*4.1.1 Flow patterns*

Typical examples of instantaneous vorticity fields are presented in **Figure 8**. In the present calculation, for the case of *L/d =* 2, when the spacing between the rod and the airfoil is small, the boundary layers separated from the rod upstream did not roll up and reattaches to the airfoil downstream. The shear layers rolled up and formed vortices on the downstream airfoil surface, and the vortices were shed and convected downstream. The Kármán-vortex street formed from the rod was suppressed for the short spacing, and this mode is called "non-shedding mode" and "the Kármán-vortex street suppressing mode" as indicated by Munetaka et al. [10] and Jiang et al. [21], respectively. On the other hand, for the case of *L/d =* 10, when the spacing between the rod and the airfoil is large, the boundary layers separated from the rod upstream rolled up and formed vortices in the region between the rod and the airfoil, and the formed vortices shed from the rod interacted with the airfoil and impinged on the leading edge of the airfoil. The impinged vortices were distorted and convected downstream. This mode is called "the Kármán-street shedding mode" as indicated by Jiang et al. [21], and also called wake-body interaction or body-vortex interaction [6, 16, 21]. Similar phenomena such as the

#### **Figure 8.**

*Vorticity in the z direction. (a)* L/d = *2; (b)* L/d = *10.*

#### **Figure 9.**

*Time-averaged velocity in the streamwise (*x*) and the vertical (*y*) directions. (a), (b) velocity in the streamwise (*x*) direction; (c), (d) velocity in the vertical (*y*) direction. (a)* L/d = *2; (b)* L/d = *10; (c)* L/d = *2; (d)* L/d = *10.*

Kármán-street shedding and suppressing modes can be seen in the flow around two obstacles in tandem [2, 4, 6].

**Figure 9(a)** and **(b)** shows the fields of time-averaged velocity *Umean* in the *x* (streamwise) direction. In the space between the rod and the airfoil, **Figure 9(a)** shows that the counter-rotating vortices near the leading edge of the airfoil lead to very small or negative values of *Umean* for the case of *L/d =* 2. The reattachments of the counter-rotating vortices to the leading edge of the airfoil due to the approaching rod slow down the flow. By contrast, for the case of *L/d =* 10, the flow behind the rod was accelerated in front of the leading edge of airfoil as shown in **Figure 9(b)**.

**Figure 9(c)** and **(d)** shows the fields of time-averaged velocity *Vmean* in the y (vertical) direction. For the case of *L/d =* 2, the positive *Vmean* appears in the *–y* region around the leading edge of the airfoil and the negative one appears in the opposite region. This is due to the reattachments of the negative and positive vortices formed from the rod upstream. For the case of *L/d =* 10, the *Vmean* distribution in the regions around the leading edge of the airfoil is contrary to that for the case of *L/d =* 2, as shown in **Figure 9(c)**. This phenomenon is indicated by Jiang et al. [21].

Finally, the fields of RMS value of the fluctuation velocity *urms* in the streamwise (*x*) direction are represented in **Figure 10** for *L/d =* 2 and *L/d =* 10. As shown in **Figure 10**, for the case of *L/d =* 2, the *urms* values behind the rod and near the central line *y =* 0 are much lower than those for the case of *L/d =* 10. This result corresponds to the suppression of the Kármán-vortex street for the case of *L/d =* 2 in the rodairfoil model, and the reattachment of the main separated vortices to the airfoil (see **Figure 8**) causes the very small or negative *Umean* generated in the space between the rod and the airfoil (see **Figure 9(a)**) in this case. On the other hand, in the case of *L/d =* 10, the *urms* values behind the rod and around the airfoil are much larger than those for the case of *L/d =* 2. Therefore the turbulent fluctuations in the space between the rod and the airfoil for the case of *L/d =* 10 seem to be much larger than those for the case of *L/d =* 2.

#### *4.1.2 Near pressure field*

Instantaneous snapshots of static pressure and snapshots of mean pressure are represented in **Figure 11**. For the case of *L/d =* 2, **Figure 11** shows that the value of the static pressure on the upstream side of the rod is larger than that on the downstream side, and that on the leading edge of the airfoil is negative and much lower than that for the case of *L/d =* 10.

**Figure 12** shows the mean pressure distribution on the surface of rod and the airfoil, where *xrod* is the coordinate in the streamwise direction and the origin of *xrod* is the located in the stagnation point of the rod. Depending on the spacing between the rod and the airfoil, the pressure behind the rod is affected by the airfoil [21]. For the case of *L/d =* 2, as shown in **Figure 12(a)**, the pressure behind the rod is much larger than that for the case of *L/d =* 10 due to the presence of the airfoil behind the rod. As mentioned in Section 3.1.1, this phenomenon is related to the reattachments of the separated shear layers from the rod to the leading edge of the airfoil, and the Kármán-vortex street formed from the rod is suppressed and the pressure behind the rod increases. **Figure 12(b)** shows that in the case of *L/d =* 2, the negative pressure occurs around the leading edge of the airfoil, which means that the negative drag force acts on the airfoil, and in the case of *L/d =* 10, the positive pressure occurs the leading edge of the airfoil and the influence of the rod on the airfoil

*Mean pressure profile on the rod and the airfoil. (a) Rod. (b) Airfoil.* L/d = *2;* L/d = *10.*

*Static pressure and mean pressure fields. (a)* L/d = *2; (b)* L/d = *10; (c)* L/d = *2; (d)* L/d = *10.*

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

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

**Figure 13** shows instantaneous snapshots of a fluctuation pressure (*dp)* and RMS of the fluctuation pressure (*dprms*) fields, and the fluctuation pressure is defined as

*dp* ¼ *p*<sup>s</sup> � *p*mean (12)

seems to be small.

follows [6].

**131**

**Figure 11.**

**Figure 12.**

**Figure 10.** *RMS value of fluctuation velocity in the streamwise (*x*) direction. (a)* L/d = *2; (b)* L/d = *10.*

*Wake-Body Interaction Noise Simulated by the Coupling Method Using CFD and BEM DOI: http://dx.doi.org/10.5772/intechopen.92783*

#### **Figure 11.**

Kármán-street shedding and suppressing modes can be seen in the flow around two

**Figure 9(a)** and **(b)** shows the fields of time-averaged velocity *Umean* in the *x* (streamwise) direction. In the space between the rod and the airfoil, **Figure 9(a)** shows that the counter-rotating vortices near the leading edge of the airfoil lead to very small or negative values of *Umean* for the case of *L/d =* 2. The reattachments of

approaching rod slow down the flow. By contrast, for the case of *L/d =* 10, the flow behind the rod was accelerated in front of the leading edge of airfoil as shown in

**Figure 9(c)** and **(d)** shows the fields of time-averaged velocity *Vmean* in the y (vertical) direction. For the case of *L/d =* 2, the positive *Vmean* appears in the *–y* region around the leading edge of the airfoil and the negative one appears in the opposite region. This is due to the reattachments of the negative and positive vortices formed from the rod upstream. For the case of *L/d =* 10, the *Vmean* distribution in the regions around the leading edge of the airfoil is contrary to that for the case of *L/d =* 2, as shown in **Figure 9(c)**. This phenomenon is indicated by Jiang

Finally, the fields of RMS value of the fluctuation velocity *urms* in the streamwise (*x*) direction are represented in **Figure 10** for *L/d =* 2 and *L/d =* 10. As shown in **Figure 10**, for the case of *L/d =* 2, the *urms* values behind the rod and near the central line *y =* 0 are much lower than those for the case of *L/d =* 10. This result corresponds to the suppression of the Kármán-vortex street for the case of *L/d =* 2 in the rodairfoil model, and the reattachment of the main separated vortices to the airfoil (see **Figure 8**) causes the very small or negative *Umean* generated in the space between the rod and the airfoil (see **Figure 9(a)**) in this case. On the other hand, in the case of *L/d =* 10, the *urms* values behind the rod and around the airfoil are much larger than those for the case of *L/d =* 2. Therefore the turbulent fluctuations in the space between the rod and the airfoil for the case of *L/d =* 10 seem to be much larger than

Instantaneous snapshots of static pressure and snapshots of mean pressure are represented in **Figure 11**. For the case of *L/d =* 2, **Figure 11** shows that the value of the static pressure on the upstream side of the rod is larger than that on the downstream side, and that on the leading edge of the airfoil is negative and much

**Figure 12** shows the mean pressure distribution on the surface of rod and the airfoil, where *xrod* is the coordinate in the streamwise direction and the origin of *xrod* is the located in the stagnation point of the rod. Depending on the spacing between the rod and the airfoil, the pressure behind the rod is affected by the airfoil [21]. For

*RMS value of fluctuation velocity in the streamwise (*x*) direction. (a)* L/d = *2; (b)* L/d = *10.*

the counter-rotating vortices to the leading edge of the airfoil due to the

obstacles in tandem [2, 4, 6].

*Vortex Dynamics Theories and Applications*

**Figure 9(b)**.

et al. [21].

those for the case of *L/d =* 2.

lower than that for the case of *L/d =* 10.

*4.1.2 Near pressure field*

**Figure 10.**

**130**

*Static pressure and mean pressure fields. (a)* L/d = *2; (b)* L/d = *10; (c)* L/d = *2; (d)* L/d = *10.*

**Figure 12.** *Mean pressure profile on the rod and the airfoil. (a) Rod. (b) Airfoil.* L/d = *2;* L/d = *10.*

the case of *L/d =* 2, as shown in **Figure 12(a)**, the pressure behind the rod is much larger than that for the case of *L/d =* 10 due to the presence of the airfoil behind the rod. As mentioned in Section 3.1.1, this phenomenon is related to the reattachments of the separated shear layers from the rod to the leading edge of the airfoil, and the Kármán-vortex street formed from the rod is suppressed and the pressure behind the rod increases. **Figure 12(b)** shows that in the case of *L/d =* 2, the negative pressure occurs around the leading edge of the airfoil, which means that the negative drag force acts on the airfoil, and in the case of *L/d =* 10, the positive pressure occurs the leading edge of the airfoil and the influence of the rod on the airfoil seems to be small.

**Figure 13** shows instantaneous snapshots of a fluctuation pressure (*dp)* and RMS of the fluctuation pressure (*dprms*) fields, and the fluctuation pressure is defined as follows [6].

$$dp = p\_s - p\_{\text{mean}} \tag{12}$$

#### **Figure 13.**

*Fluctuation pressure and RMS of fluctuation pressure fields. (a), (b) Fluctuation pressure, (c), (d) RMS of fluctuation pressure. (a)* L/d = *2; (b)* L/d = *10; (c)* L/d = *2; (d)* L/d = *10.*

*L/d =* 2, the RMS of the fluctuation pressure in the region near the rod is much lower than that near the leading edge of the airfoil due to the suppression of the vortex shedding form the rod, and in the case of *L/d =* 10, the pressure fluctuation near both the rod and the leading edge of the airfoil is much larger than that in the case of *L/d =* 2 due to the vortex shedding from the rod and the impingement of the vortices on the leading edge of the airfoil, as mentioned in **Figure 13**.

*Distributions of mean pressure and RMS of fluctuation pressure in wake region of rod along symmetry line (*y = *0) of rod and airfoil. (a) Mean pressure (b) RMS of fluctuation pressure.* L/d = *2;* L/d = *10.*

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

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

**Figure 15** shows the spectra of the SPL at the location (*x =* 0.68 m, *y =* 1.74 m) calculated by the acoustic BEM simulation using the acoustic sources extracted from the CFD results. The SPL in the case of the single airfoil is also plotted to compare the two cases with it. The SPL in the case of *L/d =* 10 is much larger than that in the case of *L/d =* 2, and the peak frequencies for the cases of *L/d =* 10 and *L/d =* 2 are 2440 and 1960 Hz, respectively. However, the peak frequency for the case of *L/d =* 2 does not appear clearly compared to that in the case of *L/d =* 10, and the

*SPL spectra at the location* x = *0.68 m,* y = *1.74 m.* L/d = *2;* L/d = *10; single airfoil.*

*4.1.3 Far acoustic pressure field*

**Figure 14.**

**Figure 15.**

**133**

Here, *ps* is the static pressure and *pmean* is the mean pressure as represented in **Figure 11(c)** and **(d)**. As shown in **Figure 13(a)** and **(b)**, in the case of *L/d =* 2, the pressure fluctuation mainly occurs around the leading edge of the airfoil, and the pressure fluctuation in the region near the rod is much lower than that near the leading edge of the airfoil. This result is related to the reattachment of the shear layers formed from the rod to the airfoil as mentioned in Section 3.1.1. **Figures 8(a), 13(a)** and **(c)** show that the suppression of the vortex shedding from the rod results in the suppression of the noise radiation from the rod-airfoil model, as mentioned by Munetaka et al. [10] and Jiang et al. [21].

On the other hand, in the case of *L/d =* 10, that is, for large spacing, the pressure fluctuation near both the rod and the leading edge of the airfoil is large, and especially the pressure fluctuation near the rod is larger than that near the leading edge of the airfoil, as shown in **Figure 13(b)** and **(d)**. **Figures 8(b), 13(b)** and **(d)** show that the pressure fluctuation behind the rod is generated by the vortex shedding from the rod. They also show that the pressure fluctuation near the leading edge of the airfoil is generated by the impingement of the vortices shed from the rod onto the leading edge of the airfoil and the distortion of the impinged vortices (wake-body interaction or body-vortex interaction), as mentioned by Boudet et al. [16]. Similar phenomenon is investigated by Inoue and Mori [6] in the simulations of the noise generated by the flow around two square cylinders in tandem. They reported that the distortion of the impinging vortices shed from the upstream cylinder onto the downstream cylinder plays an important role for the noise radiation when the spacing between the two cylinders is large [6].

The distributions of the mean pressure and RMS of the fluctuation pressure in the wake region of the rod along the symmetry line (*y =* 0) of the rod and the airfoil are represented in **Figure 14**. As shown in **Figure 14(a)**, in the case of *L/d =* 2, the mean pressure is negative in the wake region between the rod and the leading edge of the airfoil, and it leads to the negative drag of the airfoil as mentioned above. In the case of *L/d =* 10, the mean pressure behind the rod is negative and much lower than in the case of *L/d =* 2, and that near the leading edge of the airfoil is positive, which leads to the positive drag of the airfoil. **Figure 14(b)** shows that in the case of *Wake-Body Interaction Noise Simulated by the Coupling Method Using CFD and BEM DOI: http://dx.doi.org/10.5772/intechopen.92783*

**Figure 14.**

Here, *ps* is the static pressure and *pmean* is the mean pressure as represented in **Figure 11(c)** and **(d)**. As shown in **Figure 13(a)** and **(b)**, in the case of *L/d =* 2, the pressure fluctuation mainly occurs around the leading edge of the airfoil, and the pressure fluctuation in the region near the rod is much lower than that near the leading edge of the airfoil. This result is related to the reattachment of the shear

*Fluctuation pressure and RMS of fluctuation pressure fields. (a), (b) Fluctuation pressure, (c), (d) RMS of*

**Figures 8(a), 13(a)** and **(c)** show that the suppression of the vortex shedding from the rod results in the suppression of the noise radiation from the rod-airfoil model,

On the other hand, in the case of *L/d =* 10, that is, for large spacing, the pressure

The distributions of the mean pressure and RMS of the fluctuation pressure in the wake region of the rod along the symmetry line (*y =* 0) of the rod and the airfoil are represented in **Figure 14**. As shown in **Figure 14(a)**, in the case of *L/d =* 2, the mean pressure is negative in the wake region between the rod and the leading edge of the airfoil, and it leads to the negative drag of the airfoil as mentioned above. In the case of *L/d =* 10, the mean pressure behind the rod is negative and much lower than in the case of *L/d =* 2, and that near the leading edge of the airfoil is positive, which leads to the positive drag of the airfoil. **Figure 14(b)** shows that in the case of

layers formed from the rod to the airfoil as mentioned in Section 3.1.1.

radiation when the spacing between the two cylinders is large [6].

fluctuation near both the rod and the leading edge of the airfoil is large, and especially the pressure fluctuation near the rod is larger than that near the leading edge of the airfoil, as shown in **Figure 13(b)** and **(d)**. **Figures 8(b), 13(b)** and **(d)** show that the pressure fluctuation behind the rod is generated by the vortex shedding from the rod. They also show that the pressure fluctuation near the leading edge of the airfoil is generated by the impingement of the vortices shed from the rod onto the leading edge of the airfoil and the distortion of the impinged vortices (wake-body interaction or body-vortex interaction), as mentioned by Boudet et al. [16]. Similar phenomenon is investigated by Inoue and Mori [6] in the simulations of the noise generated by the flow around two square cylinders in tandem. They reported that the distortion of the impinging vortices shed from the upstream cylinder onto the downstream cylinder plays an important role for the noise

as mentioned by Munetaka et al. [10] and Jiang et al. [21].

*fluctuation pressure. (a)* L/d = *2; (b)* L/d = *10; (c)* L/d = *2; (d)* L/d = *10.*

*Vortex Dynamics Theories and Applications*

**Figure 13.**

**132**

*Distributions of mean pressure and RMS of fluctuation pressure in wake region of rod along symmetry line (*y = *0) of rod and airfoil. (a) Mean pressure (b) RMS of fluctuation pressure.* L/d = *2;* L/d = *10.*

*L/d =* 2, the RMS of the fluctuation pressure in the region near the rod is much lower than that near the leading edge of the airfoil due to the suppression of the vortex shedding form the rod, and in the case of *L/d =* 10, the pressure fluctuation near both the rod and the leading edge of the airfoil is much larger than that in the case of *L/d =* 2 due to the vortex shedding from the rod and the impingement of the vortices on the leading edge of the airfoil, as mentioned in **Figure 13**.

#### *4.1.3 Far acoustic pressure field*

**Figure 15** shows the spectra of the SPL at the location (*x =* 0.68 m, *y =* 1.74 m) calculated by the acoustic BEM simulation using the acoustic sources extracted from the CFD results. The SPL in the case of the single airfoil is also plotted to compare the two cases with it. The SPL in the case of *L/d =* 10 is much larger than that in the case of *L/d =* 2, and the peak frequencies for the cases of *L/d =* 10 and *L/d =* 2 are 2440 and 1960 Hz, respectively. However, the peak frequency for the case of *L/d =* 2 does not appear clearly compared to that in the case of *L/d =* 10, and the

**Figure 15.** *SPL spectra at the location* x = *0.68 m,* y = *1.74 m.* L/d = *2;* L/d = *10; single airfoil.*

**4.2 Airfoil-airfoil simulation results**

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

Typical examples of instantaneous vorticity fields are presented in **Figure 17**. In the present calculation, for the case of *L/c =* 0.2, when the spacing between the two airfoils is small, the boundary layers separated from the airfoil upstream did not roll up and reattach to the airfoil downstream. It seems that the reattached shear layers are oscillating at the leading edge of the downstream airfoil. The shear layers rolled up and formed vortices on the downstream airfoil surface, and the vortices were shed and convected downstream. On the other hand, in the cases of *L/c =* 0.6 and 1.0, the shear layers shed from the leading edge rolled up in front of the leading edge of the airfoil and impinged onto the leading edge of the airfoil. In the case of the single, the

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

shear layer and the vortices shed only from the trailing edge of the airfoil.

*Vorticity in the* z *direction. (a)* L/c = *0.2; (b)* L/c = *0.6; (c)* L/c = *1; (d) single airfoil.*

*Time-averaged velocity in the streamwise (*x*) direction. (a)* L/c = *0.2; (b)* L/c = *0.6; (c)* L/c = *1; (d) single*

*4.2.1 Flow patterns*

**Figure 17.**

**Figure 18.**

*airfoil.*

**135**

**Figure 16.** *Far-field sound pressure and SPL fields at the peak frequency. (a), (b) Sound pressure and (c), (d) SPL. (a)* L/d = *2, 1960 Hz; (b)* L/d = *10, 2440 Hz; (c)* L/d = *2, 1960 Hz; (d)* L/d = *10, 2440 Hz.*

peak frequency increases depending on the spacing between the rod and airfoil as reported by Jiang et al. [21]. The peak SPL in the case of single airfoil is larger than that in the case of *L/d =* 2, and the peak frequency is higher than those in the cases of *L/d =* 2 and 10. The sound radiation is mainly generated by both the vortex shedding from the rod and the impingement of the vortices shed from the rod onto the leading edge of the airfoil in the case of *L/d =* 10, as indicated by Boudet et al. [16]. According to Jiang et al. [21], when *L/d* ≥ 4, the vortex shedding from the rod and the impingement of the shed vortices onto the leading edge of the airfoil are the main generation sources of the noise radiation. As mentioned in Section 3.1.2, the suppression of the vortex shedding from the upstream rod results in the suppression of the pressure fluctuation in the region between the rod and the airfoil and the noise radiation for the case of *L/d =* 2.

**Figure 16** presents the far-field sound pressure and SPL fields at the peak frequency. The far-field sound pressure and SPL fields show a dipolar nature of the sound radiation, and the lift dipole is dominant in the fields. The magnitude of the generated noise in the case of *L/d =* 10 is much larger than that in the case of *L/d =* 2, as indicated above. In the case of the *L/d =* 2, the noise propagation on the upstream side is larger than that on the downstream side. On the other hand, in the case of the *L/d =* 10, the noise propagation on the downstream side is larger than that in the upstream direction, which means that impingement of the vortices shed from the rod onto the leading edge of the airfoil, in other words, the wake body interaction is the main generation of the far filed noise radiation.

*Wake-Body Interaction Noise Simulated by the Coupling Method Using CFD and BEM DOI: http://dx.doi.org/10.5772/intechopen.92783*
