**2.1 Jet-drive model**

#### *2.1.1 Volume-flow drive vs. pressure (momentum) drive*

Jet motion in an organ pipe model when the jet drive is operating at the steady state is depicted in **Figure 2**. The air jet smoked with incense sticks is observed by a stroboscope and recorded on a VTR (8-mm video cassette) as analog data [2]. The pipe length *L* is 500 mm, the flue-to-edge distance (cutup length) *d* 10.2 mm, and the jet thickness *h* at the flue exit 2.2 mm [2, 17]. The blowing pressure is 200 Pa (the jet velocity at the flue exit is estimated from Bernoulli's law to be 18.3 m/s). The sounding frequency is 285 Hz. One period *T* of the jet motion is divided by 9 in **Figure 2**.

As shown in **Figure 2**, the jet oscillates up and down. It does not break into vortices but keeps a diaphragm-like shape in the jet-drive operation. Large vortexlike air observed above the edge will not take a part in the sound generation. It is a kind of odds and ends of the jet driving the pipe. Also, it should be noticed that the jet behaves like an amplifying wave as inferred from the first six frames [29].

When the jet enters the pipe passing through the mouth area between the flue exit and the edge, the jet provides the pipe with the acoustic volume flow *q*(*t*) that is roughly approximated by the product of jet velocity *U*e, jet breadth *b*, and jet lateral displacement *ξ*eð Þ*t* (these quantities are given at the edge):

$$q(t) \approx -U\_\mathbf{e} b \xi\_\mathbf{e}(t),\tag{1}$$

where *t* is the time. The minus sign is needed from the definition that *ξ*eð Þ*t* is positive outward and *q t*ð Þ is positive inward. Eq. (1) defines the volume-flow model that was first proposed by Helmholtz [30] and utilized by many researchers afterwards for small amplitudes of jet oscillation [1, 2, 10–13, 20, 21, 24–27]. At the same time, the jet provides the pipe with the acoustic pressure produced by the momentum exchange with still air in the pipe:

$$p(t) = \rho U\_\mathbf{e}^2(\mathbf{S}\_\mathbf{\dot{\beta}}/\mathbf{S}\_\mathbf{\dot{p}}), \mathbf{S}\_\mathbf{\dot{\beta}} \approx b\xi\_\mathbf{e}(t),\tag{2}$$

*Δx* below the pipe edge. The turbulent mixing takes place over this control volume. The loss of jet momentum there will result in the simple pressure rise at the inner plane of the control volume. The net force on the control volume due to this pressure rise can then be equated to the rate at which jet momentum changes in the control volume [24]. In other words, just as "the momentum difference equals to the force impulse," the momentum-flow-rate difference gives the force that

*Stroboscopically visualized jet oscillation at the steady state caused by the jet drive of an organ pipe model made*

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

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

accelerates the mass of the control volume.

**Figure 2.**

**47**

*with the acrylic resin.*

where *ρ* is the air density, *S*<sup>j</sup> the temporally varying cross section of the jet entering the pipe from the edge, and *S*<sup>p</sup> the pipe cross section. Eq. (2) defines the jet pressure of the jet momentum model, which was first proposed by Rayleigh [19] and utilized by many researchers afterwards [1, 2, 10–13, 21–27]. Opposing Helmholtz, Rayleigh insisted that the momentum drive should be effective. This is based on that the pipe is open and the acoustic power is produced by the product of the acoustic particle velocity near the pipe edge and the driving pressure given by the jet. However, the acoustic pressure considerably remains near the pipe edge due to the end correction. As a result, the volume-flow drive of Helmholtz is usually predominant except for the jet drive with very high blowing pressures [1, 2, 10, 21, 24, 25, 27].

The jet-drive model based on the volume-flow drive and the pressure drive was first formulated by Elder [24] by deriving the so-called jet momentum equation and then simplified by Fletcher [25]. They assumed a small control volume with length

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

**Figure 2.**

Section 4, and some examples of jet vortices are also introduced from experiments

Jet motion in an organ pipe model when the jet drive is operating at the steady state is depicted in **Figure 2**. The air jet smoked with incense sticks is observed by a stroboscope and recorded on a VTR (8-mm video cassette) as analog data [2]. The pipe length *L* is 500 mm, the flue-to-edge distance (cutup length) *d* 10.2 mm, and the jet thickness *h* at the flue exit 2.2 mm [2, 17]. The blowing pressure is 200 Pa (the jet velocity at the flue exit is estimated from Bernoulli's law to be 18.3 m/s). The sounding frequency is 285 Hz. One period *T* of the jet motion is divided by 9 in

As shown in **Figure 2**, the jet oscillates up and down. It does not break into vortices but keeps a diaphragm-like shape in the jet-drive operation. Large vortexlike air observed above the edge will not take a part in the sound generation. It is a kind of odds and ends of the jet driving the pipe. Also, it should be noticed that the jet behaves like an amplifying wave as inferred from the first six frames [29]. When the jet enters the pipe passing through the mouth area between the flue exit and the edge, the jet provides the pipe with the acoustic volume flow *q*(*t*) that is roughly approximated by the product of jet velocity *U*e, jet breadth *b*, and jet lateral

where *t* is the time. The minus sign is needed from the definition that *ξ*eð Þ*t* is positive outward and *q t*ð Þ is positive inward. Eq. (1) defines the volume-flow model that was first proposed by Helmholtz [30] and utilized by many researchers afterwards for small amplitudes of jet oscillation [1, 2, 10–13, 20, 21, 24–27]. At the same time, the jet provides the pipe with the acoustic pressure produced by the momen-

<sup>e</sup> *S*j*=S*<sup>p</sup>

where *ρ* is the air density, *S*<sup>j</sup> the temporally varying cross section of the jet entering the pipe from the edge, and *S*<sup>p</sup> the pipe cross section. Eq. (2) defines the jet pressure of the jet momentum model, which was first proposed by Rayleigh [19] and utilized by many researchers afterwards [1, 2, 10–13, 21–27]. Opposing Helmholtz, Rayleigh insisted that the momentum drive should be effective. This is based on that the pipe is open and the acoustic power is produced by the

product of the acoustic particle velocity near the pipe edge and the driving pressure given by the jet. However, the acoustic pressure considerably remains near the pipe edge due to the end correction. As a result, the volume-flow drive of Helmholtz is usually predominant except for the jet drive with very high blowing pressures

The jet-drive model based on the volume-flow drive and the pressure drive was first formulated by Elder [24] by deriving the so-called jet momentum equation and then simplified by Fletcher [25]. They assumed a small control volume with length

*q t*ð Þ≈ � *U*e*bξ*eð Þ*t* , (1)

, *<sup>S</sup>*<sup>j</sup> <sup>≈</sup> *<sup>b</sup>ξ*eð Þ*<sup>t</sup>* , (2)

and simulations. Conclusions are given in Section 5.

*Vortex Dynamics Theories and Applications*

*2.1.1 Volume-flow drive vs. pressure (momentum) drive*

**2.1 Jet-drive model**

**Figure 2**.

**2. Models on sound generation in flue instruments**

displacement *ξ*eð Þ*t* (these quantities are given at the edge):

*p t*ðÞ¼ *<sup>ρ</sup>U*<sup>2</sup>

tum exchange with still air in the pipe:

[1, 2, 10, 21, 24, 25, 27].

**46**

*Stroboscopically visualized jet oscillation at the steady state caused by the jet drive of an organ pipe model made with the acrylic resin.*

*Δx* below the pipe edge. The turbulent mixing takes place over this control volume. The loss of jet momentum there will result in the simple pressure rise at the inner plane of the control volume. The net force on the control volume due to this pressure rise can then be equated to the rate at which jet momentum changes in the control volume [24]. In other words, just as "the momentum difference equals to the force impulse," the momentum-flow-rate difference gives the force that accelerates the mass of the control volume.

It should be noticed that there is an appreciable phase difference between the volume-flow drive and the pressure drive. This phase difference is not well understood from Eqs. (1) and (2). The acoustic impedance or admittance should be introduced to connect these equations. According to Fletcher [1], this phase difference *ϕ*, which gives the phase lag of the pressure drive, is given by:

$$\phi = -\tan^{-1}(U\_{\text{e}}/a\Delta L),\tag{3}$$

Since the cos *ωx=U*ph dependence diminishes as *μx* becomes larger than one, it may be reasonable to approximate the second term in square bracket of Eq. (5) as

**Figure 3** shows the transverse displacements of the jet oscillation and their approximated envelopes (indicated by the broken line) at the steady state which are calculated by Eqs. (4) and (6), respectively. The following parameter values are

slowdown of *U*ph is assumed so that *k* may be proportional to *x* in order to make

The jet displacements from *t* = (3/10)*T* to *t* = (8/10)*T* in **Figure 3** almost correspond to those from *t* = (2/9)*T* to *t* = (7/9)*T* in **Figure 2**, though the flue-toedge distance *d* is not the same (*d* = 10.2 mm in **Figure 2** and *d* = 15.8 mm in **Figure 3**). This good correspondence proves the effectiveness of the displacement

It is possible to directly estimate *μ* from Eq. (4) by applying it to the experimental data. However, such an approach needs exact information about *U*ph and a time reference. Another much simpler method to estimate *μ* is to apply Eq. (6) to the experimental data. The result is shown in **Figure 4** for the first mode of an organ pipe model with *L* = 50.0 cm and *d* = 10.2 mm (cf. **Figure 2**) [29]. If a high-speed digital video camera is used instead of a stroboscope, the frames showing the jet waves such as given in **Figure 2** are digitally memorized, and then the digital

*ξ*envð Þ *x* ≈ � ð Þ *u=ω* ½ � cosh ð Þ� *μx* 1 *:* (6)

), *ω*=2π ∙ 133

), *U*phð Þ¼ *x* ¼ 0 *U*<sup>0</sup> ¼ 11*:*5 (m/s). The

2 cosh ð Þ *μx* . We then obtain the following simple expression [29]:

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

supposed: the mouth-field strength *<sup>u</sup>=<sup>ω</sup>* <sup>¼</sup> 1 (mm), *<sup>μ</sup>* <sup>¼</sup> <sup>0</sup>*:*20 (mm�<sup>1</sup>

superposition of these data yields a direct superposition of jet waves.

*The jet displacements calculated at the instants from 3 T/10 to 8 T/10 and their approximated envelope*

calculation easier. For more detailed information, refer to [29].

(rad/s), *<sup>k</sup>* <sup>¼</sup> *<sup>ω</sup>=U*ph <sup>¼</sup> <sup>0</sup>*:*<sup>073</sup> <sup>þ</sup> <sup>0</sup>*:*01*<sup>x</sup>* (mm�<sup>1</sup>

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

model based on Eq. (4).

**Figure 3.**

**49**

*(the negative envelope is also indicated).*

where *ω* is the angular frequency and Δ*L* the effective mouth length including the open end correction. Since *ω*Δ*L*> *U*<sup>e</sup> in usual cases, *ϕ* is quite small. However, as mentioned above, *ϕ* becomes appreciably large when the high blowing pressure is applied. See [1, 2, 10, 24–27] for more discussion on the complex jet/pipe interaction in flue instruments and on the conditions of the phase and amplitude for sound regeneration.

#### *2.1.2 Jet wave and its amplification*

Each frame in **Figure 2** does not show the path of air particle, but corresponds to the snapshot of the position of air particle at a given instant. Therefore, each frame indicates the *streak line* in the fluid-dynamical sense. On the other hand, the jet particle pass (*pass line*) may be determined as soon as the jet issues from the flue to the acoustic field in the mouth [26]. If the jet pass line can be determined, the jet deflection shape as the streak line may be estimated by considering the transit time of the particle issued from the flue exit [2, 26]. However, in the field of musical acoustics, the image visualizing the jet oscillation at the steady state in **Figure 2** has been called the *jet wave*, which seems to give the transverse displacement (in *z* direction) of the jet (cf. **Figure 1**).

It is assumed that the jet displacement may be expressed as a superposition of a progressive wave due to the jet instability [19, 20] and a spatially uniform oscillation induced by acoustic velocity *u* through the mouth as follows [1, 31]:

$$\xi(\varkappa, t) = (\varkappa/\alpha) \left\{-\cosh\left(\mu \varkappa\right) \sin\left[\mu \left(t - \varkappa/U\_{\text{ph}}\right)\right] + \sin\left(at\right)\right\},\tag{4}$$

where *ξ*ð Þ *x*, *t* denotes the transverse displacement of the jet at distance *x* from the flue exit. Also, *μ* and *U*ph denote the amplification factor and phase speed of the instability wave, respectively, while *u* is the transverse acoustic velocity of the mouth field. The jet displacement *ξ*eð Þ*t* in Eqs. (1) and (2) is given by *ξ*ð Þ *x* ¼ *d*, *t* . It should be noted that *ξ*ð Þ *x* and *u* are positive in the external (positive *z*) direction. Both *μ* and *U*ph are functions of *x* in a rigorous sense. Also, it is important that *ω* in Eq. (4) corresponds to the fundamental of a sound generated. When very high blowing pressures are applied, the effect of the second harmonic is significant and the jet strikes the edge downward twice a period. Such an effect is excluded in Eq. (4). Also, an important parameter, jet thickness *h* is not included in Eq. (4), and a thin jet (*d*/*h* > 2) is assumed. Although Eq. (4) lacks its physical basis and experimental confirmation, it is a simple and practical representation of the jet deflection that describes our current knowledge. So, we postulate in this chapter that Eq. (4) is valid in the acoustical sense.

The envelope of positive and negative peak displacements is yielded from Eq. (4) as follows:

$$\xi\_{\rm env}(\mathbf{x}) = \pm (\boldsymbol{\mu}/\boldsymbol{\omega}) \left[ \cosh^2(\boldsymbol{\mu x}) - 2 \cosh \left( \boldsymbol{\mu x} \right) \cos \left( \boldsymbol{\alpha x}/U\_{\rm ph} \right) + 1 \right]^{1/2}.\tag{5}$$

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

Since the cos *ωx=U*ph dependence diminishes as *μx* becomes larger than one, it may be reasonable to approximate the second term in square bracket of Eq. (5) as 2 cosh ð Þ *μx* . We then obtain the following simple expression [29]:

$$\xi\_{\rm env}(\boldsymbol{\kappa}) \approx \pm \left(\boldsymbol{u}/\boldsymbol{\alpha}\right) [\cosh\left(\boldsymbol{\mu}\boldsymbol{\kappa}\right) - \mathbf{1}].\tag{6}$$

**Figure 3** shows the transverse displacements of the jet oscillation and their approximated envelopes (indicated by the broken line) at the steady state which are calculated by Eqs. (4) and (6), respectively. The following parameter values are supposed: the mouth-field strength *<sup>u</sup>=<sup>ω</sup>* <sup>¼</sup> 1 (mm), *<sup>μ</sup>* <sup>¼</sup> <sup>0</sup>*:*20 (mm�<sup>1</sup> ), *ω*=2π ∙ 133 (rad/s), *<sup>k</sup>* <sup>¼</sup> *<sup>ω</sup>=U*ph <sup>¼</sup> <sup>0</sup>*:*<sup>073</sup> <sup>þ</sup> <sup>0</sup>*:*01*<sup>x</sup>* (mm�<sup>1</sup> ), *U*phð Þ¼ *x* ¼ 0 *U*<sup>0</sup> ¼ 11*:*5 (m/s). The slowdown of *U*ph is assumed so that *k* may be proportional to *x* in order to make calculation easier. For more detailed information, refer to [29].

The jet displacements from *t* = (3/10)*T* to *t* = (8/10)*T* in **Figure 3** almost correspond to those from *t* = (2/9)*T* to *t* = (7/9)*T* in **Figure 2**, though the flue-toedge distance *d* is not the same (*d* = 10.2 mm in **Figure 2** and *d* = 15.8 mm in **Figure 3**). This good correspondence proves the effectiveness of the displacement model based on Eq. (4).

It is possible to directly estimate *μ* from Eq. (4) by applying it to the experimental data. However, such an approach needs exact information about *U*ph and a time reference. Another much simpler method to estimate *μ* is to apply Eq. (6) to the experimental data. The result is shown in **Figure 4** for the first mode of an organ pipe model with *L* = 50.0 cm and *d* = 10.2 mm (cf. **Figure 2**) [29]. If a high-speed digital video camera is used instead of a stroboscope, the frames showing the jet waves such as given in **Figure 2** are digitally memorized, and then the digital superposition of these data yields a direct superposition of jet waves.

#### **Figure 3.**

*The jet displacements calculated at the instants from 3 T/10 to 8 T/10 and their approximated envelope (the negative envelope is also indicated).*

It should be noticed that there is an appreciable phase difference between the volume-flow drive and the pressure drive. This phase difference is not well understood from Eqs. (1) and (2). The acoustic impedance or admittance should be introduced to connect these equations. According to Fletcher [1], this phase difference *ϕ*, which gives the phase lag of the pressure drive, is given by:

where *ω* is the angular frequency and Δ*L* the effective mouth length including the open end correction. Since *ω*Δ*L*> *U*<sup>e</sup> in usual cases, *ϕ* is quite small. However, as mentioned above, *ϕ* becomes appreciably large when the high blowing pressure is applied. See [1, 2, 10, 24–27] for more discussion on the complex jet/pipe interaction in flue instruments and on the conditions of the phase and amplitude for sound

Each frame in **Figure 2** does not show the path of air particle, but corresponds to the snapshot of the position of air particle at a given instant. Therefore, each frame indicates the *streak line* in the fluid-dynamical sense. On the other hand, the jet particle pass (*pass line*) may be determined as soon as the jet issues from the flue to the acoustic field in the mouth [26]. If the jet pass line can be determined, the jet deflection shape as the streak line may be estimated by considering the transit time of the particle issued from the flue exit [2, 26]. However, in the field of musical acoustics, the image visualizing the jet oscillation at the steady state in **Figure 2** has been called the *jet wave*, which seems to give the transverse displacement

It is assumed that the jet displacement may be expressed as a superposition of

where *ξ*ð Þ *x*, *t* denotes the transverse displacement of the jet at distance *x* from the flue exit. Also, *μ* and *U*ph denote the amplification factor and phase speed of the instability wave, respectively, while *u* is the transverse acoustic velocity of the mouth field. The jet displacement *ξ*eð Þ*t* in Eqs. (1) and (2) is given by *ξ*ð Þ *x* ¼ *d*, *t* . It should be noted that *ξ*ð Þ *x* and *u* are positive in the external (positive *z*) direction. Both *μ* and *U*ph are functions of *x* in a rigorous sense. Also, it is important that *ω* in Eq. (4) corresponds to the fundamental of a sound generated. When very high blowing pressures are applied, the effect of the second harmonic is significant and the jet strikes the edge downward twice a period. Such an effect is excluded in Eq. (4). Also, an important parameter, jet thickness *h* is not included in Eq. (4), and a thin jet (*d*/*h* > 2) is assumed. Although Eq. (4) lacks its physical basis and experimental confirmation, it is a simple and practical representation of the jet deflection that describes our current knowledge. So, we postulate in this chapter

The envelope of positive and negative peak displacements is yielded from

ð Þ� *μx* 2 cosh ð Þ *μx* cos *ωx=U*ph <sup>þ</sup> <sup>1</sup> <sup>1</sup>*=*<sup>2</sup>

*:* (5)

<sup>þ</sup> sin ð Þ *<sup>ω</sup><sup>t</sup>* , (4)

a progressive wave due to the jet instability [19, 20] and a spatially uniform oscillation induced by acoustic velocity *u* through the mouth as follows [1, 31]:

*ξ*ð Þ¼ *x*, *t* ð Þ� *u=ω* cosh ð Þ *μx* sin *ω t* � *x=U*ph

ð Þ *U*e*=ω*Δ*L* , (3)

*<sup>ϕ</sup>* ¼ � tan �<sup>1</sup>

regeneration.

*2.1.2 Jet wave and its amplification*

*Vortex Dynamics Theories and Applications*

(in *z* direction) of the jet (cf. **Figure 1**).

that Eq. (4) is valid in the acoustical sense.

*<sup>ξ</sup>*envð Þ¼� *<sup>x</sup>* ð Þ *<sup>u</sup>=<sup>ω</sup>* cosh <sup>2</sup>

Eq. (4) as follows:

**48**

**Figure 4.**

*Digitally superposed jet waves for different blowing pressures (a)–(c) and an illustration of how to derive the amplification factor μ and the mouth-field strength u=ω (d). The envelope of jet center-planes is estimated by the red dotted line that best fits to the green template curve with μ = 0.24 mm*�*<sup>1</sup> and u=ω = 1. The amplitude ratio of the dotted line to the green line determines u=ω = 1.3 mm.*

Digitally superposed jet waves are shown in frames (a), (b), and (c) for different blowing pressures in **Figure 4**, where the *x* axis is drawn straightforwardly from the flue center labeled as "0," and label "*e*" indicates the edge position. Frame (d) on the right of **Figure 4** illustrates how the jet envelope function is derived. The envelope of jet center-planes (indicated by dotted lines in **Figure 4**) is almost parallel to the outer fringe of the superposed jet waves as long as significant spreading of the jet can be neglected by using thin smokes for visualization. More details on the estimation of *μ* from flow visualization are given in [29].

jet flow for the measurement of *u* because *u* ≪ *U*0. Therefore, after carrying out a preliminary experiment to select proper positions of the probe, the rms acoustic particle velocity *u*rms was measured as the rms output voltage *V*rms of the hot-wire

*The amplification factor μ and the mouth-field strength u=ω of the organ pipe jet as functions of the jet velocity <sup>U</sup>*<sup>0</sup> *at the flue exit [29]. The symbols* ○ *and* □ *indicate <sup>μ</sup> and u=<sup>ω</sup> estimated from the digital superposition based on the jet visualization, respectively. The symbol* ■ *indicates u=<sup>ω</sup> measured by a hot-wire anemometer*

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

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

Comparing the values of *u=ω* measured using the hot-wire anemometer with those estimated from flow visualization, we may see a good agreement between them. This agreement implies that the method of deriving the envelope function of the jet wave is valid and sufficiently accurate. However, we had particular difficulty in obtaining a smooth jet-wave envelope near saturation, and the estimated data of *μ* and *u=ω* were lacking in **Figure 5**. This was due to other jet waves which were generated by the second harmonic and superposed upon the jet waves generated by

The origin of the jet-wave amplification is the jet instability. The applicability of the spatial and temporal theories on the jet instability [1, 32–34] to organ pipe jets can be discussed. If we assume a Poiseuille flow at the flue exit and a subsequent Bickley jet, the spatial theory [32, 33] seems to be relevant to organ pipe jets [29].

The jet-drive model described above has supposed small displacements of the jet at the pipe edge. However, as demonstrated in **Figures 2**–**4**, the jet displacement is too large to apply Eqs. (1) and (2) to the sound generation in flue instruments in rigorous sense. According to Dequand et al*.* [21] and Verge et al*.* [27], a jet-drive model reasonable for large jet displacements is explained and roughly formulated

As understood from **Figures 2** and **3**, the passage time of the jet from one side to the other side of the edge seems to be very short compared to the oscillation period. In other words, the jet seems to be instantaneously switching from the inside to the outside of the pipe and vice versa. Then, the jet volume flow may be assumed to be split into two complementary antiphase monopole sources *q*in [¼ j j *q* exp ð Þ *iωt* )] and *q*out (¼ �*q*in) whose temporal waveforms are rectangular pulses with the same

<sup>p</sup> *<sup>u</sup>*rms) [29]. The result of *<sup>u</sup>=<sup>ω</sup>* is indicated by the closed square

anemometer (*<sup>u</sup>* <sup>¼</sup> ffiffi

the fundamental.

below.

**51**

in **Figure 5**.

*and a microphone.*

**Figure 5.**

2

*2.1.3 Jet-drive model for large jet displacements*

In **Figure 4(d)** the estimated envelope is shown by dots on a template, that is, curves of jet envelope function cosh ð Þ� *μx* 1 for various *μ* values assuming that *u=ω* = 1. **Figure 4(d)** suggests a close fit of the dotted line to an envelope curve with *μ* = 0.24 mm�<sup>1</sup> when the magnitude ratio of the dotted line to that curve is about 1.3. This ratio determines *u=ω* = 1.3 mm from Eq. (6).

Flow visualization suggests the following general trends from the result summarized in **Figure 5** on a particular experimental model of the organ pipe:


It should be noted here that the estimate of *μ* based on Eq. (6) tends to be a little larger (about 10%) than that based on Eq. (4). However, this estimation error is roughly equivalent to the resolution of the experimental data [29].

In order to confirm the validity of our digital superposition explained above, *u=ω* was determined from measurements of the acoustic particle velocity *u* with a hot-wire anemometer (its sensing part is 1-mm long and 5 μm in diameter) and of the sounding frequency *ω=*2*π* (about 280 Hz in the first mode) with a microphone located inside the pipe. It is important to avoid exposing the hot-wire probe to the

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

#### **Figure 5.**

Digitally superposed jet waves are shown in frames (a), (b), and (c) for different blowing pressures in **Figure 4**, where the *x* axis is drawn straightforwardly from the flue center labeled as "0," and label "*e*" indicates the edge position. Frame (d) on the right of **Figure 4** illustrates how the jet envelope function is derived. The envelope of jet center-planes (indicated by dotted lines in **Figure 4**) is almost parallel to the outer fringe of the superposed jet waves as long as significant spreading of the jet can be neglected by using thin smokes for visualization. More

*Digitally superposed jet waves for different blowing pressures (a)–(c) and an illustration of how to derive the amplification factor μ and the mouth-field strength u=ω (d). The envelope of jet center-planes is estimated by the red dotted line that best fits to the green template curve with μ = 0.24 mm*�*<sup>1</sup> and u=ω = 1. The amplitude*

details on the estimation of *μ* from flow visualization are given in [29].

rized in **Figure 5** on a particular experimental model of the organ pipe:

2.The averaged amplification factor is roughly estimated as 0.24 mm�<sup>1</sup>

roughly equivalent to the resolution of the experimental data [29].

3.The mouth-field strength *u=ω*, which means the displacement amplitude of the acoustic field at the mouth, tends to increase and saturate to a given value as the oscillation of each mode shifts toward higher blowing velocities.

It should be noted here that the estimate of *μ* based on Eq. (6) tends to be a little larger (about 10%) than that based on Eq. (4). However, this estimation error is

In order to confirm the validity of our digital superposition explained above, *u=ω* was determined from measurements of the acoustic particle velocity *u* with a hot-wire anemometer (its sensing part is 1-mm long and 5 μm in diameter) and of the sounding frequency *ω=*2*π* (about 280 Hz in the first mode) with a microphone located inside the pipe. It is important to avoid exposing the hot-wire probe to the

1.3. This ratio determines *u=ω* = 1.3 mm from Eq. (6).

*ratio of the dotted line to the green line determines u=ω = 1.3 mm.*

*Vortex Dynamics Theories and Applications*

**Figure 4.**

**50**

data on the second mode are not sufficient.

In **Figure 4(d)** the estimated envelope is shown by dots on a template, that is, curves of jet envelope function cosh ð Þ� *μx* 1 for various *μ* values assuming that *u=ω* = 1. **Figure 4(d)** suggests a close fit of the dotted line to an envelope curve with *μ* = 0.24 mm�<sup>1</sup> when the magnitude ratio of the dotted line to that curve is about

Flow visualization suggests the following general trends from the result summa-

1.The amplification factor *μ* tends to decrease and saturate to a given value as the oscillation of each mode shifts toward higher blowing velocities, although the

.

*The amplification factor μ and the mouth-field strength u=ω of the organ pipe jet as functions of the jet velocity <sup>U</sup>*<sup>0</sup> *at the flue exit [29]. The symbols* ○ *and* □ *indicate <sup>μ</sup> and u=<sup>ω</sup> estimated from the digital superposition based on the jet visualization, respectively. The symbol* ■ *indicates u=<sup>ω</sup> measured by a hot-wire anemometer and a microphone.*

jet flow for the measurement of *u* because *u* ≪ *U*0. Therefore, after carrying out a preliminary experiment to select proper positions of the probe, the rms acoustic particle velocity *u*rms was measured as the rms output voltage *V*rms of the hot-wire anemometer (*<sup>u</sup>* <sup>¼</sup> ffiffi 2 <sup>p</sup> *<sup>u</sup>*rms) [29]. The result of *<sup>u</sup>=<sup>ω</sup>* is indicated by the closed square in **Figure 5**.

Comparing the values of *u=ω* measured using the hot-wire anemometer with those estimated from flow visualization, we may see a good agreement between them. This agreement implies that the method of deriving the envelope function of the jet wave is valid and sufficiently accurate. However, we had particular difficulty in obtaining a smooth jet-wave envelope near saturation, and the estimated data of *μ* and *u=ω* were lacking in **Figure 5**. This was due to other jet waves which were generated by the second harmonic and superposed upon the jet waves generated by the fundamental.

The origin of the jet-wave amplification is the jet instability. The applicability of the spatial and temporal theories on the jet instability [1, 32–34] to organ pipe jets can be discussed. If we assume a Poiseuille flow at the flue exit and a subsequent Bickley jet, the spatial theory [32, 33] seems to be relevant to organ pipe jets [29].

#### *2.1.3 Jet-drive model for large jet displacements*

The jet-drive model described above has supposed small displacements of the jet at the pipe edge. However, as demonstrated in **Figures 2**–**4**, the jet displacement is too large to apply Eqs. (1) and (2) to the sound generation in flue instruments in rigorous sense. According to Dequand et al*.* [21] and Verge et al*.* [27], a jet-drive model reasonable for large jet displacements is explained and roughly formulated below.

As understood from **Figures 2** and **3**, the passage time of the jet from one side to the other side of the edge seems to be very short compared to the oscillation period. In other words, the jet seems to be instantaneously switching from the inside to the outside of the pipe and vice versa. Then, the jet volume flow may be assumed to be split into two complementary antiphase monopole sources *q*in [¼ j j *q* exp ð Þ *iωt* )] and *q*out (¼ �*q*in) whose temporal waveforms are rectangular pulses with the same

amplitude j j *q* [27]. These sources are supposed to be placed at a distance *ϵ* from the edge tip at the lower and upper sides of the edge.

The acoustic pressure *p*<sup>j</sup> ð Þ*t* is derived from the potential difference across the mouth induced by two monopole sources [21, 27]:

$$p\_j(t) = -\rho \left(\delta\_l / bd\right) (\mathbf{d}q / \mathbf{d}t),\tag{7}$$

*I t*ðÞ¼ <sup>ð</sup>

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

vortical field around the edge is given by:

Eq. (10.42) in [18].

**Figure 6.**

**53**

*instruments proposed by Howe [18].*

divð Þ *ω* � *v ϕ y*

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

the integral of Eq. (10) is performed. The specific form of *ϕ y*

Howe [3, 35] then proposed the *acoustic dissipation formula*:

*Π*Dð Þ*t* ≈

Hence, the *acoustic generation formula* may be proposed:

*Π*Gð Þ*t* ≈ �

mouth opening (see **Figure 6**). That is, *v* is given by:

� �d<sup>3</sup>

*<sup>y</sup>* ¼ � <sup>ð</sup>

where the vorticity *ω* is defined as rot *v* and the velocity *v* is the superposition of the jet mean flow velocity *U* directing against the edge and the time-dependent cross-flow velocity *u* that is specified by reciprocating potential flow through the

Also, *y* denotes the source region (vortical field) of the pipe mouth over which

Therefore, if ð Þ *ω* � *v* ∙ *u* **>** 0, it may be said that the energy of the acoustic field is absorbed by the vortical field. As a result, an *acoustically induced vortex* or *acoustic vortex* is generated [16, 17]. The generation of it satisfies the phase relation in which a clockwise-rotating vortex appears above the edge when *u* directs into the pipe [see **Figure 12(b)** in Section 3.2.1]. This phase relation is contrary to that illustrated in **Figure 6**. A discrete vortex of Howe's type has not been observed in sound generation of flue instruments where jets are used in normal conditions [16, 17, 20–27, 29].

This equation determines the rate of dissipation of acoustic energy, where ∇*ϕ* in Eq. (10) is now simply denoted by *u*. Also, *V* denotes a volume enclosing the vorticity formed in the flow field. This *Π*<sup>D</sup> of Eq. (13) can be negative in oscillation systems: If the phase of vorticity production enables a steady transfer of energy to the oscillation from a mean flow, the self-sustained oscillation can be maintained.

*Conceptual sketch of a discrete-vortex model (the vortex shedding at the edge) for sound generation in flue*

According to [3], the power density supplied from the acoustic field to the

ð Þ *ω* � *v* ∙ ∇*ϕ y*

*v* ¼ *U* þ *u:* (11)

ð Þ <sup>1</sup>*=*<sup>2</sup> ð Þ *<sup>∂</sup>=∂<sup>t</sup> <sup>ρ</sup>v*<sup>2</sup> � � <sup>¼</sup> *<sup>ρ</sup>*ð Þ *<sup>ω</sup>* � *<sup>v</sup>* <sup>∙</sup> *<sup>u</sup>:* (12)

ððð *<sup>ρ</sup>*ð Þ *<sup>ω</sup>* � *<sup>v</sup>* <sup>∙</sup> *<sup>u</sup>*d*V*, *<sup>v</sup>*<sup>≈</sup> *<sup>U</sup>:* (13)

ððð *<sup>ρ</sup>*ð Þ *<sup>ω</sup>* � *<sup>U</sup>* <sup>∙</sup> *<sup>u</sup>*d*V*, *<sup>ω</sup>* <sup>¼</sup> rot*U*, (14)

� �d<sup>3</sup>

*y*, (10)

� � is given as *ϕ*<sup>∗</sup>

<sup>2</sup> *y* � � of

where *ρ* is the air density, *b* the jet (or mouth) breadth, *d* the flue-to-edge distance (or the mouth length), and *δ*<sup>j</sup> the effective distance between the two monopole sources. If *ϵ* ≪ *d*, we have from [27].

$$
\delta\_{\parallel}/d \approx (4/\pi)\sqrt{2\epsilon/d}.\tag{8}
$$

In the limit of thin jets ð Þ *d=h* ≫ 1 , where *h* denotes the jet thickness at the flue exit, ϵ ¼ *h*<sup>e</sup> (the jet thickness at the edge) is assumed [27].

The power generated by the source is calculated by assuming that the source is in phase with the acoustic volume flow *q*mðÞ¼ *t* ð Þ d*ξ*m*=*d*t db* ¼ *udb* through the mouth opening, where *ξ*<sup>m</sup> denotes the particle displacement over the mouth opening [21]. This *q*<sup>m</sup> is supposed to be a local two-dimensional incompressible flow. The above in-phase relation between the pressure source *p*<sup>j</sup> and the acoustic volume flow *q*<sup>m</sup> gives the condition for which the oscillation amplitude has a maximum as a function of the blowing pressure.

The time average over an oscillation period *T* of the power generated by the jet drive above is given as follows [21]:

$$
\langle \langle \Pi\_{\rm jet} \rangle = \left\langle p\_{\rm j} q\_{\rm m} \right\rangle \approx (8/\pi T) \rho \sqrt{2h\_{\rm e}/d} U\_0 h\_{\rm e} bd \, |\mathbf{d}\xi\_{\rm m}/\mathbf{d}t|.\tag{9}
$$

In addition to the thin jet assumption ð Þ *d=h* ≫ 1 , we have to suppose that the jet does not break down into discrete vortices. This is only reasonable for the first hydrodynamic mode (*S*<sup>t</sup> ¼ *fd=U*<sup>0</sup> <0*:*3). The validity of Eq. (9) will be discussed in Section 3 after deriving the acoustic energy loss due to vortex shedding at the edge.

#### **2.2 Discrete-vortex model**

#### *2.2.1 Discrete-vortex model based on the vortex shedding at the edge*

On the basis of the two-dimensional theory, Howe [18] proposed a discretevortex model on sound generation in flute-like instruments. He assumed that a compact vortex core appearing alternately just above and beneath the edge was created by the interaction with the acoustic cross-flow velocity *u* [d*ξ*m*=*d*t* in Eq. (9) corresponds to one-dimensional (*z* direction) component] at the mouth opening (see **Figure 6**). That is, instead of the jet oscillation over the mouth explained in the previous section, a point vortex is produced at the edge. Then, this vortex core is assumed to drive the air column in the pipe. A discrete-vortex model for thick jets assumes that a discrete vortex is generated from the flow separation at the flue exit corner [9, 10, 21], while Howe [18] attached greater importance to the flow separation (vortex shedding) at an opposing sharp edge due to the acoustic cross-flow.

The sound excitation by the periodic vortex shedding at the edge is controlled by the product of the aeroacoustic source term divð Þ *ω* � *v* and the potential function *ϕ y* � � representing the irrotational cross-flow into and out of the mouth as expressed by the following integral [18]:

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

$$I(t) = \int \text{div}(\boldsymbol{\omega} \times \boldsymbol{\nu}) \phi(\boldsymbol{\jmath}) \mathbf{d}^3 \boldsymbol{\jmath} = -\int (\boldsymbol{\omega} \times \boldsymbol{\nu}) \cdot \nabla \phi(\boldsymbol{\jmath}) \mathbf{d}^3 \boldsymbol{\jmath},\tag{10}$$

where the vorticity *ω* is defined as rot *v* and the velocity *v* is the superposition of the jet mean flow velocity *U* directing against the edge and the time-dependent cross-flow velocity *u* that is specified by reciprocating potential flow through the mouth opening (see **Figure 6**). That is, *v* is given by:

$$
\boldsymbol{\nu} = \boldsymbol{U} + \boldsymbol{\mu}.\tag{11}
$$

Also, *y* denotes the source region (vortical field) of the pipe mouth over which the integral of Eq. (10) is performed. The specific form of *ϕ y* � � is given as *ϕ*<sup>∗</sup> <sup>2</sup> *y* � � of Eq. (10.42) in [18].

According to [3], the power density supplied from the acoustic field to the vortical field around the edge is given by:

$$(\mathbf{1}/2)(\partial/\partial t)(\rho \mathbf{v}^2) = \rho(\mathbf{a}\times\mathbf{v})\,\bullet \,\mathbf{u}.\tag{12}$$

Therefore, if ð Þ *ω* � *v* ∙ *u* **>** 0, it may be said that the energy of the acoustic field is absorbed by the vortical field. As a result, an *acoustically induced vortex* or *acoustic vortex* is generated [16, 17]. The generation of it satisfies the phase relation in which a clockwise-rotating vortex appears above the edge when *u* directs into the pipe [see **Figure 12(b)** in Section 3.2.1]. This phase relation is contrary to that illustrated in **Figure 6**. A discrete vortex of Howe's type has not been observed in sound generation of flue instruments where jets are used in normal conditions [16, 17, 20–27, 29].

Howe [3, 35] then proposed the *acoustic dissipation formula*:

$$\Pi\_{\rm D}(t) \approx \iiint \rho(\boldsymbol{\omega} \times \boldsymbol{\nu}) \bullet \boldsymbol{\nu} \mathrm{d}V, \boldsymbol{\nu} \approx \boldsymbol{U}. \tag{13}$$

This equation determines the rate of dissipation of acoustic energy, where ∇*ϕ* in Eq. (10) is now simply denoted by *u*. Also, *V* denotes a volume enclosing the vorticity formed in the flow field. This *Π*<sup>D</sup> of Eq. (13) can be negative in oscillation systems: If the phase of vorticity production enables a steady transfer of energy to the oscillation from a mean flow, the self-sustained oscillation can be maintained.

Hence, the *acoustic generation formula* may be proposed:

$$\Pi\_{\mathcal{G}}(t) \approx -\iiint \rho(\boldsymbol{\omega} \times \mathbf{U}) \bullet \mathbf{u} \mathrm{d}V, \boldsymbol{\omega} = \mathrm{rot}\mathbf{U},\tag{14}$$

#### **Figure 6.**

*Conceptual sketch of a discrete-vortex model (the vortex shedding at the edge) for sound generation in flue instruments proposed by Howe [18].*

amplitude j j *q* [27]. These sources are supposed to be placed at a distance *ϵ* from the

where *ρ* is the air density, *b* the jet (or mouth) breadth, *d* the flue-to-edge distance (or the mouth length), and *δ*<sup>j</sup> the effective distance between the two

*<sup>δ</sup>*j*=d*≈ð Þ <sup>4</sup>*=<sup>π</sup>* ffiffiffiffiffiffiffiffiffiffi

In the limit of thin jets ð Þ *d=h* ≫ 1 , where *h* denotes the jet thickness at the flue

The power generated by the source is calculated by assuming that the source is in phase with the acoustic volume flow *q*mðÞ¼ *t* ð Þ d*ξ*m*=*d*t db* ¼ *udb* through the mouth opening, where *ξ*<sup>m</sup> denotes the particle displacement over the mouth opening [21]. This *q*<sup>m</sup> is supposed to be a local two-dimensional incompressible flow. The above in-phase relation between the pressure source *p*<sup>j</sup> and the acoustic volume flow *q*<sup>m</sup> gives the condition for which the oscillation amplitude has a maximum as a function

The time average over an oscillation period *T* of the power generated by the jet

<sup>≈</sup>ð Þ <sup>8</sup>*=π<sup>T</sup> <sup>ρ</sup>* ffiffiffiffiffiffiffiffiffiffiffiffi

In addition to the thin jet assumption ð Þ *d=h* ≫ 1 , we have to suppose that the jet does not break down into discrete vortices. This is only reasonable for the first hydrodynamic mode (*S*<sup>t</sup> ¼ *fd=U*<sup>0</sup> <0*:*3). The validity of Eq. (9) will be discussed in Section 3 after deriving the acoustic energy loss due to vortex shedding at the edge.

On the basis of the two-dimensional theory, Howe [18] proposed a discretevortex model on sound generation in flute-like instruments. He assumed that a compact vortex core appearing alternately just above and beneath the edge was created by the interaction with the acoustic cross-flow velocity *u* [d*ξ*m*=*d*t* in Eq. (9) corresponds to one-dimensional (*z* direction) component] at the mouth opening (see **Figure 6**). That is, instead of the jet oscillation over the mouth explained in the previous section, a point vortex is produced at the edge. Then, this vortex core is assumed to drive the air column in the pipe. A discrete-vortex model for thick jets assumes that a discrete vortex is generated from the flow separation at the flue exit corner [9, 10, 21], while Howe [18] attached greater importance to the flow separation (vortex shedding) at an opposing sharp edge due to the acoustic cross-flow. The sound excitation by the periodic vortex shedding at the edge is controlled by the product of the aeroacoustic source term divð Þ *ω* � *v* and the potential function

� � representing the irrotational cross-flow into and out of the mouth as expressed

ð Þ*t* is derived from the potential difference across the

ðÞ¼� *<sup>t</sup> ρ δ*j*=bd* � �ð Þ <sup>d</sup>*q=*d*<sup>t</sup>* , (7)

2*ϵ=d* p *:* (8)

<sup>2</sup>*h*e*=<sup>d</sup>* <sup>p</sup> *<sup>U</sup>*0*h*e*bd*j j <sup>d</sup>*ξ*m*=*d*<sup>t</sup> :* (9)

edge tip at the lower and upper sides of the edge.

mouth induced by two monopole sources [21, 27]:

monopole sources. If *ϵ* ≪ *d*, we have from [27].

*p*j

exit, ϵ ¼ *h*<sup>e</sup> (the jet thickness at the edge) is assumed [27].

*q*m D E

*2.2.1 Discrete-vortex model based on the vortex shedding at the edge*

The acoustic pressure *p*<sup>j</sup>

*Vortex Dynamics Theories and Applications*

of the blowing pressure.

**2.2 Discrete-vortex model**

by the following integral [18]:

*ϕ y*

**52**

drive above is given as follows [21]:

*Π*jet � � <sup>¼</sup> *<sup>p</sup>*<sup>j</sup>

where the vorticity *ω* is simply given as rot*U*. If the time average h i *Π*Gð Þ*t* is positive, the vorticity production from the jet flow supplies the acoustic power to the resonant pipe.

*U*<sup>c</sup> ¼ ð Þ *U*0*=*2 tanh ð Þ *πh=λ*<sup>v</sup> , (15)

<sup>0</sup>½ � *t* � ð Þ *j* � 1 *T* for ð Þ *j* � 1 *T* ≤ *t*≤*jT*, (16)

<sup>0</sup>*T* for *t*> *jT*, (17)

<sup>0</sup>*T* for *t*>ð Þ 2*j* þ 1 ð Þ *T=*2 *:* (19)

*ρ*ð Þ *ω* � *U* ∙ *u*d*V*d*t*, (20)

outð Þ*t*

outð Þ*t* the position of the *j*th vortex at the

in ð Þ*t* the position of the

outð Þ*<sup>t</sup> <sup>δ</sup> <sup>x</sup>* � *<sup>x</sup>*ð Þ*<sup>j</sup>*

h i � � , (21)

(18)

where *λ*<sup>v</sup> is the distance between successive vortices in the lower and the upper

In the case of flue instruments with thick jets (*d=h*<2) [21], a new vortex is formed at the inner shear layer (on the resonant pipe side) each time the acoustic velocity d*ξ*m*=*d*t* changes sign from directed toward the outside to directed toward the inside of the resonator (acoustic pressure in the resonator takes the minimum). A new vortex is formed at the outer shear layer half an oscillation period ð Þ *T=*2 later (acoustic pressure in the resonator takes the maximum). In the steady state of oscillation, the circulation of the *j*th vortex (*j* = 1, 2, 3, … .) at the inner shear layer

out can be written as [21].

<sup>0</sup>½ � *t* � ð Þ 2*j* � 1 ð Þ *T=*2 ,

line vortices and *h* is the distance between both lines which is equal to the jet thickness. It is also noted that the circulation *Γ* of the vortex increases linearly with

0*t*.

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

the time according to *<sup>Γ</sup>*ðÞ¼ *<sup>t</sup>* ð Þ <sup>1</sup>*=*<sup>2</sup> *<sup>U</sup>*<sup>2</sup>

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

in and at the outer shear layer *<sup>Γ</sup>*ð Þ*<sup>j</sup>*

*Γ*ð Þ*<sup>j</sup>*

in ðÞ¼ *<sup>t</sup>* ð Þ <sup>1</sup>*=*<sup>2</sup> *<sup>U</sup>*<sup>2</sup>

*Γ*ð Þ*<sup>j</sup>*

*Γ*ð Þ*<sup>j</sup>*

*Γ*ð Þ*<sup>j</sup>*

in ðÞ¼ *<sup>t</sup>* ð Þ <sup>1</sup>*=*<sup>2</sup> *<sup>U</sup>*<sup>2</sup>

for 2ð Þ *j* � 1 ð Þ *T=*2 ≤*t*≤ ð Þ 2*j* þ 1 ð Þ *T=*2

The acoustic power generated by the vortices is calculated by Eq. (14). The

ð *T*

ð

*V*<sup>S</sup>

where the volume integration is taken over the source region of volume *V*S. The vorticity field *ω* ¼ rot*U* takes into account the contribution of each vortex at the

It is first necessary to know the position and the circulation of vortices in order to calculate h i *Π*vortex from Eqs. (20) and (21). It was done by time-domain simula-

Dequand et al*.* [21] visualized the steady-state periodic flow in the mouth of the

resonator by applying a standard Schlieren technique. They used three types of flute-like mouth configuration with a common edge of 60°, a common sharp edged

tions in [38]. For the sake of paper space, see the details described in [38].

0

in ð Þ*t* � � <sup>þ</sup> *<sup>Γ</sup>*ð Þ*<sup>j</sup>*

outðÞ¼� *<sup>t</sup>* ð Þ <sup>1</sup>*=*<sup>2</sup> *<sup>U</sup>*<sup>2</sup>

outðÞ¼� *<sup>t</sup>* ð Þ <sup>1</sup>*=*<sup>2</sup> *<sup>U</sup>*<sup>2</sup>

h i *Π*vortex ¼ �ð Þ 1*=T*

in ð Þ*<sup>t</sup> <sup>δ</sup> <sup>x</sup>* � *<sup>x</sup>*ð Þ*<sup>j</sup>*

where *x* defines a two-dimensional (2-D) point (*x*, *y*), *x*ð Þ*<sup>j</sup>*

average of it over the oscillation period *T* is as follows:

*Γ*ð Þ*<sup>j</sup>*

shear layers [21]:

outer shear layer.

**55**

*<sup>ω</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>t</sup>* <sup>X</sup>

*j*

*j*th vortex at the inner shear layer, and *x*ð Þ*<sup>j</sup>*

**2.3 Flow visualization and some discussion**

*2.3.1 Jet-wave drive vs. discrete-vortex drive*

*Γ*ð Þ*<sup>j</sup>*

#### *2.2.2 Discrete-vortex model based on the vortex shedding at the flue exit*

When the jet is thick, the jet flow is not fully deflected into the resonant pipe. It is then difficult to apply the jet-drive model to thick jets. As the jet becomes thicker and thicker, the two shear layers at both sides of the jet tend to behave independently of each other. Meissner [9] described both shear layers in terms of discrete vortices (see **Figure 7**). He used the jet with *h* ¼ 2*:*7 mm (not so very thick). This jet excited a cavity resonator (its cross section: 40 mm � 28 mm; its depth: 12, 14, 16, 18, and 20 cm). The distance from the nozzle exit to the opposing orifice edge corresponding to the flue-to-edge distance *d* was set to be 8 mm (*d=h* ¼ 2*:*96). The orifice width was 28 mm, and the edge thickness 2 mm.

Meissner [9] found experimental results as follows: In stage I (5*:*9 m*=*s< *U*<sup>0</sup> <8*:*3 m*=*sÞ, the frequency increased fast with the jet speed, and a frequency increment was proportional to the jet speed just as in the edge-tone generation [36, 37]. Similar phenomenon often appears at the very first stage in sound generation of flue instruments [1, 23]. In stage II (8.3m*=*s< *U*<sup>0</sup> <16*:*1 m*=*s), an increase in the frequency was still observed, but a frequency growth was much smaller. The experimental results obtained for different cavity depths correlated reasonably well because data points corresponding to this stage approximately lay in one curve [9].

The cavity-tone generator shown in **Figure 7** can be considered as a simplified model of the ocarina. It is assumed that vorticity generation begins immediately after the jet issues from the nozzle exit due to flow separation. The vorticity of both shear layers is concentrated into line vortices traveling along straight lines with the convection velocity *U*c. In the case of asymmetric vortex formation as shown in **Figure 7**, a configuration of vortices will be similar to that in the conventional Kármán vortex street. Thus, *U*<sup>c</sup> may be approximated as that of an infinite street [9].

#### **Figure 7.**

*Vortex shedding in a cavity-tone generator [9], which can be considered as a simplified mechanical model of the ocarina. h* ¼ 2*:*7 *mm and d* ¼ 8 *mm.*

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

$$U\_{\mathfrak{c}} = (U\_0/2)\tanh\left(\pi h/\lambda\_{\mathfrak{v}}\right),\tag{15}$$

where *λ*<sup>v</sup> is the distance between successive vortices in the lower and the upper line vortices and *h* is the distance between both lines which is equal to the jet thickness. It is also noted that the circulation *Γ* of the vortex increases linearly with the time according to *<sup>Γ</sup>*ðÞ¼ *<sup>t</sup>* ð Þ <sup>1</sup>*=*<sup>2</sup> *<sup>U</sup>*<sup>2</sup> 0*t*.

In the case of flue instruments with thick jets (*d=h*<2) [21], a new vortex is formed at the inner shear layer (on the resonant pipe side) each time the acoustic velocity d*ξ*m*=*d*t* changes sign from directed toward the outside to directed toward the inside of the resonator (acoustic pressure in the resonator takes the minimum). A new vortex is formed at the outer shear layer half an oscillation period ð Þ *T=*2 later (acoustic pressure in the resonator takes the maximum). In the steady state of oscillation, the circulation of the *j*th vortex (*j* = 1, 2, 3, … .) at the inner shear layer *Γ*ð Þ*<sup>j</sup>* in and at the outer shear layer *<sup>Γ</sup>*ð Þ*<sup>j</sup>* out can be written as [21].

$$
\Gamma\_{\rm in}^{(j)}(t) = (\mathbf{1}/2)U\_0^2[t - (j-1)T] \text{ for } (j-1)T \le t \le jT,\tag{16}
$$

$$
\Gamma\_{\rm in}^{(j)}(t) = (\mathbf{1}/2)U\_0^2 T \text{ for } t > jT,\tag{17}
$$

$$\begin{aligned} I\_{\text{out}}^{(j)}(t) &= - (\mathbf{1}/2) U\_0^2 [t - (2j - \mathbf{1})(T/2)], \\ \text{for } (2j - \mathbf{1})(T/2) &\le t \le (2j + \mathbf{1})(T/2) \end{aligned} \tag{18}$$

$$
\Gamma\_{\rm out}^{(j)}(t) = - (\mathbf{1}/2) U\_0^2 T \text{ for } t > (2j+1)(T/2). \tag{19}
$$

The acoustic power generated by the vortices is calculated by Eq. (14). The average of it over the oscillation period *T* is as follows:

$$
\langle \mathcal{U}\_{\text{vortex}} \rangle = -(\mathbf{1}/T) \int\_{0}^{T} \int\_{V\_{\text{s}}} \rho (\boldsymbol{\omega} \times \mathbf{U}) \cdot \mathbf{u} \mathbf{d}V \mathbf{d}t,\tag{20}
$$

where the volume integration is taken over the source region of volume *V*S. The vorticity field *ω* ¼ rot*U* takes into account the contribution of each vortex at the shear layers [21]:

$$\rho a(\mathbf{x}, \mathbf{y}, t) = \sum\_{j} \left[ \boldsymbol{\Gamma}\_{\text{in}}^{(j)}(t) \delta \left( \mathbf{x} - \boldsymbol{\mathfrak{x}}\_{\text{in}}^{(j)}(t) \right) + \boldsymbol{\Gamma}\_{\text{out}}^{(j)}(t) \delta \left( \mathbf{x} - \boldsymbol{\mathfrak{x}}\_{\text{out}}^{(j)}(t) \right) \right], \tag{21}$$

where *x* defines a two-dimensional (2-D) point (*x*, *y*), *x*ð Þ*<sup>j</sup>* in ð Þ*t* the position of the *j*th vortex at the inner shear layer, and *x*ð Þ*<sup>j</sup>* outð Þ*t* the position of the *j*th vortex at the outer shear layer.

It is first necessary to know the position and the circulation of vortices in order to calculate h i *Π*vortex from Eqs. (20) and (21). It was done by time-domain simulations in [38]. For the sake of paper space, see the details described in [38].

#### **2.3 Flow visualization and some discussion**

#### *2.3.1 Jet-wave drive vs. discrete-vortex drive*

Dequand et al*.* [21] visualized the steady-state periodic flow in the mouth of the resonator by applying a standard Schlieren technique. They used three types of flute-like mouth configuration with a common edge of 60°, a common sharp edged

where the vorticity *ω* is simply given as rot*U*. If the time average h i *Π*Gð Þ*t* is positive, the vorticity production from the jet flow supplies the acoustic power to

When the jet is thick, the jet flow is not fully deflected into the resonant pipe. It is then difficult to apply the jet-drive model to thick jets. As the jet becomes thicker and thicker, the two shear layers at both sides of the jet tend to behave independently of each other. Meissner [9] described both shear layers in terms of discrete vortices (see **Figure 7**). He used the jet with *h* ¼ 2*:*7 mm (not so very thick). This jet excited a cavity resonator (its cross section: 40 mm � 28 mm; its depth: 12, 14, 16, 18, and 20 cm). The distance from the nozzle exit to the opposing orifice edge corresponding to the flue-to-edge distance *d* was set to be 8 mm (*d=h* ¼ 2*:*96). The

*2.2.2 Discrete-vortex model based on the vortex shedding at the flue exit*

Meissner [9] found experimental results as follows: In stage I

(5*:*9 m*=*s< *U*<sup>0</sup> <8*:*3 m*=*sÞ, the frequency increased fast with the jet speed, and a frequency increment was proportional to the jet speed just as in the edge-tone generation [36, 37]. Similar phenomenon often appears at the very first stage in sound generation of flue instruments [1, 23]. In stage II (8.3m*=*s< *U*<sup>0</sup> <16*:*1 m*=*s), an increase in the frequency was still observed, but a frequency growth was much smaller. The experimental results obtained for different cavity depths correlated reasonably well because data points corresponding to this stage approximately lay

The cavity-tone generator shown in **Figure 7** can be considered as a simplified model of the ocarina. It is assumed that vorticity generation begins immediately after the jet issues from the nozzle exit due to flow separation. The vorticity of both shear layers is concentrated into line vortices traveling along straight lines with the convection velocity *U*c. In the case of asymmetric vortex formation as shown in **Figure 7**, a configuration of vortices will be similar to that in the conventional Kármán vortex street. Thus, *U*<sup>c</sup> may be approximated as that of an infinite

*Vortex shedding in a cavity-tone generator [9], which can be considered as a simplified mechanical model of the*

orifice width was 28 mm, and the edge thickness 2 mm.

the resonant pipe.

*Vortex Dynamics Theories and Applications*

in one curve [9].

street [9].

**Figure 7.**

**54**

*ocarina. h* ¼ 2*:*7 *mm and d* ¼ 8 *mm.*

flue exit, a common flue-to-edge distance *d* = 24 mm, and different flue channel height (i.e., jet thickness) *h* = 4, 14, and 30 mm (*d/h* = 6, 1.7, and 0.8). The pipe length *L* was 552 mm. The initial jet velocity *U*<sup>0</sup> had, respectively, 16.3, 14, and 14.5 m/s (*S*<sup>t</sup> = 0.19, 0.22, and 0.22). **Figure 8** summarizes their result by rough illustration, though the flow inside the pipe is not so clear in [21].

of *p* (*t*) is delayed from *u* (*t*) by 90° at the resonance. Therefore, the positive acoustic power h i *p t*ð Þ∙ *q t*ð Þ is generated, where *q t*ð Þ denotes the acoustic volume flow into the pipe [see Eq. (1)]. The positive acoustic power is also generated at the instant given in **Figure 8(b)**. As the result, the volume-flow drive in the jet-drive

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

The discrete-vortex drive illustrated in **Figure 8(e)** indicates that the upper vortex is just created at the upper flue exit corner and the lower vortex reaches to the pipe edge in a fully developed shape. Also, **Figure 8(f)** indicates that the lower vortex is just created at the lower flue exit corner and the upper vortex is reached to the pipe edge in a fully developed shape. As a result, the positive acoustic power

Two illustrations of **Figure 8(c)** and **(d)** correspond to a boundary condition between the jet-wave drive and the discrete-vortex drive. Then, the lower and upper large vortices are located halfway between the flue exit and the edge. This is

In the discrete-vortex drive, the horizontal arrow connects the vortex creation at the flue exit and the vortex arrival at the edge as shown in **Figure 9(b)**. Since the upper vortex rotates anticlockwise, the vector direction of *ω* � *U* in Eq. (20) is upward (positive *z* direction). During the passage of the upper vortex from the flue exit to the pipe edge, the acoustic cross-flow *u* in *z* direction has negative values. Hence, h i *Π*vortex of Eq. (20) takes a positive value during the latter half of an oscillation period. Similarly, the lower vortex produces positive h i *Π*vortex during the former half of an oscillation period. As a result, discrete vortices will create positive h i *Π*vortex in an oscillation period. In other words, the vortex configurations illustrated in **Figure 8(e)** and **(f)** may create the acoustic power for sound generation in

*Phase relation between the physical quantities involved in the jet-wave drive (a) and the discrete-vortex drive (b). Their amplitudes are arbitrary. The red dot on the curve of u corresponds to the phase of u in Figure 8. The horizontal arrow in (b) connects the vortex creation at the flue exit and the vortex arrival at the pipe edge. Note that the positive direction of u and ξ*<sup>e</sup> *is upward (outward) and the phase of p is delayed from u by 90° at*

**Figure 9** summarizes the phase relation between the physical quantities involved in the jet-wave drive (a) and the discrete-vortex drive (b). The red dot on the sinusoidal curve of *u* corresponds to the phase of *u* in **Figure 8**. The time scale is converted to the integer by 12(*t*/*T*). The magnitude of physical quantities is arbitrary. The jet-wave drive indicates the antiphase relation between the jet displacement *ξ*eð Þ*t* at the edge and the acoustic pressure *p t*ð Þ in the resonant pipe. This result endorses the acoustic power generation in good manner (note that *u* and *ξ*<sup>e</sup> are

given by Eq. (20) or Eq. (14) is generated [see **Figure 9(b)** below].

probably due to an opposing effect between both drives.

model is satisfied very well.

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

defined positive upward).

flue pipes.

**Figure 9.**

*the resonance.*

**57**

Frames in the left column [(a), (c), and (e)] show flow conditions at the phase of *u* ¼ 0 (the instant from the positive to the negative), while frames in the right column [(b), (d), and (f)] show flow conditions at the next phase of *u* ¼ 0 (the instant from the negative to the positive). Note that the positive *u* indicates the upward (outward) acoustic velocity here, but it indicates the downward (inward) acoustic velocity in [21]. Also, the flow visualization of each case is shown by eight frames consisting of one oscillation period [21].

The jet-wave drive illustrated in **Figure 8(a)** reveals that the jet enters into the pipe at the instant when the acoustic pressure *p* (*t*) is maximum because the phase

#### **Figure 8.**

*Illustrations of flow visualization by Dequand et al. [21]. Top two (a) and (b) have h = 4 mm, middle two (c), and (d) h = 14 mm, and bottom two (e) and (f) h = 30 mm. The sinusoidal wave inserted in each frame depicts the acoustic velocity u in the mouth, and the dot gives its instantaneous phase. Note that u is positive upward (outward) here, but u is positive downward (inward) in [21].*

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

of *p* (*t*) is delayed from *u* (*t*) by 90° at the resonance. Therefore, the positive acoustic power h i *p t*ð Þ∙ *q t*ð Þ is generated, where *q t*ð Þ denotes the acoustic volume flow into the pipe [see Eq. (1)]. The positive acoustic power is also generated at the instant given in **Figure 8(b)**. As the result, the volume-flow drive in the jet-drive model is satisfied very well.

The discrete-vortex drive illustrated in **Figure 8(e)** indicates that the upper vortex is just created at the upper flue exit corner and the lower vortex reaches to the pipe edge in a fully developed shape. Also, **Figure 8(f)** indicates that the lower vortex is just created at the lower flue exit corner and the upper vortex is reached to the pipe edge in a fully developed shape. As a result, the positive acoustic power given by Eq. (20) or Eq. (14) is generated [see **Figure 9(b)** below].

Two illustrations of **Figure 8(c)** and **(d)** correspond to a boundary condition between the jet-wave drive and the discrete-vortex drive. Then, the lower and upper large vortices are located halfway between the flue exit and the edge. This is probably due to an opposing effect between both drives.

**Figure 9** summarizes the phase relation between the physical quantities involved in the jet-wave drive (a) and the discrete-vortex drive (b). The red dot on the sinusoidal curve of *u* corresponds to the phase of *u* in **Figure 8**. The time scale is converted to the integer by 12(*t*/*T*). The magnitude of physical quantities is arbitrary. The jet-wave drive indicates the antiphase relation between the jet displacement *ξ*eð Þ*t* at the edge and the acoustic pressure *p t*ð Þ in the resonant pipe. This result endorses the acoustic power generation in good manner (note that *u* and *ξ*<sup>e</sup> are defined positive upward).

In the discrete-vortex drive, the horizontal arrow connects the vortex creation at the flue exit and the vortex arrival at the edge as shown in **Figure 9(b)**. Since the upper vortex rotates anticlockwise, the vector direction of *ω* � *U* in Eq. (20) is upward (positive *z* direction). During the passage of the upper vortex from the flue exit to the pipe edge, the acoustic cross-flow *u* in *z* direction has negative values. Hence, h i *Π*vortex of Eq. (20) takes a positive value during the latter half of an oscillation period. Similarly, the lower vortex produces positive h i *Π*vortex during the former half of an oscillation period. As a result, discrete vortices will create positive h i *Π*vortex in an oscillation period. In other words, the vortex configurations illustrated in **Figure 8(e)** and **(f)** may create the acoustic power for sound generation in flue pipes.

#### **Figure 9.**

flue exit, a common flue-to-edge distance *d* = 24 mm, and different flue channel height (i.e., jet thickness) *h* = 4, 14, and 30 mm (*d/h* = 6, 1.7, and 0.8). The pipe length *L* was 552 mm. The initial jet velocity *U*<sup>0</sup> had, respectively, 16.3, 14, and 14.5 m/s (*S*<sup>t</sup> = 0.19, 0.22, and 0.22). **Figure 8** summarizes their result by rough

Frames in the left column [(a), (c), and (e)] show flow conditions at the phase of *u* ¼ 0 (the instant from the positive to the negative), while frames in the right column [(b), (d), and (f)] show flow conditions at the next phase of *u* ¼ 0 (the instant from the negative to the positive). Note that the positive *u* indicates the upward (outward) acoustic velocity here, but it indicates the downward (inward) acoustic velocity in [21]. Also, the flow visualization of each case is shown by eight

The jet-wave drive illustrated in **Figure 8(a)** reveals that the jet enters into the pipe at the instant when the acoustic pressure *p* (*t*) is maximum because the phase

*Illustrations of flow visualization by Dequand et al. [21]. Top two (a) and (b) have h = 4 mm, middle two (c), and (d) h = 14 mm, and bottom two (e) and (f) h = 30 mm. The sinusoidal wave inserted in each frame depicts the acoustic velocity u in the mouth, and the dot gives its instantaneous phase. Note that u is positive*

*upward (outward) here, but u is positive downward (inward) in [21].*

illustration, though the flow inside the pipe is not so clear in [21].

frames consisting of one oscillation period [21].

*Vortex Dynamics Theories and Applications*

**Figure 8.**

**56**

*Phase relation between the physical quantities involved in the jet-wave drive (a) and the discrete-vortex drive (b). Their amplitudes are arbitrary. The red dot on the curve of u corresponds to the phase of u in Figure 8. The horizontal arrow in (b) connects the vortex creation at the flue exit and the vortex arrival at the pipe edge. Note that the positive direction of u and ξ*<sup>e</sup> *is upward (outward) and the phase of p is delayed from u by 90° at the resonance.*

#### *2.3.2 Edge tone vs. pipe tone*

The edge tone is a dipole source, whose acoustic pressure directly correlates with the vortex generation. That is, when the jet impinges the edge by moving from the downward to the upward, a vortex rotating clockwise is produced just below the edge, and another vortex rotating anticlockwise exists downstream above the edge. This configuration of the vortex pair generates the maximum acoustic pressure above the edge and the minimum acoustic pressure below the edge. When the jet impinges the edge by moving from the upward to the downward, a vortex rotating anticlockwise is produced just above the edge, and another vortex rotating clockwise exists downstream below the edge. This configuration of the vortex pair generates the maximum acoustic pressure below the edge and the minimum acoustic pressure above the edge [2, 39]. Although Eq. (14) cannot be applied to the edge tone since there is no acoustic feedback (i.e., *u* = 0), the acceleration *ω* � *U* given by vortex rotation and jet velocity should be involved in the edge-tone generation. As a result, the edge tone satisfies:

$$\text{phase}\left[\left.p\_{\text{e}}(t)\right]-\text{phase}[\xi\_{\text{e}}(t)] \approx \pm \pi/2,\tag{22}$$

*<sup>P</sup>*jð Þ¼ *<sup>ω</sup> <sup>ρ</sup>hδ*j*=<sup>d</sup>* ð Þ <sup>i</sup>*<sup>ω</sup>* <sup>e</sup>*<sup>μ</sup>d*e�i*ωτ*jw *<sup>U</sup>*fð Þ *<sup>ω</sup>* , (26)

*<sup>G</sup>*ð Þ¼ *<sup>ω</sup> <sup>Y</sup>*ð Þ *<sup>ω</sup> <sup>P</sup>*jð Þ *<sup>ω</sup> <sup>=</sup>U*fð Þ *<sup>ω</sup> :* (27)

1 4 2*π*,

*τ*jw ¼ *π=ω* ¼ *T=*2 (30)

*τ*dv ¼ *d=Uc* ¼ *T=*2*:* (31)

(29)

phase½ �¼ *G*ð Þ *ω* phase½ �þ *Y*ð Þ *ω π=*2 � *ωτ*jw ¼ �2*mπ* (28)

where *U*fð Þ *ω* is the Fourier transform of *u*fð Þ*t* and *U*<sup>e</sup> ¼ *U*<sup>0</sup> is assumed by neglecting the jet spreading for simplicity [41]. The feedback loop gain *G*ð Þ *ω* is thus

Hence, the phase condition for the self-sustained (feedback) oscillation is:

*ωτ*jw ¼ phase½ �þ *Y*ð Þ *ω π=*2 þ 2*mπ*

¼ phase½ �þ *Y*ð Þ *ω m* þ

where �2*mπ* is abandoned because *ωτ*jw is always positive and *m* (= 0, 1, 2, … .) denotes the hydrodynamic mode number. Usually, sound generation in flue pipes occurs for *m* ¼ 0. Since phase½ �¼ *Y*ð Þ *ω π=*2 at the pipe resonance (see **Figure 9**), we

for the first mode *m* ¼ 0. Therefore, it may be said that flue instruments are well excited when the time delay of the jet wave is around half an oscillation period. More detailed discussion is given in [38, 41]. Although the amplitude condition for sound generation can be calculated from Eq. (27), we do not have the space enough

According to **Figure 9(b)**, the upper and lower vortices created at the flue exit

This Eq. (31) just corresponds to Eq. (30) in the jet-drive model. Therefore, both models provide the same dependence of the oscillation frequency on the jet

Although *τ*dv is easily derived from **Figure 9(b)**, it will be desirable to consider *τ*dv based on the phase balance such as in Eq. (29). The sound generation by the periodic pulse-like force (produced by each vortex arrival) at the edge will be maximum when the pulse is in phase with the maximum of acoustic velocity *u* at the edge. Since the vortex arrival occurs at the zero-crossing of *u* [see **Figure 9(b)**], the instant of the maximum *u* is given by *τ*dv � *T=*4, which should be balanced with the delay due to the input admittance *Y*ð Þ *ω* of the resonant pipe. Then, we have the

arrive at the pipe edge with a time delay of *T=*2, respectively. As explained in Section 2.3.1 the convection of these two vortices may create the acoustic power h i *Π*vortex defined by Eq. (20) for sound generation in flue pipes. Thus, the time delay

That is, the time delay *τ*jw of the jet wave must satisfy:

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

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

*2.3.4 Time delay of vortex convection in the discrete-vortex model*

of vortex convection *τ*dv in the discrete-vortex model is:

following phase balance by allowing a time delay of *m* periods:

defined as

finally have:

to do that.

velocity.

**59**

where *p*eð Þ*t* denotes the acoustic pressure below or above the edge. This phase relation is clearly different from that of the pipe tone shown in **Figure 9(a)**:

$$\text{phase}[\
p(t)] - \text{phase}[\xi\_{\text{e}}(t)] \approx \pi. \tag{23}$$

The difference between the edge tone and the pipe tone is reflected in Eqs. (22) and (23) in good manner.

#### *2.3.3 Feedback loop gain and time delay of the jet wave in the jet-drive model*

Let us consider the feedback loop to find out the time delay of the jet wave which fulfills the phase condition for sound generation. As mentioned in Section 2.1.2, the jet particle pass may be determined as soon as the jet issues from the flue to the acoustic field in the mouth [26]. At that instant, the initial transverse displacement *ξ*fð Þ*t* of the jet at the flue exit is supposed to be non-zero and related with the acoustic velocity *u*fð Þ*t* at the flue exit as follows [38, 40]:

$$
\xi\_{\mathbf{f}}(\mathbf{t})/h = \mathfrak{u}\_{\mathbf{f}}(\mathbf{t})/U\_{\mathbf{0}},\tag{24}
$$

where *h* and *U*<sup>0</sup> are the jet thickness and jet velocity at the flue exit, respectively. The starting point of the feedback loop is *u*fð Þ*t* , which creates *ξ*fð Þ*t* . The jet displacement travels to the pipe edge as the jet wave, and we have at the edge [21].

$$\xi\_{\mathbf{e}}(t) = \mathbf{e}^{\mu d} \xi\_{\mathbf{f}}(t - \tau\_{\mathbf{j}\mathbf{w}}) = (h/U\_0) \mathbf{e}^{\mu d} u\_{\mathbf{f}}(t - \tau\_{\mathbf{j}\mathbf{w}}),\tag{25}$$

which yields the acoustic pressure *p*<sup>j</sup> ð Þ*t* at the pipe entrance according to Eq. (7) in which *q t*ð Þ≈ � *U*e*bξ*eð Þ*t* from Eq. (1). Note that Eq. (25) largely simplifies Eq. (4) by considering the essential elements (spatial amplification and phase velocity) of the jet wave. The quantity *τ*jw denotes the time delay of the jet wave when it travels from the flue exit to the edge (*τ*jw ¼ *d=U*phÞ.

The acoustic pressure *p*<sup>j</sup> ð Þ*t* drives the pipe and yields its resonance. As a result, *u*fð Þ*t* at the starting point is fed back through the input admittance *Y*ð Þ *ω* of the pipe. The Fourier transform of *p*<sup>j</sup> ð Þ*t* is thus given by:

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

$$P\_{\mathbf{j}}(\boldsymbol{\alpha}) = \left(\rho h \delta\_{\mathbf{j}}/d\right) (\mathbf{i}\boldsymbol{\alpha}) \mathbf{e}^{\mu d} \mathbf{e}^{-\mathbf{i}\alpha \tau\_{\mathbf{j}\ast}} U\_{\mathbf{f}}(\boldsymbol{\alpha}),\tag{26}$$

where *U*fð Þ *ω* is the Fourier transform of *u*fð Þ*t* and *U*<sup>e</sup> ¼ *U*<sup>0</sup> is assumed by neglecting the jet spreading for simplicity [41]. The feedback loop gain *G*ð Þ *ω* is thus defined as

$$G(a) = Y(a) \left[ P\_{\mathfrak{j}}(a) / U\_{\mathfrak{f}}(a) \right]. \tag{27}$$

Hence, the phase condition for the self-sustained (feedback) oscillation is:

$$\text{phase}[G(o)] = \text{phase}[Y(o)] + \pi/2 - o\tau\_{\text{fw}} = \pm 2m\pi \tag{28}$$

That is, the time delay *τ*jw of the jet wave must satisfy:

$$\begin{split} \rho \boldsymbol{\pi}\_{\text{jw}} &= \text{phase}[Y(\boldsymbol{\alpha})] + \boldsymbol{\pi}/2 + 2m\boldsymbol{\pi} \\ &= \text{phase}[Y(\boldsymbol{\alpha})] + \left(m + \frac{1}{4}\right) 2\boldsymbol{\pi}, \end{split} \tag{29}$$

where �2*mπ* is abandoned because *ωτ*jw is always positive and *m* (= 0, 1, 2, … .) denotes the hydrodynamic mode number. Usually, sound generation in flue pipes occurs for *m* ¼ 0. Since phase½ �¼ *Y*ð Þ *ω π=*2 at the pipe resonance (see **Figure 9**), we finally have:

$$
\pi\_{\rm jw} = \pi/o = T/2\tag{30}
$$

for the first mode *m* ¼ 0. Therefore, it may be said that flue instruments are well excited when the time delay of the jet wave is around half an oscillation period. More detailed discussion is given in [38, 41]. Although the amplitude condition for sound generation can be calculated from Eq. (27), we do not have the space enough to do that.

#### *2.3.4 Time delay of vortex convection in the discrete-vortex model*

According to **Figure 9(b)**, the upper and lower vortices created at the flue exit arrive at the pipe edge with a time delay of *T=*2, respectively. As explained in Section 2.3.1 the convection of these two vortices may create the acoustic power h i *Π*vortex defined by Eq. (20) for sound generation in flue pipes. Thus, the time delay of vortex convection *τ*dv in the discrete-vortex model is:

$$
\pi\_{\rm dv} = \mathbf{d} / \mathbf{U}\_{\rm c} = \mathbf{T} / \mathbf{2}.\tag{31}
$$

This Eq. (31) just corresponds to Eq. (30) in the jet-drive model. Therefore, both models provide the same dependence of the oscillation frequency on the jet velocity.

Although *τ*dv is easily derived from **Figure 9(b)**, it will be desirable to consider *τ*dv based on the phase balance such as in Eq. (29). The sound generation by the periodic pulse-like force (produced by each vortex arrival) at the edge will be maximum when the pulse is in phase with the maximum of acoustic velocity *u* at the edge. Since the vortex arrival occurs at the zero-crossing of *u* [see **Figure 9(b)**], the instant of the maximum *u* is given by *τ*dv � *T=*4, which should be balanced with the delay due to the input admittance *Y*ð Þ *ω* of the resonant pipe. Then, we have the following phase balance by allowing a time delay of *m* periods:

*2.3.2 Edge tone vs. pipe tone*

*Vortex Dynamics Theories and Applications*

result, the edge tone satisfies:

and (23) in good manner.

The edge tone is a dipole source, whose acoustic pressure directly correlates with the vortex generation. That is, when the jet impinges the edge by moving from the downward to the upward, a vortex rotating clockwise is produced just below the edge, and another vortex rotating anticlockwise exists downstream above the edge. This configuration of the vortex pair generates the maximum acoustic pressure above the edge and the minimum acoustic pressure below the edge. When the jet impinges the edge by moving from the upward to the downward, a vortex rotating anticlockwise is produced just above the edge, and another vortex rotating clockwise exists downstream below the edge. This configuration of the vortex pair generates the maximum acoustic pressure below the edge and the minimum acoustic pressure above the edge [2, 39]. Although Eq. (14) cannot be applied to the edge tone since there is no acoustic feedback (i.e., *u* = 0), the acceleration *ω* � *U* given by vortex rotation and jet velocity should be involved in the edge-tone generation. As a

where *p*eð Þ*t* denotes the acoustic pressure below or above the edge. This phase

The difference between the edge tone and the pipe tone is reflected in Eqs. (22)

Let us consider the feedback loop to find out the time delay of the jet wave which fulfills the phase condition for sound generation. As mentioned in Section 2.1.2, the jet particle pass may be determined as soon as the jet issues from the flue to the acoustic field in the mouth [26]. At that instant, the initial transverse displacement *ξ*fð Þ*t* of the jet at the flue exit is supposed to be non-zero and related with

where *h* and *U*<sup>0</sup> are the jet thickness and jet velocity at the flue exit, respectively.

<sup>¼</sup> ð Þ *<sup>h</sup>=U*<sup>0</sup> <sup>e</sup>*<sup>μ</sup>du*<sup>f</sup> *<sup>t</sup>* � *<sup>τ</sup>*jw

displacement travels to the pipe edge as the jet wave, and we have at the edge [21].

in which *q t*ð Þ≈ � *U*e*bξ*eð Þ*t* from Eq. (1). Note that Eq. (25) largely simplifies Eq. (4) by considering the essential elements (spatial amplification and phase velocity) of the jet wave. The quantity *τ*jw denotes the time delay of the jet wave when it travels

*u*fð Þ*t* at the starting point is fed back through the input admittance *Y*ð Þ *ω* of the pipe.

ð Þ*t* is thus given by:

The starting point of the feedback loop is *u*fð Þ*t* , which creates *ξ*fð Þ*t* . The jet

relation is clearly different from that of the pipe tone shown in **Figure 9(a)**:

*2.3.3 Feedback loop gain and time delay of the jet wave in the jet-drive model*

the acoustic velocity *u*fð Þ*t* at the flue exit as follows [38, 40]:

*<sup>ξ</sup>*eðÞ¼ *<sup>t</sup>* <sup>e</sup>*<sup>μ</sup><sup>d</sup>ξ*<sup>f</sup> *<sup>t</sup>* � *<sup>τ</sup>*jw

which yields the acoustic pressure *p*<sup>j</sup>

from the flue exit to the edge (*τ*jw ¼ *d=U*phÞ.

The acoustic pressure *p*<sup>j</sup>

The Fourier transform of *p*<sup>j</sup>

**58**

phase *<sup>p</sup>*eð Þ*<sup>t</sup>* � phase *<sup>ξ</sup>*<sup>e</sup> ½ � ð Þ*<sup>t</sup>* <sup>≈</sup> � *<sup>π</sup>=*2, (22)

phase½ �� *p t*ð Þ phase *ξ*<sup>e</sup> ½ � ð Þ*t* ≈*π:* (23)

*ξ*fð Þ*t =h* ¼ *u*fð Þ*t =U*0, (24)

ð Þ*t* drives the pipe and yields its resonance. As a result,

, (25)

ð Þ*t* at the pipe entrance according to Eq. (7)

$$
\pi\_{\rm dv} - \left(m + \frac{1}{4}\right)T = \text{phase}[Y(\rho)]/\rho.\tag{32}
$$

the upper and lower vortex layers. Hence, the sound generation in flue pipes may be yielded by the interaction between the jet vortex layer and the acoustic mouth flow. In this sense, *ω* � *U* and (*ω* � *U*Þ∙ *u* may be called the *aeroacoustical source term* (with the same dimension as the acceleration) and the *acoustic generation term*, respectively. Helmholtz [30] already suggested the importance of the jet vortex layer. His vortical surface (or stratum) that has a very unstable equilibrium acts as "an accelerating force with a periodically alternating direction" to reinforce the inward and outward velocity at the pipe entrance. Interestingly enough, this physical picture of Helmholtz is very similar to the jet vortex-layer model shown in **Figure 10** [2, 28]. It should be recognized that the volume-flow drive first proposed by Helmholtz [30] is based on his physical concept of the vortex layers formed along the jet flow.

*Conceptual sketch of the vortex-layer model on sound generation in flue instruments [2, 28]: (a) the vortex layers (consisting of tiny vortices) along both sides of the jet flow and (b) the generation and cancelation of the*

*aeroacoustical source term ω* � *U. The dashed line depicts a lateral profile of the jet velocity U.*

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

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

Let us briefly discuss the mechanisms of sound dissipation (or absorption) in flue instruments because the self-sustained musical instruments must overcome the acoustic dissipations involved in them. At first let us consider within the field of

In free space, the classical sources of dissipation are internal friction and heat conduction. Both phenomena tend to equalize the local variations of the particle

**3. Vortices on sound dissipation**

**Figure 10.**

**61**

**3.1 Sound dissipations in linear acoustics**

linear acoustics and start from sound dissipation in free space.

*3.1.1 Classical absorption and molecular absorption in free space*

This equation is the same as Eq. (29) when it is divided by *ω*. A similar derivation of Eq. (32) is given in [38]. When *m* = 0, Eq. (31) is given from Eq. (32).

#### *2.3.5 Aspect ratio d/h of the jet*

The aspect ratio *d*/*h* (jet length/jet thickness) is an essential parameter that discriminates the jet-drive model from the discrete-vortex model as indicated in **Figure 8** based on [21]. The value of *d*/*h* = 1.7 set up in **Figure 8(c)** and **(d)** seems to be more favored by the discrete-vortex model. The critical aspect ratio that discriminates both models is *d*/*h* = 2.3 in more rigorous sense [21, 38].

Dequand et al. [21] calculated dj j *ξ*m*=*d*t* max*=U*<sup>0</sup> as a function of *d*/*h* for both models, and experimental data on the flue exit with chamfered edges and the pipe edge with 15° were plotted on the calculated diagram (see Figure 12 in [21]). In very rough sense, the solution curve of the jet-drive model is proportional to ð Þ *<sup>d</sup>=<sup>h</sup>* �<sup>1</sup> , and that of the discrete-vortex model is proportional to ð Þ *<sup>d</sup>=<sup>h</sup>* <sup>1</sup>*=*<sup>2</sup> . The crossing of the two curves occurs near *d*/*h* = 2.3, and the experimental data approximately fit the discrete-vortex solution below this crossing and better fit the jet-drive solution above the crossing. Also, the experimental curve of data points indicates the maximum value at the crossing or the critical aspect ratio. It was experimentally confirmed on four kinds of the flue-edge geometry (see Figure 10 in [21]). Furthermore, Auvray et al. [38] extended similar calculation for different oscillation regimes (*m* = 0, 1; *f* = *f*1, *f*2) (see Figure 8 in [38]). According to [38], the critical aspect ratio depends on the hydrodynamic jet mode *m*. For an eolian regime (*m* = 1, *f* = *f*2), the critical aspect ratio is much larger (*d=h* ffi 13).

#### **2.4 Vortex-layer model**

Howe [18] and Dequand et al*.* [21] proposed the discrete-vortex model driven by thick jets (*d/h* < 2) as explained in Section 2.2. However, there seems to be a room for the vortex even in the sound generation of flue instruments driven by thin jets that satisfy *d/h* > 2. Since an actual jet has a velocity profile as indicated by the broken line in **Figure 10(a)**, the vorticity can be formed along the boundary between the jet and the surrounding fluid. As the result, a layer (or sheet) of vorticity is organized along an immediate vicinity of the jet. The upper layer consists of the positive vorticity (the counterclockwise-rotating tiny vortices), and the lower layer consists of the negative vorticity (the clockwise-rotating tiny vortices). This physical picture depicted in **Figure 10(a)** may be called the vortex-layer model on the sound generation in musical flue instruments [2, 28].

It should be carefully noted that actual sound generation is three-dimensional (3-D) as inferred from the volume integral of Eq. (14), but our vortex-layer model illustrated in **Figure 10(a)** assumes the two-dimensionality (2-D). This 2-D assumption corresponds to the 2-D assumption of *U* and *u* that has been conventionally made in acoustical models. See Figures 10 and 11 in [28] on the 3-D nature of *U* and *u*. The PIV observation on a plane sheet by the laser (see Section 4.1 and **Figure 14**) is based on the 2-D assumption above.

Since *u* is periodic, the time average h i *Π*Gð Þ*t* should be null if *ω* � *U* is stationary in time as shown in **Figure 10(b)**. However, since the actual jet has any fluctuation, if *ω* � *U* has a component that changes temporarily in accordance with temporal change of *u*, non-zero value of h i *Π*Gð Þ*t* may be expected from the unbalance between *Vortices on Sound Generation and Dissipation in Musical Flue Instruments DOI: http://dx.doi.org/10.5772/intechopen.91258*

**Figure 10.**

*τ*dv � *m* þ

*2.3.5 Aspect ratio d/h of the jet*

*Vortex Dynamics Theories and Applications*

**2.4 Vortex-layer model**

**60**

1 4 

tion of Eq. (32) is given in [38]. When *m* = 0, Eq. (31) is given from Eq. (32).

The aspect ratio *d*/*h* (jet length/jet thickness) is an essential parameter that discriminates the jet-drive model from the discrete-vortex model as indicated in **Figure 8** based on [21]. The value of *d*/*h* = 1.7 set up in **Figure 8(c)** and **(d)** seems to be more favored by the discrete-vortex model. The critical aspect ratio that

Dequand et al. [21] calculated dj j *ξ*m*=*d*t* max*=U*<sup>0</sup> as a function of *d*/*h* for both models, and experimental data on the flue exit with chamfered edges and the pipe edge with 15° were plotted on the calculated diagram (see Figure 12 in [21]). In very rough sense, the solution curve of the jet-drive model is proportional to ð Þ *<sup>d</sup>=<sup>h</sup>* �<sup>1</sup>

curves occurs near *d*/*h* = 2.3, and the experimental data approximately fit the discrete-vortex solution below this crossing and better fit the jet-drive solution above the crossing. Also, the experimental curve of data points indicates the maximum value at the crossing or the critical aspect ratio. It was experimentally confirmed on four kinds of the flue-edge geometry (see Figure 10 in [21]). Furthermore, Auvray et al. [38] extended similar calculation for different oscillation regimes (*m* = 0, 1; *f* = *f*1, *f*2) (see Figure 8 in [38]). According to [38], the critical aspect ratio depends on the hydrodynamic jet mode *m*. For an eolian regime (*m* = 1,

Howe [18] and Dequand et al*.* [21] proposed the discrete-vortex model driven by thick jets (*d/h* < 2) as explained in Section 2.2. However, there seems to be a room for the vortex even in the sound generation of flue instruments driven by thin jets that satisfy *d/h* > 2. Since an actual jet has a velocity profile as indicated by the broken line in **Figure 10(a)**, the vorticity can be formed along the boundary between the jet and the surrounding fluid. As the result, a layer (or sheet) of vorticity is organized along an immediate vicinity of the jet. The upper layer consists of the positive vorticity (the counterclockwise-rotating tiny vortices), and the lower layer consists of the negative vorticity (the clockwise-rotating tiny vortices). This physical picture depicted in **Figure 10(a)** may be called the vortex-layer model

It should be carefully noted that actual sound generation is three-dimensional (3-D) as inferred from the volume integral of Eq. (14), but our vortex-layer model illustrated in **Figure 10(a)** assumes the two-dimensionality (2-D). This 2-D assumption corresponds to the 2-D assumption of *U* and *u* that has been conventionally made in acoustical models. See Figures 10 and 11 in [28] on the 3-D nature of *U* and *u*. The PIV observation on a plane sheet by the laser (see Section 4.1 and

Since *u* is periodic, the time average h i *Π*Gð Þ*t* should be null if *ω* � *U* is stationary in time as shown in **Figure 10(b)**. However, since the actual jet has any fluctuation, if *ω* � *U* has a component that changes temporarily in accordance with temporal change of *u*, non-zero value of h i *Π*Gð Þ*t* may be expected from the unbalance between

discriminates both models is *d*/*h* = 2.3 in more rigorous sense [21, 38].

that of the discrete-vortex model is proportional to ð Þ *<sup>d</sup>=<sup>h</sup>* <sup>1</sup>*=*<sup>2</sup>

*f* = *f*2), the critical aspect ratio is much larger (*d=h* ffi 13).

on the sound generation in musical flue instruments [2, 28].

**Figure 14**) is based on the 2-D assumption above.

This equation is the same as Eq. (29) when it is divided by *ω*. A similar deriva-

*T* ¼ phase½ � *Y*ð Þ *ω =ω:* (32)

, and

. The crossing of the two

*Conceptual sketch of the vortex-layer model on sound generation in flue instruments [2, 28]: (a) the vortex layers (consisting of tiny vortices) along both sides of the jet flow and (b) the generation and cancelation of the aeroacoustical source term ω* � *U. The dashed line depicts a lateral profile of the jet velocity U.*

the upper and lower vortex layers. Hence, the sound generation in flue pipes may be yielded by the interaction between the jet vortex layer and the acoustic mouth flow. In this sense, *ω* � *U* and (*ω* � *U*Þ∙ *u* may be called the *aeroacoustical source term* (with the same dimension as the acceleration) and the *acoustic generation term*, respectively.

Helmholtz [30] already suggested the importance of the jet vortex layer. His vortical surface (or stratum) that has a very unstable equilibrium acts as "an accelerating force with a periodically alternating direction" to reinforce the inward and outward velocity at the pipe entrance. Interestingly enough, this physical picture of Helmholtz is very similar to the jet vortex-layer model shown in **Figure 10** [2, 28]. It should be recognized that the volume-flow drive first proposed by Helmholtz [30] is based on his physical concept of the vortex layers formed along the jet flow.

## **3. Vortices on sound dissipation**

#### **3.1 Sound dissipations in linear acoustics**

Let us briefly discuss the mechanisms of sound dissipation (or absorption) in flue instruments because the self-sustained musical instruments must overcome the acoustic dissipations involved in them. At first let us consider within the field of linear acoustics and start from sound dissipation in free space.

#### *3.1.1 Classical absorption and molecular absorption in free space*

In free space, the classical sources of dissipation are internal friction and heat conduction. Both phenomena tend to equalize the local variations of the particle

velocity and temperature accompanying the acoustic wave [42]. As a result, the acoustic energy is removed from the acoustic wave.

The equations on dissipation due to internal friction were derived by G. Stokes in 1845 and those on dissipation due to heat conduction by G. Kirchhoff in 1868. A plane sound wave is exponentially damped in the direction of propagation (*x* direction): *e*�*αx*. The coefficient *α*<sup>F</sup> of the dissipation due to internal friction is [42]

$$a\_{\rm F} = \left(2a^{\rm 2}/3c^{3}\right)(\eta/\rho),\tag{33}$$

*Q*<sup>v</sup> ¼ ð Þ *ωρ=*2*η*

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

hertz) and the pipe radius *R* (in meters) [1, 42, 43]:

<sup>¼</sup> <sup>2</sup>*:*<sup>7</sup> � <sup>10</sup>�<sup>4</sup> *<sup>f</sup>*

*α*<sup>w</sup> ¼ ð Þ *ω=*2*c Q*<sup>v</sup>

source of sound dissipations in the bore.

wall boundary losses [1, 43]).

*<sup>Q</sup>*�<sup>1</sup> <sup>¼</sup> <sup>1</sup> 2*π*

Therefore,

**63**

to viscous loss is given by *Q*<sup>t</sup>

1*=*2

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

�1 *=Q*<sup>v</sup>

*<sup>R</sup>*, *<sup>Q</sup>*<sup>t</sup> <sup>¼</sup> *ωρC*p*=*2*<sup>κ</sup>* <sup>1</sup>*=*<sup>2</sup>

where *η* denotes the viscosity of the air, *C*<sup>p</sup> the specific heat at constant pressure, *κ* the thermal conductivity, and *γ* the ratio of specific heats. The ratio of thermal loss

The attenuation constant *α* in total is given by adding *Q*rad�<sup>1</sup> in Eq. (36):

�<sup>1</sup> <sup>≈</sup>3*:*<sup>1</sup> � <sup>10</sup>�<sup>5</sup> *<sup>f</sup>*

where *α*<sup>w</sup> is approximately evaluated in m�<sup>1</sup> and dBm�1. The conversion is done by the relation *<sup>α</sup>*<sup>w</sup> in dBm�<sup>1</sup> ð Þ¼ 20 log *<sup>e</sup>* � *<sup>α</sup>*<sup>w</sup> <sup>m</sup>�<sup>1</sup> ð Þ¼ <sup>8</sup>*:*<sup>68</sup> *<sup>α</sup>*<sup>w</sup> <sup>m</sup>�<sup>1</sup> ð Þ based on the exponential decay. For example, the modern flute with *R*≈10 mm indicates

*<sup>α</sup>*<sup>w</sup> <sup>≈</sup> <sup>0</sup>*:*85 dBm�<sup>1</sup> for *<sup>f</sup>* <sup>¼</sup> 1000 Hz. This attenuation is much larger than that occurs in free air, but still small enough. For example, tubes many meters long formerly were used on ships to transmit commands from the bridge to the engine room [42]. This large but still small enough magnitude of *α*<sup>w</sup> is the right reason why musical flue instruments and other wind instruments work out well. In order to suppress sound propagation in tubes (or in air conditioning systems) from the viewpoint of noise control, the tube walls should be covered with sound-absorbing material.

Since most musical flue instruments are of finite length, the sound wave that propagates in the instrument bore is reflected at both open ends. As a result, the acoustic resonance occurs if the energy enough to overcome all dissipations is supplied to the bore. The acoustical condition of the bore is characterized by the input impedance or admittance in which wall boundary losses defined by *α*<sup>w</sup> is involved. However, the reflection is not complete, and a little of the acoustic energy confined in the bore escapes to free space. This is sound radiation, which is another

If the resonance condition is given by *kL* ¼ *nπ* (*k* ¼ *ω=c* denotes the wave number and *n* ¼ 1, 2, 3, … *:*) and the source strength of radiation at each open end is

where *ω<sup>n</sup>* is the angular frequency at the *n*th mode resonance, *c*<sup>0</sup> the sound speed in free space, and *c* the sound speed in the bore (*c* is a little smaller than *c*<sup>0</sup> due to

time average of the power lost from the bore

h i *<sup>Π</sup>*rad *<sup>=</sup>*h i *<sup>Π</sup>*<sup>B</sup> <sup>¼</sup> <sup>2</sup>*πQ*rad�<sup>1</sup>

time average of the power stored in the bore *:* (38)

the same, we have the value of *Q* for sound radiation as follows [43]:

*Q*rad ¼ *n=ω<sup>n</sup>*

Generally, the loss factor or the inverse of *Q* is defined as [43, 44].

�<sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*<sup>t</sup>

1*=*2 *=R* in dBm�<sup>1</sup>

*3.1.3 Finite cylindrical pipe: acoustic resonance and sound radiation*

�<sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*46 for the air. The attenuation constant *α*<sup>w</sup> of a round pipe is a function of the frequency *f* (in

,

ð Þ *<sup>γ</sup>* � <sup>1</sup> �<sup>1</sup>

1*=*2 *=R* in m�<sup>1</sup>

<sup>2</sup> *πc*0*c=R*<sup>2</sup> , (37)

, (39)

*R*, (35)

(36)

where *c* is the propagating sound velocity, *η* the dynamic viscosity, and *ρ* the air (or generally, gas) density. The coefficient *α*<sup>H</sup> due to heat conduction in gases is of the same order of magnitude as *α*<sup>F</sup> and is proportional to *ω*<sup>2</sup> [42]. The distance within which the sound level falls by 1 dB (an amplitude decrease of about 11%) due to classical absorption is very large in air (1-kHz wave gives 5 km; 10-kHz wave 50 m) [42]. Therefore, classical dissipation is almost negligibly weak.

The major source of strong dissipation in free space is molecular sound absorption. The translational and rotational energies of gas molecule are very quickly increased by a sudden impact, while the oscillatory energy builds up gradually at the expense of the translational and rotational energies [42]. The delay in reaching thermal equilibrium is called relaxation, and its time constant is called the relaxation time *τ*<sup>R</sup> . The source of molecular absorption in air is oscillatory relaxation of oxygen. The relaxation frequency (defined as 1*=*2*πτ*R) of pure oxygen is very low (about 10 Hz). However, the water vapor content of air greatly shortens the relaxation time and shifts the absorbing range into the audio frequencies (see Figure 3.7 in [42]). The acoustic dissipation in moist air is significantly greater than the classical absorption given by *α*<sup>F</sup> þ *α*H.

#### *3.1.2 Sound dissipation at the internal wall of a long pipe*

Next let us consider the dissipation in the confined air instead of in free air. If a sound wave propagates in a long pipe where sound reflection can be neglected, it suffers additional losses because of internal friction and heat conduction in the boundary layer next to the wall. The acoustic particle velocity parallel to the pipe axis is zero at the internal wall surface because of friction (called no-slip condition). Its maximum value is not reached until the distance from the wall amounts to a quarter of viscosity wavelength *λ*vw (see Figure 3.10 in [42]). This *λ*vw characterizes the thickness of the wall boundary layer and is given by

$$
\lambda\_{\rm rev} = 2\pi\delta = 2\pi\sqrt{2\eta/a\rho} \approx \left(1.4 \text{ cm} \text{s}^{-1/2}\right) / \sqrt{f},\tag{34}
$$

where *δ* is the skin depth and the equation of the right-hand side is for the air with the kinematic viscosity *<sup>η</sup>=<sup>ρ</sup>* = 1.5 � <sup>10</sup>�<sup>5</sup> <sup>m</sup><sup>2</sup> sec�<sup>1</sup> [42].

The losses occurring in the wall boundary layer due to viscous friction and heat conduction (the wall is considered as a surface with a constant temperature and the thermal change followed by the acoustic wave should be null at the wall surface) attenuate sound waves in pipes. A parameter to appropriately express the sound attenuation in a pipe is the ratio of the pipe radius *R* to the boundary layer thickness (or the skin depth) *δ*. This ratio defines the quality factor *Q* or the inverse of the loss factor <sup>ζ</sup> (generally, *<sup>Q</sup>* <sup>¼</sup> *<sup>ζ</sup>*�<sup>1</sup> <sup>¼</sup> *<sup>R</sup>=δ*). The values of *<sup>Q</sup>* for viscous friction and heat conduction are, respectively, given as [1, 43].

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

$$\mathbf{Q\_{v}} = (a\rho/2\eta)^{1/2}\mathbf{R}, \mathbf{Q\_{t}} = \left(a\rho\mathbf{C\_{p}/2\kappa}\right)^{1/2}(\mathbf{y}-\mathbf{1})^{-1}\mathbf{R},\tag{35}$$

where *η* denotes the viscosity of the air, *C*<sup>p</sup> the specific heat at constant pressure, *κ* the thermal conductivity, and *γ* the ratio of specific heats. The ratio of thermal loss to viscous loss is given by *Q*<sup>t</sup> �1 *=Q*<sup>v</sup> �<sup>1</sup> <sup>¼</sup> <sup>0</sup>*:*46 for the air.

The attenuation constant *α*<sup>w</sup> of a round pipe is a function of the frequency *f* (in hertz) and the pipe radius *R* (in meters) [1, 42, 43]:

The attenuation constant *α* in total is given by adding *Q*rad�<sup>1</sup> in Eq. (36):

$$\begin{split} a\_{\mathbf{w}} &= (a/2c) \left( Q\_{\mathbf{v}}^{-1} + Q\_{\mathbf{t}}^{-1} \right) \approx \mathbf{3.1} \times \mathbf{10}^{-5} \left( f^{1/2}/\mathbf{R} \right) \|\, \mathbf{m}^{-1} \\ &= 2.7 \times \mathbf{10}^{-4} \left( f^{1/2}/\mathbf{R} \right) \|\, \mathbf{d} \, \mathbf{B} \mathbf{m}^{-1}, \end{split} \tag{36}$$

where *α*<sup>w</sup> is approximately evaluated in m�<sup>1</sup> and dBm�1. The conversion is done by the relation *<sup>α</sup>*<sup>w</sup> in dBm�<sup>1</sup> ð Þ¼ 20 log *<sup>e</sup>* � *<sup>α</sup>*<sup>w</sup> <sup>m</sup>�<sup>1</sup> ð Þ¼ <sup>8</sup>*:*<sup>68</sup> *<sup>α</sup>*<sup>w</sup> <sup>m</sup>�<sup>1</sup> ð Þ based on the exponential decay. For example, the modern flute with *R*≈10 mm indicates *<sup>α</sup>*<sup>w</sup> <sup>≈</sup> <sup>0</sup>*:*85 dBm�<sup>1</sup> for *<sup>f</sup>* <sup>¼</sup> 1000 Hz. This attenuation is much larger than that occurs in free air, but still small enough. For example, tubes many meters long formerly were used on ships to transmit commands from the bridge to the engine room [42]. This large but still small enough magnitude of *α*<sup>w</sup> is the right reason why musical flue instruments and other wind instruments work out well. In order to suppress sound propagation in tubes (or in air conditioning systems) from the viewpoint of noise control, the tube walls should be covered with sound-absorbing material.

#### *3.1.3 Finite cylindrical pipe: acoustic resonance and sound radiation*

Since most musical flue instruments are of finite length, the sound wave that propagates in the instrument bore is reflected at both open ends. As a result, the acoustic resonance occurs if the energy enough to overcome all dissipations is supplied to the bore. The acoustical condition of the bore is characterized by the input impedance or admittance in which wall boundary losses defined by *α*<sup>w</sup> is involved. However, the reflection is not complete, and a little of the acoustic energy confined in the bore escapes to free space. This is sound radiation, which is another source of sound dissipations in the bore.

If the resonance condition is given by *kL* ¼ *nπ* (*k* ¼ *ω=c* denotes the wave number and *n* ¼ 1, 2, 3, … *:*) and the source strength of radiation at each open end is the same, we have the value of *Q* for sound radiation as follows [43]:

$$Q\_{\rm rad} = \left(\pi/\alpha\_n^2\right) \left(\pi c\_0 c/R^2\right),\tag{37}$$

where *ω<sup>n</sup>* is the angular frequency at the *n*th mode resonance, *c*<sup>0</sup> the sound speed in free space, and *c* the sound speed in the bore (*c* is a little smaller than *c*<sup>0</sup> due to wall boundary losses [1, 43]).

Generally, the loss factor or the inverse of *Q* is defined as [43, 44].

$$Q^{-1} = \frac{1}{2\pi} \frac{\text{time average of the power lost from the bore}}{\text{time average of the power stored in the bore}}.\tag{38}$$

Therefore,

$$
\langle \Pi\_{\rm rad} \rangle / \langle \Pi\_{\rm B} \rangle = 2\pi Q\_{\rm rad}^{-1}, \tag{39}
$$

velocity and temperature accompanying the acoustic wave [42]. As a result, the

The equations on dissipation due to internal friction were derived by G. Stokes in 1845 and those on dissipation due to heat conduction by G. Kirchhoff in 1868. A plane sound wave is exponentially damped in the direction of propagation (*x* direction): *e*�*αx*. The coefficient *α*<sup>F</sup> of the dissipation due to internal

*<sup>α</sup>*<sup>F</sup> <sup>¼</sup> <sup>2</sup>*ω*<sup>2</sup>

50 m) [42]. Therefore, classical dissipation is almost negligibly weak.

*=*3*c*

where *c* is the propagating sound velocity, *η* the dynamic viscosity, and *ρ* the air (or generally, gas) density. The coefficient *α*<sup>H</sup> due to heat conduction in gases is of the same order of magnitude as *α*<sup>F</sup> and is proportional to *ω*<sup>2</sup> [42]. The distance within which the sound level falls by 1 dB (an amplitude decrease of about 11%) due to classical absorption is very large in air (1-kHz wave gives 5 km; 10-kHz wave

The major source of strong dissipation in free space is molecular sound absorption. The translational and rotational energies of gas molecule are very quickly increased by a sudden impact, while the oscillatory energy builds up gradually at the expense of the translational and rotational energies [42]. The delay in reaching thermal equilibrium is called relaxation, and its time constant is called the relaxation time *τ*<sup>R</sup> . The source of molecular absorption in air is oscillatory relaxation of oxygen. The relaxation frequency (defined as 1*=*2*πτ*R) of pure oxygen is very low (about 10 Hz). However, the water vapor content of air greatly shortens the relaxation time and shifts the absorbing range into the audio frequencies (see Figure 3.7 in [42]). The acoustic dissipation in moist air is significantly greater than the

Next let us consider the dissipation in the confined air instead of in free air. If a sound wave propagates in a long pipe where sound reflection can be neglected, it suffers additional losses because of internal friction and heat conduction in the boundary layer next to the wall. The acoustic particle velocity parallel to the pipe axis is zero at the internal wall surface because of friction (called no-slip condition). Its maximum value is not reached until the distance from the wall amounts to a quarter of viscosity wavelength *λ*vw (see Figure 3.10 in [42]). This *λ*vw characterizes

<sup>2</sup>*η=ωρ* <sup>p</sup> <sup>≈</sup> <sup>1</sup>*:*4 cms�1*=*<sup>2</sup> � �

where *δ* is the skin depth and the equation of the right-hand side is for the air

The losses occurring in the wall boundary layer due to viscous friction and heat conduction (the wall is considered as a surface with a constant temperature and the thermal change followed by the acoustic wave should be null at the wall surface) attenuate sound waves in pipes. A parameter to appropriately express the sound attenuation in a pipe is the ratio of the pipe radius *R* to the boundary layer thickness (or the skin depth) *δ*. This ratio defines the quality factor *Q* or the inverse of the loss factor <sup>ζ</sup> (generally, *<sup>Q</sup>* <sup>¼</sup> *<sup>ζ</sup>*�<sup>1</sup> <sup>¼</sup> *<sup>R</sup>=δ*). The values of *<sup>Q</sup>* for viscous friction and heat

*=* ffiffi *f* q

, (34)

<sup>3</sup> � �ð Þ *<sup>η</sup>=<sup>ρ</sup>* , (33)

acoustic energy is removed from the acoustic wave.

*Vortex Dynamics Theories and Applications*

classical absorption given by *α*<sup>F</sup> þ *α*H.

*3.1.2 Sound dissipation at the internal wall of a long pipe*

the thickness of the wall boundary layer and is given by

*<sup>λ</sup>*vw <sup>¼</sup> <sup>2</sup>π*<sup>δ</sup>* <sup>¼</sup> <sup>2</sup><sup>π</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffi

with the kinematic viscosity *<sup>η</sup>=<sup>ρ</sup>* = 1.5 � <sup>10</sup>�<sup>5</sup> <sup>m</sup><sup>2</sup> sec�<sup>1</sup> [42].

conduction are, respectively, given as [1, 43].

**62**

friction is [42]

where h i *Π*rad denotes the time average of the power lost from the bore by sound radiation and h i *Π*<sup>B</sup> the time average of the power stored in the bore. For *f* = 300 Hz and *R* = 25 and 15 mm, we have *Q*rad ¼ 164 and 454, respectively. We can thus understand that the radiated power is very little from the evaluated values h i *Π*rad *=*h i *Π*<sup>B</sup> ¼ 0*:*038 and 0*:*014. This implies that the first priority is clearly offered to the resonance in wind instruments. Sound radiation is only the faint leakage of the power stored in the bore. In spite of it, we can easily hear instrument tones.

$$a = a\_{\rm w} + a\_{\rm rad} = (w/2c) \left( Q\_{\rm v}^{-1} + Q\_{\rm t}^{-1} + Q\_{\rm rad}^{-1} \right), \tag{40}$$

150 Pa *just before* the steady state and the sounding frequency 283 Hz). These vortices are making up an acoustic dipole. We can see the same rotation of the interior vortex in **Figure 12(c)**. That vortex was recognized as acoustic vortex in [16] (the steady-state blowing pressure was 270 Pa and the sounding frequency 477 Hz). Therefore, the vortices in **Figure 12(a)** and **(b)** may be regarded as acoustic vortices, too. An interior vortex is produced in **Figure 12(a)** when the jet just crosses the edge from the outside to the inside, while that in **Figure 12(c)** is produced when the jet is deflected to the inside. This difference may be due to various causes. A long flue-to-edge distance *d* (=10.2 mm), almost null offset of the edge, a very slow buildup of the blowing pressure, and a low final blowing pressure were used in [17], while a much shorter *d*, a large offset, a quick buildup of the

*Visualized examples of acoustic vortices. (a) and (b): Visualized using a high-speed video camera and a smoked jet when the jet blowing pressure is 150 Pa and the sounding frequency 283 Hz [17]; (c): Visualized by means of a standard Schlieren technique when jet blowing pressure is 270 Pa and the sounding frequency 477 Hz [16]. Note that the jet-edge configuration and the buildup of the blowing pressure in frames (a) and (b) are quite different from those in frame (c). Also, frames (a) and (b) appear just before the steady state (at the*

blowing pressure, and a much higher final pressure were used in [16].

seems to be large enough to cause nonlinear oscillations.

*"pre-steady state" in [17]), while frame (c) appears at the steady state.*

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

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

pre-steady state or the steady state.

**65**

**Figure 12.**

These acoustic vortices shedding from the edge are considered to serve as a significant source of the sound energy dissipation in large-amplitude nonlinear oscillation [16]. According to **Figure 5** [the same organ pipe model was used in **Figures 5, 12(a)** and **(b)**], the acoustic particle velocity at the mouth is estimated as 2.3 m/s for the jet blowing velocity of 15.8 m/s (corresponding to the jet blowing pressure of 150 Pa). The acoustic velocity is thus about 15% of the flow velocity and

A physical modeling of the acoustic vortex generation in organ flue pipes is shown in **Figure 13** in comparison with the hydrodynamic vortex generation. A typical hydrodynamic vortex formed above the edge at the starting transient rotates anticlockwise as shown in **Figure 13(a)**. At this time the vorticity vector *ω* is in the negative *y* direction and *ω* � *U* is in the positive *z* direction (upwards) when the jet velocity *U* is in the positive *x* direction. On the other hand, an acoustic vortex formed above the edge [see **Figure 12(a)**] rotates clockwise as shown in

**Figure 13(b)**. Then, *ω* � *U* is in the negative *z* direction. Since the jet oscillates from the upward to the downward in **Figure 12(a)**, the acoustic particle velocity *u* takes negative maximum amplitude as known from **Figure 9(a)**. This condition is indicated by the dashed line around the mouth area in **Figure 13(b)**. As a result, the inner product ð Þ *ω* � *U* ∙ *u* becomes positive and the absorption of sound energy by the vortex is caused according to Eq. (13). Half a period later, an acoustic vortex rotating anticlockwise is formed below the edge as shown in **Figures 12(b)** and **13(c)**, and *ω* � *U* as well as *u* is in the positive *z* direction. Hence, ð Þ *ω* � *U* ∙ *u* is positive again as shown in **Figure 13(c)**, and sound energy absorption takes place at the

Although the jet deflection shown in **Figure 12(c)** is negative, the jet might be moving upward [the phase of *ξ*<sup>e</sup> may be around 12ð Þ¼ *t=T* 1 in **Figure 9(a)**] and then *u* as well as *ω* � *U* is possibly upward as shown in **Figure 13(c)**. Also, the effects of the pressure drive [cf. Eqs. (2) and (3)] should be considered because of

where *α*rad indicates the attenuation constant due to sound radiation. Also, the total *Q*�<sup>1</sup> defined by Eq. (38) is equal to *Q*<sup>v</sup> �<sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*<sup>t</sup> �<sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*rad�<sup>1</sup> . Since *<sup>Q</sup>* (called the quality factor) indicates the sharpness or height of the resonance, it is adequate to show *Q* instead of *Q*�<sup>1</sup> for wind instruments. **Figure 11** depicts *Q* as a function of *f* for the bores of the clarinet, flute, and bass flute. We may well understand that wind instruments with cylindrical bores have appreciably high Q values over their playing ranges.

#### **3.2 Acoustically induced vortices as the final dissipation agent**

The above description in 3.1 on sound dissipations is correct within the scope of linear acoustics. Then, as the input energy from the player continues to increase, the output energy (viz., the sound level) from the instrument keeps increasing. However, in actual wind instruments, the saturation of the output energy necessarily occurs. In other words, sound generation is nonlinear.

An important source of the saturation in flue instruments is acoustically induced vortices (simply, acoustic vortices) at the pipe edge. These acoustic vortices work as the final dissipation agent that determines the final amplitude of the saturated sound.

#### *3.2.1 Visualization of acoustic vortices and their modeling*

Jet and vortex behaviors during attack transients in organ pipe models were studied intensively using a high-speed video camera and a smoked jet in [17]. Experimental procedures are described in [17, 29]. **Figure 12(a)** and **(b)** is the visualization result which shows the exterior vortex (a) is rotating clockwise and the interior vortex (b) is rotating anticlockwise (the blowing pressure is about

#### **Figure 11.**

*The quality factor Q as a function of f for cylindrical bore instruments. The bore radius R = 7.3, 9.5, and 17 mm for the clarinet, flute, and bass flute, respectively.*

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

#### **Figure 12.**

where h i *Π*rad denotes the time average of the power lost from the bore by sound radiation and h i *Π*<sup>B</sup> the time average of the power stored in the bore. For *f* = 300 Hz and *R* = 25 and 15 mm, we have *Q*rad ¼ 164 and 454, respectively. We can thus understand that the radiated power is very little from the evaluated values

h i *Π*rad *=*h i *Π*<sup>B</sup> ¼ 0*:*038 and 0*:*014. This implies that the first priority is clearly offered to the resonance in wind instruments. Sound radiation is only the faint leakage of the power stored in the bore. In spite of it, we can easily hear instrument tones.

where *α*rad indicates the attenuation constant due to sound radiation. Also, the

The above description in 3.1 on sound dissipations is correct within the scope of linear acoustics. Then, as the input energy from the player continues to increase, the output energy (viz., the sound level) from the instrument keeps increasing. However, in actual wind instruments, the saturation of the output energy necessarily

An important source of the saturation in flue instruments is acoustically induced vortices (simply, acoustic vortices) at the pipe edge. These acoustic vortices work as the final dissipation agent that determines the final amplitude of the saturated sound.

Jet and vortex behaviors during attack transients in organ pipe models were studied intensively using a high-speed video camera and a smoked jet in [17]. Experimental procedures are described in [17, 29]. **Figure 12(a)** and **(b)** is the visualization result which shows the exterior vortex (a) is rotating clockwise and the interior vortex (b) is rotating anticlockwise (the blowing pressure is about

*The quality factor Q as a function of f for cylindrical bore instruments. The bore radius R = 7.3, 9.5, and*

quality factor) indicates the sharpness or height of the resonance, it is adequate to show *Q* instead of *Q*�<sup>1</sup> for wind instruments. **Figure 11** depicts *Q* as a function of *f* for the bores of the clarinet, flute, and bass flute. We may well understand that wind instruments with cylindrical bores have appreciably high Q values over their

�<sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*<sup>t</sup>

�<sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*<sup>t</sup>

�<sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*rad�<sup>1</sup> , (40)

�<sup>1</sup> <sup>þ</sup> *<sup>Q</sup>*rad�<sup>1</sup> . Since *<sup>Q</sup>* (called the

*α* ¼ *α*<sup>w</sup> þ *α*rad ¼ ð Þ *ω=*2*c Q*<sup>v</sup>

**3.2 Acoustically induced vortices as the final dissipation agent**

occurs. In other words, sound generation is nonlinear.

*3.2.1 Visualization of acoustic vortices and their modeling*

*17 mm for the clarinet, flute, and bass flute, respectively.*

total *Q*�<sup>1</sup> defined by Eq. (38) is equal to *Q*<sup>v</sup>

*Vortex Dynamics Theories and Applications*

playing ranges.

**Figure 11.**

**64**

*Visualized examples of acoustic vortices. (a) and (b): Visualized using a high-speed video camera and a smoked jet when the jet blowing pressure is 150 Pa and the sounding frequency 283 Hz [17]; (c): Visualized by means of a standard Schlieren technique when jet blowing pressure is 270 Pa and the sounding frequency 477 Hz [16]. Note that the jet-edge configuration and the buildup of the blowing pressure in frames (a) and (b) are quite different from those in frame (c). Also, frames (a) and (b) appear just before the steady state (at the "pre-steady state" in [17]), while frame (c) appears at the steady state.*

150 Pa *just before* the steady state and the sounding frequency 283 Hz). These vortices are making up an acoustic dipole. We can see the same rotation of the interior vortex in **Figure 12(c)**. That vortex was recognized as acoustic vortex in [16] (the steady-state blowing pressure was 270 Pa and the sounding frequency 477 Hz). Therefore, the vortices in **Figure 12(a)** and **(b)** may be regarded as acoustic vortices, too. An interior vortex is produced in **Figure 12(a)** when the jet just crosses the edge from the outside to the inside, while that in **Figure 12(c)** is produced when the jet is deflected to the inside. This difference may be due to various causes. A long flue-to-edge distance *d* (=10.2 mm), almost null offset of the edge, a very slow buildup of the blowing pressure, and a low final blowing pressure were used in [17], while a much shorter *d*, a large offset, a quick buildup of the blowing pressure, and a much higher final pressure were used in [16].

These acoustic vortices shedding from the edge are considered to serve as a significant source of the sound energy dissipation in large-amplitude nonlinear oscillation [16]. According to **Figure 5** [the same organ pipe model was used in **Figures 5, 12(a)** and **(b)**], the acoustic particle velocity at the mouth is estimated as 2.3 m/s for the jet blowing velocity of 15.8 m/s (corresponding to the jet blowing pressure of 150 Pa). The acoustic velocity is thus about 15% of the flow velocity and seems to be large enough to cause nonlinear oscillations.

A physical modeling of the acoustic vortex generation in organ flue pipes is shown in **Figure 13** in comparison with the hydrodynamic vortex generation. A typical hydrodynamic vortex formed above the edge at the starting transient rotates anticlockwise as shown in **Figure 13(a)**. At this time the vorticity vector *ω* is in the negative *y* direction and *ω* � *U* is in the positive *z* direction (upwards) when the jet velocity *U* is in the positive *x* direction. On the other hand, an acoustic vortex formed above the edge [see **Figure 12(a)**] rotates clockwise as shown in **Figure 13(b)**. Then, *ω* � *U* is in the negative *z* direction. Since the jet oscillates from the upward to the downward in **Figure 12(a)**, the acoustic particle velocity *u* takes negative maximum amplitude as known from **Figure 9(a)**. This condition is indicated by the dashed line around the mouth area in **Figure 13(b)**. As a result, the inner product ð Þ *ω* � *U* ∙ *u* becomes positive and the absorption of sound energy by the vortex is caused according to Eq. (13). Half a period later, an acoustic vortex rotating anticlockwise is formed below the edge as shown in **Figures 12(b)** and **13(c)**, and *ω* � *U* as well as *u* is in the positive *z* direction. Hence, ð Þ *ω* � *U* ∙ *u* is positive again as shown in **Figure 13(c)**, and sound energy absorption takes place at the pre-steady state or the steady state.

Although the jet deflection shown in **Figure 12(c)** is negative, the jet might be moving upward [the phase of *ξ*<sup>e</sup> may be around 12ð Þ¼ *t=T* 1 in **Figure 9(a)**] and then *u* as well as *ω* � *U* is possibly upward as shown in **Figure 13(c)**. Also, the effects of the pressure drive [cf. Eqs. (2) and (3)] should be considered because of

completely steady state or at the pre-steady state. Also, the acoustic vortex should be discussed from the common viewpoint of acoustic power dissipation and radiation of

Acoustic power generation by the unbalance between the upper and lower vortex layers (cf. Section 2.4) will be balanced with acoustic power dissipations by the wall boundary effects (cf. Section 3.1.2), sound radiation (cf. Section 3.1.3), and

where *Π*Gð Þ*t* is given by Eq. (14) with *ω* ¼ *ω*vl concerning the vortex layer, *Π*αð Þ*t* is the power lost from the bore that is given by the total attenuation constant of Eq. (40), and *Π*Dð Þ*t* is given by Eq. (13) with *ω* ¼ *ω*av concerning the acoustic vortex. A more exact description of *Π*Gð Þ*t* derived from the unbalance between the

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

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

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].

<sup>2</sup> � �ð*<sup>T</sup>*

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]:

If the integral in the right-hand side of Eq. (43) can be replaced with a product

� �<sup>2</sup> � *<sup>S</sup>*tð Þ *<sup>h</sup>=<sup>d</sup>* <sup>3</sup>*=*<sup>2</sup>

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

ð Þ d*ξ*m*=*d*t* max*=U*<sup>0</sup>

j j d*ξ*m*=*d*t* and an appropriate division of *T* by supposing a

0

h i *Π*Gð Þ*t* ¼ h i *Π*αð Þ*t* þ h i *Π*Dð Þ*t* , (41)

sign *q*<sup>m</sup>

ð Þ <sup>d</sup>*ξ*m*=*d*<sup>t</sup>* <sup>2</sup>

*<sup>Π</sup>*jet � � <sup>þ</sup> h i *<sup>Π</sup>*lost <sup>≈</sup> <sup>0</sup>*:* (44)

� �, (42)

j j d*ξ*m*=*d*t* d*t*, (43)

*:* (45)

high-amplitude jet noise at duct termination [3, 15, 35, 44].

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

upper and lower vortex layers will be given in Section 4.1.

Verge et al*.* [12]. This acoustic flow separation causes a free jet [45].

*<sup>p</sup>*<sup>v</sup> ¼ �ð Þ <sup>1</sup>*=*<sup>2</sup> *<sup>ρ</sup> <sup>q</sup>*m*=c*v*db* � �<sup>2</sup>

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

fluctuating pressure *p*<sup>v</sup> across the mouth [12, 21]:

� � ¼ �ð Þ <sup>1</sup>*=*2*<sup>T</sup> <sup>ρ</sup>db=c*<sup>v</sup>

*S*<sup>t</sup> ¼ *fd=U*0, and the aspect ratio *d=h* of the jet [12, 21]:

h i *Π*lost ¼ *p*v*q*<sup>m</sup>

2

of dð Þ *ξ*m*=*d*t* max

**67**

acoustic vortices in the sense of time average:

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

*3.2.2 Acoustic power balance between vortex layers and acoustic vortices*

#### **Figure 13.**

*Schematic of vortex formation in organ flue pipes. (a): A hydrodynamic vortex formed at the initial phase of the starting transient; (b) and (c): An acoustic vortex formed at the pre-steady state [17].*

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

Interestingly enough, the acoustic vortices shown in **Figure 12(a)** and **(b)** were 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 completely steady state or at the pre-steady state. Also, the acoustic vortex should be discussed from the common viewpoint of acoustic power dissipation and radiation of high-amplitude jet noise at duct termination [3, 15, 35, 44].
