**2. Analysis model**

spacing is small, the shear layer separated from the upstream cylinder does not roll up to form vortices but reattaches to the downstream of the cylinder, and the vortices are shed from the downstream only, and this flow pattern is called *Mode I*. When the spacing is large, the shear layer separated from the upstream cylinder rolls up and forms vortices in front of the downstream of the cylinder, and the rollup vortices impinge on the downstream cylinder (body-vortex interaction, BVI, wake-body interaction), and this flow pattern is called *Mode II*. Furthermore, Inoue and Mori [5] showed that in *Mode I,* the sound is generated only by the vortex shedding of the downstream cylinder, and in *Mode II,* the sound and strong pressure fluctuation around the downstream cylinder are generated mainly by BVI. The magnitude of the generated sound is much larger in *Mode II* than both *Mode I* and

Since a *von Karman* vortex street can be regarded as a gust impinging on the obstacle or blade, a rod-airfoil model is another typical one for the study of the body

immersed upstream of the blade or airfoil, the wake formed behind the rod interacts with the leading edge of the blade or airfoil. Many studies concerned with the rodairfoil model have been done both experimentally and numerically [9–21]. Jacob et al. [9] measured the flow field of the rod-NACA0012 airfoil model at the fixed spacing between the rod and the airfoil with varying the inflow velocity, and also measured the far field radiated noise spectra that are generated mainly by the bodywake interaction. They also performed numerical calculations using the Reynoldsaveraged Navier-Stokes (RANS) and the large eddy simulation (LES) approaches and compared numerical results with the measured data. Munekata et al. [10–11] measured the flow field of the rod-airfoil model to research the effects of the spacing between the rod (cylinder) and the airfoil and the characteristics of the flow-induced sound generated by the flow around rod-airfoil. They showed that the roll up of the shear layer separated from the upstream rod is suppressed when the spacing between the rod and the airfoil is small, and the interaction between the wake from the rod upstream and the airfoil downstream becomes weak and it results in decreasing the level of the noise radiation. They also showed that the attack angle of the airfoil located downstream affects the characteristics of the flowinduced sound and wake structure at a given spacing between the rod and the airfoil, and the generated sound pressure decreases with the increase of the attack angle of the airfoil. Li et al. [12] performed the experiments of the body-wake interaction noise radiated from the flow around the rod-airfoil model by focusing on the noise control using "air blowing" on the upstream rod and a soft-vane leading edge on the airfoil. Numerical investigations by using the RANS approach have been done by Casalino et al. [13], Jacob et al. [9] and Jiang et al. [14], and those by using the LES approach have been done by Casalino et al. [13], Magagnato et al. [15], Boudet et al. [16], Jacob et al. [9], Greschner et al. [17], Agrawal and Sharma [18], Giret et al. [19], Daude et al. [20], and Jiang et al. [21]. Jiang et al. [21]

performed LES simulations of the flow around the rod-airfoil for the inflow velocity

*U*<sup>∞</sup> *=* 72 m/s and a Reynolds number based on the rod diameter (*d*) 48,000 (480,000 based on the airfoil chord, *c*) to clarify the flow patterns, velocity and pressure fluctuations, and noise radiation with varying the spacing between the rod and airfoil. They varied the spacing between the rod and the airfoil, such as *L/d* = 2, 4, 6, 8, and 10. They showed that when the spacing is small (*L/d =* 2), the vortex shedding of the rod upstream, the pressure fluctuation, and the noise radiation are suppressed as shown by Munekata et al. [10–11] and when the spacing is large (*L/d* = 6, 10), the pressure fluctuation, the noise radiation, and the fluid resonant oscillation due to the feedback loop between the rod and the airfoil become

(blade) vortex interaction or wake-body interaction. In this model, a rod is

the single cylinder.

*Vortex Dynamics Theories and Applications*

stronger.

**122**

#### **2.1 Rod-airfoil flow model**

A schematic diagram of the flow model is presented in **Figure 1**. The origin is at the leading edge of the airfoil. The coordinates parallel and normal to the free stream are denoted by *x* and *y,* respectively. The coordinate in the spanwise direction is denoted by *z*. The symbol *L* denotes the spacing between the rod and the airfoil. The lengths are made dimensionless by the rod diameter *d* and the velocity is scaled by the speed of sound *c*∞*.* The normalized spacing *L/d* is prescribed to be 2 and 10. The Mach number, *M*, of a uniform flow is defined by *M* = *U*∞*/c*∞, where *U*<sup>∞</sup> denotes the velocity of the uniform flow. In this chapter, the Reynolds number is fixed to be *Red* = 28,800 or *Rec* = 288,000, and those are based on the rod diameter and the airfoil chord *c*, respectively. The spanwise length of the rod and the airfoil is 3*d*.

#### **2.2 Airfoil-airfoil flow model**

A schematic diagram of the flow model is presented in **Figure 2**. The origin is at the leading edge of the downstream airfoil. The normalized spacing *L/c* is

**Figure 1.** *Schematic diagram of rod-airfoil model. (a) Rod-airfoil model; (b) parameters.*

**Figure 2.** *Schematic diagram of arifoil-airfoil model. (a) Airfoil-airfoil model; (b) parameters.*

prescribed to be 0.2, 0.6, and 1.0, and those are equal to *L/d =* 2.0, 6.0, and 10. In this chapter, the Reynolds number is fixed to be *Rec* = 288,000, which is based on the airfoil chord *c*. The spanwise length of the airfoil and the airfoil is 0.3*c,* which corresponds to 3*d*.

> 0.6, and 1.0, respectively. **Figure 4(a)** and **(b)** shows the computational meshes near the rod-airfoil and the airfoil-airfoil models, respectively. The cell spacing adjacent to the wall is 0.2 mm (0.033*d*). Steady-state simulations were performed using Spalart-Allmaras (S-A) turbulence model and then used as initial conditions of transient LESs. The transient simulations were performed for 50,000 time steps

*Wake-Body Interaction Noise Simulated by the Coupling Method Using CFD and BEM*

Lighthill equation [24, 25] in the frequency domain is derived from the equation

where *p* is the acoustic pressure, *k* is the wave number, *c*<sup>∞</sup> is the speed of sound,

where ρ is the density and is 1.225 kg/m3, *δij* is the Kronecker delta, and *τlm* is the

*Tlm ∂xl∂xm*

*<sup>ρ</sup> <sup>δ</sup>lm* � *<sup>τ</sup>lm*, (2)

(1)

<sup>∇</sup><sup>2</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> *<sup>p</sup>* ¼ � *<sup>∂</sup>*<sup>2</sup>

*l* and *m* indicate each direction in the Cartesian coordinates, and *v* is the flow

viscous stress tensor. For a low-Mach number and high-Reynolds number flow regime, the second and third terms of Eq. (2) are negligible [26–29]. Therefore, the

To convert the acoustic source time histories into the frequency spectra, the discrete Fourier transform (DFT) has been applied. The acoustics sources are extracted from 1250 steps (from *t =* 0.05 s to 0.1 s) flow field data, the sampling

The BEM solver in commercial acoustic simulation package, WAON, is used to

solve the acoustic characteristics [30]. In a sound field that satisfies the threedimensional Helmholtz equation, the Kirchhoff-Helmholtz integral equation [31]

*Tlm* <sup>¼</sup> *<sup>ρ</sup>vlvm* <sup>þ</sup> *<sup>p</sup>* � *<sup>c</sup>*∞<sup>2</sup>

of continuity and compressible Navier-Stokes equation and as follows:

velocity. *Tlm* is the Lighthill stress tensor and as follows:

*Computational mesh. (a) Rod-airfoil model; (b) airfoil-airfoil model.*

*DOI: http://dx.doi.org/10.5772/intechopen.92783*

first term is used for the present work.

**3.3 Extraction of acoustic source**

period is 4e-5 s.

**125**

**3.4 Acoustic simulation**

with a time step size *Δt =* 2e-6 s.

**3.2 Lighthill equation**

**Figure 4.**
