*3.2.3 Acoustic losses due to vortex shedding at the edge*

In the framework of the jet-drive model, Dequand et al. [21] assumed that the separation of the acoustic flow *q*mðÞ¼ *t* ð Þ d*ξ*m*=*d*t db* occurs at the edge by following Verge et al*.* [12]. This acoustic flow separation causes a free jet [45].

Although they neglect the effects of the separation of the jet flow and their viewpoint is different from the modeling illustrated in **Figure 13**, it seems to be worth taking into consideration. The effects of vortices can be represented by a fluctuating pressure *p*<sup>v</sup> across the mouth [12, 21]:

$$p\_{\rm v} = - (\mathbf{1}/2)\rho \left(q\_{\rm m}/c\_{\rm v}db\right)^{2} \text{sign}(q\_{\rm m}),\tag{42}$$

where *c*<sup>v</sup> (= 0.6) is the vena contracta factor of the free jet. The time-averaged power losses due to the acoustic vortex shedding at the edge is then given as [21].

$$
\langle \Pi\_{\rm lost} \rangle = \langle p\_{\rm v} q\_{\rm m} \rangle = - (\mathbf{1}/2T) \left( \rho d b / c\_{\rm v}^2 \right) \int\_0^T (\mathbf{d} \xi\_{\rm m}/\mathbf{d}t)^2 |\mathbf{d} \xi\_{\rm m}/\mathbf{d}t| \, \text{d}t,\tag{43}
$$

where it is assumed that the dissipation occurs during the entire period *T*.

Therefore, the power dissipation given by Eq. (43) may be considered as an upper limit approximation, and by neglecting *Π*<sup>α</sup> in Eq. (41), it can be roughly balanced with the power generation by the jet drive given by Eq. (9) [12, 21]:

$$
\langle \Pi\_{\text{jet}} \rangle + \langle \Pi\_{\text{best}} \rangle \approx \mathbf{0}. \tag{44}
$$

If the integral in the right-hand side of Eq. (43) can be replaced with a product of dð Þ *ξ*m*=*d*t* max 2 j j d*ξ*m*=*d*t* and an appropriate division of *T* by supposing a rectangular-like waveform of *ξ*mð Þ*t* , we have the following relation between the maximum acoustic velocity dð Þ *ξ*m*=*d*t* max over the mouth, the Strouhal number *S*<sup>t</sup> ¼ *fd=U*0, and the aspect ratio *d=h* of the jet [12, 21]:

$$\left[\left(\mathbf{d}\xi\_{\mathrm{m}}/\mathbf{d}\mathbf{t}\right)\_{\mathrm{max}}/U\_{0}\right]^{2} \sim \mathcal{S}\_{\mathrm{t}}(h/d)^{3/2}.\tag{45}$$

quite high blowing pressure. The phase lag due to the pressure drive can make the acoustic velocity in the case of **Figure 12(c)** more positive as inferred from **Figure 8(a)**.

*starting transient; (b) and (c): An acoustic vortex formed at the pre-steady state [17].*

not observed at the steady state in [17]. Instead of that, we observed a steadily deflecting jet, particularly its penetration into the pipe as captured in Figure 13 in [17]. According to this result, we may consider that the acoustic vortex is formed to lead the finally saturated amplification of the jet stability wave by absorbing the final excess in the acoustic energy generation occurring at the pre-steady state. The acoustic vortex may be then conveyed by the jet flow into the region where the vorticity can no longer continue to interact with the acoustic field [35]. Since the completely steady state has already reached the energy balance, any more acoustic vortices seem to be not needed. Instead, the acoustic vortices will be strongly needed just before the

Interestingly enough, the acoustic vortices shown in **Figure 12(a)** and **(b)** were

*Schematic of vortex formation in organ flue pipes. (a): A hydrodynamic vortex formed at the initial phase of the*

Then, ð Þ *ω* � *U* ∙ *u*>0 will be realized in better fashion.

*Vortex Dynamics Theories and Applications*

**Figure 13.**

**66**

This interesting non-dimensional relation was almost confirmed by the experiment on thin jets (*d=h*>2) for the four different flue-edge geometries (see Figure 11 in [21]). The maximum of non-dimensional amplitude dð Þ *ξ*m*=*d*t* max*=U*<sup>0</sup> reached for the edge with an angle of 60° is 20% higher than that obtained for the edge with an angle of 15°. This difference in amplitude can reflect the difference between the flute and the recorder. The recorder with a sharper edge probably brings about stronger losses due to vortex shedding at the edge.

is driven at low amplitudes to satisfy the requirement of the potential and sinusoidal

*Vortices on Sound Generation and Dissipation in Musical Flue Instruments*

Since j j *u* ≪ j j *U* , it is very difficult to measure the distribution of *u* over the mouth area using the PIV when the pipe is driven by the air jet. Therefore, both measurements of *u* and *U* should be separately carried out. Of course, *U* cannot be measured without using the jet. On the other hand, *u* can be measured by resonating the pipe externally, for example, by using an inverse exponential horn [28, 49]. A larger cross section of this horn is firmly fitted to the loudspeaker diaphragm, and a smaller cross section is coupled to the pipe end with a distance larger than the end correction to maintain the resonance pattern of the pressure distribution along the air column. The loudspeaker is driven by an oscillator to generate a sinusoidal wave in the pipe with the same frequency and amplitude as those when the jet drives the pipe. The organ pipe is thus driven by this loudspeaker horn system when *u* is

Also, in order to experimentally examine the generation of the vortex sound based on Eq. (14), both measurements of *u* and *U* must be carried out at the same condition as exactly as possible. That is, these vectors must be measured at the same phase of the generating sound and at the same measurement area by using the same organ pipe. However, since *u* and *U* cannot be measured simultaneously, the jet drive and the loudspeaker horn drive must be switched as quickly as possible while maintaining the same sounding condition and the same measurement condition. The phase-locked PIV measurement on *u* and *U* (see **Figure 14**) is thus essentially important to evaluate Eq. (14). Since j j *u* ≪ j j *U* , *u* should be first measured using the horn drive at a given phase of the sound, and the *U* is measured at the same phase

PIV measurement of *u* and *U* was carried out twice (Trials 1 and 2) in [28]. The fundamental frequencies of the pipe tone were 192.0 Hz and 192.1 Hz, respectively (the cutoff frequency of the horn was designed to be about 150 Hz). The averaged sound levels were 59.0 dB and 59.3 dB, respectively. When the averaged sound level was 57.8 dB (the *pianissimo* level), the third and fifth harmonics were not negligible as compared with these trials. On the other hand, when the averaged sound level was 60.2 dB, the second harmonic was only 10 dB lower than the fundamental.

*Experimental setup based on the phase-locked PIV system (the PIV itself is manufactured by Dantec dynamics). The blower and the loudspeaker horn system are alternately used for the jet drive to measure the jet velocity and*

*for the horn drive to measure the acoustic cross-flow velocity, respectively [28].*

by quickly switching the horn drive to the jet drive.

*4.1.2 Measurement procedures*

flow on *u* simultaneously [28].

*DOI: http://dx.doi.org/10.5772/intechopen.91258*

measured.

**Figure 14.**

**69**

## **4. Vortices on sound generation**

In this section let us consider what the cause of the jet oscillation is for thin jets (*d=h*>2). Fletcher's displacement model of Eq. (4) [1, 31] has no definite physical basis, and Coltman's velocity model [22, 23] lacks in quantitative analysis. The present author proposed an acceleration model based on the pressure difference between the upper and lower surfaces of the jet [26]. Although this model could not involve the effects of the jet instability [20], it could successfully predict the possibility of underwater organ pipes [46]. Therefore, another acceleration model based on the vorticity generation is greatly expected [28].

#### **4.1 Vortex layer along the jet visualized by PIV**

A great advantage of PIV is to yield global and quantitative information on the flow-acoustic interaction. The PIV was already successfully applied to the experimental research of the edge-tone generation [6], where the complicated jet-edge interaction was investigated to accurately localize the vortex cores (dipole sources) just before the edge. Also, it was applied to measure the flow velocity and acoustic particle velocity [47]. Measurements of both quantities are required to consider sound generation based on the vortex sound theory.

#### *4.1.1 Measurement requirements*

Since the vortex sound theory hypothesizes an irrotational potential flow for *u* [18], the measurement should meet the requirement for this potential flow. However, actual acoustic cross-flow *u* in flue instruments tends to yield a non-potential flow through any nonlinear process at large-amplitude conditions. Hence, the measurement should be carried out at low amplitudes to assure the potential flow. Moreover, the PIV cannot evaluate the contribution from the harmonics but only estimates the instantaneous flow magnitude on a plane sheet illuminated by the laser. Hence, the acoustic cross-flow should be measured based on the acoustical field with a waveform as sinusoidal as possible.

Both requirements of (1) a potential flow and (2) a sinusoidal flow for *u* are not easily fulfilled at the same time in rigorous manner. A practical way seems to be a measurement at appropriate low amplitudes (the drive at *piano* level was better than that at *pianissimo* level [28]). Bamberger [48–50], who first introduced the PIV into the field of musical acoustics, carried out his measurement at *mezzo-forte* level by driving a flute at a very high pitch of about 1150 Hz. This sounding condition seems to satisfy the sinusoidal flow condition because a very high tone of the flute almost consists of only the fundamental. This is due to the cutoff frequency around 2 kHz of the column resonance of the modern flute [51]. However, it is uncertain whether the requirement of the irrotational potential flow on *u* is satisfied or not. On the other hand, an organ pipe with a low-frequency resonance (at about 195 Hz)

#### *Vortices on Sound Generation and Dissipation in Musical Flue Instruments DOI: http://dx.doi.org/10.5772/intechopen.91258*

is driven at low amplitudes to satisfy the requirement of the potential and sinusoidal flow on *u* simultaneously [28].

Since j j *u* ≪ j j *U* , it is very difficult to measure the distribution of *u* over the mouth area using the PIV when the pipe is driven by the air jet. Therefore, both measurements of *u* and *U* should be separately carried out. Of course, *U* cannot be measured without using the jet. On the other hand, *u* can be measured by resonating the pipe externally, for example, by using an inverse exponential horn [28, 49]. A larger cross section of this horn is firmly fitted to the loudspeaker diaphragm, and a smaller cross section is coupled to the pipe end with a distance larger than the end correction to maintain the resonance pattern of the pressure distribution along the air column. The loudspeaker is driven by an oscillator to generate a sinusoidal wave in the pipe with the same frequency and amplitude as those when the jet drives the pipe. The organ pipe is thus driven by this loudspeaker horn system when *u* is measured.

Also, in order to experimentally examine the generation of the vortex sound based on Eq. (14), both measurements of *u* and *U* must be carried out at the same condition as exactly as possible. That is, these vectors must be measured at the same phase of the generating sound and at the same measurement area by using the same organ pipe. However, since *u* and *U* cannot be measured simultaneously, the jet drive and the loudspeaker horn drive must be switched as quickly as possible while maintaining the same sounding condition and the same measurement condition. The phase-locked PIV measurement on *u* and *U* (see **Figure 14**) is thus essentially important to evaluate Eq. (14). Since j j *u* ≪ j j *U* , *u* should be first measured using the horn drive at a given phase of the sound, and the *U* is measured at the same phase by quickly switching the horn drive to the jet drive.

## *4.1.2 Measurement procedures*

This interesting non-dimensional relation was almost confirmed by the experiment on thin jets (*d=h*>2) for the four different flue-edge geometries (see Figure 11 in [21]). The maximum of non-dimensional amplitude dð Þ *ξ*m*=*d*t* max*=U*<sup>0</sup> reached for the edge with an angle of 60° is 20% higher than that obtained for the edge with an angle of 15°. This difference in amplitude can reflect the difference between the flute and the recorder. The recorder with a sharper edge probably brings about

In this section let us consider what the cause of the jet oscillation is for thin jets (*d=h*>2). Fletcher's displacement model of Eq. (4) [1, 31] has no definite physical basis, and Coltman's velocity model [22, 23] lacks in quantitative analysis. The present author proposed an acceleration model based on the pressure difference between the upper and lower surfaces of the jet [26]. Although this model could not involve the effects of the jet instability [20], it could successfully predict the possibility of underwater organ pipes [46]. Therefore, another acceleration model based

A great advantage of PIV is to yield global and quantitative information on the flow-acoustic interaction. The PIV was already successfully applied to the experimental research of the edge-tone generation [6], where the complicated jet-edge interaction was investigated to accurately localize the vortex cores (dipole sources) just before the edge. Also, it was applied to measure the flow velocity and acoustic particle velocity [47]. Measurements of both quantities are required to consider

Since the vortex sound theory hypothesizes an irrotational potential flow for *u* [18], the measurement should meet the requirement for this potential flow. However, actual acoustic cross-flow *u* in flue instruments tends to yield a non-potential flow through any nonlinear process at large-amplitude conditions. Hence, the measurement should be carried out at low amplitudes to assure the potential flow. Moreover, the PIV cannot evaluate the contribution from the harmonics but only estimates the instantaneous flow magnitude on a plane sheet illuminated by the laser. Hence, the acoustic cross-flow should be measured based on the acoustical

Both requirements of (1) a potential flow and (2) a sinusoidal flow for *u* are not easily fulfilled at the same time in rigorous manner. A practical way seems to be a measurement at appropriate low amplitudes (the drive at *piano* level was better than that at *pianissimo* level [28]). Bamberger [48–50], who first introduced the PIV into the field of musical acoustics, carried out his measurement at *mezzo-forte* level by driving a flute at a very high pitch of about 1150 Hz. This sounding condition seems to satisfy the sinusoidal flow condition because a very high tone of the flute almost consists of only the fundamental. This is due to the cutoff frequency around 2 kHz of the column resonance of the modern flute [51]. However, it is uncertain whether the requirement of the irrotational potential flow on *u* is satisfied or not. On the other hand, an organ pipe with a low-frequency resonance (at about 195 Hz)

stronger losses due to vortex shedding at the edge.

on the vorticity generation is greatly expected [28].

**4.1 Vortex layer along the jet visualized by PIV**

sound generation based on the vortex sound theory.

field with a waveform as sinusoidal as possible.

*4.1.1 Measurement requirements*

**68**

**4. Vortices on sound generation**

*Vortex Dynamics Theories and Applications*

PIV measurement of *u* and *U* was carried out twice (Trials 1 and 2) in [28]. The fundamental frequencies of the pipe tone were 192.0 Hz and 192.1 Hz, respectively (the cutoff frequency of the horn was designed to be about 150 Hz). The averaged sound levels were 59.0 dB and 59.3 dB, respectively. When the averaged sound level was 57.8 dB (the *pianissimo* level), the third and fifth harmonics were not negligible as compared with these trials. On the other hand, when the averaged sound level was 60.2 dB, the second harmonic was only 10 dB lower than the fundamental.

#### **Figure 14.**

*Experimental setup based on the phase-locked PIV system (the PIV itself is manufactured by Dantec dynamics). The blower and the loudspeaker horn system are alternately used for the jet drive to measure the jet velocity and for the horn drive to measure the acoustic cross-flow velocity, respectively [28].*

When the level was 69.4 dB, the second harmonic showed almost the same magnitude as the fundamental. As a result, two trials above with the *piano* level excitation seem to yield good conditions satisfying the measurement requirement for *u*. It should be noted that all these tones with their levels from 57.8 dB to 69.4 dB are produced by the same first mode resonance.

The phase lock of the PIV system is easily implemented if the external trigger signal is produced to activate the laser and the CCD camera. This is because the PIV system can set the trigger delay almost arbitrarily through the software embedded in the trigger signal production system shown in **Figure 14**. The trigger delay was set to be (1/12)*T* times *n* (*n* = 0, 1, 2, ..., 11), where *T* denotes the period of the pipe tone. For more details on the production of the external trigger, refer to [28]. As a result, the phase-locked measurement of *u* and *U* is carried out at the specific phases (Phase 0, Phase 1, Phase 2, ..., Phase 11). Note that Phase 0 is defined by the buildup of the positive trigger pulse when the trigger delay is not applied.

In the experiment a metallic organ pipe, which was made by a German organ builder, was measured [28]. Its cross-sectional structure (in *x-z* plane) around the mouth is already shown in **Figure 1**, and its important geometry is as follows: the physical pipe length *L* = 793 mm, the pipe inner diameter 2*R* = 43.6 mm, the flue-toedge distance *d* = 8.8 mm, the jet thickness *h* = 0.75 mm, and the mouth breadth *b* = 31.9 mm. The value of *d/h* is 11.7, much larger than 2. It should be noted that the edge is not very sharp like a wedge but plate-like as an extension of pipe wall, although the edge tip is 0.4 mm thick compared to the pipe wall that is 1.0 mm thick. Such an edge and flue are common in metal flue pipes as illustrated in Figure 17.6 of [1].

#### *4.1.3 Calculation of the acoustic generation formula*

The PIV can derive the vorticity map from the jet velocity distribution. The vorticity *ω* ð Þ ¼ rot *U* at a field point is calculated from the 2-D velocities at four discrete points surrounding the point of interest. Therefore, the aeroacoustical source term *ω* � *U* and the acoustic power generation term ð Þ *ω* � *U* **∙** *u* can be calculated from the measurement of velocity fields *u* and *U* (see [28] on their measurement results, which are spared in this chapter).

The vorticity map given at Trial 1 is illustrated in **Figure 15(a)**. Since the 2-D velocity *U* was measured in *x-z* plane, the vorticity vector has only *y* direction component [*<sup>ω</sup>* <sup>¼</sup> 0,*ω*y, 0 ]:

$$
\rho\_\mathbf{y}(\mathbf{x}, \mathbf{z}) = \frac{\partial U\_\mathbf{x}}{\partial \mathbf{z}}(\mathbf{x}, \mathbf{z}) - \frac{\partial U\_\mathbf{z}}{\partial \mathbf{x}}(\mathbf{x}, \mathbf{z}).\tag{46}
$$

components of *ω* � *U*, respectively. The maximum magnitude of *ω* � *U*, which is about 2 � <sup>10</sup><sup>5</sup> <sup>m</sup>*=*s2 ð Þ, can be observed near the flue. The opposing vectors of *<sup>ω</sup>* � *<sup>U</sup>* along both layers seem to almost cancel each other as depicted in **Figure 10(b)**. However, since *ω* � *U* has a very large acceleration, an imperfect cancelation (or a slight unbalance) between *ω* � *U* vectors along both layers yields a significant effect

*Aerodynamical quantities derived from the phase-locked PIV measurement at the organ pipe mouth: (a) vorticity ω; (b) aeroacoustical source term (acceleration) ω* � *U; (c) acoustic generation term* ð Þ *ω* � *U* **∙** *u. The positions of the flue exit and the edge tip correspond to (x, z) = (2.0 mm, 2.2 mm) and (x, z) = (10.5 mm,*

*Vortices on Sound Generation and Dissipation in Musical Flue Instruments*

*DOI: http://dx.doi.org/10.5772/intechopen.91258*

Acoustic generation term ð Þ� *ω* � *U u* defined by Eq. (14) is shown in **Figure 15(c)**.

Since *u* indicates the outflow at Phases 1 and 4, and inflow at Phases 7 and 10, ð Þ� *ω* � *U u* takes the opposite sign along the vortex layer between these phases. This sign inversion occurs near Phases 0 and 6 [28]. The maximum magnitude of ð Þ� *ω* � *U u* appears near 1–2 millimeters downstream from the flue at Phases 3 and 8 as about

*4.1.4 Generation of the acoustic power from the vortical field*

in acoustical events.

*3.8 mm), respectively [28].*

**Figure 15.**

**71**

The vorticity is formed along the upper and lower boundaries of the jet. The upper layer possesses the positive vorticity (the counterclockwise rotation of small vortices) and the lower layer the negative vorticity [cf. **Figure 10(a)**]. These layers may be called *vorticity layers* or simply *vortex layers*. At Phases 7 and 10, large-scaled positive vortices are indicated before and above the edge, but these do not seem to be important because the magnitudes of *U* and *u* there are very small. As a result, the effects of these vortices almost completely disappear as shown in **Figure 15(c)**. It should be correctly recognized that vortex shedding from the edge tip (cf. **Figure 6**) is never observed in **Figure 15(a)**. This implies that Howe's vortexshedding model may not be applicable to the sound generation in organ flue pipes that are usually driven by thin jets.

The resulting aeroacoustical source term *ω* � *U* is displayed in **Figure 15(b)**, where the upper and lower layers of the vorticity yield the positive *z* and negative *z* *Vortices on Sound Generation and Dissipation in Musical Flue Instruments DOI: http://dx.doi.org/10.5772/intechopen.91258*

#### **Figure 15.**

When the level was 69.4 dB, the second harmonic showed almost the same magnitude as the fundamental. As a result, two trials above with the *piano* level excitation seem to yield good conditions satisfying the measurement requirement for *u*. It should be noted that all these tones with their levels from 57.8 dB to 69.4 dB are

The phase lock of the PIV system is easily implemented if the external trigger signal is produced to activate the laser and the CCD camera. This is because the PIV system can set the trigger delay almost arbitrarily through the software embedded in the trigger signal production system shown in **Figure 14**. The trigger delay was set to be (1/12)*T* times *n* (*n* = 0, 1, 2, ..., 11), where *T* denotes the period of the pipe tone. For more details on the production of the external trigger, refer to [28]. As a result, the phase-locked measurement of *u* and *U* is carried out at the specific phases (Phase 0, Phase 1, Phase 2, ..., Phase 11). Note that Phase 0 is defined by the

In the experiment a metallic organ pipe, which was made by a German organ builder, was measured [28]. Its cross-sectional structure (in *x-z* plane) around the mouth is already shown in **Figure 1**, and its important geometry is as follows: the physical pipe length *L* = 793 mm, the pipe inner diameter 2*R* = 43.6 mm, the flue-toedge distance *d* = 8.8 mm, the jet thickness *h* = 0.75 mm, and the mouth breadth *b* = 31.9 mm. The value of *d/h* is 11.7, much larger than 2. It should be noted that the edge is not very sharp like a wedge but plate-like as an extension of pipe wall, although the edge tip is 0.4 mm thick compared to the pipe wall that is 1.0 mm thick. Such an edge and flue are common in metal flue pipes as illustrated in

The PIV can derive the vorticity map from the jet velocity distribution. The vorticity *ω* ð Þ ¼ rot *U* at a field point is calculated from the 2-D velocities at four discrete points surrounding the point of interest. Therefore, the aeroacoustical source term *ω* � *U* and the acoustic power generation term ð Þ *ω* � *U* **∙** *u* can be calculated from the measurement of velocity fields *u* and *U* (see [28] on their

The vorticity map given at Trial 1 is illustrated in **Figure 15(a)**. Since the 2-D velocity *U* was measured in *x-z* plane, the vorticity vector has only *y* direction

*<sup>∂</sup><sup>z</sup>* ð Þ� *<sup>x</sup>*, *<sup>z</sup>*

The vorticity is formed along the upper and lower boundaries of the jet. The upper layer possesses the positive vorticity (the counterclockwise rotation of small vortices) and the lower layer the negative vorticity [cf. **Figure 10(a)**]. These layers may be called *vorticity layers* or simply *vortex layers*. At Phases 7 and 10, large-scaled positive vortices are indicated before and above the edge, but these do not seem to be important because the magnitudes of *U* and *u* there are very small. As a result, the effects of these vortices almost completely disappear as shown in **Figure 15(c)**.

(cf. **Figure 6**) is never observed in **Figure 15(a)**. This implies that Howe's vortexshedding model may not be applicable to the sound generation in organ flue pipes

The resulting aeroacoustical source term *ω* � *U* is displayed in **Figure 15(b)**, where the upper and lower layers of the vorticity yield the positive *z* and negative *z*

*∂U*<sup>z</sup>

*<sup>∂</sup><sup>x</sup>* ð Þ *<sup>x</sup>*, *<sup>z</sup> :* (46)

*∂U*<sup>x</sup>

It should be correctly recognized that vortex shedding from the edge tip

buildup of the positive trigger pulse when the trigger delay is not applied.

produced by the same first mode resonance.

*Vortex Dynamics Theories and Applications*

*4.1.3 Calculation of the acoustic generation formula*

measurement results, which are spared in this chapter).

*ω*yð Þ¼ *x*, *z*

Figure 17.6 of [1].

component [*<sup>ω</sup>* <sup>¼</sup> 0,*ω*y, 0 ]:

that are usually driven by thin jets.

**70**

*Aerodynamical quantities derived from the phase-locked PIV measurement at the organ pipe mouth: (a) vorticity ω; (b) aeroacoustical source term (acceleration) ω* � *U; (c) acoustic generation term* ð Þ *ω* � *U* **∙** *u. The positions of the flue exit and the edge tip correspond to (x, z) = (2.0 mm, 2.2 mm) and (x, z) = (10.5 mm, 3.8 mm), respectively [28].*

components of *ω* � *U*, respectively. The maximum magnitude of *ω* � *U*, which is about 2 � <sup>10</sup><sup>5</sup> <sup>m</sup>*=*s2 ð Þ, can be observed near the flue. The opposing vectors of *<sup>ω</sup>* � *<sup>U</sup>* along both layers seem to almost cancel each other as depicted in **Figure 10(b)**. However, since *ω* � *U* has a very large acceleration, an imperfect cancelation (or a slight unbalance) between *ω* � *U* vectors along both layers yields a significant effect in acoustical events.

#### *4.1.4 Generation of the acoustic power from the vortical field*

Acoustic generation term ð Þ� *ω* � *U u* defined by Eq. (14) is shown in **Figure 15(c)**. Since *u* indicates the outflow at Phases 1 and 4, and inflow at Phases 7 and 10, ð Þ� *ω* � *U u* takes the opposite sign along the vortex layer between these phases. This sign inversion occurs near Phases 0 and 6 [28]. The maximum magnitude of ð Þ� *ω* � *U u* appears near 1–2 millimeters downstream from the flue at Phases 3 and 8 as about

<sup>4</sup>*:*<sup>2</sup> � <sup>10</sup><sup>4</sup> <sup>m</sup>2*=*s<sup>3</sup> ð Þ. The jet crosses the edge from the inside at Phase 3 and from the outside at Phase 8 as inferred from **Figure 15(c)**. Although the magnitude of ð Þ� *ω* � *U u* is relatively small near the edge, it should be noted that ð Þ� *ω* � *U u* originally has very large values in acoustical sense.

Since the volume integral defined by Eq. (14) is not easily executed, the acoustic power generation from the vortex layer is estimated from the following surface integral by assuming the 2-D property (see Figure 11 in [28]) of *u* and *U*:

$$
\partial \Pi\_{\mathbf{G}}(\mathbf{t})/\partial \mathbf{y} \approx -\iint \rho(\mathbf{w} \times \mathbf{U}) \cdot \mathbf{u} \mathbf{d} \mathbf{x} \,\mathrm{d}z.\tag{47}
$$

at the edge side (see Figures 5(a) and 7 in [28]). It should be then discussed which

The area for the surface integral of Eq. (47) is now set to be 2 ≤*x*≤11 mm and 1≤*z*≤6 mm. Then, this area is divided into two at *x* ¼ 7, 8, and 9 mm. Hence, we have six sub-areas with the same *z* extent. The calculation result is demonstrated in Figure 12 of [28]. A very sharp contrast is displayed between area 5 (2 ≤*x*≤9 mm) and area 6 (9 <sup>≤</sup>*x*≤11 mm): Area 5 yields larger negative values of *<sup>∂</sup>Π*Gð Þ*<sup>t</sup> <sup>=</sup>∂<sup>y</sup>* at Phases 3, 4, and 5; area 6 yields much larger positive values of *<sup>∂</sup>Π*Gð Þ*<sup>t</sup> <sup>=</sup>∂<sup>y</sup>* at the same phases. Hence, it may be concluded that such a small area as area 6 (very close to the edge) is most responsible for the acoustic power generation whose instanta-

Also, the phase relation between the jet displacement, the acoustic velocity, and the acoustic pressure at the edge can be considered based on the PIV measurement results. The result is the same as **Figure 9(a)** (see Figure 13 in [28]). The dominant sound generation in our PIV experiment occurs with a phase lag of about 60 � 90° from the jet impingement against the edge [28]. This seems to verify that our

Coltman [52] discussed the activating force for the jet wavy motion. This is the most difficult problem in the flue instrument acoustics and is defined as the problem of the *receptivity* (the generation of jet oscillation by acoustic flow perturbations at the flue exit) [10]. Our present study based on the PIV measurement demonstrated that the aeroacoustic source term *ω* � *U* (having the dimension of the acceleration) associated with the vortex-layer formation along the jet could activate the jet oscillation in an organ pipe. More precisely, an incomplete cancelation (or a net unbalance) of *ω* � *U* between both sides of the jet can activate (oscillate) the jet. Since this *ω* � *U* can also activate the jet motion in the edge-tone generation [3–6, 18], the vortex-layer formation may be regarded as the fluid-dynamical mechanism common to the edge-tone generation and the pipe-tone generation. This fluid-dynamical model, which is a leading candidate to solve the receptivity problem, may be referred to as the *acceleration unbalance model* [28]. In the acoustical framework, this model can lead the volume-flow drive in an organ pipe driven by a thin jet with relatively low blowing pressures. Helmholtz might have envisaged

side is more dominant for the acoustic power generation.

*DOI: http://dx.doi.org/10.5772/intechopen.91258*

*Vortices on Sound Generation and Dissipation in Musical Flue Instruments*

neous contributions are given from Phases 2 to 5.

such a physical picture as mentioned in Section 2.4.

tivity problem in the near future.

**73**

**4.2 Vortices from the jet visualized by direct numerical simulations**

Sound generation in flue instruments is the revelation of the fluid *compressibility* in *low* Mach number state. This is a contradicting phenomenon in fluid-dynamical sense. Because of this, direct numerical simulations based on the Navier–Stokes equation could not achieve satisfactory outcomes [2]. However, in the 2010s we had many outstanding results from various viewpoints [53–55]. They are mentioning the roles of vortices, but do not have resolution enough to discuss the almost invisible (to our naked eyes) microstructure in the jet and its boundaries, particularly vortices in jet vortex layers, which seem to be a key point to solve the recep-

For the sake of page limitation, the description here is restricted to an essential point given by Eq. (24), which manifests the importance of the acoustic velocity *u*fð Þ*t* at the flue exit. By reformulating Fletcher's displacement model given by Eq. (4), Onogi et al*.* [56] proposed another formula that decomposed the jet oscillation into hydrodynamic and acoustic displacements, which were simulated on the basis of the 3-D compressible Navier–Stokes equations. They supposed the non-zero initial amplitude at the flue exit and the variable oscillation center with the flow direction for the jet displacement, although Coltman [52] strongly denied Fletcher's

experiment satisfies the requirements for the volume-flow model.

This surface integral, which may be called the *instantaneous 2-D vortex sound power*, is carried out at each phase, and the result is represented in **Figure 16(a)** and **(b)** concerning Trials 1 and 2, respectively. The integral area is restricted to 1≤*x*≤13 mm and 1≤*z*≤9 mm to reduce the calculation error caused from the area irrelevant to the acoustic generation term. Also, another scale of the ordinate is added to the right side of **Figure 16** in order to give a rough estimate of the magnitude of *Π*Gð Þ*t* . Since the mouth breadth *b* of our organ pipe is 31.9 mm, *<sup>Π</sup>*Gð Þ*<sup>t</sup>* is estimated as [*∂Π*Gð Þ*<sup>t</sup> <sup>=</sup>∂y*]�0*:*030 (m) by assuming almost perfect 2-D property of *u* and *U*.

Significant double-peak structure of *Π*<sup>G</sup> is clearly demonstrated in **Figure 16(a)**. A larger peak is indicated at Phases 2, 3, and 4 when the jet crosses the edge from the inside and moves to the outside (cf. **Figure 15**). On the other hand, a smaller peak is shown at Phases 10 and 11 when the jet enters deeply into the pipe. It should be noted that the jet crosses the edge from the outside at Phase 8 and almost null vortex sound power is generated at Phase 8. Hence, this smaller peak occurs in a little phase delay from the impingement of the jet against the edge. The temporal average of *Π*<sup>G</sup> estimated from the 2-D vortex sound power in **Figure 16(a)** will take a definitely positive value. Therefore, it may be recognized that the acoustic power is generated from the jet vortex layers. Although **Figure 16(b)** shows the characteristics similar to those of **Figure 16(a)**, the value at Phase 3 seems to be too large and erroneous because of the instant when the jet impinges against the edge [28].

#### *4.1.5 Dominant area for the acoustic power generation and receptivity problem*

The maps of the vorticity and aeroacoustical source term definitely indicate much larger magnitudes at the flue side as shown in **Figure 15(a)** and **(b)**, respectively. On the other hand, the acoustic flow velocity takes much larger magnitudes

#### **Figure 16.**

*The 2-D vortex sound power (left ordinate) and vortex sound power (right ordinate) as a function of the phase: (a) trial 1 and (b) trial 2 [28].*

#### *Vortices on Sound Generation and Dissipation in Musical Flue Instruments DOI: http://dx.doi.org/10.5772/intechopen.91258*

at the edge side (see Figures 5(a) and 7 in [28]). It should be then discussed which side is more dominant for the acoustic power generation.

The area for the surface integral of Eq. (47) is now set to be 2 ≤*x*≤11 mm and 1≤*z*≤6 mm. Then, this area is divided into two at *x* ¼ 7, 8, and 9 mm. Hence, we have six sub-areas with the same *z* extent. The calculation result is demonstrated in Figure 12 of [28]. A very sharp contrast is displayed between area 5 (2 ≤*x*≤9 mm) and area 6 (9 <sup>≤</sup>*x*≤11 mm): Area 5 yields larger negative values of *<sup>∂</sup>Π*Gð Þ*<sup>t</sup> <sup>=</sup>∂<sup>y</sup>* at Phases 3, 4, and 5; area 6 yields much larger positive values of *<sup>∂</sup>Π*Gð Þ*<sup>t</sup> <sup>=</sup>∂<sup>y</sup>* at the same phases. Hence, it may be concluded that such a small area as area 6 (very close to the edge) is most responsible for the acoustic power generation whose instantaneous contributions are given from Phases 2 to 5.

Also, the phase relation between the jet displacement, the acoustic velocity, and the acoustic pressure at the edge can be considered based on the PIV measurement results. The result is the same as **Figure 9(a)** (see Figure 13 in [28]). The dominant sound generation in our PIV experiment occurs with a phase lag of about 60 � 90° from the jet impingement against the edge [28]. This seems to verify that our experiment satisfies the requirements for the volume-flow model.

Coltman [52] discussed the activating force for the jet wavy motion. This is the most difficult problem in the flue instrument acoustics and is defined as the problem of the *receptivity* (the generation of jet oscillation by acoustic flow perturbations at the flue exit) [10]. Our present study based on the PIV measurement demonstrated that the aeroacoustic source term *ω* � *U* (having the dimension of the acceleration) associated with the vortex-layer formation along the jet could activate the jet oscillation in an organ pipe. More precisely, an incomplete cancelation (or a net unbalance) of *ω* � *U* between both sides of the jet can activate (oscillate) the jet.

Since this *ω* � *U* can also activate the jet motion in the edge-tone generation [3–6, 18], the vortex-layer formation may be regarded as the fluid-dynamical mechanism common to the edge-tone generation and the pipe-tone generation. This fluid-dynamical model, which is a leading candidate to solve the receptivity problem, may be referred to as the *acceleration unbalance model* [28]. In the acoustical framework, this model can lead the volume-flow drive in an organ pipe driven by a thin jet with relatively low blowing pressures. Helmholtz might have envisaged such a physical picture as mentioned in Section 2.4.

#### **4.2 Vortices from the jet visualized by direct numerical simulations**

Sound generation in flue instruments is the revelation of the fluid *compressibility* in *low* Mach number state. This is a contradicting phenomenon in fluid-dynamical sense. Because of this, direct numerical simulations based on the Navier–Stokes equation could not achieve satisfactory outcomes [2]. However, in the 2010s we had many outstanding results from various viewpoints [53–55]. They are mentioning the roles of vortices, but do not have resolution enough to discuss the almost invisible (to our naked eyes) microstructure in the jet and its boundaries, particularly vortices in jet vortex layers, which seem to be a key point to solve the receptivity problem in the near future.

For the sake of page limitation, the description here is restricted to an essential point given by Eq. (24), which manifests the importance of the acoustic velocity *u*fð Þ*t* at the flue exit. By reformulating Fletcher's displacement model given by Eq. (4), Onogi et al*.* [56] proposed another formula that decomposed the jet oscillation into hydrodynamic and acoustic displacements, which were simulated on the basis of the 3-D compressible Navier–Stokes equations. They supposed the non-zero initial amplitude at the flue exit and the variable oscillation center with the flow direction for the jet displacement, although Coltman [52] strongly denied Fletcher's

<sup>4</sup>*:*<sup>2</sup> � <sup>10</sup><sup>4</sup> <sup>m</sup>2*=*s<sup>3</sup> ð Þ. The jet crosses the edge from the inside at Phase 3 and from the outside at Phase 8 as inferred from **Figure 15(c)**. Although the magnitude of ð Þ� *ω* � *U u* is relatively small near the edge, it should be noted that ð Þ� *ω* � *U u*

ðð

This surface integral, which may be called the *instantaneous 2-D vortex sound power*, is carried out at each phase, and the result is represented in

**Figure 16(a)** and **(b)** concerning Trials 1 and 2, respectively. The integral area is restricted to 1≤*x*≤13 mm and 1≤*z*≤9 mm to reduce the calculation error caused from the area irrelevant to the acoustic generation term. Also, another scale of the ordinate is added to the right side of **Figure 16** in order to give a rough estimate of the magnitude of *Π*Gð Þ*t* . Since the mouth breadth *b* of our organ pipe is 31.9 mm, *<sup>Π</sup>*Gð Þ*<sup>t</sup>* is estimated as [*∂Π*Gð Þ*<sup>t</sup> <sup>=</sup>∂y*]�0*:*030 (m) by assuming almost perfect 2-D

Significant double-peak structure of *Π*<sup>G</sup> is clearly demonstrated in **Figure 16(a)**. A larger peak is indicated at Phases 2, 3, and 4 when the jet crosses the edge from the inside and moves to the outside (cf. **Figure 15**). On the other hand, a smaller peak is shown at Phases 10 and 11 when the jet enters deeply into the pipe. It should be noted that the jet crosses the edge from the outside at Phase 8 and almost null vortex sound power is generated at Phase 8. Hence, this smaller peak occurs in a little phase delay from the impingement of the jet against the edge. The temporal average of *Π*<sup>G</sup> estimated from the 2-D vortex sound power in **Figure 16(a)** will take a definitely positive value. Therefore, it may be recognized that the acoustic power is generated from the jet vortex layers. Although **Figure 16(b)** shows the characteristics similar to those of **Figure 16(a)**, the value at Phase 3 seems to be too large and erroneous because of the instant when the jet impinges against the edge [28].

*4.1.5 Dominant area for the acoustic power generation and receptivity problem*

The maps of the vorticity and aeroacoustical source term definitely indicate much larger magnitudes at the flue side as shown in **Figure 15(a)** and **(b)**, respectively. On the other hand, the acoustic flow velocity takes much larger magnitudes

*The 2-D vortex sound power (left ordinate) and vortex sound power (right ordinate) as a function of the phase:*

Since the volume integral defined by Eq. (14) is not easily executed, the acoustic power generation from the vortex layer is estimated from the following surface integral by assuming the 2-D property (see Figure 11 in [28]) of *u* and *U*:

*ρ*ð Þ� *ω* � *U u*d*x*d*z:* (47)

originally has very large values in acoustical sense.

*Vortex Dynamics Theories and Applications*

property of *u* and *U*.

**Figure 16.**

**72**

*(a) trial 1 and (b) trial 2 [28].*

*<sup>∂</sup>Π*Gð Þ*<sup>t</sup> <sup>=</sup>∂y*<sup>≈</sup> �

displacement model. Their simulation results (see Figures 7, 9, and 10 and Table IV in [56]) seem to confirm the non-zero amplitude at the flue exit, and the acoustic feedback effects on the jet wave may be given at its starting point.
