*4.2.1 Flow patterns*

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

#### **Figure 17.**

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

*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.*

**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

noise radiation for the case of *L/d =* 2.

*Vortex Dynamics Theories and Applications*

**Figure 16.**

**134**

the main generation of the far filed noise radiation.

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

#### **Figure 18.**

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

**Figure 18(a)** and **(b)** show the fields of time-averaged velocity *Umean* in the *x* (streamwise) direction. In the spacing between the two airfoils, **Figure 18(a)** shows that the shear layers shed from the upstream airfoil near the leading edge of the airfoil lead to very small or negative values of *Umean* for the case of *L/c =* 0.2. The reattachments of the shear layer shed from the upstream airfoil to the leading edge of the airfoil due to the approaching airfoil slow down the flow. By contrast, for the cases of *L/c =* 0.6 and 1, the flow behind the upstream airfoil was accelerated in front of the leading edge of airfoil as shown in **Figure 18(b)** and **(c)**. These similar phenomena were observed in the rod-airfoil model, as explained in Section 3.1.1. In the case of the single, *Umean* contour around the airfoil shows similar one of the two airfoils in the case of *L/c =* 1. The influence of the wake from the upstream airfoil seems to become smaller with the spacing between the two airfoils increasing.

downstream airfoil is lower than that in other cases. This is due to the reattachments

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

Snapshots of mean pressure are represented in **Figure 21**. For the case of *L/c =* 0.2, **Figure 21** shows that the value of the static pressure on the leading edge of

*Mean pressure field. (a)* L/c = *0.2; (b)* L/c = *0.6; (c)* L/c = *1; (d) single airfoil.*

L/c = *0.2;* L/c = *0.6;* L/c = *1.0; single airfoil.*

*Mean pressure profile on the upstream and downstream airfoils. (a) Upstream airfoil; (b) downstream airfoil.*

Finally, the fields of RMS value of the fluctuation velocity *urms* in the streamwise (*x*) direction are represented in **Figure 20**. As shown in **Figure 20**, for the case of *L/c =* 0.2, the *urms* values around the trailing edges of the two airfoils are lower than those for the cases of *L/c =* 0.6, 1.0 and the single airfoil. This result corresponds to the reattachment of the shear layers to the airfoil (see **Figure 17(a)**) in this case. Therefore the turbulent fluctuations around the trailing edges of the two airfoils in the cases of *L/c =* 0.6 and 1.0 seem to be larger than those for the case of *L/c =* 0.2.

of the shear layers shed from the upstream airfoil.

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

*4.2.2 Near pressure field*

**Figure 21.**

**Figure 22.**

**137**

**Figure 19** shows the fields of time-averaged velocity *Vmean* in the y (vertical) direction. For the case of *L/c =* 0.2, the *Vmean* around the leading edge of the

#### **Figure 19.**

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

#### **Figure 20.**

*RMS value of fluctuation velocity in the streamwise (*x*) direction. (a)* L/c = *0.2; (b)* L/c = *0.6; (c)* L/c = *1; (d) single airfoil.*

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

downstream airfoil is lower than that in other cases. This is due to the reattachments of the shear layers shed from the upstream airfoil.

Finally, the fields of RMS value of the fluctuation velocity *urms* in the streamwise (*x*) direction are represented in **Figure 20**. As shown in **Figure 20**, for the case of *L/c =* 0.2, the *urms* values around the trailing edges of the two airfoils are lower than those for the cases of *L/c =* 0.6, 1.0 and the single airfoil. This result corresponds to the reattachment of the shear layers to the airfoil (see **Figure 17(a)**) in this case. Therefore the turbulent fluctuations around the trailing edges of the two airfoils in the cases of *L/c =* 0.6 and 1.0 seem to be larger than those for the case of *L/c =* 0.2.

#### *4.2.2 Near pressure field*

**Figure 18(a)** and **(b)** show the fields of time-averaged velocity *Umean* in the *x* (streamwise) direction. In the spacing between the two airfoils, **Figure 18(a)** shows that the shear layers shed from the upstream airfoil near the leading edge of the airfoil lead to very small or negative values of *Umean* for the case of *L/c =* 0.2. The reattachments of the shear layer shed from the upstream airfoil to the leading edge of the airfoil due to the approaching airfoil slow down the flow. By contrast, for the cases of *L/c =* 0.6 and 1, the flow behind the upstream airfoil was accelerated in front of the leading edge of airfoil as shown in **Figure 18(b)** and **(c)**. These similar phenomena were observed in the rod-airfoil model, as explained in Section 3.1.1. In the case of the single, *Umean* contour around the airfoil shows similar one of the two airfoils in the case of *L/c =* 1. The influence of the wake from the upstream airfoil seems to become smaller with the spacing between the two airfoils increasing. **Figure 19** shows the fields of time-averaged velocity *Vmean* in the y (vertical)

*Vortex Dynamics Theories and Applications*

direction. For the case of *L/c =* 0.2, the *Vmean* around the leading edge of the

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

*RMS value of fluctuation velocity in the streamwise (*x*) direction. (a)* L/c = *0.2; (b)* L/c = *0.6; (c)* L/c = *1; (d)*

**Figure 19.**

**Figure 20.**

*single airfoil.*

**136**

*airfoil.*

Snapshots of mean pressure are represented in **Figure 21**. For the case of *L/c =* 0.2, **Figure 21** shows that the value of the static pressure on the leading edge of

**Figure 21.**

*Mean pressure field. (a)* L/c = *0.2; (b)* L/c = *0.6; (c)* L/c = *1; (d) single airfoil.*

#### **Figure 22.**

*Mean pressure profile on the upstream and downstream airfoils. (a) Upstream airfoil; (b) downstream airfoil.* L/c = *0.2;* L/c = *0.6;* L/c = *1.0; single airfoil.*

the upstream airfoil is much larger than that of the downstream one, and that on the leading edge of the downstream airfoil is much lower than those for the cases of *L/c =* 0.6, 1.0 and the single airfoil.

**Figure 22** shows the mean pressure distribution on the surfaces of the two airfoils, where *xc* is the coordinate in the streamwise direction and the origin of *xc* is the located in the leading edge of the upstream airfoil. Depending on the spacing between the upstream and downstream airfoils, the pressure behind the upstream airfoil is affected by the downstream one. For the case of *L/c =* 0.2, as shown in **Figure 22(a)**, the pressure behind the upstream airfoil is larger than that for the cases of *L/c =* 0.6, 1 and single airfoil due to the presence of the downstream airfoil behind the upstream one. As mentioned in Section 3.2.1, this phenomenon is related to the reattachments of the separated shear layers from the upstream airfoil to the leading edge of the downstream airfoil, and the vortices formed from the upstream

airfoil is suppressed and the pressure behind the upstream airfoil increases. **Figure 22(b)** shows that in the case of *L/c =* 0.2, the pressure around the leading edge of the downstream airfoil is lower than that in the cases of *L/c =* 0.6, 1 and single airfoil, which means that the drag force acting on the downstream airfoil is

*SPL spectra at the location* x = *0.68 m,* y = *1.74 m.* L/c = *0.2;* L/c = *0.6;* L/c = *1.0;*

*Distributions of mean pressure and RMS of fluctuation pressure in wake region of upstream airfoil along symmetry line (*y = *0) of two airfoils. (a) Mean pressure; (b) RMS of fluctuation pressure.* L/c = *0.2;*

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

nature of the fluctuation pressure generated around the trailing edge of the upstream airfoil and the leading and trailing edges of the downstream airfoil.

**Figure 23** shows instantaneous snapshots of a fluctuation pressure (*dp*) and snapshots of RMS of the fluctuation pressure (*dprms*) fields. As shown in **Figure 23 (a)**–**(c)**, in the two airfoil cases, the pressure fluctuation occurs around the trailing edge of the upstream airfoil and the leading and trailing edges of the downstream airfoil. In the case of single airfoil, the pressure fluctuation mainly occurs near the trailing edge of the airfoil as shown in **Figure 23(d)**. These figures show the dipolar

**Figure 23(e)**–**(g)** shows that the pressure fluctuation near the leading edge of the downstream airfoil is generated by the impingement of the shear layers or

lower than that for the other cases.

**Figure 25.**

**139**

**Figure 24.**

L/c = *0.6;* L/c = *1.0.*

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

*single airfoil.*

#### **Figure 23.**

*Fluctuation pressure and RMS of fluctuation pressure fields. (a)–(d) Fluctuation pressure; (e)–(h) RMS of fluctuation pressure. (a)* L/c = *0.2; (b)* L/c = *0.6; (c)* L/c = *1; (d) single airfoil; (e)* L/c = *0.2; (f)* L/c = *0.6; (g)* L/c = *1.0; (h) single airfoil.*

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

#### **Figure 24.**

the upstream airfoil is much larger than that of the downstream one, and that on the leading edge of the downstream airfoil is much lower than those for the cases of

**Figure 22** shows the mean pressure distribution on the surfaces of the two airfoils, where *xc* is the coordinate in the streamwise direction and the origin of *xc* is the located in the leading edge of the upstream airfoil. Depending on the spacing between the upstream and downstream airfoils, the pressure behind the upstream airfoil is affected by the downstream one. For the case of *L/c =* 0.2, as shown in **Figure 22(a)**, the pressure behind the upstream airfoil is larger than that for the cases of *L/c =* 0.6, 1 and single airfoil due to the presence of the downstream airfoil behind the upstream one. As mentioned in Section 3.2.1, this phenomenon is related to the reattachments of the separated shear layers from the upstream airfoil to the leading edge of the downstream airfoil, and the vortices formed from the upstream

*Fluctuation pressure and RMS of fluctuation pressure fields. (a)–(d) Fluctuation pressure; (e)–(h) RMS of fluctuation pressure. (a)* L/c = *0.2; (b)* L/c = *0.6; (c)* L/c = *1; (d) single airfoil; (e)* L/c = *0.2; (f)* L/c = *0.6;*

*L/c =* 0.6, 1.0 and the single airfoil.

*Vortex Dynamics Theories and Applications*

**Figure 23.**

**138**

*(g)* L/c = *1.0; (h) single airfoil.*

*Distributions of mean pressure and RMS of fluctuation pressure in wake region of upstream airfoil along symmetry line (*y = *0) of two airfoils. (a) Mean pressure; (b) RMS of fluctuation pressure.* L/c = *0.2;* L/c = *0.6;* L/c = *1.0.*

**Figure 25.**

*SPL spectra at the location* x = *0.68 m,* y = *1.74 m.* L/c = *0.2;* L/c = *0.6;* L/c = *1.0; single airfoil.*

airfoil is suppressed and the pressure behind the upstream airfoil increases. **Figure 22(b)** shows that in the case of *L/c =* 0.2, the pressure around the leading edge of the downstream airfoil is lower than that in the cases of *L/c =* 0.6, 1 and single airfoil, which means that the drag force acting on the downstream airfoil is lower than that for the other cases.

**Figure 23** shows instantaneous snapshots of a fluctuation pressure (*dp*) and snapshots of RMS of the fluctuation pressure (*dprms*) fields. As shown in **Figure 23 (a)**–**(c)**, in the two airfoil cases, the pressure fluctuation occurs around the trailing edge of the upstream airfoil and the leading and trailing edges of the downstream airfoil. In the case of single airfoil, the pressure fluctuation mainly occurs near the trailing edge of the airfoil as shown in **Figure 23(d)**. These figures show the dipolar nature of the fluctuation pressure generated around the trailing edge of the upstream airfoil and the leading and trailing edges of the downstream airfoil.

**Figure 23(e)**–**(g)** shows that the pressure fluctuation near the leading edge of the downstream airfoil is generated by the impingement of the shear layers or

vortices shed from the upstream airfoil onto the leading edge of the downstream airfoil and the distortion of the impinged vortices (wake-body interaction or body-

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

The distributions of the mean pressure and RMS of the fluctuation pressure in the wake region of the upstream airfoil along the symmetry line (*y =* 0) of the two airfoils are represented in **Figure 24**. As shown in **Figure 24(a)**, the mean pressure behind the upstream airfoil is positive and lower than that near the leading edge of the downstream airfoil, and it decreases depending on the spacing between the two airfoils. On the other hand, that near the leading edge of the downstream airfoil increases depending on the spacing between the two airfoils. **Figure 24(b)** shows that the pressure fluctuation near both the trailing edge of the upstream airfoil and the leading edge of the downstream airfoil is large due to the shear layers or vortices shed from the upstream airfoil and the impingement of the shear layers or vortices onto the leading edge of the downstream airfoil, as mentioned in **Figure 23**.

**Figure 25** 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 three cases with it. The SPLs in the cases of *L/c =* 0.6, 1.0 and single airfoil are much larger than that in the case of *L/c =* 0.2, and the peak frequencies for the cases of *L/c =* 0.2, 0.6, 1.0 and single airfoil are 540, 2840, 3640, and 3640 Hz, respectively. **Figure 26** shows the far field sound pressure and SPL fields at the peak frequency. The peak frequency increases depending on the spacing between the two airfoils as reported in the rod-airfoil model. The peak SPL in the case of single airfoil is larger than that in the case of *L/c =* 0.2. The peak frequency in the case of the single airfoil is higher than those in the cases of *L/c =* 0.2 and 0.6, and same as that

In the cases of the two airfoils, the sound radiation is mainly generated by three

In this chapter, we simulated the flow around the rod-airfoil model and the noise generated by the wake-body interaction or body-vortex interaction by the coupling method using commercial CFD and acoustic BEM codes, and compared the results with those obtained by Jacob et al. [9] and Jiang et al. [21]. Then, we simulated the flow around the airfoil-airfoil model (airfoils in tandem) and the noise generation

1. In the rod-airfoil model, when the spacing between the rod and the airfoil is small, the shear layers separated from the rod upstream did not roll up and

factors: (1) vortices or shear layers shedding from the upstream airfoil, (2) the impingement of the vortices or shear layers shed from the upstream airfoil onto the leading edge of the downstream airfoil, (3) vortices or shear layers shedding from the downstream airfoil as indicated in Section 3.2.2. As mentioned in Section 3.2.2, the reattachment of the shear layers from the upstream airfoil to the leading edge of the downstream airfoil results in the suppression of the pressure fluctuation in the region between the upstream and downstream airfoils and the noise radiation for

vortex interaction), as mentioned in Section 3.1.2.

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

*4.2.3 Far acoustic pressure field*

in the case of *L/c =* 1.0.

the case of *L/c =* 0.2.

**5. Conclusions**

and propagation.

**141**

**Figure 26.**

*Far-field sound pressure and SPL fields at the peak frequency. (a)–(d) Sound pressure and (e)–(h) SPL. (a)* L/c = *0.2; (b)* L/c = *0.6; (c)* L/c = *1; (d) single airfoil; (e)* L/c = *0.2; (f)* L/c = *0.6; (g)* L/c = *1.0; (h) single airfoil.*

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

vortices shed from the upstream airfoil onto the leading edge of the downstream airfoil and the distortion of the impinged vortices (wake-body interaction or bodyvortex interaction), as mentioned in Section 3.1.2.

The distributions of the mean pressure and RMS of the fluctuation pressure in the wake region of the upstream airfoil along the symmetry line (*y =* 0) of the two airfoils are represented in **Figure 24**. As shown in **Figure 24(a)**, the mean pressure behind the upstream airfoil is positive and lower than that near the leading edge of the downstream airfoil, and it decreases depending on the spacing between the two airfoils. On the other hand, that near the leading edge of the downstream airfoil increases depending on the spacing between the two airfoils. **Figure 24(b)** shows that the pressure fluctuation near both the trailing edge of the upstream airfoil and the leading edge of the downstream airfoil is large due to the shear layers or vortices shed from the upstream airfoil and the impingement of the shear layers or vortices onto the leading edge of the downstream airfoil, as mentioned in **Figure 23**.

## *4.2.3 Far acoustic pressure field*

**Figure 25** 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 three cases with it. The SPLs in the cases of *L/c =* 0.6, 1.0 and single airfoil are much larger than that in the case of *L/c =* 0.2, and the peak frequencies for the cases of *L/c =* 0.2, 0.6, 1.0 and single airfoil are 540, 2840, 3640, and 3640 Hz, respectively. **Figure 26** shows the far field sound pressure and SPL fields at the peak frequency. The peak frequency increases depending on the spacing between the two airfoils as reported in the rod-airfoil model. The peak SPL in the case of single airfoil is larger than that in the case of *L/c =* 0.2. The peak frequency in the case of the single airfoil is higher than those in the cases of *L/c =* 0.2 and 0.6, and same as that in the case of *L/c =* 1.0.

In the cases of the two airfoils, the sound radiation is mainly generated by three factors: (1) vortices or shear layers shedding from the upstream airfoil, (2) the impingement of the vortices or shear layers shed from the upstream airfoil onto the leading edge of the downstream airfoil, (3) vortices or shear layers shedding from the downstream airfoil as indicated in Section 3.2.2. As mentioned in Section 3.2.2, the reattachment of the shear layers from the upstream airfoil to the leading edge of the downstream airfoil results in the suppression of the pressure fluctuation in the region between the upstream and downstream airfoils and the noise radiation for the case of *L/c =* 0.2.

## **5. Conclusions**

In this chapter, we simulated the flow around the rod-airfoil model and the noise generated by the wake-body interaction or body-vortex interaction by the coupling method using commercial CFD and acoustic BEM codes, and compared the results with those obtained by Jacob et al. [9] and Jiang et al. [21]. Then, we simulated the flow around the airfoil-airfoil model (airfoils in tandem) and the noise generation and propagation.

1. In the rod-airfoil model, when the spacing between the rod and the airfoil is small, the shear layers separated from the rod upstream did not roll up and

**Figure 26.**

*Vortex Dynamics Theories and Applications*

*airfoil.*

**140**

*Far-field sound pressure and SPL fields at the peak frequency. (a)–(d) Sound pressure and (e)–(h) SPL. (a)* L/c = *0.2; (b)* L/c = *0.6; (c)* L/c = *1; (d) single airfoil; (e)* L/c = *0.2; (f)* L/c = *0.6; (g)* L/c = *1.0; (h) single* reattach to the airfoil downstream, and the vortex shedding from the rod is suppressed. It leads to the suppression of the pressure fluctuation near the rod and the airfoil and the noise radiation as reported by Jiang et al. [21].

**References**

473-489

618-633

1019-1050

2003;**17**:97-113

[1] Mahir N, Rockwell D. Vortex shedding from a forced system of two cylinders. Part I: Tandem arrangement. Journal of Fluids and Structures. 1996;**9**:

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

Computational Fluid Dynamics. 2005;

[10] Munekata M, Kawahara K, Udo T, Yoshikawa H, Ohba H. An experimental study on aerodynamic sound generated from wake interference of circular cylinder and airfoil vane in tandem. Journal of Thermal Science. 2006;**15**(4):

[11] Munekata M, Koshiishi R,

2008;**17**(3):212-217

Yoshikawa H, Ohba H. An experimental study on aerodynamic sound generated from wake interaction of circular cylinder and airfoil with attack angle in tandem. Journal of Thermal Science.

[12] Li Y, Wang XN, Chen ZW, Li ZC. Experimental study of vortex-structure interaction noise radiated from rod– airfoil configurations. Journal of Fluids and Structures. 2014;**51**(3):313-325

[13] Casalino D, Jacob MC, Roger M. Prediction of rod airfoil interaction noise using the FWH analogy. AIAA

investigation on sound generation from

[15] Magagnato F, Sorgüven E, Gabi M. Far field noise prediction by large eddy simulation and Ffowcs-Williams Hawkings analogy. AIAA Journal. 2003;

[16] Boudet J, Grosjean N, Jacob MC. Wake-airfoil interaction as broadband noise source: A large-eddy simulation study. International Journal of Aeroacoustics. 2005;**4**(1):93-116

[17] Greschner B, Thiele F, Jacob MC, Casalino D. Prediction of sound

Journal. 2003;**41**(2):182-191

[14] Jiang M, Li XD, Zhou JJ. Experimental and numerical

**32**(6):765-776

**6**:2003-3206

airfoil-flow interaction. Applied Mathematics and Mechanics. 2011;

**19**(3):171-196

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

342-348

[2] Zdravkovich MM. Review of flow interference between two circular cylinders in various arrangements. Journal of Fluids Engineering. 1977;**99**:

[3] Ljungkrona L, Norberg CH,

Structures. 1991;**5**:701-727

Sunden B. Free-stream turbulence and tube spacing effects on surface pressure fluctuations for two tubes in an in-line arrangement. Journal of Fluids and

[4] Liu CH, Chen JM. Observations of hysteresis in flow around two square cylinders in a tandem arrangement. Journal of Wind Engineering and Industrial Aerodynamics. 2002;**90**(9):

[5] Fitzpatrick JA. Flow/acoustic interactions of two cylinders in crossflow. Journal of Fluids and Structures.

[6] Inoue O, Mori M, Hatakeyama N. Aeolian tones radiated from flow past two square cylinders in tandem. Physics

[7] King WFN, Pfizenmaier E. An experimental study of sound generated by flows around cylinders of different cross-section. Journal of Sound and Vibration. 2009;**328**:318-337

[8] Hutcheson FV, Brooks TF. Noise radiation from single and multiple rod configurations. International Journal of

[9] Jacob MC, Boudet J, Casalino D, Michard M. A rod–airfoil experiment as

benchmark for broadband noise modeling. Theoretical and

**143**

Aeroacoustics. 2012;**11**:291-334

of Fluids. 2006;**18**:046101

