Propagation angle of acoustic waves.
Aeroacoustic simulations are divided into hybrid and direct simulations. In this chapter, the effects of freestream Mach number on flow and acoustic fields around a two-dimensional square cylinder in a uniform flow are focused on using direct and hybrid simulations of flow and acoustic fields are performed. These results indicate the effectiveness and limit of the hybrid simulations. The Mach number M is varied from 0.2 to 0.6. The propagation angle of the acoustic waves for a high Mach number such as M = 0.6 greatly differs from that predicted by modified Curle’s equation, which assumes the scattered sound to be dominant and takes the Doppler effects into consideration. This is because the acoustic field is affected by the direct sound, which is generated by quadrupoles in the original Curle’s equation. To clarify the effects of the direct sound on the acoustic field, the scattered and direct sounds are decomposed. The results show that the direct sound is too intense to neglect for M ≥ 0.4. Moreover, acoustic simulations are performed using the Lighthill’s acoustic sources.
- aeolian tone
- direct simulation
- acoustic analogy
- Lighthill’s equation
The sound generated by a cylinder in a uniform flow is known as the aeolian tone. This sound is often radiated from flows around a cylinder. Strouhal  found that the frequency of the tone is identical to the vortex shedding frequency. Lighthill  derived the nonhomogeneous wave equation as shown in Eqs. (1) and (2) from the compressive Navier-Stokes equations.
Recently, many investigations using numerical simulations have been performed, for instance, Inoue and Hatakeyama , Gloerfelt et al. , and Liow et al. . Inoue and Hatakeyama  modified the Curle’s solution considering the Doppler effects and showed that the acoustic fields predicted by the proposed equation agree well with those predicted by their direct simulations for a low freestream Mach number
Despite many investigations into the aeolian sound around a cylinder, little attention has been given to flows around a cylinder with a high Mach number
In the present chapter, aerodynamic sound radiated from a two-dimensional square cylinder in a freestream is investigated. The flow field around a square cylinder has been investigated by many researchers [12, 13, 14]. However, little is known about the acoustic field. The hybrid and direct simulations of flow and acoustic fields are introduced. The freestream Mach number on the flow and acoustic fields are focused on. The Mach number is varied from 0.2 to 0.6. Moreover, the contributions of each term of Lighthill’s acoustic source to the acoustic field are focused on. To do this, the acoustic simulations are also performed using the Lighthill’s acoustic sources computed by the direct simulations. This method for predicting the acoustic field using the acoustic simulation is referred to as the hybrid simulation in this chapter.
2. Numerical methods
2.1. Flow configurations
The flow around a two-dimensional square cylinder, as shown in Figure 1, is investigated. To clarify the effects of the freestream Mach number on flow and acoustic fields, the computations are performed for
The fluid was assumed to be standard air, where Sutherland’s formula can be applied for the viscosity coefficient. The specific heat
2.2. Direct simulation
2.2.1. Governing equations and finite difference formulation
Both flow and acoustic fields are solved by the two-dimensional compressible Navier-Stokes equations in a conservative form, which is written as:
2.2.2. Computational grids and boundary conditions
Figure 2 shows the computational domain and boundary conditions. The coordinates originate from the center of the cylinder. Generally, the nonreflecting boundary conditions based on the characteristic wave relations [17, 18, 19] are used at the inflow, upper, and outflow boundaries along with a buffer region. The role of the buffer region is similar to that of the “sponge region” of Colonius et al. . At the wall, the nonslip and adiabatic boundary conditions are used.
For all the cases of
The spacing in the vortex region is prescribed to be fine enough to analyze the separated shear layer and the vortical structures in the wake of the cylinder. Figure 3 shows the computational grid near the cylinder. The spacing adjacent to the cylinder surface is ∆
In the sound region, the spacing is prescribed to be larger than that in the vortex region but still fine enough to capture the acoustic waves. The spacings are ∆
After many preliminary tests, grid- and domain-size independence has been established for the solutions presented in this chapter.
2.3. Hybrid simulation
2.3.1. Governing equations and discretization formulation
The two-dimensional Lighthill’s equation [Eqs. (1) and (2)] is solved based on the wave equation. Here, the open-source software, FrontFlow/blue-ACOUSTICS, was used. Here, the acoustic simulations are performed in a frequency domain using finite-element methods. A component perturbed at the frequency
Using Eq. (5), Lighthill’s equation can be written as:
2.3.2. Computational grids and boundary conditions
Figure 4 shows the computational grid for the acoustic simulations. The spacing adjacent to the cylinder surface is ∆
The reflecting conditions are adopted on the cylinder wall. On the other boundaries, the nonreflecting boundary conditions are adopted.
3. Validation of computational methods
3.1. Validation of direct simulations
Figure 5 shows the Strouhal number of vortex shedding predicted by the present direct simulations. The Strouhal number St is the frequency nondimensionalized by the freestream velocity
3.2. Validation of hybrid simulation
Figure 6 shows the polar plots of the sound pressure levels at
4. Results and discussion
4.1. Flow fields
Figure 7 shows the contours of vorticity for
Figure 8(a) shows the mean streamwise velocity at
A possible reason the Reynolds stress becomes larger is that the acoustic feedback like that in the oscillations in cavity flows  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  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.
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
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
4.3. Acoustic fields
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 . In this theory, the scattered sound in Curle’s equation  is assumed to be dominant, and the sound speed is assumed to be varied by the Doppler effects as indicated in Eq. (12).
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
Figure 15 shows the polar plots of pressure levels of the total, scattered, and direct sounds at
|Present propagation angle||75||64||62||67||80|
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
4.4. Lighthill’s acoustic sources
The right-hand term of Lighthill’s equation [Eq. (6)] can be decomposed into three components,
Figure 17 shows the contours of the total Lighthill’s acoustic sources at the frequency of the vortex shedding in (a), those of the first term in (b), and those of the second term 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.
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
Figure 19 shows the predicted sound pressure level at
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 simulations.
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
The sound pressure level at the frequency of the vortex shedding in the direction of the acoustic propagation angle is proportional to
Moreover, to clarify the contributions of each term of Lighthill’s acoustic source to the acoustic field, acoustic simulations were performed using Lighthill’s acoustic sources computed by the direct simulations. As a result, the momentum (the first term) of Lighthill’s acoustic source was found to be dominant for all the Mach numbers 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
The present study has provided useful guidelines for predicting the aerodynamic sound on the basis of the Lighthill’s acoustic analogy.
This work was supported by JSPS KAKENHI Grant Numbers JP26820044, 17 K06153 and through the application development for Post K computer (FLAGSHIP 2020) by the Ministry of Education, Culture, Sports, Science, and Technology of Japan (MEXT).
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