4. Results and discussion

### 4.1. Flow fields

Figure 7 shows the contours of vorticity for M = 0.2, 0.4, and 0.6. The periodic vortex shedding was clarified to occur. The effects of the freestream Mach number on the frequency of the vortex shedding are shown in Figure 5. As mentioned above, it was found that the Strouhal number becomes lower as the Mach number becomes higher. The Strouhal number for M = 0.2

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Figure 7. Contours of vorticity ωz/(U0/d). (a) M = 0.2, (b) M = 0.4, and (c) M = 0.6.

related to the variation of vortices with the variation of the Mach number as discussed in detail in Section 4.1. Also, based on the present results, the extrapolated Strouhal number at M = 0.0 is 0.155. This value agrees well with the past computational data [13], although it is not clear why the past experimental value [14] is slightly higher. Consequently, the present direct

Figure 6 shows the polar plots of the sound pressure levels at r/d = 30.0 predicted by direct and hybrid simulations for M = 0.4. The acoustic field by the hybrid simulation is approximately in good agreement with that by the direct simulation. It has been confirmed that the two fields also agree for other Mach numbers such as M = 0.2 and 0.6. The above-mentioned methods of

Figure 7 shows the contours of vorticity for M = 0.2, 0.4, and 0.6. The periodic vortex shedding was clarified to occur. The effects of the freestream Mach number on the frequency of the vortex shedding are shown in Figure 5. As mentioned above, it was found that the Strouhal number becomes lower as the Mach number becomes higher. The Strouhal number for M = 0.2

Figure 6. Polar plots of sound pressure levels by direct and hybrid simulations at r/d = 30.0 for M = 0.4.

simulations are confirmed to be validated.

hybrid simulation are clarified to be validated.

3.2. Validation of hybrid simulation

334 Numerical Simulations in Engineering and Science

4. Results and discussion

4.1. Flow fields

is 0.151 and that for M = 0.6 is 0.144. Here, to clarify the reason, the Strouhal number becomes lower, and the flow fields are discussed.

Figure 8(a) shows the mean streamwise velocity at x/D = 1.0 for M = 0.2, 0.4, and 0.6. Figure 8(b) shows the half-value width of that profile, dh, for M = 0.2–0.6. The half-value width is shown to increase as the freestream Mach number becomes higher. Also, Figure 9 shows the mean streamwise Reynolds stress u1rms/U0. This figure shows that the Reynolds stress becomes larger as the freestream Mach number becomes higher. This means that the velocity fluctuations of the vortices intensify. Due to this intensification, the recovery of the mean streamwise in the wake becomes more rapid and the wake becomes wider as mentioned above. This change is different

Figure 8. (a) Mean streamwise velocity at x/D = 1.0 (M = 0.2, 0.4, and 0.6) and (b) half-value width of mean streamwise velocity at x/D = 1.0 for M = 0.2–0.6.

Figure 9. Mean streamwise Reynolds stress u1rms/U<sup>0</sup> (M = 0.2, 0.4, and 0.6).

from that of the vortices in the turbulent mixing layer [21], where the vortices become weaker with compressibility for a higher Mach number.

A possible reason the Reynolds stress becomes larger is that the acoustic feedback like that in the oscillations in cavity flows [22] also exists in the present cylinder flow and the acoustic waves affect the shed vortices. In this case, as the freestream Mach number becomes higher, the acoustic wave intensifies as shown in Section 4.2 and the shed vortex intensifies due to the acoustic feedback.

Roshko [23] showed that the frequency of the vortex shedding around a bluff body is proportional to the wake width. Here, to clarify the relationship between the wake width and the frequency of the vortex shedding, the modified Strouhal number Std, which is defined by Eq. (11), was computed.

$$\mathbf{St}\_{\mathbf{d}} = f \mathbf{d}\_{\mathbf{h}} / \mathbf{U}\_0. \tag{11}$$

Figure 10. Modified Strouhal number.

Figure 11. Contours of pressure fluctuations p'/(ρ0a<sup>0</sup>

(T is the period).

2

M 3.5) (M = 0.4). (a) t = 0, (b) t/T = 1/4, (c) t/T = 1/2, and (d) t/T = 3/4

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Figure 10 shows the effects of the freestream Mach number on the modified Strouhal number Std. This figure clarifies that the modified Strouhal number is approximately independent of the Mach number. Consequently, it is confirmed that the original Strouhal number decreases because the wake becomes wider. As mentioned above, the intensification of the velocity fluctuations of the vortices widens the wake.

### 4.2. Acoustic radiation

Figure 11 shows the contours of pressure fluctuations with the time-averaged pressure subtracted for M = 0.4. For the same Mach number, Figure 12 shows the contours of the second

Figure 10. Modified Strouhal number.

from that of the vortices in the turbulent mixing layer [21], where the vortices become weaker

A possible reason the Reynolds stress becomes larger is that the acoustic feedback like that in the oscillations in cavity flows [22] also exists in the present cylinder flow and the acoustic waves affect the shed vortices. In this case, as the freestream Mach number becomes higher, the acoustic wave intensifies as shown in Section 4.2 and the shed vortex intensifies due to the

Roshko [23] showed that the frequency of the vortex shedding around a bluff body is proportional to the wake width. Here, to clarify the relationship between the wake width and the frequency of the vortex shedding, the modified Strouhal number Std, which is defined by

Figure 10 shows the effects of the freestream Mach number on the modified Strouhal number Std. This figure clarifies that the modified Strouhal number is approximately independent of the Mach number. Consequently, it is confirmed that the original Strouhal number decreases because the wake becomes wider. As mentioned above, the intensification of the velocity

Figure 11 shows the contours of pressure fluctuations with the time-averaged pressure subtracted for M = 0.4. For the same Mach number, Figure 12 shows the contours of the second

Std ¼ fdh=U0: (11)

with compressibility for a higher Mach number.

336 Numerical Simulations in Engineering and Science

Figure 9. Mean streamwise Reynolds stress u1rms/U<sup>0</sup> (M = 0.2, 0.4, and 0.6).

fluctuations of the vortices widens the wake.

acoustic feedback.

Eq. (11), was computed.

4.2. Acoustic radiation

Figure 11. Contours of pressure fluctuations p'/(ρ0a<sup>0</sup> 2 M 3.5) (M = 0.4). (a) t = 0, (b) t/T = 1/4, (c) t/T = 1/2, and (d) t/T = 3/4 (T is the period).

Figure 12. Contours of the second invariant of velocity gradient tensor (M = 0.4). (a) t = 0, (b) t/T = 1/4, (c) t/T = 1/2, and (d) t/T = 3/4 (T is the period).

4.3. Acoustic fields

captions of Figures 11 and 12.

4.3.1. Directivity of acoustic wave

quantitatively in Section 4.3.2.

The direct sound pdirect is defined as.

4.3.2. Decomposition of scattered and direct sounds

Figure 14 shows the contours of the pressure fluctuations and the propagation angle of the peak of the acoustic wave, which is referred to as the propagation angle in the following. The propagation angle is compared with the theoretical angle proposed by Inoue and Hatakeyama [6]. In this theory, the scattered sound in Curle's equation [3] is assumed to be dominant, and the sound speed is assumed to be varied by the Doppler effects as indicated in Eq. (12).

Figure 13. Time histories of pressure and density at the center of the vortex. The letters in this figure correspond to the

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>M</sup><sup>2</sup> sin <sup>2</sup><sup>θ</sup>

where θ is an angle as shown in Figure 1. For M = 0.2, the propagation angle θ = 75� is approximately in good agreement with the theoretical angle θ = 79�. However, the propagation angle θ = 80� greatly differs from the theoretical angle θ = 62� for M = 0.6. This is because the direct sound in Curle's equation [3] becomes more intense as the freestream Mach number becomes higher. The contributions of direct and scattered sounds to total sound are presented

The sound predicted by the direct simulation is decomposed into scattered and direct sounds.

� M cos θ

� �, (12)

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p

aθð Þ¼ θ a<sup>0</sup>

invariant of velocity gradient tensor <sup>q</sup> = ||Ω||<sup>2</sup> ||S||<sup>2</sup> is computed, where <sup>Ω</sup> and <sup>S</sup> are, respectively, the asymmetric and symmetric parts of the velocity gradient tensor. Regions with q > 0 represent vortex tubes. These figures show that when the vortex is shed from the cylinder, an expansion wave is radiated on that side. For example, the expansion wave is radiated from the lower side of the cylinder in Figures 11(b) and 12(b). Meanwhile, a compression wave is radiated from the other side. This relationship of the vortex shedding and acoustic radiation is consistent with the computational results of flows around a circular cylinder by Inoue and Hatakeyama [6]. The acoustic radiation mechanism is discussed in detail.

Figure 13 shows the time histories of the pressure and density at the center of a shed vortex, where the positions of the vortex center are estimated by the local maxima of the second invariant and indicated in Figure 12. The pressure and density are nondimensionalized by the values at t = 0 (the time of t = 0 corresponds to Figures 11(a) and 12(a)). Also, the density is raised to the power of the specific ratio γ. This figure shows that the variation of the density is approximately in good agreement with that of pressure. This means that these phenomena are adiabatic. Also, Figure 13 shows that both the pressure and density become lower as the vortex is developed from t =0(Figures 11(a) and 12(a)) to t = T/4 (Figures 11(b) and 12(b)). This means that the fluid in the vortex expands. As a result, an expansion wave is radiated when the vortex is shed. After the shedding, the density in the vortex becomes higher and recovers to the initial value. At this time, a compression wave is radiated between the expansion waves. This radiation mechanism is independent of the freestream Mach number.

Figure 13. Time histories of pressure and density at the center of the vortex. The letters in this figure correspond to the captions of Figures 11 and 12.

### 4.3. Acoustic fields

invariant of velocity gradient tensor <sup>q</sup> = ||Ω||<sup>2</sup> ||S||<sup>2</sup> is computed, where <sup>Ω</sup> and <sup>S</sup> are, respectively, the asymmetric and symmetric parts of the velocity gradient tensor. Regions with q > 0 represent vortex tubes. These figures show that when the vortex is shed from the cylinder, an expansion wave is radiated on that side. For example, the expansion wave is radiated from the lower side of the cylinder in Figures 11(b) and 12(b). Meanwhile, a compression wave is radiated from the other side. This relationship of the vortex shedding and acoustic radiation is consistent with the computational results of flows around a circular cylinder by Inoue and

Figure 12. Contours of the second invariant of velocity gradient tensor (M = 0.4). (a) t = 0, (b) t/T = 1/4, (c) t/T = 1/2, and

Figure 13 shows the time histories of the pressure and density at the center of a shed vortex, where the positions of the vortex center are estimated by the local maxima of the second invariant and indicated in Figure 12. The pressure and density are nondimensionalized by the values at t = 0 (the time of t = 0 corresponds to Figures 11(a) and 12(a)). Also, the density is raised to the power of the specific ratio γ. This figure shows that the variation of the density is approximately in good agreement with that of pressure. This means that these phenomena are adiabatic. Also, Figure 13 shows that both the pressure and density become lower as the vortex is developed from t =0(Figures 11(a) and 12(a)) to t = T/4 (Figures 11(b) and 12(b)). This means that the fluid in the vortex expands. As a result, an expansion wave is radiated when the vortex is shed. After the shedding, the density in the vortex becomes higher and recovers to the initial value. At this time, a compression wave is radiated between the expansion

waves. This radiation mechanism is independent of the freestream Mach number.

Hatakeyama [6]. The acoustic radiation mechanism is discussed in detail.

(d) t/T = 3/4 (T is the period).

338 Numerical Simulations in Engineering and Science

### 4.3.1. Directivity of acoustic wave

Figure 14 shows the contours of the pressure fluctuations and the propagation angle of the peak of the acoustic wave, which is referred to as the propagation angle in the following. The propagation angle is compared with the theoretical angle proposed by Inoue and Hatakeyama [6]. In this theory, the scattered sound in Curle's equation [3] is assumed to be dominant, and the sound speed is assumed to be varied by the Doppler effects as indicated in Eq. (12).

$$a\_{\theta}(\theta) = a\_0 \Big(\sqrt{1 - M^2 \sin^2 \theta} - M \cos \theta\Big),\tag{12}$$

where θ is an angle as shown in Figure 1. For M = 0.2, the propagation angle θ = 75� is approximately in good agreement with the theoretical angle θ = 79�. However, the propagation angle θ = 80� greatly differs from the theoretical angle θ = 62� for M = 0.6. This is because the direct sound in Curle's equation [3] becomes more intense as the freestream Mach number becomes higher. The contributions of direct and scattered sounds to total sound are presented quantitatively in Section 4.3.2.

### 4.3.2. Decomposition of scattered and direct sounds

The sound predicted by the direct simulation is decomposed into scattered and direct sounds. The direct sound pdirect is defined as.

where τ = t � r/a<sup>θ</sup> and F<sup>0</sup>

integration interval sufficiently.

<sup>x</sup> and F<sup>0</sup>

<sup>y</sup> are the time derivatives of the forces exerted on the fluid by

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,

341

the cylinder. Also, the start point of the time integration τ<sup>0</sup> is set to τ �10 T to enlarge the

Figure 15 shows the polar plots of pressure levels of the total, scattered, and direct sounds at r/D = 30.0 for M = 0.2, 0.4, and 0.6. Figure 16 shows the effects of the freestream Mach number on each sound pressure levels at r/D = 30.0 in the above-mentioned propagation angle as shown in Table 1. For M > 0.3, the sound pressure level of the total sound is proportional to M<sup>7</sup>

although that is proportional for M<sup>5</sup> for M ≤ 0.3. According to the two-dimensional Curle's

Figure 15. Polar plots of total, scattered, and direct acoustic fields at r/D = 30.0. (a) M = 0.2, (b) M = 0.4, and (C) M = 0.6 [24].

Figure 14. Contours of pressure fluctuations p'/(ρ0a<sup>0</sup> 2 M 3.5) and propagation angle. Here, (a) M = 0.2, (b) M = 0.4, and (c) M = 0.6.

$$p\_{\text{direct}}\left(t\right) = p\_{\text{total}}\left(t\right) - p\_{\text{scatter}}\left(t\right),\tag{13}$$

where the pscatter is the dipole sound that contains the Doppler effect [6]. Also, the acoustic wavelength is 11.6D and so is sufficiently large even for M = 0.6 to neglect the difference in the retarded time on the cylinder. The scattered sound pscatter is

$$\begin{split} p\_{\text{scatter}} &= \frac{1}{2^{3/2} \pi a\_{\theta}^{1/2} r^{1/2}} \int\_{x\_0}^{t} \frac{F'(\tau')}{\sqrt{\tau - \tau'}} d\tau', \\ F' &= F'\_x \cos \theta + F'\_y \sin \theta. \end{split} \tag{14}$$

where τ = t � r/a<sup>θ</sup> and F<sup>0</sup> <sup>x</sup> and F<sup>0</sup> <sup>y</sup> are the time derivatives of the forces exerted on the fluid by the cylinder. Also, the start point of the time integration τ<sup>0</sup> is set to τ �10 T to enlarge the integration interval sufficiently.

Figure 15 shows the polar plots of pressure levels of the total, scattered, and direct sounds at r/D = 30.0 for M = 0.2, 0.4, and 0.6. Figure 16 shows the effects of the freestream Mach number on each sound pressure levels at r/D = 30.0 in the above-mentioned propagation angle as shown in Table 1. For M > 0.3, the sound pressure level of the total sound is proportional to M<sup>7</sup> , although that is proportional for M<sup>5</sup> for M ≤ 0.3. According to the two-dimensional Curle's

Figure 15. Polar plots of total, scattered, and direct acoustic fields at r/D = 30.0. (a) M = 0.2, (b) M = 0.4, and (C) M = 0.6 [24].

pdirect ðÞ¼ t ptotal ðÞ� t pscatter ð Þt , (13)

M 3.5) and propagation angle. Here, (a) M = 0.2, (b) M = 0.4, and

,

(14)

where the pscatter is the dipole sound that contains the Doppler effect [6]. Also, the acoustic wavelength is 11.6D and so is sufficiently large even for M = 0.6 to neglect the difference in the

2

ðτ τ0

<sup>y</sup> sin θ:

F<sup>0</sup> τ<sup>0</sup> ð Þ ffiffiffiffiffiffiffiffiffiffiffiffi <sup>τ</sup> � <sup>τ</sup><sup>0</sup> <sup>p</sup> <sup>d</sup>τ<sup>0</sup>

retarded time on the cylinder. The scattered sound pscatter is

Figure 14. Contours of pressure fluctuations p'/(ρ0a<sup>0</sup>

340 Numerical Simulations in Engineering and Science

(c) M = 0.6.

F<sup>0</sup> ¼ F<sup>0</sup>

pscatter <sup>¼</sup> <sup>1</sup> 23=<sup>2</sup> πa 1=2 <sup>θ</sup> r<sup>1</sup>=<sup>2</sup>

<sup>x</sup> cos θ þ F<sup>0</sup>

∂<sup>2</sup>T~ij ∂xi∂xj

Figure 17 shows the contours of the total Lighthill's acoustic sources ∂<sup>2</sup>

the frequency of the vortex shedding in (a), those of the first term ∂<sup>2</sup>

Figure 17. Lighthill's acoustic sources for M = 0.4 (real part is shown). (a) All terms ∂<sup>2</sup>

T~2

ij=∂xi∂xj<sup>=</sup> <sup>ρ</sup>0U<sup>2</sup>

<sup>0</sup>=D<sup>2</sup> .

<sup>0</sup>=D<sup>2</sup> , and (c) second term <sup>∂</sup><sup>2</sup>

T~2

Tij

(b), and those of the second term ∂<sup>2</sup>

the first term.

∂2 T~1

ij=∂xi∂xj<sup>=</sup> <sup>ρ</sup>0U<sup>2</sup>

<sup>1</sup> <sup>¼</sup> <sup>ρ</sup>vivj, Tij

<sup>¼</sup> <sup>∂</sup><sup>2</sup>T~<sup>1</sup> ij ∂xi∂xj þ

<sup>2</sup> <sup>¼</sup> <sup>δ</sup>ij <sup>p</sup> � <sup>p</sup><sup>0</sup>

<sup>∂</sup><sup>2</sup>T~<sup>2</sup> ij ∂xi∂xj þ

� <sup>a</sup><sup>2</sup>

ij=∂xi∂xj<sup>=</sup> <sup>ρ</sup>0U<sup>2</sup>

and second terms, respectively). Here, the third term is negligibly small, so its contour is not presented here. All the contours show that the acoustic sources near the cylinder are more intense than the acoustic sources in the wake far from the cylinder. This is because the acoustic waves are radiated by the vortex shedding from the cylinder as mentioned above. Also, the intensity of the second term, which is usually neglected for the acoustic prediction using Lighthill's acoustic analogy [7, 8], was found to be comparable to that of

<sup>∂</sup><sup>2</sup>T~<sup>3</sup> ij ∂xi∂xj

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<sup>0</sup> ρ � ρ<sup>0</sup> , Tij

, (15)

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<sup>T</sup>~ij=∂xi∂xj<sup>=</sup> <sup>ρ</sup>0U<sup>2</sup>

ij=∂xi∂xj<sup>=</sup> <sup>ρ</sup>0U<sup>2</sup>

T~1

<sup>0</sup>=D<sup>2</sup> in (c) (hereafter referred to as first

<sup>T</sup>~ij=∂xi∂xj<sup>=</sup> <sup>ρ</sup>0U<sup>2</sup>

<sup>0</sup>=D<sup>2</sup> , (b) first term

<sup>3</sup> ¼ �τij: (16)

<sup>0</sup>=D<sup>2</sup> at

343

<sup>0</sup>=D<sup>2</sup> in

Figure 16. Effects of freestream Mach number on sound pressure levels of total, scattered, and direct sounds at the frequency of the vortex shedding at r/D = 30.0 in the direction of acoustic propagation angle [24].


Table 1. Propagation angle of acoustic waves.

equation introduced by Inoue and Hatakeyama [6], the pressure level of the direct sound is proportional to M<sup>7</sup> , whereas that of the scattered sound is proportional to M<sup>5</sup> . The present results clarified that sound pressure levels of scattered and direct sounds intersect around M = 0.4.

Consequently, it was confirmed that the effects of the direct sound need to be taken into consideration when predicting the sound radiating from a cylinder flow for M ≥ 0.4. Also, as shown in Figure 14, the directivity of the acoustic field for such a high Mach number cannot be predicted by the modified Curle's equation [6], which assumes the scattered sound to be dominant and takes the Doppler effects into consideration. To the authors' knowledge, this is the first time that the effects of the freestream Mach number on the contributions of the scattered and direct sounds have been quantitatively clarified for flows around a cylinder.

### 4.4. Lighthill's acoustic sources

The right-hand term of Lighthill's equation [Eq. (6)] can be decomposed into three components,

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$$\frac{\partial^2 \tilde{T}\_{\vec{\boldsymbol{\alpha}}}}{\partial \mathbf{x}\_i \partial \mathbf{x}\_{\vec{\boldsymbol{\beta}}}} = \frac{\partial^2 \tilde{T}\_{\vec{\boldsymbol{\alpha}}\vec{\boldsymbol{\beta}}}^1}{\partial \mathbf{x}\_i \partial \mathbf{x}\_{\vec{\boldsymbol{\beta}}}} + \frac{\partial^2 \tilde{T}\_{\vec{\boldsymbol{\alpha}}\vec{\boldsymbol{\beta}}}^2}{\partial \mathbf{x}\_i \partial \mathbf{x}\_{\vec{\boldsymbol{\beta}}}} + \frac{\partial^2 \tilde{T}\_{\vec{\boldsymbol{\alpha}}\vec{\boldsymbol{\beta}}}^3}{\partial \mathbf{x}\_i \partial \mathbf{x}\_{\vec{\boldsymbol{\beta}}}},\tag{15}$$

$$T\_{i\dot{\eta}}{}^1 = \rho v\_i v\_{\dot{\eta}} \quad T\_{i\dot{\eta}}{}^2 = \delta\_{\dot{\eta}}((p - p\_0) - a\_0^2(\rho - \rho\_0)), \quad T\_{\dot{\eta}}{}^3 = -\tau\_{\ddot{\eta}}.\tag{16}$$

Figure 17 shows the contours of the total Lighthill's acoustic sources ∂<sup>2</sup> <sup>T</sup>~ij=∂xi∂xj<sup>=</sup> <sup>ρ</sup>0U<sup>2</sup> <sup>0</sup>=D<sup>2</sup> at the frequency of the vortex shedding in (a), those of the first term ∂<sup>2</sup> T~1 ij=∂xi∂xj<sup>=</sup> <sup>ρ</sup>0U<sup>2</sup> <sup>0</sup>=D<sup>2</sup> in (b), and those of the second term ∂<sup>2</sup> T~2 ij=∂xi∂xj<sup>=</sup> <sup>ρ</sup>0U<sup>2</sup> <sup>0</sup>=D<sup>2</sup> in (c) (hereafter referred to as first and second terms, respectively). Here, the third term is negligibly small, so its contour is not presented here. All the contours show that the acoustic sources near the cylinder are more intense than the acoustic sources in the wake far from the cylinder. This is because the acoustic waves are radiated by the vortex shedding from the cylinder as mentioned above. Also, the intensity of the second term, which is usually neglected for the acoustic prediction using Lighthill's acoustic analogy [7, 8], was found to be comparable to that of the first term.

equation introduced by Inoue and Hatakeyama [6], the pressure level of the direct sound is

Figure 16. Effects of freestream Mach number on sound pressure levels of total, scattered, and direct sounds at the

Mach number 0.2 0.3 0.4 0.5 0.6 Present propagation angle 75 64 62 67 80

frequency of the vortex shedding at r/D = 30.0 in the direction of acoustic propagation angle [24].

results clarified that sound pressure levels of scattered and direct sounds intersect around

Consequently, it was confirmed that the effects of the direct sound need to be taken into consideration when predicting the sound radiating from a cylinder flow for M ≥ 0.4. Also, as shown in Figure 14, the directivity of the acoustic field for such a high Mach number cannot be predicted by the modified Curle's equation [6], which assumes the scattered sound to be dominant and takes the Doppler effects into consideration. To the authors' knowledge, this is the first time that the effects of the freestream Mach number on the contributions of the scattered and direct sounds have been quantitatively clarified for

The right-hand term of Lighthill's equation [Eq. (6)] can be decomposed into three compo-

, whereas that of the scattered sound is proportional to M<sup>5</sup>

. The present

proportional to M<sup>7</sup>

Table 1. Propagation angle of acoustic waves.

342 Numerical Simulations in Engineering and Science

flows around a cylinder.

4.4. Lighthill's acoustic sources

M = 0.4.

nents,

Figure 17. Lighthill's acoustic sources for M = 0.4 (real part is shown). (a) All terms ∂<sup>2</sup> <sup>T</sup>~ij=∂xi∂xj<sup>=</sup> <sup>ρ</sup>0U<sup>2</sup> <sup>0</sup>=D<sup>2</sup> , (b) first term ∂2 T~1 ij=∂xi∂xj<sup>=</sup> <sup>ρ</sup>0U<sup>2</sup> <sup>0</sup>=D<sup>2</sup> , and (c) second term <sup>∂</sup><sup>2</sup> T~2 ij=∂xi∂xj<sup>=</sup> <sup>ρ</sup>0U<sup>2</sup> <sup>0</sup>=D<sup>2</sup> .

To clarify the contributions of each term to the acoustic field, four hybrid simulations were performed for each Mach number on the basis of total Lighthill's acoustic sources computed by the direct simulation, only the first term, only the second term, and only the third term.

Figure 18 shows the polar plots of the sound pressure level at the frequency of the vortex shedding predicted at r/D = 30.0 by the hybrid simulations for M = 0.4. It was clarified that the sound pressure levels predicted by the hybrid simulation based on all terms agree well with those based on only the first term. The sound pressure level based on the second term and that on the third term is negligibly weaker than that based on the first term. Meanwhile, the intensity of the second term is in itself comparable to that of the first term as mentioned above. This indicates that the radiation efficiency of the second term is weaker than that of the first term.

Figure 19 shows the predicted sound pressure level at r/D = 30.0 in the direction of the abovementioned acoustic propagation angle. The results clarified that the first term is the dominant acoustic source for all the Mach numbers. The difference between the sound pressure level based on the first term and that based on the second or third term was more than 30 dB for all the Mach numbers. This result shows that the momentum (the first term) of Lighthill's acoustic source is the dominant acoustic source for all the Mach numbers for cylinder flows, while it has

> been clarified in the past research that the entropy (the second term) also needs to be taken into consideration for high-speed jets such as M = 0.9. Consequently, it was confirmed that only the first term needs to be taken into consideration independently of the freestream Mach number when the sound radiating from a cylinder flow is predicted on the basis of the Lighthill's

> Figure 19. Effects of Mach number on sound pressure levels predicted by hybrid simulations based on all, first, second, and third terms of Lighthill's acoustic sources at the frequency of vortex shedding at r/D = 30.0 in the direction of acoustic

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Aeroacoustic simulations composed of hybrid and direct simulations were introduced. The effects of the freestream Mach number on the flow and acoustic fields around a square cylinder were investigated. The Mach number was varied from 0.2 to 0.6. The Reynolds number based on the side length was 150. These results indicate the effectiveness and limit of the hybrid

It was found that the Strouhal number of vortex shedding, which is based on the side length, becomes lower as the freestream Mach number becomes higher. The Strouhal number for M = 0.2 is 0.151 and that for M = 0.6 is 0.144. As the Mach number increases, the velocity fluctuations of the vortices shed from the cylinder intensifies and the wake widens. The possible reason the velocity fluctuations of the vortices intensify is that the acoustic feedback exists like that in the oscillations in cavity flows. These effects can be found by the direct

The sound pressure level at the frequency of the vortex shedding in the direction of the acoustic propagation angle is proportional to M<sup>7</sup> for M > 0.3, while that is proportional to M<sup>5</sup>

acoustic analogy.

propagation angle [24].

5. Conclusion

simulations.

simulations.

Figure 18. Polar plots of sound pressure levels predicted by decoupled simulations based on all, first, second, and third terms of Lighthill's acoustic sources at the frequency of vortex shedding at r/D = 30.0 [24].

Figure 19. Effects of Mach number on sound pressure levels predicted by hybrid simulations based on all, first, second, and third terms of Lighthill's acoustic sources at the frequency of vortex shedding at r/D = 30.0 in the direction of acoustic propagation angle [24].

been clarified in the past research that the entropy (the second term) also needs to be taken into consideration for high-speed jets such as M = 0.9. Consequently, it was confirmed that only the first term needs to be taken into consideration independently of the freestream Mach number when the sound radiating from a cylinder flow is predicted on the basis of the Lighthill's acoustic analogy.
