix} 1 \\ (a-x)(b-y) \\ (b-y) \\ (b-y) \\ (a-x) \end{Bmatrix} \times \begin{Bmatrix} \sin k\_{m}x \cdot \sin k\_{n}y \\ \cos k\_{m}x \cdot \cos k\_{n}y \\ \sin k\_{m}x \cdot \cos k\_{n}y \\ \cos k\_{m}x \cdot \sin k\_{n}y \end{Bmatrix} \bar{S}\_{pp}(x,y,a) dx dy$$
 
$$\bar{\mathfrak{S}}\_{m}(x,y,a) = \text{sgn}(x,y,b,y) \text{sgn}(x,y,b,y) \text{sgn}(bx) \tag{34}$$

$$\tilde{S}\_{pp}(\mathbf{x}, \boldsymbol{y}, \boldsymbol{\omega}) = \exp\left(-\boldsymbol{\gamma}\_1 \mathbf{k}\_c \mathbf{x}\right) \exp\left(-\boldsymbol{\gamma}\_3 \mathbf{k}\_c \mathbf{y}\right) \cos\left(\mathbf{k}\_c \mathbf{x}\right) \tag{35}$$

where s1 � s2 ¼ ξx; ξ<sup>y</sup>

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

> <sup>¼</sup> <sup>1</sup> ð Þ <sup>2</sup><sup>π</sup> <sup>2</sup>

> <sup>¼</sup> <sup>1</sup> ð Þ <sup>2</sup><sup>π</sup> <sup>2</sup>

> <sup>¼</sup> <sup>1</sup> ð Þ <sup>2</sup><sup>π</sup> <sup>2</sup>

ð

ð

ð

Gð Þ¼ r; k;ω

Sppð Þ k;ω dk

ð

¼ ∑ M m¼1 ∑ N n¼1

¼ ∑ M m¼1 ∑ N n¼1

Ψ mnð Þs exp ð Þ jks ds

ðb 0 ða 0

ð Þ <sup>2</sup><sup>π</sup> <sup>2</sup>

<sup>¼</sup> <sup>ω</sup><sup>2</sup> ð Þ <sup>2</sup><sup>π</sup> <sup>2</sup> <sup>∑</sup> M m¼1 ∑ N n¼1 Ψ2

Sppð Þ <sup>k</sup>;<sup>ω</sup> j j Imnð Þ <sup>k</sup> <sup>2</sup>

<sup>Λ</sup>mð Þ¼ <sup>ω</sup> <sup>2</sup>k<sup>2</sup>

m ð<sup>∞</sup> �∞

ð

As for the Corcos model, we can obtain that

<sup>d</sup><sup>k</sup> <sup>¼</sup> <sup>4</sup>

k2 <sup>x</sup> � <sup>k</sup><sup>2</sup> m

vibration response

SWWð Þ¼ r1; r2;ω

where

Imnð Þ¼ <sup>k</sup> <sup>Ð</sup>

<sup>¼</sup> <sup>2</sup> ffiffiffiffiffi ab <sup>p</sup>

<sup>¼</sup> <sup>2</sup> ffiffiffiffiffi ab <sup>p</sup> �

SVVð Þ¼ <sup>r</sup>;<sup>ω</sup> <sup>ω</sup><sup>2</sup>

ð

where

35

� �, <sup>k</sup> <sup>¼</sup> kx; ky

� �.

ð ð 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

Sppð Þ k;ω exp ½ � �jk sð Þ <sup>1</sup> � s2 dk

Sppð Þ <sup>k</sup>;<sup>ω</sup> <sup>G</sup><sup>∗</sup> ð Þ r1; <sup>k</sup>;<sup>ω</sup> <sup>G</sup>ð Þ r2; <sup>k</sup>;<sup>ω</sup> <sup>d</sup><sup>k</sup>

Hð Þ r;s;ω exp ð Þ jks ds

km <sup>1</sup> � cosð Þ <sup>m</sup><sup>π</sup> exp jkxa � � � � k2 <sup>x</sup> � <sup>k</sup><sup>2</sup> m

Similarly, the spectral density of the vibration velocity can be obtained as

mnð Þ<sup>r</sup> j j Hmnð Þ <sup>ω</sup> <sup>2</sup>

Sppð Þ <sup>k</sup>;<sup>ω</sup> j j <sup>G</sup>ð Þ <sup>r</sup>; <sup>k</sup>;<sup>ω</sup> <sup>2</sup>

Hmnð Þ ω Ψ mnð Þr

Hmnð Þ ω Ψ mnð Þr Imnð Þ k

sin ð Þ kmx sin ð Þ kny exp j kxx <sup>þ</sup> kyy � � � � dxdy

�

dk

ð

1 � cosð Þ mπ cosð Þ kxa

ð

The formula can be obtained by substituting the cross spectral density of the

<sup>H</sup><sup>∗</sup> ð Þ r1;s1;<sup>ω</sup> exp ð Þ �jks1 <sup>d</sup>s1

ð

ð ð <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

Hð Þ r2;s2;ω exp ð Þ jks2 ds2

(42)

(43)

(44)

(45)

ð

Ψ mnð Þs exp ð Þ jks ds

kn 1 � cosð Þ nπ exp jkyb h i � �

> k2 <sup>y</sup> � <sup>k</sup><sup>2</sup> n

Sppð Þ <sup>k</sup>;<sup>ω</sup> j j Imnð Þ <sup>k</sup> <sup>2</sup>

<sup>S</sup> <sup>Φ</sup>ppð Þ <sup>ω</sup> <sup>2</sup><sup>γ</sup> ½ � <sup>1</sup>kcΛmð Þ <sup>ω</sup> <sup>2</sup><sup>γ</sup> ½ � <sup>3</sup>kcΓnð Þ <sup>ω</sup> (46)

� �<sup>2</sup> ð Þ kx � kc <sup>2</sup> <sup>þ</sup> <sup>γ</sup>ð Þ <sup>1</sup>kc <sup>2</sup> h i dkx (47)

dk

When r<sup>1</sup> = r2, the auto-spectral density of displacement response can be obtained as

$$\mathcal{L}\_{WW}(\boldsymbol{r},\boldsymbol{\alpha}) = \Phi\_{pp}(\boldsymbol{\alpha}) \sum\_{m=1}^{M} \sum\_{n=1}^{N} |H\_{mn}(\boldsymbol{\alpha})|^{2} \Psi\_{mn}^{2}(\boldsymbol{r}) I\_{mn}(\boldsymbol{\alpha}) \tag{36}$$

As for vibration (V = jωW) the auto-spectral density is

$$\begin{split} \mathbb{S}\_{VV}(\mathbf{r}, \boldsymbol{\omega}) &= \boldsymbol{\alpha}^2 \mathbb{S}\_{WW}(\mathbf{r}, \boldsymbol{\omega}) \\ &= \boldsymbol{\alpha}^2 \Phi\_{pp}(\boldsymbol{\alpha}) \sum\_{m=2n-1}^{M} \sum\_{l=1}^{N} |H\_{mn}(\boldsymbol{\alpha})|^2 \Psi\_{mn}^2(\mathbf{r}) I\_{mn}(\boldsymbol{\alpha}) \end{split} \tag{37}$$

So, vibration energy and acoustic radiation energy can be expressed as

$$
\begin{split}
\langle\mathcal{V}^2\rangle &= \frac{1}{\mathcal{S}} \int \Big\| \mathcal{S}\_{VV}(\mathbf{x}, \mathbf{y}, \boldsymbol{\omega}) d\mathcal{S} \\ &= \frac{1}{\mathcal{S}} \boldsymbol{\omega}^2 \Phi\_{pp}(\boldsymbol{\omega}) \sum\_{m=1}^M \sum\_{n=1}^N J\_{mn}(\boldsymbol{\omega}) \left| H\_{mn}(\boldsymbol{\omega}) \right|^2
\end{split}
\tag{38}
$$

$$
\begin{split}
\Pi' &= \rho\_0 c\_0 \boldsymbol{\omega}^2 \Phi\_{pp}(\boldsymbol{\omega}) \sum\_{m=1}^M \sum\_{n=1}^N \sigma\_{mn} \boldsymbol{J}\_{mn}(\boldsymbol{\omega}) \left| H\_{mn}(\boldsymbol{\omega}) \right|^2
\end{split}
\tag{39}
$$

According to the definition, the modal average acoustic radiation efficiency excited by TBL of the thin plate is

$$\sigma = \frac{\sum\_{m=1}^{M} \sum\_{n=1}^{N} \sigma\_{mn} J\_{mn}(\alpha) |H\_{mn}(\alpha)|^2}{\sum\_{m=1}^{M} \sum\_{n=1}^{N} J\_{mn}(\alpha) \left| H\_{mn}(\alpha) \right|^2} \tag{40}$$

### 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 [24].

The wavenumber-frequency spectrum of TBL satisfies the following relationship

$$\begin{split} S\_{pp}(\mathbf{s}\_1 - \mathbf{s}\_2, \boldsymbol{\omega}) &= \frac{1}{(2\pi)^2} \int \mathbf{S}\_{pp}(\mathbf{k}, \boldsymbol{\omega}) \exp\left[-j\mathbf{k}(\mathbf{s}\_1 - \mathbf{s}\_2)\right] d\mathbf{k} \\ &= \frac{1}{(2\pi)^2} \int \left[\mathbf{S}\_{pp}\left(\mathbf{k}\_\mathbf{x}, \mathbf{k}\_\mathbf{y}, \boldsymbol{\omega}\right) \exp\left[-j\left(\mathbf{k}\_\mathbf{x}\boldsymbol{\xi}\_\mathbf{x} + \mathbf{k}\_\mathbf{y}\boldsymbol{\xi}\_\mathbf{y}\right)\right] d\mathbf{k}\_\mathbf{x} d\mathbf{k}\_\mathbf{y} \end{split} \tag{41}$$

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

where s1 � s2 ¼ ξx; ξ<sup>y</sup> � �, <sup>k</sup> <sup>¼</sup> kx; ky � �.

The formula can be obtained by substituting the cross spectral density of the vibration response

$$\begin{aligned} S\_{WW}(\mathbf{r}\_1, \mathbf{r}\_2, \boldsymbol{\omega}) &= \int \left[ \int S\_{pp}(\mathbf{s}\_1, \mathbf{s}\_2, \boldsymbol{\omega}) H^\* \left( \mathbf{r}\_1, \mathbf{s}\_1, \boldsymbol{\omega} \right) H(\mathbf{r}\_2, \mathbf{s}\_2, \boldsymbol{\omega}) ds\_2 ds\_2 \\ &= \frac{1}{\left( 2\pi \right)^2} \int S\_{pp}(\mathbf{k}, \boldsymbol{\omega}) \exp \left[ -j\mathbf{k}(\mathbf{s}\_1 - \mathbf{s}\_2) \right] d\mathbf{k} \right] \int H^\* \left( \mathbf{r}\_1, \mathbf{s}\_1, \boldsymbol{\omega} \right) H(\mathbf{r}\_2, \mathbf{s}\_2, \boldsymbol{\omega}) ds\_1 ds\_2 \\ &= \frac{1}{\left( 2\pi \right)^2} \int S\_{pp}(\mathbf{k}, \boldsymbol{\omega}) d\mathbf{k} \left[ H^\* \left( \mathbf{r}\_1, \mathbf{s}\_1, \boldsymbol{\omega} \right) \exp \left( -j\mathbf{k}s\_1 \right) ds\_1 \int H(\mathbf{r}\_2, \mathbf{s}\_2, \boldsymbol{\omega}) \exp \left( j\mathbf{k}s\_2 \right) ds\_2 \right] \\ &= \frac{1}{\left( 2\pi \right)^2} \int S\_{pp}(\mathbf{k}, \boldsymbol{\omega}) G^\* \left( \mathbf{r}\_1, \mathbf{k}, \boldsymbol{\omega} \right) G(\mathbf{r}\_2, \mathbf{k}, \boldsymbol{\omega}) d\mathbf{k} \end{aligned} \tag{42}$$

where

Where

J 1 mn J 2 mn J 3 mn J 4 mn 9 >>>>>=

ða 0

~

8 >>>>><

>>>>>:

1

Boundary Layer Flows - Theory, Applications and Numerical Methods

9 >>>>>= 8 >>>>><

>>>>>:

sin kmx � sin kny cos kmx � cos kny sin kmx � cos kny cos kmx � sin kny

Sppð Þ¼ x; y;ω exp �γ ð Þ <sup>1</sup>kcx exp �γ ð Þ <sup>3</sup>kcy cosð Þ kcx (35)

j j Hmnð Þ <sup>ω</sup> <sup>2</sup>

j j Hmnð Þ <sup>ω</sup> <sup>2</sup>

Ψ2

Ψ2

Jmnð Þ <sup>ω</sup> j j Hmnð Þ <sup>ω</sup> <sup>2</sup>

<sup>σ</sup>mnJmnð Þ <sup>ω</sup> j j Hmnð Þ <sup>ω</sup> <sup>2</sup> (39)

<sup>n</sup>¼<sup>1</sup> Jmnð Þ <sup>ω</sup> j j Hmnð Þ <sup>ω</sup> <sup>2</sup> (40)

h i � �

dkxdky

(41)

9 >>>>>=

>>>>>;

~

Sppð Þ x; y;ω dxdy

mnð Þr Jmnð Þ ω (36)

mnð Þ<sup>r</sup> <sup>J</sup>mnð Þ <sup>ω</sup> (37)

(34)

(38)

>>>>>; �

When r<sup>1</sup> = r2, the auto-spectral density of displacement response can be

M m¼1 ∑ N n¼1

> M m¼1 ∑ N n¼1

So, vibration energy and acoustic radiation energy can be expressed as

Φppð Þ ω ∑ M m¼1 ∑ N n¼1

Φppð Þ ω ∑ M m¼1 ∑ N n¼1

<sup>m</sup>¼<sup>1</sup>∑<sup>N</sup>

3.2 Vibro-acoustic responses of plate solved by wavenumber domain

ary layer pulsating pressure models in the wavenumber domain [24].

∑<sup>M</sup> <sup>m</sup>¼<sup>1</sup>∑<sup>N</sup>

SVVð Þ x; y;ω dS

According to the definition, the modal average acoustic radiation efficiency

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 bound-

The wavenumber-frequency spectrum of TBL satisfies the following relationship

Sppð Þ k;ω exp ½ � �jk sð Þ <sup>1</sup> � s2 dk

Spp kx; ky;<sup>ω</sup> � � exp �j kxξ<sup>x</sup> <sup>þ</sup> kyξ<sup>y</sup>

<sup>n</sup>¼<sup>1</sup>σmn Jmnð Þ <sup>ω</sup> j j Hmnð Þ <sup>ω</sup> <sup>2</sup>

ð Þ a � x ð Þ b � y ð Þ b � y ð Þ a � x

SWWð Þ¼ r;ω Φppð Þ ω ∑

SVVð Þ¼ <sup>r</sup>;<sup>ω</sup> <sup>ω</sup><sup>2</sup>SWWð Þ <sup>r</sup>;<sup>ω</sup>

<sup>V</sup><sup>2</sup> � � <sup>¼</sup> <sup>1</sup> S ð ð

<sup>Π</sup><sup>r</sup> <sup>¼</sup> <sup>ρ</sup>0c0ω<sup>2</sup>

excited by TBL of the thin plate is

integration

34

Sppð Þ¼ s1 � s2;<sup>ω</sup> <sup>1</sup>

ð Þ <sup>2</sup><sup>π</sup> <sup>2</sup>

<sup>¼</sup> <sup>1</sup> ð Þ <sup>2</sup><sup>π</sup> <sup>2</sup>

ð

ð ð

¼ 1 S ω2

<sup>σ</sup> <sup>¼</sup> <sup>∑</sup><sup>M</sup>

As for vibration (V = jωW) the auto-spectral density is

<sup>¼</sup> <sup>ω</sup><sup>2</sup>Φppð Þ <sup>ω</sup> <sup>∑</sup>

>>>>>; ¼ ðb 0

8 >>>>><

>>>>>:

obtained as

$$\begin{aligned} G(r,\mathbf{k},\omega) &= \int H(r,\mathbf{s},\omega) \exp\left(j\mathbf{k}\mathbf{s}\right) d\mathbf{s} \\ &= \sum\_{m=1}^{M} \sum\_{n=1}^{N} H\_{mn}(\omega) \Psi\_{mn}(r) \left[\Psi\_{mn}(\mathbf{s}) \exp\left(j\mathbf{k}\mathbf{s}\right) d\mathbf{s}\right] \\ &= \sum\_{m=1}^{M} \sum\_{n=1}^{N} H\_{mn}(\omega) \Psi\_{mn}(r) I\_{mn}(\mathbf{k}) \\ &= \int \Psi\_{mn}(\mathbf{s}) \exp\left(i\mathbf{k}\mathbf{s}\right) d\mathbf{s} \end{aligned} \tag{43}$$

$$\begin{split} I\_{mn}(\mathbf{k}) &= \int \Psi\_{mn}(s) \exp\left(j\mathbf{k}s\right) ds \\ &= \frac{2}{\sqrt{ab}} \int\_{0}^{b} \int\_{0}^{a} \sin\left(k\_{\mathrm{x}}\boldsymbol{x}\right) \sin\left(k\_{\mathrm{y}}\boldsymbol{y}\right) \exp\left[j\left(k\_{\mathrm{x}}\boldsymbol{x} + k\_{\mathrm{y}}\boldsymbol{y}\right)\right] d\mathbf{x} d\boldsymbol{y} \\ &= \frac{2}{\sqrt{ab}} \cdot \frac{k\_{m} \left[1 - \cos\left(m\boldsymbol{\pi}\right) \exp\left(j k\_{\mathrm{x}}a\right)\right]}{k\_{\mathrm{x}}^{2} - k\_{m}^{2}} \cdot \frac{k\_{n} \left[1 - \cos\left(n\boldsymbol{\pi}\right) \exp\left(j k\_{\mathrm{y}}b\right)\right]}{k\_{\mathrm{y}}^{2} - k\_{n}^{2}} \end{split} \tag{44}$$

Similarly, the spectral density of the vibration velocity can be obtained as

$$\begin{split} \mathcal{S}\_{VV}(\mathbf{r}, \omega) &= \frac{\alpha^2}{\left(2\pi\right)^2} \int \mathcal{S}\_{pp}(\mathbf{k}, \omega) |G(\mathbf{r}, \mathbf{k}, \omega)|^2 d\mathbf{k} \\ &= \frac{\alpha^2}{\left(2\pi\right)^2} \sum\_{m=1}^M \sum\_{n=1}^N \Psi\_{mn}^2(\mathbf{r}) |H\_{mn}(\omega)|^2 \int \mathcal{S}\_{pp}(\mathbf{k}, \omega) |I\_{mn}(\mathbf{k})|^2 d\mathbf{k} \end{split} \tag{45}$$

As for the Corcos model, we can obtain that

$$\int \left| \mathbb{S}\_{pp}(\mathbb{k}, \alpha) \right| I\_{mn}(\mathbb{k}) \right|^2 d\mathbb{k} = \frac{4}{\mathsf{S}} \Phi\_{pp}(\alpha) [2\gamma\_1 k\_c \Lambda\_m(\alpha)] [2\gamma\_3 k\_c \Gamma\_n(\alpha)] \tag{46}$$

where

$$\Lambda\_m(a) = 2k\_m^2 \int\_{-\infty}^{\infty} \frac{1 - \cos\left(m\pi\right)\cos\left(k\_x a\right)}{\left(k\_x^2 - k\_m^2\right)^2 \left[\left(k\_x - k\_c\right)^2 + \left(\chi\_1 k\_c\right)^2\right]} dk\_x \tag{47}$$

Boundary Layer Flows - Theory, Applications and Numerical Methods

$$\Gamma\_n(\alpha) = 2k\_n^2 \int\_{-\infty}^{\infty} \frac{1 - \cos\left(n\pi\right)\cos\left(k\_\gamma b\right)}{\left(k\_\gamma^2 - k\_n^2\right)^2 \left[k\_\gamma^2 + \left(\gamma\_3 k\_c\right)^2\right]} dk\_\gamma \tag{48}$$

According to the residue theorem, Λm(ω) and Γn(ω) can be further simplified as

<sup>Λ</sup>mð Þ¼ <sup>ω</sup> <sup>2</sup>k<sup>2</sup> m ð<sup>∞</sup> �∞ 1 � cosð Þ mπ cosð Þ kxa k2 <sup>x</sup> � <sup>k</sup><sup>2</sup> m � �<sup>2</sup> ð Þ kx � kc <sup>2</sup> <sup>þ</sup> <sup>γ</sup>ð Þ <sup>1</sup>kc <sup>2</sup> h i dkx <sup>¼</sup> <sup>2</sup>πk<sup>2</sup> m a 4k<sup>2</sup> <sup>m</sup> ð Þ km <sup>þ</sup> kc <sup>2</sup> <sup>þ</sup> <sup>γ</sup>ð Þ <sup>1</sup>kc <sup>2</sup> h i <sup>þ</sup> a 4k<sup>2</sup> <sup>m</sup> ð Þ km � kc <sup>2</sup> <sup>þ</sup> <sup>γ</sup>ð Þ <sup>1</sup>kc <sup>2</sup> h i 8 < : þ 1 � cosð Þ mπ exp � j þ γ<sup>1</sup> ½ � ð Þkca <sup>2</sup><sup>γ</sup> ð Þ <sup>1</sup>kc <sup>k</sup><sup>2</sup> <sup>c</sup> <sup>1</sup> � <sup>j</sup>γ<sup>1</sup> ð Þ<sup>2</sup> � <sup>k</sup><sup>2</sup> m h i<sup>2</sup> þ 1 � cosð Þ mπ exp j � γ<sup>1</sup> ½ � ð Þkca <sup>2</sup><sup>γ</sup> ð Þ <sup>1</sup>kc <sup>k</sup><sup>2</sup> <sup>c</sup> <sup>1</sup> <sup>þ</sup> <sup>j</sup>γ<sup>1</sup> ð Þ<sup>2</sup> � <sup>k</sup><sup>2</sup> m h i<sup>2</sup> 9 >= >; (49) <sup>Γ</sup>nð Þ¼ <sup>ω</sup> <sup>2</sup>k<sup>2</sup> n ð<sup>∞</sup> �∞ <sup>1</sup> � cosð Þ <sup>n</sup><sup>π</sup> cos kyb � � k2 � �<sup>2</sup> k2 <sup>y</sup> <sup>þ</sup> <sup>γ</sup>ð Þ <sup>3</sup>kc <sup>2</sup> h i dky

$$\begin{split} \mathbf{J}^{-\infty} \left( k\_{\mathcal{Y}}^2 - k\_n^2 \right)^2 \left[ k\_{\mathcal{Y}}^2 + \left( \boldsymbol{\chi}\_3 \mathbf{k}\_c \right)^2 \right] \\ = 2\pi k\_n^2 \left\{ \frac{\mathbf{b}}{2k\_n^2 \left[ k\_n^2 + \left( \boldsymbol{\chi}\_3 \mathbf{k}\_c \right)^2 \right]} + \frac{\mathbf{1} - \cos \left( n\pi \right) \exp \left( -\boldsymbol{\chi}\_3 \mathbf{k}\_c \mathbf{b} \right)}{\left( \boldsymbol{\chi}\_3 \mathbf{k}\_c \right) \left[ k\_n^2 + \left( \boldsymbol{\chi}\_3 \mathbf{k}\_c \right)^2 \right]^2} \right\} \end{split} \tag{50}$$

By observing the above equation, it can be found that only the modal excitation

Comparison of calculation methods of the modal averaged radiation efficiency excited by TBL. Reproduced

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

The comparison of measured and predicted velocity spectral density and the radiated sound intensity of a plate (a � b = 0.62 � 0.3 m, and the thickness is 1.1 mm) is shown in Figure 4, which is only compared in narrow band. In this study, the loss factor of the plate assumes as 1.5%. The measured and predicted results for radiated sound intensity and auto spectrum of velocity have a good agreement with the frequency ranges from 100 to 3500 Hz. The agreement of the two type curves provides solid verification to test measured and predicted results.

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

> <sup>f</sup> <sup>c</sup> <sup>¼</sup> <sup>c</sup><sup>2</sup> 0 2π

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

ffiffiffiffiffi ms D r

(55)

term in the modal averaged radiation efficiency is related to turbulence.

3.3 Characteristic frequency in hydrodynamic coincidence

integrals in the wave number domain.

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

Figure 3.

from Ref. [23].

hydrodynamic coincidence frequency

37

Vibration energy and sound radiation energy are

$$
\begin{split}
\langle V^2 \rangle &= \frac{\alpha^2}{(2\pi)^2 S} \Bigg[ \int \mathbf{S}\_{\text{pp}}(\mathbf{k}, \boldsymbol{\alpha}) |G(\mathbf{r}, \mathbf{k}, \boldsymbol{\alpha})|^2 d\mathbf{k} d\mathbf{r} \\
&= \frac{\alpha^2}{(2\pi)^2 S} \sum\_{m=1}^M \sum\_{n=1}^N |H\_{mn}(\boldsymbol{\alpha})|^2 \int \mathbf{S}\_{\text{pp}}(\mathbf{k}, \boldsymbol{\alpha}) |I\_{mn}(\mathbf{k})|^2 d\mathbf{k} \\
&= \frac{1}{S} \alpha^2 \Phi\_{\text{pp}}(\boldsymbol{\alpha}) \left( \frac{4}{S} \frac{\mathbf{r}\_1 \mathbf{k}\_c \gamma\_3 \mathbf{k}\_c}{\pi} \right) \sum\_{m=1}^M \sum\_{n=1}^N \Lambda\_m(\boldsymbol{\alpha}) \Gamma\_n(\boldsymbol{\alpha}) |H\_{mn}(\boldsymbol{\alpha})|^2 \\
\varPi' &= \frac{1}{(2\pi)^2} \rho\_0 c\_0 \alpha^2 \sum\_{m=1}^M \sum\_{n=1}^N \sigma\_{mn} |H\_{mn}(\boldsymbol{\alpha})|^2 \int \mathbf{S}\_{\text{pp}}(\mathbf{k}, \boldsymbol{\alpha}) |I\_{mn}(\mathbf{k})|^2 d\mathbf{k} \\
&= \rho\_0 c\_0 \alpha^2 \Phi\_{\text{pp}}(\boldsymbol{\alpha}) \left( \frac{4}{S} \frac{\mathbf{r}\_1 \mathbf{k}\_c \gamma\_3 \mathbf{k}\_c}{\pi} \right) \sum\_{m=1}^M \sum\_{n=1}^N \sigma\_{mn} \Lambda\_m(\boldsymbol{\alpha}) \Gamma\_n(\boldsymbol{\alpha}) |H\_{mn}(\boldsymbol{\alpha})|^2 \end{split} \tag{52}
$$

Compare the above two equations, it can be seen that

$$J\_{mn}(\boldsymbol{\alpha}) = \frac{4}{\mathcal{S}} \left[ \frac{\boldsymbol{\gamma}\_1 \boldsymbol{k}\_c}{\pi} \boldsymbol{\Lambda}\_m(\boldsymbol{\alpha}) \right] \times \left[ \frac{\boldsymbol{\gamma}\_3 \boldsymbol{k}\_c}{\pi} \boldsymbol{\Gamma}\_n(\boldsymbol{\alpha}) \right] \tag{53}$$

Finally, the modal average acoustic radiation efficiency can be obtained as

$$\sigma = \frac{\sum\_{m=1}^{M} \sum\_{n=1}^{N} \sigma\_{mn} \Lambda\_m(\alpha) \Gamma\_n(\alpha) |H\_{mn}(\alpha)|^2}{\sum\_{m=1}^{M} \sum\_{n=1}^{N} \Lambda\_m(\alpha) \Gamma\_n(\alpha) \left| H\_{mn}(\alpha) \right|^2} \tag{54}$$

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

#### Figure 3.

<sup>Γ</sup>nð Þ¼ <sup>ω</sup> <sup>2</sup>k<sup>2</sup>

<sup>Λ</sup>mð Þ¼ <sup>ω</sup> <sup>2</sup>k<sup>2</sup>

m ð<sup>∞</sup> �∞

<sup>¼</sup> <sup>2</sup>πk<sup>2</sup> m

þ

<sup>Γ</sup>nð Þ¼ <sup>ω</sup> <sup>2</sup>k<sup>2</sup>

k2 <sup>x</sup> � <sup>k</sup><sup>2</sup> m

4k<sup>2</sup>

<sup>2</sup><sup>γ</sup> ð Þ <sup>1</sup>kc <sup>k</sup><sup>2</sup>

n ð<sup>∞</sup> �∞

<sup>¼</sup> <sup>2</sup>πk<sup>2</sup> n

<sup>V</sup><sup>2</sup> � � <sup>¼</sup> <sup>ω</sup><sup>2</sup>

ð Þ <sup>2</sup><sup>π</sup> <sup>2</sup> S ð ð

<sup>¼</sup> <sup>ω</sup><sup>2</sup> ð Þ <sup>2</sup><sup>π</sup> <sup>2</sup>

¼ 1 S ω2

ð Þ <sup>2</sup><sup>π</sup> <sup>2</sup> <sup>ρ</sup>0c0ω<sup>2</sup> <sup>∑</sup>

<sup>¼</sup> <sup>ρ</sup>0c0ω<sup>2</sup>Φppð Þ <sup>ω</sup> <sup>4</sup>

<sup>Π</sup><sup>r</sup> <sup>¼</sup> <sup>1</sup>

36

8 < : n ð<sup>∞</sup> �∞

Boundary Layer Flows - Theory, Applications and Numerical Methods

a

1 � cosð Þ mπ exp � j þ γ<sup>1</sup> ½ � ð Þkca

k2 <sup>y</sup> � <sup>k</sup><sup>2</sup> n � �<sup>2</sup>

2k<sup>2</sup> <sup>n</sup> <sup>k</sup><sup>2</sup>

<sup>S</sup> <sup>∑</sup> M m¼1 ∑ N n¼1

<sup>Φ</sup>ppð Þ <sup>ω</sup> <sup>4</sup>

M m¼1 ∑ N n¼1

S γ1kc π

S γ1kc π

Jmnð Þ¼ <sup>ω</sup> <sup>4</sup>

<sup>σ</sup> <sup>¼</sup> <sup>∑</sup><sup>M</sup>

Compare the above two equations, it can be seen that

S

<sup>m</sup>¼<sup>1</sup>∑<sup>N</sup>

∑<sup>M</sup> <sup>m</sup>¼<sup>1</sup>∑<sup>N</sup>

Vibration energy and sound radiation energy are

8 ><

>:

<sup>m</sup> ð Þ km <sup>þ</sup> kc <sup>2</sup> <sup>þ</sup> <sup>γ</sup>ð Þ <sup>1</sup>kc <sup>2</sup> h i <sup>þ</sup>

<sup>c</sup> <sup>1</sup> � <sup>j</sup>γ<sup>1</sup> ð Þ<sup>2</sup> � <sup>k</sup><sup>2</sup>

b

<sup>n</sup> <sup>þ</sup> <sup>γ</sup>ð Þ <sup>3</sup>kc <sup>2</sup> h i <sup>þ</sup>

Sppð Þ <sup>k</sup>;<sup>ω</sup> j j <sup>G</sup>ð Þ <sup>r</sup>; <sup>k</sup>;<sup>ω</sup> <sup>2</sup>

γ3kc π � �

<sup>σ</sup>mnj j Hmnð Þ <sup>ω</sup> <sup>2</sup>

γ3kc π � �

> γ1kc <sup>π</sup> <sup>Λ</sup>mð Þ <sup>ω</sup> � �

ð

∑ M m¼1 ∑ N n¼1

∑ M m¼1 ∑ N n¼1

Finally, the modal average acoustic radiation efficiency can be obtained as

ð

j j Hmnð Þ <sup>ω</sup> <sup>2</sup>

h i<sup>2</sup> þ

1 � cosð Þ mπ cosð Þ kxa

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

m

<sup>1</sup> � cosð Þ <sup>n</sup><sup>π</sup> cos kyb � �

k2

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

k2 <sup>y</sup> � <sup>k</sup><sup>2</sup> n � �<sup>2</sup>

According to the residue theorem, Λm(ω) and Γn(ω) can be further simplified as

4k<sup>2</sup>

<sup>2</sup><sup>γ</sup> ð Þ <sup>1</sup>kc <sup>k</sup><sup>2</sup>

dky

<sup>γ</sup>ð Þ <sup>3</sup>kc <sup>k</sup><sup>2</sup>

dkdr

Sppð Þ <sup>k</sup>;<sup>ω</sup> j j Imnð Þ <sup>k</sup> <sup>2</sup>

<sup>1</sup> � cosð Þ <sup>n</sup><sup>π</sup> cos kyb � �

k2

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

a

1 � cosð Þ nπ exp �γ ð Þ <sup>3</sup> kcb

<sup>n</sup> <sup>þ</sup> <sup>γ</sup>ð Þ <sup>3</sup>kc <sup>2</sup> h i<sup>2</sup>

dk

<sup>Λ</sup>mð Þ <sup>ω</sup> <sup>Γ</sup>nð Þ <sup>ω</sup> j j Hmnð Þ <sup>ω</sup> <sup>2</sup>

<sup>σ</sup>mnΛmð Þ <sup>ω</sup> <sup>Γ</sup>nð Þ <sup>ω</sup> j j Hmnð Þ <sup>ω</sup> <sup>2</sup>

<sup>π</sup> <sup>Γ</sup>nð Þ <sup>ω</sup> � �

<sup>n</sup>¼<sup>1</sup>Λmð Þ <sup>ω</sup> <sup>Γ</sup>nð Þ <sup>ω</sup> j j Hmnð Þ <sup>ω</sup> <sup>2</sup> (54)

dk

Sppð Þ <sup>k</sup>;<sup>ω</sup> j j Imnð Þ <sup>k</sup> <sup>2</sup>

� <sup>γ</sup>3kc

<sup>n</sup>¼<sup>1</sup>σmnΛmð Þ <sup>ω</sup> <sup>Γ</sup>nð Þ <sup>ω</sup> j j Hmnð Þ <sup>ω</sup> <sup>2</sup>

<sup>m</sup> ð Þ km � kc <sup>2</sup> <sup>þ</sup> <sup>γ</sup>ð Þ <sup>1</sup>kc <sup>2</sup> h i

1 � cosð Þ mπ exp j � γ<sup>1</sup> ½ � ð Þkca

<sup>c</sup> <sup>1</sup> <sup>þ</sup> <sup>j</sup>γ<sup>1</sup> ð Þ<sup>2</sup> � <sup>k</sup><sup>2</sup>

h i<sup>2</sup>

m

9 >=

>;

9 >= >; (49)

(50)

(51)

(52)

(53)

dky (48)

Comparison of calculation methods of the modal averaged radiation efficiency excited by TBL. Reproduced from Ref. [23].

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 (a � b = 0.62 � 0.3 m, and the thickness is 1.1 mm) is shown in Figure 4, which is only compared in narrow band. In this study, the loss factor of the plate assumes as 1.5%. The measured and predicted results for radiated sound intensity and auto spectrum of velocity have a good agreement with the frequency ranges from 100 to 3500 Hz. The agreement of the two type curves provides solid verification to test measured and predicted results.

#### 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

$$f\_c = \frac{c\_0^2}{2\pi} \sqrt{\frac{m\_s}{D}}\tag{55}$$

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

Figure 4. Measured and predicted velocity auto spectrum and the radiated sound intensity of the plate with the size of a � b = 0.62 � 0.3 m. Narrow band analysis in per Hz. Flow speed 86 m/s.

$$f\_h = \frac{U\_c^2}{2\pi} \sqrt{\frac{m\_s}{D}}\tag{56}$$

4. Effect of flow velocity and structural damping on the acoustic

radiated sound power is larger than that of the mean square velocity.

4.1 Effect of convection velocity on the modal averaged radiation efficiency

The specific parameters and dimensions used in the calculation are listed in

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

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

Plate length a 1.25 m Plate width b 1.1 m Plate thickness h 0.002 m Plate surface density ms 5.4 kg/m<sup>2</sup> Plate bending stiffness D 52 Nm Air density ρ<sup>0</sup> 1.21 kg/m<sup>3</sup> Sound speed c<sup>0</sup> 340 m/s

Effect of the convective Mach number on the modal averaged radiation efficiency of the finite aluminum plate.

radiation efficiency

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

Table 1.

Table 1.

Figure 5.

39

Reproduced from Ref. [23].

Parameters used in calculation.

Similarly, for order (m, n) mode, its critical frequency and hydrodynamic coincidence frequency are

$$f\_{c,mn} = \frac{c\_0}{2\pi} k\_{mn} \tag{57}$$

$$f\_{h,mn} = \frac{U\_c}{2\pi} k\_{mn} \tag{58}$$

In conclusion, the relationship between critical frequency and hydrodynamic coincidence frequency can be summarized as follows

$$f\_h = \mathbf{M}\_c^2 \cdot f\_c \tag{59}$$

$$\left.f\_{\;h,mn} = \mathcal{M}\_{\mathfrak{c}} \cdot f\_{\;c,mn} \right. \tag{60}$$

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