1. Introduction

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 [1] found that the frequency of the tone is identical to the vortex shedding frequency. Lighthill [2] derived the nonhomogeneous wave equation as shown in Eqs. (1) and (2) from the compressive Navier-Stokes equations.

$$\frac{\partial^2}{\partial t^2} \rho - a\_0^2 \frac{\partial^2}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} \rho = \frac{\partial^2 T\_{ij}}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} \tag{1}$$

### © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited.

where ρ is the density, a<sup>0</sup> is the freestream sound speed, and the tensor Tij is defined by:

$$T\_{\vec{\eta}} = \rho \upsilon\_{\vec{\nu}} \upsilon\_{\vec{\eta}} + \delta\_{\vec{\eta}} \left( \left( p - p\_0 \right) - a\_0^2 \left( \rho - \rho\_0 \right) \right) - \tau\_{\vec{\eta}\prime} \tag{2}$$

2. Numerical methods

from the cylinder are focused on.

2.2.1. Governing equations and finite difference formulation

equations in a conservative form, which is written as:

*M* = *U*0/*a*<sup>0</sup>

Figure 1. Configurations for flow around a two-dimensional square cylinder.

Prandtl number Pr was 0.72.

2.2. Direct simulation

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 M = 0.2, 0.3, 0.4, 0.5, and 0.6. The Reynolds number based on the freestream velocity and the side length of the cylinder is set to 150, where the three-dimensional instability does not occur [13]. Here, the two-dimensional phenomena related to the vortex shedding

The fluid was assumed to be standard air, where Sutherland's formula can be applied for the viscosity coefficient. The specific heat C was assumed to be 1004 J kg�<sup>1</sup> K�<sup>1</sup> and that the

Both flow and acoustic fields are solved by the two-dimensional compressible Navier-Stokes

where Q is the vector of the conservative variables, E and F are the inviscid fluxes, and E<sup>v</sup> and F<sup>v</sup> are the viscous fluxes. The spatial derivatives and time integration were evaluated by the sixth-order accurate compact finite difference scheme [15] and a third-order accurate Runge-Kutta method. To suppress the numerical instabilities associated with the central differencing

*r*

*θ*

in the compact scheme, we use a tenth-order accurate spatial filter shown below:

Qt þ ð Þ E � E<sup>v</sup> <sup>x</sup> þ ð Þ F � F<sup>v</sup> <sup>y</sup> ¼ 0, (3)

Direct and Hybrid Aeroacoustic Simulations Around a Rectangular Cylinder

http://dx.doi.org/10.5772/intechopen.70810

329

*d*

*y*

o

*x*

2.1. Flow configurations

where vi is velocity, p is pressure, the tensor τ is the viscous stress tensor, and ∂<sup>2</sup> Tij/∂xi∂xj is referred to as Lighthill's acoustic source. The first, second, and third terms of the Lighthill's acoustic source are related to the momentum, entropy, and viscosity, respectively. For aerodynamic sound around a body in a fluid stream of a low Mach number, Curle [3] has shown through analytical solution of Lighthill's equation [2] that the surface pressure fluctuations around the body lead to a dipole sound field. Investigations such as those by Gerrard [4] and Phillips [5] have experimentally confirmed that the acoustic field around a circular cylinder has directivity normal to the fluid stream and is closely related to the fluctuations of the lift force.

Recently, many investigations using numerical simulations have been performed, for instance, Inoue and Hatakeyama [6], Gloerfelt et al. [7], and Liow et al. [8]. Inoue and Hatakeyama [6] 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 M ≤ 0.3. Gloerfelt et al. [7] performed incompressible flow computations and acoustic computations on the basis of Lighthill's acoustic analogy [2] for the flow around a circular cylinder. The role of the acoustic scattering on the cylinder in the mechanism of the sound generation was investigated for M = 0.12 and the Reynolds number based on the diameter Red <sup>≈</sup> 1.1 � 105 . Here, the entropy (second) and viscous (third) terms in Lighthill's acoustic source Eq. (2) were neglected. This is reasonable for the flow of such a high Reynolds number and a low Mach number [9]. Liow et al. [8] also performed the incompressible flow simulations and acoustic simulations for a flow around an elongated rectangular cylinder with M ≤ 0.2. The acoustic computations are based on the Powell's theory [10], where the acoustic sources approximately correspond to the momentum (the first term) of Lighthill's acoustic source. The effects of the drag force on the acoustic field were clarified.

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 M > 0.3. For such a high Mach number, the effects of the Mach number on the flow and acoustic fields around a cylinder have not been clarified. Also, it is currently unknown whether the contribution of the second and third terms in Lighthill's acoustic source to the acoustic field can be neglected for such a high Mach number. In high-speed jets such as M = 0.9–2.0, it has been clarified that the second term needs to be taken into consideration [11].

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 [12–14]. 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.
